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Shape Optimization of Geometrically Nonlinear Modal Coupling Coefficients: An Application to MEMS Gyroscopes (2403.17679v1)

Published 26 Mar 2024 in cs.CE

Abstract: Micro- and nanoelectromechanical system (MEMS and NEMS) resonators can exhibit rich nonlinear dynamics as they are often operated at large amplitudes with high quality factors and possess a high mode density with a variety of nonlinear modal couplings. Their impact is strongly influenced by internal resonance conditions and by the strength of the modal coupling coefficients. On one hand, strong nonlinear couplings are of academic interest and promise novel device concepts. On the other hand, however, they have the potential to disturb the linear system behavior on which industrial devices such as gyroscopes and micro mirrors are based on. In either case, being able to optimize the coupling coefficients by design is certainly beneficial. A main source of nonlinear modal couplings are geometric nonlinearities. In this work, we apply node-based shape optimization to tune the geometrically nonlinear 3-wave coupling coefficients of a MEMS gyroscope. We demonstrate that individual coupling coefficients can be tuned over several orders of magnitude by shape optimization, while satisfying typical constraints on manufacturability and operability of the devices. The optimized designs contain unintuitive geometrical features far away from any solution an experienced human MEMS or NEMS designer could have thought of. Thus, this work demonstrates the power of shape optimization for tailoring the complex nonlinear dynamic properties of MEMS and NEMS resonators.

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References (35)
  1. Mesoscopic physics of nanomechanical systems. Reviews of Modern Physics 94, 045005 (2022). [2] Eichler, A. & Zilberberg, O. Classical and quantum parametric phenomena (Oxford University Press, New York, 2023). [3] Nayfeh, A. H. & Mook, D. T. Nonlinear oscillations (John Wiley & Sons, 2008). [4] Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Eichler, A. & Zilberberg, O. Classical and quantum parametric phenomena (Oxford University Press, New York, 2023). [3] Nayfeh, A. H. & Mook, D. T. Nonlinear oscillations (John Wiley & Sons, 2008). [4] Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nayfeh, A. H. & Mook, D. T. Nonlinear oscillations (John Wiley & Sons, 2008). [4] Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. 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[24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nayfeh, A. H. & Mook, D. T. Nonlinear oscillations (John Wiley & Sons, 2008). [4] Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. 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[17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. 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[26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
  3. Nonlinear oscillations (John Wiley & Sons, 2008). [4] Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Kerschen, G. (ed.) Modal Analysis of Nonlinear Mechanical Systems Vol. 555 (Springer, Vienna, 2014). [5] Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touzé, C. & Amabili, M. Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298, 958–981 (2006). [6] Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. 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Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. 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[26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Opreni, A., Vizzaccaro, A., Frangi, A. & Touzé, C. Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance. Nonlinear Dynamics 105, 1237–1272 (2021). [7] Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. 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An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Sarrafan, A., Azimi, S., Golnaraghi, F. & Bahreyni, B. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
  7. A Nonlinear Rate Microsensor utilising Internal Resonance. Scientific Reports 9, 8648 (2019). [8] Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). 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[17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). 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KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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[24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. 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International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). 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Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. 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Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. 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  8. Nitzan, S. H. et al. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Scientific Reports 5, 9036 (2015). [9] Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Antonio, D., Zanette, D. H. & Lopez, D. Frequency stabilization in nonlinear micromechanical oscillators. Nature Communications 3, 806 (2012). [10] Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. 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Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Lamprecht, L., Mehner, J. E., Zimmermann, A. & Degenfeld-Schonburg, P. Parametric amplification of broadband vibrational energy harvesters for energy-autonomous sensors enabled by field-induced striction. Mechanical Systems and Signal Processing 139, 106642 (2020). [11] Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. 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A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Ganesan, A., Do, C. & Seshia, A. Phononic Frequency Comb via Intrinsic Three-Wave Mixing. Physical Review Letters 118, 033903 (2017). [12] Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. 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Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Curcic, M., Mehner, J. E. & Degenfeld-Schonburg, P. Nonlinear Dynamical System Model for Drive Mode Amplitude Instabilities in MEMS Gyroscopes (2019). [13] Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. 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Nabholz, U., Schatz, F., Mehner, J. E. & Degenfeld-Schonburg, P. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
  13. Spontaneous Parametric Down-Conversion Induced by Non-Degenerate Three-Wave Mixing in a Scanning MEMS Micro Mirror. Scientific Reports 9, 3997 (2019). [14] Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nabholz, U., Stockmar, F., Mehner, J. E. & Degenfeld-Schonburg, P. Validating the Critical Point of Spontaneous Parametric Downconversion for Over 600 Scanning MEMS Micromirrors on Wafer Level. IEEE Sensors Letters 4, 1–4 (2020). [15] Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. 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[32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lifshitz, R. & Cross, M. C. Nonlinear dynamics of nanomechanical and micromechanical resonators. Reviews of nonlinear dynamics and complexity 1 (2008). [16] Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. 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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. 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A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. 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The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Bonaccorsi, G. & Braghin, F. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. 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Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
  16. Size optimization of MEMS gyroscopes using substructuring. European Journal of Mechanics - A/Solids 84, 104045 (2020). [17] Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Braghin, F. & Aage, N. Topology optimization of 2D in-plane single mass MEMS gyroscopes. Structural and Multidisciplinary Optimization 62, 2069–2089 (2020). [18] Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. 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Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Giannini, D., Aage, N. & Braghin, F. Topology optimization of MEMS resonators with target eigenfrequencies and modes. European Journal of Mechanics - A/Solids 91, 104352 (2022). [19] Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Shin, D. et al. Spiderweb Nanomechanical Resonators via Bayesian Optimization: Inspired by Nature and Guided by Machine Learning. Advanced Materials 34, 2106248 (2022). [20] Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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[26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Hoj, D. et al. Ultra-coherent nanomechanical resonators based on inverse design. Nature Communications 12, 5766 (2021). [21] Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. 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Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. 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International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). 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Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). 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[22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. 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[25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. 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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). 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KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. 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Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
  21. Pedersen, N. Designing plates for minimum internal resonances. Structural and Multidisciplinary Optimization 30, 297–307 (2005). [22] Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Schiwietz, D., Hörsting, M., Weig, E. M., Degenfeld-Schonburg, P. & Wenzel, M. Shape Optimization of Eigenfrequencies in MEMS Gyroscopes. arXiv preprint arXiv:2402.05837 (2024). [23] Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Dou, S., Strachan, B. S., Shaw, S. W. & Jensen, J. S. Structural optimization for nonlinear dynamic response. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, 20140408 (2015). [24] Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Li, L. L. et al. Tailoring the nonlinear response of MEMS resonators using shape optimization. Applied Physics Letters 110, 081902 (2017). [25] Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Wriggers, P. Nonlinear Finite Element Methods (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008). [26] Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Touze, C., Vizzaccaro, A. & Thomas, O. Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques. Nonlinear Dynamics 105, 1141–1190 (2021). [27] Svanberg, K. The method of moving asymptotes-a new method for structural optimization. International Journal for Numerical Methods in Engineering 24, 359–373 (1987). [28] Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. 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An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. 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Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. 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A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Lee, T. H. An adjoint variable method for structural design sensitivity analysis of a distinct eigenvalue problem. KSME International Journal 13, 470–476 (1999). [29] Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
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Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Tcherniak, D. Topology optimization of resonating structures using SIMP method. International Journal for Numerical Methods in Engineering 54, 1605–1622 (2002). [30] Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Nelson, R. B. Simplified calculation of eigenvector derivatives. AIAA Journal 14, 1201–1205 (1976). [31] Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M., Cardanobile, S., Nagel, C., Degenfeld-Schonburg, P. & Mehner, J. Simulation and Modelling of the Drive Mode Nonlinearity in MEMS-gyroscopes. Procedia Engineering 168, 950–953 (2016). [32] Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Incorporating geometrical nonlinearities in reduced order models for MEMS gyroscopes (2017). [33] Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. A static approach for the frequency shift of parasitic excitations in MEMS gyroscopes with geometric nonlinear drive mode (2017). [34] Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Predicting the Resonance Frequencies in Geometric Nonlinear Actuated MEMS. Journal of Microelectromechanical Systems 27, 954–962 (2018). [35] Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018). Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).
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  35. Putnik, M. et al. Simulation methods for generating reduced order models of MEMS sensors with geometric nonlinear drive motion (2018).

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