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Variational approximation for a non-isothermal coupled phase-field system: Structure-preservation & Nonlinear stability (2312.14566v2)

Published 22 Dec 2023 in math.NA, cs.NA, and math.AP

Abstract: A Cahn-Hilliard-Allen-Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem w.r.t. the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

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References (49)
  1. Energy-decaying extrapolated RK–SAV methods for the Allen–Cahn and Cahn–Hilliard equations. SIAM J. Sci. Comput., 41(6):A3703–A3727, 2019.
  2. H. W. Alt and I. Pawlow. Dynamics of non-isothermal phase separation. In K.-H. Hoffmann and J. Sprekels, editors, Free Boundary Value Problems: Proceedings of a Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, July 9–15, 1989, pages 1–26. Birkhäuser Basel, Basel, 1990.
  3. H. W. Alt and I. Pawlow. A mathematical model and an existence theory for non-isothermal phase separation. In P. Neittaanmäki, editor, Numerical Methods for Free Boundary Problems: Proceedings of a Conference held at the Department of Mathematics, University of Jyväskylä, Finland, July 23–27, 1990, pages 1–32. Birkhäuser Basel, Basel, 1991.
  4. H. W. Alt and I. Pawlow. A mathematical model of dynamics of non-isothermal phase separation. Physica D, 59(4):389–416, 1992.
  5. Temperature dependent extensions of the Cahn-Hilliard equation. arXiv, 2022.
  6. A phase field model with thermal memory governed by the entropy balance. Math. Models Methods Appl. Sci., 13(11):1565–1588, 2003.
  7. G. Boussinot and E. A. Brener. Interface kinetics in phase-field models: Isothermal transformations in binary alloys and step dynamics in molecular-beam epitaxy. Phys. Rev. E, 88:022406, 2013.
  8. E. A. Brener and G. Boussinot. Kinetic cross coupling between nonconserved and conserved fields in phase field models. Phys. Rev. E, 86:060601, 2012.
  9. A second-order fully-balanced structure-preserving variational discretization scheme for the cahn–hilliard–navier–stokes system. Math. Models Methods Appl. Sci., 33(12):2587–2627, 2023.
  10. Stability and discretization error analysis for the cahn–hilliard system via relative energy estimates. ESAIM: M2AN, 57(3):1297–1322, 2023.
  11. G. Caginalp. An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal., 92(3):205–245, 1986.
  12. C. Charach and P. C. Fife. On thermodynamically consistent schemes for phase field equations. Open Syst. Inf. Dyn, 5(2):99–123, 1998.
  13. C. Chen and X. Yang. Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard model. Comput. Methods Appl. Mech. Eng., 351:35–59, 2019.
  14. R. Chen and S. Gu. On novel linear schemes for the Cahn–Hilliard equation based on an improved invariant energy quadratization approach. J. Comput. Appl. Math., 414:114405, 2022.
  15. Global smooth solution to the standard phase-field model with memory. Adv. Differ. Equ., 2(3):453 – 486, 1997.
  16. Well-posedness of the weak formulation for the phase-field model with memory. Adv. Differ. Equ., 2(3):487 – 508, 1997.
  17. Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen–Cahn type. Math. Models Methods Appl. Sci., 20(04):519–541, 2010.
  18. On a Penrose-Fife phase-field model with nonhomogeneous Neumann boundary conditions for the temperature. Differ. Integral Equ., 17(5-6):511 – 534, 2004.
  19. On a cahn-hilliard system with source term and thermal memory. arXiv, 2022.
  20. P. Colli and K.-H. Hoffmann. A nonlinear evolution problem describing multi-component phase changes with dissipation. Numer. Funct. Anal. Optim., 14(3-4):275–297, 1993.
  21. P. Colli and P. Laurençot. Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws. Physica D, 111(1-4):311–334, 1998.
  22. Quantitative phase-field model of alloy solidification. Phys. Rev. E, 70(6):061604, 2004.
  23. A thermodynamic approach to non-isothermal phase-field evolution in continuum physics. Physica D, 214(2):144–156, 2006.
  24. Numerical Analysis of Compressible Fluid Flows. Springer International Publishing, 2021.
  25. R. German. Sintering: from empirical observations to scientific principles. Butterworth-Heinemann, 2014.
  26. Thermodynamic model formulations for inhomogeneous solids with application to non-isothermal phase field modelling. J. Non-Equilib. Thermodyn., 41(2):131–139, 2016.
  27. Y. Gong and J. Zhao. Energy-stable runge–kutta schemes for gradient flow models using the energy quadratization approach. Appl. Math. Lett., 94:224–231, 2019.
  28. O. Gonzalez. Time integration and discrete Hamiltonian systems. J. Nonlinear Sci., 6:449–467, 1996.
  29. A thermodynamically consistent numerical method for a phase field model of solidification. Commun. Nonlinear Sci. Numer. Simul., 19(7):2309–2323, 2014.
  30. F. Guillén-González and J. V. Gutiérrez-Santacreu. Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model. ESAIM: Math. Model. Numer., 43(3):563–589, 2009.
  31. Z. Guo and P. Lin. A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. J. Fluid Mech., 766:226–271, 2015.
  32. S.-J. Kang. Sintering: Densification, Grain Growth and Microstructure. Elsevier, 2004.
  33. Generalized phase field approach for computer simulation of sintering: incorporation of rigid-body motion. Scr. Mater., 41(5):487–492, 1999.
  34. N. Kenmochi and M. Niezgódka. Evolution systems of nonlinear variational inequalities arising from phase change problems. Nonlinear Anal. Theory Methods Appl., 22(9):1163–1180, 1994.
  35. Y. Li and J. Yang. Consistency-enhanced SAV BDF2 time-marching method with relaxation for the incompressible Cahn–Hilliard–Navier–Stokes binary fluid model. Commun. Nonlinear Sci. Numer., 118:107055, 2023.
  36. A. Marveggio and G. Schimperna. On a non-isothermal cahn-hilliard model based on a microforce balance. J. Differ. Equ., 274:924–970, 2021.
  37. Geometric integration using discrete gradients. R. Soc. Lond. Philos. Trans. Ser. A: Math. Phys. Eng. Sci., 357:1021–1045, 1999.
  38. Variational quantitative phase-field modeling of nonisothermal sintering process. Phys. Rev. E, 108:025301, Aug 2023.
  39. I. Pawłow. A thermodynamic approach to nonisothermal phase-field models. Appl. Math, pages 1–63, 2016.
  40. O. Penrose and P. C. Fife. Thermodynamically consistent models of phase-field type for the kinetic of phase transitions. Physica D, 43(1):44–62, 1990.
  41. The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys., 353:407–416, 2018.
  42. J. Shen and X. Yang. Numerical approximations of allen-cahn and cahn-hilliard equations. Discrete Contin. Dyn. Syst., 28(4):1669–1691, 2010.
  43. Structure-preserving numerical approximations to a non-isothermal hydrodynamic model of binary fluid flows. J. Sci. Comput., 83(3), 2020.
  44. Y. U. Wang. Computer modeling and simulation of solid-state sintering: A phase field approach. Acta Mater., 54(4):953–961, 2006.
  45. X. Yang and G.-D. Zhang. Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn–Hilliard and Allen–Cahn equations with general nonlinear potential. J. Sci. Comput., 82(3), 2020.
  46. Investigation on temperature-gradient-driven effects in unconventional sintering via non-isothermal phase-field simulation. Scr. Mater., 186:152–157, 2020.
  47. 3D non-isothermal phase-field simulation of microstructure evolution during selective laser sintering. npj Comput. Mater., 5(1):81, 2019.
  48. A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws. Electron. Res. Arch., 30(2):701–714, 2022.
  49. S. M. Zheng. Global existence for a thermodynamically consistent model of phase field type. Differ. Integral Equ., 5(2):241 – 253, 1992.
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