Nesterov's acceleration for level set-based topology optimization using reaction-diffusion equations

Abstract: This paper discusses level set-based structural optimization. Level set-based structural optimization is a method used to determine an optimal configuration for minimizing an objective functional by updating level set functions characterized as solutions to partial differential equations (PDEs) (e.g., Hamilton-Jacobi and reaction-diffusion equations). In this study, based on Nesterov's accelerated method, a nonlinear (damped) wave equation will be derived as a PDE satisfied by level set functions and applied to a minimum mean compliance problem. Numerically, the method developed in this study will yield convergence to an optimal configuration faster than methods using only a reaction-diffusion equation, and moreover, its FreeFEM++ code will also be described.

• The paper introduces a novel level set-based approach that incorporates Nesterov’s acceleration to improve convergence in topology optimization.
• The method applies reaction-diffusion equations to yield smooth geometrical configurations and reduce sensitivity to initial conditions.
• Numerical experiments demonstrate faster convergence on benchmark problems, highlighting its practical advantages in structural design.

Essay: Nesterov's Acceleration for Level Set-Based Topology Optimization Using Reaction-Diffusion Equations

The paper discusses an advanced approach to level set-based structural optimization, focusing on deriving and applying a nonlinear (damped) wave equation rooted in Nesterov's accelerated gradient method for topology optimization. This study addresses structural optimization by determining an optimal material configuration that minimizes objective functionals using levels set functions, such as solutions to partial differential equations (PDEs). The optimization method utilizes the reaction-diffusion equations combined with Nesterov's acceleration and demonstrates faster convergence toward optimal configurations compared to methods solely relying on reaction-diffusion.

Key Aspects of the Paper

1. Level Set-Based Optimization: The level set method transforms the topology optimization problem into finding an optimal configuration by tracking the evolution of the zero level set of a higher-dimensional function. The method tackles the issues of grayscale domains and dependency on initial configurations.
2. Reaction-Diffusion Equation: This approach incorporates a reaction-diffusion equation, aimed at minimizing mean compliance problems undervolved with structural optimization. Through a reaction-diffusion equation, the optimization process gains robustness against initial configuration dependency and yields geometrically smooth results.
3. Nesterov's Accelerated Gradient: The study introduces Nesterov's accelerated gradient method, enhancing convergence speed in topology optimization. This technique modifies the standard gradient descent update with an inertia term for accelerated optimization, effectively translating to a nonlinear (damped) wave equation in the PDE context.
4. Numerical Experiments: The paper contains an extensive number of numerical examples illustrating the application of the proposed method across two-dimensional and three-dimensional cases. Results underscored convergence improvements over traditional methods.

Numerical Results and Observations

The numerical findings presented in this paper are indicative of the faster convergence achieved by employing Nesterov's accelerated gradient method compared to conventional reaction-diffusion-based approaches. One notable observation shows that the optimization of a cantilever model, which initially traverses through a complex configuration, converges substantially faster using the proposed method. The implementation is demonstrated using the FreeFEM++ software, highlighting the practical applicability of the formulation.

Implications and Future Directions

The incorporation of Nesterov's acceleration into topology optimization represents a significant advance in PDE-driven design methods. While the paper demonstrates the computational superiority of this approach in case studies of structural components, future research can focus on extending these techniques to more complex optimization scenarios, including multi-material and dynamic loading cases. Moreover, addressing theoretical concerns such as the mathematical justification for replacing topological derivatives with Fréchet derivatives remains an important open problem. However, given the promising results, exploring more sophisticated PDEs and optimization methods can further enhance the efficiency and scope of topology optimization in engineering applications.

In conclusion, this study provides a valuable contribution to the field of structural optimization, showcasing the efficacy of combining state-of-the-art mathematical techniques for enhancing the convergence and robustness of level set-based topology optimization. The adaptation of theoretical advancements such as Nesterov's method within this engineering context holds considerable potential for future research and practical applications.