Topology optimization method with nonlinear diffusion

TL;DR

This paper introduces a novel topology optimization method using nonlinear diffusion equations with reaction terms, enhancing convergence speed and stability without relying on topological derivatives.

Contribution

It develops a new level set-based topology optimization approach employing nonlinear diffusion with reaction terms, relaxing sensitivity analysis requirements.

Findings

Demonstrates fast convergence of configurations.

Shows damping oscillations on boundary structures.

Validates the method through numerical experiments.

Abstract

This paper is concerned with topology optimization based on a level set method using (doubly) nonlinear diffusion equations. Topology optimization using the level set method is called level set-based topology optimization, which is possible to determine optimal configurations that minimize objective functionals by updating level set functions. In this paper, as an update equation for level set functions, (doubly) nonlinear diffusion equations with reaction terms are derived, and then the singularity and degeneracy of the diffusion coefficient are applied to obtain fast convergence of configurations and damping oscillation on boundary structures. In particular, the reaction terms in the proposed method do not depend on the topological derivatives, and therefore, sensitivity analysis to determine a descent direction for objective functionals is relaxed. Furthermore, a numerical algorithm for the proposed method is constructed and applied to typical minimization problems to show numerical validity. This paper is a justification and generalization of the method using reaction-diffusion equations developed by one of the authors in Yamada et al. (2010).