Numerical analysis of the SIMP model for the topology optimization problem of minimizing compliance in linear elasticity

TL;DR

This paper analyzes finite element approximations of the SIMP method for topology optimization in linear elasticity, demonstrating convergence to all isolated local minimizers and eliminating checkerboarding artifacts.

Contribution

It proves strong convergence of finite element local minimizers to all isolated minimizers, improving understanding of SIMP's numerical behavior and regularization effects.

Findings

Strong convergence to all isolated minimizers

Existence of unfiltered distributions without checkerboarding

Enhanced theoretical understanding of SIMP approximations

Abstract

We study the finite element approximation of the solid isotropic material with penalization method (SIMP) for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: $W^{1,p}$-type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to \emph{all} the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding.