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

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.

Figures (3)

• Figure 1: Setup of the MBB beam. The remaining unlabeled boundary conditions are $(\pmb{\mathsf{S}} \boldsymbol{\hat{n}})_t = 0$ on $\Gamma^n_{D_1} \cup \Gamma^n_{D_2}$ and $\pmb{\mathsf{S}} \boldsymbol{\hat{n}} = (0,0)^\top$ on $\partial \Omega \backslash (\Gamma^n_{D_1} \cup \Gamma^n_{D_2} \cup \Gamma^n_{N_1})$.
• Figure 2: The material distribution of two (local) minima of the MBB beam problem Papadopoulos2021a. In white regions $\rho=1$ whereas in black regions $\rho = 0$. The Ginzburg--Landau parameters, as defined in \ref{['eq:ginzburg-landau']}, are $\epsilon = 1.90 \times 10^{-2}$, $\beta = 9 \times 10^{-3}$. Moreover, $\gamma = 0.535$, $\epsilon_{\text{SIMP}} = 10^{-5}$, $p_s = 3$, and the Lamé coefficients are $\mu = 75.38$ and $\lambda = 64.62$.
• Figure 3: A summary of the proof of \ref{['prop:FE:filtering-rho']}.

Theorems & Definitions (58)

• Remark 2.1
• Definition 2.1: $W^{1,p}$-type penalty methods
• Example 2.1: MBB beam
• Definition 2.2
• Proposition 2.1
• Proposition 2.2: Existence & uniqueness with a fixed $\rho$
• proof
• Theorem 2.1: Existence of a local minimizer to \ref{['cto']}
• proof
• Definition 2.3: Isolated local minimizer of \ref{['cto']}