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

• Published in: Numerische Mathematik Vol. 157; no. 1; pp. 213 - 248
• Main Author: Papadopoulos, Ioannis P. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2025; Springer Nature B.V
• Subjects: Convergence; Elastic analysis; Isotropic material; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Regularization; Simulation; Theoretical; Topology optimization; 65K10; 90C26; 74F99; 65N30
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-024-01438-3

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 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.