Mathematical Sciences Department Numerical Methods Seminar - Dohyun Kim, Brown University

Title: SiMPL Method for Density Based Topology Optimization

Abstract: We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). Density-based topology optimization is a design process that seeks to find the optimal distribution of material in a selected physical domain subject to multiple constraints. The density field represents the design, taking values zero in the void regions and one in the parts of the domain occupied with solid material. The SiMPL method is inspired by the proximal Galerkin method for variational inequalities introduced in [1]. The optimization method utilizes a so-called mirror descent algorithm tailored to the bound constraints on the density field. We update the design based only on first-order derivative information of the objective, significantly simplifying practical implementations. The bounds on the density field are enforced with the help of the (negative) Fermi-Dirac entropy used to define the Bregman divergence in the mirror descent updates. The update rule is simplified with the help of a latent variable. In this talk, we analyze the SiMPL method in infinite dimensional non-reflexive Banach spaces. In addition, numerical experiments demonstrate apparent mesh independent convergence of the algorithm and superior performance over the two most popular first-order methods in topology optimization: OC and MMA.

[1] B. Keith and T. M. Surowiec, Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints, 2023, https://arxiv.org/abs/2307.12444

[2] B. Keith, D. Kim, B. Lazarov, and T. M. Surowiec, Analysis of SiMPL method for density-based topology optimization, 2024, https://arxiv.org/abs/2409.19341