Topology optimization concerning the mass distribution via filtered gradient flows on the Wasserstein space

TL;DR

This paper introduces a novel approach to topology optimization by formulating it as a minimization problem on the Wasserstein space, utilizing filtered gradient flows to improve mass distribution design.

Contribution

It develops a relaxed formulation of the non-convex optimization problem on Wasserstein space using the Neumann heat semigroup and introduces filtered Wasserstein gradient flows with error estimates.

Findings

Existence of minimizers for relaxed problems is proven.

A new numerical method for optimal mass distribution is proposed.

Error bounds between original and filtered gradient flows are established.

Abstract

In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the Wasserstein space by using the Neumann heat semigroup and prove the existence of minimizers of relaxed problems. Furthermore, we introduce the filtered Wasserstein gradient flow and derive the error estimate between the original Wasserstein gradient flow and the filtered one in terms of the Wasserstein distance. We also construct a candidate for the optimal mass distribution for a given fixed total mass and simultaneously obtain the shape of the material by the numerical calculation of filtered Wasserstein gradient flows.