On sensitivities regarding shape and topology optimization as derivatives on Wasserstein spaces

TL;DR

This paper explores the use of optimal transport and Wasserstein space to model shape and topology optimization problems, linking differential calculus on these spaces to sensitivities in design, and framing optimization as gradient flows.

Contribution

It introduces a novel approach by applying Wasserstein space differentials to shape and topology optimization, connecting optimal transport theory with design sensitivities.

Findings

Establishes a framework linking Wasserstein differentials to shape sensitivities

Reframes optimization as gradient flows on Wasserstein space

Provides theoretical insights into optimal transport-based design methods

Abstract

In this paper, we apply the framework of optimal transport to the formulation of optimal design problems. By considering the Wasserstein space as a set of design variables, we associate each probability measure with a shape configuration of a material in some ways. In particular, we focus on connections between differentials on the Wasserstein space and sensitivities in the standard setting of shape and topology optimization in order to regard the optimization procedure of those problems as gradient flows on the Wasserstein space.