A novel $W^{1,\infty}$ approach to shape optimisation with Lipschitz domains

TL;DR

This paper introduces a new $W^{1, ext{infinity}}$-based shape optimisation method for Lipschitz domains, demonstrating its effectiveness and superiority over traditional Hilbert space approaches through numerical experiments.

Contribution

The paper develops a novel shape derivative approach in the $W^{1, ext{infinity}}$ topology for Lipschitz domains, including existence proofs and numerical validation.

Findings

The $W^{1, ext{infinity}}$ approach effectively handles shapes with corners.

Numerical experiments show superiority over Hilbert space methods.

The method relates to the $ ext{infinity}$-Laplacian and optimal transport.

Abstract

This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape functional in the $W^{1,\infty}-$ topology. The idea of our approach is demonstrated for shape optimisation of $n$-dimensional star-shaped domains, which we represent as functions defined on the unit $(n-1)$-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the $W^{1,\infty}-$ topology. We also note that shape optimisation in this context is closely related to the $\infty-$Laplacian, and to optimal transport, where we highlight the latter in the numerics section. We present several numerical experiments in two dimensions illustrating that our approach seems to be superior over a widely used Hilbert space method in the considered examples, in particular in developing optimised shapes with corners.