PDE constrained shape optimisation with first-order and Newton-type methods in the $W^{1,\infty}$ topology

TL;DR

This paper introduces a shape optimisation framework using $W^{1, Infty}$ topology with steepest descent and Newton algorithms, applied to PDE constrained problems, offering a novel approach compared to classical methods.

Contribution

It develops a $W^{1, Infty}$ topology-based shape optimisation method with new algorithms, extending previous work on star-shaped domains to PDE constrained problems.

Findings

The proposed algorithms outperform classical Hilbert space methods.

Comparison shows advantages of $W^{1, Infty}$ approach over $p$-approximations.

Numerical results demonstrate effectiveness in PDE constrained shape optimisation.

Abstract

We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape optimisation problems. Our work is built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a $W^{1,\infty}$ framework for star-shaped domains is proposed. To illustrate our approach we present a selection of PDE constrained shape optimisation problems and compare our findings to results from so far classical Hilbert space methods and recent $p$-approximations.