Efficient PDE constrained shape optimization based on Steklov-Poincar\'e type metrics

TL;DR

This paper introduces a Steklov-Poincaré type metric for PDE constrained shape optimization, harmonizing boundary and domain integral approaches to develop efficient algorithms that simplify derivative derivation.

Contribution

It proposes a novel Steklov-Poincaré type intrinsic metric derived from an outer domain metric, improving efficiency and reducing analytical complexity in shape optimization.

Findings

Develops a new intrinsic metric for shape optimization

Enhances computational efficiency of PDE constrained shape optimization

Simplifies the derivation process of shape derivatives

Abstract

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincar\'e type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.