Computational comparison of surface metrics for PDE constrained shape optimization

TL;DR

This paper compares different surface metrics for PDE-constrained shape optimization, analyzing their impact on convergence and mesh quality through computational experiments on a Stokes flow problem.

Contribution

It introduces a computational comparison of Laplace-Beltrami and Steklov-Poincaré metrics within a shape optimization framework involving PDE constraints.

Findings

Steklov-Poincaré metrics improve convergence stability.

Mesh quality is better maintained with certain metrics.

Different metrics influence the efficiency of the shape optimization process.

Abstract

We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace-Beltrami type based metrics are compared with Steklov-Poincar\'e type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.