Shape optimisation with nonsmooth cost functions: from theory to numerics

TL;DR

This paper develops a theoretical and numerical framework for shape optimization involving nonsmooth cost functions constrained by quasi-linear PDEs, highlighting new methods for computing derivatives and demonstrating improved convergence rates.

Contribution

It introduces a novel approach to derive the Eulerian semi-derivative for nonsmooth cost functions and provides practical algorithms for steepest descent directions in shape optimization.

Findings

Derived the Eulerian semi-derivative using averaged adjoint approach

Characterized stationary points for nonsmooth cost functions

Numerical results show higher convergence rates in the nonsmooth case

Abstract

This paper is concerned with the study of a class of nonsmooth cost functions subject to a quasi-linear PDE in Lipschitz domains in dimension two. We derive the Eulerian semi-derivative of the cost function by employing the averaged adjoint approach and maximal elliptic regularity. Furthermore we characterise stationary points and show how to compute steepest descent direc- tions theoretically and practically. Finally, we present some numerical results for a simple toy problem and compare them with the smooth case. We also compare the convergence rates and obtain higher rates in the nonsmooth case.