Topological derivative for PDEs on surfaces

TL;DR

This paper develops a method using topological derivatives and level set techniques to optimize material distribution on surfaces constrained by PDEs, enabling numerical solutions for surface topology optimization.

Contribution

It introduces a novel approach combining topological derivatives with level set methods for PDE-constrained shape optimization on surfaces.

Findings

Derived the topological derivative for PDEs on surfaces.

Proposed a numerical scheme using level set methods.

Demonstrated the approach on example problems.

Abstract

In this paper we study the problem of the optimal distribution of two materials on smooth submanifolds $M$ of dimension $d-1$ in $\mathbf R^d$ without boundary by means of the topological derivative. We consider a class of shape optimisation problems which are constrained by a linear partial differential equation on the surface. We examine the singular perturbation of the differential operator and material coefficients and derive the topological derivative. Finally, we show how the topological derivative in conjunction with a level set method on the surface can be used to solve the topology optimisation problem numerically.