Topological derivative for PDEs on surfaces

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.