Topological Derivative for PDEs on Surfaces

In this paper we study the problem of the optimal distribution of two materials on C2 submanifolds M of dimension d-1 in Rd by means of the topological derivative. We consider a class of shape optimization problems which are constrained by a linear partial differential equation on the surface. We examine the configurational perturbation of the differential operator and material coefficients and derive the corresponding 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 optimization problem numerically.