The Topological Asymptotic for the Helmholtz Equation with Dirichlet Condition on the Boundary of an Arbitrarily Shaped Hole

The aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in the domain. In this paper, such an expansion is obtained for the Helmholtz equation, in two and three space dimensions, with a Dirichlet condition on the boundary of an arbitrarily shaped hole. In this case, the main difficulty is related to the nonhomogeneous symbol of the Helmholtz operator. In the numerical part of this work, we will show that the topological sensitivity method is very promising for solving shape inverse problems in electromagnetic applications.