On the second order topological asymptotic expansion

Abstract

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal (non smooth) domain perturbation at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. However, for practical applications, we need to insert perturbations of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we calculate first as well as second order topological derivatives for the total potential energy associated to the Laplace’s equation, when the domain is perturbed with a hole. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). In the Neumann’s case, the second order topological derivative depends explicitly on higher-order gradients of the state solution and also implicitly on the point where the hole is nucleated through the solution of an auxiliary problem. On the other hand, in the Dirichlet’s case, the first order topological derivative depends explicitly on the state solution as well as implicitly through the solution of an auxiliary problem, and the second order topological derivative depends only explicitly on the solution associated to the original problem. Finally, we present two simple examples showing the influence of both terms in the second order topological asymptotic expansion for each case of boundary condition on the hole.