ON THE SECOND ORDER TOPOLOGICAL ASYMPTOTIC EXPANSION A.A. NOVOTNY AND J. ROCHA DE FARIA 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. 1. Introduction The topological sensitivity analysis gives the topological asymptotic expansion of a shape functional with respect to an infinitesimal domain perturbation, like the insertion of holes, inclusions or source term [3, 12]. The second term of this expansion provides the topological derivative, which has been applied in several problems, such as topology optimization, image processing and inverse problems. Analogously to the classical Taylor’s theorem, we can consider new terms in the topological asymptotic expansion of a smooth enough shape functional. As would be expected, we define the next one as the second order topological derivative. This procedure allows to deal with perturbation of finite size, which is an important requirement for practical applications. In our previous work [5] we have extended the method proposed in [11] to calculate the second order topological asymptotic expansion for the Laplace’s equation; considering the total potential energy as the shape functional, the state equation as the constraint and taking into account two different homogeneous boundary conditions on the hole: Neumann and Dirichlet. In particular, we will demonstrate that the second order topological derivative associated to the Neumann’s case 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. Concerning 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. However, in this last case, the second order topological derivative is given explicitly in terms of the state solution. Furthermore, for the sake of simplicity, in [5] we have disregarded all these implicit terms, leading to a discrepancy in the topological asymptotic expansion. Therefore, in the present paper we calculate the complete second order topological asymp- totic expansion for the two cases under consideration. Then, we present two simple examples with analytical solutions, where we discuss the effects of the ad hoc approximations adopted Key words and phrases. topological asymptotic expansion, (first order) topological derivative, second order topological derivative. 1