Asymptotic Models for the Topological Sensitivity Versus the Topological Derivative

An approach, based on the refined method of matched asymptotic expansions, is proposed for the construction of asymptotic models for the topological sensitivity of the energy functional with respect to the creation of a small hole in the geometrical domain. It is shown that the asymptotic model provides more information for calculations than the topological derivative. INTRODUCTION The shape optimization theory [1] provides well established techniques for the investigation of shape optimization problems when the topology class of the geometrical domain under consideration is supposed to be fixed. At the same time, the shape optimization methods cannot produce useful criteria whether a topological change (for instance, the creation of a hole in the interior of the geometrical domain) will lead to a decreasing value of the shape functional or not. One such criterion [2] is based on the notion of the topological derivative whose importance in the topology optimization is now widely recognized. The present paper is devoted to analyzing the application of the topological derivative in shape and topology optimization problems which take into consideration the question of changing the topology class of the geometrical domain. More precisely, the so-called asymptotic model based on the refined asymptotic expansion is presented for the topological sensitivity of the Dirichlet integral in a special case of nucleation of a hole with the homogeneous Neumann boundary condition imposed on its boundary. It is known that the aim of the topological sensitivity analysis (see, for example, [3,4]) is to obtain the so-called topological asymptotic expansion of a given shape functional ) , ( v J with respect to the creation of a small opening ) ( 0 x of diameter ) ( O with the center at a given point 0 x in the geometrical domain ( < 0 is a small parameter). Let ) ( 0 x T be the ratio between the difference ) , ( ) , ( 0 v J u J , where u is the solution of the boundary value problem defined on the singularly perturbed domain ) ( \ 0 x = , and the area ) ( 0 x of the small