Topological sensitivity analysis for time-dependent problems

¹ Laboratoire d'analyse non-linéaire et géométrie, Faculté des sciences, 33 rue Louis Pasteur, 84000 Avignon, France; samuel.amstutz@univ-avignon.fr
² Institut Élie Cartan de Nancy, Nancy-Université, CNRS, INRIA, BP 239, 54506 Vandœuvre-lès-Nancy cedex, France; Takeo.Takahashi@iecn.u-nancy.fr
³ Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria; boris.vexler@oeaw.ac.at

Received: 26 June 2006
Revised: 5 December 2006

Abstract

The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.