Removing holes in topological shape optimization

¹ MIP, UMR 5640, INSA Département de Mathématiques Complexe Scientifique de Rangueil, 31077 Toulouse Cedex 4, France; philippe.guillaume@insa-toulouse.fr
² ENIT-LAMSIN et Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia; maatoug.hassine@enit.rnu.tn

Received: 14 December 2005
Revised: 2 June 2006
Revised: 19 July 2006

Abstract

The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical examples are presented: the first one compares topological optimization with standard shape optimization, and the second one, issued from a lake oxygenation problem, illustrates the use of the new asymptotic expansion.