The gradient based topological optimization tools introduced during thelast 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 remainunchanged, at least during the topological phase of the optimizationalgorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not forimproving 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.


In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized lake.