Abstract : In shape optimization, the main results concerning the case of domains with
smooth boundaries and smooth perturbations of these domains are well-known, whereas the
study of non-smooth domains, such as domains with cracks for instance, and the study of singular perturbations such as the creation of a hole in a domain is more recent and complex. This
new field of research is motivated by multiple applications, since the smoothness assumptions
are not fulfilled in the general case. These singular perturbations can be handled now with
new and efficient tools like topological derivative.
In the first part, the structure of the shape derivative for domains with cracks is studied.
In the case of a smooth domain, with boundary of class C1 or lipschitzian for instance, the
derivative depends only on the perturbations of the boundary of the domain in the normal
direction. This structure theorem is no longer valid for domains with cracks. We extend here
the structure theorem to domains with cracks in any dimension for the first and second derivatives. In dimension two, we get the usual result, i.e. the shape derivative depends also on
the tangential components of the deformation at the tips of the crack. In higher dimension, a
new term appears in addition to the classical one, coming from the boundary of the manifold
representing the crack.
In the second part, the singular perturbation of a domain is approximated by using self
adjoint extensions of operators. This approximation is first described, then it is applied to a
shape optimization problem. An approximated energy functional can be defined for this model problem, and we obtain in particular the usual formula of the topological derivative.
In the third part, a numerical application of the topological and shape derivatives is proposed for a non-linear problem. The problem consists in maximizing the energy associated to
a Signorini problem in a domain . The evolution of the domain is done with the help of a
levelset method to handle easily topological changes.