Problems involving cracks are of particular importance in structural mechanics, and gave rise to many interesting mathematical techniques to treat them. The difficulties stem from the singularities of domains, which yield lower regularity of solutions. Of particular interest are techniques which allow us to identify cracks and defects from the mechanical properties. Long before advent of mathematical modeling in structural mechanics, defects were identified by the fact that they changed the sound of a piece of material when struck. These techniques have been refined over the years.

This volume gives a compilation of recent mathematical methods used in the solution of problems involving cracks, in particular problems of shape optimization. It is based on a collection of recent papers in this area and reflects the work of many authors, namely Gilles Frémiot (Nancy), Werner Horn (Northridge), Jiri Jarusek (Prague), Alexander Khludnev (Novosibirsk), Antoine Laurain (Graz), Murali Rao (Gainesville), Jan Sokolowski (Nancy) and Carol Ann Shubin (Northridge).

> On analysis of boundary value problems in nonsmooth domains

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive set, the original problem is reformulated as a shape optimization problem. The topological sensitivity of the new objective functional is used to estimate the "topology" of the active set. Then, for local correction purposes near the interface, a level set based shape sensitivity technique is employed. A numerical algorithm is devised, and a report on numerical test runs ends the paper.

> A shape and topology optimization technique for solving a class of linear complementary problems in function space

The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Soko?owski and Zochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.

> A level set method in shape and topology optimization for variational inequalities

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 of my thesis, the structure of the shape derivative for domains with cracks is studied. In the case of a smooth domain, 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. It is possible to model the perturbation of the solution of an elliptic partial differential equation due the creation of a small hole in its domain of definition by a perturbation of the elliptic operator as shown by Serguei Nazarov. In the Thesis, the model 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.

> Singularly perturbed domains in shape optimization (French)