ANTOINE LAURAIN
PHD IN MATHEMATICS
SHAPE AND TOPOLOGY OPTIMIZATION

Keywords: shape optimization, topology optimization, asymptotic analysis, shape and topological derivative, free boundary problems

## Shape Optimization for Superconductors Governed by H(Curl)-Elliptic Variational Inequalities

This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization problem is to find an optimal superconductor shape which minimizes a certain cost functional under a given target on the electric field over a specific domain of interest. For the governing PDE-model, we consider an elliptic curl-curl variational inequality (VI) of the second kind with an L1-type nonlinearity. In particular, the non-smooth VI character and the involved H(curl)-structure make the corresponding shape sensitivity analysis challenging. To tackle the non-smoothness, a penalized dual VI formulation is proposed, leading to the Gateaux differentiability of the corresponding dual variable mapping. This property allows us to derive the distributed shape derivative of the cost functional through rigorous shape calculus on the basis of the averaged adjoint method. Based on the achieved theoretical findings, we propose 3D numerical solutions, realised using a level set algorithm and a Newton method with the Nédélec edge element discretization. Numerical results indicate a favourable and efficient performance of the proposed approach for a specific HTS application in superconducting shielding. > Shape Optimization for Superconductors Governed by H(Curl)-Elliptic Variational Inequalities

## Analysis of boundary value problems in nonsmooth domains

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

## Linear complementary problems

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

## Level set method and variational inequalities

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

## PHD Thesis: Singularly perturbed domains in shape optimization

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. > Singularly perturbed domains in shape optimization (French)