The main purpose of this thesis is to propose a method for structural optimization which combines the accuracy of featuring an exact description of shapes (i.e. with a mesh) at each iteration of the process with the versatility of the level set method for tracking their evolution. Independently, we also study two problems related to modeling in structural optimization. In the first, bibliographical part, we present several classical notions, together with some recent developments about the three main issues of this thesis - namely level set methods (Chapter 1), shape optimization (Chapter 2), and meshing (Chapter 3). The second part of this manuscript deals with two issues in shape optimization, that of the optimal repartition of several materials within a fixed structure (Chapter 4), and that of the robust optimization of functions depending on the domain when perturbations are expected over the considered mechanical model. In the third part, we study the design of numerical schemes for performing the level set method on simplicial (and possibly adapted) computational meshes. The computation of the signed distance function to a domain is investigated in Chapter 6, and the resolution of the level set advection equation is presented in Chapter 7. The fourth part (Chapter 8) is devoted to the meshing techniques introduced in this thesis. Eventually, the last part (Chapter 9) describes the proposed strategy for mesh evolution in the context of shape optimization, relying on the numerical ingredients introduced in Chapters 7, 8, 9.
