Shape optimization, level set methods on unstructured meshes and mesh evolution
Charles Dapogny, ALLAIRE, Grégoire, FREY, Pascal · OpenGrey (Institut de l'Information Scientifique et Technique) · 2013
The main purpose of this thesis is to propose a method for structural optimization which combines theaccuracy of featuring an exact description of shapes (i.e. with a mesh) at each iteration of the process withthe versatility of the level set method for tracking their evolution. Independently, we also study two problemsrelated to modeling in structural optimization.In the first, bibliographical part, we present several classical notions, together with some recent developmentsabout 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 optimalrepartition of several materials within a fixed structure (Chapter 4), and that of the robust optimization offunctions 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 onsimplicial (and possibly adapted) computational meshes. The computation of the signed distance functionto a domain is investigated in Chapter 6, and the resolution of the level set advection equation is presentedin 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 contextof shape optimization, relying on the numerical ingredients introduced in Chapters 7, 8, 9.