Shape optimization – SHAPO

This project comes within the scope of shape optimization, which can be seen as the study of optimization problems whose unknown is a set, and arise generally from applied fields like physics, biology and engineering.

We propose to deal with four main objectives, whose common vision is to reduce the gap between academic models, highly developed (especially by members of the project) in recent years, and more realistic models, whose caracteristics were put aside so far :
- the first objective deals with the energy functional itself, whose academic versions have often been reduced to the study of the Laplace operator with Dirichlet boundary condition. Even though this study was very productive, it leads to simplifying hypotheses. We would like to confront the usual questions of optimization (study of symmetries, existence and regularity theories) to more realistic energies, for example involving drifted or non-local operators, or in view of industrial applications, functionals taking into account uncertainties on the data and the manufacturing processes.
- the second point focuses on the constraints on admissible shapes; from an academic point of view, we can quote connectedness, convexity, or diameter constraints, which lead to difficulties in the analysis of optimality conditions (non-locality, non-differentiability) and in their numerical treatment as well. From the point of view of applications, these questions are naturally related to the recent and strong development of additive manufacturing (also known as "3d printing") as the layer by layer construction process can be seen also as a non-local constraint.
- the third objective consists in the development of a fairly new area of research, taking a step back from optimization itself. Indeed, we would like to develop a theory of gradient flows for shapes, generalizing the mean curvature flow, that can be seen as a gradient flow for the perimeter functional. The link with shape optimization lies for example within the Almgren-Taylor-Wang scheme approach, and we will bring our expertise in existence and regularity theories in shape optimization to develop such theory, with the new difficulty of analyzing the apparition of singularities along the flow.
- the last objective is transverse to the rest of the project, and involves the numerical resolution of the problems linked to the previous objectives. These new models will bring new numerical difficulties we want to deal with. We see this objective transversally as the problems under study will often require a parallel study of theoretical and numerical aspects: for example, numerical results will allow to measure the behavior of optimal shapes with the parameters, and suggest reasonable questions from the theoretical point of view. We also plan to tackle some conjectures (as the Polya conjecture about the Faber-Krahn inequality among polygonal domains) with an hybrid theoretical-numerical approach.

The project is composed by 22 members (disseminated on 4 centers and 10 laboratories), and 10 PhD/Post-doc students currently advised by the members of the project and whose subject is directly linked with the aforementioned objectives. We bring various expertise (either with different theoretical and numerical aspects of shape optimization, or with related fields as calculus of variations, non-smooth analysis, optimal control...) in order to bring at the same time a strong basis in the field, and new methods and ideas.