Broadly speaking, shape optimization is the discipline of optimizing the design of a domain in two or three space dimensions, with respect to an objective and under certain constraints, which are formulated as functions of the domain.
In applications, these functions depend on the shape via to solution to a partial differential equation encoding the physical aspects of the problem under scrutiny, which is complemented with boundary conditions accounting for the effects the exterior medium. Thus, a mechanical structure is characterized by its displacement, solution to the linear elasticity system, equipped with boundary conditions of homogeneous Dirichlet (modeling the fixation regions of the structure), homogeneous Neumann (for the traction-free boundaries), or inhomogeneous Neumann (boundaries where loads are applied) types.
Most often, only one part of the boundary of the shape is optimized -- typically, the traction-free boundary in structural mechanics. The aim of this work is, on the contrary, to consider the optimization of these regions bearing the boundary conditions of the physical problem at play. This question is considered from two complementary viewpoints. - On the one hand, we investigate the shape derivative of a function of the domain in the sense of Hadamard, when the involved deformations do not vanish where the boundary conditions change types: this allows to optimize how the regions bearing boundary conditions may ``slide'' along the boundary of the shape. - On the other hand, we investigate the sensitivity of the solution to a physical problem (and that of a related quantity of interest) when a small region bearing a certain type of boundary conditions (typically, of homogeneous Dirichlet type) is nucleated within a region bearing other conditions (e.g. of Neumann type). This paves the way to a notion of ``topological derivative'' describing the change of boundary conditions on the boundary of a given shape.
Several numerical applications of these developments will be discussed.These works have been realized in collaboration with Eric Bonnetier, Carlos Brito-Pacheco, Nicolas Lebbe, Edouard Oudet and Michael Vogelius.
Romain Duboscq, David Lafontaine