Shape and Topological Optimization: Analysis and Differential Calculus

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The theme of this program is the study of the mathematical aspects of shapes and topologies in the modeling, design, control, and identification of physical, technological, and biomedical systems. Fundamental ideas come from systems theory, optimization, and control theory, but the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object or simply a subset of the Euclidean space. This area of research has an important potential in applications and in responding to challenging issues in many different areas: optimal design of mechanical parts (automotive industry), positioning of sensors and actuators, control of the position of the free boundary in material sciences, active control of noise, image processing, free and moving boundary problems, design of medical devices, drug release, design and control of thin structures, control of the drag by small changes in the shape of the wing of an aircraft, optimal swimming. To consider optimization/design problems and their numerical simulation/solution, a good analytical framework is essential. It must be sufficiently broad to accommodate sets that are not locally identifiable to the Euclidean space with a smooth structure. In this spirit two types of metric spaces have been constructed: spaces of diffeomorphisms corresponding to images of a fixed set (e.g. Courant metrics) and spaces of set-parametrized functions such as the characteristic, distance (Hausdorff metric), or oriented distance functions that allow topological changes and cracks. The spaces of the first type are infinite dimensional Finsler manifold while those of the second type are of a more complex nature. Most of the known compactness theorems are related to spaces of the second type (e.g. uniform cone or cusp property, Caccioppoli sets). The next ingredient to characterize optimal sets is a good differential calculus that includes the chain rule. The Eulerian derivative (shape derivative) is a differential that meets the expectations for the spaces of the first type. The tangent space to spaces of the second type is not linear and only semi-differentials such as the Hadamard semi-differential can be obtained. For instance, the Abelian group of characteristic functions of Lebesgue measurable sets contains measures and distributions that are semi-tangents. The measures are associated with the so-called topological derivative obtained from the dilatation of a point. This notion extends to the dilatation of rectifiable sets and sets of positive reach of Federer via the d-dimensional Minkowski content. They provide additional necessary optimality conditions. Extensive numerical computations show its pertinence in Mechanics. Finally, theorems on the parametric derivative of the minimax of a Lagrangian are used for state constrained objective functions.