Shape Analysis and Optimization in Distributed Parameter Systems

Abstract

The originality in Shape Optimization and Control problems is that the design or control variable is no longer a vector of parameters or functions but the shape of a geometric domain. They include engineering applications to Shape and Structural Optimization, but also original applications to Image Segmentation, Control Theory (optimal location of the geometric support of sensors and actuators), Stabilization of membranes and plates by boundary variations, and World Health Problems such as the control of larvae of black flies in rivers and running waters which are the carrier of devastating diseases in West Africa.The characterization and efficient computation of optimal shapes require a Shape Calculus which differs from its analog in vector spaces. It is necessary to make sense of the notions of "Shape Gradients" and eventually "Shape Hessians" which are the basic tools to obtain necessary and/or sufficient conditions or to do efficient computations. One important mathematical development was the introduction of a method based on deformations of the domain by a Velocity Field. With this approach a precise mathematical meaning was given to the notions of Shape Gradients and Hessians. The interest for this type of analysis increased when it was found that Discrete Gradients (in a Finite Element Problem) could be obtained from Continuous Gradients by an appropriate choice of Velocity Fields. This unified continuous and discrete approaches, considerably simplified the computation of discrete shape gradients and made it possible to deal with parametrized shapes all in the same framework. Finally the use of theorems on the differentiability of the minimum or the saddle point of a functional with respect to a parameter provided very efficient and powerful tools to easily obtain Shape gradients and Hessians without the usual associated study of the Shape derivative of the state.In this paper we survey selected elements of the theory and applications of Shape Sensitivity Analysis and Optimization.