Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût

Abstract

This paper is devoted to the optimum design problems of distributedsystems. The general formulation is the following: If y\sb{\Omega}denotes the solution of a P.D.E. in a domain Ω, we want tominimize a functional J(Ω,y\sb{\Omega}) with respect to aclass of open sets Ω. \par The main purpose of the paper isthe presentation of techniques to compute the gradient of the functionalwith respect to the domain Ω. First the case of a problemdescretized by a finite element method is considered. Second thecontinuous case is studied, in particular by introducing a Lagrangianfunction. Several examples illustrate this method. Finally the casewhen the domain Ω is depending on a set of parameters is considered.Different examples are explained (polar coordinates, cartesian coordinates,splines,...). \par The article includes a complete bibliography onoptimum design problems.