Shape optimization for non-smooth geometry in two dimensions

The derivative of a functional J(u, Omega) with respect to the domain Omega , where y is solution of a boundary value problem in Omega, is broadly stud ied when Omega is a smooth domain. Let Omega be a non-smooth domain in R-n (n = 2) such that its boundary Gamma presents singularities at some points a(i), and let u be the solution of a boundary value problem in Omega. We wi ll see in this paper how to differentiate such functional, and deduce the c ontributions coming from the signularities. The principal objective of this gradient is to avoid the computation of the curvature of the boundary aris ing generally in the continuous gradient. From a numerical point of view, t his method allows us to compute the gradient of a Functional with respect t o discrete domain, whose boundary in R-2 is polygonal, used in a finite ele ment approximation. From a practical point of view, one generally considers the continuous gradient on a smooth domain, that one will discretize, or t he gradient with respect to the nodes, which is the discrete gradient on a discrete domain. The gradient we will develop is between these two gradient s. It's a continuous gradient on a discrete domain. (C) 2000 Published by E lsevier Science S.A. All rights reserved.