Shape optimization for non-smooth geometry in two dimensions Original Research Article

The derivative of a functional J(u,Ω) with respect to the domain Ω, where y is solution of a boundary value problem in Ω, is broadly studied when Ω is a smooth domain. Let Ω be a non-smooth domain in Rn (n=2) such that its boundary Γ presents singularities at some points ai, and let u be the solution of a boundary value problem in Ω. We will see in this paper how to differentiate such functional, and deduce the contributions coming from the signularities. The principal objective of this gradient is to avoid the computation of the curvature of the boundary arising generally in the continuous gradient. From a numerical point of view, this method allows us to compute the gradient of a functional with respect to discrete domain, whose boundary in R2 is polygonal, used in a finite element approximation. From a practical point of view, one generally considers the continuous gradient on a smooth domain, that one will discretize, or the gradient with respect to the nodes, which is the discrete gradient on a discrete domain. The gradient we will develop is between these two gradients. Itʹs a continuous gradient on a discrete domain.