Shape optimization for non-smooth geometry in two dimensions
The derivative of a functional \(J(u,\Omega)\) with respect to the domain \(\Omega\), where \(u\) is the solution of a boundary value problem in \(\Omega\), was broadly studied when \(\Omega\) is a smooth domain. Let \(\Omega\) be a non-smooth domain in \(\mathbb{R}^2\) such that its boundary \(\Gamma\) exhibits singularities at points \(a_i\), and let \(u\) be the solution of a boundary value problem in \(\Omega\). This paper shows how to differentiate such functional, and how to deduce the contributions coming from singularities. The main advantage of this expression for the functional gradient is to avoid the computation of the curvature of the boundary, arising generally for the gradient for non-singular domains. From the numerical point of view, this method allows to compute the gradient of a functional for polygonal domains in \(\mathbb{R}^2\), using a finite element approximation. Practically, one considers a continuous gradient for a smooth domain, that one will discretize, or a gradient with respect to nodes, which is a discrete gradient for a discrete domain. Then the desired gradient lies between these two gradients, being a continuous gradient for a discrete domain.
scientific article; zbMATH DE number 1506859