Adaptive Shape Optimization Method

Abstract

The principal objective of this work is to introduce an adaptive strategy to monitor the convergence rate of a Newton-like method in shape optimization. Shape optimization problems are characterized by a cost function and a partial differential equation (P.D.E.) which depend both of the geometrical domain. Typically we want to compute a shape * such that $$ \Omega ^ * = \arg \min \left\{ {G\left( {\Omega ,\varphi _\Omega } \right):\Omega \in \mathcal{O}} \right\} $$ The scalar potential is the solution in a Sobolev space H of an elliptic problem, the state equation. The function \( G\left( {\Omega ,\varphi _\Omega } \right):\mathcal{O} \times \mathcal{H} \to \mathbb{R} \) is the cost function. The set of admissible domains O is characterized by geometrical and regularity constraints, for analytical calculus we consider domains of class C 2.