Fixed domain approaches in shape optimization problems

Abstract

This work is a review of results in the approximation of optimal design problems, defined in variable/unknown domains, based on associated optimization problems defined in a fixed 'hold-all' domain, including the family of all admissible open sets. The literature in this respect is very rich and we concentrate on three main approaches: penalization-regularization, finite element discretization on a fixed grid, controllability and control properties of elliptic systems. Comparison with other fixed domain approaches or, in general, with other methods in shape optimization is performed as well and several numerical examples are included.

A Penalization Method for the Elliptic Bilateral Obstacle Problem

“…Our approach is inspired from shape optimization techniques, but no shape optimization problem is used here although this is a known method in free boundary problems, [2]. One may compare the present approach to the recent works [8,19,22]. An efficient Lagrangian method together with a primal-dual active set strategy with regularization is studied in [13].…”

A direct algorithm in some free boundary problems

“…One may compare the present approach to the recent works [14,24,28]. An efficient Lagrangian method together with a primal-dual active set strategy with regularization is studied in [17] by using two perturbation parameters (except in the infeasible case) and nonlinear equations.…”

Shape optimization for the generalized Graetz problem

“…2 The sectional geometry of the generalized Graetz problem This requires a more subtle treatment than the cases where the outer boundary of the domain is varying. Shape sensitivity in the case of an interface between two materials was already studied, see Hettlich and Rundell (1998), Bernardi and Pironneau (2003), Pantz (2005), Conca et al (2009), Allaire et al (2009) and Neittaanmäki and Tiba (2012) for a recent review article. To our knowledge the eigenvalue problem for the Graetz operator, where some coefficient of the operator depends on an auxiliary partial differential equation, was not addressed previously.…”

“…Homogenization theory (Murat and Tartar 1978;Čerkaev and Kohn 1997) ensures that T * verifies the equation…”