Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions

Abstract

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also 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].…”

Optimization of a plate with holes

“…Example 1. This is inspired by the example 2 from [13], but the second order elliptic equation is replaced by (2.1)-(2.2). We have D =] − 1, 1[×] − 1, 1[, the load f = 3, the cost function j(g) = 1 2 16 .…”

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.…”

“…The algorithm uses just linear elliptic equations in the whole domain D. The type of penalization term from Step 2 may be compared with the approach developed in shape optimization problems in [24]. A classical nonlinear penalization term for solving the obstacle problem is [12].…”