On Convergence in Elliptic Shape Optimization

On Convergence in Elliptic Shape Optimization

This paper is aimed at analyzing the existence and convergence of approximate solutions in shape optimization. Two questions arise when one applies a Ritz-Galerkin discretization to solve the necessary condition: does there exists an approximate solution and how good does it approximate the solution of the original infinite dimensional problem? We motivate a general setting by some illustrative...

Shape optimization of a nonlinear elliptic system

Shape optimization problem for a nonlinear elastic plate governed by von Karman equation is considered. Using material derivative method, sensitivity analysis of the solution to the von Karman system with respect to the variation of the domain is performed and a necessary optimality condition is derived.

Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems

In this paper, several efficient rearrangement algorithms are proposed to find the optimal shape and topology for elliptic eigenvalue problems with inhomogeneous structures. The goal is to solve minimization and maximization of the k−th eigenvalue and maximization of spectrum ratios of the second order elliptic differential operator. Physically, these problems are motivated by the frequency con...

Shape Optimization Problems for Eigenvalues of Elliptic Operators

We consider a general formulation for shape optimization problems involving the eigenvalues of the Laplace operator. Both the cases of Dirichlet and Neumann conditions on the free boundary are studied. We survey the most recent results concerning the existence of optimal domains, and list some conjectures and open problems. Some open problems are supported by efficient numerical computations.

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)