Numerical Approximation of Optimal Convex Shapes

Abstract

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations. A numerical algorithm is devised that iteratively solves the discrete formulation. Numerical experiments show that optimal convex shapes are generally non-smooth and that three-dimensional problems require an appropriate relaxation of the convexity condition.

Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation

“…To prove existence, we adapt the strategy from [8]. However, due to the lack of homogeneous Dirichlet boundary conditions, which allow for trivial extensions in H 1 ( Q), we need to incorporate a convergence result for special functions of bounded variations.…”

“…We adopt an approach similar to [8], where the admissible domains are obtained from a discrete deformation of a given convex reference domain. A convex polygonal domain ω h with a regular triangulation T h is optimized by moving the vertices of the triangulation.…”

“…Rather than deforming the entire triangulation, we deform the boundary Γ out of ω h , and then generate a triangulation of ω h , to calculate the objective values or to find the deformation field. Equivalently, we could also say that we add remeshing of the domain to the deformed triangulations of [8]. So rather than trying to solve (P h ), we instead solve the problem as follows.…”

“…This comes with a higher computational cost, due to the regular generation of the triangulation. Since the deformation of the entire triangulation (as in [8]) has often led to a degeneration of the triangulation and the boundary nodes in the conducted experiments, a frequent generation of a new triangulation was often necessary in either versions.…”

“…We prove existence of an optimal domain in Section 3 and derive the two-dimensional problem and its numerical approximation and comment on the stability of the numerical scheme in Section 4. In Section 5 we establish a framework for the numerical approximation of optimal convex domains described in [8] but adjusted for rotational symmetry, which can be applied to different shape optimization problems as well. The numerical experiments are evaluated in Section 6.…”