Numerical approximation of optimal convex shapes

TL;DR

This paper develops a numerical method for approximating optimal convex shapes in PDE-constrained shape optimization, demonstrating convergence in 2D and revealing the non-smooth nature of solutions.

Contribution

It introduces a convergent discretization approach for convex shape optimization problems with PDE constraints and explores the effects of convexity relaxation in 3D.

Findings

Optimal convex shapes are generally non-smooth.

Convergence of discretizations is proven in two dimensions.

3D problems require relaxed convexity conditions.

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.