Numerical Approximation of Optimal Convex Shapes in $\mathbb{R}^3$

TL;DR

This paper develops two numerical methods for approximating optimal convex shapes in three-dimensional space, addressing the challenges of convex domain approximation under PDE constraints without relying on symmetry assumptions.

Contribution

It introduces a discrete convexity approach and a finite element-based method for conformal approximation of convex domains in 3D, expanding beyond previous symmetry-restricted techniques.

Findings

Both algorithms achieved similar results in shape optimization problems.

The methods effectively approximate convex domains without symmetry restrictions.

Numerical tests demonstrate the feasibility of the proposed approaches.

Abstract

In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape optimization problems to a two-dimensional setting. In the current research, two approaches for the approximation in $\mathbb{R}^3$ are considered. First, a notion of discrete convexity allows for a nearly convex approximation with polyhedral domains. An alternative approach is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains. The proposed algorithms were tested on shape optimization problems constrained by a Poisson equation and both algorithms achieved similar results.