A Discretize-Then-Optimize Approach to PDE-Constrained Shape Optimization

TL;DR

This paper introduces a discretize-then-optimize method for PDE-constrained shape optimization using triangular meshes, incorporating mesh quality penalization to ensure solvability and optimality, supported by theoretical existence proofs and numerical comparisons.

Contribution

It proposes a new penalized formulation for shape optimization on triangular meshes that guarantees the existence of a global optimum and analyzes different Riemannian metrics for mesh deformation.

Findings

The penalized problem admits a global optimal solution.

Different Riemannian metrics impact the steepest descent method.

Numerical experiments compare Euclidean, elasticity, and a novel complete metric.

Abstract

We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see arXiv:2012.05624. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of a globally optimal solution for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.