Higher-Order Moving Mesh Methods for PDE-Constrained Shape Optimization

Alberto Paganini,Florian Wechsung,Patrick E Farrell

doi:10.1137/17m1133956

Abstract

We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of PDE-constraints. This shape optimization method is based on discretized deformation diffeomorphisms and allows for arbitrarily high resolution of shapes with arbitrary smoothness. Numerical experiments show that this method allows the solution of PDE-constrained shape optimization problems to high accuracy.

Highlights

Shape optimization problems are optimization problems where the control to be optimized is the shape of a domain
The shape functional depends on the shape of a domain Ω ⊂ Rd, and on the solution u of a boundary value problem (BVP) posed on Ω, in which case (1) becomes find Ω∗ ∈ argmin J (Ω, uΩ) subject to uΩ ∈ V (Ω), aΩ(uΩ, v) = fΩ(v) for all v ∈ W (Ω), where (2b) represents the variational formulation of a BVP that acts as a PDEconstraint
The function uΩ cannot be computed analytically. Even approximating it with a numerical method is challenging because the computational domain of the PDE-constraint is the unknown variable to be solved for in the shape optimization problem

Summary

Introduction

Shape optimization problems are optimization problems where the control to be optimized is the shape of a domain. The resulting method is formally equivalent to the one presented in this work, but implies hard-coding of geometric transformation into shape functionals and PDE-constraints (which is problem dependent), and requires the derivation of Frechet derivatives in Lagrangian coordinates (which are usually not considered in the shape optimization literature). The authors of [3, 24] suggest the use of linear Lagrangian finite elements built on two nested meshes: a coarser one to discretize the geometry and a finer one to solve the state equation They report that this reduces the presence of spurious oscillations in the optimized shape. We consider a PDE-constrained shape optimization problem that admits stable minimizers We use this test case to investigate the approximation properties of the algorithm presented in section 6 for different discretizations of control and state variables.

Misfit functional J

Jerr Jerr

Mesh linear

Ωo Γ

Conclusion