PDE-Constrained High-Order Mesh Optimization

TL;DR

This paper introduces a comprehensive PDE-constrained high-order mesh optimization framework that enhances mesh quality and solution accuracy through a convex combination of quality metrics, error measures, and regularization, applicable to various PDEs.

Contribution

It develops a novel optimization-based approach integrating mesh quality, PDE accuracy, and regularization, adaptable to any PDE with adjoint operators, improving high-order mesh optimization.

Findings

Mesh quality improved by up to 10 times.

Error reduction demonstrated for Poisson and elastostatic problems.

Framework is general for various PDEs and mesh types.

Abstract

We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regularization. We adopt the Target-Matrix Optimization Paradigm to control geometric properties across the mesh, independent of the PDE of interest. To incorporate the accuracy of the PDE solution, we introduce error measures that control the finite element discretization error. The implicit dependence of these error measures on the mesh nodal positions is accurately captured by adjoint sensitivity analysis. Additionally, a convolution-based gradient regularization strategy is used to ensure stable and effective adaptation of high-order meshes. We demonstrate that the proposed framework can improve mesh quality and reduce the error by up to 10 times for the solution of Poisson and linear elasto-static problems. The approach is general with respect to the dimensionality, the order of the mesh, the types of mesh elements, and can be applied to any PDE that admits well-defined adjoint operators.