The Target-Matrix Optimization Paradigm for High-Order Meshes

TL;DR

This paper introduces an advanced framework for optimizing high-order finite element meshes using the Target-Matrix Optimization Paradigm, enabling precise local quality control and global improvements, applicable to complex 2D and 3D meshes.

Contribution

It extends TMOP to high-order meshes, addressing isoparametric mappings, quality metrics, and practical concerns, with open-source implementation.

Findings

Enhanced mesh quality control in high-order applications

Successful application to complex test problems

Open-source implementation available

Abstract

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of Knupp. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Eulerian-Lagrangian (ALE) application. Our implementation is freely available in an open-source library form.