Mesh Quality Metrics for Isogeometric Bernstein-B\'ezier Discretizations

TL;DR

This paper introduces new shape regularity metrics for curvilinear meshes in high-order finite element methods, supported by theoretical bounds, and proposes an optimization procedure to improve mesh quality for enhanced accuracy.

Contribution

It provides a formal definition of shape regularity for curvilinear meshes, derives computable bounds for quality metrics using Bernstein-Bézier form, and introduces a novel mesh optimization method.

Findings

Shape regularity significantly impacts high-order finite element accuracy.

The proposed quality metrics can be computed using Bernstein-Bézier representations.

Numerical results show the effectiveness of the mesh optimization procedure.

Abstract

High-order finite element methods harbor the potential to deliver improved accuracy per degree of freedom versus low-order methods. Their success, however, hinges upon the use of a curvilinear mesh of not only sufficiently high accuracy but also sufficiently high quality. In this paper, theoretical results are presented quantifying the impact of mesh parameterization on the accuracy of a high-order finite element approximation, and a formal definition of shape regularity is introduced for curvilinear meshes based on these results. This formal definition of shape regularity in turn inspires a new set of quality metrics for curvilinear finite elements. Computable bounds are established for these quality metrics using the Bernstein-B\'ezier form, and a new curvilinear mesh optimization procedure is proposed based on these bounds. Numerical results confirming the importance of shape regularity in the context of high-order finite element methods are presented, and numerical results demonstrating the promise of the proposed curvilinear mesh optimization procedure are also provided. The theoretical results in this paper apply to any piecewise-polynomial or piecewise-rational finite element method posed on a mesh of polynomial or rational mapped simplices and hypercubes. As such, they apply not only to classical continuous Galerkin finite element methods but also to discontinuous Galerkin finite element methods and even isogeometric methods based on NURBS, T-splines, or hierarchical B-splines.