Convergence in the incompressible limit of new discontinuous Galerkin methods with general quadrilateral and hexahedral elements

TL;DR

This paper demonstrates that under-integration in interior penalty discontinuous Galerkin methods effectively prevents locking on general quadrilateral and hexahedral meshes, ensuring accurate displacement and stress approximations in nearly incompressible materials.

Contribution

It extends the use of under-integration in IP discontinuous Galerkin methods to general quadrilateral and hexahedral meshes, ensuring convergence in the incompressible limit.

Findings

Under-integration eliminates locking on general quadrilateral meshes.

Uniform convergence with respect to the compressibility parameter is achieved.

Postprocessed stress approximations show good convergence in the incompressible limit.

Abstract

Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here it is demonstrated through an extensive numerical investigation that the effect of using under-integration carries over successfully to meshes of more general quadrilateral elements, as would likely be used in practical applications, and results in accurate displacement approximations. Uniform convergence with respect to the compressibility parameter is shown numerically. Additionally, a stress approximation obtained here by postprocessing shows good convergence in the incompressible limit.