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.
Highlights
The finite element method is well established as a method for solving boundary value problems approximately
A new method, in which selected edge terms of the interior penalty (IP) formulation are under-integrated, circumvents the problem, as shown through an analytical proof that the new method is locking-free for rectangular elements, and through numerical examples that demonstrate the optimal performance of this formulation
Displacement approximations use the nodal solution values, while for those of the stress approximations, stress values are calculated at quadrature points and a projection is done over the domain, onto the nodal points, giving continuous stress fields
Summary
The finite element method is well established as a method for solving boundary value problems approximately. A new method, in which selected edge terms of the IP formulation are under-integrated, circumvents the problem, as shown through an analytical proof that the new method is locking-free for rectangular elements, and through numerical examples that demonstrate the optimal performance of this formulation. In a more recent computational paper, Bayat et al [3] have shown several variants of IIPG to be volumetrically locking-free when selected edge terms are under-integrated These authors consider limited test cases only and their results are inconclusive regarding the general case. As a second component of this work we study the accuracy of the stress field approximation obtained from the new IP methods, considering both error convergence rates and approximation quality at individual refinement levels
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