On the use of reduced integration in combination with discontinuous Galerkin discretization: application to volumetric and shear locking problems

Hamid Reza Bayat,Stefanie Reese,Steffen Kastian,Stephan Wulfinghoff

doi:10.1186/s40323-018-0103-x

Hamid Reza Bayat, Stefanie Reese + Show 2 more

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https://doi.org/10.1186/s40323-018-0103-x

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Abstract

In the present work, the discontinuous Galerkin (DG) method is applied to linear elasticity for two-dimensional and three-dimensional settings. A locking-free element formulation based on reduced integration and physically-based hourglass stabilization (Q1SP) is coupled for the first time with the DG framework. The incomplete interior penalty Galerkin method is chosen, being one example of different variations of DG methods. Several 2D and 3D typical benchmark problems of linear elasticity are investigated. A selection of numerical integration schemes for the boundary terms is presented, namely reduced and mixed integration schemes. The treatment of the surface terms by means of different rules of integration shows a significant influence on the performance of the resulting DG method in combination with the standard Q1 element. This intelligent treatment of the surface part leads to a DG variant with very good convergence properties.

Highlights

IntroductionReed and Hill [1] (see LeSaint and Raviart [2]) were among the first authors to introduce a discontinuous Galerkin methods (DG) method
In the last decades, finite element-based discontinuous Galerkin methods (DG) have been established as good alternative to standard continuous finite element formulations.Reed and Hill [1] were among the first authors to introduce a DG method
Douglas and Dupont [8], Wheeler [9] and Arnold [10] introduced the interior penalty (IP) variant of the DG method to extend the range of applicability of the method

Summary

Introduction

Reed and Hill [1] (see LeSaint and Raviart [2]) were among the first authors to introduce a DG method. The method was set up for hyperbolic PDEs—to solve the problem of neutron transport. In order to stabilize the solution, a penalty term due to Nitsche [3] is frequently added on the element boundaries. Douglas and Dupont [8], Wheeler [9] and Arnold [10] introduced the interior penalty (IP) variant of the DG method to extend the range of applicability of the method (see [11,12,13]). The first application of DG for a fourth order problem was carried out by Baker [14]. Bassi and Rebay [15] solved the compressible Navier-Stokes equations by a DG method

Objectives

Conclusion