On an equivalence between a discontinuous Galerkin method and reduced integration with hourglass stabilization for finite elasticity

Abstract

Discontinuous Galerkin methods and methods of reduced integration with hourglass stabilization have different points of origin. Whereas the discontinuous Galerkin idea was developed to investigate a wide range of applications, among them problems of electrodynamics, fluid and solid mechanics, the reduced integration technology started out from the observation of the so-called locking problem in linear elasticity. Today, the two methodologies are still considered separately from each other although it has already been recognized that discontinuous Galerkin methods can be also used to circumvent locking phenomena in elasticity.The idea of the present contribution is to find a direct equivalence between a discontinuous Galerkin method and a finite element technology based on reduced integration with hourglass stabilization. The transfer between the two methods is carried out in the context of finite elasticity. Thus the derivation is valid for arbitrarily large deformations. The main advantage of this analysis is that the choice of the “free” parameter in Nitsche’s method (often also called stabilization parameter) is no longer unclear but prescribed by the connection to the hourglass stabilization. Further important in this context is the replacement of the scalar parameter by a symmetric 2×2 matrix (if two-dimensional problems are considered which is here the case).The result of the derivation is a discontinuous Galerkin formulation which shows very good convergence behaviour — not only in the limit of incompressibility but also for bending of thin structures.