Stabilization of bi‐linear mixed finite elements for tetrahedra with enhanced interpolation using volume and area bubble functions

Abstract

AbstractIn order to overcome the oscillatory effects of the mixed bi‐linear Galerkin formulation for tetrahedral elements, a stabilization approach is presented. To this end the mixed method of incompatible modes and the mixed method of enhanced strains are reformulated, thus giving both the interpretation of a mixed finite element method with stabilization terms. For non‐linear problems, these are non‐linearly dependent on the current deformation state and therefore are replaced by linearly dependent stabilization terms. The approach becomes most attractive for the numerical implementation, since the use of quantities related to the previous Newton iteration step, typically arising for mixed‐enhanced elements, is completely avoided. The stabilization matrices for the mixed method of incompatible modes and the mixed method of enhanced strains are obtained with volume and area bubble functions. Various numerical examples are presented, which illustrate successfully the stabilization effect for bi‐linear tetrahedral elements. Copyright © 2007 John Wiley & Sons, Ltd.