Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

Locking-Free Finite Element Methods Based on the Hu-Washizu Formulation for Linear and Geometrically Nonlinear Elasticity

Barbara Wohlmuth

Universität Stuttgart

Abstract:   It is well-known that the low-order elements based on four-noded quadrilaterals or eight-noded hexahedra have two drawbacks in finite element computation. The first one is the locking effect in nearly incompressible case; in other words, they do not converge uniformly with respect to the Lam´e parameter λ. The second one is that these standard elements lead to poor accuracy in bending-dominated problems when coarse meshes are used. In the first part of the talk, we examine the classical Hu-Washizu mixed formulation for plane and three dimensional problems in linear elasticity with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu-Washizu problems parameterized by a scalar for which = λ/µ corresponds to the classical formulation, with λ and µ being the Lam´e parameters. We discuss the uniform well-posedness in the incompressible limit of the continuous problem for α≠-1. Finite element approximations are based on the choice of piecewise bilinear or trilinear approximations for the displacements on quadrilateral or hexahedral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains that include well-known bases as well as newly constructed ones. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, we establish a λ-independent optimal a priori error estimates for the is placement and for the postprocessed stress. Furthermore, starting from a suitable three-field formulation we introduce a two-field mixed formulation for geometrically nonlinear elasticity with Saint-Venant Kirchho law. In the second part, we demonstrate the performance of our approach through different numerical examples.