Abstract:
It is wellknown that the loworder elements based on fournoded quadrilaterals or eightnoded 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 bendingdominated problems
when coarse meshes are used. In the first part of the talk, we examine the classical HuWashizu 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 HuWashizu 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 wellposedness 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 wellknown
bases as well as newly constructed ones. Though a discrete version of the spherical part of the stress exhibits
checkerboard modes, we establish a λindependent optimal a priori error estimates for the is placement and for the
postprocessed stress. Furthermore, starting from a suitable threefield formulation we introduce a twofield mixed
formulation for geometrically nonlinear elasticity with SaintVenant Kirchho law. In the second part, we demonstrate
the performance of our approach through different numerical examples.
