Mixed finite elements for elasticity

Abstract

There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.

Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

“…Applying the mixed formulation, the local mixed stiffness matrix K AB mixed of one bar element is a 4 × 4 matrix compose by terms affecting degrees of freedom of displacements and strains. The sub-matricesM ,G, G τ and K τ defined in (32)(33)(34)(35) for one mixed bar element arȇ…”

Inf–sup conditions for twofold saddle point problems

“…First Inf-Sup Condition: The issue of finding conforming finite elements for symmetric tensors satisfying an inf-sup condition of the form (4.2) is well-documented ( [14,32,4,5]). We consider the finite element pairs (S h , U h ) of symmetric tensors and vectors constructed by Arnold and Winther [7,1] which satisfy the inf-sup condition.…”

“…Figure 4.1 gives a diagram of the degrees of freedom on each triangle for the lowest order Arnold-Winther (S h , u h ) pair (k = 1) in two dimensions. In [7,1] an interpolation operator…”

A mixed nonconforming finite element for linear elasticity

“…This is due to the symmetry constraint on the stress tensor. Arnold and Winther in [12] only recently constructed the first family of stable conforming elements in two dimensions on a triangular tessellation. Besides its complication, this family has yet to be extended to three dimensions and to types of nonsimplicial meshes often favored by practitioners.…”