Symmetric Matrix Fields in the Finite Element Method

Abstract

The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However the symmetry of the stress tensor which expresses the conservation of angular momentum had been a challenge for finite element computations. We review in this paper approaches based on mixed finite element methods.

Two Remarks on Rectangular Mixed Finite Elements for Elasticity

“…As noted for example in [12], high order mixed finite elements for elasticity usually require a high number of degrees of freedom and typically require to find the dimension and a base for the space {τ ∈ Σ(K), div σ = 0, σn = 0, on ∂Ω}.…”

“…Starting with the seminal work of Arnold and Winther, [8,9], for triangular elements, extended to tetrahedral elements in [1,4,18], conforming and nonconforming mixed finite elements for elasticity on rectangular meshes have been constructed by several authors [3,11,27,28,20,21,15]. We refer to [8,11,7,12] and the references therein for alternative approaches to mixed formulations of the elasticity equations. In particular, we note advances on mixed finite elements for elasticity where the symmetry of the stress field is enforced weakly using Lagrange multipliers, [2,5,24,26,25,22,6,7,14,19,17,16,10].…”

Rectangular mixed elements for elasticity with weakly imposed symmetry condition

“…We notice that for q ∈ h , for the surjectivity assumption to hold, the following degrees of freedom were not used: f q 12 n 3 − q 13 n 2 dx f = f (q ∧ n) 13 , f q 23 n 1 − q 21 n 3 dx f = f (q ∧ n) 12 , f q 31 n 2 − q 32 n 1 dx f = f (q ∧ n) 32 . However since the faces of a rectangle are parallel to the axes, one of these degrees of freedom is identically zero for each face, hence two degrees of freedom per face are unnecessary.…”

“…Nonconformity can be introduced by weakening the symmetry condition or by weakening the requirement that the stress field is L 2 integrable. We refer to [12] for a review on nonconforming elements with symmetric stress fields and other approaches to linear elasticity.…”