Two Remarks on Rectangular Mixed Finite Elements for Elasticity

Abstract

The lowest order nonconforming rectangular element in three dimensions involves 54 degrees of freedom for the stress and 12 degrees of freedom for the displacement. With a modest increase in the number of degrees of freedom (24 for the stress), we obtain a conforming rectangular element for linear elasticity in three dimensions. Moreover, unlike the conforming plane rectangular or simplicial elements, this element does not involve any vertex degrees of freedom. Second, we remark that further low order elements can be constructed by approximating the displacement with rigid body motions. This results in a pair of conforming elements with 72 degrees of freedom for the stress and 6 degrees of freedom for the displacement.

Nonconforming mixed finite elements for linear elasticity on simplicial grids

“…Arnold and Winther [8] designed the first family of conforming mixed finite element methods in two dimensions, based on polynomial shape function spaces. Analogous results on tetrahedral grids can be found in [9,10], and on rectangular and cuboid grids in [11][12][13]. Recently, Hu and Zhang [14,15] and Hu [16] proposed a new family of conforming mixed elements on simplicial grids and the latter is for any dimension.…”

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.…”

“…Indeed for several decades before the work of Arnold and Winther [10,11] the existence of such elements was an open problem. These elements have been extended to rectangular meshes in two dimensions [3,19], three dimensions [13] and on tetrahedral meshes [1,5]. Despite their relative complexity, mixed finite elements with symmetric stress fields are useful in certain situations [27].…”

Symmetric Matrix Fields in the Finite Element Method

“…This lead to a different family with the lowest order element having 17 degrees of freedom for the stress field and 4 degrees of freedom for the displacement. A low order conforming rectangular elements in three dimensions has recently been constructed in [26] with 72 degrees of freedom for the stress and 12 degrees of freedom for the displacement.…”