Two remarks on rectangular mixed finite elements for elasticity

TL;DR

This paper introduces new conforming rectangular finite elements for 3D linear elasticity, reducing degrees of freedom and avoiding vertex degrees, with potential for further low-order element development.

Contribution

It presents a conforming rectangular element with fewer degrees of freedom and no vertex degrees, and discusses constructing even lower order elements using rigid body motions.

Findings

A conforming 3D rectangular element with 24 stress and no vertex degrees.

A lower order element with 72 stress and 6 displacement degrees of freedom.

Potential for further low order element development.

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.