Nonconforming tetrahedral mixed finite elements for elasticity

TL;DR

This paper introduces a new nonconforming mixed finite element method for elasticity on tetrahedral meshes, offering a simpler, stable, and linearly convergent approach for stress and displacement approximation in three dimensions.

Contribution

It develops a novel nonconforming finite element for symmetric tensor approximation in 3D elasticity, extending a 2D element and maintaining linear convergence with reduced displacement space.

Findings

The method is stable and linearly convergent for stress and displacement.

It is significantly simpler than existing conforming methods.

A variant with reduced displacement approximation also achieves linear convergence.

Abstract

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.