Conforming mixed triangular prism and nonconforming mixed tetrahedral elements for the linear elasticity problem

TL;DR

This paper introduces new conforming mixed triangular prism elements and nonconforming mixed tetrahedral elements for 3D linear elasticity, achieving optimal convergence and reducing degrees of freedom.

Contribution

It presents novel conforming and nonconforming mixed finite elements with improved properties for solving linear elasticity problems in three dimensions.

Findings

Conforming mixed triangular prism elements are well-posed for all k ≥ 1.

The lowest order conforming element has 126 degrees of freedom per element.

New nonconforming tetrahedral elements with distinct stress shape functions.

Abstract

We propose two families of mixed finite elements for solving the classical Hellinger-Reissner mixed problem of the linear elasticity equations in three dimensions. First, a family of conforming mixed triangular prism elements is constructed by product of elements on triangular meshes and elements in one dimension. The well-posedness is established for all elements with $k\geq1$, which are of $k+1$ order convergence for both the stress and displacement. Besides, a family of reduced stress spaces is proposed by dropping the degrees of polynomial functions associated with faces. As a result, the lowest order conforming mixed triangular prism element has 93 plus 33 degrees of freedom on each element. Second, we construct a new family of nonconforming mixed tetrahedral elements. The shape function spaces of our stress spaces are different from those of the elements in literature.