Four-Field Mixed Finite Element Methods for Incompressible Nonlinear Elasticity

TL;DR

This paper develops conformal mixed finite element methods for 2D and 3D incompressible nonlinear elasticity, enabling accurate independent computation of strain and stress with stability analysis and numerical validation.

Contribution

It introduces new finite element discretizations for nonlinear elasticity that satisfy stability conditions and allow separate approximation of strain and stress.

Findings

28 stable element choices in 2D

6 stable element choices in 3D

Numerical results confirm stability and accuracy

Abstract

We introduce conformal mixed finite element methods for $2$D and $3$D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the $\mathrm{curl}$ and the $\mathrm{div}$ operators are used to discretize strain and stress, respectively. These choices of elements follow from the strain compatibility and the momentum balance law. Some inf-sup conditions are derived to study the stability of methods. By considering $96$ choices of simplicial finite elements of degree less than or equal to $2$ in $2$D and $3$D, we conclude that $28$ choices in $2$D and $6$ choices in $3$D satisfy these inf-sup conditions. The performance of stable finite element choices are numerically studied. Although the proposed methods are computationally more expensive than the standard two-field methods for incompressible elasticity, they are potentially useful for accurate approximations of strain and stress as they are independently computed in the solution process.