Three-field mixed finite element methods for nonlinear elasticity

TL;DR

This paper develops three innovative mixed finite element methods for nonlinear elasticity, extending previous techniques to handle nonlinear materials with improved accuracy and efficiency, demonstrated through numerical experiments.

Contribution

Introduces three new mixed finite element methods for nonlinear elasticity using the Hu-Washizu principle, with local elimination of stress and strain variables for simplified computations.

Findings

Methods show high accuracy in numerical tests

Efficient local elimination reduces computational complexity

Applicable to various nonlinear elastic materials

Abstract

In this paper, we extend the tangential-displacement normal-normal-stress continuous (TDNNS) method from [26] to nonlinear elasticity. By means of the Hu-Washizu principle, the distibutional derivatives of the displacement vector are lifted to a regular strain tensor. We introduce three different methods, where either the deformation gradient, the Cauchy-Green strain tensor, or both of them are used as independent variables. Within the linear sub-problems, all stress and strain variables can be locally eliminated leading to an equation system in displacement variables, only. The good performance and accuracy of the presented methods are demonstrated by means of several numerical examples (available via www.gitlab.com/mneunteufel/nonlinear_elasticity).