# Compatible-Strain Mixed Finite Element Methods for 3D Compressible and Incompressible Nonlinear Elasticity

## TL;DR

This paper introduces compatible-strain mixed finite element methods (CSFEMs) for 3D nonlinear elasticity, effectively modeling large strains and complex geometries in both compressible and incompressible regimes with high accuracy and stability.

## Contribution

The paper develops a new family of mixed finite element methods using compatible strains, stabilizing terms, and conforming interpolation for nonlinear elasticity in three dimensions, ensuring stability and efficiency.

## Key findings

- Accurate stress and pressure approximation in large strain regimes
- No observed numerical artifacts such as locking or hourglass instability
- Effective modeling of heterogeneous solids and complex geometries

## Abstract

A new family of mixed finite element methods$-$compatible-strain mixed finite element methods (CSFEMs)$-$are introduced for three-dimensional compressible and incompressible nonlinear elasticity. A Hu-Washizu-type functional is extremized in order to obtain a mixed formulation for nonlinear elasticity. The independent fields of the mixed formulations are the displacement, the displacement gradient, and the first Piola-Kirchhoff stress. A pressure-like field is also introduced in the case of incompressible elasticity. We define the displacement in $H^1$, the displacement gradient in $H(curl)$, the stress in $H(div)$, and the pressure-like field in $L^2$. In this setting, for improving the stability of the proposed finite element methods without compromising their consistency, we consider some stabilizing terms in the Hu-Washizu-type functional that vanish at its critical points. Using a conforming interpolation, the solution and the test spaces are approximated with some piecewise polynomial subspaces of them. In three dimensions, this requires using the N\'ed\'elec edge elements for the displacement gradient and the N\'ed\'elec face elements for the stress. This approach results in mixed finite element methods that satisfy the Hadamard jump condition and the continuity of traction on all internal faces of the mesh. This, in particular, makes CSFEMs quite efficient for modeling heterogeneous solids. We assess the performance of CSFEMs by solving several numerical examples, and demonstrate their good performance for bending problems, for bodies with complex geometries, and in the near-incompressible and the incompressible regimes. Using CSFEMs, one can capture very large strains and accurately approximate stresses and the pressure field. Moreover, in our numerical examples, we do not observe any numerical artifacts such as checkerboarding of pressure, hourglass instability, or locking.