A mixed method for 3D nonlinear elasticity using finite element exterior calculus

TL;DR

This paper introduces a novel mixed finite element method for 3D nonlinear elasticity that leverages finite element exterior calculus to improve performance and address issues like locking and checkerboarding in benchmark problems.

Contribution

It develops a new mixed FE approach based on a Hu-Washizu variational principle using differential forms, enhancing accuracy and computational stability in nonlinear elasticity.

Findings

Outperforms traditional displacement-based methods on benchmark problems.

Effectively mitigates locking and checkerboarding issues.

Demonstrates superior convergence and stability in 3D nonlinear elasticity simulations.

Abstract

This article discusses a mixed FE technique for 3D nonlinear elasticity using a Hu-Washizu (HW) type variational principle. Here, the deformed configuration and sections from its cotangent bundle are taken as additional input arguments. The critical points of the HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. The present FE approximation distinguishes a vector from a 1-from, a feature not commonly found in FE approximations. This point of view permits us to construct finite elements with vastly superior performance. Discrete approximations for the differential forms appearing in the variational principle are constructed with ideas borrowed from finite element exterior calculus. The discrete equations describing mechanical equilibrium, compatibility and constitutive rule, are obtained by extemizing the discrete functional with respect to appropriate DoF, which are then solved using the Newton's method. This mixed FE technique is then applied to benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding.