Achieving Material Robustness via Symmetric Stress Finite Element Discretizations

TL;DR

This paper demonstrates that enforcing stress tensor symmetry strongly in finite element discretizations leads to more accurate and robust results across various materials, compared to weak enforcement methods.

Contribution

It introduces a unifying theory explaining why strong symmetry enforcement in finite element methods enhances material robustness in continuum mechanics simulations.

Findings

Strong symmetry enforcement yields accurate stress approximations regardless of material law.

Weak symmetry enforcement can produce arbitrarily poor stress results, even in zero-stress states.

Theoretical analysis explains the observed robustness difference.

Abstract

When discretizing symmetric stress tensors in variational problems arising in continuum mechanics, one has to choose how to enforce the symmetry of the stress tensor: (i) strongly by requiring the discrete tensors to be pointwise symmetric or (ii) weakly by introducing a Lagrange multiplier. For $H(\mathrm{div})$-conforming finite element discretizations of Hellinger--Reissner elasticity and velocity--stress formulations of incompressible flow, where symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum, we show that this choice may substantially impact the accuracy of the numerical scheme. Through a series of benchmark problems featuring anisotropic constitutive laws inspired by fiber reinforced material, liquid crystal polymer networks, and polar fluids, we show that schemes enforcing symmetry weakly can yield arbitrarily poor stress approximations -- even for zero-stress configurations. However, schemes enforcing symmetry strongly deliver accurate stress approximations independently of the constitutive law, a property we term material robustness. We present a unifying theory that rigorously explains this behavior.