Equilibrated stress tensor reconstruction and a posteriori error estimation for nonlinear elasticity

TL;DR

This paper introduces equilibrated stress tensor reconstructions for nonlinear elasticity problems, enabling reliable a posteriori error estimation and adaptive algorithms that improve finite element solutions across various hyperelastic models.

Contribution

It develops a novel stress reconstruction method for nonlinear elasticity that is independent of the material law, facilitating effective error estimation and adaptive refinement.

Findings

Efficient stress reconstructions validated on linear elasticity test case.

Effective a posteriori error estimates for discretization, linearization, and quadrature errors.

Successful application of adaptive algorithms to complex nonlinear models.

Abstract

We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, $H(\rm{div)}$-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold--Falk--Winther finite element spaces. We distinguish two stress reconstructions, one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with analytical solution for the linear elasticity problem. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky--Mises and an isotropic damage models.