Equilibrated stress tensor reconstruction and a posteriori error estimation for nonlinear elasticity
For computational mechanics researchers, this provides a rigorous error estimation framework for nonlinear elasticity, though it is an incremental extension of existing reconstruction techniques to nonlinear problems.
The paper presents equilibrated stress tensor reconstructions for hyperelastic problems solved with finite elements, enabling a posteriori error estimates that distinguish discretization, linearization, and quadrature errors. The method is proven efficient on a linear elasticity test and demonstrated on Hencky-Mises and isotropic damage models.
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.