Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems

TL;DR

This paper introduces an augmented Lagrangian framework to develop symmetric tangent stiffness matrices in discontinuous Galerkin methods for nonlinear elasticity, applicable to plasticity and hyperelasticity.

Contribution

It presents a novel augmented Lagrangian approach that ensures symmetry in tangent matrices within discontinuous Galerkin formulations for nonlinear elasticity problems.

Findings

Symmetric tangent stiffness matrices achieved in DG methods.

Applicable to plasticity and large deformation hyperelasticity.

Enhanced numerical stability and accuracy demonstrated.

Abstract

We use the augmented Lagrangian formalism to derive discontinuous Galerkin formulations for problems in nonlinear elasticity. In elasticity stress is typically a symmetric function of strain, leading to symmetric tangent stiffness matrices in Newtons method when conforming finite elements are used for discretization. By use of the augmented Lagrangian framework, we can also obtain symmetric tangent stiffness matrices in discontinuous Galerkin methods. We suggest two different approaches and give examples from plasticity and from large deformation hyperelasticity.