BR2 discontinuous Galerkin methods for finite hyperelastic deformations

TL;DR

This paper develops a discontinuous Galerkin framework using BR2 formulation for simulating large hyperelastic deformations, incorporating advanced stabilization, boundary condition handling, and efficient parallel solution strategies.

Contribution

It introduces a novel BR2 discontinuous Galerkin method for nonlinear hyperelasticity, including adaptive stabilization and parallel implementation, for large deformation problems.

Findings

Effective stabilization with higher-order lifting operators.

Robust parallel solution with $h$-multigrid preconditioners.

Insights into boundary conditions' impact on Newton's method.

Abstract

In this work we introduce a dG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilization term with an adaptive stabilization strategy which relies on the BR2 Laplace operator stabilization and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based $h$-multigrid preconditioners and the code implementation supports distributed memory execution on modern parallel architectures. Several benchmark test cases are proposed in order to investigate some relevant computational aspects, namely the performance of the $h$-multigrid iterative solver varying the stabilization parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy.