The enriched Galerkin method for linear elasticity and phase field fracture propagation

Abstract

This thesis focuses on the application of the discontinuous Galerkin (DG) and enriched Galerkin (EG) methods to the problems of linear elasticity and phase field fracture propagation. The use of traditional and popular continuous Galerkin method (CG) for linear elasticity has posed some challenges. For example, nonphysical stress oscillations often occur in CG solutions for linearly elastic, nearly incompressible materials. Furthermore, CG solutions produce discontinuous stresses at the finite element boundaries which need to be post-processed. Based on the success of the DG methods in solving these challenges, we attempt resolution of the same problems with the yet untested EG method. For phase field fracture propagation, the CG method has been ubiquitously used in the literature. Since the phase field displacement solution is essentially discontinuous across the crack, we hypothesize that the discontinuous DG and EG methods could offer some advantages when applied to the fracture problem. We then perform a comparative analysis of CG, DG and EG applied to the phase field equations to determine if this is indeed the case.

We begin by applying a family of DG and EG methods, including Nonsymmetric Interior Penalty Galerkin (NIPG), Symmetric Interior Penalty Galerkin (SIPG), and Incomplete Interior Penalty Galerkin (IIPG) to 2D linear elasticity problems. It is shown that the EG methods are simple and robust for dealing with linear elasticity. They are also shown to converge at the same rates as the corresponding DG methods. A detailed comparison of the performance of NIPG, IIPG, and SIPG is also made. We then propose a novel monolithic scheme with an augmented-Lagrangian method for phase field fracture propagation. We apply CG, DG and EG methods to the scheme and establish convergence in space and time through numerical studies. It is shown that the Newton method used for solving the system of nonlinear equations converges faster for DG and EG than it does for CG.