A simple, first-order, well-conditioned, and optimally convergent Generalized/eXtended FEM for two- and three-dimensional linear elastic fracture mechanics

Abstract

This paper proposes a first-order Generalized/eXtended Finite Element Method (G/XFEM) for 2-D and 3-D linear elastic fracture mechanics problems. The conditioning of the method is of the same order as the standard FEM and it is robust with respect to the position of the mesh relative to 2-D or 3-D fractures. The method achieves an optimal rate of convergence in energy norm even when the solution of the problem in the neighborhood of the fracture front is not contained in the spaces spanned by the adopted singular enrichments. Control of conditioning associated with Heaviside enrichment functions is achieved by shifting them by their nodal values, combined with a diagonal pre-conditioner. In the case of singular enrichment functions, a simple extension of the enrichment shifting concept, denoted as discontinuous-shifting, is adopted. These enrichment modifications are space-preserving—the solution space spanned by the modified enrichments is the same as the space spanned by the unmodified ones. This is often not the case for conditioning control strategies proposed in the literature. The computational cost of these enrichment modifications is negligible and their implementation in existing G/XFEM software is straightforward. The optimal convergence, well-conditioning, and robustness of the method are numerically illustrated with the aid of representative 2-D and 3-D problems, including the case of a non-planar fracture with a curved fracture front.