A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics

摘要

In this paper, we investigate the accuracy and conditioning of the Stable Generalized FEM (SGFEM) and compare it with standard Generalized FEM (GFEM) for a 2-D fracture mechanics problem. The SGFEM involves localized modifications of enrichments used in the GFEM and the conditioning of the stiffness matrix in this method is of the same order as in the FEM. Numerical experiments show that using the SGFEM with only the modified Heaviside functions, which are used as enrichments in the GFEM, to approximate the solution of fracture problems in 2-D, gives inaccurate results. However, the SGFEM using an additional set of enrichment function yields accurate results while not deteriorating the conditioning of the stiffness matrix.Rules for the selection of the optimal set of enrichment nodes based on the definition of enrichment functions used in the SGFEM are also presented. This set leads to optimal convergence rates while keeping the number of degrees of freedom equal to or close to the GFEM. We show that it is necessary to enrich additional nodes when the crack line is located along element edges in 2-D. The selection of these nodes depends on the definition of the enrichment functions at the crack discontinuity. A simple and yet generic implementation strategy for the SGFEM in an existing GFEM/XFEM software is described. The implementation can be used with 2-D and 3-D elements. It leads to an efficient evaluation of SGFEM enrichment functions.