A stable generalized/eXtended FEM with discontinuous interpolants for fracture mechanics

Abstract

This paper presents numerical studies with three classes of quadratic Generalized FEM (GFEM) approximations and shows that all of them lead to errors that are orders of magnitude smaller than the FEM with quarter-point elements, provided that appropriate enrichments are selected. However, all of them lead to severely ill-conditioned systems of equations, with a condition number up to O(h−10) in the case of the GFEM based on a quadratic partition of unity. Enrichment modifications able to address the ill-conditioning of quadratic GFEM approximations while preserving their optimal convergence are proposed. A robust enrichment modification strategy based on a discontinuous FE interpolant is proposed to control the conditioning of branch function enrichment. The discontinuous FE interpolant is a generalization of the continuous one used with the Stable GFEM (SGFEM). We show that SGFEM spaces based on p-hierarchical FEM enrichments are the same as their GFEM counterparts. This guarantees that both GFEM and SGFEM spaces will lead to the same solution. This is not the case for the other classes of second-order spaces. The robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated.