Stable generalized finite element methods (SGFEM) for interfacial crack problems in bi-materials

Abstract

The poor conditioning in generalized or extended finite element methods (GFEM/XFEM) for crack problems in homogeneous materials has been studied extensively. However, few research efforts have been devoted to resolving the conditioning difficulty of GFEM/XFEM for bi-material crack problems. The conditioning of these methods for a bi-material crack problem is poorer than that for a homogeneous material crack problem; this is because more complex enrichments, including Heaviside functions, distance functions, and singular functions characterizing radial and oscillatory singularities, are involved in the former. This study addresses the conditioning difficulty for bi-material crack problems by proposing a stable GFEM (SGFEM), which (a) reaches optimal convergence O(h), (b) has a scaled condition number O(h−2) having the same order as that of the standard FEM, and (c) achieves convergence and conditioning without deterioration as the interface lines approach the boundaries of the elements. The proposed SGFEM is based on two stability techniques, namely changing the partition of unity functions and conducting local principal component analysis of multi-fold enrichments at one particular node. Numerical experiments suggest that in comparison with the conventional GFEM/XFEM for bi-material crack problems in the literature, the proposed SGFEM achieves all features (a)–(c). In addition, the effect of the ratio of material coefficients on conditioning is investigated, revealing that the conditioning of the proposed SGFEM is insensitive to the material coefficients.