On the enrichment zone size for optimal convergence rate of the Generalized/Extended Finite Element Method

Singular enrichment functions are broadly used in Generalized or Extended Finite Element Methods (GFEM/XFEM) for linear elastic fracture mechanics problems. These functions are used at finite element nodes within an enrichment zone around the crack tip/front in 2- and 3-D problems, respectively. Small zones lead to suboptimal convergence rate of the method while large ones lead to ill-conditioning of the system of equations and to a large number of degrees of freedom. This paper presents an a priori estimate for the minimum size of the enrichment zone required for optimal convergence rate of the GFEM/XFEM. The estimate shows that the minimum size of the enrichment zone for optimal convergence rate depends on the element size and polynomial order of the GFEM/XFEM shape functions. Detailed numerical verification of these findings is also presented.