An explicit residual-type error estimator for Q1-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics

摘要

This paper deals with explicit residual a posteriori error estimation analysis for Q1-quadrilateral extended finite element method (XFEM) discretizations applied to the two-dimensional problem of linear elastic fracture mechanics. The result is twofold. First, to enable estimation procedures with application to XFEM, a specific quasi-interpolation operator of averaging type is constructed. The main challenge here arises from the different types of enrichments implemented, and hence, to impose the constant-preserving property of the interpolation operator on an element, we use the idea of an extension operator. An upper bound on the discretization error measured in the energy norm and associated local error indicators are then constructed and analyzed. The second result follows from the error analysis and concerns an alternative choice of branch functions used in XFEM applications. In particular, the branch functions have to be chosen to fulfill the divergence-free conditions within the crack tip element and traction-free boundary conditions on the crack faces. Then, the corresponding XFEM solution gains a better accuracy with less degrees of freedom. Finally, numerical examples are provided with comparative results.