XFEM crack-tip enrichment using locally smoothed branch functions within blending elements

The Extended Finite Element Method (XFEM) is currently the mainstream numerical method for crack simulations in engineering. Its improvement, the Corrected XFEM, significantly enhances computational accuracy while introducing relatively severe ill-conditioned stiffness matrix issues. To address this, we investigate the spatial distribution of branch function-enriched terms describing the crack-tip singular displacement field in the XFEM displacement approximation. We note that the improvement of the Corrected XFEM essentially lies in adjusting the branch function-enriched displacement fields in the blending-element zone toward a specific smooth decay pattern. Based on this, we propose a simplified improvement method (the local smoothing method): within the standard XFEM framework, targeted scaling based on the enrichment status of nodes in the blending-element zone is applied exclusively to the branch functions in the same region. This scaling renders the function graph approximate a horizontal plane. Since the smoothing process solely employs scaling functions built upon element shape functions, its implementation into existing programs is straightforward. Numerical examples containing both crack-tip singular and non-singular fields demonstrate that compared to the Corrected XFEM, the proposed method exhibits comparable computational accuracy and a slightly higher convergence rate, along with significantly better numerical stability comparable to that of the standard XFEM.