Although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization, XFEM still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs. An improved XFEM (View the MathML source) is developed to overcome those difficulties from three aspects: (1) to eliminate the linear dependence and the ill-conditioning issues of the standard and the corrected XFEMs; (2) to get rid of extra dofs in crack tip enrichment to facilitate optimal   mass lumping in dynamic analyses; (3) to be interpolating at enriched nodes to enable direct essential/contact boundary treatments. The core of View the MathML source is a globally interpolating approximation based on local least-squares fitting. The new method is numerically examined with comprehensive comparisons with the standard XFEM and the corrected XFEM (Fries (2008)). The new method’s excellent performance in terms of accuracy, convergence, and conditioning/stability are demonstrated.


  • Selectively interpolating (moving) least squares;
  • Partition of unity;
  • XFEM;
  • Crack

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