Correction of incompleteness of XFEM approximation (1st report: General formulation and theoretical error analyses)

The XFEM is a numerical method which employs the local enrichment considering a priori knowledge of the solution in the framework of the FEM. The XFEM has an essential problem in the approximation of the partially enriched 'blending elements', which causes a lack of accuracy. Using the weighted XFEM, the numerical accuracy was effectively improved for the problem. It was however found that the influence of the blending elements still remains by the detailed examination in the numerical results. In the present paper, the incompleteness of the weighted XFEM is proved through a theoretical error analysis. Then, a 'PU-XFEM' is proposed in order to correct the incompleteness in the existing XFEM approximations. The PU-XFEM is formulated as the exact development of the PUFEM with local enrichment. As a result of the error analysis, it is found that the PU-XFEM is a proper XFEM without problem of the blending elements.