The extended finite element method and its applications ------ areview

The extended finite element method (XFEM)  originallyproposed in 1999  is very powerful for discontinuous problems inmechanics, such as crack growth, complex fluid, interface, and so on.The major difference between the XFEM and the conventional finite elementmethod (CFEM) is that the mesh  in XFEM is independent of theinternal geometry and physical interfaces, such that meshing and re-meshingdifficulties in discontinuous problems can be overcome. Based on thepartition of unity concept, the XFEM relaxes the prohibitive requirementsfor  meshdensity by improving the shape functions with the basic knowledge ofdiscontinuous problems. The XFEM retains all advantages of the CFEM,such as thesingle-field variational principle, symmetric banded and sparse systemmatrices, the ease of application to non-linear problems, anisotropic materialsand arbitrary geometries. This paper presents an overview and comments onthe XFEM, and is organized as follows. The partition of unity method (PUM) andLevel Set Method (LSM) are briefly introduced in sections 2 and 3,respectively. Basic theory, implementation procedures and formulations ofthe XFEM are described  in detail in sections 4 and 5, together with correction toseveral inaccurated points in literature. The future investigationson XFEM are finally recommended in section 6.