XFEM

The extended finite element method (XFEM) classified, one of the partition of unity method (PUM), allows discontinuities to be simulated independently of the mesh. This is possible by adding appropriate functions to the FE approximation basis, for example, the Heaviside function. The discontinuities can evolve in time, without a need for a conforming mesh. A MATLAB implementation of the XFEM written by VP Nguyen, is given here. The interaction of cracks and crack-inclusion interaction is modelled with XFEM framework. The elements intersected by discontinuity surface are sub-divided into quadrature subcells aligned with the discontinuity and higher order quadrature is adopted.

The implementation is described in the following article:

Meshless methods: a review and computer implementation aspects
VP Nguyen, T Rabczuk, S Bordas, M Duflot, Mathematics and computers in simulation 79 (3), 763-813.