When dealing with discontinuous issues such as fracture and crack in geo-engineering

conventional finite element method (CFEM) need to refine mesh in the local zone including crack tip; furthermore

the mesh must be reconfigured and re-partitioned once the crack propagation happened. The extended finite element method (XFEM)

based on the idea of partition of unit method

can successfully solve these problems easily

which may hardly be deposed by CFEM. Usually

contact problem of crack surface must be considered when studying some compound fracture. Based on the complementary theory

the complementary equation between relative displacement and contact force of the crack surface can be established

and solved by Newton's method without considering opening and closing iteration. Finally

as a numerical example

a plate with crack is compressed to indicate the effectiveness of this method. The results indicate that the method can prevent mesh penetration from one side of the crack into the other side effectively; and also obtain the stress distribution that is consistent with the actual on the crack surfaces.