Abstract: Extended element-free Galerkin method (XEFG) is proposed to solve inhomogeneous materials with inclusions based on moving Kriging interpolation. Level set functions are used to represent the geometric interfaces of inclusions and to enrich moving Kriging shape functions in constructing a discontinuous displacement field. The displacement boundary condition can be enforced exactly as the shape functions constructed from the moving Kriging interpolation possess the Kronecker delta property, compared with traditional moving least square shape functions. The key techniques of XEFG are presented, including the construction of displacement pattern, and the establishment of the displacement governing equation. Finally, the examples of single inclusion and multiple inclusions show that XEFG method have higher accuracy and better convergence, compared with the extended finite element method.