Abstract: This paper mainly studies the effects of the enrichment functions on the numerical precision of XFEM when the method is used to model weak discontinuities problems. The dominant reasons for the effects of the enrichment functions on the numerical precision of XFEM are discussed in detail. In the corrected-XFEM, the set of enriched nodes equals to the set of enriched nodes in the standard-XFEM plus their neighboring nodes. The fact is thereby indicated that the increase of enriched nodes can improve the numerical precision of the corrected-XFEM. In this paper, the proposed method is to expand the enriched nodes of the corrected-XFEM, and the expanding enriched nodes is the set of enriched nodes in the corrected-XFEM plus their neighboring nodes. Then, the level set method is used for the description of the inner discontinuous interfaces. The governing equation of XFEM is deduced. A numerical integration scheme is elaborated to integrate the stiffness matrix of enrichment elements. Also a method is introduced for the acquirement of accurate stresses in enrichment elements. The numerical results of two illustrations with single inclusion and multi-inclusions both show that the proposed method of expanding the enriched nodes of the corrected-XFEM can improve evidently the numerical precision of XFEM.