## 1 Introduction  
In the field of fracture mechanics, the stress field near the crack tip is usually described by the SIFs. For brittle materials, it can also be used to describe the non-singular stresses on the crack surface. The SIF is of great significance for the research of material fracture and crack propagation in practical engineering problems. At present, there have been many numerical methods which were applied to solve the SIFs, such as the traditional finite element method (FEM) and boundary element method (BEM), extended finite element method (XFEM), meshless method (MLM), EMM, the improvement of the FEM, the particular weight function method and NMM, etc. After many efforts of numerous researchers, many methods were also used to extract the SIFs, such as the extrapolation method, interaction integral method and J integral method, etc.  
In the past twenty years, the XFEM has mostly been used to analyze fracture mechanics problems. The XFEM is based on the partition of the unity finite element method, adding the step function and the asymptotic field of crack tip displacement to describe the crack, and has achieved fruitful results in the analysis and research of discontinuous problems such as crack and crack propagation [1; 2; 3; 4; 5]. The EMM [6] converts the linear, elastic, two-dimensional and steady-state equations of the crack problem into a Hamiltonian system. By changing the angular stress to the radial stress, the eigensolutions satisfying the adjoint symplectic orthogonality are obtained, and then the desired solution of the problem is obtained by using the linear combination of these eigensolutions. The improvedfinite element method [7] divides the whole crack region into two sub-regions, namely the complementary energy sub-region around the crack tip and the potential energy sub-region of the rest regions, which is an improved method to extract the SIF by using the FEM. The particular weight function method [8] uses the specified reference loading conditions on both sides of the crack for finite element analysis, deduces the coefficients of each weight function, and then computes the SIF under loading by using the weight function.  
In the past few decades, the P-FEM and the CIM have been applied to extract the SIFs of the materials and structures, but most of the research is to use separately the P-FEM or using traditional low-order FEM and the CIM or the CIM in combination with other methods such as the displacement discontinuity method [9]. Currently, there are a few works for the P-FEM combined with the CIM to extract SIFs [10; 11].  
This study aimed to provide a simple, direct, and accurate numerical method to extract the SIFs for the single and multiple crack problems. The effectiveness and performance of this method are verified by numerical examples. In this paper, the commercial finite element software StressCheck developed by the American ESRD (Engineering Software Research and Development) company was used as a simulation tool. Numerical models will demonstrate that the present method is effective not only for the single-crack problems but also for multiple-crack problems in the analysis of fracture mechanic problems. The present method works as follows: firstly, the P-FEM is employed to obtain a highly accurate displacement field and stress field, and then SIFs are derived through the super-convergent CIM. The SIFs for a central straight crack plate, a slanted single-edge crack plate, the offset double-edge cracks in the rectangular plates, a branched crack in an infinite plate and a crucifix crack in a square plate were evaluated using the present method, and compared with the published results using the NMM, improved approach finite element method, particular weight function method and EMM in the literature.