## 5 Conclusions  
The SIFs of several classical LEFM models were evaluated based on the P-FEM and CIM, and compared with the results in the literature to verify the effectiveness and accuracy of the present method. The following conclusions can be drawn:  
* The computation based on the P-FEM and CIM only needs a fewer number of meshes (less preprocessing), has a faster convergence rate, which can not only be used to analyze the single-crack problems, but also can be used to calculate the SIFs of multiple-crack problems. Furthermore, the present method can also be used to analyze three-dimensional fracture problems and simulate the crack growth path.
* The CIM is a super-convergent method based on the [PERSON]'s theorem of reciprocal works. The SIFs are derived by using the displacements and stresses of higher accuracy on the integral path. Therefore, the CIM can obtain results of relatively higher precision.
* The P-FEM does not need fine meshes, has a faster convergence rate and higher accuracy. However, in the study of crack propagation behavior, it is inevitable to meet the problem of remeshing. On the other hand, the XFEM need not consider the internal geometry of structures and materials; it employs the partition of unity (PU) concept to simulate fracture and crack propagation. The XFEM uses the enrichment functions to reflect the local characteristics of the crack tip and avoids remeshing of meshes. The high-order extended finite element method (high-order XFEM), a combination of the P-FEM with XFEM, will probably have the advantages of both: avoiding remeshing as well as obtaining higher accuracy. The study of the high-order XFEM will be an interesting issue.  
Conceptualization, J.Z.; methodology, J.Z.; software, W.Y. and J.C.; resources, R.X.; Data curation, W.Y.; Funding acquisition, J.Z.; Investigation, J.C.; Validation, W.Y. and R.X.; Writing--original draft, J.Z.; writing--review and editing, W.Y.; visualization, W.Y. and J.C.; supervision, J.Z.; project administration, J.Z.; All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 51769011.  
Not applicable.  
Data Availability Statement:Data is contained within the present article. Other data presented in this research are available in [6; 7; 8; 37; 38; 39; 40; 41; 42; 43].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Model 5** \\ \hline Present & 252 & 39 & 142 & 329 & 580 \\ Ref [37] & 4410 & — & — & — & — \\ Ref [6] & — & — & — & — & — \\ Ref [38] & — & 423 & — & — & — \\ Ref [7] & — & — & 2494 & — & — \\ Ref [8] & — & — & — & — & — \\ Ref [40] & — & — & — & 3678 nodes & — \\ Ref [41] & — & — & — & 2489 nodes & — \\ Ref [42] & — & — & — & — & 3600 collocation points \\ Ref [43] & — & — & — & — & 1858 physical patches \\ \hline \hline \end{tabular} Model 1: A central straight crack plate; Model 2: A slanted single-edge crack plate; Model 3: The offset double-edge cracks plate; Model 4: A branched crack in an infinite plate; Model 5: A crucifix crack in a square plate.  
\end{table}
Table 1: The numbers of elements used for different numerical methods.  
**Acknowledgments:** This research was supported by the National Natural Science Foundation of China (51769011).  
**Conflicts of Interest:** The authors declare no conflict of interest.