## 3 Numerical Examples  
### Numerical Examples of Homogeneous Material  
To verify the accuracy of the method proposed in this paper, some typical numerical examples are selected. (a) Homogeneous Rectangular Plate with Edge Crack under Unidirectional Tensile Load  
As shown in Figure 5a, a rectangular plate with length \(h\) = 100 mm and width \(b\) = 50 mm contains a horizontal edge crack of length \(a\). The rectangular plate is subjected to uniform uniaxial tensile loads \(\sigma\ =\ 10\) MPa on upper and lower sides, with an elastic modulus and Poisson's ratio of \(E\ =\ 10^{5}\) MPa and \(\
u\ =\ 0.2\), respectively. The crack tip was implemented and defined using XFEM tools. As shown in Figure 5b, the finite element mesh of the rectangular plate is presented for \(a/b\) = 0.5. Considering the singularity at the crack tip, CPS3 elements are utilized in the crack tip region to improve computational accuracy, while CPS4 elements are used in other areas. The entire mesh consists of 48,325 quadrilateral elements, 126 triangular elements, 48,451 elements in total and 48,919 nodes. The integral domain is selected as the region enclosed by a circle, with the crack tip as the center and a radius of 10 mm. The normalized factor is selected as \(K_{0}\ =\ \sigma\sqrt{\pi a}\). According to the crack state and load distribution of the model mentioned above, it is a mode I fracture problem. The theoretical solution of the mode I stress intensity factor is as follows:  
\[K_{\rm I}\ =\ F\sigma\sqrt{\pi a}, \tag{27}\]  
\[F\ =\ 1.12-0.23\frac{a}{b}+10.6\Big{(}\frac{a}{b}\Big{)}^{2}-21.7\Big{(}\frac{a }{b}\Big{)}^{3}+30.4\Big{(}\frac{a}{b}\Big{)}^{4}, \tag{28}\]  
Table 2 presents the normalized stress intensity factors from the present method, ABAQUS, and theoretical calculations at varying crack lengths (\(a/b\)). The results from the present method closely match those from ABAQUS and the theoretical solution, with relative errors below 0.8%. This provides evidence of the accuracy of the method. Figure 5: (**a**) Homogeneous rectangular plate with edge crack under unidirectional tensile load; (**b**) model meshes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline _a/b_ & **Present** & **Theoretical** & **ABAQUS** & **Error A 1 [\%]** & **Error B 2 [\%]** \\ \hline
0.1 & 1.176 & 1.184 & 1.171 & 0.7 & 0.4 \\
0.2 & 1.362 & 1.373 & 1.360 & 0.8 & 0.1 \\
0.3 & 1.653 & 1.665 & 1.651 & 0.7 & 0.1 \\
0.4 & 2.098 & 2.113 & 2.101 & 0.7 & 0.1 \\
0.5 & 2.821 & 2.843 & 2.823 & 0.8 & 0.1 \\ \hline \hline \end{tabular}  
1  
2  
\end{table} 
Table 2: Normalized stress intensity factor of a homogeneous rectangular plate with an edge crack under a unidirectional tensile load. Figure 6: (a) Homogeneous rectangular plate with a center-inclined crack under unidirectional tensile load; (b) model meshes. To verify the mesh convergence of the present method in calculating fracture parameters at the crack tip, the example in Figure 5a is used to study the impact of overall and local mesh density near the crack tip on the convergence of the results. As shown in Figure 7a, mesh density is varied by changing the number of elements (N1 and N2) along the length and width of the rectangular plate, and Nr denotes the number of radial elements in the crack tip refinement zone. The normalized factor was selected as \(K_{0}~{}=~{}\sigma\sqrt{\pi a}\). First, Nr was fixed at 20 to analyze the impact of N1 \(\times\) N2 on convergence. N1 \(\times\) N2 were set at 8 \(\times\) 16, 10 \(\times\) 20, 34 \(\times\) 66, 100 \(\times\) 200, and 166 \(\times\) 334. The integral domain was the area enclosed by a circle with a radius of 10 mm centered at the crack tip. Table 4 shows the normalized stress intensity factor results obtained under different mesh densities. When the mesh density N1 \(\times\) N2 increases from 34 \(\times\) 66 to 166 \(\times\) 334, the stress intensity factor remains nearly unchanged, with a relative variation within \(\pm\)0.07%.