## 3 Results and Discussions
### Three-Point Bending of a Beam with Three Circular Holes
#### 3.1.1 Case I  
In this particular specimen, the initial crack length was set to 38.1 mm, and its position was located 127 mm away from the mid-span of the specimen. These specific values were chosen to investigate the crack behavior under the given conditions and assess the influence of the crack's initial length and position on the interaction with the middle hole. It is clearly observable that the crack is noticeably attracted towards the middle hole, indicating a strong tendency to move towards it. As the crack continues to propagate, it ultimately reaches the left side of the middle hole, leading to a complete cessation of crack growth. Figure 4 illustrates the direction of crack growth for case I. Subsequently, the direction of crack growth observed in this study is compared with both experimental crack trajectories conducted by [PERSON] et al. [45], and numerical results obtained by [PERSON] and [PERSON] [46], who utilized an adaptive finite element approach on a multifunctional super singular element. Additionally, a comparison is made with the numerical method using the floating node method, which was combined with the symplectic analytical singular element by [PERSON] et al. [47]. The crack growth path predicted in the current study demonstrated greater accuracy in reaching the middle hole compared to the predicted crack path obtained by [PERSON] et al. [48], which employed a polygonal XFEM with new numerical integration. Furthermore, the results obtained using a coupled extended meshfree-smoothed meshfree method by [PERSON] et al. [49] were also used for comparison. To enhance the visualization of the accuracy of the predicted crack growth path in comparison to the experimental path obtained by [PERSON] et al. [45], Figure 4g displays the crack tip coordinate throughout the crack growth process, which closely aligns with the experimental data. This similarity between the predicted and experimental crack tip coordinates highlights the effectiveness and reliability of the crack growth prediction in the analysis. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Case No.** & **Crack Length, a (mm)** & **Crack Location, b (mm)** & **Hole Configuration** \\ \hline I & 38.1 & 127 & With hole \\ II & 38.1 & 127 & Without hole \\ III & 63.5 & 152.4 & With hole \\ IV & 63.5 & 152.4 & Without hole \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulated three-point bending beam configurations. Figure 3: Finite element meshes for the three-point bending beam (**a**) without hole and (**b**) with three holes. In order to assess the dimensionless SIF (\(\overline{K_{I}}\)), a comparative analysis was conducted between the results obtained in this study and the results obtained by [PERSON] et al. [50]. [PERSON] et al. [50] utilized XFEM and a smooth nodal stress method for their investigation. The comparison, as shown in Figure 5, demonstrates a high degree of similarity between the present study and theirs. Figure 6 shows a comprehensive comparison of the von Mises stress distribution results for case I in the present study with the corresponding findings obtained by [PERSON] and [PERSON] [43] using a mesh-independent computational algorithm that was specifically developed and integrated into the widely used commercial finite element software ABAQUS. The figure provides a visual representation of the stress distribution patterns and highlights the agreement between the two studies. Notably, there is an excellent level of concordance observed in both the stress distribution and the stress values. Figure 7 shows the crack growth path predicted by ABAQUS standard software without a mesh-independent computational algorithm, as reported by [PERSON] and [PERSON] [43]. Notably, no modifications or enhancements were made to the software for this analysis. However, in this specific instance, the predicted crack path intersects the bottom hole, in contrast to the expected behavior observed in the experimental results where the crack is anticipated to pass the hole and reach the middle hole.