## 4 Validation of New AI-FEM Multiple Cracks Coalescence Criteria
### Results
#### 4.4.2 Comparison of FCG Results from AI-FEM, Experiment, and Codes  
Figure 9 compares the natural multiple FCG results using AI-FEM with experimental results and the FCG results applying the crack coalescence criteria from ASME Section XI. In Figure 9a, the crack depth is compared over the cycles, while, in Figure 9b, the fracture surfaces from the experimental results and AI-FEM FCG results are compared over the cycles. Additionally, Figure 9c compares the normalized crack area over the cycles between the ASME Section XI and AI-FEM FCG results. \(A_{c}\) represents the crack area, while \(A_{plate}\) represents the area of the plate cross-section. Figure 9d compares the fracture surfaces over the cycles between ASME Section XI and AI-FEM results. Figure 8: Comparison of the SIF values derived along the crack front obtained by AI-FEM with Raju–Newman SIF values. \begin{table}
\begin{tabular}{c c} \hline \hline
**Maximum Bending Moment** & **Minimum Bending Moment** \\
**[N-mm]** & **[N-mm]** \\ \hline \hline
1.3 \(\times\) 10\({}^{6}\) & 1.3 \(\times\) 10\({}^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Loading conditions for simulating FCG. Table 6 summarizes the quantitative analysis of differences in the cycle at which crack coalescence occurs, the final cycle coalesced area, and the final cycle crack depth to  
Figure 9: Comparison of FCG results: (**a**) FCG rate in the depth direction of the crack over cycles; (**b**) fatigue fracture surfaces over cycles between AI-FEM and experiments; (**c**) crack area over cycles between AI-FEM and ASME Section XI; (**d**) fatigue fracture surfaces over cycles between AI-FEM and ASME Section XI. quantify the conservatism of the ASME Section XI crack coalescence criteria. The equation representing these differences is as follows:  
\[Diff.=\frac{R_{ASME}-R_{AI-FEM}}{R_{ASME}} \tag{6}\]  
where \(R_{ASME}\) represents the crack growth results obtained using the ASME Section XI coalescence criteria, and \(R_{AI-FEM}\) represents the crack growth results obtained using AI-FEM. The results derived by each methodology are quantitatively analyzed for differences in terms of coalesced cycles, normalized crack area, and maximum crack depth. Figure 9a,b compares the FCG rate in the crack depth direction and the fatigue fracture surface formed by the beach mark method, as provided in the FCG experiment conducted by [PERSON] et al. [5], with the AI-FEM results. The experimental data for the FCG rate in the crack depth direction were reconstructed by digitizing the graph provided in the literature. Additionally, the fatigue fracture surface provided in the experiment was re-expressed in the same form to ensure a clearer comparison, as the fracture surface was not clearly visible due to experimental limitations. In Figure 9a, the experimental results (Exp.) are shown with black symbols, the FCG results from AI-FEM are shown with red open symbols, and the FCG results using the ASME Section XI crack coalescence criteria are shown with blue open symbols. The crack depth at the deepest point of the left crack (multiple crack 1) is represented by circular symbols, the crack depth at the deepest point of the right crack (multiple crack 2) is shown with triangular symbols, and the crack depth in the re-entrant region formed by the coalescence of the two multiple cracks is represented by square symbols. As shown in Figure 9a, the experimental results overall matched well with the FCG results using AI-FEM. The cycles at which crack coalescence occurred were also the same, at approximately 210,000 cycles. The maximum error in crack depth was 7.8% at around 220,000 cycles. This trend was also confirmed in Figure 9b, where the experimental results (solid line) and FCG results using AI-FEM (dashed line) showed the same shape at the same cycle. Therefore, applying the proposed plastic zone size-based crack coalescence criteria to AI-FEM is expected to simulate experimental results well, providing a realistic model for crack growth.