## 4 Validation of New AI-FEM Multiple Cracks Coalescence Criteria
### Results  
Figure 12 compares the FCG results of multiple cracks in a pipe over loading cycles using two different methods: the natural FCG results simulated using AI-FEM and the FCG results obtained by applying the crack coalescence methodology from ASME Section XI. To evaluate the conservatism of crack coalescence criteria of ASME Section XI, the results from both methods were quantitatively analyzed based on three factors: coalescence distance, coalescence shape, and maximum crack depth. The conservatism inherent in the ASME Section XI coalescence criteria, specifically the conservatism related to the distance criterion, was assessed by comparing the cycles at which crack coalescence occurred. The conservatism of the coalesced crack shape was evaluated by normalizing and comparing the difference in the crack area (\(A_{c}\)) formed at the final cycle for each method against the total pipe cross-sectional area (\(A_{pipe}\)). Lastly, the conservatism due to differences in both distance and shape criteria was investigated by comparing the maximum crack depth at the final cycle. Figure 12a presents a comparison of the fatigue fracture surfaces over cycles for the multiple crack configuration with a \(c/a=3\). The quantitative analysis results for investigating conservatism are summarized in Table 8. In the FCG results obtained using AI-FEM, coalescence occurred at 28,000 cycles, whereas, when applying the ASME Section XI methodology, coalescence occurred slightly earlier, at 24,000 cycles, resulting in a 14% difference. When comparing the normalized area of the two cracks at the final cycle, AI-FEM shows a value of 0.08, whereas ASME Section XI shows a value of 0.12, indicating that the crack grows approximately 33% larger. Due to the conservative nature of the distance and shape criteria in ASME Section XI, the crack depth at the final cycle is 0.91, while  
Figure 11: FE model and boundary conditions of pipe geometry. the crack depth determined by AI-FEM at the final cycle is 0.72, showing a difference of about 20%. Figure 12b presents a comparison of the fatigue fracture surfaces over cycles for the multiple crack configuration with a \(c/a=1\). The quantitative analysis results for investigating conservatism are summarized in Table 9. In the FCG results obtained using AI-FEM, coalescence occurred at 40,000 cycles, whereas, when applying the ASME Section XI methodology, coalescence occurred slightly earlier, at 36,000 cycles, resulting in an 11% difference. When comparing the normalized area of the two cracks at the final cycle, AI-FEM shows a value of 0.043, whereas ASME Section XI shows a value of 0.066, indicating that the crack grows approximately 35% larger. Due to the conservative nature of the distance and shape criteria in ASME Section XI, the crack depth at the final cycle is 0.87, \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Coalesced** & **Normalized** & **Normalized** \\  & **Cycles** & **Crack Area** & **Crack Depth** \\  & **(Cycles)** & **(\(A_{c}\)/\(A_{pipe}\))** & **(\(a\)/\(t\))** \\ \hline AI-FEM & 28,000 & 0.08 & 0.72 \\ ASME Section XI & 24,000 & 0.12 & 0.91 \\ Maximum difference & 16\% & 33\% & 20\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Quantitative comparison of FCG results between AI-FEM and ASME Section XI for \(c/a=3\) multiple crack configuration. Figure 12: Comparison fatigue fracture surface over cycles: (**a**) \(c/a=3\); (**b**) \(c/a=1\). while the crack depth determined by AI-FEM at the final cycle is 0.65, showing a difference of about 25%. Based on the plastic zone size coalescence criteria of AI-FEM and the coalescence criteria of ASME Section XI, FCG was analyzed for both methods. The FCG results derived from each methodology were quantitatively compared in terms of coalescence cycles, normalized crack area, and maximum crack depth.