## 3 Development of AI-FEM Multiple Cracks Growth Algorithm
### Plastic Zone Size-Based New AI-FEM Multiple Cracks Coalescence Criteria  
Linear elastic fracture mechanics (LEFM) is generally used when simulating crack growth, as the plastic zone near the crack tip is small, allowing elastic analysis to sufficiently approximate crack behavior. Elastic analysis is also relatively simple and computationally efficient, which contributes to its widespread use. However, when multiple cracks are present in the structure, especially when the distance between them decreases, significant  
Figure 3: Visualized element type in the crack-tip area generated by AI-FEM. plastic behavior may occur due to interactions between the crack tips. In such cases, the linear elastic assumption may not accurately simulate crack growth behavior. Plastic deformation affects the stress field at the crack-tip and can alter the crack growth rate and direction. Several researchers have addressed this issue using different approaches. [PERSON] et al. [26] utilized the Kachanov method [27], and Mises yield criterion to analyze crack interactions in plates, focusing on the influence of crack geometry and spacing on plastic zone development. [PERSON] [28] employed a strip yield model to study crack coalescence in riveted stiffened sheets, treating multiple cracks as a single fictitious crack. [PERSON] et al. [29] primarily focused on predicting the plastic zone size ahead of crack-tips using a numerical analysis method based on [PERSON]'s model [30]. This study reveals that the relative position and arrangement of multiple cracks are major factors in determining the plastic zone size. Their study revealed that the relative position and arrangement of multiple cracks are major factors in determining the plastic zone size. While these methods provide valuable insights into the plastic effects of crack coalescence and multiple cracks, they often come with limitations. Many are computationally intensive or limited in their applicability to complex geometries, making them less suitable for practical engineering applications, especially in cases involving multiple crack-growth simulations. AI-FEM performs iterative finite element analyses to simulate crack growth, making it essential to balance computational cost with the accuracy of capturing plastic effects. To address this challenge, this study proposes a crack coalescence criterion based on the plastic zone correction method presented by [PERSON] et al. [31]. [PERSON]'s approach offers a compromise between computational efficiency and accuracy by adjusting the stress intensity factor (SIF) to account for the plastic zone size, thereby improving the accuracy of elastic analysis without significantly increasing computational demands. The plastic zone size calculation formula proposed by [PERSON] et al. [31] was used, and it is as follows:  
\[r_{p}=\frac{1}{3\pi}\bigg{(}\frac{K}{\sigma_{y}}\bigg{)}^{2} \tag{3}\]  
where \(r_{p}\) represents the secondary plastic zone size, \(K\) represents the SIF, and \(\sigma_{y}\) represents the yield strength of the materials. Furthermore, the critical distance (\(S_{proposed}\)) of the newly proposed coalescence criteria for multiple cracks, which applies the concept of plastic zone size, is defined as follows in Equation (4):  
\[S_{proposed}=r_{p.crack1}+r_{p.crack2} \tag{4}\]  
where \(r_{p.crack1}\) and \(r_{p.crack2}\) are the secondary plastic zone sizes derived from the SIFs at the surface points of two adjacent cracks. \(S_{proposed}\) is defined as the sum of these two plastic zone sizes. If \(S\leq S_{proposed}\) (coalescence occurs if the sum of the plastic zone sizes exceeds the distance between cracks), the two cracks are combined, and the remaining sections are removed as the cracks are coalesced into a single crack. Figure 4 schematically illustrates crack growth and coalescence based on the plastic zone size criterion previously presented. First, Figure 4a illustrates the process of evaluating whether multiple cracks will coalesce into a single crack.