## 3 Results of Global-Local Analysis
### Studied Cases  
To evaluate the influence of the size of the local nonlinear problem, three cases were analyzed:  
* Case I: the local problem covers 50% of the total thickness of the plate with a length of 30 mm, including 10 mm of pre-crack, Figure 3a.
* Case II: the local problem covers the entire thickness of the test, with the same length of 30 mm, including 10 mm of pre-crack, Figure 3b.
* Case III: the local problem covers the entire thickness of the test, with a length of 35 mm, including 15 mm of pre-crack, Figure 3c.  
For all the cases, three different meshes are used for the local problems, defined by the size of the element, 1 mm, 0.625 mm and 0.5 mm, giving 2, 3 and 4 elements on the thickness of each plate. For global problems, the same size of 2 mm is used for the elements, which gives one element on the thickness of each plate. In order to connect both models, the global one has a refinement on the nodes in contact with the local model in order to obtain compatible meshes.  
Figure 2: Details and dimensions of model to be analyzed: (**a**) 3D model dimensions. (**b**) Thickness of model.  
The number of nodes for each model is presented in Table 1. For all three analyzed cases, the global problem has very few nodes, the local model has a larger number of nodes, and the reference model, the monolithic model, has an even larger number of nodes.  
Additionally, the mesh quality was evaluated using the aspect ratio parameter. Table 2 presents the corresponding values for each region of the meshes. For all monolithic models, an average aspect ratio of approximately 3.5 was observed in the plates, while values exceeding 10 were found in the cohesive zone. This pattern is consistent across all local models. In contrast, the global models exhibit improved mesh quality in the plates, with aspect ratios below 1.5; however, poor mesh quality persists in the cohesive zone.  
Although the mesh quality in the cohesive zone shows high aspect ratio values, the focus of the present study was on assessing the influence of the local model size rather than optimizing element quality.  
The solutions of the global/local models are compared with the reference problem using: deformation, nodal forces, stress distribution, displacement of the crack tip, and computation times.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case/Model (Mesh Size)**} & \multicolumn{4}{c}{**Number of Nodes**} \\ \cline{2-5}  & **1 mm** & **0.625 mm** & **0.5 mm** \\ \hline \multicolumn{2}{c}{Monolithic} & 19,656 & 65,928 & 128,010 \\ \hline \multirow{2}{*}{Case I} & Global & 288 & 288 & 288 \\  & Local & 3224 & 12,054 & 18,666 \\ \hline \multirow{2}{*}{Case II} & Global & 256 & 256 & 256 \\  & Local & 4836 & 16,072 & 31,110 \\ \hline \multirow{2}{*}{Case III} & Global & 256 & 256 & 256 \\  & Local & 5616 & 18,696 & 36,210 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Number of nodes for every model.  
Figure 3: Different cases analyzed with their corresponding meshes. (**a**) case I, (**b**) case II and (**c**) case III.  
As a reference solution, the complete problem is solved using a 3D mesh and a cohesive zone model, employing the same mesh size as the corresponding local model. For each of the three local mesh sizes considered, a reference model is generated, referred to as'monolithic', since the solution involves a single global model of the structure without any decomposition.