## 2 Materials and Methods
### Numerical Setup
#### 2.4.2 Structural Model  
Mesh tests for the structural domain were based on the element size. The geometry was the same as that used in previous tests, but computational structural dynamics simulations were performed with Ansys Transient Structural to solve the corresponding associated equations via the finite element method (FEM). The linear shell element was used since the wall thicknesses of the artery and the aneurysm were considerably smaller than the diameter of the artery. The wall thickness is commonly set to 10% of the artery diameter. In  
Figure 5: Artery surfaces (in green) created with a wall thickness of 0.35 mm and the aneurysm (in gray) with a thickness of 0.35, 0.2, or 0.10 mm.  
this case, considering the average diameter of the selected geometries, the wall thickness was set to 0.35 mm. This thickness was used for both the aneurysm and the artery, and it was kept for all cases to show the size effect.  
The element size was 0.15 to 0.50 mm. An external pressure of 1650 Pa, representing the action of cerebrospinal fluid for a healthy adult [32], was used, and the internal pressure was set to 15 kPa, representing the action of the blood flow. The inlet and outlets were assumed to have fixed support. A five-parameter Mooney-Rivlin hyperelastic model was used for both the mesh tests and FSI simulations. Its elastic energy density function \(W\) can be seen in Equation (4):  
\[W=c_{10}(I_{1}-3)+c_{01}(I_{2}-3)+c_{11}(I_{1}-3)\left(I_{2}-3\right)+c_{20}(I _{1}-3)^{2}+c_{02}(I_{2}-3)^{2} \tag{4}\]  
where \(I_{1}\) and \(I_{2}\) are the strain invariants, and \(c_{10}\), \(c_{01}\), \(c_{11}\), \(c_{20}\), and \(c_{02}\) are model constants obtained from the literature [33]. Although the wall of an aneurysm is anisotropic, in this work, it was modeled as an isotropic material, which is common practice since both approaches yield similar results [34].  
The considered results were the total displacement and the von Mises stress at the aneurysm. The results and percentage differences with respect to the mesh with the smallest element (\(D_{015}\)) can be seen in Table 2. For the von Mises stress, differences of greater than 5% were observed, so we chose to use a maximum size element of 0.20 mm, since the differences were below that value.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Element Size** & **Displacement** & **D015 [\%]** & **Von Mises** & **D015 [\%]** \\
**[mm]** & **[mm]** & & **Stress [MPa]** & **D015 [\%]** \\ \hline
0.50 & 0.39 & 2.63 & 0.13 & \(-\)7.14 \\
0.40 & 0.39 & 2.63 & 0.14 & 0.00 \\
0.35 & 0.39 & 2.63 & 0.13 & \(-\)7.14 \\
0.30 & 0.39 & 2.63 & 0.14 & 0.00 \\
0.25 & 0.39 & 2.63 & 0.14 & 0.00 \\
0.20 & 0.39 & 2.63 & 0.14 & 0.00 \\
0.15 & 0.38 & - & 0.14 & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Displacement and von Mises stress evolution obtained for the mesh tests and their percentage differences with respect to the 0.15 mm element size.  
Figure 6: Average WSS variation in the aneurysm at diastole (DWSS) and systole (SWSS), obtained in the mesh convergence test based on the mesh density with linear and quadratic tetrahedral elements.