## 2 Materials and Methods
### Numerical Methods and Setup  
In this study, the 3 pressure conditions described before were simulated in each of the 6 brain aneurysm models, and all with the same velocity condition at inlet, thus totaling 18 FSI simulations. During initial conditions, at the inlet of each model we used the corresponding velocity inlet value at time t = 0, V(t) = 0.278 m/s and an initial gauge pressure P(t) = 0 Pa.  
We configured a quadratic order unstructured tetrahedral mesh for the fluid domain and an unstructured triangles mesh using shell elements for the structural domain. During this step we used ANSYS(r) Meshing software. For the mesh size, we carried out a mesh or grid convergence study using the smallest aneurysm model (A-4): for the fluid domain, wall shear stress (WSS) at aneurysm surface was chosen for analysis because it is one of the main indicators on hemodynamics studies, and the parameters at study were based on it. Additionally, the static pressure at aneurysm surface was chosen, but we focused on WSS since it is more sensitive to changes in mesh size. Six element sizes were chosen: 0.50, 0.40, 0.30, 0.20, 0.10, and 0.05 mm, with differences in WSS of 2% between sizes 0.05 and 0.10 and below 1% in pressure, thus determining an optimal element size of 0.10 mm at the aneurysm area. For the structural or solid domain, von Mises Stress and displacement at the aneurysm were chosen. Four element sizes were chosen: 0.25, 0.20, 0.15 and 0.10 mm, with differences in von Mises stresses of 8% between the sizes 0.10 and 0.15 and below 1% in displacement. Although our study is FSI, we only analyzed the hemodynamics effects (fluid), thus we determined an optimal element size of 0.15 mm. The described results for the fluid and solid domain, along with the selected element size are shown in Figure 5a,b. It is important to note that an important factor in this selection was the available computational resources. The configured mesh for A-6 model is shown in Figure 6a (fluid domain) and 6b (solid domain), along with its boundary conditions. Simultaneously to the convergence study, we performed a mesh quality analysis for the fluid domain in every model, and we evaluated the maximum or minimum values of the following parameters: maximum skewness, minimum orthogonal quality, and maximum aspect ratio. Table 4 shows these parameters for the mesh with the selected element size. As it can be observed, the values meet the requirements and recommendations for tetrahedral elements of the ANSYS(r) Fluent Theory Guide. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Condition** & \(R_{p}\)[N\(\cdot\)s/m\({}^{5}\)] & \(R_{d}\)[N\(\cdot\)s/m\({}^{5}\)] & \(C\)[m\({}^{5}\)/N] \\ \hline Normal & \(1.40\times 10^{10}\) & \(1.66\times 10^{9}\) & \(2.32\times 10^{-10}\) \\ Hypertension & \(1.80\times 10^{9}\) & \(2.00\times 10^{10}\) & \(9.50\times 10^{-9}\) \\ ISH & \(2.00\times 10^{9}\) & \(2.00\times 10^{10}\) & \(9.50\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RCR Windkessel model constants for each pressure condition. The implementation of each fluid-structural simulation was carried out by a two-way coupling between ANSYS(r) Fluent (CFD) and ANSYS(r) Mechanical (FEA), solving the transient Navier-Stokes equations coupled with the structural equations through System Coupling component. For the fluid domain, the finite volume method (FVM) was used, with a dynamic meshing and a smoothing method (constant spring factor equal to 0.1). For the structural domain, the finite element method (FEM) was used and considering large deflections. On the other hand, for the fluid domain, the Pressure-Implicit with Splitting of Operators (PISO) algorithm was used to solve Navier-Stokes equations, with a spatial discretization using a least squares cell based on the gradient, second-order for pressure, and second-order upwind for momentum. For the transient formulation, the time was discretized by a first-order implicit formulation. The residuals of the momentum and continuity equations were kept below 0.001.