## 2 Materials and Methods
The governing equations are  
\[\frac{\partial\rho}{\partial t}+\rho\
abla\cdot\mathbf{u}=0 \tag{1a}\] \[\rho(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}(\
abla\cdot \mathbf{u}))=-\
abla p+\mu\
abla^{2}\mathbf{u}+pg \tag{1b}\]  
In this study, the Eulerian approach has been adopted. The numerical method used to solve the non-linear coupled equations in the fluid region and in the vessel region  
Figure 4: Patient-specific stent-graft proposed by the authors. Figure 5: Conventional graft (**a**) and patient-specific graft (**b**). involves a preliminary subdivision in a sequence of volumes of the tetrahedral shape of the two volumes. As the subdivision is provided, a linear approximation of the displacement and velocity fields in each discretized volume region has been introduced in the model depending explicitly on the values of the solutions at the corners of the tetrahedron. As the weak formulation of the governing equations has been enforced along the whole domain, the non-linear partial differential problem was formulated in terms of a non-linear ordinary differential problem. The solution has been obtained by means of the implicit Runge-Kutta method introducing a fourth-order polynomial description of the acceleration in each time step. The progressive reduction of time step intervals was involved as a preselected tolerance was not reached in each time step. The three fluid models (i.e., aneurismatic aorta, conventional, and patient-specific graft) were imported in.STL format in ICEM CFD 19.2 (Ansys Inc.) to discretize the volume in small elements and realize the mesh. For each model, tetrahedral elements in the core region and prismatic elements in the boundary layer (3 layers) were used. The created mesh consisted of about 1 million elements, as shown in Figure 3.0 Mesh quality was acceptable, and a verification operation was also performed in order to eliminate the unconnected vertices. Fluid dynamics was evaluated by a transient CFD simulation performed using FLUENT v19.2 (Ansys Inc.). A CFD solver was used to obtain an approximate solution of the Navier-Stokes differential equation that governs the fluid flow. The assumption and the analysis set-up were the same for the three fluid models (aneurismatic aorta, conventional, and patient-specific grafts). The blood flow was assumed to be compressible and Newtonian with a density of 1060 kg/m\({}^{3}\) and viscosity of 0.003 Pa \(\ast\) s. With a diameter greater than 0.5 mm, an assumption of Newtonian flow through the aorta is reasonable due to the fact that blood viscosity is relatively constant at the high rates of shear (100/s) typically found in the aorta [48]. The pressure velocity coupling algorithm SIMPLE was used as a solution method, and residuals in the order of 10\({}^{4}\) have been fixed for equation convergence. Six boundary surfaces were defined in all three fluid models; in particular, the inlet, four outlets: abdominal, branch1, branch2 and Mesenteric, and the wall. The boundary conditions used for the model are time-dependent. The velocity inlet boundary conditions were taken from color Doppler measurements of the aortic blood average velocity, previously adopted by [48]. Pressure outlet conditions were based on [49], in which branch pressure waveforms were assumed to follow the aorta waveform with scale down amplitudes, following [PERSON] et al. [50]. A velocity profile [51] for inlet and pressure [49] at outlet surfaces was imposed. The mixed-type boundary condition represents physiological conditions more accurately because when the heart ventricles contract, it induces a change in volume that causes a pressure gradient and forces the blood out [52]. A no-slip condition was assumed at the wall. Below, we present the velocity in Figure 7a and the different pressure profiles in Figure 7c,d fixed as boundary conditions. Two cardiac cycles have been simulated in order to achieve a solution that is independent of the initial conditions.