# Numerical Simulation and Analysis of Heart-Aorta Fluid-Structure Interaction Based on S-ALE Method
## 3 Results and Discussion
### Validation of the FSI Numerical Model  
The heart-aorta FSI model based on S-ALE method was validated for the hemodynamics of the aorta under a normal physiological state (blood viscosity 0.0035 Pa\(\cdot\)s and peak LV displacement 5.1 mm) to ensure the simulation's reliability. Hemodynamic parameters (blood velocity, flow rate, and WSS) for aortic sections 1-4 were validated against the published ALE method [23; 24] and data measured by CMR experiments [3; 4] in the literature.  
Figure 7 demonstrates the blood flow and WSS at section 1. The average blood velocity time history curve is shown in Figure 7a, which is obtained by calculating the arithmetic mean of all nodal velocities in the section. Figure 7b shows the velocity distribution at the point A (0.168 s). From the streamline diagram (Figure 7c), it can be observed that there is turbulence in the blood at the aortic arch. Numerical simulation results indicated that the peak blood velocity was 0.42 m/s at section 1, while the peak velocities at the other sections, 2-4, ranged from 0.46 to 0.83 m/s. Figure 7d displays the flow rate variation curve, which is calculated by multiplying the section-averaged velocity with the real-time area variation. The peak flow rate at section 1 was 283.4 mL/s and the other sections, 2-4, were 238.1-253.7 mL/s.  
Figure 7e demonstrates the section-averaged WSS of the aorta. The calculation of WSS is based on the local blood velocity gradient and blood viscosity, which are solved by the  
Figure 7: Blood flow and WSS at section 1 [3; 23; 24]. (**a**) Section-averaged blood velocity; (**b**) blood velocity distribution at the moment of point A; (**c**) partial blood flow stream line in the LV and aorta; (**d**) section-averaged blood flow rate; (**e**) section-averaged WSS.  
gradient tensor method. \(WSS=\mu\cdot(dv/dx)\), where \(\mu\) is the blood viscosity, \(\
u\) is blood velocity along the direction of vessel, and \(x\) is the distance from the wall along its inward normal direction. The results showed that the peak WSS at section 1 was 0.47 Pa and the other sections, 2-4, were 0.40-0.93 Pa. The hemodynamic parameters of sections 2-4 are shown in Figure 8.  
The comparison of peak hemodynamic parameters between numerical simulation and experimental data is shown in Figure 9. The analysis results indicated that the peak blood velocities obtained from the simulation were highly consistent with the experiment measurements, with a maximum relative error of less than 6%. In terms of section parameter comparison, except for the peak flow rate and WSS of section 2, which deviated from the experimental data by slightly more than 20%, the differences between the parameters of all other sections and the experimental results were within 20%.  
Figure 8: Blood flow and WSS at sections 2–4 [3,4,23]. (**a**,**d**,**g**) Section-averaged blood velocity; (**b**,**e**,**h**) section-averaged blood flow rate; (**c**,**f**,**i**) section-averaged WSS.  
The S-ALE method used in this study had good consistency with the data measured by CMR experiments. However, the hemodynamic parameters of section 1 were significantly different from the results obtained by the ALE method used by [PERSON] et al. [23]. The blood velocity and WSS simulated by the ALE method were almost twice that of the S-ALE method, and the reason for this phenomenon was the difference in aortic root diameter (Table 1). In their study, the finite element model of the aorta was obtained by scaling, and its diameter had a larger difference compared to the actual anatomical structure. The analysis of the simulation results shows that the heart-aorta FSI model developed based on the S-ALE method has good reliability and can provide support for subsequent studies.