# Numerical Simulation and Analysis of Heart-Aorta Fluid-Structure Interaction Based on S-ALE Method
## 2 Materials and Methods
### Establishment of Heart-Aorta FSI Model  
The heart-aorta finite element model used in the study was established from previous computed tomography scans [26]. The model was developed through 3D reconstruction technology to obtain detailed anatomical structural features, as shown in Figure 2. Taking the aorta as an example (Figure 2a), Mimics 10.0 (Materialize Inc., Leuven, Belgium) was used to extract the point cloud from the CT image data. Then, the point cloud data were imported into Geomagic Studio 12 (3D Systems Inc., Rock Hill, SC, USA) for geometric encapsulation. Finally, Hypermesh 14.0 (Altair Engineering Inc., Troy, MI, USA) was utilized to mesh the geometric model of the aorta. The established aortic section diameters have good consistency with the data reported in the literature, as shown in Table 1.  
Figure 2b shows the finite element model of the heart-aortic system. The aorta and LV were meshed using 4-node shell elements and connected by means of a shared node. The number of elements and nodes were 2100 and 2107, respectively. The wall thicknesses of the aorta and LV were 1.8 mm and 1.0 mm, respectively. The heart and pulmonary arteries were discretized using 4-node solid elements, and the numbers of elements and nodes were  
Figure 2: Establishment of FE model for structural domain. (**a**) Reverse modeling of FE model of aorta; (**b**) FE model of the heart–aorta system. 27,686 and 6456, respectively. The LV was in the end-diastolic state and its volume was 120.2 mL within the normal physiologic range [27]. The aorta was employed with isotropic and incompressible linear elastic material [12; 20; 28], with a density of 1100 kg/m\({}^{3}\) and a Young's modulus and Poisson's ratio of 2 MPa and 0.45 [29], respectively. Similarly, the LV was used with a linear elastic material with a density of 1000 kg/m\({}^{3}\) and [PERSON]'s modulus and Poisson's ratio of 1 MPa and 0.45, respectively. Heart tissue and pulmonary arteries were characterized using a viscoelastic constitutive model with a density of 1000 kg/m\({}^{3}\) and a bulk modulus of 2.6 MPa [30]. The expression for its mechanical behavior is given by the following:  
\[G(t)=G_{\infty}+(G_{0}-G_{\infty})e^{-\beta t} \tag{7}\]  
where \(G_{0}\) and \(G_{\infty}\) are the short and long shear moduli with values of 0.44 MPa and 0.15 MPa, respectively; and \(\beta\) is the attenuation coefficient with a value of 0.25. Figure 3 illustrates the established finite element model of the fluid domain. The fluid domain space had a length of 124 mm, a width of 120 mm, and a height of 192 mm, as shown in Figure 3a. In the S-ALE method, structured three-dimensional orthogonal mesh generation was achieved by defining the number of nodes along three coordinate axes. The keyword *ALE_STRUCTURED_MESH_CONTROL_POINT was used to control the number of nodes in the x, y, and z directions, thereby determining the spatial resolution of the computational mesh. The keyword *ALE_STRUCTURED_MESH was implemented to generate the fluid domain mesh automatically. A segment was created on the surface of the LV and aorta, while *ALE_STRUCTURED_MESH_VOLUME_FILLING was utilized to realize adaptive generation of intracavitary blood. The blood model is shown in Figure 3b. The fluid domain outside of blood was defined as vacuum. This study conducted a mesh size sensitivity analysis of the average maximum blood velocity at the aortic root using four fluid domain meshes with different resolutions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Sinus of Valsalva** & **Sinotubular Junction** & **Mid Ascending** & **Mid Descending** & **Diaphragm** \\ \hline [PERSON] et al. 2011 [31] & 31.1–33.2 & 26.7–28.5 & 27.2–29.1 & 20.3–22.1 & 19.3–20.8 \\ [PERSON] et al. 2019 [23] & 27.4 & 27.8 & 27.1 & 20.8 & 18.1 \\ Section diameters & 32.3 \(\pm\) 0.8 & 27.1 \(\pm\) 0.3 & 26.8 \(\pm\) 0.6 & 21.2 \(\pm\) 0.5 & 19.4 \(\pm\) 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of aortic section diameters with data from the literature (mm).