## 2 Materials and Methods  
The three-dimensional models of the aortic arch repaired with two graft shapes (i.e., the half-moon and crescent) are illustrated in Figure 2. Both grafts were named by their anterior shape. The grafts were highlighted in a grey color. Models were reconstructed from MRIs using the commercial software package Mimics (Materialise, Belgium). The two graft shapes were constructed using Hypermesh software (Altair Engineering, Michigan, USA). The chord length of the aortic arch, measured from the echocardiography, was used to tailor the shape of both grafts. Figure 2a,b illustrate both graft shapes at a blood pressure of 80 mmHg. It is obvious that the crescent graft matched better with the native aortic arch. The centerline of the repaired aorta was used for anatomical measurements. The cross-section A-A\({}^{\prime}\) represents the ascending aorta near the graft site, which was 38.27 mm away for the inlet along the centerline. The vertical cross-section B-B\({}^{\prime}\), as shown in Figure 2c,d, was 55.76 mm away from the inlet. The centerline angle (\(\theta\)) at plane B-B\({}^{\prime}\) was measured between two tangential vectors \(\overset{\rightarrow}{a}\) and \(\overset{\rightarrow}{b}\) using ImageJ freeware [19]. The geometric quantifications are summarized in Table 1.  
Figure 2: Reconstructed aortic arch repaired with (**a**) a half-moon shaped graft and (**b**) a crescent graft. B-B\({}^{\prime}\) cross-section in (**c**) a half-moon shape model and (**d**) a crescent model. A mesh convergence study was performed for both the solid and fluid domain, as shown in Figure 3. The aortic wall, with a thickness of 2 mm [20], was meshed into 31,081 hybrid hexahedral elements (C3D8H). The aortic lumen (i.e., the blood domain) was meshed with 166,245 tetrahedral elements. Both grafts and aortas were assumed to be isotropic elastic materials and nearly incompressible (i.e., Poisson ratio of 0.45) [21; 22]. The stiffness of the native aortic tissue was adopted as 0.84 MPa [23]. The stiffness values of the three graft materials (i.e., PTFE, CorMatrix, and pulmonary homograft) were 1000 MPa, 30 MPa and 3 MPa, respectively [24; 25; 26]. There were a total of five model configurations in terms of graft materials and shapes. For the material mismatch cases, the control configuration was the aorta with native tissue properties. For the shape effect, the control configuration was the crescent graft with a pulmonary homograft. The Arbitrary Lagrangian-Eulerian (ALE) method was employed to capture the interaction between the aorta wall and blood flow [27]. The two-way coupled FSI numerical model was implemented using a commercial software Mpci 4.5 (Fraunhofer SCAI, Germany) [28]. The passing parameters between fluid domain and solid domain were the boundary-relative force vector and nodal position (i.e., displacement) at the integration point of each element [29; 30]. The neighborhood search technique (i.e., the closest element of a source mesh to each node of a target mesh) and the \begin{table}
\begin{tabular}{c c c c} \hline  & **A-A\({}^{\prime}\)** & **B-B\({}^{\prime}\)** & **Centerline Angle** \\ \hline Half-moon & 429.3 mm\({}^{2}\) & 444.1 mm\({}^{2}\) & 155.85\({}^{\circ}\) \\ Crescent & 431.4 mm\({}^{2}\) & 335.3 mm\({}^{2}\) & 139.74\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric difference between half-moon and crescent models in terms of cross-section areas at A-A\({}^{\prime}\) and B-B\({}^{\prime}\) as well as the centerline angle. Figure 3: Mesh convergence in solid (**a**) and fluid (**b**) domains. shape function mapping [28] were adopted for non-matching meshes between fluid and solid domains. The coupling was enforced at every 0.05 s. The Gauss-Seidel algorithm for the iterative coupling scheme between the solid and fluid domains was adopted. A partitioned approach was used to solve the governing equations sequentially. The solid mechanics analysis was carried out through ABAQUS 2017 (Dassault Systemes Simulia Corp., Providence, RI, USA).