## 3 Results
### Influence of Graft Shapes  
The impact of the graft shape (i.e., a classical half-moon and our proposed crescent) was evaluated using the pulmonary homograft. The repaired aortic arch by either the half-moon or crescent graft was quantified using two representative cross-sectional areas and the centerline angle, as listed in Table 1. It was clear that the half-moon arch repair, compared with the crescent one, exhibited a wider cross-sectional area as well as a larger centerline angle. For the half-moon arch, the cross-sectional area at the aortic arch (B-B\({}^{\prime}\)) was 24.49% larger than that of the crescent one. The centerline angle of the half-moon graft was 11.52% larger than that of the crescent graft. These geometric differences were hypothesized to affect their graft mechanics. The von Mises stress distributions for both the half-moon and crescent grafts are shown in Figure 5. Both grafts demonstrated larger stress than the adjacent native tissue, and the stress concentrated on the lower border of the aortic arch. In addition, the stress level in the crescent graft was smaller than the one in the half-moon graft. The pathline across the graft zone along the lower border of the arch was used to capture the peak von Mises stress of the graft (Figure 5c). It was observed that the maximum von Mises stresses on the half-moon and crescent grafts were 0.25 MPa and 0.06 MPa, respectively. For the half-moon graft, the stress concentrated at the interface between the graft and the native tissue. In contrast, there was not much stress variation along the pathline of the crescent graft. The blood flow dynamics at 0.4 s (peak pressure, systole) and 0.6 s (early diastole) for both graft repairs are shown in Figure 6. Streamlines illustrate the helical blood motion while y-direction velocity vectors, without any projection at two representative cross sections (ascending aorta A-A\({}^{\prime}\) and aortic arch B-B\({}^{\prime}\)), demonstrate the rotational flow. It is clear that flow disturbance was higher at the diastolic phase. The positive velocity magnitude represents the forward flow, and the negative one represents the reversed flow. The minimal flow disturbance was observed at the ascending aorta only at the diastolic phase. However, at the aortic arc, the reversed flow was observed during both the diastolic and systolic phases. The reversed flow was much more obvious in the half-moon graft than the crescent one. The reversed flow zone was minimal in the crescent graft. In addition, the reversed flow occurred around the inner curvature of the graft. Figure 5: Stress distributions at t = 0.4 s (peak systolic pressure) on the (**a**) half-moon and (**b**) crescent pulmonary grafts; (**c**) von Mises stress across the graft zone along the lower border of the arch. Figure 6: _Cont._  
The crescent graft was quantitatively compared with the half-moon one at the aforementioned cross-sections in terms of the area, the maximum velocity, and the mass flow rate at both phases (Table 2). Regardless of the graft shapes, both the cross-sectional areas and maximum velocity were larger at the systolic phase than the diastolic one. At the ascending aorta (A-A\({}^{\prime}\)), the crescent graft exhibited a relatively larger cross-sectional area at both phases, but the difference in the maximum velocity for both graft shapes was minimal. This could be attributed to the alterations in both the mass flow rate and cross-sectional area. At the ascending aorta (A-A\({}^{\prime}\)), the mass flow rates of the crescent graft were 11.2% and 8.7% higher than that of the half-moon one at 0.4 s and 0.6 s, respectively. The corresponding cross-sectional areas of the crescent graft were 5.6% and 6.1% larger than that of the half-moon one, respectively. At the aortic arch (B-B\({}^{\prime}\)), the cross-sectional area of the crescent graft was 31.9% smaller than that of the half-moon one during systolic phase (0.4 s).