## 2 Materials and Methods
### Blood Property and Simulation Condition Settings  
This study employs COMSOL Multiphysics 6.2 to conduct coupled simulations of the fluid and solid domains, thereby achieving fluid-structure interaction analysis. The time step is set to 0.001 s, and the maximum residual for convergence is set to 10\({}^{-5}\). The numerical simulations are performed for three cardiac cycles, and all the results are obtained from the third cycle. Blood is regarded as an isotropic, homogeneous, incompressible, inviscid non-Newtonian fluid, with its density and viscosity coefficient set at 1060 kg/m\({}^{3}\) and 0.0035 kg/m-s, respectively. The specific viscosity of blood is characterized by shear stress and shear rate. The relationship between the viscous shear stress \(\tau\) in any flow direction, viscosity \(\mu\), and shear rate \(\partial u/\partial y\) is as follows:  
\[\tau=\mu\frac{\partial u}{\partial y} \tag{7}\]  
A non-Newtonian fluid is defined as a fluid in which the relationship between shear stress and shear rate is not linear. When \(\partial u/\partial y>100\) s\({}^{-1}\), it behaves as a Newtonian fluid. The flow behavior and shear rate of blood in rheology differ from those of a Newtonian fluid, which has a significant impact on the biomechanical analysis of AD and aneurysms. Therefore, it is not appropriate to simply regard blood as a Newtonian fluid. The non-Newtonian fluid Carreau-Yasuda equation is as follows:  
\[\mu=\mu_{\infty}+(\mu_{0}-\mu_{\infty})\Big{[}1+(\dot{\gamma}\lambda)^{a} \Big{]}^{(n-1)/a} \tag{8}\]  
In the Carreau-Yasuda equation:  
\(\mu_{0}\) is the zero-shear viscosity in mPas, \(\mu_{\infty}\) is the infinite-shear viscosity in mPas, \(\lambda\) is the relaxation time in seconds, \(n\) is the power-law index where n \(<\) 1, and a is the Yasuda index with a \(\leq\) 2. To determine whether a flow is laminar or turbulent, the Reynolds number \(Re\) is introduced, defined as follows:  
\[Re=\frac{\rho_{f}Vd}{\mu}=\frac{\rho_{f}V^{2}}{\mu V/d} \tag{9}\]  
In this formula:  
\(\rho_{f}\) denotes fluid density, kg/m\({}^{3}\); \(V\) represents the mean velocity across the cross-section, m/s; d is the aortic diameter, m; and \(\mu\) stands for dynamic viscosity of the fluid, kg/m-s. In this study, the Womersley number of the fluid domain model is 19.18, corresponding to a Reynolds number range of 4795-19,180. Given the maximum Reynolds number of 5100, which is within the turbulent threshold, the blood flow is modeled as turbulent. The shear stress transport k-\(\epsilon\) model is used, with a turbulence intensity of 1.5% at the vessel inlet [26]. Momentum and mass conservation equations are applied across the fluid domain. The corresponding equations are as follows:  
\[\rho_{f}\frac{\partial v_{f}}{\partial t}+\rho_{f}(v_{f}\cdot\
abla)v_{f}- \
abla\cdot[-pI+K]-F=0 \tag{10}\]  
\[\rho\
abla\cdot v_{f}=0 \tag{11}\]  
\[K=(\mu+\mu_{T})\Big{(}\
abla u+(\
abla u)^{T}\Big{)} \tag{12}\]  
The equation for turbulent kinetic energy (k) is given by:  
\[\rho_{f}\frac{\partial k}{\partial t}+\rho_{f}(v_{f}\cdot\
abla)k-\
abla\bigg{[} \Big{(}\mu+\frac{\mu_{T}}{\sigma_{t}}\Big{)}\
abla\varepsilon\bigg{]}-p_{k}+ \rho\varepsilon=0 \tag{13}\]  
The equation for the turbulent dissipation rate (\(\varepsilon\)) is given by:\[\rho_{f}\frac{\partial k}{\partial t}+\rho_{f}(v_{f}\cdot\
abla)k-\
abla\left[ \left(\mu+\frac{lt_{T}}{\sigma_{t}}\right)\
abla\epsilon\right]-C_{\epsilon 1}\frac{\epsilon}{k}p_{k}+C_{\epsilon 2}\rho\frac{\epsilon^{2}}{k}=0 \tag{14}\]  
In this formula:  
\(v_{f}\) is the velocity vector of the blood, in m/s; \(\rho_{f}\) is the density of the blood, in kg/m\({}^{3}\); \(p\) is the absolute pressure, in MPa; and \(C_{\epsilon 1}\) and \(C_{\epsilon 2}\) are empirical parameters, with values of 1.44 and 1.92, respectively [27].The non-Newtonian fluid properties of blood are shown in Table 3.  
Based on clinical data [19], boundary conditions for the vessel inlet, outlet, and surrounding tissues were set.