## 2 Bird Strike and Rotor Dynamic Modeling  
### Bird Strike Modeling  
According to the statistical report from FAA [1], during 1990-2019, mallards struck aircraft a total of 1146 times, making it one of the birds with the highest frequency of collisions with aero-engines. Therefore, the mallard was taken as the prototype bird model. The weight of the selected mallard was 820 g with 40 cm in body length and a 45 cm wingspan. The mallard bird was scanned by the CT scan device at a medical imaging center. A total of 1566 DICOM (Digital Imaging and Communications in Medicine) images were obtained. DICOM images and the SPH method were adopted to model the mallard. During the modeling process, the internal cavities of the mallard bird were considered, and no SPH elements were used in such areas. The final model composed of 41,685 SPH elements each weighing 0.0191 g (Figure 2). The bird was endowed with a Null material model with Gruneisen equation of state (EOS) [5; 40]; the viscous stress was calculated by providing the intrinsic relationship of the model. The state equation was used to calculate the pressure, with the material initial density of 938 kg/m\({}^{3}\)[40]. Based on the EOS [41], the pressure of the compressed material is defined as follows:  
\[p=\frac{\rho_{0}C^{2}\mu\left[1+(1-\frac{\gamma_{0}}{2})\mu-\frac{\theta}{2} \mu^{2}\right]}{\left[1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{3}\frac{\mu ^{3}}{(\mu+1)^{2}}\right]}+(\gamma_{0}+a\mu)E \tag{1}\]For expansion of the material, the pressure is expressed as:  
\[p=\rho_{0}C^{2}\mu+(\gamma_{0}+a\mu)E \tag{2}\]  
where \(C\) is the intercept of the velocity curve; \(S_{1}\), \(S_{2}\), and \(S_{3}\) are the slope coefficients of the velocity curve; \(\gamma_{0}\) is the Gruneisen constant; \(a\) is the first-order volume correction to \(\gamma_{0}\); \(E\) is the Elastic Modulus of material; and \(\rho_{0}\) is the reference density. The parameter \(\mu\) is defined by:  
\[\mu=\frac{\rho}{\rho_{0}}-1 \tag{3}\]  
where \(\rho\) is density of the material. Based on Ref [42], the EOS's parameters of \(C\) = 1480, \(S_{1}\) = 1.92, \(S_{2}\) = 0, \(S_{3}\) = 0, and \(\gamma_{0}\) = 0.1 were used. A type of aero turbofan engine fan system is demonstrated in Figure 3. The fan system consisted of 24 narrow chord blades. The height of each blade was 603.2 mm, and the initial torsion angle was 61.3\({}^{\circ}\). The blade was connected to the hub with a dovetail and tongue-and-groove mating method. The entire fan system was discretized using 1,520,208 hexahedral solid elements (3 mm in size), see Figure 3. The material properties of Ti-6 Al-4 V alloy with plastic dynamics model (MAT_PLASTIC_KINEMATIC) was employed for the fan blade (Table 1). When a bird hits the fan blade, there is an uncertainty in the impact location, and different impact locations led to different damage effects [43]. To better reflect the effect of \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Mass** & **Elastic** & **Poisson’s** & **Tried** & **Tangent** & **Hardening** & **Strain Rate** & **Strain Rate** & **Effective** \\  & **Density** & **log/m\({}^{3}\)** & **Modulus** & **GPa** & **Stress** & **Modulus** & **Paramement** & **Parameter** & **Parameter** & **Parameter** & **Plastic** \\  & & & & & & & & & & \\ \hline Ti-6 Al-4V & 4500 & 112.5 & 0.33 & 958 & 1.139 & 0.2 & 200 & 15 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Blade material parameters. Figure 3: Finite element model of turbofan engine fan. Figure 2: Mallard bird model used in numerical simulation. impact forces on the rotor system, a single impact location was selected for the simulations. The same applies to the selection of the impact orientations. Five impact positions: 1/6, 1/3, 1/2, 2/3, and 5/6 fan blade height, and 15 types of impact orientations were considered. Based on the previous studies [44], the bird impact force is pronounced at 1/2 blade height, which leads to significant impact on the rotor system.