# Shock Initiation of a Satellite Tank under Debris Hypervelocity Impact
## 4 Initiation of the Satellite Tank under Debris Impact
### Verification of the Simulation Model  
To verify the validity of the simulation model, HYP-2 test is simulated using ANSYS/LS-DYNA software and the simulation results are compared with the Zeus calculation. The simulation model consists of the projectile, vessel, nitrogen and liquid hydrazine, as displayed in Figure 4. Lagrange meshs are constructed for projectile and vessel, whilst Euler meshs are constructed for nitrogen and liquid hydrazine. These two meshs are coupled through Arbitrary Lagrange-Euler (ALE) algorithm [21]. Moreover, a multi-material ALE formulation with a second-order accurate advection has been selected to treat nitrogen and liquid hydrazine. The liquid hydrazine is allowed to flow into the nitrogen region by sharing nodes on their interfaces. The interaction between the projectile and vessel wall is achieved through the eroding surface-to-surface contact, and the interaction between projectile and liquid hydrazine is achieved through the automatic surface-to-surface contact. The boundary conditions of the simulation model are summarized in Table 3.  
The mesh density affects the discretization degree of the simulation model and the accuracy of the simulation results. Generally speaking, the smaller the mesh size is, the greater the discretization degree is, and the more accurate the simulation results are. Beyond a certain limit, the mesh size continues to decrease, the simulation results do not change significantly. The simulation results are independent of the mesh density. For this simulation model, the shock wave pressure is related to the mesh density of liquid hydrazine, so it is necessary to determine the appropriate mesh size of liquid hydrazine. Numerical simulations were carried out by setting the mesh sizes  
Figure 4: The simulation model of HYP-2 test device and the right side is the inside view with the vessel wall removed. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Initial Velocity** & **Constraints** & **Contact Definition** \\ \hline Projectile & translational velocity & — & eroding/automatic surface-to-surface \\ Vessel & — & non-reflection boundary & — \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Boundary conditions of the simulation model. 0.01 cm, 0.02 cm, 0.05 cm, 0.1 cm, 0.2 cm, and 0.3 cm. The attenuation curves of shock wave pressure along the impact line at different mesh sizes are shown in Figure 5. When the mesh size is smaller than 0.1 cm, the attenuation curves of shock wave pressure are very close, indicating that the pressure is not affected by the mesh size. Therefore, the mesh size of liquid hydrazine is set to 0.1 cm. The mesh sizes of the projectile, vessel wall and nitrogen are 0.05 cm, 0.05 cm, and 0.1 cm, respectively. The computing time step is 0.02 us. The nitrogen is modelled using the constitutive equation MAT_NULL and the state equation EOS_LINEAR_POLYNOMIAL. The pressure in linear polynomial equation is defined as follows [22]:  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+(C_{4}+C_{5}\mu+C_{6}\mu^{2})E, \tag{45}\]  
where \(C_{0}\)-\(C_{6}\) are the polynomial equation coefficients, \(\mu=(\rho/\rho_{0})-1\), \(\rho/\rho_{0}\) is the ratio of current density to initial density. The nitrogen is considered as ideal gas by setting \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0\). Since the density, boiling point and critical temperature of liquid hydrazine are within 2% of the values for water [19], the constitutive equation MAT_NULL and the state equation EOS_GRUNESEN of water are used to describe liquid hydrazine. The Gruneisen state equation defines pressure as follows:  
\[P=\frac{\rho_{0}C^{2}\mu\Big{[}1+\big{(}1-\frac{\gamma_{0}}{2}\big{)}\mu- \frac{\mu}{2}\mu^{2}\Big{]}}{\Big{[}1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1} -S_{3}\frac{\mu^{3}}{(\mu+1)^{2}}\Big{]}^{2}}+(\gamma_{0}+a\mu)E, \tag{46}\]  
where \(S_{1}\)-\(S_{3}\) are the slope coefficients of the \(u_{s}-u_{p}\) curve; \(\gamma_{0}\) is the Gruneisen gamma. The properties and material parameters of nitrogen and liquid hydrazine are listed in Table 4[22].