## 3 Numerical-Simulation Study
### Validation of Numerical Simulation
However, it is difficult to achieve the continuous change in material wave impedance in both the simulation and the material preparation. The continuous change in wave impedance can only be simulated by increasing the number of material layers with different wave-impedance gradients. In the simulation study of working conditions 1-4, four materials with different wave-impedance characteristics were selected to ensure that the number of material layers was continuously increased under the condition that the total area density of the protective screen was the same, so as to simulate the continuous change in wave impedance and analyze the impact of this change on the hypervelocity-impact response of the projectile. Compared with working condition 1, working condition 5 reduced the nylon material and changed the back material into magnesium alloy, thus reducing the total gradient-attenuation value of the material, which was used to analyze the impact of the total gradient attenuation value on the protection performance. Compared with condition 5, condition 6 reduced the titanium alloy, which was used to analyze the influence of the surface-material-wave-impedance on the protection performance. The same projectile-impact parameters and protective-screen-area density were adopted for the simulation calculation conditions. The calculation-model parameters are shown in Table 4. The calculation model of calculation conditions 1-4 is shown in Figure 4. In order to obtain the shock-wave pressure at the center of the projectile, gauce points were added to the simulation model, as shown in Figure 4a, so as to record the change in the shock-wave amplitude in the projectile. Figure 3: Comparison of numerical-simulation results and experimental results for protective structures under the same working conditions (5 km/s, 20 us time). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(A_{1}\)**/GPa** & \(A_{2}\)**/GPa** & \(A_{3}\)**/GPa** & **Grüuneisen Coefficient** & **Expansion Coefficient** & **Sublimation Energy/(J kg\({}^{-1}\))** \\ \hline Magnesium alloy & 103 & 50 & 0.5 & 0.6 & 5 & \(7.0\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of Puff state equation for magnesium. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Working** & **Composition of** & **Thickness of Each** & **Surface Density** & **Impact Velocity of** & **Pellet Diameter** \\
**Condition** & **Protective Screen** & **Layer (mm)** & **of Protective** & **Projectile (km/s)** & **(mm)** \\ \hline
1 & Ti-Al-Mg-Ny & 0.4 + 0.3 + 0.3 + 0.9 & & & \\
2 & Ti-Al-Ny & 0.3 + 0.4 + 1.5 & & & \\
3 & Ti-Ny & 0.5 + 1.6 & & & \\
4 & Ti-Mg-Ny & 0.4 + 0.9 + 0.7 & 0.417 & 3, 4, 5, 6 & 6 \\
5 & Ti-Al-Mg & 0.4 + 0.3 + 0.9 & & & \\
6 & Al-Mg & 0.8 + 1.1 & & & \\
7 & Al & 1.5 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation-model parameters. Figure 4: Calculation model for working conditions 1–4: (**a**) Ti-Al-Mg-Ny protective screen; (**b**) Ti-Al-Ny protective screen; (**c**) Ti-Mg-Ny protective screen; (**d**) Ti-Ny protective screen. Figure 5 shows the impact-wave-amplitude changes at different positions in the projectile under the impact of four calculation models at 5 km/s. It can be seen from the figure that the shock wave of the projectile started at the contact surface of the projectile target, and the amplitude gradually attenuated with the increase in the propagation distance. In order to accurately compare the impact of the continuous gradient change on the shock-wave amplitude, Figure 6 shows the corresponding projectile shock waves of the Ti-Ny, Ti-Al-Ny, and Ti-Al-Mg-Ny shielding materials at the impact speed of 5 km/s. It can be seen that the initial pressure of the shock wave corresponding to the three materials was essentially the same, because the surface material impacted by the projectile target was titanium alloy.