## 3 Numerical-Simulation Study
### Validation of Numerical Simulation
The shock-wave amplitude of the three materials decreased with the increase in time, and the decreasing amplitude was Ti-Ny > Ti-Al-Ny > Ti-Al-Mg-Ny, which indicated that the attenuation amplitude of the shock wave in the projectile decreased obviously with the continuous change in the wave impedance, which was consistent with the theoretical analysis. Figure 5: Shock-wave amplitudes at different positions in the projectile: (**a**) Ti-Al-Mg-Ny protective screen; (**b**) Ti-Al-Ny protective screen; (**c**) Ti-Mg-Ny protective screen; (**d**) Ti-Ny protective screen. In the process of the hypervelocity impact, part of the kinetic energy of the projectile is converted into the kinetic energy of the debris cloud, and the other part is converted into the internal energy of the material. The internal-energy-conversion ability of the protective-screen material is an important criterion for judging the performance of the protective material. The higher the internal-energy conversion, the better the protective performance. Therefore, the analysis of the internal-energy-conversion characteristics during the interaction between the projectile and the wave-impedance gradient material can be used to effectively judge whether the continuous change in the wave impedance is conducive to improvements in the protection ability; it can also be used to examine the influence of the surface-material wave impedance and the total gradient attenuation value on the protection performance, so as to guide the design of the wave-impedance gradient material. Table 5 shows the corresponding converted internal energy of the five shielding materials, Ti-Ny, Ti-Al-Ny, Ti-Al-Mg-Ny, Ti-Al-Mg, and Al-Mg, under different impact speeds. From the results shown in Figure 7, it can be found that under the same impact velocity, the converted internal energy Ti-Al-Mg-Ny > Ti-Al-Ny > Ti-Ny; that, with the continuous change in the wave-impedance gradient, the internal-energy conversion showed a gradually increasing trend. This shows that under the same impact conditions, the continuous change in the wave-impedance gradient of a wave-impedance material with the same surface density can effectively improve the internal-energy conversion during the impact process, thereby effectively reducing the velocity of the debris cloud and reducing the ability of the debris cloud to damage the rear wall. In working condition 4, the internal  
Figure 6: The change in the projectile-shock-wave amplitude corresponding to the three protective screen materials. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Working Condition**} & \multirow{2}{*}{**Material**} & \multicolumn{5}{c}{**Internal Energy (J)**} \\ \cline{3-6}  & & **3.0 km/s** & **4.0 km/s** & **5.0 km/s** & **6.0 km/s** \\ \hline
1 & Ti-Al-Mg-Ny & 165 & 286 & 443 & 619 \\
2 & Ti-Al-Ny & 163 & 278 & 422 & 596 \\
3 & Ti-Ny & 161 & 271 & 408 & 567 \\
4 & Ti-Al-Mg & 152 & 269 & 398 & 543 \\
5 & Al-Mg & 128 & 216 & 309 & 441 \\
6 & Al & 114 & 193 & 275 & 394 \\ \hline \end{tabular}
\end{table} 
Table 5: Corresponding internal-energy conversion of five protective-screen materials at different impact speeds. energy conversion of the Ti-Al-Mg material was smaller than that of Ti-Al-Mg-Ny, Ti-Al-Ny, and Ti-Ny, which indicated that the increase in the total gradient-attenuation value is conducive to the conversion of kinetic energy to internal energy during hypervelocity impact. Working condition 5 further reduced the wave impedance of the surface materials and the gradient attenuation of the total wave impedance, resulting in the reduced conversion of internal energy. This proves that it is important to comprehensively consider the wave impedance of surface materials and the gradient-attenuation value of total wave impedance when the total area density of materials is fixed in the design process of wave-impedance gradient materials.