## 3 Numerical-Simulation Study  
### Validation of Numerical Simulation  
Experimental research is the most direct and effective method to solve the problem. However, due to the limitations of the experimental conditions, the data available from the hypervelocity impact experiment are limited. In order to better analyze and obtain its laws, numerical simulation has become a necessary means. In recent years, meshless methods have been developed rapidly, among which the SPH algorithm is typical and has been widely used in the numerical simulation of hypervelocity collisions [25; 26; 27; 28; 29; 30]. In this paper, the AUTODYN dynamic simulation program was used. In order to expand the variation range of the simulated wave-impedance gradient as fully as possible, titanium alloy, aluminum alloy, magnesium alloy, and nylon were selected as the commonly used engineering-protection materials for the numerical simulation. A 2D model was used for the model verification, and the calculation speed and accuracy were considered. The particle size was 0.1 mm. Tillotson equation of state was used in the modeling of Al and Ti, where A, a, and b were the fitting constants, B and e\({}_{0}\) were the adjustment parameters, and \(\alpha\) and \(\beta\) were the material constants. The Steinberg Guinan constitutive model was used, where Y\({}_{0}\), Y\({}_{\text{max}}\), b, h, \(\beta\), G\({}_{0}\), and T\({}_{\text{m}}\) were the material parameters. The state equation of Puff was used for the Mg material, where A\({}_{1}\), A\({}_{2}\), and A\({}_{3}\) were the material parameters, and the Von Mises constitutive model was used. The shock-state equation was selected for the nylon material, and the Von Mises constitutive model was also adopted. The parameters of the material model are shown in Tables 1-3. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Material** & **\(\mathbf{Y_{0}}\)/GPa** & **\(\mathbf{Y_{\text{max}}}\)/GPa** & **\(\mathbf{b}\)** & **\(\mathbf{h}\)** & \(\mathbf{\beta}\) & \(\text{G}_{0}\)/GPa** & **\(\mathbf{T_{m}}\)/K** \\ \hline Titanium alloy & 1.33 & 2.12 & 0.48 & 0.1 & 12 & 41.9 & 2110 \\ Aluminum alloy & 0.29 & 0.68 & 1.86 & 0.185 & 310.0 & 27.6 & 1220 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of Steinberg Guinan model for titanium and aluminum. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Material** & **\(\mathbf{Y_{0}}\)/GPa** & **\(\mathbf{Y_{\text{max}}}\)/GPa** & **\(\mathbf{b}\)** & **\(\mathbf{h}\)** & \(\mathbf{\beta}\) & \(\text{G}_{0}\)/GPa** & **\(\mathbf{T_{m}}\)/K** \\ \hline Titanium alloy & 1.33 & 2.12 & 0.48 & 0.1 & 12 & 41.9 & 2110 \\ Aluminum alloy & 0.29 & 0.68 & 1.86 & 0.185 & 310.0 & 27.6 & 1220 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of Tillotson state equation for titanium and aluminum. Figure 2: Variation trend of total transmission coefficient with gradient layer. During numerical simulations, the selection of the algorithm, material model, and parameters greatly affect the accuracy of the results. Therefore, the correctness of the numerical simulation method should be verified according to the experimental results [31; 32]. Figure 3 shows the comparison of the numerical-simulation results and experimental photographs under the same working conditions. The deviation between the numerical simulation results and the experimental results of the debris-cloud-head-extension-distance gradient material was about 3.2%. The deviation between the numerical-simulation results and the experimental results was about 4.9%. The above comparison shows that the selected calculation model and parameters are reasonable. Characteristics of Shock Pressure and Internal Energy Transformation under Different Wave-Inpedance Distributions  
The previous theoretical analysis showed that the continuous change in wave impedance is conducive to reducing the attenuation amplitude of the stress and maintaining the stress at a higher level.