## 3 Numerical Simulation Analysis
### Model Verification  
The Forrestal penetration formula, numerical simulation calculations and experimental tests of projectile penetration into limestone are introduced and the research data are shown in Figure 7 and Table 7. After the experimental tests, to study the internal crack distribution of the rock target, cross-sectional target analysis was carried out for the red sandstone target with a projectile impact speed of 1107 m/s, as shown in Figure 8. In addition, cross-sectional target analysis was carried out for the limestone target with an impact speed of 913 m/s, as shown in Figure 9.  
Figure 7 shows the effect of the projectile impact velocity on the penetration depth of the red sandstone and limestone targets. The blue curve is the result calculated by the Forrestal penetration formula, the black square scatter represents the test data, the red triangle is the numerical simulation data and the red curve is the result obtained by fitting the numerical simulation data. With increasing impact velocity, the penetration depth gradually increased, and the curve was slightly concave. When the projectile penetrated the red sandstone target at low speed, the numerical simulation results and Forrestal penetration calculation results were in good agreement with the experimental data. When the speed reached medium and high speeds, the error became larger, and the trends of the Forrestal curve and numerical simulation curve were similar. The experimental data of the projectile penetrating the limestone target were almost consistent with the calculation curve of the Forrestal penetration formula. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **V(M/S)** & \(P_{1}\)**/CM** & \(P_{2}\)**/CM** & \(P_{3}\)**/CM** & \(\tau_{1}\)**/\%** & \(\tau_{2}\)**/\%** & \(\tau_{3}\)**/\%** \\ \hline \multirow{2}{*}{Red} & 593 & 47 & 46.25 & 43 & 1.60 & 1.30 & 0.03 \\  & 890 & 81 & 90.95 & 83.3 & 12.28 & 2.8 & 8.4 \\  & 1107 & 1 * & 129.58 & 1 * & – & – & – \\ \hline \multirow{3}{*}{Limestone} & 602 & 33.5 & 35.3 & 31 & 5.37 & 7.46 & 12.18 \\  & 913 & 70 & 69.72 & 68.2 & 0.4 & 2.57 & 2.18 \\  & 1141 & 98 & 100.61 & 94 & 2.66 & 4.08 & 6.57 \\ \hline \hline \end{tabular} Note: \(P_{1}\) is the test result; \(P_{2}\) is the calculation result of Forrestal formula; \(P_{3}\) is the numerical simulation calculation result; * indicates that the target is penetrated; \(\tau_{1}\) is the error between \(P_{1}\) and \(P_{3}\); \(\tau_{3}\) is the error between \(P_{2}\) and \(P_{3}\); – means no data.  
\end{table} 
Table 7: Red sandstone and limestone penetration depth data. Figure 8: Internal crack of red sandstone target at impact velocity of 1107 m/s: (**a**) experimental result; (**b**) numerical simulation results. Figure 9: Internal crack of the limestone target at a projectile impact speed of 913 m/s: (**a**) experimental result; (**b**) numerical simulation results. Table 7 shows the results of the experimental tests, numerical simulation and Forrestal penetration formula of projectile penetration into the red sandstone target and limestone target as well as the error analysis among them. The error between the calculation results of the experimental tests and numerical simulation was less than 10%. Through experimental verification, the correctness of the numerical simulation calculation was proven. Additionally, the Forrestal penetration formula was also applicable to the penetration calculation of red sandstone and limestone. Figure 8 shows the distribution of internal cracks in the target after the projectile with an impact velocity of 1107 m/s penetrated the red sandstone. Figures 8a and 9a show the test results, and Figures 8b and 9b illustrate the numerical simulation results. Figure 9 shows the distribution of internal cracks in the target after the projectile with an impact velocity of 913 m/s penetrated the red sandstone. Figure 8a,b are divided into five areas.