## 5 Conclusions  
In this study, to acquire the parameters of the J-C constitutive model of powder liner in shaped charge and the parameters of the HJC constitutive model of hard rock in the reservoir, a penetrating steel target experiment and a simulated reservoir penetrating hard rock target experiment were first carried out, and the finite element software ANSYS/LS-DYNA (16.0)was used to reproduce the process of a jet penetrating a steel target and hard rock, respectively. Then, the expression of target variables (perforation depth \(L\) and perforation aperture \(D\)) regarding the parameters in the constitutive model was obtained through the utilization of the response surface method. Finally, a multi-objective genetic algorithm was employed to solve the display expression iteratively. We identified the Pareto optimal solution that closely approximated the experimental mean value by comparing it with experimental data. Then, we derived effective J-C constitutive parameters for powder liner and HJC constitutive parameters for hard rock. Figure 21: Perforation results: (**a**) numerical simulation results based on known parameters; (**b**) numerical simulation results based on inversion parameters; (**c**) schematic diagram of experimental results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data Source** & **Perforation Depth (cm)** & **Perforation** & **Depth Error** & **Depth Error** & **Aperture Error (cm)** & **Aperture Error** \\ \hline Experiment & 32.00 & 1.08 & - & - & - & - \\ \hline Numerical simulation (known parameters) & 27.05 & 1.76 & 4.95 & 15.47\% & 0.68 & 62.96\% \\ \hline Numerical simulation (inversion parameters) & 32.14 & 1.13 & 0.14 & 0.44\% & 0.05 & 4.63\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison table between experimental data and numerical simulation results. The numerical simulation results, based on the reconstructed J-C constitutive parameters and HJC constitutive parameters, demonstrate that the perforation depth and aperture resemble the experimental results. This suggests that the constitutive parameters of the inversion offer a more accurate depiction of the mechanical behavior of powder liner and hard rock during actual perforation processes. Simultaneously, it was demonstrated that the perforation simulation based on the inversion of constitutive parameters could provide a more precise prediction of the damage induced by the metal jet formed through powder liner on hard rock. This finding establishes a fundamental basis for further investigation into both powdered liner and the process of hard rock perforation. At the same time, by combining numerical simulation and experimental data, the method overcomes the limitations of traditional experimental methods and reduces the dependence on experiments, thus reducing the time and economic cost that may be brought by experiments. The constitutive parameters of powder liner and hard rock can be obtained more efficiently in a short time, which also makes the results more applicable. In addition, this method not only provides a new theoretical framework for the current constitutive parameter inversion of powder liner and hard rock but also provides a new reference and inspiration for researchers in related fields in material behavior prediction, energy conversion, and transfer, etc., providing a new method for structural optimization in related engineering fields, which has great application potential. It further promotes the combination of basic research and engineering application. Conceptualization, G.Y. ; Methodology, K.M. ; Software, K.M. ; Writing--original draft, K.M. ; Visualization, K.M. ; Supervision, G.Y. ; Project administration, G.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by [National Natural Science Foundation of China] grant number [11972376], [Natural Science Foundation of Shandong Province] grant number [ZR2019 MA007], [Basic Research Project of the Central Universities] grant number [22 CX03014A].