## 3 The Inversion of J-C Constitutive Parameters for Powder Liner
### Multi-Objective Optimization  
The response surface method is an efficient approach for fitting polynomial functions to accurately simulate the real limit state. It enables the representation of the target variable as a display expression of the design variable, making it a commonly employed technique for solving problems involving multiple variables [25]. The specific form is presented below:  
\[y(x)=\varphi_{0}+\sum_{j=1}^{N}\varphi_{j}x_{j}+\sum_{j=1}^{N}\varphi_{ij}x^{2} +\sum_{i\
eq j}^{N}\varphi_{ij}x_{j} \tag{5}\]  
where \(x_{j}\) is the design variable, \(N\) is the number of targets, the regression coefficient \(\varphi\) can be obtained by expression \((N+1)(N+2)/2\), \(\varphi_{0}\) stands for constant coefficient, and \(\varphi_{j}\), \(\varphi_{ij}\), \(\varphi_{ij}\) represent the primary coefficient, the pure quadratic coefficient, and the mixed quadratic coefficient, respectively. Figure 7: Influence of constitutive parameter values on perforation depth and aperture: \(A\) initial yield strength, \(B\) strain hardening coefficient, \(m\) temperature correlation coefficient, \(C\) strain rate sensitivity coefficient, \(n\) strain hardening index. The inversion variables were determined as parameters \(A\), \(B\), and \(m\) based on the results of sensitivity analysis in the previous section. Meanwhile, the target variables were identified as perforation depth \(L\) and perforation aperture \(D\). Subsequently, a central composite design was employed to simulate various combinations of these parameters, enabling us to obtain a fitting surface for the target variables (\(L\), \(D\)) and the inversion variables (\(A\), \(B\), \(m\)). The fitting surface of perforation depth and perforation aperture under the interaction of parameters \(B\) and \(m\) is demonstrated in Figure 8 as illustrative examples. It can be observed that the data points are accurately fitted onto the surface with small local fluctuations, thereby providing a reliable basis for subsequent display expressions. Explicit expressions (6) and (7) of the target variable about the design variable can be further obtained from the fitting surface. \[ \begin{array}{ll}L=&44.99+1.81\times 10^{-4}A+0.032B-3.95m-1.6\times 10^{-4}AB \\ &-2.79\times 10^{-2}mA-0.015 mB+1.2\times 10^{-4}A^{2}-2\times 10^{-4}B^{2}\\ &+2.23m^{2}+2.24\times 10^{-4}mAB\\ \end{array}  \tag{6}\]  
\[ \begin{array}{ll}D=&0.83+0.022A-1.13\times 10^{-3}B+5.33\times 10^{-2}m-9.2 \times 10^{-5}AB-\\ &2.17\times 10^{-3}mA-2.6\times 10^{-4}A^{2}+2.9\times 10^{-5}B^{2}-9.9\times 10^{- 4}m^{2}+\\ &4.10\times 10^{-7}A^{2}B+1.2\times 10^{-5}mA^{2}+1.65\times 10^{-7}AB^{2}+ \\ &9.58\times 10^{-7}A^{3}-8.86\times 10^{-8}B^{3}\\ \end{array}  \tag{7}\]  
To assess the reliability of this display expression, additional determination coefficients \(R^{2}\) are introduced for further evaluation, whose expression is as follows:  
\[R^{2}=\frac{\sum\limits_{i=1}^{Z}\left(\hat{y}_{i}-\overline{y}_{i}\right)^{2} }{\sum\limits_{i=1}^{Z}\left(y_{i}-\overline{y}_{i}\right)^{2}} \tag{8}\]  
where \(Z\) represents the number of test points, \(y_{i}\) denotes the true value of the test point, \(\hat{y}_{i}\) signifies the predicted value derived from the fitted curve, and \(\overline{y}_{i}\) corresponds to the average of true values at each respective test point. After performing calculations, it is evident that the multiple determination coefficients of explicit expressions (6) and (7) are \(R^{2}{}_{L}=0.956\) and \(R^{2}{}_{D}=0.932\), respectively. Notably, both coefficients closely approach unity. This shows that the obtained display expression from fitting the surface demonstrates a higher level of reliability. The comparison between the simulated value and the  
Figure 8: Fitting surface of perforation depth and aperture under the action of parameters \(B\) and \(m\).