## 4 Case Study
\[R^{2}=\frac{\sum\limits_{i=1}^{n}{{{({{{\hat{y}}_{i}}}-{{\overline{y}}_{i}})}^{2}} }}{\sum\limits_{i=1}^{n}{{{({y_{i}}-{{\overline{y}}_{i}})}^{2}}}}=1-\frac{\sum \limits_{i=1}^{n}{{{({y_{i}}-{{\hat{y}}_{i}})}^{2}}}}{\sum\limits_{i=1}^{n}{{{ ({y_{i}}-{{\overline{y}}_{i}})}^{2}}}} \tag{12}\]  
where \(y_{i}\) is the obtained sealing performance index value using the R-FE model, and \({{\hat{y}}_{i}}\) is the fitting values using the established Kriging models. In order to achieve the aim of unbiased estimation, the \(R\) square value should be kept at a maximum; however, its value should be smaller than 1. Nine groups of sealing performance indices were selected randomly according to the comparison from Table 2. Then, corresponding sealing performance indices values \({{\hat{y}}_{i}}\) were obtained after nine groups of design variables were substituted to the Kriging models of sealing performance. The \(R\) square value of the \(\sigma_{m}\) and the \(S^{2}\) Kriging models were 0.963 and 0.945, calculated in Equation (10), respectively. These results indicate that the fitting accuracy of the Kriging models is suitably high, and that the models can be used to fit the sealing performance indices of the VX gasket. A multiobjective optimization model of the VX gasket is proposed in Equation (11), in which the design variables are structure parameters \(\alpha\) and \(b\), and the objective functions are the established Kriging models of \(\sigma_{m}\) and \(S^{2}\). \[\left\{ \begin{array}{l}\text{find}:\alpha,\;b\\ \text{max}:{{\hat{y}}_{\sigma_{m}}}(\alpha,b)=\sum\limits_{i=1}^{n}{{{\omega}_{ \sigma_{m}}^{i}}{{y}_{\sigma_{m}}^{i}}}\\ \text{min}:{{\hat{y}}_{S^{2}}}(\alpha,b)=\sum\limits_{i=1}^{n}{{{\omega}_{S^{ 2}}^{i}}{{y}_{S^{2}}^{i}}}\\ \text{s.t.}\\ \alpha\in[\alpha^{l},\;\alpha^{u}],\;b\in[b^{l},\;b^{u}]\\ \end{array} \right. \tag{13}\]  
where, \({{\hat{y}}_{\sigma_{m}}}(\alpha,b)=\sum\limits_{i=1}^{n}{{{\omega}_{\sigma_{m} }^{i}}{{y}_{\sigma_{m}}^{i}}}\) is the \(\sigma_{m}\) Kriging model, \({{\hat{y}}_{S^{2}}}(\alpha,b)=\sum\limits_{i=1}^{n}{{{\omega}_{S^{2}}^{i}}{{y }_{S^{2}}^{i}}}\) is the \(S^{2}\) Kriging model, and \(\alpha^{l},\alpha^{u},\,b^{l}\), and \(b^{u}\) are the lower and upper bounds of the design variables shown in Table 2.  
The NSGA-II algorithm parameters are: the amount of initial population size F\({}_{0}\), 44; the number of evolutional generations \(N\), 50; the mutation ratio, 0.02; and the maximum crossover ratio \(\upsilon\), 0.9. The Pareto front was finally obtained (Figure 8). The data show that in the AB section of the Pareto front set, an increase in the maximum contact stress \(\sigma_{m}\) did not cause an obvious rise in the contact stress variance \(S^{2}\); in the CD section, great changes in \(S^{2}\) occurred, although the  
Figure 7: The sealing performance Kriging models of the VX gasket: (**a**) \(\sigma_{m}\) Kriging model, and (**b**) \(S^{2}\) Kriging. increase was small; therefore, all solutions both in the AB and CD sections were not the optimal results. The optimum solution can be selected in section BC by the designer, considering the values of \(\sigma_{m}\) and \(S^{2}\). The final selected optimum Pareto solution of the sealing performance indices were \(\sigma_{m}=544.7\) MPa and \(v_{\sigma}=107.02\times 10^{2}\), and the corresponding optimum structure parameters were \(\alpha=25.10^{\circ}\) and \(b=29.11\) mm, respectively. To check the convergence of the results searched by NSGA-II, the Hypervolume (HV) indicator [9] was applied to show the convergence of the results every 100 interpolations. A schematic illustration of HV and the concrete tendency of HV values with interpolation increases are illustrated in Figure 9a,b, separately. The HV value was kept at 0.85 after 1500 interpolations, which meant that the optimization results searched by NSGA-II converged after 1500 interpolations.