## 4 Case Study  
In this section, we used a subsea Christmas tree wellhead connector matching the 20\" casing head as an example to validate the presented optimization approach. The concrete initial structure parameters of the VX gasket are shown in Table 1. The mean value of the oil-gas pressure of the subsea connector was 34.5 MPa, and the random model of the oil-gas pressure was established based on the method presented in Step 1, in which the relative parameters were: \(\mu\) = 34.5 and \(cv\) = 0.05. Then, the concrete form was illustrated using Equation (11). Eventually, the random load model was loaded onto the connector's inner surface, and the stress-strain curve of the VX gasket material illustrated in Figure 6 was applied to simulate the plastic's deformation behavior. \[P\
eg N(34.5,1.725^{2}),p\in[29.325,39.675], \tag{11}\]  
As mentioned in Step 3 of Section 3, groups of design variables required selection using the experimental method design in modeling the Kriging model. The optimal Latin hypercube sampling method [31] was used in this paper to sample the design variables. According to the determining method of the upper and lower bounds of the design variables in Section 2, the corresponding parameters were: \(\alpha^{l}\) = 11.5\({}^{\circ}\), \(\alpha^{u}\) = 34.5\({}^{\circ}\), \(b^{l}\) = 15 mm, and \(b^{u}\) = 45 mm, separately. The sampling ranges of the design variables are listed in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Structure Parameters (unit)** & **Initial Value** \\ \hline \(\alpha\) (\({}^{\circ}\)) & 23 \\ \(b\) (mm) & 30 \\ \(B\) (mm) & 15 \\ \(h\) (mm) & 102 \\ \(D_{m}\) (mm) & 531.4 \\ \(D_{i}\) (mm) & 479 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The initial structure parameters of the VX gasket. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables (unit)** & **Upper Bound** & **Lower Bound** \\ \hline \(\alpha\) (\({}^{\circ}\)) & 11.5 & 34.5 \\ \(b\) (mm) & 15 & 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The sampling ranges of the design variables. Figure 6: The stress–strain curve of the VX gasket material. In order to guarantee the fitting accuracy of a Kriging model, an appropriate number of training samples should be determined. In general, training samples should be 3 times the number of a quadratic polynomial coefficient \(k\), where \(k=(n+1)(n+2)/2\), and \(n\) is the number of design variables [32]. As the number of structure parameters in this case is 2, the number of training samples is set to 18. Therefore, 18 groups of structure parameters were selected using the optimal Latin hypercube method, and the corresponding values of the maximum contact stress \(\sigma_{m}\) and contact stress variance \(S^{2}\) were simulated using the R-FE model proposed in Step 3 of Section 3. Then, the training sample, provided in Table 3, was formed by combining 18 groups of maximum contact stress \(\sigma_{m}\) and contact stress variance \(S^{2}\) with the corresponding design variables. The sealing performance Kriging models of the VX gasket were established based on the training sample provided in Table 2, using the construction illustrated in Figure 5. The contours of the sealing performance Kriging models are shown in Figure 7. \begin{table}
\begin{tabular}{c c|c c} \hline \multicolumn{2}{c|}{**Design Variables**} & \multicolumn{2}{c}{**Sealing Indices**} \\ \hline \(\alpha\) **(\({}^{\circ}\))** & \(b\) **(mm)** & \(\sigma_{m}\) **(MPa)** & \(\mathbf{S^{2}\) **(10\({}^{2}\))**} \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: The training samples of the sealing performance Kriging model. Figure 7: _Cont._  
To test the fitting error of the established sealing performance Kriging models, the \(R\) square coefficient as shown in Equation (10) [10] was used to evaluate the fitting accuracy of the Kriging models.