## 3 Optimization Based on the Kriging Surrogate Model-NSGA-II Algorithm
#### Step 3: Construction of Kriging surrogate models of Sealing Performance Indices  
(1) Basic form of the Kriging surrogate model  
The Kriging surrogate model is a type of interpolation method. The interpolation result is a linear weighted sum of the training samples response value as shown in Equation (6). \[\hat{y}(x)=\sum_{i=1}^{n}\omega^{i}y^{i} \tag{6}\]  
where \(y_{i}\) is the \(i\)-th response value of the \(i\)-th training sample, and \(x\) is the variables of the training sample. Therefore, the fitting result of any point in the design space can be calculated after the weighted coefficient is determined. To calculate the weighted coefficient \(\omega\), the Gaussian stationary random process is introduced to fit the response surface function, and the concrete definition is presented in Equation (7). \[Y(x)=\beta_{0}+Z(x), \tag{7}\]  
where \(\beta_{0}\) is the trend model, which represents the mathematical expectation of \(Y(x)\); \(Z(\cdot)\) is a stationary random process, with a mean value of 0; and variance is \(\sigma^{2}\). There is a relative correlation of different random variables, which can be calculated using Equation (8). \[\text{Cov}[Z(x),Z(x^{\prime})]=\sigma^{2}R(x,x^{\prime}), \tag{8}\]  
where \(R(x,x^{\prime})\) is the correlation model, which is used to describe the correlations between any two points \(x\) and \(x^{\prime}\), and it meets the assumption that the value is 1 when the distance of any two points is 0, while the value is 0 when the distance is \(\infty\). The optimum weighted coefficient \(\omega\) ensures that the mean square error shown in Equation (9) is minimum, and the unbiased interpolation condition shown in Equation (10) should also be satisfied. \[MSE[\hat{y}(x)]=E[(\omega^{T}Y_{S}-Y(x))^{2}] \tag{9}\]  
\[E[\sum_{i=1}^{n}\omega^{i}Y(x^{i})]=E[Y(x)] \tag{10}\]  
Then, the optimum weighted coefficient \(\omega\) can be calculated using the Lagrange multiplier method, and the Kriging model is constructed. The concrete derivation will not be presented in this paper, and the details can be found in the original paper published by [PERSON] et al. (2018). (2) Kriging surrogate model of different sealing performance indices  
To establish Kriging models of gasket sealing performance, we took the gasket taper \(\alpha\) and interface width \(b\) as design variables to design the experimental method; e.g., the Latin Hypercube sampling method ([PERSON] et al., 2018) was selected as the design of experiment (DOE) method to obtain different groups of sealing structure parameters, and the corresponding sealing performance indices \(\sigma_{m}\) and  
Figure 4: The R-FE model of the VX gasket. were simulated using the established R-FE model proposed in Step 2. Then, the Kriging models of sealing performance were established, in which the training samples represented the obtained design variables and the corresponding sealing performance indices. The concrete construction is illustrated in Figure 5, depicting the method where an increase in the number of samples is the key step to ensure the accuracy of the final constructed Kriging surrogate model. The mean square error based adaptive sampling method [30] was applied to obtain the new samples, and to improve the accuracy of the Kriging surrogate model, considering the unique capability of error estimation in any unknown point for a Kriging model. _Step 4_: _Create a Random Initial Population_  
Individuals within the population represent values of the design variables and sealing performance indices. The initial population of size \(F_{0}\) was randomly designed. _Step 5_: _Compute the Sealing Performance Indices of Population \(P_{t}\) and Obtain a Progeny Population \(Q_{t}\)_  
Use the constructed Kriging surrogate model to compute the values of the sealing performance index for the current population \(P_{t}\). Non-dominated sorting and a crowding distance calculation were executed for the current parent population \(P_{t}\), and the parent population was selected based on the calculated order and the crowding distance.