## 2 Problem Description  
The VX gasket is an important part of the wellhead connector, which seals the gap between the Christmas tree body and wellhead. It prevents leakage of the high-pressure medium in the wellhead, and prevents seawater entering the pipeline. The concrete structure of a subsea Christmas tree wellhead connector is presented in Figure 1.  
During setup, the VX gasket and sealing groove formed by the Christmas tree body and wellhead are connected under an interference fit to produce radial deformation, which can enforce the gasket sticks onto the sealing groove, thereby forming an effective seal interface [7]. During the working process, the high-pressure medium with random pressure value fluctuations acts on the gasket to produce a radial force. The corresponding compacting force exerted onto the sealing surface is further increased, the plastic deformation appears in the gasket, and a type of self-tightening seal structure is ultimately formed [14]. As shown in Figure 2, the VX gasket taper \(\alpha\) and the interface width \(b\) are two main structure parameters that affect the sealing performance of a VX gasket [15; 16]. The maximum value and distribution uniformity of contact stress are two important indices for evaluating a gasket's sealing performance [17; 18]. We, therefore, selected the sealing surface angle \(\alpha\) and the sealing surface width \(b\) as the optimization design variables. To obtain a large enough sample to cover all of the optimal structure parameters, upper and lower bounds were determined for 50% fluctuations above and below the initial value of the design variables. The maximum value of contact stress \(\sigma_{m}\) and the contact stress variance \(S^{2}\) were used to evaluate sealing performance. The corresponding calculation formulas for \(\sigma_{m}\) and \(S^{2}\) are presented in Equations (1) and (2). \[\sigma_{m}=\max(\sigma_{i}) \tag{1}\]  
\[S^{2}=\frac{1}{n}\sum_{n}^{i=1}\left(\sigma_{i}-\overline{\sigma}\right) \tag{2}\]  
where \(\sigma_{i}\) is the contact stress value of the \(i\)-th node in the contact area of the gasket, \(\overline{\sigma}\) is the mean value of the sum of all the nodal contact stress values in the contact area, and \(n\) is the number of the nodes in the contact line. Figure 1: A subsea Christmas tree wellhead connector: (**a**) sealing structure; and (**b**) sealing mechanism. The working process of the VX gasket contains two phases: pre-tightening and working pressure acting. In the pre-tightening phase, a preload acts on the gasket to produce a radial force, and the initial seal is formed. In the working pressure acting phase, the high-pressure medium enforces the gasket sticks on the sealing groove, and the contact force between sealing interfaces increase. Production complexity in the well leads to random fluctuations in the working load, which lead to sealing performance reductions. Ultimately, the designed solution is not optimum if the randomness of the working load is not considered during VX gasket structure optimization [19]. Therefore, working load randomness should be considered during the VX gasket structure's design optimization. Based on the descriptions above, this paper aims to obtain the optimal design variables \(\alpha\) and \(b\) to guarantee that the subsea connector has the best sealing performance, i.e., the maximum value of \(\sigma_{m}\) and the minimum value of \(S^{2}\). The randomness of the working load should be considered in the optimization process. The problem could be described in Equation (3) as follows:  
\[\left\{ \begin{array}{l}\text{find}:\alpha,b\\ \text{maximum}:\sigma_{m}\\ \text{minimum}:S^{2}\\ s.t.:\alpha\in[\alpha^{l},\alpha^{u}];b\in[b^{l},b^{u}]\\ \text{the working load randomness need to be considered}\end{array} \right. \tag{3}\]  
Optimization efficiency is relatively low due to the high computation cost during sealing performance index simulation using traditional methods, such as the finite element (FE) model.