## 4 OGVs Parameterisation and Its Optimisation Method
### Optimal Design of MRGP Combined with Sobol Method  
The axial distance \(\sigma_{\rm a}\) from the OGVs to the moving blade, the radius \(R\), the circular horizontal coordinate \(x\), and the circular vertical coordinate \(y\) are used as input variables, the total pressure and the noise value of the fan are used as output values, the corresponding proxy model is established using a multidimensional response Gaussian model, and the sensitivity analysis of the variables is carried out using the Sobol method to establish a combined MRGP. This is used to select a combined MRGP and Sobol method of active learning sensitivity for nuclear class axial fans.  
The MRGP model extends the Kriging model, which can only deal with a single response output for multiple variables. In contrast, for a multi-dimensional response output, multiple Kriging models need to be constructed, which does not consider the coupling of response values. Therefore, MRGP models for multivariate multi-response problems are gaining attention.  
The MRGP model was proposed by [PERSON] [29]. M functions are united into a specific implementation of a multi-response Gaussian process:  
\[G(x)=f(x)^{\rm T}B+Z(x) \tag{9}\]  
The polynomial regression function \(f(x)\) is the same as the Kriging model, but the regression coefficient \(B\) is an \(n\times m\) matrix, and \(Z(x)\) is expanded to a \(1\times m\) row vector. \(Z(x)\) covariance is represented by an \(m\times m\) covariance matrix, i.e., \(\Sigma^{m}\):  
\[\sum^{m}=\left[\begin{array}{ccc}\sigma^{2}(g_{1},g_{2})&\cdots&\sigma^{2}( g_{1},g_{m})\\ \vdots&\ddots&\vdots\\ \sigma^{2}(g_{m},g_{1})&\cdots&\sigma^{2}(g_{m},g_{m})\end{array}\right] \tag{10}\]  
In this study, the MRGP model is selected as the approximation function, using 65 sets of variables obtained through Opt LHD combined with numerical simulation calculations with the sample data of the corresponding responses to build the approximation model, of which the first 60 sets are used as the sample matrix data and the last five sets as the test data.  
To verify the model's accuracy, it is necessary to check the model's accuracy before solving it by means of the relevant accuracy-test formula. This study uses a global sensitivity analysis based on the Sobol method, a Monte Carlo method based on analysis of variance (ANOVA), which expresses the degree of influence of all parameters on the model results in terms of the total variance \(D\), which can be described as:  
\[D=\int f^{2}(x)\mathrm{d}x-f_{0}^{2} \tag{11}\]  
Partial variance \(D_{\rm i}\) is used to indicate the influence degree of a single parameter or the influence of parameters on the model results, which can be expressed as:  
\[D_{\rm i_{1},i_{2},\cdots,\,i_{t}}=\int f_{i_{1},i_{2},\cdots,\,i_{3}}^{2}\; \mathrm{d}x_{i_{1}}\mathrm{d}x_{i_{2}}\cdots\mathrm{d}x_{i_{s}} \tag{12}\]  
By solving the first-order and global sensitivity parameters of each influencing factor, the Sobol method is used to evaluate the sensitivity of the approximate function obtained by MRGP. The first-order sensitivity indicates the influence of a single factor on the total pressure and noise of the fan. The overall sensitivity suggests the result of a single factor on the total pressure and noise of the fan and reflects the interaction with other influencing factors. Therefore, when the first-order sensitivity of a variable is very different from the overall sensitivity, it can be concluded that the factor has a significant interaction. The sensitivity meter are shown in Tables 5 and 6.