## 3 Updating Method for FE Model of Antenna
### Parameter Sensitivity Analysis  
The most common method for assessing the sensitivity of model output parameters to changes in input parameters is parameter sensitivity analysis [27]. The Sobol method is a Monte Carlo method based on variance statistics [28], which requires multiple independent repeated calculations to obtain accurate sensitivity analysis results.  
Combining Section 3.1 with Section 3.2, we first fit the surrogate model used for the parametric sensitivity analysis and then perform the parametric sensitivity analysis of the antenna finite element model. The input variables of the surrogate model are the modulus of elasticity of the 52 rods, and the output variable is the RMS of the displacement of the upper chord nodes of the BUS.  
Assume that the RBF model \(\hat{y}=f(\mathbf{x})\), where \(f(\mathbf{x})\) represents a square-integrable function on the domain, \(x=[x_{1},x_{2},\ldots,x_{d}]\) represents a \(d\)-dimensional vector composed of input parameters, where each component \(x_{i}\) represents independently subject to the uniform distribution on the interval [0,1], expressed as \(x\sim\) U (0,1). For any \(\forall i\
eq j\), \(x_{i}\) and \(x_{j}\) are independent of each other. We will use the multiple integral method to decompose the function \(f(\mathbf{x})\) into the sum of \(2^{d}\) unrelated sub-terms:  
\[f(\mathbf{x})=f_{0}+\sum_{i=1}^{d}f_{i}(x_{i})+\sum_{1\leq i<j\leq d}f_{ij}(x _{i},x_{j})+\cdots+\text{f}_{1,2,\cdots,d}(x_{1},x_{2},\cdots,x_{d}) \tag{21}\]  
where \(f_{0}\) represents the constant term, \(f_{i}(x_{i})\) represents the \(i\)-th input variable effect on the output response, \(f_{ij}(x_{i},x_{j})\) represents the interaction between the \(i\)-th and \(j\)-th input variables on the output response, and so on for the other higher-order terms. For each subset \(S\subseteq\{x_{1},\ x_{2},\ldots,x_{d}\}\) in Formula (21),  
\[\text{E}[f_{S}(\mathbf{x}_{S})]=0 \tag{22}\]  
Formula (22) implies that the expectation of all terms except the constant term \(f_{0}\) must be 0. For any two different subsets of parameters \(S\), \(T\subseteq\{x_{1},x_{2},\ldots,x_{d}\}\), the following orthogonality conditions apply:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Radial Function** & **Gauss** & **Multi-Quadratic** & **Inverse Multi-Quadratic** & **Logarithmic Path** \\ \hline functional form & exp(-\(cr^{2}\)) & \((r^{2}+c^{2})^{\frac{1}{2}}\) & \((r^{2}+c^{2})^{-\frac{1}{2}}\) & \(r^{2}\text{log}(1+cr^{2})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The common radial functions and forms.  
\[\mathbb{E}[f_{S}(\mathbf{x}_{S})f_{T}(\mathbf{x}_{T})]=0,\forall S\
eq T \tag{23}\]  
according to Formulas (22) and (23), we can decompose the RBF model total variance into the sum of the variances of each sub-function:  
\[D=\sum_{i=1}^{d}D_{i}+\sum_{1\leq i<j\leq d}D_{ij}+\cdots+D_{1,2,\cdots,d} \tag{24}\]  
where \(D\) represents the total variance of the function, \(D_{i}\) represents the single input variable \(x_{i}\) influence on the total variance, \(D_{ij}\) represents the effect of interaction between two input variables on total variance, and so on; \(D_{1,2,\ldots,d}\) represents the effect of interaction between all input variables on total variance.  
The Sobol method evaluates a single parameter's influence on the output variance by calculating the first-order and global sensitivity coefficients. The coefficient values are in the interval of [0,1], and the higher the value, the larger the contribution to the function output variance. The first-order sensitivity index's calculation formula is  
\[S_{i}=\frac{D_{i}}{D} \tag{25}\]  
and the calculation formula of the global sensitivity index is  
\[S_{T_{i}}=1-\frac{D_{\sim i}}{D} \tag{26}\]  
where \(D_{\sim i}\) represents the variance generated by the interaction of other input variables except \(x_{i}\).