## 3 Updating Method for FE Model of Antenna
### RBF Surrogate Model  
The multiple sets of input parameters and output responses of the antenna's original FE model are utilized to fit the RBF surrogate model [23], which is _used to replace the original FE model for the calculation_. Figure 8: Backup structural bars’ classification and numbering: (**a**) main radial beam; (**b**) secondary radial beam; (**c**) ring beam; (**d**) inclined beam. Figure 7: Classification of antenna BUS beams: (**a**) the BUS; (**b**) ring beams; (**c**) radial beams; (**d**) inclined beams. _In conjunction with Section 3.1, the antenna original FE model is utilized with the LHS experimental design method to obtain \(p\) sets of sample points \((x_{i},y^{i})\), where \(x_{i}=[x_{1},x_{2},\ldots,x_{d}]\) is the \(i\)-th set input parameter and \(y^{i}\) is the corresponding output response. The RBF surrogate model is fitted through these sample points to replace the FE model for calculation. The primary forms are as follows:_  
\[\hat{y}(\mathbf{x})=\sum_{i=1}^{n}w_{i}\varphi(r_{i})=\mathbf{w}\mathbf{\varphi}^{T} \tag{11}\]  
_where \(\hat{y}(\mathbf{x})\) represents the RBF model predicted value at the point to be measured \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{d}]\), \(\mathbf{w}=[w_{1},w_{2},\ldots,w_{n}]\) is the weight coefficient's vector, \(\mathbf{\varphi}=[\varphi(r_{1}),\varphi(r_{2}),\ldots,\)\(\varphi(r_{n})]\) represents the vector of radial functions, and \(r_{i}\) represents the Euclidean distance between \(\mathbf{x}\) and \(\mathbf{x}_{i}\) of the following form:_  
\[r_{i}=\|\mathbf{x}-\mathbf{x}_{i}\|=\sqrt{\sum_{j=1}^{d}\left(x_{j}-x_{ij}\right)^{2}} \tag{12}\]  
Table 1 shows the common radial function [24]. Because the MQ (multi-quadratic) function has good local and global estimation characteristics, we choose the MQ function as the basis function to fit the RBF surrogate model. We minimize the sum of the error's squares between \(\hat{y}(\mathbf{x})\) and \(y^{i}\) by least squares and solve for the weight coefficient vector \(\mathbf{w}\) of the RBF surrogate model. The sum of squared errors is  
\[J(\mathbf{w})=\sum_{i=1}^{p}\left(\hat{y}(\mathbf{x}_{i})-y^{i}\right)^{2}=\sum_{i=1}^{ p}\left(\sum_{j=1}^{p}w_{i}\varphi(\|\mathbf{x}_{i}-\mathbf{x}_{j}\|)-y^{i}\right)^{2}=\| \mathbf{w}\mathbf{\Phi}-\mathbf{y}\|^{2} \tag{13}\]  
where \(\mathbf{y}=\left[y^{1},y^{2},\ldots,y^{d}\right]\), and matrix \(\mathbf{\Phi}\) is  
\[\mathbf{\Phi}= \begin{bmatrix}\varphi\parallel x_{1}-x_{1}\parallel&\varphi \parallel x_{1}-x_{2}\parallel&\ldots&\varphi\parallel x_{1}-x_{p}\parallel \end{bmatrix}  \tag{14}\]  
Formula (13) is expanded as follows:  
\[J(\mathbf{w})=(\mathbf{w}\mathbf{\[PERSON]}-\mathbf{y})^{T}(\mathbf{w}\mathbf{\Phi}-\mathbf{y})=\mathbf{\Phi}^{T} \mathbf{w}^{T}\mathbf{w}\mathbf{\Phi}-2\mathbf{\Phi}^{T}\mathbf{w}^{T}\mathbf{y}+\mathbf{y}^{T}\mathbf{y} \tag{15}\]  
The derivation of \(\mathbf{w}\) on both sides of Formula (15) with the derivative set as 0 results in the following:  
\[\frac{\partial}{\partial\mathbf{w}}J(\mathbf{w})=\frac{\partial}{\partial\mathbf{w}}(\mathbf{ \Phi}^{T}\mathbf{w}^{T}\mathbf{w}\mathbf{\Phi}-2\mathbf{\Phi}^{T}\mathbf{w}^{T}\mathbf{y}+\mathbf{y}^{T} \mathbf{y})=0 \tag{16}\]  
The following normal equation is obtained:  
\[\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{w}=\mathbf{\Phi}^{T}\mathbf{y} \tag{17}\]  
When \(p\) sampling points are not equal to each other, the row vector (or column vector) in the matrix is linearly independent and invertible, and the optimal weight coefficient \(\mathbf{w}\) can be obtained:  
\[\mathbf{w}=\left(\mathbf{\Phi}^{T}\mathbf{\Phi}\right)^{-1}\mathbf{\Phi}^{T}\mathbf{y} \tag{18}\]  
The optimal weight vector \(\mathbf{w}\) is substituted into the Formula (11) to obtain the RBF surrogate model:  
\[\hat{y}(\mathbf{x})=\sum_{i=1}^{p}\left[\left(\mathbf{\Phi}^{T}\mathbf{\Phi}\right)^{-1} \mathbf{\Phi}^{T}\mathbf{y}\right]_{i}(\|\mathbf{x}-\mathbf{x}_{i}\|+c^{2})^{\frac{1}{2}} \tag{19}\]In Formula (19), \(c\) is a given constant greater than 0.