# Real-Time State Evaluation System of Antenna Structures in Radio Telescopes Based on a Digital Twin
## 4 Real-Time Simulation Construction of Antenna Physical Field
### Training Sample Construction  
The data training samples are obtained for specific working conditions. Considering the influence of three external conditions of wind speed, wind direction, and elevation angle on the structural deformation of the radio telescope, the wind speed range is defined as 0-15 m/s and the characteristic working conditions are selected every 5 m/s for a total of four kinds. Considering the symmetry of the structure, the range of the wind direction is defined as 0\({}^{\circ}\)-180\({}^{\circ}\), and the characteristic conditions are selected every 45\({}^{\circ}\) for a total of five kinds; the elevation range of the primary reflector is 5\({}^{\circ}\)-90\({}^{\circ}\) and seven elevation angles of 90\({}^{\circ}\), 75\({}^{\circ}\), 60\({}^{\circ}\), 45\({}^{\circ}\), 30\({}^{\circ}\), 15\({}^{\circ}\), and 5\({}^{\circ}\) are selected. Therefore, based on the above description, a total of 140 sets of working conditions are generated as the original training data set \(P1\).  
The original data set \(P1\) contains the node coordinates of the original finite element mesh and the corresponding mechanical performance data of the node. After the original mesh simplification process, its vertex position changes, so it is necessary to update the point cloud information, that is, to obtain the mechanical performance value corresponding to the vertex of the reduced model. Therefore, this paper uses the K-nearest neighbor (KNN) and the radial basis function (RBF) interpolation algorithm to construct the matching model of the original finite element mesh and the lightweight mesh [22], and the construction process is shown in Figure 8.  
Figure 7: Axial deformation cloud diagram of back-up structure: (**a,b**) indicate that the structure is subjected to wind loads only, (**c,d**) indicate coupled wind and self-weight loads.  
That is, \(P1\) is used as the basic sample, and the three-dimensional coordinates of the lightweight mesh are used as input. The Ball tree search method in the KNN algorithm is used to search and calculate the Euclidean distance \(d(x,y)\) between the center point and its nearest n \(P1\) sample data points. The distance is arranged from small to large, and \(K\) instances with the smallest distance are found in the data set. At the same time, the eigenvalues of the \(K\) instances are used to construct an RBF interpolation function \(\hat{f}(x)\) with the three-dimensional coordinates of the \(P1\) node as the input and the equivalent mechanical data corresponding to the lightweight mesh node as the output. The kernel function uses the Gaussian function \(\phi(r)\), and the equation is provided below:  
\[d(x,y)=\sqrt{\sum_{i=1}^{n}{(x_{i}-y_{i})}^{2}} \tag{8}\]  
\[\hat{f}(x)=\sum_{i=1}^{N}{w_{i}\varphi(r)} \tag{9}\]  
\[\phi(r)=e^{-(r)^{2}} \tag{10}\]  
where \(r\) represents the Euclidean distance between the two position vectors, and \(w_{i}\) represents the weight of the original data point relative to the target data point.  
After the above process, the point cloud sample data set \(P2\) of the mechanical performance of the antenna primary reflector structure after data dimensionality reduction is obtained.