Finite Element Model Updating Method for Radio Telescope Antenna Based on Parameter Optimization with Surrogate Model

Abstract

There are deviations between the radio telescope antenna finite element (FE) model, founded on the design stage, and the actual working antenna structure. The original FE model cannot accurately describe the antenna structure deformation characteristics under the environmental load, thereby compromising the accuracy of the active structural compensation. This article proposes an antenna FE model updating method founded on parameter optimization with a surrogate model. The updating method updates the modulus of elasticity parameters of different components of the antenna backup structure (BUS) to obtain finite element analysis (FEA) results consistent with the actual measurement of the antenna reflector surface shape. The surrogate model founded on the multi-quadratic radial basis function (RBF) improves the computational efficiency of FE model updating, replacing the complex and time-consuming finite element analysis and calculation process. This method is implemented on a radio telescope antenna with an aperture of 25 m. The results show a significant reduction in the mismatch between the antenna and the updated FE model. This method’s calculation time is significantly reduced compared with the updating method without using the surrogate model, with the RBF surrogate model taking 1% of the time of the finite element model in the FEA calculations. The proposed method can improve the antenna FE model calculation accuracy and significantly enhance the efficiency of FE model updating calculations. Thus, it can offer a reference for antenna engineering practice.

1. Introduction

The radio telescope is essential for receiving radio waves from celestial bodies. Its most apparent feature is its large dual-reflector antenna structure [1]. Environmental factors such as gravity, temperature, wind, and inertial loads [2] can deform the antenna structure, resulting in a deviation from the required surface shape. This deformation reduces the antenna gain, making the radio telescope unable to meet the accuracy requirements of high-frequency operation [3,4]. The conformal design of the antenna can reduce this deformation to a certain extent, but it cannot eliminate it [5].

Finite element analysis technology [6] can analyze the mechanical properties of the antenna structure. It is essential to establish an FE model that meets the accuracy requirements and can accurately reflect the structure’s mechanical properties for evaluating the structure’s mechanical properties [7]. For antenna structures, uncertainties such as manufacturing errors, assembly defects, and model simplification will cause errors between the FE model based on design data and the calculation results of the actual structure [8,9], and these will affect the evaluation of antenna performance and active structural compensation [10].

A technique for updating the FE model [11] can effectively address this issue. Current methods for updating the FE model are mainly matrix and parametric parameter methods. The matrix method mainly updates the FE model by updating the critical matrices of the FE model, such as stiffness, mass, and damping matrices [12]. The parameter method mainly updates the physical parameters, such as geometric dimensions and material properties in the model, to align the FE model calculation results closer to the measured response [13]. Compared to the matrix method, the parametric method holds greater physical significance and is more readily applicable in engineering and subsequent analysis. Therefore, the parameter method is widely used in engineering [14]. However, when updating FE models using traditional parametric methods, the numerous model parameters make the computational cost of identifying key parameters and optimizing the model too high.

The development of computer technology has effectively mitigated these problems. However, for problems with large, complex structures, thousands of simulations are often required to construct a suitable solution due to the high fidelity requirements of their FE models [15]. Surrogate models are usually introduced in engineering calculations to approximate the structural behavior and reduce the computational burden, thus effectively reducing the computational complexity in finite element analysis and making analysis and design more efficient [16].

Currently, there are relatively few studies on updating FE models for ground antenna main reflector deformation problems. In Reference [17], the FE model of a 64 m aperture radio telescope antenna was updated by updating the load vector in the stiffness matrix based on the deformation data of the antenna’s primary reflector surface obtained by photogrammetry. This method is classified as a matrix method, and it can make the results calculated by the FE model of the antenna approximate the actual measured results. However, the physical meaning of the model updating needs to be improved. Reference [18] proposed a method of updating the antenna FE model based on a hypothetical experiment. It updates the material property parameters in the FE model of the antenna of a 1.2 m radio telescope and reduces the deformation error of the main reflecting surface of the antenna’s FE model under temperature and gravity loads. This method is a parametric method with a clear physical meaning. However, for FE model updating of a large-aperture radio telescope antenna composed of complex structures, it will take much time to use this method.

This paper proposes an FE model updating method founded on parameter optimization for the 25 m aperture radio telescope antenna shown in Figure 1. We use the RBF surrogate model technology, combined with the antenna primary reflector surface deformation data obtained by photogrammetry technology, optimize the elastic modulus parameters of the different components of the antenna FE model BUS, update the original FE model, and reduce the impact of the BUS’s deformation on the antenna primary reflector surface deformation error. In the model updating, the RBF surrogate model is used instead of the FE model to perform parameter sensitivity analysis and optimization. Finally, the updated FE model’s analysis results are closer to the antenna primary reflector surface deformation results than the original FE model.

2. Finite Element Analysis of Antenna Structures

As shown in Figure 1b, the whole structure of the radio telescope antenna can be briefly separated into two parts: the reflector and the alidade. Above the alidade is the antenna’s reflector, which mainly includes the primary reflector surface, the sub-reflector, the BUS, the quadruped, the elevation gear, and the umbrella structure. These structures are primarily composed of a large number of bars. According to the antenna’s motion characteristics, in the antenna structure finite element analysis, the antenna structural bar can be discretized into bar elements or beam elements [19] and solved by the finite element stiffness equation.

We illustrate the basic idea of antenna finite element analysis with a beam element as an example. In finite element analysis, Hooke’s law can be used to calculate the elongation of the elastic beam under tensile load or the shortening of the rod due to compressive load. The law is

$$\mathsf{\sigma }=Eϵ$$

(1)

where $\mathsf{\sigma }$ represents the stress, $E$ represents the elastic modulus, and $ϵ$ represents the strain. The space beam element can bear the action of axial force, torque, and bending moment. Figure 2 shows a space beam element in a local coordinate system oxyz, with a cross-sectional area of A, a length of l, an elastic modulus of E, a cross-sectional moment inertia around the neutral axis parallel to the y-axis, a cross-sectional moment inertia around the neutral axis parallel to the z-axis, and a torsional moment of inertia J on the cross-section. It is assumed that each node has six degrees of freedom in the element compositions of two nodes, and its nodal displacement vector is in the local coordinate system:

$${\mathit{q}}^e={[u_1 v_1 w_1 {\theta }_{x1} {\theta }_{y1} {\theta }_{z1} u_2 v_2 w_2 {\theta }_{x2} {\theta }_{y2} {\theta }_{z2}]}^{\mathrm{T}}$$

(2)

where u, v, and w represent the node’s displacement components in triaxial directions in the local coordinate system, respectively, and ${\theta }_x$, ${\theta }_y$, and ${\theta }_z$ represent the angles of rotation of the node in triaxial directions.

The stiffness matrix corresponding to the nodal axial displacement $u_1$ and $u_2$ generated by the axial force is

$${\mathit{K}}_{u_1{u}_2}^e=\frac{EA}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$$

(3)

The stiffness matrix corresponding to the nodal displacements ${\theta }_{x1}$ and ${\theta }_{x2}$ generated by the torsion problem is

$${\mathit{K}}_{{\theta }_{x1}{\theta }_{x2}}^e=\frac{\mathit{GJ}}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$$

(4)

where J represents the torsional moment inertia on the beam’s cross-section. G represents the shear modulus; according to elastic mechanics, its expression is

$$G=\frac{E}{2(1+v)}$$

(5)

where v represents the Poisson’s ratio. On the oxy plane, the stiffness matrix corresponding to the nodal displacement $v_1$, ${\theta }_{z1}$, $v_2$, and ${\theta }_{z2}$ generated by the bending problem is

$${\mathit{K}}_{\mathit{oxy}}^e=\frac{EA}{l^3}\left[\begin{array}{cccc}12& 6l& -12& 6l\\ 6l& 4l^2& -6l& 2l^2\\ -12& -6l& 12& -6l\\ 6l& 2l^2& -6l& 4l^2\end{array}\right]$$

(6)

By replacing $I_z$ in Formula (6) with $I_y$, the stiffness matrix expressions corresponding to the nodal displacement $w_1$, ${\theta }_{y1}$, $w_2$, and ${\theta }_{y2}$ generated by the bending problem on the oxz plane can be obtained.

The elements in the torque stiffness matrix, bending moment stiffness matrix, and axial force stiffness matrix are assembled based on the order of node displacement in Equation (1), and the beam element global stiffness matrix can be obtained:

$${\mathit{K}}^e={\mathit{K}}_{u_1{u}_2}^e\oplus {\mathit{K}}_{{\theta }_{x1}{\theta }_{x2}}^e\oplus {\mathit{K}}_{\mathit{oxy}}^e\oplus {\mathit{K}}_{\mathit{oxz}}^e$$

(7)

where $\oplus$ represents the appropriate combination of matrices to ensure that the corresponding influence of the degree of freedom of each node is correctly considered. From the Formulas (1), (2), and (7), the beam element stiffness equation can be obtained as follows:

$${\mathit{K}}^e{\mathit{q}}^e={\mathit{P}}^e$$

(8)

where ${\mathit{q}}^e$ and ${\mathit{P}}^e$ are the node force vector and displacement vector of the element respectively. The beam element stiffness equation in the global coordinate system can be obtained using the direction cosine between the local coordinate oxyz’s axis and the overall coordinate OXYZ’s axis.

When the element’s stiffness equations are solved, these stiffness matrixes are assembled according to the determined connection relationship, and the antenna FE model’s stiffness equation is obtained:

$$\mathit{Kq}=\mathit{P}$$

(9)

where $\mathit{q}$ represents the displacements at each node in the overall coordinate system, K represents the global stiffness matrix, and $\mathit{P}$ represents each node’s nodal force vector.

Considering the influence of uncertainties such as manufacturing errors and assembly defects on the antenna structure’s deformation characteristics, the antenna structure global stiffness equation of the antenna is as follows:

$$(\mathit{K}+∆\mathit{K})\mathit{u}=\mathit{f}$$

(10)

where $∆\mathit{K}$ represents the stiffness matrix error between the FE model and the antenna. According to the Formula (10), when the node force vector f of the FE model in the global coordinate system remains constant, the overall stiffness matrix K’s change will affect the node displacement vector. Therefore, updating the global stiffness matrix can increase the closeness between the FE model deformation characteristics and the antenna.

3. Updating Method for FE Model of Antenna

The BUS is a typical three-dimensional truss system that directly supports the primary reflector surface of the antenna. The truss and related supports are composed of members mainly subjected to axial force in the truss structure. Based on the linear elastic deformation framework in Section 2, it can be seen that in the antenna FE model, the main material property parameter that affects the BUS deformation is the modulus of elasticity of the member. In order to solve the error caused by the difference in deformation between the antenna BUS and the FE model BUS, we evaluate the deformation of the antenna’s primary surface using the FE method. We have designed a method for updating the large-aperture reflector antenna FE model, as illustrated in Figure 3. The method uses the Root Mean Square (RMS) of the primary reflector surface node’s half-path length error to represent the antenna’s primary reflector surface’s deformation [20]. The antenna’s original FE model is updated by optimizing the rod’s elastic modulus in the BUS.

We analyzed the effect of the BUS deformation on the primary reflector surface deformation in terms of the antenna original FE model BUS upper chord node displacement. The whole flow consists of the following two main parts:

(1)

Parameter sensitivity analysis

(a)

The antenna’s original FE model is established based on the design information.

(b)

The model input parameters are set to the elastic modulus of d different BUS bars. The output response is set as the RMS of the displacement of the BUS upper chord nodes.

(c)

The Latin hypercube sampling (LHS) experimental design method [21] is utilized to obtain p sets of random input parameter samples. Then, the FE model is employed to acquire p output responses.

(d)

The RBF surrogate model is obtained by using p sets of input and output responses, and it is used to replace the FE model for the parameter sensitivity analysis.

(e)

The sensitivity of these d input parameters to the output response is calculated using the RBF model and combined with the Sobol method to obtain d’ sensitive parameters.

(2)

Parameter optimization

(a)

The original FE model input parameters are set to the d’ sensitivity parameters obtained by sensitivity analysis, and the output response is the RMS of the relative displacement error between the antenna’s nodes and the original FE model’s corresponding nodes.

(b)

The Latin hypercube sampling experimental design method is utilized to obtain q sets of random input parameter samples. Then, the FE model is employed to acquire q output responses.

(c)

The RBF surrogate model is obtained by using q sets of input and output responses, and it is used to replace the FE model for the parameter optimization.

(d)

The d’ parameters were optimized using an agent model and combined with a genetic algorithm.

(e)

The optimized parameters are used in place of the original parameters in the FE model to calculate the RMS of the BUS upper chord nodes’ half-path length errors, which is compared with the RMS of the antenna primary reflector surface nodes’ half-path length errors to test the model updating effect [22].

3.1. Establishment of Original FE Model and Analysis of Model Error

Based on the antenna design diagram and ANSYS APDL 2022 software, we established the parametric original FE model of the antenna reflector shown in Figure 4a, which includes 28,132 elements and 24,135 nodes. Modeling omits the alidade, uses a mass element to equate the primary reflector surface, with the mass element being set on the upper chord node of the BUS, and uses beam and shell elements to simulate the rest of the antenna’s structure, simplifying the connecting parts between the components and applying full-degree-of-freedom constraints at the nodes at the ends of the pitch axis. After applying the gravity load to the model, we calculated the deformation for an elevation angle of 35° and updated the FE model for this condition.

As shown in Figure 5, we measured the deformation of the primary reflecting surface by photogrammetry. Firstly, we arranged the marking points on the antenna reflecting surface. Then, we used a 25-ton crane to suspend the basket to carry the measurement personnel holding a camera with a measurement accuracy of 5 μm + 5 μm/m to take pictures of the antenna in front of the reflecting surface. Finally, we measured the displacement data of the marking points distributed on the primary reflecting surface under the action of gravity load. The number of these marker points is 1640.

The deformation data of 1640 points randomly distributed on the main reflector surface under gravity load were obtained by the photogrammetric method under a 35° elevation angle working condition. The RMS of the antenna primary reflector surface nodes’ half-path length errors is 0.321 mm, and the distributions of errors are as shown in Figure 6a. We use the deformation data and the cubic polynomial interpolation method to process 424 upper chord nodes in the original FE model BUS. The RMS’s error between the original FE model and the antenna is 0.019 mm, as shown in Figure 6b.

It is not difficult to see from Figure 6b that the positions of more significant errors are located on the primary reflector’s outer side, which indicates that the antenna backup part shown in Figure 7a significantly influences the deformation of the primary reflector. This part of the backup is a double-layer structure in which the bars are divided into ring, radiation, and inclined beams, as shown in Figure 7b–d. The initial values of the different bars’ elastic modulus parameters are 206 GPa.

Due to the rotationally symmetric nature of the BUS, we chose 1/16 of the entire BUS to show its division. As shown in Figure 8, we divided the backup structure shown in Figure 7 into 52 parts based on the bars’ different locations, cross-sectional sizes, and lengths. We set the bars’ elastic modulus as the input variables of the original finite element model of the antenna, which has 52 in total. In the FEA model, we take as output the displacements of the upper chord nodes of the backstay structure, as shown in Figure 8a. The number of these nodes is 424.

3.2. 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.

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 (${\mathit{x}}_i,y^i$), where ${\mathit{x}}_i=[x_1,x_2,\dots ,x_d]$ 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:

$$\widehat{y}(\mathit{x})=\sum _{i=1}^n{w}_i\phi (r_i)=\mathit{w}{\mathit{\phi }}^T$$

(11)

where $\widehat{y}(\mathit{x})$ represents the RBF model predicted value at the point to be measured $\mathit{x}=[x_1,x_2,\dots ,x_d]$, $\mathit{w}=[w_1,w_2,\dots ,w_n]$ is the weight coefficient’s vector, $\mathit{\phi }=[\phi (r_1),\phi (r_2),\dots ,\phi (r_n)]$ represents the vector of radial functions, and $r_i$ represents the Euclidean distance between x and ${\mathit{x}}_i$ of the following form:

$$r_i=‖\mathit{x}-{\mathit{x}}_i‖=\sqrt{\sum _{j=1}^d{(x_j-x_{\mathit{ij}})}^2}$$

(12)

Table 1. The common radial functions and forms.

Radial Function	Gauss	Multi-Quadratic	Inverse Multi-Quadratic	Logarithmic Path
functional form	exp(-$c{r}^2$)	${(r^2+c^2)}^{\frac{1}{2}}$	${(r^2+c^2)}^{-\frac{1}{2}}$	$r^2\log (1+{\mathit{cr}}^2)$

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 $\widehat{y}(\mathit{x})$ and $y^i$ by least squares and solve for the weight coefficient vector w of the RBF surrogate model. The sum of squared errors is

$$J(\mathit{w})={\sum }_{i=1}^p{(\widehat{y}({\mathbf{x}}_i)-y^i)}^2={\sum }_{i=1}^p{\left({\sum }_{j=1}^p{w}_i\phi (‖{\mathit{x}}_i-{\mathit{x}}_j‖)-y^i\right)}^2={‖\mathit{w}\mathit{\Phi }-\mathit{y}‖}^2$$

(13)

where$\mathit{y}=[y^1,y^2,\dots ,y^d]$, and matrix $\mathit{\Phi }$ is

$$\mathit{\Phi }=\left[\begin{array}{cccc}\phi \parallel x_1-x_1\parallel & \phi \parallel x_1-x_2\parallel & \dots & \phi \parallel x_1-x_p\parallel \\ \phi \parallel x_2-x_1\parallel & \phi \parallel x_2-x_2\parallel & \dots & \phi \parallel x_2-x_p\parallel \\ ⋮& ⋮& \ddots & ⋮\\ \phi \parallel x_p-x_1\parallel & \phi \parallel x_p-x_2\parallel & \dots & \phi \parallel x_p-x_p\parallel \end{array}\right]$$

(14)

Formula (13) is expanded as follows:

$$J(\mathit{w})=(\mathit{w}\mathit{\Phi }-\mathit{y}{)}^T(\mathit{w}\mathit{\Phi }-\mathit{y})={\mathit{\Phi }}^T{\mathit{w}}^T\mathit{w}\mathit{\Phi }-2{\mathit{\Phi }}^T{\mathit{w}}^T\mathit{y}+{\mathit{y}}^T\mathit{y}$$

(15)

The derivation of w on both sides of Formula (15) with the derivative set as 0 results in the following:

$$\frac{\partial }{\partial \mathit{w}}J(\mathit{w})=\frac{\partial }{\partial \mathit{w}}({\mathit{\Phi }}^T{\mathit{w}}^T\mathit{w}\mathit{\Phi }-2{\mathit{\Phi }}^T{\mathit{w}}^T\mathit{y}+{\mathit{y}}^T\mathit{y})=0$$

(16)

The following normal equation is obtained:

$${\mathit{\Phi }}^T\mathit{\Phi }\mathit{w}={\mathit{\Phi }}^T\mathit{y}$$

(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 w can be obtained:

$$\mathit{w}={({\mathit{\Phi }}^T\mathit{\Phi })}^{-1}{\mathit{\Phi }}^T\mathit{y}$$

(18)

The optimal weight vector w is substituted into the Formula (11) to obtain the RBF surrogate model:

$$\widehat{y}(\mathit{x})=\sum _{i=1}^p{\left[({{\mathit{\Phi }}^T\mathit{\Phi })}^{-1}{\mathit{\Phi }}^T\mathit{y}\right]}_i{(‖\mathit{x}-{\mathit{x}}_i‖+c^2)}^{\frac{1}{2}}$$

(19)

In Formula (19), c is a given constant greater than 0. Due to the lack of reference for selecting c in practical applications, we use the golden section method to optimize c [25] and verify the RBF surrogate model accuracy through the determination coefficient $R^2$ method [26]. Firstly, we train m sets of random input parameter samples and then use the original FE model and the RBF surrogate model to calculate the corresponding output response. With c as the variable and minimum error $f=1-R^2$ as the objective function, the optimization model is as follows:

$$\left\{\begin{array}{c}\begin{array}{c}\mathrm{find}c\hfill \\ \min f=1-R^2=1-\left(1-\frac{\sum _{\mathrm{k}=1}^m{(y^k-\stackrel{\mathrm{-}}{y})}^2}{\sum _{k=1}^m{(y^k-{\widehat{y}}^k)}^2}\right)\hfill \end{array}\\ \mathrm{s}.\mathrm{t}.0<c<\stackrel{\mathrm{-}}{c}\hfill \\ 0<f<\stackrel{\mathrm{-}}{f}\hfill \end{array}\right.$$

(20)

where $y^k$ represents the result computed by the original FE model, ${\widehat{y}}^k$ represents the result computed by the RBF surrogate model, $\stackrel{\mathrm{-}}{y}$ represents the average of the results computed by m sets of the FE model, $\stackrel{\mathrm{-}}{c}$ represents the constant’s upper limit, and $\stackrel{\mathrm{-}}{f}$represents the objective function’s upper limit.

3.3. 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 $\widehat{y}=f(\mathit{x})$, where $f(\mathit{x})$ represents a square-integrable function on the domain, $x=[x_1,x_2,\dots ,x_d]$ 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 ~ U (0,1). For any $\forall i\ne j$, $x_i$ and $x_j$ are independent of each other. We will use the multiple integral method to decompose the function$f(\mathit{x})$ into the sum of $2^d$ unrelated sub-terms:

$$f(\mathit{x})=f_0+\sum _{i=1}^d{f}_i(x_i)+\sum _{1\le i<j\le d}{f}_{\mathit{ij}}(x_i,x_j)+\cdots +{\mathrm{f}}_{1,2,\cdots ,d}(x_1,x_2,\cdots ,x_d)$$

(21)

where $f_0$ represents the constant term, $f_i(x_i)$ represents the i-th input variable effect on the output response, $f_{\mathit{ij}}(x_i,x_j)$ 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 $S\subseteq \{x_1,$ $x_2,\dots ,x_d\}$in Formula (21),

$$\mathrm{E}\left[f_S({\mathit{x}}_S)\right]=0$$

(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 $S,T\subseteq \left\{x_1,x_2,\dots ,x_d\right\}$, the following orthogonality conditions apply:

$$\mathrm{E}\left[f_S({\mathit{x}}_S)f_T({\mathit{x}}_T)\right]=0,\forall S\ne T$$

(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:

$$D=\sum _{i=1}^d{D}_i+\sum _{1\le i<j\le d}{D}_{\mathit{ij}}+\cdots +D_{1,2,\cdots ,d}$$

(24)

where D represents the total variance of the function, $D_i$ represents the single input variable $x_i$ influence on the total variance, $D_{\mathit{ij}}$ represents the effect of interaction between two input variables on total variance, and so on; $D_{1,2,\dots ,d}$ 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

$$S_i=\frac{D_i}{D}$$

(25)

and the calculation formula of the global sensitivity index is

$$S_{T_i}=1-\frac{D_{~i}}{D}$$

(26)

where $D_{~i}$ represents the variance generated by the interaction of other input variables except $x_i$.

3.4. Parameter Optimization

We update the original FE model of the antenna by establishing an optimization function. Similar to Section 3.3, the surrogate model for parameter optimization is fitted before parameter optimization is performed. We used the d’ sensitivity parameters obtained from the sensitivity analysis as input variables for parameter optimization. Then, we use the cubic polynomial interpolation method to interpolate the displacements of the 424 upper chord nodes of the backup structure to obtain the displacements matching the 1640 marker points on the main surface of the actual antenna.

$(u_k^x,u_k^y,u_k^z)$ represents the displacement of the actual antenna primary reflector surface node measured by the photogrammetry method, and $(v_k^x,v_k^y,v_k^z)$ represents the node displacement of the FE model obtained by the interpolation method relative to the actual antenna measurement point node; the relative displacement error between the antenna’s node and the original FE model’s corresponding nodes is calculated as

$${\Delta }_k=\sqrt{(u_k^x-v_k^x)+(u_k^y-v_k^y)+(u_k^z-v_k^z)}$$

(27)

to minimize the relative displacement error’s RMS as the optimization objective, and the objective function is

$$\begin{array}{c}f=min\sqrt{\frac{\sum _{k=1}^M{\Delta }_k}{M}}\end{array}$$

(28)

The optimization problem for the antenna primary reflector surface deformation is expressed as follows:

$$\left\{\begin{array}{c}find[E_1,\dots ,E_{d’}]\hfill \\ \min f=\sqrt{\frac{\sum _{k=1}^M{\Delta }_k}{M}}\hfill \\ \mathrm{s}.\mathrm{t}.\underset{\_}{E}<E<\overline{E}\hfill \\ \underset{\_}{f}<f<\overline{f}\hfill \end{array}\right.$$

(29)

where $\underset{\_}{E}$ and $\overline{E}$ represent the elastic modulus parameter E upper and lower bounds, and $\underset{\_}{f}$ and $\overline{f}$ represent the objective function $f$ upper and lower bounds. Finally, the established optimization model is solved using the genetic algorithm.

4. Simulation Experiment Example of FE Model Updating of Antenna

We combined the content of Section 3 and the deformation data of the antenna primary reflector surface nodes under the conditions of a 35° elevation angle and gravity load. The 25 m radio telescope antenna primary reflector original FE model is updated.

The modulus of elasticity parameters of the 52 types of bars, which are classified according to the different cross-sectional dimensions and lengths of the bars, are set as input variables to the antenna original FE model. The RMS value of the BUS upper chord node displacement is taken as the output response for parameter sensitivity analysis. According to the method shown in Section 3.2, the initial value of constant c is 1, and the random sampling range of elastic modulus parameters is 190–220 GPa. Using the original FE model of the antenna and the LHS experimental design method, 200 sets of input and output samples are obtained to fit the RBF surrogate model, which is used for parameter sensitivity analysis.

According to the method shown in Section 3.2, 10 sets of samples are re-extracted, and the prediction results are calculated by using the antenna original FE model and the RBF surrogate model, respectively. The constant c is optimized with the objective function f function value less than 0.01. The optimization process and results are shown in

Table 2. Optimization results of constant c of RBF model for parameter sensitivity analysis.

	Variable c	Objective Function Values
initial value	1	0.74
optimal values	69.4	0.61 × 10−2

and Figure 9. After 20 iterations, the RBF surrogate model meets the accuracy requirements when the value of c is 69.4 and the value of function f is 0.61 × 10⁻². Fitting the surrogate model for parameter sensitivity analysis took 1546 s.

Using the RBF surrogate model and Sobol method to calculate 20,000 sets of random input and output samples, the sensitivity of these 52 elastic modulus parameters to the BUS deformation is obtained, as shown in Figure 10. It can be seen that the elastic modulus parameter of the bar numbered 50 is the most sensitive to the antenna primary reflector surface deformation. However, the antenna’s primary reflector surface consists of multiple panels, each supported by multiple bars, and updating this parameter is not enough to correct the deformation of the entire surface. Compared with other bars, the elastic modulus of the seven bars, numbered 2, 4, 18, 20, 33, 48, and 50, is the most sensitive to the antenna primary reflector surface deformation. Therefore, we choose these seven parameters as optimization variables for parameter optimization. In order to clearly show the elastic modulus parameters’ position of these seven bars in the backup, according to the rotational symmetry of the antenna BUS, 1/16 of the entire structure is selected to show the position distribution of these bars, as shown in Figure 11.

The other settings of the fitting process of the parameter-optimized RBF surrogate model and the verification process of the calculation accuracy are the same as those of the parameter sensitivity analysis. After 17 iterations, the surrogate model meets the accuracy requirements when the value of c is 57.2 and the value of function f is 0.83 × 10⁻², as shown in Figure 12 and

Table 3. Optimization results of constant c of RBF model for parameter optimization.

	Variable c	Objective Function Values
initial value	1	0.38
optimal values	57.2	0.83 × 10−2

. Fitting the surrogate model for parameter optimization took 1517 s.

We use the genetic algorithm and the fitted RBF surrogate model to calculate the parameter optimization. The population number is set to 50, and the genetic generation is 100. After 5000 calculations, the relative displacement error RMS between the antenna’s nodes and the original FE model’s corresponding nodes is reduced from 0.0258 mm to 0.0186 mm. The optimization process and results are shown in Figure 13 and

Table 4. Parameter optimization results.

The Number of Parameters	2	4	18	19	33	49	50
Elastic modulus/GPa	218.3	190.4	191.1	190.3	190.4	190.7	190.5

, respectively.

After the model updating was completed, we calculated the BUS model half-path length errors of the chord node RMS in the updated FE model under gravity loading and at a 35° elevation angle and compared them with the RMS of the antenna at a 35° elevation angle. The results are shown in Figure 14.

From Figure 6b and Figure 14, it can be seen that the error between the FE model’s RMS and the antenna’s RMS at a 35° elevation angle is reduced from 0.019 mm to 0.013 mm, which is 36.8% lower than the original error.

Then, we calculate the RMS of the BUS upper chord nodes’ half-path length errors of the original and updated FE models under gravity loading under 10° and 70° conditions and compare them with the RMS of the antenna primary reflector surface nodes half-path length errors under these two elevation angle conditions. The results are shown in Figure 15 and Figure 16. The error between the RMS of the FE model and the antenna at a 10° elevation angle is reduced from 0.051 mm to 0.04 mm, 21.6% lower than the original error. The error between the RMS of the FE model and the antenna at a 70° elevation angle is reduced from 0.049 mm to 0.045 mm, which is 8.1% lower than the original error.

As shown in

Table 5. Time comparison.

	Number of Calculations	Time-Consuming/s
FE model	1	7.73
RBF model	1	0.081

, in the whole model updating process, the FE model is used to calculate once, with an average time of 7.73 s. The RBF surrogate model calculates once, with an average time of 0.081 s. The calculation time of the RBF model is 1% of that of the FE model.

5. Conclusions

We propose a parameter updating method for an antenna FE model founded on the RBF surrogate model, which aims at the BUS deformation’s difference between a large-aperture radio telescope antenna and an FE model. Under a 35° elevation angle, the antenna FE model is optimized and updated by combining the primary reflector surface deformation data for a 25 m aperture radio telescope antenna under gravity load. Firstly, according to the design data, the 25 m antenna original FE model is established, and the parameter sensitivity analysis is carried out. The elastic modulus parameters of different BUS bars with high sensitivity to the antenna primary reflector surface deformation are selected as the optimization variables. The relative displacement error RMS between the antenna’s nodes and the corresponding nodes in the original FE model is taken as the optimization target, and the antenna FE model is optimized and updated by a genetic algorithm. Due to the complex structure of the antenna, it is very time-consuming to use the FE model for analysis and calculation. Therefore, in the modified parameter sensitivity analysis and optimization part, the RBF surrogate model is used instead of the FE model for calculation. We compare the calculation time of the FE model and the RBF surrogate model and find that the calculation efficiency is improved by about 99%, significantly saving the FE model updating process’s calculation cost. The RMS’s error between the antenna and the FE model is reduced by 36.8%, which preliminarily shows that the method is feasible. To further verify the method feasibility, compared with the photogrammetry data under 10° and 70° working conditions, the error of the RMS between the antenna and the updated antenna FE model primary surface is reduced by 21.6% and 8.1%, respectively, which again shows that the updating method is feasible.