# Influence of Optimal Adjustment of FAST Active Reflector on Receiving Efficiency

Authors: Jianjie Zhao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, how to optimize and adjust the five hundred-meter aperture spherical radio telescope—FAST—active reflector and the impact on the receiving efficiency after optimization are discussed. The shape adjustment of the active reflector of FAST can be taken as the deformation of a continuous and ideal surface. The surface equation with parameter idealization is established. Meanwhile, the ideal surface equation is obtained and the corresponding evaluation function is given to evaluate the ideal parabolic equation. Furthermore, the ideal surface equation is discretized, and the discrete points are taken as the position to which the actual main cable node should be displaced by the down-tie cable. Finally, a receiving efficiency model is established to compare the actual receiving efficiency of the previous and after reflection panels. The accuracy similar to the ideal paraboloid of the reflector shape in the working area previous and after deformation is given. The results show that our optimization and adjustment model can improve the receiving efficiency of FAST reflective panel with small deformation.


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## Body

## 1. Introduction

Five hundred-meter aperture spherical radio telescope—FAST—is the world's largest single dish spherical radio telescope, which is mainly composed of reflection panels, feed cabin, main cable nodes, down-tie cable, back frames, and actuators. The main cable net is installed on the inner side of the ring beam and divided by geodesic grid. There are 4450 reflective surface elements installed on its 2225 movable nodes. Corresponding down-tie cable and actuators are connected below each node [1–5]. The reflecting panel is used to reflect electromagnetic waves from the universe. The feed cabin is driven by six cables and servo mechanism to receive reflected signals. All reflective panels of FAST form a spherical structure with radius (R) of 300 m and aperture of 500 m, there is a feed cabin above it for receiving reflected electromagnetic waves, and the feed cabin moves on a focal sphere surface with a focal length of [6–9] 0.466 times R concentric with the FAST sphere. During the operation of FAST, in the direction of observation angle, the reflection panels in a partial sphere with an aperture of 300 m on the FAST sphere change the shape of its integral surface through the actuator and the down-tie cable, so that it changes from a sphere to a paraboloid with the focus on the surface of the feed cabin [10–14] [15]. There have been some researches on the radio telescope and FAST. The research on the reception and fixed structure of huge radio telescope has been comprehensively introduced [16–20]. FAST project faces many challenges, like the commissioning challenges survey [21–25]. And the development of FAST undoubtedly has a deeper understanding and cognition of the research of radio telescope [5]. In order to reduce the construction cost of radio telescopes and improve their reliability, the requirements for the environment are also harsh. Whether the radio telescope can effectively receive electromagnetic waves is related to the overall shape of its reflecting surface. The requirement for the shape error of the reflector is related to the received electromagnetic wave length—reflector shape error must be at most one-twentieth of the accepted wavelength of the electromagnetic wave in order for its telescope's directional coefficient to exceed 67% of the ideal reflector [26–29]. The frequency range of the electromagnetic wave received by the radio telescope is 70 MHz to 10 GHz [28]; that is, the wavelength is 30 mm to 4.3 m. According to the principle, the minimum error accuracy required is RMS = 1.5 mm. In reality, the telescope can reduce the deformation error caused by gravity by using allogeneic isomorphic design, and the influence of thermal and wind deformation can be reduced by selecting appropriate materials and suitable environment, which makes the final limit wavelength reach 7 mm or even 3 mm. The actual acceptance efficiency of FAST is between 1400 and 1450 MHz, and the light wave acceptance rate is 83% ± 2% [29].Therefore, in this study, the method of how to optimize and adjust the reflection panel of FAST to minimize the displacement and overall deformation of the main cable node and improve the efficiency is discussed. We establish the mathematical model of the ideal paraboloid within the specified 300 m aperture, and optimize and adjust its parameters in the established multi-objective.In section2, the mathematical model of constraints is established from four aspects, and the objective function is given by synthesizing the four constraints. An ideal parabolic equation is obtained in section 2. In Section 3, the ideal paraboloid given is evaluated, and the evaluation function is given in three respects. How to adjust FAST in practice is discussed in section 4. Discretize the ideal paraboloid and adjust the main cable nodes by fitting the ideal paraboloid to obtain the actual results of the adjustment of the main cable nodes. Finally, in section 5, a mathematical model for calculating the acceptance efficiency is established and the accuracy of the actual and ideal paraboloid is given.


## 2. Mathematical Model and Calculation Method

Based on the requirements of engineering practice, the mathematical model of the adjustment theoretical surface and related optimization methods are analyzed and established, which lays a foundation for the establishment and calculation of the relative coordinates and the efficiency model of the actual adjusted main cable nodes given by FAST. As shown in Figure1, the displacement of each main cable node can be regarded as two main directions, radial and latitudinal directions, and the displacement in the radial and latitudinal directions is also indicator for evaluation and optimization.Figure 1
Working diagram of FAST.


### 2.1. Model Determination of the Ideal Paraboloid for Work

Figure1 shows an ideal sphere and an ideal working paraboloid. There are many main cable nodes on the real working surface, and the spherical surface changes into a working paraboloid by moving the main cable nodes. We can simplify the model first and turn the originally limited main cable nodes into infinite main cable nodes, all of which are on a continuous and ideal working surface, so that we can discuss and analyze each point on the ideal paraboloid in general. Therefore, the ideal working paraboloid can be obtained from the ideal spherical surface.Figure2 shows the ideal working sphere and the ideal working paraboloid after the change. Due to the rotational symmetry, the paraboloid and the sphere can be discussed on the same plane; that is, the curve equations of the circle and the parabola are established first. The focus of the parabola equation is at point Q, and the ideal paraboloid can be obtained by using the rotation of the curve, so we can set it at the parabola equation and the circle equation established in the ZOX plane rectangular coordinate system:(1)x2=2pz+2p0.534⋅R+p2,x2+z2=R2.where p is the focal length, R is the ideal spherical radius of FAST, and the constant 0.534 is obtained by dividing the distance between the focal plane and the origin of coordinates by the radius R.Figure 2
Schematic diagram of radial and latitudinal displacement diagram.In order to ensure the required ideal working parabola and fit the spherical surface as much as possible to ensure that the displacement of the main cable node is as small as possible, we need to compare the fit degree of the parabola and the spherical surface on the XOZ plane, for which we need a series of mathematical models and indicators are established to ensure that the overall deformation is within the minimum range to the parabola with the optimal focus at pointQ.


#### 2.1.1. Radial Displacement

Figure2 illustrates the direction of node displacement. The displacement of all nodes can be divided into radial displacement direction and zonal displacement direction. As shown in Figure 2, when comparing parabola and circular curve, the radial displacement of each node can be regarded as the displacement in the direction of passing through the origin, that is, the ball center O, and the radial displacement can be obtained by comparing the straight-line distance between the front and rear two points. The sum of the displacement of the main cable nodes on all paraboloids is called radial and displacement. Because the nodes are very dense, it can be expressed by integral. The radial displacement of any point on the XOZ plane can be expressed as(2)ΔLA=x22p−0.534R+p22+x2−R.The sum of the radial movements of all points inside can be expressed as(3)f1p=2∫0150x22p−0.534R+p22+x2−R2dx.


#### 2.1.2. Zonal Displacement

Figure3 shows the latitudinal direction. It can be seen from the figure that the displacement in this plane can be expanded uniformly, and the optimization equation is(4)f2p=2π∫-R−32R2pz+2p0.534R+p2−R2−z2dz.Figure 3
Parallel and displacement diagram.


#### 2.1.3. Overall Adjustment and Optimization

In the process of overall adjustment and optimization, we can start from two aspects: one can start from the variation of the overall arc length, and the other can consider the integrity and smoothness of the overall working face and nonworking face.In terms of arc length, the change of arc length before and after the arc is made as small as possible under the whole, and the optimization equation can be written as(5)f3p=∫-150+1501+xp2dx−π3R.All the above optimization equations are based on the working face, without considering the integrity and continuity between the working face and the nonworking face. From a practical point of view, the outermost main cable node of the reference sphere is fixed on the ring beam on, not connected to the actuator, no radial displacement. Therefore, in order to maintain the smoothness and integrity of the working paraboloid as much as possible, it is required that in the process of adjusting the shape of the reflecting surface, the radial displacement of the paraboloid edge nodes should be as small as possible; that is, the working paraboloid at the endpoint should be as close as possible to the reference spherical surface. The optimization model is established as follows:(6)f4p=15022p−0.534R+p2−−R2−1502.


### 2.2. Integration of Optimization Equations and Determination of Objective Function

From all the optimization equations obtained above, the determined optimization equations are all cost-type optimization equations. On the basis of the above equations, we can establish a multi-objective nonlinear programming model. The most important thing for a nonlinear multi-objective programming model is how to establish its objective function. The difference from above is that a forced adjustment factor is added to the radial displacement, in order to force the range to be controlled within a certain range in the multi-objective planning to ensure the established corresponding objective function. The displacement of main cable must be forced to satisfy the corresponding range and converge to the minimum value. The optimization equation system is as follows:(7)f1p=2∫0150x22p−0.534R+p22+x2−R2dx,f2p=2π∫−R−32R2pz+2p0.534R+p2+R2−z2dz,f3p=∫-150+1501+xp2dx−π3R,f4p=15022p−0.534R+p2−−R2−1502,ΔLA=x22p−0.534R+p22+x2−R∈0,+0.6,where p is the focal length of the ideal paraboloid, R is the radius of the ideal base circle, and both z and x are variables that establish a Cartesian coordinate system.Since the establishment of the corresponding objective functions is all cost-type objectives, the cost objectives can be superimposed according to weights, and the objective function can be minimized. The formula is as follows:(8)minfp=a1f1p+a2f2p+a3f3p+a4f4p,where(9)a1+a2+a3+a4=1.Among them are the weights of each cost target obtained and normalized.


## 2.1. Model Determination of the Ideal Paraboloid for Work

Figure1 shows an ideal sphere and an ideal working paraboloid. There are many main cable nodes on the real working surface, and the spherical surface changes into a working paraboloid by moving the main cable nodes. We can simplify the model first and turn the originally limited main cable nodes into infinite main cable nodes, all of which are on a continuous and ideal working surface, so that we can discuss and analyze each point on the ideal paraboloid in general. Therefore, the ideal working paraboloid can be obtained from the ideal spherical surface.Figure2 shows the ideal working sphere and the ideal working paraboloid after the change. Due to the rotational symmetry, the paraboloid and the sphere can be discussed on the same plane; that is, the curve equations of the circle and the parabola are established first. The focus of the parabola equation is at point Q, and the ideal paraboloid can be obtained by using the rotation of the curve, so we can set it at the parabola equation and the circle equation established in the ZOX plane rectangular coordinate system:(1)x2=2pz+2p0.534⋅R+p2,x2+z2=R2.where p is the focal length, R is the ideal spherical radius of FAST, and the constant 0.534 is obtained by dividing the distance between the focal plane and the origin of coordinates by the radius R.Figure 2
Schematic diagram of radial and latitudinal displacement diagram.In order to ensure the required ideal working parabola and fit the spherical surface as much as possible to ensure that the displacement of the main cable node is as small as possible, we need to compare the fit degree of the parabola and the spherical surface on the XOZ plane, for which we need a series of mathematical models and indicators are established to ensure that the overall deformation is within the minimum range to the parabola with the optimal focus at pointQ.


### 2.1.1. Radial Displacement

Figure2 illustrates the direction of node displacement. The displacement of all nodes can be divided into radial displacement direction and zonal displacement direction. As shown in Figure 2, when comparing parabola and circular curve, the radial displacement of each node can be regarded as the displacement in the direction of passing through the origin, that is, the ball center O, and the radial displacement can be obtained by comparing the straight-line distance between the front and rear two points. The sum of the displacement of the main cable nodes on all paraboloids is called radial and displacement. Because the nodes are very dense, it can be expressed by integral. The radial displacement of any point on the XOZ plane can be expressed as(2)ΔLA=x22p−0.534R+p22+x2−R.The sum of the radial movements of all points inside can be expressed as(3)f1p=2∫0150x22p−0.534R+p22+x2−R2dx.


### 2.1.2. Zonal Displacement

Figure3 shows the latitudinal direction. It can be seen from the figure that the displacement in this plane can be expanded uniformly, and the optimization equation is(4)f2p=2π∫-R−32R2pz+2p0.534R+p2−R2−z2dz.Figure 3
Parallel and displacement diagram.


### 2.1.3. Overall Adjustment and Optimization

In the process of overall adjustment and optimization, we can start from two aspects: one can start from the variation of the overall arc length, and the other can consider the integrity and smoothness of the overall working face and nonworking face.In terms of arc length, the change of arc length before and after the arc is made as small as possible under the whole, and the optimization equation can be written as(5)f3p=∫-150+1501+xp2dx−π3R.All the above optimization equations are based on the working face, without considering the integrity and continuity between the working face and the nonworking face. From a practical point of view, the outermost main cable node of the reference sphere is fixed on the ring beam on, not connected to the actuator, no radial displacement. Therefore, in order to maintain the smoothness and integrity of the working paraboloid as much as possible, it is required that in the process of adjusting the shape of the reflecting surface, the radial displacement of the paraboloid edge nodes should be as small as possible; that is, the working paraboloid at the endpoint should be as close as possible to the reference spherical surface. The optimization model is established as follows:(6)f4p=15022p−0.534R+p2−−R2−1502.


## 2.1.1. Radial Displacement

Figure2 illustrates the direction of node displacement. The displacement of all nodes can be divided into radial displacement direction and zonal displacement direction. As shown in Figure 2, when comparing parabola and circular curve, the radial displacement of each node can be regarded as the displacement in the direction of passing through the origin, that is, the ball center O, and the radial displacement can be obtained by comparing the straight-line distance between the front and rear two points. The sum of the displacement of the main cable nodes on all paraboloids is called radial and displacement. Because the nodes are very dense, it can be expressed by integral. The radial displacement of any point on the XOZ plane can be expressed as(2)ΔLA=x22p−0.534R+p22+x2−R.The sum of the radial movements of all points inside can be expressed as(3)f1p=2∫0150x22p−0.534R+p22+x2−R2dx.


## 2.1.2. Zonal Displacement

Figure3 shows the latitudinal direction. It can be seen from the figure that the displacement in this plane can be expanded uniformly, and the optimization equation is(4)f2p=2π∫-R−32R2pz+2p0.534R+p2−R2−z2dz.Figure 3
Parallel and displacement diagram.


## 2.1.3. Overall Adjustment and Optimization

In the process of overall adjustment and optimization, we can start from two aspects: one can start from the variation of the overall arc length, and the other can consider the integrity and smoothness of the overall working face and nonworking face.In terms of arc length, the change of arc length before and after the arc is made as small as possible under the whole, and the optimization equation can be written as(5)f3p=∫-150+1501+xp2dx−π3R.All the above optimization equations are based on the working face, without considering the integrity and continuity between the working face and the nonworking face. From a practical point of view, the outermost main cable node of the reference sphere is fixed on the ring beam on, not connected to the actuator, no radial displacement. Therefore, in order to maintain the smoothness and integrity of the working paraboloid as much as possible, it is required that in the process of adjusting the shape of the reflecting surface, the radial displacement of the paraboloid edge nodes should be as small as possible; that is, the working paraboloid at the endpoint should be as close as possible to the reference spherical surface. The optimization model is established as follows:(6)f4p=15022p−0.534R+p2−−R2−1502.


## 2.2. Integration of Optimization Equations and Determination of Objective Function

From all the optimization equations obtained above, the determined optimization equations are all cost-type optimization equations. On the basis of the above equations, we can establish a multi-objective nonlinear programming model. The most important thing for a nonlinear multi-objective programming model is how to establish its objective function. The difference from above is that a forced adjustment factor is added to the radial displacement, in order to force the range to be controlled within a certain range in the multi-objective planning to ensure the established corresponding objective function. The displacement of main cable must be forced to satisfy the corresponding range and converge to the minimum value. The optimization equation system is as follows:(7)f1p=2∫0150x22p−0.534R+p22+x2−R2dx,f2p=2π∫−R−32R2pz+2p0.534R+p2+R2−z2dz,f3p=∫-150+1501+xp2dx−π3R,f4p=15022p−0.534R+p2−−R2−1502,ΔLA=x22p−0.534R+p22+x2−R∈0,+0.6,where p is the focal length of the ideal paraboloid, R is the radius of the ideal base circle, and both z and x are variables that establish a Cartesian coordinate system.Since the establishment of the corresponding objective functions is all cost-type objectives, the cost objectives can be superimposed according to weights, and the objective function can be minimized. The formula is as follows:(8)minfp=a1f1p+a2f2p+a3f3p+a4f4p,where(9)a1+a2+a3+a4=1.Among them are the weights of each cost target obtained and normalized.


## 3. Fitting and Simulation Results

For each of the above established optimization functions, the final objective function is obtained through the normalized weighting method. In this regard, considering all aspects of balance, we make the effect of each optimization function consistent and equalize the weights. The ideal paraboloid derived from the base sphere simulation is determined, simulated, and analyzed.


### 3.1. Selection of Working Base Sphere

The coordinate expression for an ideally working base sphere would be(10)z=−R2−x2+y2x2+y2≤1502.Among them,R is the radius of the base sphere, and x and y are only working surfaces within a circle with a radius of 150 m.The selected ideal working spherical radiusR = 300 m. For intuition, since both the ideal base sphere and the fitted ideal working paraboloid have rotational symmetry, they can be discussed on the ZOX plane.


### 3.2. Determination of Ideal Working Paraboloid

For the general formula of the ideal working paraboloid, you can refer to formula (1), and the ideal working paraboloid can be gotten by rotating the ideal working parabola:(11)z=x2+y22p−0.534⋅R+p2x2+y2≤1502.The determination of the ideal working paraboloid is mainly to determine the parameterp, and the parameter determination of the p focal length is determined by the objective function, namely, formulas (8) and (9). In the determined range, the solution has a single extreme value. We use the simulated annealing algorithm to solve the solution. It is found that the p value is 280.36 m.


### 3.3. Comparison and Analysis of Fitting Results

#### 3.3.1. Fitting Results

By comparing the FAST ideal spherical surface with the given ideal paraboloid, you can see that the absolute difference between the two is not large, and the difference is hardly visible with respect to the 300 m caliber. It means that the position of the main cable node needs to be changed and is small enough. And from the results, we can see that the middle fitting is better than the fitting at the edge.


#### 3.3.2. Absolute Comprehensive Change

In order to quantitatively distinguish the base sphere and the ideal working paraboloid, we can find the error between sphere and paraboloidΔe, and its expression is as follows:(12)Δe=x2+y22p−0.534⋅R+p2−−R2−x2+y2.The difference between the paraboloid and the base sphere can be seen quantitatively, and theΔe-function image is shown in Figure 4.Figure 4
The error between sphere and paraboloid.It can be quantitatively seen from the image that the absolute error function does not exceed 0.45 m, and the largest error occurs at the center and edge of the paraboloid, where the absolute error of the paraboloid edge is 0.45 m. Ideally, this fitting is within the allowable range, so it is consistent in terms of fitting error.


#### 3.3.3. Length Change of Main Cable

In addition to the requirements for the displacement index of the actuator, there are also requirements for the change of the length of the main cable. In order to reflect the change of the length of the main cable more reasonably, we use the relative radial changeΔa of the adjacent main cable nodes, and its formula is as follows:(13)Δa=∫−150+1501+x/p2dxRL⋅π/3−1L×100%.Besides, there is also the relative latitudinal variation, the formula of which is as follows:(14)Δo=max−p/tanθ+p2/tan2θ+2p0.534R+p/2−RsinθRsinθ,θ∈0,π6×100%,where L is the length of adjacent main cable nodes, p is the focal length of the working paraboloid, and R is the radius of the base sphere. After calculation, the relative radial variation is 0.011% and the relative latitudinal variation is 0.13%. It can be seen from this that the relative change obtained is very small. However, it should be noted that the relative change in the latitudinal direction is not on the same order of magnitude as the relative change in the radial direction. Therefore, it can adjust the direction of the down-tie cable of the actuator properly to compensate for the change in the latitudinal direction.


## 3.1. Selection of Working Base Sphere

The coordinate expression for an ideally working base sphere would be(10)z=−R2−x2+y2x2+y2≤1502.Among them,R is the radius of the base sphere, and x and y are only working surfaces within a circle with a radius of 150 m.The selected ideal working spherical radiusR = 300 m. For intuition, since both the ideal base sphere and the fitted ideal working paraboloid have rotational symmetry, they can be discussed on the ZOX plane.


## 3.2. Determination of Ideal Working Paraboloid

For the general formula of the ideal working paraboloid, you can refer to formula (1), and the ideal working paraboloid can be gotten by rotating the ideal working parabola:(11)z=x2+y22p−0.534⋅R+p2x2+y2≤1502.The determination of the ideal working paraboloid is mainly to determine the parameterp, and the parameter determination of the p focal length is determined by the objective function, namely, formulas (8) and (9). In the determined range, the solution has a single extreme value. We use the simulated annealing algorithm to solve the solution. It is found that the p value is 280.36 m.


## 3.3. Comparison and Analysis of Fitting Results

### 3.3.1. Fitting Results

By comparing the FAST ideal spherical surface with the given ideal paraboloid, you can see that the absolute difference between the two is not large, and the difference is hardly visible with respect to the 300 m caliber. It means that the position of the main cable node needs to be changed and is small enough. And from the results, we can see that the middle fitting is better than the fitting at the edge.


### 3.3.2. Absolute Comprehensive Change

In order to quantitatively distinguish the base sphere and the ideal working paraboloid, we can find the error between sphere and paraboloidΔe, and its expression is as follows:(12)Δe=x2+y22p−0.534⋅R+p2−−R2−x2+y2.The difference between the paraboloid and the base sphere can be seen quantitatively, and theΔe-function image is shown in Figure 4.Figure 4
The error between sphere and paraboloid.It can be quantitatively seen from the image that the absolute error function does not exceed 0.45 m, and the largest error occurs at the center and edge of the paraboloid, where the absolute error of the paraboloid edge is 0.45 m. Ideally, this fitting is within the allowable range, so it is consistent in terms of fitting error.


### 3.3.3. Length Change of Main Cable

In addition to the requirements for the displacement index of the actuator, there are also requirements for the change of the length of the main cable. In order to reflect the change of the length of the main cable more reasonably, we use the relative radial changeΔa of the adjacent main cable nodes, and its formula is as follows:(13)Δa=∫−150+1501+x/p2dxRL⋅π/3−1L×100%.Besides, there is also the relative latitudinal variation, the formula of which is as follows:(14)Δo=max−p/tanθ+p2/tan2θ+2p0.534R+p/2−RsinθRsinθ,θ∈0,π6×100%,where L is the length of adjacent main cable nodes, p is the focal length of the working paraboloid, and R is the radius of the base sphere. After calculation, the relative radial variation is 0.011% and the relative latitudinal variation is 0.13%. It can be seen from this that the relative change obtained is very small. However, it should be noted that the relative change in the latitudinal direction is not on the same order of magnitude as the relative change in the radial direction. Therefore, it can adjust the direction of the down-tie cable of the actuator properly to compensate for the change in the latitudinal direction.


## 3.3.1. Fitting Results

By comparing the FAST ideal spherical surface with the given ideal paraboloid, you can see that the absolute difference between the two is not large, and the difference is hardly visible with respect to the 300 m caliber. It means that the position of the main cable node needs to be changed and is small enough. And from the results, we can see that the middle fitting is better than the fitting at the edge.


## 3.3.2. Absolute Comprehensive Change

In order to quantitatively distinguish the base sphere and the ideal working paraboloid, we can find the error between sphere and paraboloidΔe, and its expression is as follows:(12)Δe=x2+y22p−0.534⋅R+p2−−R2−x2+y2.The difference between the paraboloid and the base sphere can be seen quantitatively, and theΔe-function image is shown in Figure 4.Figure 4
The error between sphere and paraboloid.It can be quantitatively seen from the image that the absolute error function does not exceed 0.45 m, and the largest error occurs at the center and edge of the paraboloid, where the absolute error of the paraboloid edge is 0.45 m. Ideally, this fitting is within the allowable range, so it is consistent in terms of fitting error.


## 3.3.3. Length Change of Main Cable

In addition to the requirements for the displacement index of the actuator, there are also requirements for the change of the length of the main cable. In order to reflect the change of the length of the main cable more reasonably, we use the relative radial changeΔa of the adjacent main cable nodes, and its formula is as follows:(13)Δa=∫−150+1501+x/p2dxRL⋅π/3−1L×100%.Besides, there is also the relative latitudinal variation, the formula of which is as follows:(14)Δo=max−p/tanθ+p2/tan2θ+2p0.534R+p/2−RsinθRsinθ,θ∈0,π6×100%,where L is the length of adjacent main cable nodes, p is the focal length of the working paraboloid, and R is the radius of the base sphere. After calculation, the relative radial variation is 0.011% and the relative latitudinal variation is 0.13%. It can be seen from this that the relative change obtained is very small. However, it should be noted that the relative change in the latitudinal direction is not on the same order of magnitude as the relative change in the radial direction. Therefore, it can adjust the direction of the down-tie cable of the actuator properly to compensate for the change in the latitudinal direction.


## 4. Simulation and Adjustment Results of Actual Work

### 4.1. Determination of the Actual Working Mode Model

In practice, celestial objects are not always located directly above the FAST radio telescope, but at an angle to the FAST symmetry axis. Since every celestial body has a different angle with the direction above the FAST, the position of the corresponding adjustment surface is also distinct in actual observation. Therefore, in practice, it is necessary to determine different working planes and working methods.


### 4.2. Determination of Ideal Paraboloid in Practical Work

#### 4.2.1. Conversion of Cartesian Coordinate System

Although the position of the celestial body is not directly above FAST, but at a certain angle, because the sphere has axisymmetric and rotational symmetry, the overall Cartesian coordinates of FAST can be converted, so that the required paraboloid is normalized to get the standard paraboloid equation, simplifying the problem. The transformed coordinates are shown in Figure5.Figure 5
Schematic diagram after coordinate transformation.


#### 4.2.2. Determination of Transformation Matrix

As shown in Figure6 the azimuth angle of the celestial body S is the angle α between AC and the x-axis, and the elevation angle is the angle β between CS and CA. Taking the azimuth and elevation angles of the celestial body S as the benchmark, the movable coordinate system C−xyz fixed to the spherical center C is transformed to obtain the coordinate system C−x″y′z′.Figure 6
Cartesian coordinate transformation.
(a)(b)After the coordinate systemC−xyz is rotated counterclockwise by an angle a with the z-axis as the central axis of rotation, as shown in Figure 6(a), the coordinate changes of the points in the intermediate reference coordinate system C−x′y′z are obtained. The y-axis is the central axis of rotation. After rotating the coordinate system C−x′y′z clockwise by angle b, as shown in Figure 6(b), the coordinates C−x″y′z′ of the points in the coordinate system are obtained.Through two coordinate changes, the matrix operation can finally be transformed from coordinateC−xyz to coordinate C−x″y′z′:(15)p⟶″=A⋅p⟶.Among them,p⟶ is the coordinate vector before transformation, p⟶″ is the coordinate vector after transformation, and A is the total transformation matrix multiplied by the front and rear transformation matrices; that is, A=A2⋅A1 is(16)A=cosβ−90°cosαcosβ−90°sinαsinβ−90°−sinαcosα0−sinβ−90°cosα−sinβ−90°cosαcosβ−90°.


#### 4.2.3. Transformation Simulation Results

By transforming the actual main cable node from the coordinates in the coordinate systemC−xyz to the coordinates in the coordinate system C−x″y′z′ through formula (15), the coordinates of the main cable node in the coordinate system C−x″y′z′ are obtained as shown in Figure 7. After the transformation, the center of the working area of the coordinate system passes through the vertical axis of the coordinate system and takes the vertical axis as the axis of symmetry, so that the equation of the working paraboloid is the standard equation in this coordinate system.Figure 7
Coordinate diagram of each main index node after transformation.


#### 4.2.4. Determination of Ideal Spherical Equation

Since the positions of the main cable nodes are not standard on the same standard spherical surface, the ideal spherical surface can be determined according to the least squares method. The principle is shown in Figure8. The absolute value of the difference between the z-axis coordinates of each main cable node and the point corresponding to the ideal spherical surface is added, and the ideal spherical surface is obtained which makes the sum of the absolute values of each node difference the smallest, which is(17)min∑i=1Nzi′−z,z=−R′−x″i2+y′i2..Figure 8
Least squares method to find ideal sphere.Thus, the radius of the ideal sphere isR  = 300.8 m. Thus, the coordinate equation of the ideal sphere is obtained.


#### 4.2.5. Determination of Ideal Working Paraboloid

On the working surface of the ideal spherical surface with the parameters determined by formula (7), the simulated annealing algorithm is used to solve the ideal working paraboloid and the parameters can be obtained to obtain the corresponding ideal working paraboloid equation:(18)z′=x″2+y′2562.4−301.2272x″2+y′2≤1502.Among them, the focal length of the working paraboloid isp = 281.2 m, and the ideal radius of the base sphere is R’ = 300.8 m.All the main cable nodes of FAST and the ideal working paraboloid after the parameters are determined are shown in Figure9. From the figure, it can be roughly judged that the fitting is very close.Figure 9
All main cable nodes of FAST and ideal working paraboloid.


### 4.3. How the Actuator Works

Figure10 shows the working schematic diagram of the actuator; the main cable node can only move up and down in the pull-down direction of the actuator. The actuator adjusts the position of the main cable node in the pull-down direction to adjust the deformation of the reflective panel, and the length of the main cable is only slightly deformed.Figure 10
Schematic diagram of the actuator.The corresponding discrete paraboloid of actual work can be obtained by pulling the corresponding main cable node as far as possible into the plane through which the determined ideal working parabola passes through the pull-down cable.


## 4.1. Determination of the Actual Working Mode Model

In practice, celestial objects are not always located directly above the FAST radio telescope, but at an angle to the FAST symmetry axis. Since every celestial body has a different angle with the direction above the FAST, the position of the corresponding adjustment surface is also distinct in actual observation. Therefore, in practice, it is necessary to determine different working planes and working methods.


## 4.2. Determination of Ideal Paraboloid in Practical Work

### 4.2.1. Conversion of Cartesian Coordinate System

Although the position of the celestial body is not directly above FAST, but at a certain angle, because the sphere has axisymmetric and rotational symmetry, the overall Cartesian coordinates of FAST can be converted, so that the required paraboloid is normalized to get the standard paraboloid equation, simplifying the problem. The transformed coordinates are shown in Figure5.Figure 5
Schematic diagram after coordinate transformation.


### 4.2.2. Determination of Transformation Matrix

As shown in Figure6 the azimuth angle of the celestial body S is the angle α between AC and the x-axis, and the elevation angle is the angle β between CS and CA. Taking the azimuth and elevation angles of the celestial body S as the benchmark, the movable coordinate system C−xyz fixed to the spherical center C is transformed to obtain the coordinate system C−x″y′z′.Figure 6
Cartesian coordinate transformation.
(a)(b)After the coordinate systemC−xyz is rotated counterclockwise by an angle a with the z-axis as the central axis of rotation, as shown in Figure 6(a), the coordinate changes of the points in the intermediate reference coordinate system C−x′y′z are obtained. The y-axis is the central axis of rotation. After rotating the coordinate system C−x′y′z clockwise by angle b, as shown in Figure 6(b), the coordinates C−x″y′z′ of the points in the coordinate system are obtained.Through two coordinate changes, the matrix operation can finally be transformed from coordinateC−xyz to coordinate C−x″y′z′:(15)p⟶″=A⋅p⟶.Among them,p⟶ is the coordinate vector before transformation, p⟶″ is the coordinate vector after transformation, and A is the total transformation matrix multiplied by the front and rear transformation matrices; that is, A=A2⋅A1 is(16)A=cosβ−90°cosαcosβ−90°sinαsinβ−90°−sinαcosα0−sinβ−90°cosα−sinβ−90°cosαcosβ−90°.


### 4.2.3. Transformation Simulation Results

By transforming the actual main cable node from the coordinates in the coordinate systemC−xyz to the coordinates in the coordinate system C−x″y′z′ through formula (15), the coordinates of the main cable node in the coordinate system C−x″y′z′ are obtained as shown in Figure 7. After the transformation, the center of the working area of the coordinate system passes through the vertical axis of the coordinate system and takes the vertical axis as the axis of symmetry, so that the equation of the working paraboloid is the standard equation in this coordinate system.Figure 7
Coordinate diagram of each main index node after transformation.


### 4.2.4. Determination of Ideal Spherical Equation

Since the positions of the main cable nodes are not standard on the same standard spherical surface, the ideal spherical surface can be determined according to the least squares method. The principle is shown in Figure8. The absolute value of the difference between the z-axis coordinates of each main cable node and the point corresponding to the ideal spherical surface is added, and the ideal spherical surface is obtained which makes the sum of the absolute values of each node difference the smallest, which is(17)min∑i=1Nzi′−z,z=−R′−x″i2+y′i2..Figure 8
Least squares method to find ideal sphere.Thus, the radius of the ideal sphere isR  = 300.8 m. Thus, the coordinate equation of the ideal sphere is obtained.


### 4.2.5. Determination of Ideal Working Paraboloid

On the working surface of the ideal spherical surface with the parameters determined by formula (7), the simulated annealing algorithm is used to solve the ideal working paraboloid and the parameters can be obtained to obtain the corresponding ideal working paraboloid equation:(18)z′=x″2+y′2562.4−301.2272x″2+y′2≤1502.Among them, the focal length of the working paraboloid isp = 281.2 m, and the ideal radius of the base sphere is R’ = 300.8 m.All the main cable nodes of FAST and the ideal working paraboloid after the parameters are determined are shown in Figure9. From the figure, it can be roughly judged that the fitting is very close.Figure 9
All main cable nodes of FAST and ideal working paraboloid.


## 4.2.1. Conversion of Cartesian Coordinate System

Although the position of the celestial body is not directly above FAST, but at a certain angle, because the sphere has axisymmetric and rotational symmetry, the overall Cartesian coordinates of FAST can be converted, so that the required paraboloid is normalized to get the standard paraboloid equation, simplifying the problem. The transformed coordinates are shown in Figure5.Figure 5
Schematic diagram after coordinate transformation.


## 4.2.2. Determination of Transformation Matrix

As shown in Figure6 the azimuth angle of the celestial body S is the angle α between AC and the x-axis, and the elevation angle is the angle β between CS and CA. Taking the azimuth and elevation angles of the celestial body S as the benchmark, the movable coordinate system C−xyz fixed to the spherical center C is transformed to obtain the coordinate system C−x″y′z′.Figure 6
Cartesian coordinate transformation.
(a)(b)After the coordinate systemC−xyz is rotated counterclockwise by an angle a with the z-axis as the central axis of rotation, as shown in Figure 6(a), the coordinate changes of the points in the intermediate reference coordinate system C−x′y′z are obtained. The y-axis is the central axis of rotation. After rotating the coordinate system C−x′y′z clockwise by angle b, as shown in Figure 6(b), the coordinates C−x″y′z′ of the points in the coordinate system are obtained.Through two coordinate changes, the matrix operation can finally be transformed from coordinateC−xyz to coordinate C−x″y′z′:(15)p⟶″=A⋅p⟶.Among them,p⟶ is the coordinate vector before transformation, p⟶″ is the coordinate vector after transformation, and A is the total transformation matrix multiplied by the front and rear transformation matrices; that is, A=A2⋅A1 is(16)A=cosβ−90°cosαcosβ−90°sinαsinβ−90°−sinαcosα0−sinβ−90°cosα−sinβ−90°cosαcosβ−90°.


## 4.2.3. Transformation Simulation Results

By transforming the actual main cable node from the coordinates in the coordinate systemC−xyz to the coordinates in the coordinate system C−x″y′z′ through formula (15), the coordinates of the main cable node in the coordinate system C−x″y′z′ are obtained as shown in Figure 7. After the transformation, the center of the working area of the coordinate system passes through the vertical axis of the coordinate system and takes the vertical axis as the axis of symmetry, so that the equation of the working paraboloid is the standard equation in this coordinate system.Figure 7
Coordinate diagram of each main index node after transformation.


## 4.2.4. Determination of Ideal Spherical Equation

Since the positions of the main cable nodes are not standard on the same standard spherical surface, the ideal spherical surface can be determined according to the least squares method. The principle is shown in Figure8. The absolute value of the difference between the z-axis coordinates of each main cable node and the point corresponding to the ideal spherical surface is added, and the ideal spherical surface is obtained which makes the sum of the absolute values of each node difference the smallest, which is(17)min∑i=1Nzi′−z,z=−R′−x″i2+y′i2..Figure 8
Least squares method to find ideal sphere.Thus, the radius of the ideal sphere isR  = 300.8 m. Thus, the coordinate equation of the ideal sphere is obtained.


## 4.2.5. Determination of Ideal Working Paraboloid

On the working surface of the ideal spherical surface with the parameters determined by formula (7), the simulated annealing algorithm is used to solve the ideal working paraboloid and the parameters can be obtained to obtain the corresponding ideal working paraboloid equation:(18)z′=x″2+y′2562.4−301.2272x″2+y′2≤1502.Among them, the focal length of the working paraboloid isp = 281.2 m, and the ideal radius of the base sphere is R’ = 300.8 m.All the main cable nodes of FAST and the ideal working paraboloid after the parameters are determined are shown in Figure9. From the figure, it can be roughly judged that the fitting is very close.Figure 9
All main cable nodes of FAST and ideal working paraboloid.


## 4.3. How the Actuator Works

Figure10 shows the working schematic diagram of the actuator; the main cable node can only move up and down in the pull-down direction of the actuator. The actuator adjusts the position of the main cable node in the pull-down direction to adjust the deformation of the reflective panel, and the length of the main cable is only slightly deformed.Figure 10
Schematic diagram of the actuator.The corresponding discrete paraboloid of actual work can be obtained by pulling the corresponding main cable node as far as possible into the plane through which the determined ideal working parabola passes through the pull-down cable.


## 5. The Receiving Efficiency of the Actual Working Paraboloid

In the case of an ideal base sphere and a working paraboloid, the electromagnetic waves reflected in the working paraboloid should all converge on the feed cabin. Under ideal conditions, all electromagnetic waves in the ideal working paraboloid should be absorbed by the feed cabin, but this is not the case in reality. As shown in Figure11, the working paraboloid is not ideally smooth and continuous, but discrete and close to the ideal working paraboloid, which is suitable for practical applications. For the discrete case, we have to define a model for evaluating the receiving efficiency to compare the receiving efficiency before and after deformation.Figure 11
Schematic diagram of actual reception.


### 5.1. The Actual Working Paraboloid Receiving Efficiency Model

The location information of each store is obtained by fitting the ideal paraboloid obtained, and the ratio of the signal received from the reflective panel in the effective area with a diameter of 1 m of the feed cabin and the signal reflected by the reflective surface with a diameter of 300 m is calculated. The acceptance ratio of the working parabolic reflective area to the reference reflective spherical area is compared. As shown in Figure12, the receiving ratio I is the ratio of the electromagnetic wave intensity received by the effective range of the feed cabin to the total intensity. In order to simplify the model, we assume that the received parallel electromagnetic wave intensity is uniform, and there is no energy consumption in the process of propagation and reflection. Then, the total reflected parallel electromagnetic waves are also uniform. Assume that the intensity surface density on the vertical plane parallel to the reflection is c. Assume that on the electromagnetic wave reflected by a certain reflector, since the intensity is uniform, c is a constant, so there is a feedback. The intensity received in the effective area of the source cabin is(19)Ii=c⋅Smix⋅cosθi,where Smix represents the effective total equivalent receiving area, which represents the intersection area of the effective range and the reflected parallel light. Stotal is the projected area of the intersection area in the direction of the electromagnetic wave. And the total intensity in the projection direction is(20)Iit=c⋅Stotal,where Stotal represents the total area projected by the electromagnetic wave in the projection direction. Combining Ii and Iit, the receiving ratio Q can be obtained from the geometric properties, and the normal vector n⟶i and the direction vector λ⟶i about each surface can be easily obtained from Figure 12, where b⟶ and λ⟶i are the direction vectors of the incoming and outgoing electromagnetic waves, respectively, n⟶0 and n⟶i are the normal vectors of the receiving plane and the reflector, respectively, and p⟶1i and p⟶2i are any two pointing vectors of the three endpoints of the i-th plane. Among them, the directions of b⟶ and n⟶i are parallel, and the angle between the reflector and the receiving plane and the outgoing electromagnetic wave can be obtained. The actual working paraboloid receiving efficiency model can be obtained:(21)Q=∑i=1nπr2cosθi∑i=1nSΔcosαi,where(22)cosθi=n⟶0λ⟶in⟶0×λ⟶i,cosαi=n⟶iλ⟶in⟶i×λ⟶i..Figure 12
Schematic diagram of the actual receiving efficiency ratio.The actual receiving efficiency of FAST before and after deformation can be obtained by traversing and solving. When solving, the pointing vectors ofp⟶1i and p⟶2i can be obtained by traversing the coordinates of each fitting point obtained by each solution. Afterward, cosθi and cosαi can be found using formula (22). Finally, the receiving ratio Q can be obtained.For the spherical surface, the spherical coordinates are very similar to the parabolic coordinates due to the fitting results. Therefore, although the above are all methods for solving the acceptance ratioQ based on the fitting parabolic coordinates, they can still be used to bring the reference spherical coordinates into the fitting solution.The acceptance ratio for fitting parabolic coordinates isQ1 = 0.0159.The acceptance ratio for the base spherical coordinates isQ2 = 0.0059.This fitted paraboloid can obtain the largest possible acceptance ratio within the smallest deformation. Although the basic shape is not displaced, the relative increase in the acceptance ratio over the reference sphere is evident.


### 5.2. The Accuracy of the Overall Shape of the Reflector Is Improved

Theoretically, the whole working surface is a sphere before deformation and a paraboloid after deformation. But in reality, neither is an ideal spherical or parabolic surface, but is similar to a parabolic surface. For the sake of quantitative description, the distance between the discrete point previous deformation and the corresponding point on the ideal paraboloid in the actual working face can be divided by the distance between the discrete point after deformation and the point. In this way, the relative accuracy improvement value can be obtained. It can be found that the overall accuracy after the actual deformation is improved by 23.1% by averaging the relative accuracy improvement value in the working face. The absolute precision of the working area after deformation is within 0.014 m. This clearly conforms to the twenty times wavelength principle. Figure13 shows the accuracy improvement of each plane. It can be seen from the figure that the accuracy improvement of the central area is not very high, most of which are only about 10%, because the ideal sphere and the ideal paraboloid are very close. The point where the efficiency is abnormally high is due to the fact that some of the discrete main cable nodes deviate severely from the ideal sphere, which is an accidental elevation anomaly. In most of the peripheral areas, the accuracy is improved by about 20%, and the improvement effect at the edge is the best, up to about 30%.Figure 13
Distribution grayscale image of accuracy improvement in the working area.


## 5.1. The Actual Working Paraboloid Receiving Efficiency Model

The location information of each store is obtained by fitting the ideal paraboloid obtained, and the ratio of the signal received from the reflective panel in the effective area with a diameter of 1 m of the feed cabin and the signal reflected by the reflective surface with a diameter of 300 m is calculated. The acceptance ratio of the working parabolic reflective area to the reference reflective spherical area is compared. As shown in Figure12, the receiving ratio I is the ratio of the electromagnetic wave intensity received by the effective range of the feed cabin to the total intensity. In order to simplify the model, we assume that the received parallel electromagnetic wave intensity is uniform, and there is no energy consumption in the process of propagation and reflection. Then, the total reflected parallel electromagnetic waves are also uniform. Assume that the intensity surface density on the vertical plane parallel to the reflection is c. Assume that on the electromagnetic wave reflected by a certain reflector, since the intensity is uniform, c is a constant, so there is a feedback. The intensity received in the effective area of the source cabin is(19)Ii=c⋅Smix⋅cosθi,where Smix represents the effective total equivalent receiving area, which represents the intersection area of the effective range and the reflected parallel light. Stotal is the projected area of the intersection area in the direction of the electromagnetic wave. And the total intensity in the projection direction is(20)Iit=c⋅Stotal,where Stotal represents the total area projected by the electromagnetic wave in the projection direction. Combining Ii and Iit, the receiving ratio Q can be obtained from the geometric properties, and the normal vector n⟶i and the direction vector λ⟶i about each surface can be easily obtained from Figure 12, where b⟶ and λ⟶i are the direction vectors of the incoming and outgoing electromagnetic waves, respectively, n⟶0 and n⟶i are the normal vectors of the receiving plane and the reflector, respectively, and p⟶1i and p⟶2i are any two pointing vectors of the three endpoints of the i-th plane. Among them, the directions of b⟶ and n⟶i are parallel, and the angle between the reflector and the receiving plane and the outgoing electromagnetic wave can be obtained. The actual working paraboloid receiving efficiency model can be obtained:(21)Q=∑i=1nπr2cosθi∑i=1nSΔcosαi,where(22)cosθi=n⟶0λ⟶in⟶0×λ⟶i,cosαi=n⟶iλ⟶in⟶i×λ⟶i..Figure 12
Schematic diagram of the actual receiving efficiency ratio.The actual receiving efficiency of FAST before and after deformation can be obtained by traversing and solving. When solving, the pointing vectors ofp⟶1i and p⟶2i can be obtained by traversing the coordinates of each fitting point obtained by each solution. Afterward, cosθi and cosαi can be found using formula (22). Finally, the receiving ratio Q can be obtained.For the spherical surface, the spherical coordinates are very similar to the parabolic coordinates due to the fitting results. Therefore, although the above are all methods for solving the acceptance ratioQ based on the fitting parabolic coordinates, they can still be used to bring the reference spherical coordinates into the fitting solution.The acceptance ratio for fitting parabolic coordinates isQ1 = 0.0159.The acceptance ratio for the base spherical coordinates isQ2 = 0.0059.This fitted paraboloid can obtain the largest possible acceptance ratio within the smallest deformation. Although the basic shape is not displaced, the relative increase in the acceptance ratio over the reference sphere is evident.


## 5.2. The Accuracy of the Overall Shape of the Reflector Is Improved

Theoretically, the whole working surface is a sphere before deformation and a paraboloid after deformation. But in reality, neither is an ideal spherical or parabolic surface, but is similar to a parabolic surface. For the sake of quantitative description, the distance between the discrete point previous deformation and the corresponding point on the ideal paraboloid in the actual working face can be divided by the distance between the discrete point after deformation and the point. In this way, the relative accuracy improvement value can be obtained. It can be found that the overall accuracy after the actual deformation is improved by 23.1% by averaging the relative accuracy improvement value in the working face. The absolute precision of the working area after deformation is within 0.014 m. This clearly conforms to the twenty times wavelength principle. Figure13 shows the accuracy improvement of each plane. It can be seen from the figure that the accuracy improvement of the central area is not very high, most of which are only about 10%, because the ideal sphere and the ideal paraboloid are very close. The point where the efficiency is abnormally high is due to the fact that some of the discrete main cable nodes deviate severely from the ideal sphere, which is an accidental elevation anomaly. In most of the peripheral areas, the accuracy is improved by about 20%, and the improvement effect at the edge is the best, up to about 30%.Figure 13
Distribution grayscale image of accuracy improvement in the working area.


## 6. Discussion

The influence of the optimal adjustment of FAST active reflector on the receiving efficiency is discussed. And a mathematical model for optimizing FAST working paraboloid is established, including considering radial displacement, zonal displacement, overall continuity and integrity, and the optimization of overall results. In the process of optimization, firstly, in the ideal case, the optimal working paraboloid in all aspects is considered, the constraints of nonlinear programming in all aspects are given, and the objective function is established through the actual requirements. The nonlinear programming model is solved by simulated annealing algorithm to obtain the equation of the ideal working paraboloid. Secondly, after obtaining the ideal working paraboloid, the comprehensive indexes in the ideal state are evaluated and the optimization suggestions are given. The results show that the relative changes of radial displacement and latitudinal displacement of the ideal working paraboloid are very small, but the relative changes of latitudinal displacement are larger than those of radial displacement, which indicates that in the actual situation, a certain compensation from latitudinal displacement to radial displacement is needed. Thirdly, from the practical point of view, the actual main cable node is fitted. The ideal working surface is generalized and discussed in the transformed Cartesian coordinate system, and then the actual basic spherical equation is determined by the least square method. The ideal working paraboloid equation is determined by the established optimization model of working paraboloid. Using the established actual ideal working paraboloid equation and the working mode of the actual actuator, the actual discrete working paraboloid and the relative position of each main cable node are obtained. Finally, in order to evaluate the reception efficiency, a model for determining the actual FAST reception efficiency ratio is established and measured the improvement of the accuracy close to the determined ideal parabolic before and after deformation.The analysis results show that the efficiency of the work area before and after deformation is improved. The efficiency simulation results show that the receiving efficiency is only 5.9% when the position of the main cable node is unchanged. After the position of the main cable node is changed according to the fitting ideal paraboloid method, the total receiving efficiency of all the reflectors is 15.9%. The previous and after reception efficiency which are 5.9% and 15.9% can be the Q value in equation20. According to the actual simulation, it can be seen that even when the relative position of the main cable node is basically unchanged, the factual receiving efficiency after optimization is still much higher than that of the working face before optimization. The accuracy of the deformed surface in the working area relative to the ideal paraboloid is within 0.014 m, and this error is within the limit of the 20 times wavelength principle. The actual acceptance efficiency of FAST is about 80%, which is different from the acceptance model established in this study. The reason is guessed as the inconsistent direction of the model used to measure the efficiency, and the shape of the FAST reflector and various details related to the improvement of efficiency are not fully taken into account in this model. In summary, this study provides a feasible mathematical model to some extent, but there are still some difficulties in using the model established in this paper for practical purposes, and there are still many problems to be solved and faced.


## 7. Conclusion

In this study, the ideal polishing surface of deformation in the working area is obtained by establishing a mathematical model. Based on the actual situation, the method of distortion in the working area is given. The simulation is carried out, and a model for calculating the acceptance efficiency is established. The actual acceptance efficiency and the accuracy of fitting the ideal paraboloid to the distorted working surface are calculated according to the actual working surface. The conclusions are as follows:(1)
The focal length of the ideal paraboloid is p 280.36 m whenR = 300 m(2)
The acceptance efficiency of all reflective surfaces of FAST is Q1 5.9% before deformation andQ2 15.9% after deformation(3)
The accuracy of fitting the ideal paraboloid after deformation is improved by 23.1% compared with that before deformation(4)
The absolute value of the accuracy after deformation in the working area is within 0.014 m



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*Source: 8342015-2022-09-01.xml*