Thermal deformation optimization method for carbon fiber parabolic reflector with honeycomb sandwich structure

Abstract

The invention provides a thermal deformation optimization method for a carbon fiber parabolic reflector of a honeycomb sandwich structure. The thermal deformation optimization method comprises the following implementation steps: constructing a reflector model; determining a constraint interval of structural parameters of the reflector model; calculating the root-mean-square of thermal deformation of the working surface of the reflector model under the high-temperature and low-temperature working conditions; and constructing a reflector optimization model and solving the reflector optimization model based on an ant lion optimization algorithm. According to the parameterized model constructed by the method, the time for re-modeling in the optimization process is shortened, and the optimization efficiency is improved; the reflector optimization model is solved based on the ant lion optimization algorithm to obtain an optimal solution, the defect that in the prior art, the optimization precision of an optimization result obtained by changing a single structure parameter for multiple times of calculation and then taking heat deformation to achieve the minimum root mean square is low is overcome, and the optimization precision is improved; and determining the constraint interval of the structural parameters of the parameterized model of the reflector, thereby avoiding the defects of overlarge optimization variable range and low optimization precision in the prior art, and improving the optimization efficiency and precision.

Description

Technical Field

[0001] The invention belongs to the technical field of aerospace reflectors and relates to a method for optimizing thermal deformation of a carbon fiber parabolic reflector with a honeycomb sandwich structure, which can be applied to satellite antennas. Background Art

[0002] With the continuous development of satellite communication technology, in order to meet the needs of observations in farther and smaller spaces, people's requirements for the accuracy of satellite antennas are also increasing day by day. However, as people's demand for antenna accuracy increases, the aperture of optical telescopes will become larger and larger, and the corresponding volume and weight will continue to increase. When subjected to other external loads such as gravity, temperature, and radiation, they will undergo large deformations, thereby increasing the difficulty of controlling the surface accuracy. In recent years, with the continuous development of materials science, carbon fiber composite materials have gradually been used in various space satellites as honeycomb sandwich structures of high-precision antenna structures due to their high specific strength and high specific modulus.

[0003] The carbon fiber parabolic reflector with honeycomb sandwich structure is a high-precision solid surface reflector, which includes a honeycomb core composed of multiple honeycomb core cells, a skin covering the inside and outside of the honeycomb core, and a film bonding the honeycomb core and the inner and outer skins. The skin material is generally a carbon fiber composite material, and the honeycomb core includes an aluminum honeycomb core and a carbon fiber honeycomb core. In order to meet the high-precision requirements of satellite-borne antennas, scholars have done a lot of research on the thermal deformation optimization analysis of carbon fiber parabolic reflectors with honeycomb sandwich structures.

[0004] At present, the technical idea of ​​thermal deformation optimization of carbon fiber parabolic reflector with honeycomb sandwich structure is to build a reflector model, optimize the structural parameters of the reflector model with the goal of minimizing the root mean square of thermal deformation of the reflector working surface under high and low temperature conditions, and finally obtain a smaller root mean square of thermal deformation. The method of building a reflector model is often a general modeling method such as directly modeling in finite element software or modeling in professional modeling software and then importing it into finite element analysis software. Subsequent analysis and calculation require meshing of the model, setting loads and boundary conditions, and often selecting structural parameters such as the side length and height of the honeycomb core cell, the number of skin layers and the angle of the layers as optimization variables. The optimization method is often to change a single structural parameter of the model for multiple groups of calculation experiments, and take the structural parameters when the root mean square of thermal deformation is optimal as the optimization result. However, each change in the structural parameters requires re-modeling, the optimization efficiency is low, and multiple structural parameters are not optimized as optimization variables at the same time, and the value range of the optimization variables is not properly constrained, which affects the improvement of optimization accuracy and efficiency. Summary of the invention

[0005] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and propose a thermal deformation optimization method for a carbon fiber parabolic reflector with a honeycomb sandwich structure to solve the technical problems of low optimization accuracy and efficiency in the prior art.

[0006] To achieve the above object, the technical solution adopted by the present invention includes the following steps:

[0007] (1) Constructing a parameterized reflector model:

[0008] A carbon fiber parabolic honeycomb sandwich structure consisting of an intermediate layer and skins bonded to the inner and outer surfaces of the intermediate layer by adhesive films distributed in a spatial rectangular coordinate system O-XYZ is constructed, and its working surface is meshed, and then the structural parameters and mechanical parameters of multiple regular hexagonal carbon fiber honeycomb core cells constituting the intermediate layer and multiple plies constituting the inner and outer skins, as well as the coordinates of N working surface nodes after meshing are written into an inp file format to form a reflector parameterized model;

[0009] (2) Determine the constraint interval of the reflector parameterized model structure parameters:

[0010] Determine the upper limit l of the side length l of the honeycomb core cell in the reflector parameterized model under the condition of pressure P uper and the lower limit l lower , the upper limit of height h under gravity conditions uper and lower limit h lower , and the angles of the Q layers ply=[ply1,ply2…,ply q ,…,ply Q ], where ply q is the ply angle of the qth ply;

[0011] (3) Calculate the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions:

[0012] Calculate at high temperature T h Low temperature is T l RMS of thermal deformation of the working surface of the reflector parameterized model under working conditions h and RMS l ;

[0013] (4) Constructing reflector optimization model:

[0014] Construct the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions h and RMS l As the optimization target, a reflector optimization model is established with the structural parameters of the honeycomb sandwich structure as optimization variables and the constraint interval determined in step (2) as a condition;

[0015] (5) Obtain the optimization results of the reflector optimization model:

[0016] The reflector optimization model is solved based on the ant lion optimization algorithm, and the minimum root mean square of thermal deformation of the working surface under high and low temperature conditions is obtained as RMS hu and RMS lu Carbon fiber parabolic reflector.

[0017] Compared with the prior art, the present invention has the following advantages:

[0018] (1) The present invention constructs a reflector parameterized model in which the structural parameters and mechanical parameters of a plurality of regular hexagonal carbon fiber honeycomb core cells constituting an intermediate layer and a plurality of plies constituting inner and outer skins, as well as the coordinates of the working surface nodes after meshing are written in the inp file format. The reflector model with different parameters can be quickly rebuilt by changing the input parameters of the inp file, thereby reducing the time for remodeling during the optimization process and effectively improving the optimization efficiency.

[0019] (2) The present invention solves the reflector optimization model based on the ant lion optimization algorithm to obtain the optimal solution, avoiding the defect of the prior art that the optimization result is obtained by changing a single structural parameter, performing multiple calculations and then taking the minimum root mean square of thermal deformation, that is, the low optimization accuracy caused by artificially given experimental values, and effectively improves the optimization accuracy.

[0020] (3) The present invention determines the constraint range of the structural parameters of the reflector parameterized model through the solidification deformation and self-weight deformation of the reflector parameterized model and the actual engineering situation, thereby narrowing the range of optimization variables and thus reducing the optimization workload. At the same time, it avoids the defects of the prior art that the optimization variable range is too large and the optimization accuracy is not high, and further improves the optimization efficiency and accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a flow chart for implementing the present invention.

[0022] Figure 2 A line graph showing the maximum deformation of the skin of the reflector model constructed in the present invention changing with the side length of the honeycomb core cell under the condition of a curing pressure of 0.15 MPa.

[0023] Figure 3 The line graph of the reflector model constructed in the present invention showing the root mean square of the self-weight deformation of the working surface under gravity conditions as the height of the honeycomb core cell changes.

[0024] Figure 4 This is a graph showing how the fitness of each ant changes with the number of iterations when the ant lion optimization algorithm of the present invention optimizes the root mean square of thermal deformation of the reflector working surface. DETAILED DESCRIPTION

[0025] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0026] Reference Figure 1 , the present invention comprises the following steps:

[0027] Step 1) Build a reflector model:

[0028] A carbon fiber parabolic honeycomb sandwich structure consisting of an intermediate layer and skins bonded to the inner and outer surfaces of the intermediate layer by adhesive films distributed in a spatial rectangular coordinate system O-XYZ is constructed, and its working surface is meshed. Then, the structural parameters and mechanical parameters of multiple regular hexagonal carbon fiber honeycomb core cells constituting the intermediate layer and multiple plies constituting the inner and outer skins, as well as the coordinates of the N working surface nodes after meshing are written into an inp file format through Python language to form a parametric model of the reflector; wherein, the structural parameters include the side length l and height h of the honeycomb core cell, the number of plies Q and the ply angle ply of the plies in the inner and outer skins; the mechanical parameters include the density ρ of the honeycomb core cell core and the density of the inner and outer skins ρ sheet .

[0029] This embodiment:

[0030] A carbon fiber parabolic honeycomb sandwich structure consisting of an intermediate layer and skins bonded to the inner and outer surfaces of the intermediate layer by adhesive films distributed in a spatial rectangular coordinate system O-XYZ is constructed, and the working surface is divided into tetrahedral grids. Then, the structural parameters and mechanical parameters of multiple regular hexagonal carbon fiber honeycomb core cells constituting the intermediate layer and multiple plies constituting the inner and outer skins, as well as the coordinates of the N working surface nodes after grid division are written into inp file format through Python language to form a parametric model of the reflector; wherein, the structural parameters include the side length l and height h of the honeycomb core cell, the number of plies Q and the ply angle ply of the plies in the inner and outer skins; the mechanical parameters include the density ρ of the honeycomb core cell core and the density of the inner and outer skins ρ sheet , inp file can realize the rapid reconstruction of reflector models with different parameters by changing the input parameters.

[0031] Step 2) Determine the constraint interval of the reflector model structural parameters:

[0032] Determine the upper limit l of the side length l of the honeycomb core cell in the reflector parameterized model under the condition of pressure P uper and the lower limit l lower , the upper limit of height h under gravity conditions uper and lower limit h lower , and the angles of the Q layers ply=[ply1,ply2…,ply q ,…,ply Q ], where plyq is the ply angle of the qth ply. The inner and outer skins are designed for symmetrical ply laying, that is, the ply angles of the 2Q inner and outer skins are [ply1, ply2…, ply q ,…,ply Q ,ply Q ,…,ply q ,…ply2,ply1];

[0033] The finite element software Abaqus6.14 is used to calculate the J sides of the honeycomb core cell in the reflector parameterized model under the working condition of pressure P, which are l=[l1,l2…,l j ,…,l J ] corresponds to the maximum deformation of the skin Δu=[Δu1,Δu2Ω,Δu j ,…,Δu J ], and draw a line graph of Δu changing with l, and then use the minimum side length l1 as the lower limit l lower , the side length value l of the honeycomb core cell when Δu increases sharply in the line graph uper As an upper limit, where Δu j Indicates l j The corresponding maximum deformation of the skin.

[0034] The K heights of the honeycomb core cells in the parametric model of the reflector under gravity conditions are calculated by finite element calculation software h=[h1,h2Ω,h k ,…,h K ]Corresponding working surface self-weight deformation root mean square RMS z =[RMS z1 ,RMS z2 Ω,RMS zk ,…,RMS zK ] and plot the RMS z A line chart that changes with h1, and then RMS in the line chart z Equal to the profile test requirement RMS cl The height of the honeycomb core cell is h lower As the lower limit, the upper limit of the working surface density design index ρ ZBsurface The calculated maximum height is taken as the upper limit h uper , where RMS zk Indicates h k The corresponding working surface self-weight deformation root mean square, h uper The calculation formula is:

[0035] h uper =max(h)

[0036]

[0037] M=ρZBsurface ·S

[0038]

[0039] V sheet =0.52·Q·t m 10 7

[0040] Where M and S are the weight and aperture area of ​​the reflector respectively, V sheet is the total volume of the inner and outer skins, ρ * is the equivalent density of the honeycomb core cell, V core is the envelope volume of the honeycomb core cell, t is the wall thickness of the honeycomb core cell, t m The thickness of each ply of the skin.

[0041] The angle of Q plies, where the ply angle of each q plies is q The value of is determined through quasi-isotropic ply design principles and engineering manufacturing practice.

[0042] This embodiment:

[0043] The finite element software Abaqus6.14 is used to calculate the maximum deformation of the skin Δu = [Δu1, Δu2, Δu3, Δu4, Δu5, Δu6] corresponding to the six increasing side lengths l = [5, 11, 17, 23, 29, 34] (mm) of the honeycomb core cell in the reflector parametric model under the working condition of 0.15 MPa, and a line graph of Δu changing with l is drawn as shown below: Figure 2 , and then take the minimum side length of 5mm as the lower limit l lower , 5mm is the minimum side length of the honeycomb core cell in practical engineering applications. The side length of the honeycomb core cell when Δu increases sharply in the line graph is 20mm as the upper limit l uper .

[0044] The finite element software Abaqus6.14 is used to calculate the root mean square RMS of the self-weight deformation of the working surface corresponding to the six heights h = [15, 25, 35, 45, 55, 65] (mm) of the honeycomb core cells in the parametric model of the reflector under gravity conditions. z =[RMS z1 ,RMS z2 ,RMS z3 ,RMS z4 ,RMS z5 ,RMS z6 ] and plot the RMS z The line graph that changes with h is as follows Figure 3 , and then put RMS in the line graph z Equal to the profile test requirement (RMS cl=2.1μm) when the height of the honeycomb core cell is 45mm as the lower limit h lower The maximum height calculated by the upper limit of the surface density design index is taken as the upper limit h uper =197.4mm,h uper The calculation formula is:

[0045] h uper =max(h)

[0046]

[0047] M=ρ ZBsurface ·S

[0048]

[0049] V sheet =0.52·Q·t m 10 7

[0050] Substitute the specific value t m =0.1mm, Q=8, t=0.4mm, 5mm≤l≤20mm, ρ ZBsurface =10kg / m 2 , S=2.591m 2 , ρ sheet =1.7*10 -9 t / mm 3 , ρ core =1.58*10 -9 t / mm 3 , when l is 20mm, h is the largest, that is, h uper =max(h)=197.4mm; finally determine the upper limit of the side length l of the honeycomb core cell in the reflector parameterized model under the working condition of 0.15MPa pressure uper =20mm and lower limit l lower =5mm, the upper limit of height h under gravity condition uper =148.3mm and lower limit h lower =45mm.

[0051] The angle of Q layers ply=[ply1,ply2…,ply q ,…,ply Q ], where the ply angle of each q plies is qThe value of is determined by the quasi-isotropic ply design principle and engineering manufacturing practice. Quasi-isotropy means that the material is isotropic only in a certain plane, that is, the material strength and stiffness are equal in all directions in the plane of the part, but there may be differences on the vertical plane. In order to achieve quasi-isotropy, the ply angles are usually 0°, 90°, +45° and -45°. The 0° layer provides axial strength and stiffness, the + / -45° layer provides shear and torsional strength and stiffness, and the 90° layer provides lateral strength and stiffness. At the same time, combined with engineering manufacturing practice, the number of layers and angles of the skin ply of the reflector parameterized model are restricted as follows:

[0052] The angle of the two layers ply=[ply1,ply2]=[0,90],

[0053] The angles of the three layers ply=[ply1,ply2,ply3]=[0,60,-60],

[0054] The angles of the four layers ply = [ply1, ply2, ply3, ply4] = [0, 45, -45, 90],

[0055] The angles of the 5 layers ply=[ply1,ply2,ply3,ply4,ply5]=[45,-45,0,-45,45],

[0056] The angles of the 6 layers ply=[ply1,ply2,ply3,ply4,ply5,ply6]=[0,60,-60,-60,60,0],

[0057] The angles of the 7 layers ply=[ply1,ply2,ply3,ply4,ply5,ply6,ply7]=[0,60,-60,0,-60,60,0],

[0058] The angles of the 8 plies are ply=[ply1,ply2,ply3,ply4,ply5,ply6,ply7,ply8]=[0,45,-45,90,90,-45,45,0]; the inner skin and outer skin are designed with symmetrical plies. For example, when the inner and outer skins are both 3-ply plies, the 6 ply angles are [ply1,ply2,ply3,ply3,ply2,ply1]=[0,60,-60,-60,60,0].

[0059] Step 3) Calculate the root mean square of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions:

[0060] Calculate at high temperature T h Low temperature is T l RMS of thermal deformation of the working surface of the reflector model under working conditionsh and RMS l ; The finite element software Abaqus6.14 was used to calculate the high temperature T h Low temperature is T l The node coordinates of the working surface of the reflector parameterized model before and after thermal deformation under working conditions are fitted by the least square method for focal length of each node coordinate of the working surface before and after thermal deformation, and then the high temperature is T h Low temperature is T l Normal distance d from the nth node of the working surface to the best fitting surface after thermal deformation of the reflector parameterized model under working conditions hn and d ln Calculate the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions h and RMS l :

[0061]

[0062] Step 4) Construct the reflector optimization model:

[0063] Construct the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions h and RMS l As the optimization target, the structural parameters of the honeycomb sandwich structure are used as optimization variables, and the reflector optimization model is conditioned by the constraint interval determined in step (2). Its expression is:

[0064]

[0065] Step 5) Obtain the optimization results of the reflector optimization model:

[0066] The reflector optimization model is solved based on the ant lion optimization algorithm, and the minimum root mean square of thermal deformation of the working surface under high and low temperature conditions is obtained as RMS hu and RMS lu Carbon fiber parabolic reflector.

[0067] Initialize the positions of ants and ant lions, and assign 4 variables to the ants and ant lions in the ant lion algorithm, namely the structural parameters l, h, Q and ply in the inp file. The structural parameters l, h, Q and ply in the inp file are updated by the 4 variable values ​​of the position parameters of the ant lion after initialization, and the structural parameters l, h, Q and ply of the honeycomb sandwich structure in the inp file are updated by the 4 variable values ​​of the position parameters of each ant in each iteration. Then, the root mean square of the thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions in each iteration is calculated by the updated structural parameters until the maximum number of iterations is reached, and the minimum root mean square of the thermal deformation of the working surface under high and low temperature conditions is obtained as RMS hu and RMSlu The carbon fiber parabolic reflector. When the Ant Lion Optimization Algorithm optimizes the RMS thermal deformation of the reflector working surface, the fitness of each ant changes with the number of iterations. Figure 4 In the iterative process, the fitness of each ant is calculated according to the position of the ant. It can be seen that as the optimization proceeds, the positions of all ants gradually approach each other, and the activity range of the ants is constantly decreasing, which indicates the convergence of the ant lion algorithm. In each iteration, the position of the ant with the highest fitness will update the position of the elite ant lion, and all individual positions will gradually approach each other, thus obtaining the optimal solution of the objective function.

Claims

1. A method for optimizing thermal deformation of a carbon fiber parabolic reflector with a honeycomb sandwich structure, characterized in that: The steps include: (1) Constructing a reflector parameterized model: A carbon fiber parabolic honeycomb sandwich structure consisting of an intermediate layer and skins bonded to the inner and outer surfaces of the intermediate layer by adhesive films distributed in a spatial rectangular coordinate system O-XYZ is constructed, and its working surface is meshed, and then the structural parameters and mechanical parameters of multiple regular hexagonal carbon fiber honeycomb core cells constituting the intermediate layer and multiple plies constituting the inner and outer skins, as well as the coordinates of N working surface nodes after meshing are written into an inp file format to form a reflector parameterized model; (2) Determine the constraint interval of the reflector parameterized model structure parameters: Determine the upper limit l of the side length l of the honeycomb core cell in the reflector parameterized model under the condition of pressure P uper and the lower limit l lower , the upper limit of height h under gravity conditions uper and lower limit h lower , and the angles of the Q layers ply=[ply1,ply2…,ply q ,…,ply Q ], where ply q is the ply angle of the qth ply; (3) Calculate the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions: Calculate at high temperature T h Low temperature is T l RMS of thermal deformation of the working surface of the reflector parameterized model under working conditions h and RMS l ; (4) Constructing reflector optimization model: Construct the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions h and RMS l As the optimization target, a reflector optimization model is established with the structural parameters of the honeycomb sandwich structure as optimization variables and the constraint interval determined in step (2) as a condition; (5) Obtain the optimization results of the reflector optimization model: The reflector optimization model is solved based on the ant lion optimization algorithm, and the minimum root mean square of thermal deformation of the working surface under high and low temperature conditions is obtained as RMS hu and RMS lu Carbon fiber parabolic reflector.

2. The method according to claim 1, characterized in that: The structural parameters and mechanical parameters described in step (1), wherein the structural parameters include the side length l and height h of the honeycomb core cell, the number of layers Q and the ply angle ply in the inner and outer skins; the mechanical parameters include the density ρ of the honeycomb core cell core and the density of the inner and outer skins ρ sheet .

3. The method according to claim 2, characterized in that The upper limit l of the side length l of the honeycomb core cell described in step (2) under the working condition of pressure P is uper and the lower limit l lower , the determination method is: The finite element calculation software is used to calculate the J sides of the honeycomb core cell in the reflector parameterized model under the working condition of pressure P, which are l=[l1,l2…,l j ,…,l J ] corresponds to the maximum deformation of the skin Δu=[Δu1,Δu2…,Δu j ,…,Δu J ], and draw a line graph of Δu changing with l, and then use the minimum side length l1 as the lower limit l lower , the side length value l of the honeycomb core cell when Δu increases sharply in the line graph uper As an upper limit, where Δu j Indicates l j The corresponding maximum deformation of the skin.

4. The method according to claim 2, characterized in that: The upper limit h of the height h under gravity conditions described in step (2) is uper and lower limit h lower , the determination method is: The K heights h=[h1,h2…,h k ,…,h K ]Corresponding working surface self-weight deformation root mean square RMS z =[RMS z1 ,RMS z2 …,RMS zk ,…,RMS zK ] and plot the RMS z A line chart that changes with h1, and then RMS in the line chart z Equal to the profile test requirement RMS cl The height of the honeycomb core cell is h lower As the lower limit, the upper limit of the working surface density design index ρ ZBsurface The calculated maximum height is used as the upper limit h uper , where RMS zk Indicates h k The corresponding root mean square of the self-weight deformation of the working surface, h uper The calculation formula is: h uper =max(h) M=ρ ZBsurface ·S V sheet =0.52·Q·t m ·10 7 Where M and S are the weight and aperture area of ​​the reflector respectively, V sheet is the total volume of the inner and outer skins, ρ * is the equivalent density of the honeycomb core cell, V core is the envelope volume of the honeycomb core cell, t is the wall thickness of the honeycomb core cell, t m The thickness of each ply of the skin.

5. The method according to claim 2, characterized in that: The angles of the Q plies in step (2), wherein the ply angle of each q plies is q The value of is determined through quasi-isotropic ply design principles and engineering manufacturing practice.

6. The method according to claim 2, characterized in that The calculation described in step (3) is performed at a high temperature T h Low temperature is T l RMS of thermal deformation of the working surface of the reflector parameterized model under working conditions h and RMS l , the calculation method is: By finite element software calculation at high temperature T h Low temperature is T l The node coordinates of the working surface of the reflector parameterized model before and after thermal deformation under working conditions are fitted by the least square method for focal length of each node coordinate of the working surface before and after thermal deformation, and then the high temperature is T h Low temperature is T l Normal distance d from the nth node of the working surface to the best fitting surface after thermal deformation of the reflector parameterized model under working conditions hn and d ln Calculate the RMS of thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions h and RMS l :

7. The method according to claim 6, characterized in that The reflector optimization model described in step (4) is expressed as: min:RMS h ,RMS l Find:l,h,Q,ply.

8. The method according to claim 7, characterized in that The basis pair reflector optimization model described in step (5) is solved by: Initialize the positions of ants and ant lions, assign 4 variables to the ants and ant lions in the ant lion algorithm, namely the structural parameters l, h, Q and ply in the inp file, update the structural parameters l, h, Q and ply in the inp file through the position parameters of the initialized ant lion, and update the structural parameters l, h, Q and ply of the honeycomb sandwich structure in the inp file through the position parameters of each ant in each iteration, and then calculate the root mean square of the thermal deformation of the working surface of the reflector parameterized model under high and low temperature conditions in each iteration through the updated structural parameters until the maximum number of iterations is reached, and the minimum root mean square of the thermal deformation of the working surface under high and low temperature conditions is obtained as RMS hu and RMS lu Carbon fiber parabolic reflector.

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