A method for calibrating structural deformation of a phased array antenna under typical thermal load

Abstract

A method for calibrating the structural deformation of a phased array antenna under typical thermal loads belongs to the field of electromechanical-thermal integrated lightweight design for spaceborne phased array antennas. Based on the actual temperature conditions and the antenna laminate structure, the mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters are determined. Based on the temperature boundary conditions, geometric dimensions, and thermal parameters, the temperature field function along the thickness direction of the antenna laminate structure is determined. Using the reciprocal theorem of work, the undetermined deflection surface function of the antenna laminate structure is obtained. The deflection of the antenna laminate structure under a unit concentrated load is used as the fundamental solution and substituted into the undetermined deflection surface function. Combined with the mechanical boundary conditions, the undetermined coefficients are solved to obtain the deflection surface function of the antenna laminate structure. The mechanical parameters of the structure are adjusted according to the deflection surface function until the structural deformation calibration of the phased array antenna under typical thermal loads is achieved. This invention can rapidly predict structural deformation within a predetermined accuracy range and has the advantages of high calibration accuracy and high efficiency.

Description

Technical Field

[0001] This invention belongs to the field of lightweight electromechanical-thermal integration technology for spaceborne phased array antennas, and relates to a method for calibrating structural deformation of phased array antennas under typical thermal loads. Background Technology

[0002] Active phased array antennas are extremely flexible in their operation and increasingly widely used. This leads to increasingly stringent electrical performance requirements, demanding higher precision in their mechanical structures. The heat flux density of the antennas increases dramatically, making heat dissipation design more difficult, and resulting in a tighter coupling between the antenna's mechanical structure, thermal, and electromagnetic performance. Among these, thermal characteristics are particularly important. The temperature distribution generated by numerous heat-generating components on the phased array surface causes deformation of the array structure. Simultaneously, changes in the array's temperature distribution lead to a decline in the performance of temperature-sensitive transducer / receiver (T / R) components. These factors cause offsets and deflections of antenna elements on the array surface, as well as fluctuations in the amplitude and phase of the excitation current. This, in turn, causes directional distortion within the antenna element array, alters the mutual coupling effect between elements, and ultimately degrades the overall antenna performance, rendering it unusable or even causing it to malfunction.

[0003] A phased array antenna structure is a typical laminated structure. Generally, the antenna structure has a small thickness and a large side length, and can be regarded as a laminated thin plate structure. Under the action of temperature field, due to the different thermal expansion coefficients of the materials in each layer, thermal stress and thermal torque will be generated under the constraint of boundary conditions. This will cause the antenna structure to bend and deform, which will in turn cause changes in the position of the antenna array elements, change the phase distribution of the antenna aperture, and also cause deformation of the radiating elements, resulting in changes in the coupling between elements and changes in the radiation pattern, thus affecting the overall performance of the antenna.

[0004] Conducting flexural deformation analysis of phased array antennas under typical thermal loads can accurately determine the deformation of the antenna array surface under typical operating conditions, thus providing a basis for the debugging, calibration, and optimization design of phased array antennas. This is of great significance in the field of lightweight electromechanical-thermal integration of spaceborne phased array antennas.

[0005] Currently, the structural deformation problem of phased array antennas under typical thermal loads is mainly addressed through finite element numerical simulation. This method consumes a lot of computational resources and time, resulting in significant time and resource costs for the structural optimization design of phased array antennas. Therefore, a structural deformation calibration method for phased array antennas under typical thermal loads is needed, which can quickly calculate the structural deformation within a certain accuracy range, facilitating rapid debugging, calibration, and optimization design of antenna structural deformation. Summary of the Invention

[0006] To address the shortcomings of the existing technology, the purpose of this invention is to provide a method for calibrating the structural deformation of a phased array antenna under typical thermal loads. Based on the fundamental laws of thermal conduction and the reciprocal theorem of work, this method evaluates and calibrates the structural deformation of a phased array antenna under typical thermal loads, thereby improving the accuracy and efficiency of the evaluation and calibration.

[0007] The objective of this invention is achieved through the following technical solution:

[0008] This invention discloses a method for calibrating the structural deformation of a phased array antenna under typical thermal loads, comprising the following steps:

[0009] Step 1: Determine the mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters based on the actual temperature conditions and antenna laminate structure.

[0010] The actual temperature conditions refer to the typical thermal loads experienced by the phased array antenna in a near-Earth orbit environment. The antenna laminate structure mainly consists of the antenna array surface, chip layer, network layer, and cold plate, which are sequentially numbered 1, 2, ..., k, ..., l from the antenna array surface to the cold plate. The antenna laminate structure is considered as a laminated plate structure in which the principal directions of the material coincide with the x and y coordinate axes, and the number of structural layers is denoted as l. The temperature boundary conditions include the antenna array surface temperature T1 and the heat source intensity q in each layer. vk (k = 1, 2, ..., l), coolant temperature T f With the convective heat transfer coefficient h f Mechanical boundary conditions include the deflection w| at the boundary. x＝0,a and w| y＝0,b Corner at the boundary and In the antenna laminate structure, the mechanical boundary conditions are fixed-support mechanical boundary conditions, where the deflection and rotation at the boundary are both equal to 0. The geometric dimensions include the length *a* along the x-axis, the width *b* along the y-axis, and the thickness *δ* of each layer along the height direction. k (k = 1, 2, ..., l); mechanical parameters include the equivalent Young's modulus E of each layer. k (k = 1, 2, ..., l), equivalent shear modulus G k (k = 1, 2, ..., l) and equivalent Poisson's ratio v k (k = 1, 2, ..., l); thermal parameters include the thermal expansion coefficients of each layer [α]. xx ,α yy ,α xy ] k (k = 1, 2, ..., l) and thermal conductivity κ k (k = 1, 2, ..., l).

[0011] Step 2: Determine the temperature field function along the thickness direction of the antenna laminate structure based on the temperature boundary conditions, geometric dimensions, and thermal parameters;

[0012] The solution to the temperature field function along the thickness direction of the antenna laminate structure includes the following steps:

[0013] Step 2.1: Under typical thermal loads, the temperature field of each layer of the antenna laminate structure satisfies a one-dimensional steady-state thermal conductivity differential equation, the expression of which is:

[0014]

[0015] In the formula: T k (z,T k1 ,T k2 ) represents the piecewise division of the undetermined temperature field function of the k-th layer, z is the coordinate along the thickness direction, and T k1 and T k2 Let represent the upper and lower surface temperatures of the k-th layer, respectively, which are unknowns. The piecewise expression of the undetermined temperature field function of the k-th layer can be derived from the one-dimensional steady-state heat conduction differential equation:

[0016]

[0017] In the formula: C k1 and C k2 z is a coefficient k1 Let z be the z-coordinate of the upper surface of the k-th layer. k2 Let z be the z-coordinate of the lower surface of the k-th layer;

[0018] Step 2.2: Using Fourier's law, divide the undetermined temperature field function of the k-th layer into piecewise T segments. k (z,T k1 ,T k2 Substituting the values ​​into the equation, we obtain the heat flux density q of the k-th layer. k (z,T k1 ,T k2 );

[0019]

[0020] Step 2.3: Obtain the convective heat flux density q using Newton's law of cooling. f The expression is:

[0021]

[0022] Step 2.4: Under steady-state heat transfer, when the heat source layer is in the r-th layer, due to the internal heat source intensity q... vk When z is zero, the heat flux density is a constant function, and its value is independent of z. Therefore, the heat flux density should be written as q. k (T k1 ,T k2 ), while the heat source intensity q within the heat source layer vkSince the value of heat flux density is not zero, it changes with z, so heat flux density is still written as q. k (z,T k1 ,T k2 );

[0023] According to the first law of thermodynamics, selecting a boundary on one side of the heat source layer, when the boundary is at z r1 At the location, the heat flux density q in the continuous region outside this boundary k (T k1 ,T k2 It is equal everywhere and equal to the heat source layer z. r1 Heat flux density q at location r (z r1 ,T r1 ,T r2 The same applies to the other side of the heat source layer boundary. However, when there is a convective heat transfer boundary condition, the heat flux density q k (T k1 ,T k2 ) and convective heat transfer heat flux density q f They are also equal, and thus a system of linear equations is established to solve for the temperature T on the upper surface of the k-th layer. k1 and the temperature T of the lower surface k2 :

[0024]

[0025] In the formula: q1(T 11 ,T 12 ) represents the heat flux density of the first layer, q2(T) 21 ,T 22 ) represents the heat flux density of the second layer, q r+1 (T (r+1)1 ,T (r+1)2 Let q be the heat flux density of the (r+1)th layer. l (T l1 ,T l2 q is the heat flux density of the l-th layer. r (z r1 ,T r1 ,T r2 q is the heat flux density on the upper surface of the heat source layer. r (z r2 ,T r1 ,T r2 () represents the heat flux density on the lower surface of the heat source layer;

[0026] Step 2.5: Calculate the upper surface temperature T. k1 and lower surface temperature T k2 Substituting into equation (2) yields the piecewise T of the temperature field function of the k-th layer of the antenna laminate structure. k (z):

[0027]

[0028] In the formula: all parameters except the independent variable z are known quantities. The temperature field function of each layer of the antenna laminate structure is divided into pieces T. k (z) Summarizing, we obtain the temperature field function T(z) along the thickness direction of the antenna laminate structure:

[0029]

[0030] Step 3: Using the reciprocal theorem of work, obtain the undetermined deflection surface function of the antenna laminate structure;

[0031] The solution to the undetermined deflection surface function of the antenna laminate structure includes the following steps:

[0032] Step 3.1: Consider the antenna laminate structure subjected to typical thermal loads as laminate plate A, and have a laminate plate B with the same structure and mechanical boundary conditions as the antenna laminate. Apply a unit concentrated load P at point (ξ,η) on plate B, where P = 1. Calculate the work W(σ',ε”) done by the stress of plate A on the deformation of plate B and the work W(σ”,ε’) done by the stress of plate B on the deformation of plate A. The corresponding expressions are:

[0033]

[0034] In the formula: σ' and ε' are the stress and strain of plate A, σ” and ε” are the stress and strain of plate B, the subscript ij is the dummy index of Einstein's summation convention, in the calculation of this formula, ij takes the three sets of subscripts xx, yy and xy in sequence, and k is the layer number;

[0035] Step 3.2: When considering temperature changes, the stress-strain relationship of the k-th layer of the laminate is expressed as follows:

[0036]

[0037] In the formula: [σ xx σ yy σ xy ] k T Let [ε] be the stress array of the k-th layer. xx ε yy ε xy ] k T Let [α] be the strain matrix of the k-th layer. xx α yy α xy ] k T Let be the thermal expansion coefficient array for the k-th layer. Since the principal directions of the material coincide with the x and y axes of the coordinate system, temperature changes do not cause shear strain, therefore αxy =0,T k (z) represents the piecewise temperature field function of the k-th layer of the antenna laminate structure, where T0 is the initial temperature value. Let Q be the transformation matrix of the two-dimensional stiffness matrix in the principal direction of the k-th layer, and its corresponding expression is:

[0038]

[0039] In the formula: E is Young's modulus, G is shear modulus, v is Poisson's ratio, and subscripts 1 and 2 represent the principal directions of the material. Since the principal directions of the material coincide with the x and y axes of the coordinate system, Q... 16 =Q 26 =0;

[0040] Step 3.3: Transform the two-dimensional stiffness matrix Q into a transformation matrix. Substituting into equation (9), we get:

[0041]

[0042] Substituting equation (11) into equation (8), we get:

[0043]

[0044] In the formula: T k '(z) represents the piecewise temperature field function of the k-th layer of plate A, T0' represents the initial temperature of plate A, and T k "(z) is the piecewise temperature field function of the kth layer of plate B, T0" is the initial temperature of plate B, and k is the layer number;

[0045] According to the reciprocal theorem of work, we can establish an equation, and the corresponding expression is:

[0046]

[0047] Step 3.4: Integrate both sides of equation (13) simultaneously, and the corresponding expression is:

[0048]

[0049] In the formula: P' is the surface force acting on plate A, P” is the surface force acting on plate B, Q' is the volume force acting on plate A, Q” is the volume force acting on plate B, U' is the displacement vector of plate A, U” is the displacement vector of plate B, Ω is the surface area of ​​plate A, the area of ​​plate B is the same as that of plate A, and V is the volume of plate A, the volume of plate B is the same as that of plate A.

[0050] Step 3.5: Since plate B is an isothermal structure, the temperature field function T(z) of plate B is always equal to the initial temperature value T0". Therefore, the piecewise T(z) of the temperature field function of each layer of plate B is... k"(z) are all equal to the initial temperature T0 of plate B". Rewriting equation (14) into equation (15) yields the undetermined deflection surface function w(ξ,η,M) of the antenna laminate structure. m M n ):

[0051]

[0052] In the formula: w*(ξ,η,x,y) refers to the deflection of plate B at point (ξ,η) when subjected to a unit concentrated load P, which is called the fundamental solution, M m and M n All are undetermined coefficients related to the mechanical boundary conditions of plate A, m and n are summation parameters, a is the length in the x direction, and b is the width in the y direction;

[0053] The mechanical boundary conditions of plate A are fixed boundary conditions. We transform them into simply supported boundary conditions with bending moments. The expression for the bending moment acting on plate A after the transformation is:

[0054]

[0055] In the formula: parameter M m and M n The meaning of the parameters is consistent with that in equation (15).

[0056] The deflection of the antenna laminate under a unit concentrated load is taken as the basic solution and substituted into the undetermined deflection surface function in step three. The undetermined coefficients are solved by combining the mechanical boundary conditions in step one, and the deflection surface function of the antenna laminate is obtained.

[0057] The process of substituting the fundamental solution into the undetermined deflection surface function and combining it with mechanical boundary conditions to obtain the deflection surface function of the antenna laminate structure includes the following steps:

[0058] Step 4.1, for the basic solution w * The derivation is performed using (ξ,η,x,y). Since the mechanical boundary conditions of plate A have been transformed from fixed boundary conditions to simply supported boundary conditions with bending moments, the mechanical boundary conditions of plate B are also simply supported boundary conditions. The deflection surface function of plate B is w. * The corresponding expressions for the differential equation of the laminated plate and the simply supported boundary conditions are as follows:

[0059]

[0060] In the formula: q(x,y) is the lateral load on plate B, and D 11 D 12 D 22 D 66 Let D be the stiffness coefficient in the bending stiffness matrix. The corresponding expression for the bending stiffness matrix D is:

[0061]

[0062] In the formula: Q ij The transformation matrix of the two-dimensional stiffness matrix Q The stiffness coefficient, z k1 Let z be the z-coordinate of the upper surface of the k-th layer. k2 Let z be the z-coordinate of the lower surface of the k-th layer, where k is the layer number;

[0063] Step 4.2, Plate B Deflection Surface Function w * Both (x,y) and the lateral distributed load q(x,y) adopt the form of a double trigonometric series, and the corresponding expressions are:

[0064]

[0065] In the formula: w mn The coefficients are undetermined. Since the lateral distributed load q(x,y) is known, it represents the unit concentrated load applied at point (ξ,η). mn It can be directly obtained from the following formula:

[0066]

[0067] Step 4.3: Apply the deflection surface function w to plate B. * Substituting (x,y) and the lateral distributed load q(x,y) into both sides of the differential equation (17), we obtain the coefficient w. mn The expression is:

[0068]

[0069] Since plate B is subjected to a unit concentrated load, the lateral distributed load q(x,y) is replaced by a uniformly distributed load 1 / dxdy on the differential area dxdy. At this point, q(x,y) is equal to zero at all positions except at the point (ξ,η) where it equals 1 / dxdy on the differential area. Therefore, the coefficient w... mn The expression is rewritten as:

[0070]

[0071] Step 4.4: Substitute equation (22) into equation (19) to obtain the plate B deflection surface function w. * (x,y), that is, the basic solution w is obtained. * (ξ,η,x,y), the expression is:

[0072]

[0073] Step 4.5: Apply the basic solution w obtained from equation (23) *Substituting (ξ,η,x,y) into equation (15), we obtain the deflection surface function w(ξ,η) of the antenna laminate structure, and the corresponding expression is:

[0074]

[0075] Where: undetermined coefficient M m and M n Solve the equations using the fixed boundary conditions, and obtain the equations and the coefficients M obtained from the solution. m and M n The corresponding expression is:

[0076]

[0077]

[0078] Step 5: Use the deflection surface function obtained in Step 4 to determine whether the degree of structural deformation meets the preset requirements. If it does, output the result to achieve structural deformation calibration; if it does not, modify the mechanical parameters of the structure and return to Step 2.

[0079] Beneficial effects:

[0080] 1. The present invention discloses a structural deformation calibration method for a phased array antenna under typical thermal loads. Based on the actual temperature boundary conditions, geometric dimensions and thermal parameters, the temperature field function along the thickness direction of the antenna laminate structure can be determined, which can effectively prevent the deterioration of the antenna's electromagnetic performance due to the temperature drift of the thermal electronic components (such as phase shifters).

[0081] 2. Based solely on temperature boundary conditions, mechanical boundary conditions, relevant geometric dimensions of the laminate, mechanical parameters, and thermal parameters, this invention can accurately obtain the deflection deformation of an antenna laminate under typical thermal loads through theoretical calculations. It can quickly predict structural deformation within a predetermined accuracy range, facilitating rapid debugging, calibration, and optimization of antenna structural deformation, and saving significant human and material resources.

[0082] 3. The present invention discloses a structural deformation calibration method for phased array antennas under typical thermal loads. Based on the fundamental laws of thermal conduction and the reciprocal theorem of work, the structural deformation calibration method can improve the accuracy and efficiency of evaluation and calibration of the structural deformation of phased array antennas under typical thermal loads, and provides a basis for guiding the assembly optimization design of heat-dissipating components.

[0083] 4. The present invention discloses a structural deformation calibration method for a phased array antenna under typical thermal loads. The method uses a laminated thin plate structure model to analyze the flexural deformation of the antenna under typical thermal loads. Based on the structural deformation calibration results of the present invention, the structural optimization of the phased array antenna under typical thermal loads is realized. Based on the optimization results, the electromechanical and thermal integration and lightweighting of the spaceborne phased array antenna are achieved. Attached Figure Description

[0084] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0085] Figure 1 A flowchart of a structural deformation calibration method for a phased array antenna under typical thermal loads according to the present invention;

[0086] Figure 2 A schematic diagram of a phased array antenna transmitting array layered structure in an embodiment of the method of the present invention;

[0087] Figure 3 A schematic diagram of the equivalent mechanical boundary conditions in the embodiments of the present invention. Detailed Implementation

[0088] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and examples.

[0089] refer to Figure 2 This embodiment uses a simplified scenario of overall structural deformation of a phased array antenna transmitter array under typical operating thermal load as an example for specific illustration. The specific implementation steps of the structural deformation calibration method for a phased array antenna under typical thermal load are as follows:

[0090] Step 1: Based on the actual temperature conditions and antenna laminate structure, determine the mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters. The actual temperature conditions refer to the typical thermal loads experienced by the phased array antenna in a near-Earth orbit environment. The antenna laminate structure mainly consists of the antenna array surface, chip layer, network layer, and cold plate, sequentially numbered 1, 2, ..., k, ..., l from the antenna array surface to the cold plate. The antenna laminate structure is considered as a laminated plate structure where the principal material directions coincide with the x and y coordinate axes, and the number of layers is denoted as l. The temperature boundary conditions include the antenna array surface temperature T1 and the heat source intensity q within each layer. vk (k = 1, 2, ..., l), coolant temperature T f With the convective heat transfer coefficient h f Mechanical boundary conditions include the deflection w| at the boundary. x＝0,a and w| y＝0,b Corner at the boundary and In the antenna laminate structure, the mechanical boundary conditions are fixed-support mechanical boundary conditions, where the deflection and rotation at the boundary are both equal to 0. The geometric dimensions include the length *a* along the x-axis, the width *b* along the y-axis, and the thickness *δ* of each layer along the height direction. k (k = 1, 2, ..., l); mechanical parameters include the equivalent Young's modulus E of each layer. k (k = 1, 2, ..., l), equivalent shear modulus G k (k = 1, 2, ..., l) and equivalent Poisson's ratio v k (k = 1, 2, ..., l); thermal parameters include the thermal expansion coefficients of each layer [α]. xx ,α yy ,α xy ] k (k = 1, 2, ..., l) and thermal conductivity κ k (k = 1, 2, ..., l).

[0091] In this embodiment, the maximum deflection is required to be no more than 0.5 mm. The mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters are shown in Table 1 below.

[0092] Table 1. Mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters.

[0093]

[0094]

[0095] Step 2: Determine the temperature field function along the thickness direction of the antenna laminate structure. Based on the conditions in Step 1, the expression for the temperature field function is obtained as follows:

[0096]

[0097] Step 3: Using the reciprocal theorem of work, obtain the undetermined deflection surface function of the antenna laminate structure;

[0098] The first step is to consider the antenna laminate structure subjected to typical thermal loads as laminate plate A, and to have a laminate plate B with the same structure and mechanical boundary conditions as the antenna laminate. A unit concentrated load P is applied at point (ξ,η) on plate B, where P = 1. The work W(σ',ε”) done by the stress in plate A on the deformation of plate B and the work W(σ”,ε’) done by the stress in plate B on the deformation of plate A are calculated. The corresponding expressions are:

[0099]

[0100] in:

[0101]

[0102] The second step is to establish an equation relationship between the two equations based on the reciprocal theorem of work. After integrating both sides of the equation, since plate B is an isothermal structure, the temperature field function T(z) of plate B is always equal to the initial temperature value T0". Therefore, the piecewise T(z) of the temperature field function of each layer of plate B is... k "(z) are all equal to the initial temperature T0 of plate B", from which the undetermined deflection surface function w(ξ,η,M) of the antenna laminate structure can be written. m M n ):

[0103]

[0104] Where: w*(ξ,η,x,y) refers to the deflection of plate B at point (ξ,η) under a unit concentrated load P, called the fundamental solution, M m and M n All are undetermined coefficients related to the mechanical boundary conditions of plate A, m and n are summation parameters, a is the length in the x direction, and b is the width in the y direction;

[0105] The mechanical boundary conditions of plate A are fixed boundary conditions. We transform them into simply supported boundary conditions with bending moments. The expression for the bending moment acting on plate A after the transformation is:

[0106]

[0107] Where: parameter M m and M n The meaning of the parameters is consistent with that in equation (5).

[0108] Step 4: Take the deflection of the antenna laminate structure under a unit concentrated load as the basic solution, substitute it into the undetermined deflection surface function of Step 3, and solve for the undetermined coefficients in combination with the mechanical boundary conditions of Step 1 to obtain the deflection surface function of the antenna laminate structure.

[0109] The first step is to write the deflection surface function w of plate B. * The differential equation and simply supported boundary conditions for the laminated plate corresponding to (x,y) are expressed as follows:

[0110]

[0111] Where q(x,y) is the transverse load on plate B, i.e., the unit concentrated load P, and the expression for the bending stiffness matrix D is as follows:

[0112]

[0113] The second step is to solve the differential equation to obtain the deflection surface function of plate B, i.e., the fundamental solution w. * (ξ,η,x,y), the expression is:

[0114]

[0115] The third step is to apply the basic solution w obtained from equation (9) * Substituting (ξ,η,x,y) into equation (5), we obtain the deflection surface function w(ξ,η) of the antenna laminate structure, and the corresponding expression is:

[0116]

[0117] in:

[0118]

[0119] Step 5: Determine whether the degree of structural deformation meets the preset requirements. The maximum deflection occurs at point (127.5, 127.5). Substituting the coordinates of this point, the maximum deflection is 0.289 mm. After verification, the error of this prediction result is within the allowable range of engineering and less than 0.5 mm. The degree of deformation meets the preset requirements. Output the structural deformation calibration result to realize the structural deformation calibration of the phased array antenna under typical thermal load.

[0120] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calibrating the structural deformation of a phased array antenna under typical thermal loads, characterized in that: Includes the following steps, Step 1: Determine the mechanical boundary conditions, temperature boundary conditions, geometric dimensions, mechanical parameters, and thermal parameters based on the actual temperature conditions and antenna laminate structure. Step 2: Determine the temperature field function along the thickness direction of the antenna laminate structure based on the temperature boundary conditions, geometric dimensions, and thermal parameters; The specific implementation method of step two includes the following steps: Step 2.1: Under typical thermal loads, the temperature field of each layer of the antenna laminate structure satisfies a one-dimensional steady-state thermal conductivity differential equation, the expression of which is: （1） In the formula: T k (z, T k1 , T k2 ) represents the piecewise division of the undetermined temperature field function of the k-th layer, z is the coordinate along the thickness direction, and T k1 and T k2 Let represent the upper and lower surface temperatures of the k-th layer, respectively, which are unknowns. The piecewise expression of the undetermined temperature field function of the k-th layer is derived from the one-dimensional steady-state heat conduction differential equation: （2） In the formula: C k1 and C k2 z is a coefficient k1 Let z be the z-coordinate of the upper surface of the k-th layer. k2 Let z be the z-coordinate of the lower surface of the k-th layer; Step 2.2: Using Fourier's law, divide the undetermined temperature field function of the k-th layer into piecewise T segments. k (z, T k1 , T k2 Substituting the values ​​into the equation, we obtain the heat flux density q of the k-th layer. k (z, T k1 , T k2 ); （3） Step 2.3: Obtain the convective heat flux density q using Newton's law of cooling. f The expression is: （4） Step 2.4: Under steady-state heat transfer, when the heat source layer is in the r-th layer, due to the internal heat source intensity q... vk When z is zero, the heat flux density is a constant function, and its value is independent of z. Therefore, the heat flux density should be written as q. k (T k1 , T k2 ), while the heat source intensity q within the heat source layer vk Since the value of heat flux density is not zero, it changes with z, so heat flux density is still written as q. k (z, T k1 , T k2 ); According to the first law of thermodynamics, selecting a boundary on one side of the heat source layer, when the boundary is at z r1 At the location, the heat flux density q in the continuous region outside this boundary k (T k1 , T k2 It is equal everywhere and equal to the heat source layer z. r1 Heat flux density q at location r (z r1 , T r1 ,T r2 The same applies to the other side of the heat source layer boundary. However, when there is a convective heat transfer boundary condition, the heat flux density q k (T k1 , T k2 ) and convective heat transfer heat flux density q f They are also equal, and thus a system of linear equations is established to solve for the temperature T on the upper surface of the k-th layer. k1 and the temperature T of the lower surface k2 : （5） In the formula: q1(T 11 , T 12 ) represents the heat flux density of the first layer, q2(T) 21 , T 22 ) represents the heat flux density of the second layer, q r+1 (T (r+1)1 ,T (r+1)2 Let q be the heat flux density of the (r+1)th layer. l (T l1 , T l2 Let q be the heat flux density of the l-th layer. r (z r1 , T r1 , T r2 q is the heat flux density on the upper surface of the heat source layer. r (z r2 , T r1 , T r2 () represents the heat flux density on the lower surface of the heat source layer; Step 2.5: Calculate the upper surface temperature T. k1 and lower surface temperature T k2 Substituting into equation (2) yields the piecewise T of the temperature field function of the k-th layer of the antenna laminate structure. k (z): （6） In the formula: all parameters except the independent variable z are known quantities. The temperature field function of each layer of the antenna laminate structure is divided into pieces T. k (z) Summarizing, we obtain the temperature field function T(z) along the thickness direction of the antenna laminate structure: （7） Step 3: Using the reciprocal theorem of work, obtain the undetermined deflection surface function of the antenna laminate structure; Step 4: Take the deflection of the antenna laminate structure under a unit concentrated load as the basic solution, substitute it into the undetermined deflection surface function of Step 3, and solve for the undetermined coefficients in combination with the mechanical boundary conditions of Step 1 to obtain the deflection surface function of the antenna laminate structure. Step 5: Use the deflection surface function obtained in Step 4 to determine whether the degree of structural deformation meets the preset requirements. If it does, output the result to achieve structural deformation calibration; if it does not, modify the mechanical parameters of the structure and return to Step 2.

2. The structural deformation calibration method for a phased array antenna under typical thermal loads as described in claim 1, characterized in that: The actual temperature conditions mentioned in step one refer to the typical thermal loads experienced by the phased array antenna in a near-Earth orbit environment. The antenna laminate structure consists of an antenna array surface, a chip layer, a network layer, and a cold plate, which are sequentially numbered 1, 2, …, k, …, l from the antenna array surface to the cold plate. The antenna laminate structure is considered as a laminated plate structure in which the principal directions of the material coincide with the x and y coordinate axes, and the number of structural layers is denoted as l. The temperature boundary conditions include the antenna array surface temperature T1 and the heat source intensity q within each layer. vk k = 1,2,…,l, coolant temperature T f With the convective heat transfer coefficient h f Mechanical boundary conditions include the deflection w| at the boundary. x=0,a and w| y=0,b The corner at the boundary is ∂w / ∂x| x=0,a and ∂w / ∂y| y=0,b In the antenna laminate structure, the mechanical boundary conditions are fixed-support mechanical boundary conditions, where the deflection and rotation at the boundary are both equal to 0. The geometric dimensions include the length a along the x-axis, the width b along the y-axis, and the thickness δ of each layer along the height direction. k k = 1,2,…,l; mechanical parameters include the equivalent Young's modulus E of each layer of the structure. k k = 1,2,…,l, equivalent shear modulus G k k = 1,2,…,l and equivalent Poisson ratio v k k = 1,2,…,l; thermal parameters include the thermal expansion coefficients of each layer [α]. xx , α yy , α xy ] k k = 1,2,…,l and thermal conductivity κ k , k = 1,2,…,l.

3. The structural deformation calibration method for a phased array antenna under typical thermal loads as described in claim 2, characterized in that: Step 3 includes the following steps: Step 3.1: Consider the antenna laminate structure subjected to typical thermal loads as laminate plate A, and have a laminate plate B with the same structure and mechanical boundary conditions as the antenna laminate. Apply a unit concentrated load P at point (ξ, η) on plate B. Obviously, P=1. Calculate the work W(σ) done by the stress of plate A on the deformation of plate B. ' , ε '' The work W(σ) done by the stress in plate B on the deformation of plate A is due to the stress in plate B. '' , ε ' The corresponding expression is: （8） Where: σ ' and ε ' Let σ be the stress and strain of plate A. '' and ε '' Let be the stress and strain of plate B, and let ij be the dummy index of Einstein's summation convention. In the calculation of this formula, ij takes three sets of subscripts xx, yy, and xy in sequence, and k is the layer number. Step 3.2: When considering temperature changes, the stress-strain relationship of the k-th layer of the laminate is expressed as follows: （9） In the formula: [σ xx σ yy σ xy ] k T Let [ε] be the stress array of the k-th layer. xx ε yy ε xy ] k T Let [α] be the strain matrix of the k-th layer. xx α yy α xy ] k T Let be the thermal expansion coefficient array for the k-th layer. Since the principal directions of the material coincide with the x and y axes of the coordinate system, temperature changes do not cause shear strain, therefore α xy =0,T k (z) represents the piecewise temperature field function of the k-th layer of the antenna laminate structure, where T0 is the initial temperature value. The two-dimensional stiffness matrix of the k-th principal direction The transformation matrix, and its corresponding expression is: （10） In the formula: E is Young's modulus, G is shear modulus, v is Poisson's ratio, and subscripts 1 and 2 represent the principal directions of the material. Since the principal directions of the material coincide with the x and y axes of the coordinate system, ; Step 3.3: Convert the two-dimensional stiffness matrix... Transformation matrix Substituting into equation (9), we get: （11） Substituting equation (11) into equation (8), we get: （12） In the formula: T k '(z) represents the piecewise temperature field function of the k-th layer of plate A, T0' represents the initial temperature of plate A, and T k ''(z) represents the piecewise temperature field function of the kth layer of plate B, T0'' represents the initial temperature of plate B, and k represents the layer number; According to the reciprocal theorem of work, we can establish an equation, and the corresponding expression is: （13） Step 3.4: Integrate both sides of equation (13) simultaneously, and the corresponding expression is: （14） In the formula: P' is the surface force acting on plate A, P'' is the surface force acting on plate B, and Q' is the volume force acting on plate A. Let U' be the volume force acting on plate B, U' be the displacement vector of plate A, U'' be the displacement vector of plate B, Ω be the surface area of ​​plate A, the area of ​​plate B is the same as that of plate A, and V be the volume of plate A, the volume of plate B is the same as that of plate A. Step 3.5: Since plate B is an isothermal structure, the temperature field function T''(z) of plate B is always equal to the initial temperature value T0''. Therefore, the piecewise T of the temperature field function of each layer of plate B is... k ''(z) are all equal to the initial temperature T0 of plate B. Rewriting equation (14) as equation (15) yields the undetermined deflection surface function w(ξ, η, M) of the antenna laminate structure. m M n ): （15） In the formula: w*(ξ, η, x, y) refers to the deflection of plate B at point (ξ, η) under a unit concentrated load P, which is called the fundamental solution. m and M n All are undetermined coefficients related to the mechanical boundary conditions of plate A, m and n are summation parameters, a is the length in the x direction, and b is the width in the y direction; The mechanical boundary conditions of plate A are fixed boundary conditions. We transform them into simply supported boundary conditions with bending moments. The expression for the bending moment acting on plate A after the transformation is: （16） In the formula: parameter M m and M n The meaning of the parameters is consistent with that in equation (15).

4. The structural deformation calibration method for a phased array antenna under typical thermal loads as described in claim 3, characterized in that: Step four is implemented using the following steps: Step 4.1, for the basic solution w * The derivation is based on (ξ, η, x, y). Since the mechanical boundary conditions of plate A have been transformed from fixed boundary conditions to simply supported boundary conditions with bending moments, the mechanical boundary conditions of plate B are also simply supported boundary conditions. The deflection surface function of plate B is w. * The corresponding expressions for the differential equation of the laminated plate and the simply supported boundary conditions for the four sides (x, y) are: （17） In the formula: q(x, y) is the lateral load on plate B, and D 11 D 12 D 22 D 66 Let D be the stiffness coefficient in the bending stiffness matrix. The corresponding expression for the bending stiffness matrix D is: （18） In the formula: Two-dimensional stiffness matrix Transformation matrix The stiffness coefficient, z k1 Let z be the z-coordinate of the upper surface of the k-th layer. k2 Let z be the z-coordinate of the lower surface of the k-th layer, where k is the layer number; Step 4.2, Plate B Deflection Surface Function w * Both (x, y) and the lateral distributed load q(x, y) are expressed in a double trigonometric series form, and the corresponding expressions are as follows: （19） In the formula: w mn The coefficient q is an undetermined coefficient. Since the lateral distributed load q(x, y) is known, it represents the unit concentrated load applied at point (ξ, η). mn It can be directly obtained from the following formula: （20） Step 4.3: Apply the deflection surface function w to plate B. * Substituting (x, y) and the lateral distributed load q(x, y) into both sides of the differential equation (17), we obtain the coefficient w. mn The expression is: （21） Since plate B is subjected to a unit concentrated load, the lateral distributed load q(x, y) is replaced by a uniformly distributed load 1 / dxdy on the differential area dxdy. At this point, q(x, y) is equal to zero at all positions except for 1 / dxdy on the differential area at point (ξ, η). Therefore, the coefficient w... mn The expression is rewritten as: （22） Step 4.4: Substitute equation (22) into equation (19) to obtain the plate B deflection surface function w * (x, y), thus obtaining the basic solution w * (ξ, η, x, y), the expression is: （23） Step 4.5, take the basic solution w obtained from equation (23) * Substituting (ξ, η, x, y) into equation (15), we obtain the deflection surface function w(ξ, η) of the antenna laminate structure, and the corresponding expression is: （24） Where: undetermined coefficient M m and M n Solve the equations using the fixed boundary conditions, and obtain the equations and the coefficients M obtained from the solution. m and M n The corresponding expression is: （25） （26）。

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