An antenna substrate vibration fundamental frequency optimization method based on machine learning
By employing a dimensionality reduction optimization method based on surrogate models and active subspaces, combined with Bayesian optimization algorithms, the frequency optimization challenge of large-size antenna substrate structures was solved, achieving efficient and accurate vibration fundamental frequency optimization and improving satellite stability and computational efficiency.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- HARBIN INST OF TECH
- Filing Date
- 2023-12-25
- Publication Date
- 2026-05-08
AI Technical Summary
Frequency optimization of large-size antenna substrate structures involves a large scale and excessively high design space dimensions. Furthermore, it requires consideration of various linear and nonlinear constraints, resulting in high computational load, low efficiency, and impact on the functional stability of satellite payloads.
An efficient optimization method based on a surrogate model is adopted, which combines the active subspace method and the adaptive Kriging method to reconstruct the design space by reducing its dimension. The Bayesian optimization algorithm is used to optimize the antenna substrate in the reconstructed space, and a Kriging surrogate model is constructed for sensitivity analysis and global search.
It significantly improved optimization efficiency, reduced computational load, enhanced the optimization effect of antenna substrate vibration fundamental frequency, solved optimization problems under high-dimensional and complex constraints, and improved the stability of satellite payload.
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Figure CN117763910B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of vibration optimization technology for large-area thin-walled plates, specifically involving a method for optimizing the fundamental frequency of antenna substrate vibration based on machine learning. Background Technology
[0002] Currently, with satellite launches becoming routine and large-scale, the substrates for carrying antennas are also trending towards larger sizes and lighter weights. Traditional aerospace components are constrained by launch vehicles, requiring strict folding and storage designs and limiting substrate thickness to maximize efficiency under the weight constraints of large systems like launch vehicles. Therefore, a common approach is to connect unit substrates with hinges, unfolding them into a pre-defined structure upon reaching the designated orbit. However, the presence of hinges makes this large, thin-plate structure highly flexible. Vibrations generated during on-orbit missions can easily cause attitude movements in the satellite's central rigid body, affecting satellite payload functionality and potentially leading to mission failure.
[0003] Therefore, research on vibration control and optimization methods for large-size antenna substrate structures with hinged connections is of great significance. However, frequency optimization for large-size antenna substrate structures presents significant challenges. The main problem is the excessively large scale of the optimization problem or the dimensionality of the design space, as the increased structural size corresponds to a greater number of hinges and panels. Furthermore, various linear and nonlinear constraints must be considered throughout the optimization process. For example, considering transmission cost limitations, the structural mass must be kept below a specified threshold; and considering structural safety and deployment stability, the maximum displacement of the antenna substrate structure under thermal loads should be taken into account. Simultaneously, calculating the frequency and complex constraints of the antenna substrate structure requires a large number of high-precision samples.
[0004] To address these issues, this invention employs an efficient optimization and active subspace (AC) method based on a surrogate model. The active subspace method, combined with adaptive kriging (step four) and global sensitivity analysis (GSA), is used to reconstruct the design space through dimensionality reduction. Then, an efficient optimization algorithm is used to optimize the antenna substrate within the reconstructed design space, ultimately yielding the optimal fundamental frequency. Furthermore, compared to optimization of the original space, the optimization of the reconstructed dimensionality-reduced space is faster and more efficient. Summary of the Invention
[0005] The purpose of this invention is to address the aforementioned problems in the background art by providing a machine learning-based method for optimizing the fundamental frequency of antenna substrate vibration. The method of this invention is an efficient dimensionality reduction optimization method based on a surrogate model and active subspace (AC).
[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0007] A machine learning-based method for optimizing the fundamental frequency of antenna substrate vibration, the method comprising the following steps:
[0008] Step 1: Identify active subspaces
[0009] N0 samples are sampled from the initial design space of the antenna substrate to form a set, denoted as the initial sample set S0; then N1 samples are selected from S0, denoted as S1; the gradient vector of the fundamental frequency of the antenna substrate vibration of the samples in set S1 is estimated using finite element simulation and finite difference method.
[0010] By using the gradient vectors of these S1 sample points The elements of the covariance matrix C are estimated by calculating the average value, and eigenvalues and eigenvectors are obtained by performing eigenvalue decomposition on the covariance matrix C. Active subspaces are identified by analyzing the relative magnitudes of the eigenvalues. After finding the active subspaces, the mapping between the original input variable x0 and the variables in the active subspace is approximated as follows:
[0011] x0=W1u+W2z≈W1u (13)
[0012] In equation (13), W1 and W2 are the active subspace vector and the inactive subspace vector, respectively, and u and z are the active variable and the inactive variable, respectively;
[0013] Step 2: Adaptively construct the proxy model within the active subspace
[0014] Adaptively construct a Kriging surrogate model for the fundamental frequency of antenna substrate vibration within the active subspace. Using the unobserved point MSE estimated by the Kriging surrogate model, the new sample point u new The point corresponding to the largest MSE:
[0015]
[0016] In equation (14), S 0,u This represents the mapping of the initial sample set S0 within the subspace.
[0017] Step 3: Use the Kriging surrogate model to estimate the sensitivity index.
[0018] Introducing a quasi-Monte Carlo method using a constructed Kriging proxy model Sensitivity analysis is performed, and the initial design space is reconstructed based on the results. Sensitive variables are selected to form a new design space, while insensitive variables are always fixed at the mean.
[0019] Step 4: Optimize and reconstruct the antenna substrate within the design space
[0020] Bayesian optimization is used to optimize the antenna substrate within the reconstruction space, aiming to improve the global search of the sampling function of the EI sampling criterion to detect unexplored but promising regions, while improving prediction accuracy. The sampling function of the EI sampling criterion is expressed in closed form:
[0021]
[0022] In equation (1), EI(x) is the EI function value corresponding to the observation point x, Φ(·) and φ(·) are the cumulative distribution and probability density function of the standard normal distribution, respectively, and y min To evaluate the minimum value in the sample, For the Kriging surrogate model predictions, Predict the square root of the variance for the Kriging surrogate model;
[0023] For constrained optimization problems, the constraints include structural weight and thermally induced displacement of the structure. A surrogate model G for the constraint function is established. j (x), j = 1, ..., n g , where n g Let G represent the number of constraint functions. Suppose the random variable G corresponding to the j-th constraint function is... j (x) follows the mean of The standard deviation is s g,j If (x) follows a normal distribution, then the probability P[G] satisfies the expectation. j [x)≥0] is calculated using the following formula (2):
[0024]
[0025] The constrained EI sampling function CEI(x) is expressed as:
[0026]
[0027] Furthermore, in step two, the process of adaptively constructing the Kriging proxy model within the active subspace is as follows:
[0028] Step 21: Map all samples in S0 to the active subspace u using the relation in formula (13), and denote the new sample set as S. 0,u From S 0,u We extract N2 points as initial samples. For the k-th initial sample u k We can map it back to the initial variable space using formula (13) to obtain x. k The fundamental frequency y of the antenna substrate was obtained through finite element analysis. k Thus forming the initial training dataset T = {u k ,y k};
[0029] Step 22: Train the Kriging surrogate model using the initial training dataset T to obtain the surrogate model.
[0030] Steps 2 and 3: Using the proxy model Estimate S 0,u The MSE of all samples in the sample is used to identify new points using formula (14);
[0031] Step Two Four: The convergence criterion is defined as follows:
[0032]
[0033] In the formula, maxMSE represents S 0,u The maximum MSE of all samples, where the superscript indicates the number of iterations; Cr is the convergence threshold, which is 5 × 10⁻⁶. -6 ;
[0034] When the convergence criterion is not met, u new Mapped to the initial input space, the fundamental vibration frequency y of the antenna substrate is calculated by calling the finite element method. new Then the new training point {u new ,y new} Add to T for updating, and retrain the Kriging surrogate model in step 22; if the convergence criterion is met, the Kriging surrogate model is considered to meet the accuracy requirement.
[0035] Compared with the prior art, the beneficial effects of the present invention are:
[0036] First, existing optimization methods for antenna substrate structures mostly employ traditional algorithms such as genetic algorithms and particle swarm optimization. However, for structures like antenna substrates, which have high nonlinearity and complex multidisciplinary constraints, the computational load is often unacceptable. Therefore, this invention uses a Bayesian optimization method (step four) to progressively update the surrogate model and find the optimal solution in the global space using a sampling function, thereby overcoming the problem of excessive computational load.
[0037] Secondly, for complex antenna substrate structures with numerous design parameters, the optimization process faces the "curse of dimensionality." This is because numerous design parameters reduce the prediction accuracy of the surrogate model and increase model training time; furthermore, the high-dimensional design space poses a significant challenge to the global optimization ability of the optimization algorithm. Therefore, this invention utilizes a sensitivity analysis method based on active subspaces to screen design parameters, selecting those that have a greater impact on the antenna substrate structure's response to reconstruct the design space, thereby overcoming the aforementioned challenges. Attached Figure Description
[0038] Figure 1This is a schematic diagram of the overall structure of the antenna substrate used in this invention;
[0039] Figure 2 This is a schematic diagram of a cross-section along the thickness direction of a honeycomb panel.
[0040] Figure 3 This is a schematic diagram of the honeycomb core of a honeycomb panel;
[0041] Figure 4 This is a schematic diagram of the on-orbit thermal environment of the antenna substrate structure;
[0042] Figure 5 This is a graph showing the temperature change of the upper and lower panels of the honeycomb panel over time.
[0043] Figure 6 This is a diagram of thermally induced structural bending deformation of a spacecraft.
[0044] Figure 7 This is a flowchart of the active subspace method for optimizing antenna substrates;
[0045] Figure 8 This is a schematic diagram of the most active one-dimensional subspace;
[0046] Figure 9 These are the first three variables that affect the first natural frequency, namely, the schematic diagram of the cellular geometry parameters;
[0047] Figure 10 This is a diagram illustrating the number of iterations and the optimal result after dimensionality reduction optimization. Figure 10 (a) represents the original space; Figure 10 (b) represents the reconstruction space. Detailed Implementation
[0048] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.
[0049] Specific Implementation Method 1: This implementation method describes a method for optimizing the fundamental frequency of antenna substrate vibration based on machine learning. The method includes the following steps:
[0050] Step 1: Identify active subspaces
[0051] Within the initial design space of the antenna substrate (using sampling techniques such as low-difference Sobol sequences), N0 samples are sampled to generate a set, denoted as the initial sample set S0; then, N1 samples are selected from S0, denoted as S1; the gradient vector of the fundamental frequency of the antenna substrate vibration of the samples in set S1 is estimated using finite element simulation and the finite difference method.
[0052] By using the gradient vectors of these S1 sample points The elements of the covariance matrix C are estimated by calculating the average value, and eigenvalues and eigenvectors are obtained by performing eigenvalue decomposition on the covariance matrix C. Active subspaces are identified by analyzing the relative magnitudes of the eigenvalues. After finding the active subspaces, the mapping between the original input variable x0 and the variables in the active subspace is approximated as follows:
[0053] x0=W1u+W2z≈W1u (13)
[0054] In equation (13), W1 and W2 are the active subspace vector and the inactive subspace vector, respectively, and u and z are the active variable and the inactive variable, respectively;
[0055] Step 2: Adaptively construct the proxy model within the active subspace
[0056] Adaptively construct a Kriging surrogate model for the fundamental frequency of antenna substrate vibration within the active subspace. (where it is built within the active subspace) This replaces the existing model f(x). It's worth noting that the Kriging surrogate model of this invention is not constructed all at once within the active subspace, but rather each new sample point is selected through an adaptive sampling function. The unobserved point MSE (mean squared error) estimated by the Kriging surrogate model is used to estimate the new sample point u. new The point corresponding to the largest MSE:
[0057]
[0058] In equation (14), S 0,u This represents the mapping of the initial sample set S0 within the subspace.
[0059] Step 3: Use the Kriging surrogate model to estimate the sensitivity index.
[0060] In parametric studies, sensitivity analysis can determine the extent of a factor's local or global influence on the output response. A quasi-Monte Carlo (QMC) method is introduced using a constructed Kriging surrogate model. Sensitivity analysis is performed (this invention uses Sobol variance analysis to determine the degree of influence of each input variable on the fundamental frequency of antenna substrate vibration); the initial design space is reconstructed based on the sensitivity analysis results, sensitive variables are selected to form a new design space, while insensitive variables are always fixed at the mean;
[0061] Step 4: Optimize and reconstruct the antenna substrate within the design space
[0062] Step three significantly reduces the dimensionality of the variable space, substantially lowering the optimization difficulty. To further reduce the computational cost, this step uses Bayesian optimization to optimize the antenna substrate within the reconstruction space. As one of the most popular methods in Bayesian optimization, it aims to improve the global search of the sampling function of the EI sampling criterion to detect unexplored but promising regions, while also improving prediction accuracy. The sampling function of the EI sampling criterion is represented in closed-form:
[0063]
[0064] In equation (1), EI(x) is the EI function value corresponding to the observation point x, Φ(·) and φ(·) are the cumulative distribution and probability density function of the standard normal distribution, respectively, and y min To evaluate the minimum value in the sample, For the Kriging surrogate model predictions, Predict the square root of the variance for the Kriging surrogate model;
[0065] For constrained optimization problems, the constraints include structural weight and thermally induced displacement of the structure. A surrogate model G for the constraint function is established. j (x), j = 1, ..., n g , where n g Let G represent the number of constraint functions. Suppose the random variable G corresponds to the j-th constraint function. j (x) follows the mean of The standard deviation is s g,j If (x) follows a normal distribution, then the probability P[G] satisfies the expectation. j [x)≥0] is calculated using the following formula (2):
[0066]
[0067] The constrained EI sampling function CEI(x) is expressed as:
[0068]
[0069] Specific Implementation Method Two: This implementation method is a further explanation of Specific Implementation Method One. In step two, the process of adaptively constructing the Kriging proxy model within the active subspace is as follows:
[0070] Step 21: Use the relation in formula (13) to map all samples in S0 to the active subspace u, and denote the new sample set as S. 0,u From S 0,u We extract N2 points as initial samples. For the k-th initial sample u k We can map it back to the initial variable space using formula (13) to obtain x. kThe fundamental frequency y of the antenna substrate was obtained through finite element analysis. k Thus forming the initial training dataset T = {u k ,y k};
[0071] Step 22: Train the Kriging surrogate model using the initial training dataset T to obtain the surrogate model.
[0072] Steps 2 and 3: Using the proxy model Estimate S 0,u The MSE of all samples in the sample is used to identify new points using formula (14);
[0073] Step Two Four: The convergence criterion is defined as follows:
[0074]
[0075] In the formula, maxMSE represents S 0,u The maximum MSE of all samples, where the superscript indicates the number of iterations; Cr is the convergence threshold, which is 5 × 10⁻⁶. -6 ;
[0076] When the convergence criterion is not met (i.e., formula (15) does not hold), u new Mapped to the initial input space, the fundamental vibration frequency y of the antenna substrate is calculated by calling the finite element method. new Then the new training point {u new ,y new} Add to T for updating, and retrain the Kriging surrogate model in step 22; if the convergence criterion (i.e., formula (15) holds) is met, the Kriging surrogate model is considered to meet the accuracy requirement.
[0077] Example 1:
[0078] This embodiment discloses a machine learning-based method for optimizing the fundamental frequency of antenna substrate vibration. The method includes the following steps:
[0079] Step 1: Select the structural components of the antenna substrate as the design variables to be optimized, provide the structural geometry data and boundary conditions related to the design variables, and set the constraints related to optimization. For the specified design variables, within a given variable range, use some existing efficient sampling methods, such as Sobol sampling, to construct a large initial sample space. Randomly select a finite number of samples from the initial sample space, and use the finite difference method to estimate the gradient vector of the first-order fundamental frequency of the selected sample points. Identify active subspaces according to active subspace theory, and establish a mapping relationship between the solar panel variables and the original input space within the subspaces.
[0080] Step Two: Within the active subspace identified in Step One, an adaptive Kriging surrogate model for the first-order fundamental frequency of the antenna substrate is constructed. Unlike traditional methods for constructing surrogate models, this invention uses an adaptive sampling function to select the point with the largest prediction variance among the unobserved points provided by the Kriging surrogate model as a new sample point, and then reconstructs a new Kriging surrogate model. This iterative process is considered to have met the accuracy requirements of the Kriging surrogate model for the first-order fundamental frequency of the antenna substrate once the convergence condition is met, and the model update stops.
[0081] Step 3: Using the Kriging surrogate model of the antenna substrate's first-order fundamental frequency constructed in Step 2, global sensitivity analysis is performed using the Sobol variance analysis method. To improve the computational efficiency of the sensitivity analysis, a quasi-Monte Carlo method is used. After obtaining the sensitivity analysis results of the antenna substrate's first-order fundamental frequency, the sensitivity index values are sorted, and the reconstruction space is determined based on the differences between the index values.
[0082] Step 4: In the reconstruction space of Step 3, the first-order fundamental frequency of the antenna substrate is optimized. This invention utilizes the EI sampling criterion, which is the most widely used in Bayesian optimization, to select the point in the reconstruction space that maximizes the expected improvement of the optimization objective. This global search detects unexplored but promising regions and improves prediction accuracy. The sampling function of the EI sampling criterion can be expressed in closed form, as shown in Equation (1):
[0083]
[0084] In equation (1), EI(x) is the EI function value corresponding to the observation point x, Φ(·) and φ(·) are the cumulative distribution and probability density function of the standard normal distribution, respectively, and y min To evaluate the minimum value in the sample, For the Kriging surrogate model predictions, The square root of the variance predicted for the Kriging surrogate model.
[0085] For the constrained optimization problem in this invention, a surrogate model of the constraint function can be established. Where, n g Let G represent the number of constraint functions, and assume that the random variable corresponding to each constraint function is G. j (x) follows the mean of The standard deviation is s g,j If (x) follows a normal distribution, then the probability P[G] satisfies the expectation. j [x)≥0] can be calculated using the following formula (2):
[0086]
[0087] Furthermore, the constrained EI sampling function CEI(x) is expressed as:
[0088]
[0089] Once the termination condition is met, the optimal design point corresponding to the first-order fundamental frequency of the antenna substrate is considered to have been obtained, and the optimization stops at this point.
[0090] The active subspace theory used in step one is as follows:
[0091] For an n-dimensional problem, the active subspace is represented by the eigenvectors of an n×n symmetric positive semidefinite matrix:
[0092]
[0093] In equation (4), C is the covariance matrix. It is the gradient vector, ρ(x) is the probability density function (PDF) of x, and W = [w1,...,w n ] is the orthogonal matrix of eigenvectors, Λ=diag(λ1,...,λ n ) is a diagonal matrix composed of eigenvalues in descending order, and the superscript T indicates transpose.
[0094] This invention uses the first-order finite difference method to estimate the gradient. After estimating with N1 initial samples, the covariance matrix C is calculated as follows:
[0095]
[0096] In equation (5), the superscript “^” indicates the estimated value of the quantity corresponding to equation (4).
[0097] The criteria for selecting the value of N1 are as follows:
[0098] N1=αmlog(n) (6)
[0099] In equation (6), α is the sampling factor, and its recommended value is selected between 2 and 10; m is the estimated dimension of the active subspace, and n is the dimension of the original input space. The computational cost of estimating the covariance matrix C using the finite difference method is N1(n+1).
[0100] In steps two and four above, the Kriging proxy model is constructed based on the following sub-steps:
[0101] Gaussian processes (GPs) are unbiased estimation models designed to minimize the variance of the estimates.
[0102] Y(x)=β0+Z(x) (7)
[0103] In equation (7), Y(x) is an unknown Gaussian process model, β0 is an unknown constant, Z(x) is a stationary random process with zero mean, and its covariance is:
[0104] Cov[(Z(x),Z(x'))]=σ 2 R(x,x') (8)
[0105] In equation (8), σ 2 Z(x) represents the process variance; R(x,x') is the kernel function, which depends only on the distance between the two design points x and x'.
[0106] Given sample points X = {x1, x2, ..., x...} n} and its response f(X)={f(x1),f(x2),…,f(x n If )}, then the predicted response and prediction variance can be expressed as:
[0107]
[0108] In equation (9), β0 is the predicted mean; r is the correlation vector between the predicted points and the sampled points; R is the correlation matrix between the sampled points.
[0109]
[0110] In the formula, To predict variance.
[0111] The following describes the model of Example 1 in detail, including structure, materials, optimization objectives, constraints, and design variables, but does not involve algorithm content:
[0112] More specifically, for such Figure 1 The diagram illustrates a large-area antenna substrate structure, comprising unit substrates, hinges, and a satellite center rigid body. It consists of 152 unit substrates, each 650mm in length; each unit substrate is constructed from two layers of boards sandwiching an aluminum honeycomb panel. Figure 2 , Figure 3 As shown, an aluminum honeycomb panel with height H consists of components with height H... c The cellular core unit 1 and the height of H f The panel 2 is composed of a honeycomb core unit l, which is a regular hexagon, where L is the length of the honeycomb wall, and δ... c The thickness of the honeycomb wall; two symmetrically arranged hinges connect adjacent unit substrates, each hinge being 30mm long. All hinges in this structure have a cylindrical cross-section, with an outer radius of R. h The unit is mm, and the wall thickness is δ. h Unit: mm.
[0113] (1) Equivalent material parameters of the honeycomb core unit
[0114] The material properties of the antenna substrate are shown in Table 1. Specifically, the honeycomb core unit 1 is made of aluminum, and its equivalent material properties can be calculated using the Gibson formula. For a regular hexagonal honeycomb core unit 1, the equivalent material parameters of the honeycomb core unit 1 are expressed as follows:
[0115]
[0116] In equation (11), E cx E cy E cz G represents the equivalent modulus of the honeycomb core in the x, y, and z directions, respectively; cxy G cxz G cyz μ1, μ2, and μ3 represent the shear modulus of the honeycomb core in the xy, xz, and yz planes, respectively; μ1, μ2, and μ3 represent the equivalent Poisson's ratios of the honeycomb core in the x, y, and z directions, respectively; t and l represent the wall thickness and wall length of the honeycomb core, respectively; ρ and ρ c These represent the density of the honeycomb core material and the equivalent density of the honeycomb core, respectively.
[0117] Furthermore, the thermal conductivity of the honeycomb core unit 1 is related to its structure. Ignoring radiative heat transfer within the panel 2, the effective thermal conductivity is expressed as:
[0118]
[0119] In equation (12), k represents the thermal conductivity of the honeycomb core cell material.
[0120] (2) Optimize objectives, constraints and design variables
[0121] Considering the relatively high first-order natural frequency freq 1st It is beneficial for the stability control of spacecraft, therefore it is chosen as the objective function. The objective function is defined as follows:
[0122] Furthermore, the design of large antenna substrates must also meet structural and performance constraints. The first constraint is that the structural mass must be less than the permissible value. Additionally, when operating in orbit, especially when the spacecraft enters or exits Earth's shadow, the structure will be subjected to periodic or uneven lighting, such as… Figure 4 As shown, the bending moment generated by the heat flow (thermal bending moment) causes a temperature gradient in the cross-section, leading to out-of-plane deformation in large-span structures. Therefore, the second constraint is that the structural deformation caused by the thermal bending moment must be less than the permissible value.
[0123] Structural mass can be obtained through simple linear addition, while structural thermal deformation is obtained through thermal-structural analysis. Structural thermal analysis can be divided into two steps: (1) Calculate the transient temperature field based on the heat flux and the equivalent thermal conductivity of the honeycomb core unit 1 material; (2) Calculate the structural thermal deformation by using the transient temperature field as a thermal load.
[0124] Assume the spacecraft is initially located within the sunlit region at time t0 = 0; after orbiting for t1 = 1829 s, the spacecraft transitions into the semi-shadowed region, as... Figure 4 As shown, the spacecraft receives only partial sunlight in the semi-shaded area; at t2 = 1837.5s, it enters a completely shaded area; approximately 2117s later, it leaves the shaded area and re-enters the penumbra at t3 = 3954.5s; at t4 = 3963s, it is fully exposed to sunlight again. One orbit takes 5792 seconds. In near-Earth orbit, the spacecraft is affected by the thermal environment, with a solar radiation heat flux of approximately 1350 Wm². -2 Earth's radiation is 138.57 Wm. -2 Assuming the initial temperature of the structure is 293.15 K, the ambient temperature is 4 K, and the Stephan-Boltzmann constant is set to 5.67 x 10⁻⁶. -8 Wm -2 K -4 .
[0125] Figure 5 The thermal radiation period of the upper and lower surfaces is shown. Figure 6 The diagram shows thermally induced structural bending deformation, in which thermally induced vibrations are clearly observed, with the maximum amplitude being approximately equal in different periods. Therefore, considering computational efficiency, the thermal displacement of the structure is calculated only in the first period.
[0126] Secondly, select the hinge distribution location and the hinge outer circle radius r. h Wall thickness δ h The height H of the cellular core unit c Panel height H f The length L and thickness δ of the honeycomb wall c As the initial design parameters, and considering the overall structure, all honeycomb panels are required to have the same design, and all hinges are required to have the same geometric parameters. Taking into account the overall symmetry of the structure and the symmetry of the hinges on the same side, 1 / 4 of the total number of honeycomb panels and 1 / 8 of the total number of hinges are selected as design variables.
[0127] In summary, the design space contains a total of 80 dimensions, of which 74 dimensions define the position of the hinge, 2 dimensions describe the geometry of the hinge, and 4 dimensions describe the geometry of the honeycomb panel.
[0128] like Figure 7 As shown in the figure, this embodiment describes a method for optimizing the fundamental frequency of antenna substrate vibration based on machine learning. First, the estimated value of the covariance matrix is obtained according to formula (5). Eigenvalue decomposition yielded the most active eigendirections and their corresponding eigenvalues, which are arranged in descending order as follows: Figure 8 As shown. An adaptive Kriging surrogate model is constructed in a one-dimensional active subspace, and global sensitivity analysis is performed using the Kriging surrogate model. The results are as follows. Figure 9 As shown, it is clear that the height of the honeycomb core, the thickness of the honeycomb wall, and the length are the top three variables affecting the first natural frequency.
[0129] The height of the honeycomb core, the thickness and length of the honeycomb wall are selected as the reconstruction space, and the boundaries of other variables are shown in Table 2. Bayesian optimization is performed on the fundamental frequency of the antenna substrate within the reconstruction space, considering structural weight and maximum displacement constraints. To compare the effectiveness of the proposed method, optimization is also performed in the original space. The results of the two optimizations are as follows: Figure 10 As shown, the first-order natural frequency of the original model is 0.4572 Hz, and after dimensionality reduction optimization, the first-order natural frequency is 0.6308 Hz, an improvement of 37.97%. Furthermore, from... Figure 10 It can be seen that the number of iterations after dimensionality reduction is reduced from 658 to 50, and the cost is only 50% of the original. The optimized cellular geometry data is shown in Table 3.
[0130] This invention utilizes an active subspace for dimensionality reduction, adaptively constructs a surrogate model of the antenna substrate within the active space, and performs global sensitivity analysis to reconstruct the dimension (variable selection). The final optimized configuration is then obtained based on a Bayesian method. The proposed antenna substrate vibration fundamental frequency optimization method significantly improves optimization efficiency while overcoming the premature convergence problem that may arise from high-dimensional spaces, making it significant for solving the optimization design of large-scale antenna substrates with complex constraints.
[0131] Table 1 (Properties of different materials for antenna substrates);
[0132]
[0133]
[0134] Table 2 (Design Variable Limit Values);
[0135]
[0136] Table 3 (Optimized Cellular Geometric Parameters)
[0137]
[0138] This invention combines sensitivity analysis based on active subspace and adaptive point-addition strategy with Bayesian optimization to establish a machine learning-based global optimization framework for the fundamental frequency of antenna substrate vibration. This framework can efficiently complete the global optimization design of the fundamental frequency of large antenna substrates. Optimization verification on large-scale antenna substrates with complex constraints demonstrates that the method overcomes the problem of premature convergence in traditional methods and achieves a better fundamental frequency design with less computational time, significantly improving the efficiency of fundamental frequency optimization for antenna substrates.
[0139] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.
[0140] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.
Claims
1. A method for optimizing the fundamental frequency of antenna substrate vibration based on machine learning, characterized in that: The method includes the following steps: Step 1: Identify active subspaces N0 samples are sampled from the initial design space of the antenna substrate to form a set, denoted as the initial sample set S0; then N1 samples are selected from S0, denoted as S1; the gradient vector of the fundamental frequency of the antenna substrate vibration of the samples in set S1 is estimated using finite element simulation and finite difference method. By using the gradient vectors of these S1 sample points The elements of the covariance matrix C are estimated by calculating the average value, and eigenvalues and eigenvectors are obtained by performing eigenvalue decomposition on the covariance matrix C. Active subspaces are identified by analyzing the relative magnitudes of the eigenvalues. After finding the active subspaces, the mapping between the original input variable x0 and the variables in the active subspace is approximated as follows: x0=W1u+W2z≈W1u (13) In equation (13), W1 and W2 are the active subspace vector and the inactive subspace vector, respectively, and u and z are the active variable and the inactive variable, respectively; Step 2: Adaptively construct the proxy model within the active subspace Adaptively construct a Kriging surrogate model for the fundamental frequency of antenna substrate vibration within the active subspace. Using the unobserved point MSE estimated by the Kriging surrogate model, the new sample point u new The point corresponding to the largest MSE: In equation (14), S 0,u This represents the mapping of the initial sample set S0 within the subspace. Step 3: Use the Kriging surrogate model to estimate the sensitivity index. Introducing a quasi-Monte Carlo method using a constructed Kriging proxy model Sensitivity analysis is performed, and the initial design space is reconstructed based on the results. Sensitive variables are selected to form a new design space, while insensitive variables are always fixed at the mean. Step 4: Optimize and reconstruct the antenna substrate within the design space Bayesian optimization is used to optimize the antenna substrate within the reconstruction space, aiming to improve the global search of the sampling function of the EI sampling criterion to detect unexplored but promising regions, while improving prediction accuracy. The sampling function of the EI sampling criterion is expressed in closed form: In equation (1), EI(x) is the EI function value corresponding to the observation point x, Φ(·) and φ(·) are the cumulative distribution and probability density function of the standard normal distribution, respectively, and y min To evaluate the minimum value in the sample, For the Kriging surrogate model predictions, Predict the square root of the variance for the Kriging surrogate model; For constrained optimization problems, the constraints include structural weight and thermally induced displacement of the structure. A surrogate model G for the constraint function is established. j (x), j = 1, ..., n g , where n g Let G represent the number of constraint functions. Suppose the random variable G corresponds to the j-th constraint function. j (x) follows the mean of The standard deviation is s g,j If (x) follows a normal distribution, then the probability P[G] satisfies the expectation. j [x)≥0] is calculated using the following formula (2): The constrained EI sampling function CEI(x) is expressed as:
2. The method for optimizing the fundamental frequency of antenna substrate vibration based on machine learning according to claim 1, characterized in that: In step two, the process of adaptively constructing the Kriging proxy model within the active subspace is as follows: Step 21: Use the relation in formula (13) to map all samples in S0 to the active subspace u, and denote the new sample set as S. 0,u From S 0,u We extract N2 points as initial samples. For the k-th initial sample u k We can map it back to the initial variable space using formula (13) to obtain x. k The fundamental frequency y of the antenna substrate was obtained through finite element analysis. k Thus forming the initial training dataset T = {u k ,y k }; Step 22: Train the Kriging surrogate model using the initial training dataset T to obtain the surrogate model. Steps 2 and 3: Using the proxy model Estimate S 0,u The MSE of all samples in the sample is used to identify new points using formula (14); Step Two Four: The convergence criterion is defined as follows: In the formula, maxMSE represents S 0,u The maximum MSE of all samples, where the superscript indicates the number of iterations; Cr is the convergence threshold, which is 5 × 10⁻⁶. -6 ; When the convergence criterion is not met, u new Mapped to the initial input space, the fundamental vibration frequency y of the antenna substrate is calculated by calling the finite element method. new Then the new training point {u new ,y new } Add to T for updating, and retrain the Kriging surrogate model in step 22; if the convergence criterion is met, the Kriging surrogate model is considered to meet the accuracy requirement.