Efficient global optimization method for aviation reducers assisted by partial least squares Kriging model

Abstract

The present invention proposes an efficient global optimization method for aviation reducers assisted by a partial least squares Kriging model, performs initial test design; uses a computer to perform simulation calculations, obtains the target response value of the total weight of the gearbox and the constraint response values ​​of the gear, tooth root, and tooth surface strength, and establishes a DS database; performs normalization processing and partial least squares transformation on the DS database, determines the number of principal components to construct a partial least squares Kriging kernel function; constructs a feasibility probability strategy to implement sample filling, or constructs a maximized partial least squares constraint weight expectation improvement criterion to implement spatial sample filling, and places the filled sample data and its corresponding simulation output target value and response value in the DS database; iteratively updates the DS database until the maximum number of iterations is reached or the threshold of the maximum constraint expectation improvement criterion is met. The present invention can realize the efficient construction and adaptive adjustment of the Kriging proxy model under multivariable conditions, achieve rapid convergence of the global optimal solution, and reduce expensive simulation costs.

Description

Technical Field

[0001] The invention relates to computer simulation and engineering optimization calculation, and in particular to an efficient global optimization method for aviation reducers assisted by a partial least squares Kriging model. Background Art

[0002] In the context of global competition, enterprise product development and product verification are increasingly dependent on computer experiments. Efficient design methods, simulation technologies and optimization methods have become a strong support for enterprises to occupy the market. Although the application of high-precision simulation technology has effectively improved the quality and reliability of computer experiments, complex product design and system optimization problems often require a lot of simulation time. In addition, the complex inner code of high-precision simulation models and the closed nature of commercial simulation software make the input and output present a "black box" functional relationship, which makes traditional meta-heuristic methods such as differential evolution algorithms and particle swarm algorithms difficult to implement. Therefore, relying on limited test data to establish a high-precision approximate proxy model to achieve efficient global optimization of constrained optimization problems has become an important research direction.

[0003] The surrogate optimization method refers to establishing a surrogate model of the objective function and constraint function, driving new samples to fill in the space through historical data, and constantly updating the surrogate model until the generated sample point sequence approaches the local or global optimal solution. The key to success lies in the selection of sample points, the form of the model and the point addition criteria. The gradual increase in the complexity of modern products has made high-dimensional (multi-independent variable) problems more and more common, resulting in feature mapping technologies such as principal component analysis (PCA), partial least squares (PLS) and kernel PLS. Among them, the PLS method concentrates on the characteristics of principal component analysis, canonical correlation analysis and linear regression analysis, and its unique dimensionality reduction characteristics are applied to the construction of surrogate model kernel functions.

[0004] The surrogate optimization method has improved the frequency of use and solution efficiency of the Kriging model in engineering optimization problems, but in the process of high-dimensional data modeling, the training time is too long due to too many hyperparameters. In addition, for the Kriging model, the selection of the kernel function has a crucial impact on the prediction effect. Therefore, the development of efficient kernel functions can help improve the model prediction accuracy and Kriging modeling efficiency, while the development of efficient filling criteria can help promote the practical application of the EGO method in high-dimensional optimization problems. An algorithm for dimensionality reduction of the initial test sample point set through partial least squares transformation needs to be developed urgently. Summary of the invention

[0005] The purpose of the present invention is to propose an efficient global optimization method for aviation reducers assisted by a partial least squares Kriging model to solve the problems of long training time, expensive simulation time-consuming and low computing resource utilization in the high-dimensional data modeling process in the design of aviation reducers.

[0006] The technical solution to achieve the purpose of the present invention is: a partial least squares Kriging model-assisted efficient global optimization method for aviation reducers, comprising the following steps:

[0007] Step 1: For a total of 7 design parameters of the aviation reducer, namely, tooth surface meshing degree, gear module, pinion module, bearing spacing, and gear and pinion diameters, the maximum and minimum Latin hypercube sampling method is used to perform initial experimental design to obtain an initialization parameter sample;

[0008] Step 2: Based on the information of the aviation reducer problem, the optimization target is to minimize the total weight of the gearbox, and the strength of the gear, tooth root, and tooth surface is used as a constraint to construct an aviation reducer optimization model. The computer is used to perform simulation calculations to obtain the target response value of the total weight of the gearbox and the constraint response values ​​of the gear, tooth root, and tooth surface strength. The initialization parameter samples and the obtained target and constraint response values ​​are saved to establish a DS database.

[0009] Step 3, normalize the data in the DS database, perform partial least squares transformation on the normalized parameter samples and response values, determine the number of principal components based on the cumulative probability, and then construct the partial least squares Kriging kernel function;

[0010] Step 4: Based on the partial least squares Kriging kernel function, the partial least squares Kriging surrogate models of the target and constraints are constructed respectively to obtain the predicted mean and predicted variance of the target response and the constraints;

[0011] Step 5, determine whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, construct a feasible probability strategy to achieve sample filling. Otherwise, construct a maximum partial least squares constraint weight expectation improvement criterion to achieve spatial sample filling, and use a computer to obtain the target and constraint response values ​​corresponding to the new sample, and place the filled sample data and its corresponding simulation output target value and response value in the DS database;

[0012] Step 6, iterate steps 3-5 in a loop to update the DS database until the maximum number of iterations is reached or the threshold of the maximum constraint expected improvement criterion is met, thereby obtaining the minimum target value and the corresponding optimal parameter combination.

[0013] Further, in step 2, taking the lightest total weight of the gearbox as the optimization goal and the gear, tooth root and tooth surface strength as constraints, an aviation reducer optimization model is constructed, and a computer is used to perform simulation calculations to obtain the target response value of the total weight of the gearbox and the constraint response values ​​of the gear, tooth root and tooth surface strength, and the initialization parameter samples and the obtained target and constraint response values ​​are saved to establish a DK database, in which the aviation reducer optimization model is expressed as:

[0014] min y(x)

[0015] stg i (x)≤0,i=1,2,…,r

[0016] x∈D

[0017] Where y(x) represents the total weight of the gearbox; g i (x) represents the i-th constraint; r represents the total number of constraints; D = [x l ,x r ] is the variable design space, x l ,x r Represent the left and right boundary vectors of each design parameter x respectively; finite element simulation software is used to perform simulation calculations to obtain the target response value set y and the constraint response value set g i , establish DS database.

[0018] Furthermore, in step 3, the data in the DS database are normalized, the normalized parameter samples and response values ​​are transformed by partial least squares, the number of principal components is determined according to the cumulative probability, and the partial least squares Kriging kernel function is constructed. The specific method is:

[0019] Perform normalization:

[0020]

[0021] Among them, x min ,x max Indicates the minimum and maximum values ​​of the design parameter x; x s To obtain the standardized data of the design parameter x, the target response value y is processed in the same way, that is, the standardized data of the target response value y is obtained;

[0022] Perform a partial least squares transformation:

[0023]

[0024]

[0025] in, and Respectively represent the cumulative contribution of the principal components to the variation of E0 and F0;

[0026] c x ,c y Indicates the contribution rate of a single variation explanation; E0, F0 are the standardized matrices of parameter sample X matrix and target response sample y; t ncomp is the nth comp principal components; For nth comp regression coefficient vectors; l is the number of principal components, determined based on the cumulative probability being greater than 0.7 and the contribution of a single variable being greater than 0.05;

[0027] Construct the partial least squares Kriging kernel function;

[0028]

[0029] Among them, η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of principal component systems selected after partial least squares transformation; x i is the i-th component of the design variable x; The weight coefficient of the i-th component of the design variable x after partial least squares transformation.

[0030] Furthermore, in step 4, based on the partial least squares Kriging kernel function, the partial least squares Kriging proxy models of the target and constraints are constructed respectively to obtain the predicted mean and predicted variance of the target response and the constraint. The specific method is as follows:

[0031] Assume that the parameter matrix X is initialized after normalization s =[x 1s ,x 2s ,…,x ms ] T The target response value matrix y=[y1,y2,…,y m ] T It is known that the partial least squares Kriging model is expressed as:

[0032] y(x)=μ+z(x s ),

[0033] Among them, μ is the trend term; z(x s ) is the estimated correction for the prediction, which has a mean of 0 and a covariance of Gaussian process; represents the Gaussian process z(x s ) is the variance of R(x,x m )=R(x,x m|θ) is the partial least squares Kriging kernel function selected for modeling;

[0034] Construct a partial least squares Kriging proxy model for the total weight of the aviation reducer:

[0035]

[0036]

[0037] in, and They represent the predicted mean and predicted variance of the total weight target response of the aviation reducer respectively; 1 represents a unit vector with 1 as its element; is the estimated value of the undetermined trend term coefficient μ; y=[y1,y2,…,y m ] T is the target response vector of the total weight of the reducer; m is the number of samples in the DS database; r(x s )=[R(x s ,x 1s ),R(x s ,x 2s ),…,R(x s ,x ms )] T represents the design vector determined by the regression basis function; Variance The estimated value of ; R is the correlation coefficient matrix, which is expressed as follows:

[0038]

[0039] Construct a constrained partial least squares Kriging surrogate model:

[0040]

[0041]

[0042] in, and They represent the predicted mean and predicted variance of the ith constraint respectively; g i is the response vector of the ith constraint; R gi is the correlation coefficient matrix of the i-th constraint; is the constraint undetermined trend term coefficient The estimated value of ; obtained by generalized least squares estimation, is the estimated variance of the i-th constrained Gaussian process distribution.

[0043] Further, step 5 determines whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, a criterion for maximizing the feasibility probability is constructed to achieve sample filling. Otherwise, an improved criterion for maximizing the partial least squares constraint weight expectation is constructed to achieve sample filling, and a computer is used to perform simulation calculations, and the filled sample data and its corresponding simulation output target value and response value are placed in the DS database. The specific method is:

[0044] Step 51: According to the target data y=[y1,y2,…,y n ] T , calculate the current optimal target value y min =min{y1,y2,…,y n};

[0045] Step 52: construct a feasibility probability criterion based on the predicted mean and predicted variance of the constraint, and construct a maximization constraint expected improvement criterion based on the predicted mean and predicted variance of the target response and the constraint;

[0046] assumed The feasibility probability function is expressed as follows:

[0047]

[0048] Where Φ(·) is the cumulative probability density function; r is the number of constraints; Respectively represent the response mean and standard deviation of the i-th constraint;

[0049] The criterion for maximizing feasibility probability is expressed as follows:

[0050]

[0051] Among them, x m+1 Represents the m+1th input sample parameter combination;

[0052] According to the target response and the predicted mean and predicted variance of the constraints, a maximization constraint expectation improvement criterion is constructed to obtain reference sample points for subsequent sample filling;

[0053]

[0054] Among them, x R is the new sample reference point, assuming The total weight target of the aviation reducer is to improve the increment Under these conditions, the criterion for maximizing EI points is expressed by the following function:

[0055]

[0056] Among them, y min is the current optimal target value, and represents the predicted value and predicted standard deviation, Φ(·) and φ(·) are the cumulative distribution function and the distribution probability density function; and Respectively characterize the local search and global exploration capabilities of the EI criterion;

[0057] According to the obtained reference points and the predicted mean and predicted variance provided by the Kriging model, two improvement criteria for maximizing the constraint weight expectation are constructed, which are expressed as follows:

[0058]

[0059] and

[0060]

[0061] in, and represents the expected improvement criterion of partial least squares constrained weights; represents the partial least squares weighted expected improvement function, represents the partial least squares pseudo expectation improvement function, PoF represents the probability of calculating the feasibility of the current test point; Kriging kernel function hyperparameters derived from Kriging modeling; d is the dimension of design variables; represents the kth dimension of the reference point; is the partial least squares weight function; η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of transformed principal components;

[0062] Step 53: Determine whether there is a feasible solution that satisfies all constraints in the DS database. If there is no feasible solution, sample filling is achieved according to the maximization of feasibility probability. Otherwise, a reference point is obtained according to the maximization constraint expectation improvement criterion, and then a new sample is obtained according to any maximization partial least squares constraint weight expectation improvement criterion to achieve sample filling.

[0063] Step 54: Use a computer to perform simulation calculations, and place the (m+1)th filled sample data and its corresponding simulation output aviation reducer total weight target value and corresponding constraint response values ​​in the DS database, and update the DS database.

[0064] A partial least squares Kriging model-assisted aviation reducer proxy optimization and system realizes aviation reducer proxy optimization design based on the partial least squares Kriging model-assisted aviation reducer efficient global optimization method.

[0065] A computer device comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the aircraft reducer optimization design is realized based on the efficient global optimization method of the aircraft reducer assisted by the partial least squares Kriging model.

[0066] A computer-readable storage medium stores a computer program. When the computer program is executed by a processor, the aircraft reducer is optimized based on the efficient global optimization method of the aircraft reducer assisted by the partial least squares Kriging model.

[0067] Compared with the prior art, the present invention has the following significant advantages: 1) The proposed efficient global optimization method assisted by the partial least squares Kriging model can be used for complex black-box constrained optimization problems, and has certain advantages in feasibility, modeling efficiency and robustness of solutions. In high-dimensional problems, the number of experiments can be effectively reduced, the calculation efficiency is faster, and it is more suitable for high-dimensional constrained optimization problems with black-box characteristics; 2) The partial least squares weighted expectation improvement criterion has good exploration capabilities for potential areas, and the selection of new sample points is better than the current optimal value and the best solution. By selecting the previous reference point as a calibration, fast search and global optimization are achieved; 3) The present invention constructs two partial least squares expectation filling criteria through the partial least squares method, so that it can effectively improve the model prediction accuracy and Kriging modeling efficiency when solving multi-parameter black-box optimization problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 This is a flow chart of the PKriging-assisted aviation reducer proxy optimization method described in the present invention.

[0069] Figure 2 This is a comparison chart of hyperparameters of the Kriging model and the PKriging model for the test problem of the present invention.

[0070] Figure 3 This is an optimization comparison chart of the KEGO and PKEGO methods for the test problem of the present invention.

[0071] Figure 4 A simplified schematic diagram of a pressure vessel constructed for the present invention. DETAILED DESCRIPTION

[0072] In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0073] The present invention sets the parameters of the optimization model and performs the maximum and minimum Latin hypercube (LHS) initial sampling through the black box problem information of the object to be optimized, obtains the target and constraint response values ​​through computer simulation, and stores the data in the DS data set; normalizes the data in the DS data set, performs partial least squares transformation, and determines the number of partial least squares principal components; constructs the partial least squares Kriging kernel function and the PKriging model; constructs the partial least squares weighted expected improvement criterion according to the PKriging model information and maximizes the criterion; judges whether the optimization process meets the set termination condition, and when it does not meet the condition, selects the optimal point as a new sample according to the number of added points information included in the parameter setting, and performs parallel evaluation on the new sample through the black box problem to obtain the new sample response value and update the DS data set; updates the PKriging model according to the updated DS data set, repeats the above steps until the optimization process meets the termination condition, and obtains the optimal target value and the best design parameter of the expensive constrained optimization problem.

[0074] It is worth noting that if there is no feasible solution in the initial experimental design, the sub-optimization process is implemented by maximizing the feasibility probability criterion until the feasible solution conditions are met and then switching to the partial least squares constrained weighted expectation improvement criterion for maximization; during the optimization process, the PKriging proxy model is used to replace time-consuming computer simulation, which can effectively improve the design efficiency and significantly reduce the computational cost.

[0075] like Figure 1 As shown in FIG. 1 , the efficient global optimization method of aviation reducer assisted by partial least squares Kriging model specifically includes the following steps:

[0076] Step 1: For the seven parameters of the aviation reducer, namely, the tooth surface meshing degree b, the large gear module n, the small gear module m, the bearing spacing l1, l2, and the large and small gear diameters d2, d1, the maximum and minimum LHS method is used for initial experimental design, and 5d design parameter samples are extracted;

[0077] The LHS method is a stratified sampling technique that approximates random sampling and is the most widely used in computer experimental design. In the implementation of the LHS method, sampling must meet the following requirements: (1) Each sample point is evenly distributed within a small interval; (2) All sample points are projected into any dimension. Suppose n sample points need to be selected, and the sample point x∈[0,1] d There is only one sample point in each of the n subintervals on the dimension. Divide the d-dimensional design space into n subspaces evenly along each dimension of the design variable: n subspaces are combined into d n subspace, and randomly select 1 sample in each d-dimensional subspace.

[0078] The maximum minimum LHS method selects the minimum distance of the other previous sample points as the characteristic distance of each extracted sample point, and maximizes the sample distance as much as possible. Make all the extracted sample points fill the entire sample space. Suppose the real sample x∈[x l ,x r ], then use the maximum and minimum LHS to extract all sample points x∈[0,1] d , so the following transformation is required to transform it into a real sample:

[0079] x=x l +(x r -x l )x

[0080] Step 2: Based on the parameter samples extracted in step 1, a functional evaluation is performed using computer simulation software to obtain the target response value of the total weight y(x) of the gearbox and the response values ​​of the gear, tooth root, and tooth surface strength constraints, and the initialized design parameter samples and the obtained total cost target and the response values ​​of each constraint are saved to establish a DS database;

[0081] The optimization model of aviation reducer is expressed as:

[0082] min y(x)

[0083] stg i (x)≤0,i=1,2,…,r

[0084] x∈[x l ,x r ]

[0085] Where y(x) represents the total weight of the gearbox; g i (x) represents the i-th constraint; r represents the total number of constraints; x l ,x r represent the left and right boundary vectors of each design parameter x respectively.

[0086] Finite element simulation software is used to perform simulation calculations, and the target response value set y and the constraint response value set g are obtained. i , establish DS database.

[0087] Step 3: Normalize the data in the DS database, set the cumulative probability threshold as the criterion for determining the number of principal components, perform partial least squares transformation on the normalized input parameter sample data and simulation output response data, and determine the number of principal components based on the cumulative probability. The specific method is:

[0088] Normalize the data in the DS database:

[0089]

[0090] Among them, x min ,x max Indicates the minimum and maximum values ​​of data x; x s is the standardized data of data x.

[0091] Perform partial least squares transformation on the data in the normalized DS database

[0092]

[0093]

[0094] in, and Respectively represent the cumulative contribution of the principal components to the variation of E0 and F0; c x ,c y Indicates the contribution rate of a single variation explanation; E0, F0 are the standardized matrices of parameter sample X matrix and target response sample y; t ncomp is the nth comp principal components; For nth comp regression coefficient vector; l is the number of principal components, which is determined based on the cumulative probability being greater than 0.7 and the contribution of a single variable being greater than 0.05.

[0095] Step 4: Determine the transformation coefficient and the number of principal components according to the changes in step 3, and construct the partial least squares Kriging kernel function;

[0096]

[0097] Among them, η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of principal component systems selected after partial least squares transformation; x i is the i-th component of the design variable x; The weight coefficient of the i-th component of the design variable x after partial least squares transformation.

[0098] Step 5: Establish a partial least squares Kriging surrogate model based on the data in the DS database to obtain the predicted mean and predicted variance of the target response and constraints. The specific method is as follows:

[0099] Assume that the parameter matrix X is initialized after normalization s =[x 1s ,x 2s ,…,x ms ] T The target response value matrix y=[y1,y2,…,y m ] TIt is known that the partial least squares Kriging model can be expressed as:

[0100] y(x)=μ+z(x s ),

[0101] Among them, μ is the trend term; z(x s ) is the estimated correction for the prediction, which has a mean of 0 and a covariance of Gaussian process; represents the variance of the Gaussian process z(x); R(x,x m )=R(x,x m |θ) is the partial least squares Kriging kernel function selected for modeling;

[0102] Construct a partial least squares Kriging proxy model for the total weight of the aviation reducer:

[0103]

[0104]

[0105] in, and They represent the predicted mean and predicted variance of the total weight target response of the aviation reducer respectively; 1 represents a unit vector with 1 as its element; is the estimated value of the undetermined trend term coefficient μ; y=[y1,y2,…,y m ] T is the target response vector of the total weight of the reducer; m is the number of samples in the DS database; r(x s )=[R(x s ,x 1s ),R(x s ,x 2s ),…,R(x s ,x ms )] T represents the design vector determined by the regression basis function; Variance The estimated value of ; R is the correlation coefficient matrix, which is expressed as follows:

[0106]

[0107] Construct a constrained partial least squares Kriging surrogate model:

[0108]

[0109]

[0110] in, and They represent the predicted mean and predicted variance of the ith constraint respectively; g i is the response vector of the ith constraint; R gi is the correlation coefficient matrix of the i-th constraint; is the constraint undetermined trend term coefficient The estimated value of ; obtained by generalized least squares estimation, is the estimated variance of the i-th constrained Gaussian process distribution.

[0111] Step 6: Determine whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, construct a feasibility probability criterion based on the constrained predicted mean and predicted variance to achieve sample filling. Otherwise, construct a maximization partial least squares constrained weight expectation improvement criterion to achieve sample filling, and use a computer to perform simulation calculations, and place the filled sample data and its corresponding simulation output target value and response value in the DS database. The specific method is:

[0112] Step 61: According to the target data y=[y1,y2,…,y n ] T , calculate the current optimal target value y min =min{y1,y2,…,y n};

[0113] Step 62: Construct a feasibility probability criterion based on the constrained forecast mean and forecast variance, assuming The feasibility probability function is expressed as:

[0114]

[0115] Where Φ(·) is the cumulative probability density function; r is the number of constraints; Respectively represent the response mean and standard deviation of the i-th constraint;

[0116] The criterion for maximizing feasibility probability can be expressed as follows:

[0117]

[0118] Among them, x m+1 Represents the m+1th input sample parameter combination;

[0119] Step 63: construct a maximization constraint expectation improvement criterion based on the target response and the predicted mean and predicted variance of the constraint, obtain reference sample points and use them to implement subsequent sample filling;

[0120]

[0121] Among them, x R is the new sample reference point, assuming The total weight target of aviation reducers is improving Under these conditions, the criterion for maximizing EI points can be expressed by the following function:

[0122]

[0123] Among them, y min is the current optimal target value, and represents the predicted value and the predicted standard deviation. Φ(·) and φ(·) are the cumulative distribution function and the distribution probability density function; and Respectively characterize the local search and global exploration capabilities of the EI criterion.

[0124] Step 64: Based on the obtained reference points and the predicted mean and predicted variance provided by the Kriging model, two maximization constraint weight expectation improvement criteria can be obtained, which are expressed as follows:

[0125]

[0126] and

[0127]

[0128] in, and represents the expected improvement criterion of partial least squares constrained weights; represents the partial least squares weighted expected improvement function, represents the partial least squares pseudo expectation improvement function, PoF represents the probability of calculating the feasibility of the current test point; Kriging kernel function hyperparameters derived from Kriging modeling; d is the dimension of design variables; represents the kth dimension of the reference point; is the partial least squares weight function; η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of transformed principal components;

[0129] Step 65: Determine whether there is a feasible solution that satisfies all constraints in the DS database. If there is no feasible solution, sample filling is implemented according to the maximum PoF criterion. Otherwise, the reference point x is obtained according to the maximum constraint expectation improvement criterion. R , and then obtain new samples to achieve sample filling according to any maximization partial least squares constraint weight expectation improvement criterion;

[0130] Step 66: Use a computer to perform simulation calculations, and place the (m+1)th filled sample data and its corresponding simulation output aviation reducer total weight target value and corresponding constraint response values ​​in the DS database, and update the DS database.

[0131] Step 7: The termination condition can be set according to the computing resources or according to the threshold of the maximum constraint expected improvement criterion. If it is set according to the computing resources in the project, let T max It represents the maximum number of iterations set in the optimization process. The DS database is updated in the iterative loop steps 3-6 until the maximum number of iterations is reached or the threshold of the maximum constraint expected improvement criterion is met. Based on this, the minimum target value of the aviation reducer and the corresponding optimal parameter combination x are obtained. * If the termination condition is set according to the threshold of the maximum constraint expectation improvement criterion, it can be expressed as:

[0132] CEI max ≤10 -5 ,

[0133] Among them, 10 -5 Improvement threshold set for CEI max The maximum value of the constraint expectation improvement criterion.

[0134] It is worth noting that the proxy model referred to in the present invention is the PKriging model, and the proposed optimization strategy is also applicable to other proxy models and prediction problems under the Bayesian framework; the evolutionary algorithm of the sub-optimization process of the maximization optimization criterion of the present invention can select methods such as particle swarm and differential evolution. In addition, methods such as grid search and NAGA-II can also be used to solve the sub-optimization problem; the modeling process of the present invention constructs a partial least squares Gaussian kernel function based on the Gaussian kernel function, and it can also extend this method to other functions such as cubic spline kernel function, linear kernel function and Mátern series kernel function; the present invention mainly involves the application field of expensive constrained optimization problems in engineering problems without explicit expressions, and it is also applicable to unconstrained expensive optimization problems.

[0135] Example

[0136] In order to verify the effectiveness of the solution of the present invention, the following experiment was conducted.

[0137] In this embodiment, Figure 4 The aviation reducer design case shown in the figure is tested, and the efficient global optimization method of aviation reducer assisted by partial least squares Kriging model specifically includes the following steps:

[0138] Step 1, according to Figure 4The design diagram of the aviation reducer shown in the figure adopts the maximum and minimum LHS sampling technology to perform initial space filling design on 7 parameters including tooth surface meshing degree b, large gear module n, small gear module m, bearing spacing l1, l2 and large and small gear diameters d2, d1, and extract 5×7=35 design parameter samples;

[0139] Step 2: Use computer simulation modeling technology to perform functional evaluation to obtain the total weight y of the gearbox w (x) target response value and gear tooth root and tooth surface strength constraint response value, and save the initialization parameter sample and the obtained target and constraint response values ​​to establish a sample database DS;

[0140] Step 3: Normalize the data in the DS database and set the cumulative probability c x =c y >0.05 is used as the criterion for determining the principal components. The input parameter sample data and the simulation output response data are transformed by partial least squares, and the number of principal components is determined based on the cumulative probability;

[0141] Step 4, determine the transformation coefficient and the number of principal components according to the changes in step 3, and construct the partial least squares Kriging kernel function;

[0142] Step 5: Establish a partial least squares Kriging surrogate model based on the data in the DS database to obtain the predicted mean and predicted variance of the target response and constraints;

[0143] In order to better test the prediction accuracy of the PKriging model, the maximum absolute error (MAE), average absolute error (AAE) and root mean square error (RMSE) are selected as model regression (prediction) indicators:

[0144]

[0145] Among them, n e is the number of test samples; y i , Point x i The true response value at and the model predicted response value.

[0146] The accuracy changes when the initial sample number is 35 and after increasing the sample number by 5, respectively, the PKriging and Kriging models of the aviation reducer are established, and their prediction accuracy and the number of hyperparameters are compared. The specific results are shown in Table 1

[0147] Table 1 Comparison of regression (prediction) results of Kriging modeling with different kernel functions

[0148]

[0149]

[0150] It can be seen from Table 1 that the indicators of the base PKriging model are better than those of the Kriging model, and the number of hyperparameter estimates is reduced from 7 to 1. The above results show that the PKriging model can effectively realize model prediction and is more suitable for high-dimensional (multivariable) expensive constrained optimization problems. In addition, by comparing the data before and after the addition of the test points, the space filling strategy can effectively improve the model prediction accuracy.

[0151] Step 6, determine whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, construct a feasible probability strategy to achieve sample filling. Otherwise, construct a maximum partial least squares constraint weight expectation improvement criterion to achieve spatial sample filling, and use a computer to obtain the target and constraint response values ​​corresponding to the new sample, and place the filled sample data and its corresponding simulation output target value and response value in the DS database;

[0152] Step 7: Set the total number of simulation tests T according to the computing resources. max =100, when T≥T max When satisfied, the method stops, 10 -5 To set the improvement threshold, CEI max To constrain the maximum value of the expected improvement criterion, when CEI max ≤10 -5 When it is satisfied, the method stops; otherwise, the DS database is updated by loop iteration steps 3-6 until the maximum number of iterations is reached or the threshold of the maximum constraint expected improvement criterion is met, thereby obtaining the minimum target value and the corresponding optimal parameter combination.

[0153] In order to better illustrate the proposed strategy, the classical efficient global optimization method (EGO) and the PKriging assisted EGO method (PKEGO) are selected to optimize the aviation reducer problem. The worst (max y), average (max y) and average (max y) values ​​of 30 test results of aviation reducer engineering examples are statistically analyzed. and standard deviation (σ y ), see Table 2

[0154] Table 2 Comparison of statistical results of optimal solutions of aviation reducer examples in different optimization methods

[0155]

[0156] From the results in Table 2, we can see that: comparing the three indicators of different methods to obtain the approximate optimal value of the engineering case, it can be seen that the PKEGO method is better than the EGO method in terms of accuracy and robustness in obtaining the approximate optimal solution; the PKEGO method can also efficiently obtain the optimal solution and has certain advantages. In addition, considering the black box characteristics and expensive simulation costs in actual engineering problems, the PKEGO method is more suitable for solving expensive black box constrained optimization problems.

[0157] In order to better illustrate the effectiveness of the weighted expectation filling criterion, G7 with 10 variables, Hesse with 6 variables and the first 7 test functions of CEC2017 are selected for experimental verification. The forms of G7 and Hesse test functions are as follows:

[0158]

[0159] st:g1(x)=-105+4x1+5x2-3x7+9x8≤0;

[0160] g2(x)=10x1-8x2-17x7+2x8≤0;

[0161] g3(x)=-8x1+2x2+5x9-2x 10 -12≤0;

[0162]

[0163]

[0164]

[0165]

[0166] g8(x)=-3x1+6x2+12(x9-8) 2 -7x 10 ≤0, -10≤x i ≤0(i=1,2,…,10)

[0167] Hesse: min f(x) = -25(x1-2) 2 -(x2-2) 2 -(x3-1) 2 -(x4-4) 2 -(x5-1) 2 -(x6-4) 2 ,

[0168] st:g1(x)=2-x1-x2≤0; g2(x)=x1+x2-6≤0;

[0169] g3(x)=-x1+x2-2≤0; g4(x)=x1-3x2-2≤0;

[0170] g5(x)=4-(x3-3) 2 -x4≤0; g6(x)=4-(x5-3) 2 -x6≤0;

[0171] Based on the initial experimental design, Kriging model and PKriging model were established for the above test functions, and the number of hyperparameters was compared. The results are shown in Figure 2

[0172] Depend on Figure 2 The results show that the number of hyperparameters of the PKriging model is lower than that of the Kriging model, indicating that the Kriging model after partial least squares transformation helps to improve modeling efficiency. When the number of variables is larger, the advantage of this method is more obvious.

[0173] Under the same test environment, two Kriging models were constructed and proxy optimization was performed. The 9 test functions all adopted the CEI strategy to implement the EGO method and the PKEGO method. The average value of the optimization results of the 9 test functions was obtained by statistically analyzing 30 tests and plotting the iterative convergence curve. Figure 3

[0174] Depend on Figure 3 It can be seen that: (1) the test function converges to the approximate optimal solution as the number of iterations increases, indicating that both the EGO and PKEGO methods can achieve efficient global optimization of expensive constraint problems; (2) when the number of variables is large, the PKEGO method has a lower average optimal value and reaches the optimal solution faster, indicating that the PKEGO method converges faster than the standard EGO method and has a higher accuracy in approximate optimal solutions.

[0175] The average value and standard deviation of the optimal value obtained from 30 experiments were used as accuracy indicators. The termination condition was modified to reach 1% absolute value error or the maximum number of iterations. The results are shown in Table 3.

[0176] Table 3 Comparison of optimization results of different methods

[0177]

[0178] From the results in Table 5, we can see that the “standard deviation” and “average result” of the approximate optimal values ​​obtained by the PKEGO method for the nine test functions are lower than those of the EGO method, indicating that the PKEGO method is superior to the EGO method and has a higher accuracy and is more robust in obtaining the optimal value.

[0179] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0180] The above-described embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the present application. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the attached claims.

Claims

1. A partial least squares Kriging model-assisted efficient global optimization method for aviation reducers, characterized in that: The steps include: Step 1: For a total of 7 design parameters of the aviation reducer, namely, tooth surface meshing degree, gear module, pinion module, bearing spacing, and gear and pinion diameters, the maximum and minimum Latin hypercube sampling method is used to perform initial experimental design to obtain an initialization parameter sample; Step 2: Based on the information of the aviation reducer problem, the optimization target is to minimize the total weight of the gearbox, and the strength of the gear, tooth root, and tooth surface is used as a constraint to construct an aviation reducer optimization model. The computer is used to perform simulation calculations to obtain the target response value of the total weight of the gearbox and the constraint response values ​​of the gear, tooth root, and tooth surface strength. The initialization parameter samples and the obtained target and constraint response values ​​are saved to establish a DS database. Step 3, normalize the data in the DS database, perform partial least squares transformation on the normalized parameter samples and response values, determine the number of principal components based on the cumulative probability, and then construct the partial least squares Kriging kernel function; Step 4: Based on the partial least squares Kriging kernel function, the partial least squares Kriging surrogate models of the target and constraints are constructed respectively to obtain the predicted mean and predicted variance of the target response and the constraints; Step 5, determine whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, construct a feasible probability strategy to achieve sample filling. Otherwise, construct a maximum partial least squares constraint weight expectation improvement criterion to achieve spatial sample filling, and use a computer to obtain the target and constraint response values ​​corresponding to the new sample, and place the filled sample data and its corresponding simulation output target value and response value in the DS database; Step 6, iterate steps 3-5 in a loop to update the DS database until the maximum number of iterations is reached or the threshold of the maximum constraint expected improvement criterion is met, thereby obtaining the minimum target value and the corresponding optimal parameter combination.

2. The partial least squares Kriging model-assisted efficient global optimization method for aviation reducers according to claim 1 is characterized in that: Step 2, taking the lightest total weight of the gearbox as the optimization goal, and taking the gear, tooth root, and tooth surface strength as constraints, construct an aviation reducer optimization model, use a computer to perform simulation calculations, obtain the target response value of the total weight of the gearbox and the constraint response values ​​of the gear, tooth root, and tooth surface strength, and save the initialization parameter samples and the obtained target and constraint response values ​​to establish a DK database, where the aviation reducer optimization model is expressed as: min y(x) s.t.g i (x)≤0,i=1,2,…,r x∈D Where y(x) represents the total weight of the gearbox; g i (x) represents the i-th constraint; r represents the total number of constraints; D = [x l ,x r ] is the variable design space, x l ,x r Represent the left and right boundary vectors of each design parameter x respectively; finite element simulation software is used to perform simulation calculations to obtain the target response value set y and the constraint response value set g i , establish DS database.

3. The partial least squares Kriging model-assisted efficient global optimization method for aviation reducers according to claim 1 is characterized in that: Step 3: Normalize the data in the DS database, perform partial least squares transformation on the normalized parameter samples and response values, determine the number of principal components based on the cumulative probability, and construct the partial least squares Kriging kernel function. The specific method is as follows: Perform normalization: Among them, x min ,x max Indicates the minimum and maximum values ​​of the design parameter x; x s To obtain the standardized data of the design parameter x, the target response value y is processed in the same way, that is, the standardized data of the target response value y is obtained; Perform a partial least squares transformation: in, and Respectively represent the cumulative contribution of the principal components to the variation of E0 and F0; c x ,c y Indicates the contribution rate of a single variation explanation; E0, F0 are the standardized matrices of parameter sample X matrix and target response sample y; t ncomp is the nth comp principal components; For nth comp regression coefficient vectors; l is the number of principal components, determined based on the cumulative probability being greater than 0.7 and the contribution of a single variable being greater than 0.05; Construct the partial least squares Kriging kernel function; Among them, η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of principal component systems selected after partial least squares transformation; x i is the i-th component of the design variable x; The weight coefficient of the i-th component of the design variable x after partial least squares transformation.

4. The partial least squares Kriging model-assisted efficient global optimization method for aviation reducers according to claim 1 is characterized in that: Step 4: Based on the partial least squares Kriging kernel function, the partial least squares Kriging surrogate models of the target and constraints are constructed respectively to obtain the predicted mean and predicted variance of the target response and constraints. The specific method is as follows: Assume that the parameter matrix X is initialized after normalization s =[x 1s ,x 2s ,…,x ms ] T The target response value matrix y=[y1,y2,…,y m ] T It is known that the partial least squares Kriging model is expressed as: y(x)=μ+z(x s ), Among them, μ is the trend term; z(x s ) is the estimated correction for the prediction, which has a mean of 0 and a covariance of Gaussian process; represents the variance of the Gaussian process z(x); R(x,x m )=R(x,x m |θ) is the partial least squares Kriging kernel function selected for modeling; Construct a partial least squares Kriging proxy model for the total weight of the aviation reducer: in, and They represent the predicted mean and predicted variance of the total weight target response of the aviation reducer respectively; 1 represents a unit vector with 1 as its element; is the estimated value of the undetermined trend term coefficient μ; y=[y1,y2,…,y m ] T is the target response vector of the total weight of the reducer; m is the number of samples in the DS database; r(x s )=[R(x s ,x 1s ),R(x s ,x 2s ),…,R(x s ,x ms )] T represents the design vector determined by the regression basis function; Variance The estimated value of ; R is the correlation coefficient matrix, which is expressed as follows: Construct a constrained partial least squares Kriging surrogate model: in, and They represent the predicted mean and predicted variance of the ith constraint respectively; g i is the response vector of the ith constraint; R gi is the correlation coefficient matrix of the i-th constraint; is the constraint undetermined trend term coefficient The estimated value of ; obtained by generalized least squares estimation, is the estimated variance of the i-th constrained Gaussian process distribution.

5. The partial least squares Kriging model-assisted efficient global optimization method for aviation reducers according to claim 1 is characterized in that: Step 5, determine whether there is a feasible solution for the sample in the DS database. If there is no feasible solution, construct a criterion for maximizing the feasibility probability to achieve sample filling. Otherwise, construct an improved criterion for maximizing the partial least squares constraint weight expectation to achieve sample filling, and use a computer to perform simulation calculations, and place the filled sample data and its corresponding simulation output target value and response value in the DS database. The specific method is: Step 51: According to the target data y=[y1,y2,…,y m ] T , calculate the current optimal target value y min =min{y1,y2,…,y m }; Step 52: construct a feasibility probability criterion based on the predicted mean and predicted variance of the constraint, and construct a maximization constraint expected improvement criterion based on the predicted mean and predicted variance of the target response and the constraint; assumed The feasibility probability function is expressed as follows: Where Φ(·) is the cumulative probability density function; r is the number of constraints; Respectively represent the response mean and standard deviation of the i-th constraint; The criterion for maximizing feasibility probability is expressed as follows: Among them, x m+1 Represents the m+1th input sample parameter combination; According to the target response and the predicted mean and predicted variance of the constraints, a maximization constraint expectation improvement criterion is constructed to obtain reference sample points for subsequent sample filling; Among them, x R is the new sample reference point, assuming The total weight target of the aviation reducer is to improve the increment Under these conditions, the criterion for maximizing EI points is expressed by the following function: Among them, y min is the current optimal target value, and represents the predicted value and predicted standard deviation, Φ(·) and φ(·) are the cumulative distribution function and the distribution probability density function; and Respectively characterize the local search and global exploration capabilities of the EI criterion; According to the obtained reference points and the predicted mean and predicted variance provided by the Kriging model, two improvement criteria for maximizing the constraint weight expectation are constructed, which are expressed as follows: and in, and represents the expected improvement criterion of partial least squares constrained weights; represents the partial least squares weighted expected improvement function, represents the partial least squares pseudo expectation improvement function, PoF represents the probability of calculating the feasibility of the current test point; Kriging kernel function hyperparameters derived from Kriging modeling; d is the dimension of design variables; represents the kth dimension of the reference point; is the partial least squares weight function; η l is the sum of the weighted coefficients of each principal component after partial least squares transformation; l is the number of transformed principal components; Step 53: Determine whether there is a feasible solution that satisfies all constraints in the DS database. If there is no feasible solution, sample filling is achieved according to the maximization of feasibility probability. Otherwise, a reference point is obtained according to the maximization constraint expectation improvement criterion, and then a new sample is obtained according to any maximization partial least squares constraint weight expectation improvement criterion to achieve sample filling. Step 54: Use a computer to perform simulation calculations, and place the (m+1)th filled sample data and its corresponding simulation output aviation reducer total weight target value and corresponding constraint response values ​​in the DS database, and update the DS database.

6. An aviation reducer design proxy optimization system under the partial least squares constraint weight expectation criterion, characterized in that: Based on the aviation reducer design proxy optimization method under the partial least squares constraint weight expectation criterion as described in any one of claims 1 to 5, the aviation reducer proxy optimization design is realized.

7. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, the aircraft reducer design optimization method based on the aircraft reducer design proxy optimization method under the partial least squares constraint weight expectation criterion as described in any one of claims 1 to 5 is implemented.

8. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the aircraft reducer design optimization design is achieved based on the aircraft reducer design proxy optimization method under the partial least squares constraint weighted expectation criterion as described in any one of claims 1 to 5.

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