CN116933449B - Transmission gear robust design optimization method based on gradient auxiliary learning function - Google Patents
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Abstract
The invention discloses a transmission gear robust design optimization method based on a gradient auxiliary learning function, which is used for developing a self-adaptive updating framework formed by a learning function based on a prediction gradient and a stopping criterion, and is used for solving the problem of transmission gear robust design optimization based on an optimization mechanism and model precision; the learning function based on the prediction gradient is used for selecting high-value samples for targeted training of the proxy model, and the stopping criterion is used for efficiently stopping the training process. The method directly establishes the Kriging model between the design variable of the transmission gear and the steady design target thereof, successfully converts the low-efficiency double-circulation structure of the steady design optimization problem into a high-efficiency single-circulation structure, and greatly relieves the calculation burden of the steady design optimization of the transmission gear from the bottom structure.
Description
Technical Field
The invention relates to the technical field of aerospace, in particular to a transmission gear robust design optimization method based on a gradient auxiliary learning function.
Background
Uncertainty objectively exists in real world and engineering problems, and product performance is fluctuated due to uncertainty such as material difference, geometric deviation and manufacturing tolerance. The performance of the optimal design scheme of the product obtained by the deterministic optimization method can deviate or lose efficacy under the influence of uncertainty. Therefore, uncertainty factors are considered in the product design process, and uncertainty analysis and design optimization of the product are very important. Robust design optimization is an uncertainty optimization method that considers uncertainty in the design process, seeking a design solution that is insensitive to uncertainty disturbances. The robust design optimization takes the mean value and standard deviation of the product performance as optimization targets, and uncertainty analysis is required to be continuously carried out in the optimization process. However, in engineering practice, which uses time-consuming tools such as finite element analysis as a common auxiliary technique, the computational resources required for uncertainty analysis are very large. The proxy model technique can approximate a real physical model in a sample space, provides a cost-effective alternative, and greatly reduces the calculation time of uncertainty analysis in engineering practice.
The optimization target and constraint of the transmission gear robust design optimization problem usually show characteristics of complex form, strong nonlinearity, multiple peaks and the like in actual engineering, so that the use of a proxy model can have problems of precision and efficiency. If a single modeling method of the proxy model is adopted, the accuracy of the constructed proxy model is seriously dependent on selected sample points in the design space. Too little sample size, the goals or constraints of the robust design optimization problem cannot be accurately represented in the design space; the sample size is too large, and the global approximation precision of the agent model to the robust design optimization problem can be improved, but the required calculation cost is obviously increased.
Disclosure of Invention
In order to overcome the technology and the defects of the existing method and solve the problem of the steady design optimization of the transmission gear, the invention provides a steady design optimization method of the transmission gear based on a gradient auxiliary learning function.
The technical scheme of the invention is as follows:
a transmission gear robust design optimization method based on a gradient auxiliary learning function comprises the following steps:
step 1, determining a robust design optimization model of a transmission gear;
step 2, generating a sample pool of design variables by using Latin hypercube sampling on the design variablesAnd from->Is selected randomly->A plurality of initial samples;
step 3, calculating the mean value and standard deviation of the performance function corresponding to the initial sample of the design variable based on the finite element model of the transmission gear robust design, so as to form a training sample set;
step 4, constructing a Kriging model based on the design variable and the corresponding performance function response value, and preliminarily giving a gradient direction included angle threshold value and a satisfaction rate threshold value of the Kriging model;
step 5, obtaining a real gradient corresponding to the training sample set by adopting a finite difference method, calculating a direction included angle between the real gradient and a prediction gradient of the constructed Kriging model, counting the proportion of design variable samples with included angles smaller than a given included angle threshold in the training sample set, judging whether the proportion is smaller than a satisfaction rate threshold or not, if yes, turning to step 7, otherwise turning to step 6;
step 6, fusing gradient, uncertainty and distance items, providing a gradient auxiliary learning function, identifying an updated sample of the design variable based on the learning function, turning to step 3, and adding the updated sample into a training sample set;
and 7, obtaining a transmission gear robust design optimization Kriging model meeting a training stop criterion, solving the transmission gear robust design optimization model by adopting an intelligent optimization algorithm, and obtaining an optimal solution.
Preferably, considering that most stopping criteria ignore the value of the updated sample, wasting its gradient information, fully utilizing the updated sample here, then designing a stopping criterion based on the angle between the predicted gradient direction and the actual gradient direction to effectively terminate the updating process of the robust design objective function kriging model, wherein in step 5, the actual gradient of the training sample is obtained by calculation using a finite difference method, expressed as:
，
to update the sample set->For the ith component of the training sample, +.>In steps.
Preferably, in step 5, a stopping criterion is designed based on the difference between the Kriging model predicted gradient direction and the true function gradient direction, specifically, the sample predicted gradientAnd true gradient->The included angle between the two is:
，
in the method, in the process of the invention,is the modulus of the vector, included angle->Representing the degree of directional deviation between the predicted gradient and the true gradient, whose value is less than a certain threshold, it is considered that the Kriging model at the sample is sufficient to replace the original function, namely:
，
in the middle ofFor the gradient direction angle threshold, the recommended value range is +.>。
Preferably, in step 5, when the proportion of the design variable samples with the included angle between the predicted gradient and the real gradient in the updated sample set smaller than the given included angle threshold meets the stopping criterion, the Kriging model training is considered to be completed, and the stopping criterion is expressed as:
wherein,for updating the number of samples with included angles smaller than a given included angle threshold value in the sample set; />To update the number of samples of the sample set; />For the gradient direction included angle to meet the rate threshold, the value range is suggested to be +.>。
Preferably, in step 6, the gradient auxiliary learning function proposed by the comprehensive gradient, uncertainty and distance term is used for measuring the selection priority of the design variable candidate sample, the gradient is the most intuitive reaction function, the gradient is the intensity of change at one point along a certain direction, the Kriging model can be utilized to predict the response of an unknown sample point and simultaneously provide the prediction gradient at the corresponding sample point, and the gradient term is used as the core foundation of the learning function and is used for marking the sample in the area with the intense change; the uncertainty term is used for measuring the credibility of the Kriging prediction of the sample, and measuring the uncertainty of the Kriging model prediction according to the square root of the mean square error prediction value at the sample point; excessive concentration of samples can cause the learning criterion to fall into a local area for repeated selection, which is unfavorable for the advantage of the training samples to show uniformity globally, so that distance items are used for preventing clustering of the training samples so as to enhance the global point selection capability of the gradient-assisted learning function.
Preferably, the robust design target region having strong nonlinearity and larger uncertainty robust design target region having strong nonlinearity, the larger uncertainty robust design target region, and the target variation alleviation region are divided according to gradient terms, and different regions correspond to different sample update criteria.
Preferably, for a robust design target region with strong nonlinearity and large uncertainty, the sample update criteria for the learning function formed by combining the gradient term with the uncertainty term is:
，
in the method, in the process of the invention,is the normalized prediction gradient at the sampling point, +.>For normalizing the prediction standard deviation, P is a sample pool;
for a robust design target variation mitigation region with large uncertainty and local optimization, the sample update criteria for the learning function formed by combining the gradient term with the uncertainty term is:
，
introducing distance items into the learning function of the updated sample, avoiding repeated selection of the updated sample in the local area, and realizing the global point selection capability of the learning function in the design space;
finally, the learning function of the robust design of the transmission gear is obtained as follows:
，
，
in the method, in the process of the invention,is the normalized distance.
Advantageous effects
1. The method directly establishes the Kriging model between the design variable of the transmission gear and the steady design target thereof, successfully converts the low-efficiency double-circulation structure of the steady design optimization problem into a high-efficiency single-circulation structure, and greatly relieves the calculation burden of the steady design optimization of the transmission gear from the bottom structure;
2. the invention develops a novel learning function based on a prediction gradient, and can improve the Kriging model precision of a steady design objective function of a transmission gear. Firstly, marking a sample in a region with a strong change and a mild change of an objective function by a gradient item, wherein the overall trend and the local optimum of the objective function are reflected; secondly, the distance item prevents the aggregation of training samples and improves the global point selection capability of the proposed learning function; finally, the reliability of the target function Kriging predictor is measured by the uncertainty term. The gradient, uncertainty and distance item cooperate to identify a Kriging model of a high-value updated sample refinement objective function;
3. the invention discloses a new stopping criterion based on the intersection angle between a predicted gradient and a true gradient, which can efficiently terminate the self-adaptive updating process of the steady design optimization of a transmission gear and also has the capability of preventing premature convergence.
Drawings
FIG. 1 is a flow chart of a method for optimizing a robust design of a drive gear based on a gradient assist function in accordance with an embodiment of the present invention;
FIG. 2 is a diagram of a helical gear geometry model according to one embodiment of the present invention;
fig. 3 is a finite element model diagram of a drive gear according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The invention develops a self-adaptive updating framework formed by a learning function and a stopping criterion based on a prediction gradient based on an optimization mechanism and model precision and aims at the problem of steady design and optimization of a transmission gear. The learning function based on the prediction gradient is used for selecting high-value samples for targeted training of the proxy model, and the stopping criterion is used for efficiently stopping the training process.
A transmission gear robust design optimization method based on gradient auxiliary learning function includes determining a transmission gear robust optimization model, generating a sample pool of design variables by using Latin hypercube sampling on the design variables, and calculating performance function mean value and standard deviation corresponding to initial samples of the design variables by using a gear finite element model to form a training sample set; the built Kriging model is provided with a gradient direction included angle threshold value and a satisfaction rate threshold value preliminarily; obtaining a real gradient corresponding to a training sample set by adopting a finite difference method, calculating a direction included angle between the real gradient and a prediction gradient of the constructed Kriging model, counting the proportion of design variable samples with included angles smaller than a given included angle threshold in the training sample set, and judging whether the proportion meets a given updating stop criterion or not; integrating gradient, uncertainty and distance items, establishing a proposed gradient auxiliary learning function, identifying an updated sample of a design variable, and adding the updated sample into a training sample set; finally, after a drive gear robust design optimization Kriging model meeting a training stop criterion is obtained, an intelligent optimization algorithm is adopted to solve the problem to obtain an optimal solution. In particular, the method comprises the steps of,
and (3) designing a transmission gear:
the gear is used as a key component for power transmission of equipment such as an airplane, a ship and the like, and the performance of the gear has an important influence on the power level of the equipment. As shown in fig. 2, which is a model diagram of an input bevel gear, the maximum stress of the bevel gear is notable in the transmission process, and the maximum stress of the transmission gear is expressed as:
，
wherein,indicating maximum stress->And->Geometrical variables respectively representing the end face width and the modulus of the gear;represents the modulus of elasticity>Representing poisson's ratio; />Load variables representing equivalent torque, which are input random variables subjected to normal distribution, and detailed parameter information thereof are shown in table 1:
table 1 parameter information of input random variables
。
And carrying out parameterized finite element analysis on the bevel gear according to a sample of the input random variable, wherein a finite element model of the transmission gear is shown in figure 3.
Step 1, a robust design target is the sum of the mean value and the standard deviation of the maximum stress of the pair of gears, and a mathematical model for robust design optimization of the transmission gear is as follows:
，
wherein μ is the mean value, F is the mean value of the maximum stress of the gear and the sum of the standard deviations of the maximum stress、Standard deviation, subscript x 1 ~ x 5 For the design point, f is the maximum stress.
Step 2, generating sample size as follows by using Latin hypercube sampling technologyDesign variable of (2)Is selected randomly from the sample pool P of (2)>Initial samples. And sets the gradient direction included angle threshold value +.>Meet rate threshold->。
Step 3, setting an input random variable sampleThe size of the model is 10, and then the average value and standard deviation of the performance function corresponding to the initial sample of the design variable are calculated by using the gear finite element model, so that an initial training sample set is formed.
And 4, establishing a Kriging model of the relation between the design variable of the transmission gear and the steady design optimization target thereof according to the training sample set in the step 3.
And 5, obtaining a real gradient corresponding to the updated sample set by adopting a finite difference method, calculating a direction included angle between the real gradient and a predicted gradient of the constructed Kriging model, counting the proportion of design variable samples with included angles smaller than a given included angle threshold in the updated sample set, judging whether the design variable samples meet a given training stop criterion, if so, turning to the step 7, otherwise, turning to the step 6.
And 6, introducing gradient, distance between samples and uncertainty, and fusing the gradient, the uncertainty and the distance item to form the proposed gradient auxiliary learning function. The gradient is the intensity of the most intuitive reaction function changing along a certain direction at one point, and can be used for function optimization, and at least a locally optimal design solution can be found when the gradient is zero. These self-properties of the gradient are well suited for guiding the improvement of proxy model accuracy in RDO problems. Introducing a distance between samples to form a distance item: the distances between the updated samples and the initial samples are very close, and the situation that the samples are excessively concentrated occurs, which means that the learning criteria of the formulas can fall into the local area to be repeatedly selected, and the advantage that the training samples show uniformity globally is not facilitated. Introducing uncertainty: the square root of the mean square error prediction value is used for predicting uncertainty and measuring the reliability of the Kriging model prediction.
Taking the gradient term as a core foundation of a learning function, and marking samples in a region with drastic changes, wherein the overall trend and the local optimum of the objective function are reflected; the uncertainty item is used for measuring the reliability of the sample Kriging prediction, the uncertainty exists in the Kriging model prediction itself, and if the uncertainty at a sample point is large, the reliability of prediction information provided by the Kriging model is low, so that the prediction uncertainty of the Kriging model is measured according to the square root of a mean square error prediction value at the sample point; excessive concentration of samples can cause the learning criterion to fall into a local area for repeated selection, which is unfavorable for the advantage of the training samples to show uniformity globally, so that distance items are used for preventing clustering of the training samples so as to enhance the global point selection capability of the gradient-assisted learning function. And the gradient, uncertainty and distance item cooperate to identify a Kriging model of the high-value updated sample refinement objective function.
The larger the gradient value, the larger the nonlinearity and the more intense the change; the larger the square root of the mean square error, the larger the region uncertainty. Greater uncertainty and the existence of locally optimal robust design target variation mitigation regions: the square root of the mean square error is large and the local gradient region 0. Robust design target area with strong nonlinearity and large uncertainty: the values of both the gradient and the square root of the mean square error are large.
For a robust design target region with strong nonlinearity and large uncertainty, the sample update criteria for the learning function formed by combining the gradient term with the uncertainty term is:
，
in the method, in the process of the invention,is the normalized prediction gradient at the sampling point, +.>For normalized prediction standard deviation, P is the sample pool.
For a robust design target variation mitigation region with large uncertainty and local optimization, the sample update criteria for the learning function formed by combining the gradient term with the uncertainty term is:
；
and introducing distance items into the learning function of the updated sample, avoiding repeated selection of the updated sample in the local area, and realizing the global point selection capability of the learning function in the design space.
Finally, the learning function of the robust design of the transmission gear is obtained as follows:
in the method, in the process of the invention,is the normalized distance.
And (3) identifying an updated sample of the design variable according to the gradient auxiliary learning function, turning to step (3) to calculate a response value of the performance function corresponding to the design variable, and adding the updated sample into the training sample set.
And 7, finally, searching a robust optimal solution applied to the transmission gear engineering by utilizing a genetic algorithm based on the transmission gear robust design optimization Kriging model meeting the stopping criterion, wherein the result is shown in the following table 2.
TABLE 2 robust optimal solution for drive gears
。
From the above table, it can be seen that only 1700 finite element model calls are required to obtain a robust optimal solution for the drive gear design.
To make the robust optimal solution in table 2 more convincing, 10 validation samples of the input random variables were generated from the robust optimal solution for finite element analysis. Table 3 gives the finite element results for 10 validation samples, whose mean and standard deviation are 332.4902 and 3.7479, respectively, as seen in table 3. This indicates that the transmission gear scheme obtained by this method has a lower stress level and is insensitive to variations in the input random variable. Furthermore, the error of the drive gear robust design objective of the proposed method is only 1.27% compared to the results of the validation sample calculation.
Table 3 10 validation samples
。
Finally, it should be noted that: the foregoing description is only a preferred embodiment of the present invention, and the present invention is not limited thereto, but it is to be understood that modifications and equivalents of some of the technical features described in the foregoing embodiments may be made by those skilled in the art, although the present invention has been described in detail with reference to the foregoing embodiments. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. The transmission gear robust design optimization method based on the gradient auxiliary learning function is characterized by comprising the following steps of:
step 1, determining design variables of a robust design of a transmission gear, and constructing a robust design optimization model of the transmission gear;
step 2, generating a sample pool N of the design variables by using Latin hypercube sampling on the design variables P ；
Step 3, establishing a finite element model of the transmission gear, and calculating a performance function response value corresponding to a design variable sample based on the finite element model so as to form a training sample set;
step 4, constructing a Kriging model based on the design variable and the corresponding performance function response value thereof, and giving a gradient direction included angle threshold epsilon of the Kriging model θ Meeting the rate threshold ε c ；
Step 5, obtaining a real gradient corresponding to the training sample set by adopting a finite difference method, calculating a direction included angle theta (w) between the real gradient and a predicted gradient of the constructed Kriging model, and when theta (w) is less than or equal to epsilon θ When the Kriging model at the sample is used for replacing the original function;
statistical training sample set accords with theta (w) less than or equal to epsilon θ The proportion of the design variable sample of (2) is greater than or equal to the satisfaction rate threshold epsilon c Judging that the updating stop criterion is met, and turning to the step 7, otherwise turning to the step 6;
step 6, fusing gradient, uncertainty and distance items to provide a gradient auxiliary learning function, identifying an updated sample of the design variable based on the learning function, turning to step 3, adding the updated sample into a training sample set,
wherein regions of strong nonlinearity, greater uncertainty, and target change mitigation are partitioned according to gradient terms, wherein regions of strong nonlinearity, greater uncertainty correspond to learning function sample update criteria that combine gradient terms with uncertainty termsLearning function sample updating criterion of combining target change alleviation region corresponding gradient term and uncertainty term +.>Introducing distance items into the learning function to obtain a learning function of the steady design of the transmission gear:
in the method, in the process of the invention,is the normalized prediction gradient at the sampling point, +.>For normalizing the prediction standard deviation, P is the sample pool,is the normalized distance;
and 7, obtaining a transmission gear robust design optimization Kriging model meeting a training stop criterion, and solving the transmission gear robust design optimization model by adopting an intelligent optimization algorithm to obtain an optimal solution.
2. The method for optimizing the robust design of the transmission gear based on the gradient auxiliary learning function according to claim 1, wherein in the step 1, the robust design optimization model of the transmission gear takes the maximum stress of the transmission gear as an objective function, and the robust design objective is the sum of the average value of the maximum stresses of the pair of gears and the standard deviation of the maximum stresses.
3. The method of optimizing a robust design of a drive gear based on a gradient-assisted learning function of claim 2, wherein the robust design optimization model of the drive gear is expressed as:
min F＝μ f (μ x )+σ f (μ x )
wherein mu is the mean value, F is the mean value of the maximum stress of the gear, the sum of the standard deviations of the maximum stress, sigma is the standard deviation, and the subscript x 1 ～x 5 For the design point, f is the maximum stress and g is the constraint.
4. The method for optimizing the robust design of a transmission gear based on a gradient-assisted learning function according to claim 1, wherein in step 5, the true gradient of the training sample is calculated by a finite difference method, expressed as:
to update the sample set, F is the average of the maximum stresses of the gears, the sum of the standard deviations of the maximum stresses, w i For the ith component of the training sample, Δ is the step size.
5. The method for optimizing transmission gear robust design based on gradient-assisted learning function according to claim 4, wherein in step 5, the sample prediction gradient G (w) and the true gradient G real The included angle between (w) is:
in the formula, I and II are modes of vectors, and an included angle theta represents the degree of direction deviation between a predicted gradient and a real gradient.
6. The method for optimizing the robust design of a transmission gear based on a gradient-assisted learning function according to claim 5, wherein in step 5, θ (w). Ltoreq.ε is satisfied θ The ratio of the number of design variable samples to the number of update samples is greater than or equal to the satisfaction rate threshold epsilon c At this time, the Kriging model training was considered complete, expressed as:
wherein θ (w) is the angle between the sample predicted gradient and the true gradient, ε θ For a given angle threshold, N c To update sample set to satisfy theta (w). Ltoreq.epsilon. θ Number of design variable samples, N u To update the number of samples of the sample set ε c The gradient direction included angle meets the rate threshold.
7. The method for optimizing the robust design of a transmission gear based on a gradient-assisted learning function according to any one of claims 1 to 6, wherein in step 6, the gradient-assisted learning function proposed by the comprehensive gradient, uncertainty and distance term is used to measure the selection priority of candidate samples of the design variables, the Kriging model is used to predict the responses of unknown sample points while providing the predicted gradient at the corresponding sample points, and the gradient term is used as the core basis of the learning function for marking the samples in the region with drastic changes; the uncertainty term is used for measuring the credibility of the Kriging prediction of the sample, and measuring the uncertainty of the Kriging model prediction according to the square root of the mean square error prediction value at the sample point; distance terms are used to prevent clustering of training samples to enhance the global point selection ability of the gradient-assisted learning function.
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