Mechanical system reliability analysis method based on multi-population genetic optimization response surface

Abstract

The application is a mechanical system reliability analysis method based on multi-population genetic optimization response surface, and belongs to the technical field of mechanical system reliability analysis and design. In order to solve the problems of high calculation cost or large calculation result error of the existing response surface model, the method comprises the following steps: converting the response surface cross term combination into a binary vector and a decimal number, setting the initial parameters and the iteration number of the multi-population genetic algorithm, sampling to generate a test sample set, a verification sample set and initial population individuals, taking the prediction error rate of the test sample set as a dynamic constraint condition, combining a cost function and a comprehensive index of the prediction error rate to establish an optimization objective function, iteratively optimizing the optimization objective function to obtain an optimal cross term combination, then obtaining an optimal cross term polynomial response surface model, calculating the failure probability and the confidence interval by using the optimal cross term polynomial response surface model, and obtaining the reliability analysis result of the mechanical system. The method considers the calculation cost and fitting accuracy, and ensures the reliability evaluation of the mechanical system.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of mechanical system reliability analysis and design, and particularly relates to a mechanical system reliability analysis method based on multi-population genetic optimization response surface. BACKGROUND

[0002] In the mechanical system reliability analysis, for the mechanical system with simple structure, the input-output relationship can be usually described by establishing an explicit limit state function, and then the classical methods such as the matrix method, the progressive integral method or the Monte Carlo numerical simulation method are used for reliability calculation. However, when a complex mechanical system is involved, the input-output relationship often presents significant nonlinear characteristics, and it is difficult to construct an explicit limit state function by an analytical method. In this context, the surrogate method based on the proxy model emerges as the times require, and the proxy models represented by the quadratic polynomial response surface method, the Kriging model and the support vector machine (SVM) have been widely used in the reliability analysis of complex mechanical systems.

[0003] The core advantage of the proxy model method is to approximate the implicit limit state function by constructing an explicit surrogate model, thereby providing an effective solution for the reliability analysis of complex systems. As one of the earliest developed proxy model methods, the polynomial response surface method realizes model construction through the following systematic process: first, representative sample points are selected, then an iterative optimization strategy is used to determine the polynomial coefficients, and finally a mathematical model with an explicit expression is established. This method assumes that the implicit limit state function can be expressed in the form of a polynomial containing unknown parameters, and uses the least squares method to estimate the parameters. On the premise of ensuring the calculation accuracy, the efficiency of solving the reliability problem of complex implicit function is significantly improved. The simplicity of this algorithm makes this method show good applicability in engineering practice.

[0004] In fact, the selection of the form of the response surface function is the core problem of this method. For the implicit limit state function with unknown form and order, it is difficult to construct a response surface model with strong universality. Early researches mostly used quadratic polynomials with cross terms. Bucher et al. proposed a form without cross terms and pointed out that the traditional experimental design (centered on the mean point) leads to insufficient accuracy of the response surface on the failure boundary. Therefore, Bucher et al. introduced an iterative strategy: first, train the initial response surface with samples near the mean, calculate the design points, and then use linear interpolation to generate new sample points closer to the failure boundary for iterative updating. Liu et al. further improved this method to multiple iterations, and through repeated updating of design points and sample points until the reliability index converges, the approximation accuracy of the model in the failure region is effectively improved.

[0005] The response surface form without cross terms has the advantages of simple structure and high calculation efficiency, but cannot reflect the interaction between variables, and the accuracy is significantly reduced in the problem of strong nonlinear implicit limit state function. Research shows that increasing the number and number of cross terms can improve the accuracy of the response surface method, but it also requires higher computational cost, and unreasonable introduction can make the result deviate from the true solution. In view of this problem, researchers have proposed different improvement strategies: Wong and Rajashekher consider introducing all quadratic cross terms, and combining the iterative strategy to solve the unknown coefficients of the quadratic polynomial with cross terms. Although the quadratic polynomial response surface with cross terms is directly used to calculate the high accuracy, with the increase of the to-be-solved coefficients, the required calculation cost is also increasing. Therefore, Zheng et al. proposed a method of gradually modifying the response surface, which gradually adds quadratic terms or cross terms on the basis of the linear response surface. Yu et al. proposed a selective step-by-step response surface method based on Zheng's method. Fan Wenliang et al. proposed a cross term selection criterion through mathematical derivation to solve the problem of reasonable selection of cross terms, and proposed an adaptive response surface method considering cross terms.

[0006] In the above prior art for model reliability evaluation, there are still problems such as large error in the calculation result of the quadratic response surface model without cross terms, or high calculation cost caused by too many cross terms, or deviation from the true solution caused by unreasonable introduction of cross terms. In view of this, a method for reliability analysis of mechanical systems is researched, which can consider the calculation cost and ensure the fitting accuracy, and can have a positive effect and influence on industry research and practical application. SUMMARY

[0007] In view of the defects in the above background art that have not been overcome, the present application proposes a mechanical system reliability analysis method based on multi-population genetic optimization response surface, which aims to obtain the optimal response surface cross term combination mode through a multi-population genetic algorithm under the premise of considering the calculation cost and fitting accuracy, and to perform high-precision reliability evaluation of the related structure of the mechanical system.

[0008] The method of the present application represents the combination form of the polynomial response surface cross term as a binary vector, and converts it into a decimal parameter as an optimization parameter. The prediction error rate of the test sample set is used as an optimization constraint condition, and the comprehensive index of the calculation cost function considering the number of cross terms and the prediction error rate is used as an optimization objective function. The multi-population genetic algorithm is used to search the optimal cross term response surface form in parallel. This method considers the coupling factors of random variables, avoids the selection of cross term combination mode by human, and therefore can reduce the subjective error and improve the calculation accuracy compared with the traditional quadratic response surface model without cross terms.

[0009] In order to achieve the above technical purposes, the technical scheme of the present application is as follows:

[0010] The mechanical system reliability analysis method based on multi-population genetic optimization response surface includes the following steps:

[0011] S1: Determine the initial response surface form, generate one or more decimal numbers as optimization parameters, and convert the decimal numbers into binary vectors in the form of cross-term combinations;

[0012] S2: Set the initial parameters and iteration times of the multi-population genetic algorithm, sample to generate a test sample set, a verification sample set, and initial population individuals of the multi-population genetic algorithm;

[0013] S3: Fit the weighted response surface model in the corresponding form according to the cross-term combination mode represented by the initial population individuals, and set the prediction error rate of the test sample set as the optimization objective function T(X) ;

[0014] S4: Use the multi-population genetic algorithm to iteratively optimize the optimization objective function T(X) , select the optimal population individual, update the constraint threshold, and determine whether the iteration times converge to obtain the optimal cross-term combination;

[0015] S5: Calculate the polynomial coefficient matrix based on the optimal cross-term combination to obtain the optimal cross-term polynomial response surface model;

[0016] S6: Input the verification sample set into the optimal cross-term polynomial response surface model to calculate the failure probability, and calculate the confidence interval of the failure probability based on the binomial distribution method of Monte Carlo simulation to obtain the reliability result of the mechanical system.

[0017] As a further preferred embodiment of the above scheme: in step S1, the specific steps of generating one or more decimal numbers as optimization parameters and converting the decimal numbers into binary vectors in the form of cross-term combinations are as follows:

[0018] S101: Determine the initial response surface form as a quadratic response surface function , and represent the cross-term combination mode of the quadratic response surface function in decimal numbers , wherein:

[0019] ;

[0020] In the formula, is the undetermined polynomial coefficient, and is the constant term, and are the linear term coefficients, is the quadratic term coefficient, is the independent variable, n is the number of independent variables, ​is 0 or 1, indicating the cross term whether there is;

[0021] S102: convert the decimal number into a binary vector to obtain the corresponding response surface cross term vector, that is,

[0022] ;

[0023] ;

[0024] ;

[0025] wherein, is the response surface cross term vector, the value range of , is the maximum value of , is the decimal to binary algorithm, is the binary to decimal algorithm.

[0026] As a further preferred embodiment of the above scheme: the initial value of the number of iterations set in step S2 is Gen=0, the initial value of the number of individuals kept in the best population is Gen0=0, and the maximum value of the number of individuals kept in the best population is Gen0_max=5.

[0027] As a further preferred embodiment of the above scheme: in step S3, the predicted error rate The calculation process includes:

[0028] S301: calculate the polynomial coefficient matrix B based on the weighted response surface model, and the calculation formula is:

[0029] ;

[0030] ;

[0031] ;

[0032] wherein, P is the total number of terms including the constant term, the first order term and the cross term, and when there are independent variables, its length is , denotes a diagonal matrix with P diagonal elements, A is a regression matrix composed of m sample points when the response surface is a quadratic polynomial containing cross terms, is a true limit state value vector of the sample points, is a weight matrix, For the first sample assigned weight, the weight coefficient assignment rule is: if the absolute value of the true limit state value of the sample is smaller, the sample is closer to the failure boundary, and a higher weight should be assigned to improve the fitting accuracy of the failure region, that is, the weight coefficient is larger, and vice versa;

[0033] S302: Obtain the corresponding weighted response surface function B according to the response surface coefficient matrix ;

[0034] S303: Substitute the test sample set into the weighted response surface function , calculate the limit state fitting value of the test sample set to obtain the predicted error rate , and take the predicted error rate as a dynamic optimization constraint condition. The expression of the predicted error rate is:

[0035] ;

[0036] ;

[0037] In the formula, is the ith sample of the test sample set, has a corresponding true limit state value and a limit state fitting value , is an indicator function. When and have the same sign , otherwise 0, h is the number of test samples.

[0038] Further preferred solution: in step S3, the construction process of the optimization objective function includes:

[0039] (1) Considering the influence of introducing cross terms on the calculation of the predicted error rate , a cost function related to the number of cross terms is established to measure this index, whose expression is as follows:

[0040] ;

[0041] In the formula, the value of indicates the number of "1"s in , and ​the first value of the cross term vector, i k the length of the vector;

[0042] (2) based on the prediction error rate and the cost function related to the number of cross terms introduced establish an optimization objective function The expression of the optimization objective function is as follows:

[0043] ;

[0044] In the formula, is the weight coefficient of the cost function related to the number of cross terms introduced , and .

[0045] As a further preferred embodiment of the above scheme: in step S4, the optimization objective function is iteratively optimized using a multi-population genetic algorithm. The steps include:

[0046] S401: Calculate the optimization objective function fitness value of all gene individuals in the initial population;

[0047] S402: According to the fitness value, select the best gene individual in each population, and combine the gene codes two by two to form a new gene individual;

[0048] S403: Set the mutation probability to 0.01, change the value of a certain position in the gene code according to the mutation probability, and calculate the prediction error rate and the fitness value of the optimization objective function again;

[0049] S404: Migrate the optimal gene individual in each population to other populations step by step, replace the worst gene individual in the corresponding population, and after all populations are replaced, manually select the optimal individual in all populations to form an elite population, and then select the optimal individual from the elite population as the optimal gene individual for this iteration, obtain the optimal cross term combination mode, record the best fitness value of the optimization objective function , the minimum and the best gene individual, and update to the minimum .

[0050] Further preferred scheme: in step S4, determine whether the iteration number converges according to the best fitness value, the process is as follows:

[0051] ​​​If the optimal fitness value changes, the optimal number of times Gen0 is kept is 0, otherwise Gen0 = Gen0 + 1;

[0052] If Gen0>Gen0_max, the optimization iteration is considered to have converged and the optimization program is exited; otherwise, the number of iterations Gen=Gen+1, and the process goes to step S3.

[0053] As a further preferred embodiment of the above scheme: the step of obtaining the optimal cross-term polynomial response surface model in step S5 includes:

[0054] S501: After the optimization iteration converges, the decimal number of the best cross-term combination is output 、 , and convert the decimal number 、 Convert to optimal binary vector ;

[0055] S502: Combined optimal binary vector , solve the polynomial coefficient matrix at this time according to the weighted response surface, and obtain the optimal cross-term polynomial response surface model at this time.

[0056] As a further preferred implementation of the above scheme: in step S6, the confidence interval of the failure probability calculated based on the binomial distribution method of Monte Carlo simulation is:

[0057] ;

[0058] Where, N is the number of random samples, f is the number of failed samples, r is the number of reliable samples, is the significance level, which is usually set to 0.05. The inverse cumulative distribution function of the Beta distribution returns the quantile of the Beta distribution corresponding to the given probability value.

[0059] Compared with the prior art, the present invention has the following beneficial effects:

[0060] The method of the present application proposes a new response surface cross term determination method under the premise of considering the influence of weighted response surface cross term on calculation amount and calculation accuracy. By converting the optimization parameters into binary vectors, an optimization objective function related to the calculation cost and calculation fitting accuracy is constructed, the determination of the cross term number is converted into a constrained optimization problem, and a multi-population genetic optimization algorithm is used to solve the problem. The appropriate cross term is introduced for the weighted response surface, while ensuring that the calculation amount is not too large, and the implicit limit state function is better fitted. The method of the present application can obtain the accurate solution of the cross term for simple and low nonlinearity problems, and can select the appropriate cross term while considering the calculation amount for high nonlinearity problems, improve the fitting accuracy of failure probability, and the effectiveness of the method is verified, which provides a new method and new idea for selecting the cross term of the response surface method. BRIEF DESCRIPTION OF DRAWINGS

[0061] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings needed in the embodiments will be briefly introduced as follows.

[0062] Figure 1 The flow chart of the mechanical system reliability analysis method based on the multi-population genetic optimization response surface proposed by the present application;

[0063] Figure 2 The schematic diagram of the simplified headless rivet riveting process in the embodiments of the present application;

[0064] Figure 3 The cross term vector splitting and decimal number conversion schematic diagram of the present application;

[0065] Figure 4 The optimal polynomial vector structure composition schematic diagram of the present application;

[0066] Figure 5 The regression diagram of the response surface without cross term, with full cross term and optimized cross term of the present application on the test sample set;

[0067] Figure 6 The failure probability calculation structure of the optimized cross term response surface of the present application on the prediction sample set and the relative error diagram compared with the Monte Carlo method. DETAILED DESCRIPTION

[0068] In order to make the purpose, technical scheme and advantages of the present application more clear, the technical scheme in the embodiments will be clearly and completely described below in combination with the drawings of the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments, not all the embodiments. Other embodiments can be obtained by those skilled in the art without creative labor.

[0069] REFERENCE Figures 1-6The present invention proposes a mechanical system reliability analysis method based on a multi-population genetic optimization response surface, comprising the following steps:

[0070] S1: Determine the initial response surface form, generate one or more decimal numbers as optimization parameters, and convert the decimal numbers into binary vectors in the form of cross-term combinations;

[0071] Specifically, according to the complexity of the problem, such as problems with fewer independent variables and low nonlinearity, the quadratic response surface function can be selected. , the quadratic response surface function The cross term combinations are expressed in decimal numbers Indicates that:

[0072] ;

[0073] Where, are the coefficients of the unknown polynomial, and is a constant term, and is the coefficient of the first-order term, is the coefficient of the quadratic term, is the independent variable, n is the number of independent variables, The value is 0 or 1, indicating the cross term exists;

[0074] Convert decimal numbers Convert it into a binary vector and get the corresponding response surface cross term vector, that is:

[0075] ;

[0076] ;

[0077] ;

[0078] Where, is the response surface cross term vector, The value range is , for The maximum value of is the decimal to binary algorithm, It is the binary to decimal conversion algorithm.

[0079] In this step, for a polynomial response surface with multiple independent variables, the vector of cross terms is too long. Too big, The range of values ​​is too large, and the optimization calculation is not easy to converge. In order to avoid this situation, the present invention considers generating two decimal numbers 、 , respectively corresponding to two binary cross term sub-vectors , , at this time the cross term vector , , The value range of , , , Corresponding , The decimal number when all 1 in .

[0080] S2: set the initial parameters and iteration times of the multi-population genetic algorithm, sample to generate the test sample set, the verification sample set and the initial population individuals of the multi-population genetic algorithm;

[0081] In this step, the Latin hypercube sampling is used to generate the test sample set and the verification sample set, and the initial population individuals of the genetic algorithm are randomly generated in the value range , , set the initial value of the iteration times Gen=0, the initial value of the best population individual retention times Gen0=0, and the maximum value of the best population individual retention times Gen0_max=5.

[0082] S3: according to the cross term combination mode represented by the initial population individuals, fit the weighted response surface model of the corresponding form, and take the prediction error rate As a dynamic optimization constraint condition, the optimization objective function Considering the cost function related to the number of cross terms introduced And the prediction error rate ;

[0083] Specifically, based on the weighted response surface model (WRSM), the polynomial coefficient matrix In the traditional response surface calculation, the unknown polynomial coefficients can be obtained by the least square method; in order to improve the accuracy of the response surface fitting failure probability, the present application gives higher weight to the points close to the failure boundary surface, so that these points play a more important role in fitting, improve the fitting accuracy of the response surface at the failure boundary surface, and calculate the polynomial coefficient matrix B Its solution formula is as follows:

[0084] ;

[0085] ;

[0086] ;

[0087] In the formula, PLet be the total number of terms vector containing constant term, linear term and cross term, its length is when there are , , P , A is a diagonal matrix with diagonal elements , is a regression matrix composed of m sample points when the response surface is a quadratic polynomial containing cross terms, is a true limit state value vector of sample points, is a weight matrix, is the weight given to the th sample The weight coefficient assignment rule is: if the absolute value of the true limit state value of the th sample is smaller, the sample is closer to the failure boundary, and a higher weight should be given to improve the fitting accuracy of the failure region, that is, ,

[0088] According to the response surface coefficient matrix B , the weighted response surface function is obtained, the test sample set is substituted into the weighted response surface function , the limit state fitting value of the test sample set is calculated, and the predicted error rate is further obtained, and the expression of the predicted error rate is as follows:

[0089] ;

[0090] ;

[0091] In the formula, h is the number of test sample sets, for the sample in the test sample set, there is a corresponding true limit state value and a limit state fitting value , is an indicator function, assuming that the limit state function value greater than 0 indicates failure, and less than 0 is reliable, for the sample , when and have the same sign, it means that the prediction of the response surface to the failure state of the sample is the same as the true situation, and the prediction result meets the expectation, that is, , otherwise it is 0.

[0092] The value of the predicted error rate is between 0 and 1, the smaller, the closer the failure classification of the response surface to the test sample set to the true situation, 0 indicates that the response surface is completely correct for the failure classification of the test sample set, A value of 1 indicates a complete error. Therefore, the prediction error rate of the test sample set is Set it as a dynamic optimization constraint and continuously update the constraint threshold in subsequent optimization iterations.

[0093] Furthermore, considering the impact of the cross term on the prediction error rate The impact of computational effort is established by establishing a cost function related to the number of cross terms introduced. To measure this indicator, The expression is as follows:

[0094] ;

[0095] Where, is the cross term vector, is the cross term vector i values, k for The length of the vector; the more cross terms are introduced, The larger it is, the greater the computational cost will be.

[0096] Based on the prediction error rate and the cost function related to the number of cross terms introduced Establishing the optimization objective function , optimize the objective function The smaller the value, the higher the degree of fitting of the failure probability of the test sample, and the less computational effort, the better the optimization objective function. The expression is as follows:

[0097] ;

[0098] Where, is the weight coefficient of the cost function, Pick .

[0099] At this point, the optimization mathematical model for optimizing the cross-term response surface can be obtained:

[0100] ;

[0101] S4: Using multi-population genetic algorithm to optimize the objective function T(X) Iterate and optimize, select the best population individual, update the constraint threshold, and determine whether the number of iterations converges to obtain the optimal cross-term combination;

[0102] The present invention is based on a multi-population genetic algorithm to optimize the best response surface form. The important process of the multi-population genetic algorithm is as follows:

[0103] 1) Fitness calculation: according to the set fitness function, the fitness value of all gene individuals in the current population is calculated;

[0104] 2) Selection: according to the specified fitness criterion, the best gene individual is selected, and the poor gene individual is removed;

[0105] 3) Crossover: the selected good gene code is combined to form a new gene individual;

[0106] 4) Mutation: according to a certain mutation probability, the value of a certain position in the gene code is changed;

[0107] 5) Migration: the best individual in one population can migrate to another population to replace the worst gene individual in the population;

[0108] 6) Artificial selection: after all populations complete the above operations, the best gene individual in all populations is selected to form an elite population, and the best one is selected from the elite population as the best gene individual of this iteration.

[0109] In an embodiment of the application, the optimization objective function of all gene individuals in the initial population is calculated The fitness value;

[0110] Then, according to the fitness value, the best gene individual in each population is selected, and the two gene codes are combined to form a new gene individual;

[0111] The mutation probability is set to 0.01, the value of a certain position in the gene code is changed according to the mutation probability, and the prediction error rate And the fitness of the optimization objective function ;

[0112] Finally, the best gene individual in each population is gradually migrated to other populations to replace the worst gene individual in the corresponding population, and after all populations complete the replacement, the best individual in all populations is selected to form an elite population, and the best one is selected from the elite population as the best gene individual of this iteration, to obtain the optimal crossover item combination mode, record the best fitness value of the optimization objective function The minimum And the best gene individual, update The minimum .

[0113] Further, according to the best fitness value, it is judged whether the iteration number is converged, if the best fitness value changes, the optimal retention number Gen0=0, otherwise Gen0=Gen0+1;

[0114] If Gen0>Gen0_max, the optimization iteration is considered to have converged and the optimization program is exited; otherwise, the number of iterations Gen=Gen+1, and the process goes to step S3.

[0115] S5: Calculate the polynomial coefficient matrix at this time based on the optimal cross-term combination to obtain the optimal cross-term polynomial response surface model;

[0116] After the optimization iteration converges, the decimal number of the best cross-term combination is output 、 , and convert the decimal number 、 Convert to optimal binary vector ; Combined optimal binary vector , solve the polynomial coefficient matrix at this time according to the weighted response surface, and obtain the optimal cross-term polynomial response surface model at this time.

[0117] S6: Input the validation sample set into the optimal cross-term polynomial response surface model to calculate the failure probability, and calculate the confidence interval of the failure probability based on the binomial distribution method of Monte Carlo simulation to obtain the reliability results of the mechanical system; the failure probability confidence interval is:

[0118] ;

[0119] Where, N is the number of random samples, f is the number of failed samples, r is the number of reliable samples, is the significance level, which is usually set to 0.05. The inverse cumulative distribution function of the Beta distribution returns the quantile of the Beta distribution corresponding to the given probability value.

[0120] Example: Take headless rivets in aviation engineering as an example. Thin-walled parts are usually connected by riveting with headless rivets. During the riveting process, if the extrusion pressure is too high, the headless rivet may fail in strength, which will seriously threaten flight safety. In order to establish an explicit numerical relationship between the extrusion stress and the rivet size, this example assumes the following ideal conditions to simplify the riveting process: 1) The rivet hole does not expand during the riveting process; 2) The volume of the rivet does not change substantially during the riveting process; 3) After the riveting is completed, the rivet head is cylindrical. Simplify the riveting process of headless rivets Figure 2 As shown:

[0121] Reference Figure 2 The riveting process can be simplified into three states of rivet deformation: First, before formal riveting, the rivet is in Figure 2 In the middle (a) state, there is a small gap between the rivet and the rivet hole. The rivet shape is diameter d and height a cylinder; then, the rivet is forced to deform, and the rivet is in the state (b) in which the rivet is in a cylinder with a diameter of D and a height of H, and the rivet is in the state (c) in which the rivet is in a cylinder with a diameter of D and a height of H, and the rivet is in the state (d) in which the rivet is in a cylinder with a diameter of D and a height of H. Figure 2 , and the rivet is in the state (c) in which the upper and lower heads of the rivet are in cylinders with a diameter of D and a height of H, and the hole is in a cylinder with a diameter of D and a height of H. Thus, the volume of the rivet in the three states should be: Figure 2

[0122]

[0123] Since it is assumed that the volume of the rivet does not change during the riveting process, the strain in the horizontal direction is , the strain in the horizontal direction is Figure 2 , and the strain in the horizontal direction is Figure 2 , and the true strain of the rivet during the entire riveting process is , and it is known that: Figure 2 Figure 2

[0124]

[0125] The true strain is derived from the logarithmic strain superposition principle:

[0126]

[0127] According to the hardening strength theory, the maximum stress in the horizontal direction of the rivet is:

[0128]

[0129] In the formula, σ is the strength factor of the rivet material, and n is the hardening index of the rivet. The rivet material selected in this embodiment is 2017-T4, and its hardening index n = 0.15 is a constant. The rivet head H is the required value of the rivet, which is 2.2 mm. According to the material manual, the extrusion strength of the rivet is 580 MPa, and when the maximum extrusion stress K of the rivet is greater than the extrusion strength, the rivet fails in strength. Therefore, the function function of the rivet strength failure can be expressed as:

[0130] ​​​​​​​​​​​​​​​​​​​

[0131] wherein, , , , , are random variables, and it is assumed that each random variable is independent and subject to a normal distribution as shown in Table 1.

[0132] Table 1: Random variable distribution parameters

[0133]

[0134] Next, the method of the present application is used to carry out reliability analysis based on a multi-population genetic optimization response surface for rivet strength failure, as follows:

[0135] (1) There are 5 random variables in the analysis of this problem, and the response surface form is determined to be a quadratic response surface with cross terms, with 10 cross terms, to obtain two cross term vectors of length 5 , , and , as shown in Figure 3 .

[0136] (2) The Latin hypercube sampling method is used to generate 2000 samples as a test sample set, and sample points are extracted as a validation sample set, and initial population individuals are randomly extracted within the value range , , for a total of 3 populations, each with 30 individual values .

[0137] (3) The individual values are converted into corresponding cross term vectors , and a weighted nonlinear response surface is used to regress the response surface form, to calculate the polynomial coefficient matrix , as follows:

[0138] ;

[0139] ;

[0140] ;

[0141] (4) The optimization constraint value of the prediction error rate is calculated, and the result of the optimization objective function is calculated. After all population individuals are calculated, the minimum constraint value is found as the constraint threshold for the next iteration; the minimum optimization objective function value and the corresponding cross term form are found.

[0142] (5) Multi-population genetic algorithm optimization, if the best fitness changes, the optimal number of times Gen0 = 0, otherwise Gen0 = Gen0 + 1. If Gen0 > Gen0_max, it is considered that the optimization iteration converges, and the optimization program is exited, otherwise the iteration number Gen = Gen + 1.

[0143] (6) After the optimization iteration converges, output the decimal number of the best cross-term combination mode 、 , and convert the decimal number 、 into the optimal binary vector ; combine the obtained optimal binary vector , as shown in Figure 4 , solve the polynomial coefficient matrix at this time according to the weighted response surface, and obtain the optimal cross-term polynomial response surface model at this time.

[0144] (7) Through optimization iteration calculation, the optimal cross-term vector is obtained as:

[0145] ;

[0146] (8) The corresponding optimal cross-term polynomial response surface model is:

[0147] ;

[0148] The polynomial coefficients of the optimal cross-term polynomial response surface model are shown in Table 2:

[0149] Table 2: Polynomial coefficients of the optimal cross-term polynomial response surface model

[0150]

[0151] The regression fitting results of the verification sample set are shown in Figure 5 , and from Figure 5 , it can be seen that the predicted value of the optimized cross-term response surface is closer to the sample points of the verification sample set.

[0152] Using the Monte Carlo method (MC) to calculate the failure probability of the verification sample set, the full cross-term, the cross-term-free, and the optimized cross-term response surfaces are shown in Table 3, where is the significance level, and the failure probability interval is calculated based on the binomial distribution method of Monte Carlo simulation, and the results are shown in Figure 6 .

[0153] Table 3: MC, failure probability and confidence interval of full cross-term, cross-term-free, and optimized cross-term response surfaces

[0154]

[0155] From the analysis results of the embodiment, for a problem with high nonlinearity, the relative error of the failure probability calculated by using the response surface without cross terms and the response surface with full cross terms is relatively large; and the multi-population genetic optimization response surface method considering the calculation cost and the prediction accuracy of the test sample set has relatively high calculation accuracy, the failure probability and the confidence interval are basically coincident with the failure probability and the confidence interval calculated by the Monte Carlo method, indicating the effectiveness of the method, which not only improves the fitting accuracy of the response surface method to the failure probability, but also takes into account the calculation cost.

[0156] It should be clear that the above detailed description of the embodiments of the application provided in the drawings is only used for further illustration of the application, and is not intended to limit the scope of the claimed application, but only represents selected embodiments of the application. Based on the embodiments of the application, some non-essential improvements and adjustments made by those skilled in the art according to the above content of the application are within the protection scope of the application.

Claims

1. A mechanical system reliability analysis method based on multi-population genetic optimization response surface, characterized in that, The steps are as follows: S1: determine the initial response surface form, generate one or more decimal numbers as optimization parameters, and convert the decimal numbers into binary vectors in the form of cross-term combinations; S101: Determine the initial response surface form as a quadratic response surface function The cross term combination of the quadratic response surface function is expressed by a decimal number wherein: ； wherein are undetermined polynomial coefficients, and is a constant term, and is a first-order term coefficient, is a second-order term coefficient, is an independent variable, n is the number of independent variables, and i = 1, 2... n , is 0 or 1, indicating whether a cross term exists; S102: Convert the decimal number to a binary vector, obtaining the corresponding response surface cross-term vector, i.e.: ； ； ； wherein, is a response surface cross term vector, has a value range of , is the maximum value of is a decimal to binary algorithm, is a binary to decimal algorithm; S2: set the initial parameters and iteration times of the multi-population genetic algorithm, sample to generate a test sample set, a validation sample set, and an initial population of the multi-population genetic algorithm; S3: fitting the weighted response surface model of corresponding form according to the combination mode of cross terms represented by the initial population individuals to test the prediction error rate of the sample set As a dynamic optimization constraint, a cost function considering the number of cross terms is established Prediction error rate Optimization objective function T(X) ; S4: using multi-population genetic algorithm to optimize objective function T(X) Iterative optimization, selecting the optimal population individual, updating the constraint threshold, and judging whether the iteration number converges, to obtain the optimal cross-term combination; S5: calculate the polynomial coefficient matrix at this time based on the optimal cross-term combination, and obtain the optimal cross-term polynomial response surface model; S6: input the validation sample set into the optimal cross-term polynomial response surface model to calculate the failure probability, and calculate the failure probability confidence interval based on the binomial distribution method of Monte Carlo simulation to obtain the reliability result of the mechanical system.

2. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 1, characterized in that, The initial value of the iteration times set in step S2 is Gen=0, the initial value of the best population individual retention times is Gen0=0, and the maximum value of the best population individual retention times is Gen0_max=5.

3. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 2, characterized in that, In step S3, the prediction error rate The calculation process includes: S301: based on the weighted response surface model, the response surface cross term vector is calculated as the response surface coefficient matrix at the time B ; S302: Obtain the response surface coefficient matrix according to the response surface function B corresponding weighted response surface function ; S303: substituting the test sample set into the weighted response surface function , calculating the limit state fitting value of the test sample set to obtain a predicted error rate , and taking the predicted error rate as a dynamic optimization constraint condition, the expression of the predicted error rate is: ； ； Where, The first sample set of the test i samples, There is a corresponding real limit state value and limit state fitting values , is the characteristic function, when and When the signs of , otherwise 0, h is the number of test sample sets.

4. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 3, characterized in that, In step S3, the objective function is optimized The construction process includes: (1) Consider the introduction of cross terms on the prediction error rate The impact of the amount of calculation, the cost function related to the number of cross terms introduced This index, The expression is as follows: ； wherein the value of the number of "1"s in is the cross term vector, is the i value of the cross term vector, k is the length of the vector; (2) Based on the prediction error rate and the cost function related to the number of introduced cross terms Establishing the optimization objective function , the optimization objective function The expression is as follows: ； wherein is the cost function related to the number of introduced cross terms is the weight coefficient of the cost function takes the value .

5. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 4, characterized in that, In step S4, the multi-population genetic algorithm is used to optimize the objective function The iterative optimization step includes: S401: Calculate the optimization objective function of all gene individuals in the initial population fitness value; S402: select the best gene individual in each population according to the fitness value, and combine the two gene codes to form a new gene individual; S403: set the mutation probability as 0.01, change the value of a position in the gene code according to the mutation probability, and calculate the prediction error rate again and the optimization objective function the fitness value of the optimization objective function; S404: The optimal gene individual in each population is gradually migrated to other populations to replace the worst gene individual of the corresponding population, after all populations complete the replacement, the optimal individual in all populations is artificially selected to form an elite population, and the optimal individual in the elite population is selected as the optimal gene individual of the current iteration to obtain the optimal cross term combination mode, and the optimal fitness value of the optimization objective function of this iteration, the minimum , and the optimal gene individual are recorded . The minimum is updated to the minimum .

6. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 5, characterized in that, In step S404, whether the iteration times converge is judged according to the best fitness value, and the process is as follows: If the best fitness value changes, the optimal retention times Gen0=0, otherwise Gen0=Gen0+1; If Gen0>Gen0_max, it is considered that the optimization iteration converges, and the optimization program is exited; Otherwise, the iteration times Gen=Gen+1, and step S3 is turned to.

7. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 6, characterized in that, The steps of obtaining the optimal cross-term polynomial response surface model in step S5 include: S501: output the decimal number of the best cross term combination mode after the optimization iteration converges , , and convert the decimal number , into the optimal binary vector ; S502: combine the obtained optimal binary vector The polynomial coefficient matrix at this time is solved according to the weighted response surface, and an optimal cross-term polynomial response surface model at this time is obtained.

8. The mechanical system reliability analysis method based on multi-population genetic optimization response surface according to claim 7, characterized in that, In step S6, the failure probability confidence interval calculated based on the binomial distribution method of Monte Carlo simulation is: ； wherein N is the number of failure samples, f is the number of failure samples, r is the number of reliable samples, is the significance level, is the inverse cumulative distribution function of the Beta distribution.

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