A double-layer adaptive RVM reliability analysis method for small failure probability

Abstract

The application discloses a double-layer adaptive RVM reliability analysis method for small failure probability, first, a first-layer adaptive RVM model is constructed by combining a Harris Hawks optimization algorithm and an adaptive RVM, and an important sampling sample is generated at a design point according to an iterative updating strategy to approximate the design point; then, the idea of active learning is utilized, a learning function is adopted to continuously update the RVM model, and a failure probability is solved after convergence. The application greatly improves the accuracy of reliability failure probability calculation results, reduces the calculation times, saves the calculation cost, and improves the ability of calculating small failure probability reliability.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of reliability, and particularly relates to a double-layer adaptive RVM reliability analysis method. BACKGROUND

[0002] In various engineering problems, uncertain factors often lead to functional failure of structures and cause serious accidents, which are often small failure probability problems. For reliability analysis, traditional surrogate model methods such as AK-MCS often cause waste of computing resources due to too large candidate sample points, and the importance sampling method solves this problem. However, the traditional importance sampling method solves the design point by using the first-order second-moment method, which cannot solve implicit reliability analysis problems. The importance sampling method based on Markov chain is suitable for implicit reliability analysis problems, but a design point needs to be given in advance to start the algorithm. In the case of extremely small failure probability, it is extremely difficult to obtain a failure sample point. SUMMARY

[0003] In order to overcome the shortcomings of the prior art, the application provides a double-layer adaptive RVM reliability analysis method for small failure probability. Firstly, a first-layer adaptive RVM model is constructed by combining a Harris Hawks optimization algorithm and an adaptive RVM, and an importance sampling sample is generated at the design point according to an iterative updating strategy. Then, the RVM model is continuously updated by using a learning function based on the idea of active learning, and the failure probability is solved after convergence. The application greatly improves the accuracy of reliability failure probability calculation results, reduces the number of calculations, saves calculation cost, and improves the ability to calculate small failure probability reliability.

[0004] The technical solution adopted by the application to solve the technical problems comprises the following steps:

[0005] Step 1: constructing an initial model;

[0006] According to the original distribution density function f(x) of the input variable, an initial sample point is extracted, and the true function function value of the initial sample point is calculated to form a training set DoE. An initial RVM model of the response is constructed from the DoE X

[0007] Step 2: finding the design point on the Harris Hawks optimization algorithm MPP ＊ ;

[0008] Step 3: generating an importance sampling sample pool at MPP ＊ ＊ ;

[0009] ​​​Step 4: Based on the learning function U, from S ＊ Select the point with the greatest uncertainty and add it to the DoE. Determine whether the following stopping criteria are met. If they are met, proceed to step 5; otherwise, return to step 2.

[0010] The following two stopping criteria must be met simultaneously. The stopping criteria are:

[0011] A. The failure probability error between two consecutive design points does not exceed a%.

[0012]

[0013] Among them, P f1 and P f2 This represents the failure probability error corresponding to any two design points;

[0014] B. The MPP currently found ＊ The value of the learning function U is greater than or equal to 2:

[0015]

[0016] in, This represents the mean of the MPP points predicted by the current proxy model. This represents the standard deviation of the MPP points predicted by the current surrogate model;

[0017] Step 5: At the design point MPP that satisfies the two stopping criteria in Step 4 ＊ An important sample is generated at point S and labeled as S;

[0018] Step 6: Select the sample that contributes the most to improving the failure probability from S according to the learning equation (3) and add it to DoE, and retrain the RVM model;

[0019]

[0020] in, l(x) = min||xx i ||, This represents the mean of the RVM predictions. Let represent the standard deviation provided by RVM, l(x) be the minimum distance between a sample in the candidate sample point and a sample in the DoE, and l represents half the minimum distance between samples in the DoE; x represents a sample in the training pool. i x j Indicates different samples in the DoE;

[0021] Step 7: Determine if the accuracy has reached the set value. If it has, proceed to the next step; otherwise, return to step 6.

[0022] Step 8: Calculate the failure probability and the coefficient of variation;

[0023] The formula for calculating the failure probability is:

[0024]

[0025] Wherein, represents the indicator function, when the prediction value of the RVM is less than or equal to 0, 1, otherwise 0; represents the sampling function, N Is represents the number of important sampling samples;

[0026] The formula for calculating the coefficient of variation is:

[0027]

[0028] Wherein, h X (x) represents the important sampling function, f X (x) represents the original sampling function;

[0029] Step 9: If the coefficient of variation is less than or equal to b%, the result is accurate, otherwise, return to step 5, increase the number of important sampling samples.

[0030] Preferably, a = 5, b = 5.

[0031] The beneficial effects of the present application are as follows:

[0032] The present application proposes a double-layer agent model combining the Harris Hawks optimization algorithm adaptive RVM and the MCS-based adaptive RVM model to solve the problems of high calculation cost and low calculation efficiency of the failure probability calculation for small failure probability. The first layer adaptive RVM model is constructed by a small number of initial samples, and the Harris Hawks optimization algorithm is used to find the maximum design point, and then the important sampling samples are generated, which greatly reduces the calculation time and improves the calculation efficiency. The second layer MCS-based adaptive RVM model actively learns the candidate sample points generated in the first step based on the learning function proposed in the present application, which greatly improves the accuracy of the reliability failure probability calculation result and reduces the calculation times, saves the calculation cost, and improves the calculation ability of small failure probability reliability. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 It is the overall technical framework diagram of the present application.

[0034] Figure 2 It is the schematic diagram of the oil pipeline of the embodiment of the present application DETAILED DESCRIPTION

[0035] The application will be further described below in combination with the accompanying drawings and examples.

[0036] In order to improve the low efficiency and many calculation limitations of the existing reliability calculation with small failure probability, the application aims to provide a reliability analysis method which does not need to give failure samples in advance and can solve implicit function functions, and provide theoretical guidance for complex engineering problems.

[0037] The overall technical framework of the application is shown in Figure 1 .

[0038] Step 1: Constructing an initial model. According to the original distribution density function f X (x) of the input variable, a small number of initial sample points are extracted, and the true function function value is calculated to form a training set (Design of Experiment, DoE). The initial RVM model of g(X) is constructed from the DoE

[0039] Step 2: Find the design point on the Harris Hawks optimization algorithm , marked as MPP ＊

[0040] Step 3: Generate an important sampling sample pool at MPP ＊ , marked as S ＊ .

[0041] Step 4: According to the U learning function, pick out the point with the maximum uncertainty from S ＊ , add it to the DoE, and judge whether the stopping condition is reached. If it is reached, proceed to the next step, otherwise return to step 2.

[0042] The following two stopping criteria must be met at the same time, and the stopping criterion is:

[0043] A. The failure probability error corresponding to the two design points does not exceed 5%:

[0044]

[0045] B. The MPP found at present satisfies the value of the U learning function greater than or equal to 2:

[0046]

[0047] Step 5: Generate important sampling samples at the design points that meet the above criteria, marked as S.

[0048] Step 6: According to the learning equation proposed, select the point with the largest contribution to improving the failure probability from S and add it to the DoE, and retrain the proxy model.

[0049] The proposed learning equation is:

[0050]

[0051] wherein, l(x) = min ||x - x i ||, represents the mean value of RVM prediction, represents the standard deviation provided by RVM, l(x) is the minimum distance of the sample in the training pool and the sample in the DoE, represents half of the minimum distance of the sample in the DoE.

[0052] Step 7: Add the sample selected in step 6 to the DoE to retrain the RVM model, and determine whether the accuracy is sufficient. If sufficient, execute the next step, otherwise, return to step 6.

[0053] Step 8: Calculate the failure probability and the coefficient of variation. The calculation formula of the failure probability is The calculation formula of the coefficient of variation is

[0054] Step 9: If the COV is less than or equal to 5%, the result is considered accurate, otherwise, return to step 5 to increase the number of important sampling samples. Specific embodiments:

[0056] As shown in Figure 2 , for an axial functionally graded material pipe, the size of the pipe is fixed, and the pipe is used to transport oil. The density of the oil is 760 Kg / m 3 , according to the "Oil Pipeline Design and Management": the economic flow rate of the product pipeline in China is 2.0 m / s. The external excitation of the oil pipeline is 111 Hz. The density of the oil, the flow rate of the oil and the external excitation of the oil pipeline are random variables, and the standard deviation is 0.05 times the mean value, and they obey the normal variable.

[0057] The function function of the first-order resonance failure of the oil pipeline is y = |N F -E F | / N F -0.05. N F is the natural frequency of the oil pipeline, E F is the external excitation of the oil pipeline, and when y≤0, it is determined that the oil pipeline has resonance failure. The calculation results of several different algorithms are shown in Table 1.

[0058] Table 1 Analysis results of examples

[0059]

[0060] The Monte Carlo simulation (MCS) results are used as reference values, and the AK-MCS and AK-MCIS methods are used for comparison with the proposed method. Since this is a reliability problem with an implicit performance function, the AK-IS method cannot be used for computation. The results show that the proposed method has high computational efficiency, and only 42 performance function evaluations are needed to obtain a relatively accurate failure probability value. In addition, in order to start the Markov chain importance sampling algorithm, a failure sample needs to be given in advance. When the candidate sample size is 1 x 10 7 530, it is difficult to find a failure sample among only 530 failure samples, while the RVM-HIS does not need to give a failure sample in advance, greatly improving the automation of reliability analysis.

Claims

1. A double-layer adaptive RVM reliability analysis method for small failure probability, characterized in that, Applied to engineering structures, the engineering structures including a first order resonance failure of structures including oil pipelines, comprising the following steps: Step 1: Construct an initial model; According to the original distribution density function of the input variable Extracting initial sample points and calculating the real function function value of the initial sample points, and constructing a training set DoE; constructing an initial RVM model of response from DoE ; wherein the function function is the function function of the first order resonance failure of the oil pipeline, and the formula of the function function is as follows: ； wherein, is the natural frequency of the oil pipeline, is the external excitation of the oil pipeline, when resonance failure of the oil pipeline is determined to have occurred; Step 2: Find the design point on the Harris Hawks optimization algorithm, labeled MPP ＊ ;​ Step 3: Generate a pool of importance sampling samples at MPP ＊ marked as S ＊ ; Step 4: pick a point with the largest uncertainty from S ＊ according to the U-learning function, add it to the DoE, judge whether the following stopping criterion is reached, if reached, go to Step 5, otherwise return to Step 2; The following two stopping criteria must be met simultaneously, the stopping criteria are: A. The failure probability error corresponding to the design points of adjacent two iterations does not exceed a%: （1） wherein, and denotes the error in failure probability corresponding to any two design points; B. The MPP currently being sought ＊ The value of the U-learning function is greater than or equal to 2: （2） wherein, represents the mean of the MPP points predicted by the current proxy model, represents the standard deviation of the MPP points predicted by the current proxy model; Step 5: Generate an importance sampling sample at the design point MPP that satisfies both stopping criteria of Step 4, labeled S. ＊ Step 5: Generate an importance sampling sample at the design point MPP that satisfies both stopping criteria of Step 4, labeled S. Step 6: According to the learning equation (3), the sample with the largest contribution to improving the failure probability is selected from S and added to DoE, and the RVM model is retrained; （3） wherein, , , ; denotes the mean of the RVM prediction, denotes the standard deviation provided by the RVM, is the minimum distance of a sample in the candidate sample points and a sample in the DoE, denotes half of the minimum distance of a sample in the DoE; denotes a sample in the training pool, , denotes a different sample in the DoE; Step 7: Determine whether the accuracy reaches the set value, if yes, execute the next step, otherwise, return to step 6; Step 8: Calculate the failure probability and the coefficient of variation; The calculation formula of the failure probability is: wherein, represents an indicator function, when the prediction value of the RVM is less than or equal to 0, is 1, otherwise 0; represents a function to be sampled, represents the number of importance sampling samples; The calculation formula of the coefficient of variation is: wherein denotes the importance sampling function, denotes the original sampling function; Step 9: If the coefficient of variation is less than or equal to b%, then the result is accurate, otherwise, go to step 5 and increase the number of importance sampling samples.

2. The double-layer adaptive RVM reliability analysis method for small failure probability according to claim 1, characterized in that, The a=5, b=5.

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