CN114169185A - System reliability analysis method under random and interval uncertainty mixing based on Kriging - Google Patents

Abstract

Aiming at the problem that most of the existing reliability analysis methods based on the Kriging model are only suitable for random uncertainty, the invention discloses a system reliability analysis method based on the Kriging under the condition of mixing random uncertainty and interval uncertainty, which comprises the following steps: determining a training sample set of a Kriging model corresponding to each failure mode of the current iteration: calculating response values of the sample points in the training sample set, and respectively constructing n Kriging models according to the training sample set and the response values; respectively predicting response mean value and variance information of MCS sample points by using the n Kriging models, and selecting an optimal sample point by using a learning function according to the response mean value and variance information; judging whether the current convergence criterion is met according to the response mean and variance information at the optimal sample point; and (4) according to the Kriging model obtained by the last iteration, combining the MCS to simulate and calculate the failure probability or reliability of the system.

Description

System reliability analysis method under mixed random and interval uncertainty based on Kriging

technical field

The invention belongs to the technical field of reliability engineering, and relates to a Kriging-based system reliability analysis method under mixed random and interval uncertainty.

Background technique

With the rapid development of science and technology and computers, equipment and systems are becoming larger and more complex, and it is particularly important whether the equipment and systems can operate reliably. Once a failure occurs during operation, it will lead to economic losses in light, and casualties in serious cases. Therefore, reliability analysis and evaluation of equipment and systems is indispensable. Existing structural reliability analysis methods include FORM (first-order reliability method), SORM (second-order reliability method), MCS (Monte Carlo simulation), methods based on RSM (response surface method), and methods based on adaptive Kriging. method etc.

FORM and SORM approximate the limit state equation through first-order and second-order Taylor expansions, respectively, and then solve the reliability index. However, these two methods are only suitable for the problem of a single MPP point (design check point), and cannot be applied to the problem with multiple MPP points. In addition, when the limit state equation is an implicit function (the mathematical expression is not shown), the above two methods are difficult to implement due to the need to obtain the first and second order derivatives of the limit state equation.

The application of MCS method in the field of reliability largely makes up for the deficiencies of FORM and SORM in the application of implicit functions. In the reliability analysis of the implicit function problem, the MCS method often needs to use the finite element model to realize the reliability analysis of the implicit function problem. However, reliability analysis often requires a large number of samples. For a complex system, it takes hours, days or even a month for a single call to calculate the finite element model. The time cost required for a large number of repeated calls is unacceptable. of.

The RSM-based reliability analysis method calculates the response of a small number of sample points through the finite element model, constructs a quadratic response surface function to approximate the implicit limit state equation, and then combines MCS, FORM or SORM and other methods for reliability analysis. This kind of method greatly reduces the time-consuming invocation of the simulation model, and makes the reliability analysis easy to implement while improving the efficiency. However, the RSM-based method has relatively poor accuracy for highly nonlinear problems.

At present, adaptive Kriging has received extensive attention from scholars in the field of reliability. As an interpolation method, Kriging constructs a Kriging model through a small number of sample points and their response values, and uses the model to predict the response of unknown sample points. Compared with the deterministic prediction of methods such as response surface, Kriging provides not only the mean value of the predicted response, but also the variance of the predicted response, which means that Kriging's prediction is a random variable. The existing reliability analysis methods based on adaptive Kriging have the following characteristics: 1) Most methods are proposed for the input of random variables, and for the problem of mixed variables (for example, the mixture of random variables and interval variables) Few studies; 2) Most methods are only applicable to components or systems with a single failure mode, and there are relatively few studies on reliability methods for components or systems with multiple failure modes; 3) In the construction process, most methods only focus on the response whether the sign of is predicted correctly.

SUMMARY OF THE INVENTION

Aiming at the problem that most current reliability analysis methods based on Kriging model are only suitable for random uncertainty, the present invention discloses a Kriging-based system reliability analysis method under mixed random and interval uncertainty, which can be applied to random uncertainty coupled with cognitive uncertainty.

The present invention is realized by the following technical solutions.

A Kriging-based system reliability analysis method under mixed random and interval uncertainty, including:

Determine the training sample set of the Kriging model corresponding to each failure mode of the current iteration:

Calculate the response values of the sample points in the training sample set, and then construct n Kriging models according to the training sample set and the response values respectively;

Utilize the n Kriging models to predict the response mean and variance information of the MCS sample points respectively, and use the learning function to select the best sample point according to the response mean and variance information;

According to the response mean and variance information at the best sample point, it is judged whether the current convergence criterion is satisfied. If it is satisfied, the iteration is stopped. If it is not satisfied, the Kriging model corresponding to the failure model that needs to be iterated is determined, and the failure mode is adopted. Calculate the response of the optimal sample point with the finite element model under , and iterate the Kriging model;

According to the Kriging model obtained in the last iteration, the failure probability or reliability of the system is calculated in combination with the MCS simulation.

Beneficial effects of the present invention:

By constructing a Kriging model, the present invention can greatly reduce the time-consuming and repeated calls to finite element and other models; at the same time, by constructing a learning function under random and interval variables, the constructed Kriging model can accurately estimate multiple failure modes The upper and lower bounds of the failure probability of the system. The invention can be applied to the reliability analysis under the mixture of random and interval variables, especially when the reliability analysis is performed on the engineering problem of the implicit function, and can better balance the relationship between the result accuracy and efficiency of the reliability analysis.

Detailed ways

The present invention will be described in detail below.

The Kriging-based system reliability analysis method under mixed random and interval uncertainty of this specific embodiment specifically includes:

Step 1. Determine the training sample set of the Kriging model corresponding to each failure mode of the current iteration:

where t∈{1, 2,...,n}, T _t is the training sample set of the Kriging model corresponding to the t-th failure mode, and t _n is the number of sample points in the training set samples;

Step 2, calculating the response value G _t of the sample points in the training sample set, and then constructing n Kriging models according to the training sample set and the response value respectively;

In this embodiment, the finite element simulation model is used to calculate the response value G _t of the sample points in the training sample set T _t corresponding to each failure mode, where _t ∈ {1, 2, . Obtained through field tests;

In specific implementation, if the current iteration is the first iteration, the process of constructing the initial Kriging model is as follows:

Determine the current training sample set for each Kriging model according to Latin hypercube sampling:

T _t = [(X ₁ , Y ₁ ); (X ₂ , Y ₂ ); ...; (X _N , Y _N )]

Among them, t∈{1,2,…,n}, N is 12;

然后基于当前的训练样本集T_t和响应集G_t通过 MATLAB工具箱DACE分别构建得到n个Kriging模型。The response of the failure mode corresponding to each training sample set is calculated through the finite element simulation model

Then, based on the current training sample set T _t and response set G _t , n Kriging models are constructed respectively through the MATLAB toolbox DACE.

Step 3, using the n Kriging models to predict the response mean and variance information of the MCS sample points respectively, and using the learning function to select the best sample point according to the response mean and variance information;

In this embodiment, the learning function is used to select the best sample point in the following manner:

确定最佳样本点：Use the current n Kriging models to predict the MCS sample points respectively, determine the predicted state of the system at each sample point according to the prediction information of each failure mode at each sample point, and then calculate the state error of the system at each sample point predicted expected rate

Determine the best sample point:

Step 4: Determine whether the current convergence criterion is satisfied according to the response mean and variance information at the best sample point. If it is satisfied, stop the iteration, if not, determine the Kriging model corresponding to the failure model that needs to be iterated. The finite element model under this failure mode calculates the response of the optimal sample point, and iterates the Kriging model;

In this embodiment, the Kriging model corresponding to the failure mode that needs to be iterated adopts the following methods:

其中j表示第j个失效模式。After determining the best sample point, select the model with the largest error prediction expectation at this point for iterative update, that is, we have

where j represents the jth failure mode.

In this embodiment, the convergence criterion is specifically:

After determining the best iteration sample points (X _best , Y _best ) and the Kriging model θ to be iterated, the value of the U function is calculated by the following formula:

为第θ个Kriging模型在最佳迭代样本点的U函数值，

和

分别为第θ个Kriging模型在最佳迭代样本点的响应均值和方差。本实施例中所指的U函数是现有方法的AK-MCS的学习函数。in,

is the U function value of the θth Kriging model at the best iterative sample point,

and

are the response mean and variance of the θth Kriging model at the best iterative sample point, respectively. The U function referred to in this embodiment is the learning function of the AK-MCS of the existing method.

时，收敛准则不成立，则将最佳样本点添加到第θ个模型的训练样本集T_θ＝[T_θ；X_best，Y_best]，1≤θ≤n中迭代该模型，重复上述过程；当

时，则停止迭代，并将当前的Kriging模型用于可靠性分析。when

When the convergence criterion does not hold, the best sample point is added to the training sample set of the θth model T _θ = [T _θ ; X _best , Y _best ], and the model is iterated in 1≤θ≤n, and the above process is repeated; when

, the iteration is stopped and the current Kriging model is used for reliability analysis.

Step 5. According to the Kriging model obtained in the last iteration, the failure probability or reliability of the system is calculated in combination with the MCS simulation.

Embodiment 1:

S1, assign the initial value. Determine the number n of failure modes in the system; set θ=0; determine the value of MCS sample capacity N _mc and the initial training sample set capacity N according to practical problems, generally speaking, N _mc ≥ 10 ⁵ , N=12.

表征系统的认知不确定性并建模。其中α为随机变量的个数，β为区间变量的个数，

分别表示区间变量Y_j的上下界。S2. Use random variables X _i (i=1, 2,..., a) to characterize the random uncertainty of the system, interval variables

Characterize and model the cognitive uncertainty of a system. where α is the number of random variables, β is the number of interval variables,

represent the upper and lower bounds of the interval variable _Yj , respectively.

The input variables of each failure mode of the system can be determined according to the system's design standards, specifications, working environment, and operating specifications, and the distribution types of the corresponding variables can be fitted according to historical data or test data, and the corresponding parameters can be estimated.

S3. Generate an MCS simulation sample point set. Denoted as S={(X _i , Y _i )}, i=1, 2, . . . , N _mc , where N _mc is the total number of sample points. For random variables, sample points are generated according to their probability distribution functions; for interval variables, Latin hypercube sampling is used to generate sample points, so that they evenly cover the entire interval.

S4. Generate the same initial training sample set for each failure mode according to Latin hypercube sampling. Denoted as: T _t = [(X ₁ , Y ₁ ); (X ₂ , Y ₂ ); ...; (X _N , Y _N )], where t∈{1, 2, ..., n}, N takes the value is 12. For random variables, sample points are generated according to their distribution information; for interval variables, sample points are uniformly generated according to the upper and lower bounds of the interval.

其中，t∈{1，2，…，n}。S5. Calculate the response of each sample point in the training sample set according to the simulation model (for example, a finite element model). Record as:

where t∈{1,2,…,n}.

S6. Iterate the Kriging model. If θ=0, construct each Kriging model according to T _t and G _t to obtain the model g _t , where t=1, 2, ..., n; if θ≠0, iterate the θth Kriging model g according to T _θ and G _{θ θ} .

和方差

S7. Predict the response information of the unknown sample point. Use the model obtained in S6 to predict the response information of each sample point in the MCS sample set, including the response mean

and variance

确定最佳迭代样本点(X_best，Y_best)以及需要迭代的Kriging模型。学习函数

通过以下过程确定：S8. According to the predicted response information and learning function

Determine the best iterative sample points (X _best , Y _best ) and the Kriging model that needs to be iterated. learning function

Determined by the following process:

定义响应的符号预测错误的可能取值为R_(X，Y)＝max(g_(X，Y)-0，0)，响应的符号预测正确的可能取值为 S_(X，Y)＝max(0-g_(X，Y)，0),其期望分别为：For a single failure mode, when the predicted mean

Define the possible value of the wrong symbol prediction of the response as R _{(X, Y)} = max(g _{(X, Y)} -0, 0), and the possible value of the correct symbol prediction of the response as S _{(X, Y)} = max (0-g _{(X, Y)} , 0), the expectations are:

Among them, Φ and Φ are the probability distribution function and probability density function of standard normal variables, respectively.

定义响应的符号预测错误的可能取值为 R_(X，Y)＝max(0-g_(X，Y)，0)，响应的符号预测正确的可能取值为S_(X，Y)＝max(g_(X，Y)-0，0)，其期望分别为：When predicting the mean

Define the possible value of the wrong symbol prediction of the response as R _{(X, Y)} = max(0-g _{(X, Y)} , 0), and the possible value of the correct symbol prediction of the response to be S _{(X, Y)} = max (g _{(X, Y)} -0, 0), which are expected to be:

Combining the above four equations, we get:

Since the magnitude of the response of each sample point may vary greatly in value, in order to avoid the influence of the magnitude difference, the above two formulas are processed as follows:

和

其中 t＝1，2，…，n。由于系统的状态是根据各失效模式的预测响应的符号确定的，因此可根据

和

计算系统的错误预测期望率

where ER _(X,Y) and ES _(X,Y) are defined as the expected rate of wrong prediction and the expected rate of correct prediction, respectively. Generally, there are many failure modes in the system. For a system with n failure modes, there are correspondingly n Kriging models. So at each unknown sample point (X, Y), we have

and

where t=1,2,...,n. Since the state of the system is determined according to the sign of the predicted response of each failure mode, it can be determined according to the

and

Calculate the expected rate of misprediction of the system

错误预测的期望率为：For systems in which the failure modes are in series, it is assumed that the failure modes are independent of each other. If the state of the system is predicted to be normal, the state of the system is correctly predicted if and only if the response symbols for all failure modes are correctly predicted. Therefore, the expected rate of the correct prediction of the state of the system is

The expected rate of misprediction is:

If the state of the system is predicted to be faulty, then the state of the system is mispredicted only when all the failure modes with a negative response sign are mispredicted and all the failure modes with a positive response are correctly predicted. Therefore, the expected rates of incorrect prediction and correct prediction of the state of the system are:

where k is the number of failure modes whose response sign is predicted to be negative. Unifying the above two formulas are:

Similarly, for a system whose failure mode is in parallel, the expectation of system state prediction error is:

where k is the number of failure modes whose response sign is predicted to be positive.

For a system with complex connection failure mode, it can be transformed into a series-parallel system through the idea of structure function and minimum circuit set. Therefore, combining the expression of the expected rate of misprediction in series and parallel, the misprediction expression of any system can be obtained as:

Among them, m is the number of the minimum way sets, q represents the qth smallest way set, p is the number of the smallest way sets in the system without failure, n _q is the number of failure modes in the qth smallest way set, and k _q is the th th smallest way set. The response sign in the set of q smallest ways is predicted to be a negative number of failure modes.

的值应逐渐在减小。因此，MCS仿真样本点中，具有最大

值的样本点应确定为最佳迭代样本点，也即有：In the iterative process, as the iteration progresses,

should gradually decrease. Therefore, among the MCS simulation sample points, the maximum

The sample point of the value should be determined as the best iterative sample point, that is:

The above formula is the learning function in the present invention. After determining the optimal iterative sample point, it is necessary to determine the iterative Kriging model, which is determined according to the following formula:

是否成立。若成立，则执行S10，若不成立，则置T_θ＝[T_θ；(X_best，Y_best)]，

返回执行S6。S9. Judging Convergence Criterion

is established. If so, execute S10; if not, set T _θ =[T _θ ; (X _best , Y _best )],

Return to S6.

After determining the best iteration sample points (X _best , Y _best ) and the Kriging model θ to be iterated, the value of the U function is calculated by the following formula:

为第θ个Kriging模型在最佳迭代样本点的U函数值，

和

分别为第θ个Kriging模型在最佳迭代样本点的预测均值和标准差。U函数是现有方法的AK-MCS的学习函数，此处不再赘述。当

时，表明当前该样本点的响应符号错误预测的概率小于0.0228。in,

is the U function value of the θth Kriging model at the best iterative sample point,

and

are the predicted mean and standard deviation of the θth Kriging model at the best iterative sample point, respectively. The U function is the learning function of the AK-MCS of the existing method, which will not be repeated here. when

, indicating that the probability of wrong prediction of the response symbol of the current sample point is less than 0.0228.

S10, stop the iteration, and use the current Kriging model for reliability analysis. According to the n Kriging models obtained from the final iteration, combined with the structure function, the upper and lower bounds of the failure probability are estimated. The formula is as follows:

Among them, Ψ(·) is the structural function of the system, which is the basic knowledge in structural reliability, and will not be repeated here. Y ^I =[Y ^L , Y ^U ] represents the interval vector Y=[Y ₁ , Y ₂ , ..., Y _n ] upper and lower bounds of the interval.

To sum up, the above are only preferred examples of the present invention, and are not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims (6)

1. A system reliability analysis method under the mixing of random uncertainty and interval uncertainty based on Kriging is characterized by comprising the following steps:

determining a training sample set of a Kriging model corresponding to each failure mode of the current iteration:

calculating response values of the sample points in the training sample set, and respectively constructing n Kriging models according to the training sample set and the response values;

respectively predicting response mean value and variance information of MCS sample points by using the n Kriging models, and selecting an optimal sample point by using a learning function according to the response mean value and variance information;

judging whether the current convergence criterion is met or not according to the response mean value and variance information of the optimal sample point, if so, stopping iteration, if not, determining a Kriging model corresponding to a failure model needing iteration, calculating the response of the optimal sample point by adopting a finite element model in the failure mode, and iterating the Kriging model;

and (4) according to the Kriging model obtained by the last iteration, combining the MCS to simulate and calculate the failure probability or reliability of the system.

2. The method for mixed stochastic and interval uncertainty based system reliability analysis of claim 1 wherein finite element simulation models are used to calculate response values for sample points in the training sample set corresponding to each failure mode.

3. The method of claim 1 or 2, wherein if the current iteration is the first iteration, the process of constructing the initial Kriging model is as follows: determining a training sample set of each current Kriging model according to Latin hypercube sampling, calculating the response of a failure mode corresponding to each training sample set, and then respectively constructing n Kriging models through an MATLAB tool box DACE based on the current training sample set and the response set.

4. The method for Kriging-based mixed random and interval uncertainty reliability analysis of system as claimed in claim 1 or 2, wherein the learning function is used to select the best sample point by: and predicting the MCS sample points by using the current n Kriging models respectively, determining the prediction state of the system at each sample point according to the prediction information of each failure mode at each sample point, calculating the expected rate of state error prediction of the system at each sample point, and determining the optimal sample point.

5. The method for analyzing system reliability under the mixture of random and interval uncertainties based on Kriging as claimed in claim 1 or 2, wherein the Kriging model corresponding to the failure mode determined to be iterated adopts the following mode: after the best sample point is determined, the model with the largest expectation of error prediction at the point is selected for iterative updating.

6. The method for analyzing system reliability under a mixture of random and interval uncertainties based on Kriging as claimed in claim 1 or 2, wherein the convergence criterion is specifically:

after determining the best iteration sample point and the Kriging model to iterate, the value of the U function is calculated: when the value of the U function is smaller than 2 and the convergence criterion is not established, adding the optimal sample point to the training sample set of the theta model to iterate the model, and repeating the process; and when the value of the U function is more than or equal to 2, stopping iteration and using the current Kriging model for reliability analysis.