Reliability Analysis Method Based on Group Monte Carlo and Active Learning Kriging

Abstract

The present invention discloses a reliability analysis method based on group Monte Carlo and active learning kriging, including: obtaining random variables that affect the reliability and service life of product structures and converting them into standard normal variables; using the Latin hypercube algorithm to generate an initial training point set and constructing a kriging model; using a probability classification function to construct an approximate optimal importance sampling auxiliary sampling function; using the group Monte Carlo algorithm and taking the approximate optimal importance sampling auxiliary sampling function as the goal, iteratively generating an importance sampling auxiliary density function; using the importance sampling auxiliary density function to generate several importance sampling sample points; solving the stopping condition based on the kriging prediction value and the kriging variance; and solving the structural failure probability and coefficient of variation. Through the above scheme, the present invention has the advantages of simple logic, accuracy and reliability, and has high practical value and promotion value in the field of structural reliability analysis and assessment technology.

Description

Technical Field

[0001] The present invention relates to the technical field of structural reliability analysis and evaluation, in particular to a reliability analysis method based on group Monte Carlo and active learning Kriging. Background Art

[0002] Research has found that the design of mechanical products affects product quality by 70%-80%. During the design phase, controlling product performance, safety, and reliability is crucial for its safe service. Uncertainties are prevalent in every stage of engineering product design and use, and the coupling of multiple uncertainties can damage load-bearing structures. There are numerous uncertainties in the design and manufacturing of mechanical structures. For example, when a rail train runs on a line, the loads on its load-bearing structure depend on the line's operating conditions, actual operating speed, suspension parameters, and the driver's personal habits. These uncertainties cannot be fully understood during the design phase of the load-bearing structure. In welded structures, weld joint defects are also cognitive uncertainties, while the stress-life relationship of welds is a random uncertainty. These uncertainties all require a reasonable mathematical model to express.

[0003] In practical engineering applications, real-world performance functions are often complex "black-box" models, such as finite element methods. Performing a single analysis is time-consuming, and both sampling and analytical methods are required to calculate failure probabilities. This multitude of performance function calls makes structural reliability analysis nearly impossible.

[0004] Minimizing the number of calls to functional functions has become an important issue in reliability analysis. Among them, reliability methods based on agent models have made certain progress. At present, Kriging models with active learning functions are widely used in the fields of structural reliability analysis and structural optimization. In addition, the AK-MCS method, which combines the active learning Kriging model with the classic Monte Carlo method, is the most commonly used reliability analysis method. Mechanical structures are high-reliability and long-life products, and their failure probability is usually small (such as less than). When using the AK-MCS method for analysis, at least simulation samples are required to ensure that the coefficient of variation of the failure probability is less than 5%, which will consume huge computing resources.

[0005] For example, the weighted exponential Monk Carlo method (AK-WMCS) proposed by Hu Zhipeng in "Research on Reliability Design Optimization Methods for Uncertain Mechanical Structures" discretizes random variables and uses the Simpson integral formula to convert the failure probability from an integral form to the sum of a series of quadratic function integrals. This method requires discretizing random variables in the entire variable space, and there is still a certain problem of sample point waste. At the same time, since this method is a non-parametric method, it is impossible to evaluate the coefficient of variation of the failure probability, and the unbiasedness of this method cannot be determined.

[0006] Therefore, there is an urgent need to propose a reliability analysis method based on group Monte Carlo and active learning Kriging that is simple in logic, accurate and reliable. Summary of the Invention

[0007] In view of the above problems, the present invention aims to provide a reliability analysis method based on group Monte Carlo and active learning kriging. The technical solutions adopted by the present invention are as follows:

[0008] The reliability analysis method based on Group Monte Carlo and Active Learning Kriging includes the following steps:

[0009] Obtain the random variable x that affects the reliability and service life of the product structure, and use the equal probability algorithm to convert the random variable x into a standard normal variable u;

[0010] The Latin hypercube algorithm is used to generate the initial training point set and calculate the true response value;

[0011] Construct a Kriging model;

[0012] Using the probability classification function to construct the approximate optimal importance sampling auxiliary sampling function;

[0013] Using the group Monte Carlo algorithm and aiming at approximating the optimal importance sampling auxiliary sampling function, the importance sampling auxiliary density function is iteratively generated.

[0014] Generate several importance sampling sample points using the importance sampling auxiliary density function;

[0015] Solve the stopping condition based on the Kriging prediction value and Kriging variance;

[0016] If the stopping condition is met, stop learning; otherwise, update the Kriging model;

[0017] Solve for the structural failure probability and coefficient of variation.

[0018] Furthermore, the Latin hypercube algorithm is used to generate the initial training point set {u} ini , and inversely transform the initial training point set into {x} by the equal probability change method ini, and bring it into the real function to calculate the real response value, the expression is:

[0019] G(x),x∈{x} ini

[0020] Among them, x is the real variable and G is the real functional function.

[0021] Furthermore, the probability classification function is used to construct an auxiliary sampling function that approximates the optimal importance sampling Its expression is:

[0022]

[0023]

[0024] in, represents the Kriging prediction value; represents the kriging standard deviation; Φ represents the standard normal distribution cumulative distribution function; π c (u) represents the Kriging probability classification function; φ(u) represents the standard normal distribution probability density function.

[0025] Preferably, the iterative process of the importance sampling auxiliary density function h(u) is as follows:

[0026] Step S51, initialize N po Normal distribution and serves as the suggested distribution for the group Monte Carlo algorithm; the N po is a natural number greater than 1; t represents the number of iterations, and t=0 at the beginning of the algorithm; Represents the mean parameter, the initial value of which is a random number generated arbitrarily; Represents the variance parameter, the initial value is the unit matrix;

[0027] Step S52, from any normal distribution Generate k po samples; that is: Recorded as sample set

[0028] Step S53, calculate samples The corresponding Kriging prediction value and Kriging variance;

[0029] Step S54, obtain the sample set The weight of any sample Its expression is:

[0030]

[0031] in, represents the auxiliary sampling function of the approximately optimal importance sampling of the jth sample in the i-th normal distribution;

[0032] Step S55: According to the weight and sample Perform polynomial resampling to obtain resampled samples Update the recommended distribution mean, which is expressed as:

[0033]

[0034] Step S56: Get the updated (t+1)th generation normal distribution according to the recommended distribution mean And from any (t+1) generation normal distribution Generate k po new samples and form a sample set

[0035] Step S57, calculate the new sample set The Kriging prediction value of any sample in and Kriging variance And get the number of points n in the failure domain ρ , whose expression is:

[0036]

[0037] Among them, I F represents the failure indicator function;

[0038] When n ρ Greater than ρ×N po ×k po When , the normal distribution As the importance sampling auxiliary density function h(u); otherwise, return to step S51.

[0039] Furthermore, the failure indication function I F The expression is:

[0040]

[0041] in, Represents the Kriging prediction value of any point u

[0042] Furthermore, the stopping condition is calculated based on the Kriging prediction value and the Kriging variance; the expression of the Ratio value of the stopping condition is:

[0043]

[0044] Among them, eps represents a very small number that avoids the denominator being 0. represents the kriging standard deviation.

[0045] Furthermore, a threshold of a preset stop condition is included; the threshold is 0.98; if the Ratio value is greater than the threshold, learning is stopped.

[0046] Preferably, the sampling space of the standard normal variable u is [-6, 6].

[0047] Furthermore, the expression of the structural failure probability is:

[0048]

[0049] Among them, I F (u) represents the failure indicator function of the standard normal variable u; f(u) represents the density function; R n Represents an n-dimensional variable space.

[0050] Furthermore, the coefficient of variation The expression is:

[0051]

[0052]

[0053] Among them, N IS represents the number of samples, represents the failure probability variance.

[0054] Compared with the prior art, the present invention has the following beneficial effects:

[0055] The present invention cleverly employs the AK-IS method, which combines an active learning kriging model with an importance sampling method, to effectively address the analysis of small failure probabilities. The AK-IS method uses the maximum failure point (MPP) closest to the coordinate origin on the proxy limit state surface as the sampling center, and calculates the failure probability using importance sampling. Because determining the MPP location requires solving complex constrained optimization equations, the present invention employs a reliability solution based on the group Monte Carlo method. The group Monte Carlo method is a variant of the importance sampling method. After generating importance samples, each sample is assigned a weight coefficient. By resampling the weight coefficients, sample points closer to the proxy limit state surface are more likely to be resampled. New sampling centers are calculated based on the resampled samples. After several iterations, the group Monte Carlo method will obtain samples that fill the failure region; ultimately, the failure probability and coefficient of variation are obtained. In summary, the present invention has the advantages of simple logic, accuracy, and reliability, and has high practical value and promotional value in the field of structural reliability analysis and assessment technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following is a brief introduction to the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and therefore should not be regarded as limiting the scope of protection. For those skilled in the art, other relevant drawings can be obtained based on these drawings without paying any creative work.

[0057] Figure 1 It is a logic flow chart of the present invention.

[0058] Figure 2 This is a practical application example diagram of the present invention. DETAILED DESCRIPTION

[0059] To make the purpose, technical solutions, and advantages of this application more clear, the present invention is further described below with reference to the accompanying drawings and examples. Implementation methods of the present invention include, but are not limited to, the following examples. All other embodiments obtained by persons of ordinary skill in the art based on the examples in this application without creative effort are within the scope of protection of this application.

[0060] In this embodiment, the term "and / or" is merely a description of the association relationship between associated objects, indicating that three relationships may exist. For example, A and / or B can represent three situations: A exists alone, A and B exist at the same time, and B exists alone.

[0061] In the description and claims of this embodiment, the terms "first" and "second" are used to distinguish different objects rather than to describe a specific order of objects. For example, a first target object and a second target object are used to distinguish different objects rather than to describe a specific order of objects.

[0062] In the embodiments of this application, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of this application should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0063] In the description of the embodiments of this application, unless otherwise specified, "multiple" means two or more. For example, "multiple processing units" means two or more processing units; "multiple systems" means two or more systems.

[0064] like Figure 1As shown, this embodiment provides a reliability analysis method based on group Monte Carlo and active learning Kriging. This embodiment uses the Kriging method and the importance sampling method to introduce it, so as to understand how the disclosed technical method combines the Kriging model with the importance sampling method to solve the failure probability of the structure and perform reliability analysis on the structure. The specific steps are as follows:

[0065] Assume there are n0 initial design points The true response values ​​of all design points The structural form of the Kriging proxy model can be expressed as:

[0066]

[0067] in, represents the mean of the Kriging model, and z(u) represents a stationary Gaussian process with a mean of 0 and a constant variance.

[0068] The importance sampling method shifts the sampling center from the origin to the area that has a significant contribution to the failure probability estimation. This strategy can effectively reduce the number of simulation samples and improve sampling efficiency. By constructing the auxiliary density function h(u), the estimated value of the failure probability is:

[0069]

[0070] Where f(u) is the density function; R n is an n-dimensional variable space; I F (u) is the failure indicator function. h(u) is the core of the importance sampling method. The general method for h(u) is to solve the design point on the proxy limit state surface through optimization methods, and construct an appropriate h(u) with the design point as the sampling center. However, using optimization methods to solve the design point often requires a lot of solution time. The present invention proposes to use the group Monte Carlo method to iteratively derive h(u), thereby avoiding the use of optimization methods to solve the failure point problem.

[0071] The reliability analysis method based on Group Monte Carlo and Active Learning Kriging in this embodiment is mainly as follows:

[0072] The first step is to obtain the random variable x that affects the reliability of the product structure and convert it into a standard normal variable u using an equal probability algorithm.

[0073] In the second step, the Latin hypercube algorithm is used to extract n0 samples of standard normal variables u as the initial training point set; the sampling space of the standard normal variable u is [-6,6]. is the input variable, and the corresponding true response value In this embodiment, the value of n0 is 12.

[0074] The third step is to construct a Kriging model based on the initial training point set. In this embodiment, constructing a Kriging model is a conventional technique that can be accomplished using the DACE toolbox in MATLAB or the third-party module smt in Python. This will not be described in detail here.

[0075] The fourth step is to set N po A normal distribution will be N po Normal distribution As the suggested distribution of the group Monte Carlo method, where t is the number of iterations; is the mean parameter; is the variance parameter. Initial parameters Randomly generated, is the unit matrix. The number of distributions N po It is recommended to take a larger value as possible to ensure the exploration ability of the group Monte Carlo method in the state space.

[0076] Step 5: From each distribution Generate k po samples, namely: Recorded as

[0077] Step 6: Calculate Corresponding predicted values ​​and Kriging variance

[0078] Step 7: Construct an approximate optimal auxiliary density function It is expressed as:

[0079]

[0080] in,

[0081] Step 8: Calculate the weight of each sample Its expression is:

[0082]

[0083] Step 9, yes Normalize. The normalized weight is:

[0084]

[0085] The tenth step is to sort the samples according to the weights. Perform polynomial resampling to obtain particles Update the proposed distribution mean:

[0086]

[0087] Step 11: From each distribution Generate k po particles, denoted as

[0088] Step 12: Calculate the response value of each sample and Kriging variance The number of points n in the failure domain is determined by the formula ρ ,

[0089]

[0090] Among them, I F is the failure indicator function, and its expression is:

[0091]

[0092] When nρ is greater than ρ×N po ×k po When As h(u), and generate important sample points {u (i) ,i=1,...,n IS}. Proceed to step 13; otherwise, proceed to step 5. The parameter value ρ is the key to determining whether the group Monte Carlo method converges. This paper recommends ρ = 0.5.

[0093] Step 13, calculate {u (i) ,i=1,...,n IS}Corresponding predicted value and Kriging variance Calculate the stopping condition Ratio value. If it is less than 0.98, proceed to step S15; otherwise, proceed to step 4.

[0094]

[0095] Step 14: Calculate {u (i) ,i=1,...,n IS}, select the minimum point u of the U function value * The U function can be expressed as:

[0096]

[0097] Step 15: u * The inverse transformation is x through the equal probability change method * , and calculate its true corresponding value G(x * ), will (u * ,G(x * ))Add to the training point set and update the Kriging model to proceed to the fourth step.

[0098] Step 16: Using the simulation sample {u (i) ,i=1,...,n IS}Calculate the failure probability of the structure and the coefficient of variation of the failure probability

[0099]

[0100]

[0101]

[0102] This embodiment uses a 6-dimensional application example to illustrate the present invention; this embodiment specifies the structure to be analyzed, the roof truss structure such as Figure 2 As shown. The top and compression rods of the structure are reinforced with concrete, while the bottom and tension rods are reinforced with steel. A uniformly distributed load q acts on the roof, which is converted into a node concentrated load P = ql / 4. The random variables are the uniformly distributed load q, the cross-sectional area of ​​the concrete rod A, and the C , cross-sectional area of ​​steel rod A S , elastic modulus of cement rod E C and the elastic modulus E of the steel member S .

[0103] The performance function of this embodiment is that the maximum vertical displacement of the structure does not exceed 3cm, that is,

[0104]

[0105] The following table is a table of all random variable parameters;

[0106] Table 1 Random variable distribution table of the embodiment

[0107]

[0108] The random variables (q,l,E s ,E C ,A s ,A c ) is transformed into a standard normal random variable (u1,u2,u3,u4,u5,u6).

[0109] Table 2 shows the calculation results for the Monte Carlo method (MCS), the first-order second moment method (FORM), the importance sampling method with active learning kriging (AK-IS), the modified importance sampling method with active learning kriging (AK-MIS), and the method of the present invention. Using the MCS results as a reference, the failure probability estimates obtained by the present invention are closest to those calculated by the reference method, with a relative error of only 0.03%, which is smaller than the other three methods. Furthermore, the number of function calls, 76, is also the lowest among all methods. These results demonstrate that the method proposed by the present invention is more accurate than the FORM, AK-IS, and AK-MIS.

[0110] Table 2 Comparison of calculation results

[0111]

[0112]

[0113] The above embodiments are only preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any changes that adopt the design principles of the present invention and any changes made through non-creative work on this basis should fall within the scope of protection of the present invention.

Claims

1. A reliability analysis method based on group Monte Carlo and active learning kriging, characterized by: The following steps are involved: Obtain the random variable x that affects the reliability and service life of the product structure, and use the equal probability algorithm to convert the random variable x into a standard normal variable u; The Latin hypercube algorithm is used to generate the initial training point set and calculate the true response value; Construct a Kriging model; Using the probability classification function to construct the approximate optimal importance sampling auxiliary sampling function; Using the group Monte Carlo algorithm and aiming at approximating the optimal importance sampling auxiliary sampling function, the importance sampling auxiliary density function is iteratively generated. Generate several importance sampling sample points using the importance sampling auxiliary density function; Solve the stopping condition based on the Kriging prediction value and Kriging variance; If the stopping condition is met, stop learning; otherwise, update the Kriging model; Calculate the structural failure probability and coefficient of variation; Among them, the probability classification function is used to construct an approximate optimal importance sampling auxiliary sampling function Its expression is: Where, represents the Kriging prediction value; represents the kriging standard deviation; Φ represents the standard normal distribution cumulative distribution function; π c (u) represents the Kriging probability classification function; φ(u) represents the standard normal distribution probability density function; The iterative process of the importance sampling auxiliary density function h(u) is as follows: Step S51, initialize N po Normal distribution and serves as the suggested distribution for the group Monte Carlo algorithm; the N po is a natural number greater than 1; t represents the number of iterations; represents the mean parameter; represents the variance parameter; Step S52, from any normal distribution Generate k po samples; that is: Recorded as sample set Step S53, calculate samples The corresponding Kriging prediction value and Kriging variance; Step S54, obtain the sample set The weight of any sample Its expression is: Where, represents the auxiliary sampling function of the approximately optimal importance sampling of the jth sample in the i-th normal distribution; Step S55: According to the weight and sample Perform polynomial resampling to obtain resampled samples Update the recommended distribution mean, which is expressed as: Step S56, the updated (t+1)th generation normal distribution is And from any (t+1) generation normal distribution Generate k po (t+1) generation samples and form a new sample set Step S57, calculate the sample set Kriging prediction value of any resampled sample in and Kriging variance And get the number of points n in the failure domain ρ , whose expression is: Where, I F represents the failure indicator function; When n ρ Greater than ρ×N po ×k po When , the normal distribution As the importance sampling auxiliary density function h(u); otherwise, return to step S51.

2. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 1 is characterized in that: The Latin hypercube algorithm is used to generate the initial training point set {u} ini , and inversely transform the initial training point set into {x} by the equal probability transformation method ini , and bring it into the real function to calculate the real response value, the expression is: G(x),x∈{x} ini Among them, x is the real variable and G is the real functional function.

3. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 1 is characterized in that: The failure indication function I F The expression is: in, Represents the Kriging prediction value of any point u.

4. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 3 is characterized in that: The stopping condition is calculated based on the Kriging prediction value and the Kriging variance; the expression of the Ratio value of the stopping condition is: Among them, eps represents a very small number that avoids the denominator being 0; represents the kriging standard deviation.

5. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 4 is characterized in that: It also includes a threshold value of a preset stop condition; the threshold value is 0.98; if the Ratio value is greater than the threshold value, learning is stopped.

6. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 1 is characterized in that: The sampling space of the standard normal variable u is [-6, 6].

7. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 4 is characterized in that: The expression of the structural failure probability is: Among them, I F (u) represents the failure indicator function of the standard normal variable u; f(u) represents the density function; R n Represents an n-dimensional variable space.

8. The reliability analysis method based on group Monte Carlo and active learning Kriging according to claim 7, characterized in that: The coefficient of variation The expression is: Among them, N IS represents the number of samples; represents the failure probability variance.

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