A conjugate first-order reliability analysis method coupled with the active Kriging algorithm

Abstract

The present invention discloses a conjugate first-order reliability analysis method coupled with an active Kriging algorithm, which is applied to the field of structural reliability analysis and aims at the problem of low efficiency existing in the prior art. The present invention uses an active Kriging surrogate model to model the performance function of the structure whose reliability index is to be evaluated, adopts a resampling strategy to update the sample points, uses the conjugate first-order reliability method to solve the structural reliability analysis, and proposes an approximate numerical differentiation method, which solves the problem of large errors in the calculation of derivative information by the Kriging surrogate model and can effectively calculate the derivative of the conjugate first-order reliability method.

Description

Technical Field

[0001] The invention belongs to the field of structural reliability analysis, and in particular relates to a structural reliability analysis method for evaluating reliability indicators by combining a conjugate first-order reliability analysis method with an active Kriging algorithm. Background Art

[0002] With the rapid development and progress of modern aviation, aerospace, navigation and other industrial fields, engineering components are developing towards precision, complexity, high speed, large scale and high performance. The safety and reliability of such products are of great significance and value to the development of my country's economy, industry and national defense. However, uncertainty is inevitable in the design, manufacturing and service of products. At the same time, when products face harsh operating conditions and harsh service environments, once a failure occurs, it will cause economic and property losses at the least, or even catastrophic accidents and threaten life safety. Therefore, it is of great significance to analyze the sources of uncertainty of engineering components, clarify the key factors affecting the structural reliability of products, conduct structural reliability analysis of engineering components, and determine the structural reliability indicators of engineering components to meet their normal use and maintenance, which is of great significance to ensure the safe service of engineering components. The structural reliability analysis method based on probability theory and mathematical statistics still has problems in the efficient solution of nonlinear, low failure probability and high computational cost structural reliability analysis in actual engineering, and the related needs are becoming more and more urgent.

[0003] So far, there has been a deep accumulation of research on structural reliability analysis. Researchers hope to find a structural reliability analysis method that can achieve low computational cost and high computational efficiency for practical engineering problems. The main methods currently include Monte Carlo simulation and the variance reduction technology developed on this basis; approximate analytical methods based on Taylor series expansion; however, the former requires a high number of calculations for system response (i.e., the specific results of the functional function obtained by random variables in a specific structural reliability analysis problem), and requires a large sample size to achieve appropriate accuracy, while the latter has low convergence for the iterative process of complex nonlinear functional functions. The surrogate model method developed through machine learning theory can effectively reduce the computational cost of system response in structural reliability analysis, but the existing surrogate model methods mainly focus on the application of simulation methods, and such methods also have a high number of calls to the surrogate model. Especially for problems with low failure probability, the efficiency of such methods still has major problems. Summary of the invention

[0004] To solve the above technical problems, the present invention proposes a conjugate first-order reliability analysis method coupled with an active Kriging algorithm, which constructs a sample pool of the active Kriging algorithm by using a resampling strategy, establishes a surrogate model of the performance function through a U learning function, uses the obtained Kriging model for the conjugate first-order reliability method, calculates the conjugate gradient by an approximate numerical differentiation method, and finally solves the reliability index.

[0005] The technical solution adopted by the present invention is: a conjugate first-order reliability analysis method coupled with an active Kriging algorithm, including:

[0006] S1. Analyze the composition, functions, and working conditions of the structure whose reliability index is to be evaluated, determine the failure mode of the structure and the corresponding performance function, and obtain the random variables affecting the performance function of the structure and their distribution information;

[0007] S2. According to the performance function, random variables, and their distribution information determined in step S1, establish a Kriging surrogate model y(X);

[0008] According to the distribution information of the random variables in step S1, use the uniform sampling method to obtain candidate sample points; then go to step S3;

[0009] S3. Take the current candidate sample points as the input of the structure whose reliability index is to be evaluated, and obtain the mean and variance of the Kriging surrogate model y(X);

[0010] S4. Substitute the current mean and variance into the U learning function. If the convergence condition is not satisfied, find the sample point X U at which the U learning function U(X * ) reaches the maximum value. Calculate the output G(X * ) of the structure whose reliability index is to be evaluated according to the performance function in step S1, establish a Kriging model through the existing input-output data, and jump to step S5; if the convergence condition is satisfied, jump to step S6;

[0011] S5. Centered on the sample point X * at which the U learning function reaches the maximum value obtained in step S4, use the uniform sampling method to obtain a new candidate sample space according to the distribution information of the random variables in step S1; go to step S3;

[0012] S6. According to the final Kriging surrogate model y(X), calculate the gradient information of the function by an approximate numerical differentiation method, and solve the reliability index through the conjugate first-order reliability method.

[0013] The establishment of the initial Kriging surrogate model y(X) described in step S1 includes the following process:

[0014] The initial sample size is determined according to the dimension of the random variable in step S1, and the initial sample space X is obtained by uniform sampling based on the initial sample size and the distribution information of the random variable in step S1. m , based on the initial sample space X m , according to the performance function in step S1, the corresponding system response G(X m );

[0015] The initial sample space X m As the input of the structure to be evaluated for reliability indicators, X m The corresponding system response G(X m ) is used as the output of the structure of the reliability index to be evaluated, and the Kriging proxy model y(X) is established.

[0016] Beneficial effects of the present invention: The present invention is a conjugate first-order reliability analysis method coupled with an active kriging algorithm, which adopts a resampling strategy to construct a sample pool of the active kriging algorithm, and establishes a proxy model of the functional function through a U learning function, and uses the obtained kriging model for the conjugate first-order reliability method, and calculates the conjugate gradient through an approximate numerical differential method, and finally solves the reliability index. It has the following advantages:

[0017] (1) Active Kriging proxy modeling can accurately construct the limit state surface of the functional function through learning functions, and at the same time can reduce the number of calls to functional functions with high computational costs, such as the number of calls to finite element simulation, thereby improving the overall efficiency of the structural reliability analysis method;

[0018] (2) The conjugate first-order reliability method has higher convergence than the traditional first-order reliability method when solving structural reliability analysis problems, which improves the robustness of the algorithm; the analytical method is more efficient in solving structural reliability analysis problems than the simulation method;

[0019] (3) The uniform sampling method can cover the sample space well, making the establishment of the proxy model more accurate;

[0020] (4) The newly proposed resampling strategy reduces the number of times the proxy model is called each time the sample points are updated, thereby improving the efficiency of the algorithm.

[0021] (5) A new approximate numerical differentiation method is proposed to solve the problem of large errors in the calculation of derivative information of the Kriging proxy model, and can effectively calculate the derivative of the conjugate first-order reliability method.

[0022] (6) A new hybrid method for structural reliability analysis combining the Kriging algorithm and the conjugate first-order reliability method is proposed. This method has low requirements on the number of times the system response is called and the number of times the proxy model is called, which makes the overall efficiency of the present invention higher. Brief Description of the Drawings

[0023] Figure 1 is the flowchart of the method of the present invention;

[0024] Figure 2 is the schematic diagram of uniform sampling and resampling of the method of the present invention;

[0025] Figure 3 is the schematic diagram of the sample space update of the active Kriging algorithm of the present invention;

[0026] Figure 4 is the iterative process of the conjugate first-order reliability method in the present invention;

[0027] Among them, Figure 4 (a) is the iterative convergence process of the sample space, Figure 4 (b) is the convergence process diagram of the reliability index.

[0028] Figure 5 is the schematic diagram of the roof truss structure in Embodiment 2 of the present invention. Detailed Embodiments

[0029] To facilitate the understanding of the technical content of the present invention by those skilled in the art, the content of the present invention will be further explained below with reference to the accompanying drawings.

[0030] The following illustrates the implementation manners of the present invention through specific specific embodiments. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in the specification. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all the embodiments. The present invention can also be implemented or applied through other different specific implementation manners. Various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0031] It should be noted that the following describes various aspects of the embodiments within the scope of the appended claims. Obviously, the aspects described herein can be embodied in a wide variety of forms, and any specific structure and / or function described herein is illustrative only. Based on the present invention, those skilled in the art should understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects described herein can be used to implement the device and / or practice the method.

[0032] It should also be noted that the illustrations provided in the following embodiments only schematically illustrate the basic concept of the present invention. The illustrations only show the components related to the present invention, rather than being drawn according to the number, shape, and size of the components in the implementation. In actual implementation, the form, quantity, and proportion of each component can be arbitrarily changed, and the layout form of its components can also be more complex.

[0033] In addition, in the following description, specific details are provided to facilitate a thorough understanding of the examples. However, those skilled in the art will understand that the aspects can be practiced without these specific details.

[0034] An embodiment of the present invention provides a conjugate first-order reliability analysis method based on a coupling active Kriging method and a resampling strategy. To make the purpose, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0035] As Figure 1 shown, a conjugate first-order reliability analysis method using a coupling active Kriging algorithm proposed by the present invention includes the following steps:

[0036] 1. Analyze the composition, function, and working conditions of the structure based on information such as the structure's instruction manual for the reliability index to be evaluated, design standards, expert opinions, historical data, etc., determine the failure mode of the structure and the corresponding performance function G(X), and obtain the random variable X affecting the structure's performance function and its distribution information. The structure failure can be expressed as: G(X) ≤ 0;

[0037] where X = (x1, x2, …, x n ); n represents the dimension of the structure performance function;

[0038] When determining the variables affecting the performance function, such as the size of the structure, material properties, working conditions load, etc., it should be understood that due to randomness and uncertainty, the values of the variables are usually random, and a probability distribution is needed to describe their uncertainty. As for the modeling of the probability distribution of the variables, it can be obtained based on existing data or engineering experience. This belongs to the existing general technology and will not be elaborated in the present invention.

[0039] When determining the performance function of the structure, it should be understood that the performance function is divided into explicit and implicit. In general engineering applications, for implicit performance functions, finite element analysis technology is generally needed to solve and represent them.

[0040] 2. Determine the initial sample size according to the dimension n of the random variable: m = 5 * n, and obtain its initial sample space X using the uniform sampling method according to the distribution function of the random variable in step 1m , and obtain the corresponding system response G(X according to the functional function in Step 1 m ).

[0041] According to the existing input-output data, that is, the initial sample space X m and the corresponding system response G(X m ), establish the initial Kriging surrogate model y(X).

[0042] Use the uniform sampling method to sample the random variables according to the distribution function of the random variables in Step 1 to obtain candidate sample points X U , where the sample size N U The larger the value, the more accurate the result, but the computational cost will increase. In this embodiment, it is recommended that the sample size N U = 1000. Generally, the larger the sample size, the more accurate the result, but the computational cost will also increase. Considering the balance between computational cost and accuracy, the sample size in this embodiment is 1000.

[0043] For the above uniform sampling method, its sampling formula is:

[0044] X = F -1 (p)+[F -1 (1 - p)-F -1 (p)]*rand

[0045] where F -1 (·) represents the inverse function of the cumulative distribution function of the random variable X; p = 1×10 -6 is the interval boundary probability; rand is a random number between 0 and 1.

[0046] 3. According to the sample input with a sample size of N U in Step 2, obtain the mean μ y (X U ) and variance of the Kriging model y(X

[0047] 4. Substitute the mean μ y (X U ) and variance obtained in Step 3 into the U learning function to determine whether the convergence condition is satisfied: If the convergence condition is not satisfied, find the sample point X U corresponding to the maximum value of the U learning function U(X * ), calculate the system output G(X * ) according to the functional function in Step 1, establish a Kriging model through the existing input-output data, and jump to Step 5; if the convergence condition is satisfied, then jump to Step 7;

[0048] where the U learning function is:

[0049]

[0050] Further, |·| represents taking the absolute value, and the convergence condition described in step 4 is min(U(X U )) > 2.

[0051] 5. Based on the sample point X of the maximum value of the U learning function obtained in step 4 * , centered on this sample point, using the uniform sampling method according to the distribution function of the random variable in step 1, resample a sample space X with a sample size of N R = 1000; R ;

[0052] For the resampling strategy, the sampling formula is:

[0053]

[0054]

[0055] X' = Lower + (Upper - Lower) * rand

[0056] where X * is the sample point of the maximum value of the U learning function; X' is the resampled sample point; Upper and Lower are the upper and lower boundaries of resampling.

[0057] 6. Based on the sample X in step 5 R obtain the mean μ y (X R ) and variance of the Kriging model y(X). Calculate the U learning function U(X y ) according to the mean μ R and variance , and determine whether U(X R ) converges. If it does not converge, find the sample point X of the maximum value of the U function R , calculate the system output G(X * ) according to the performance function in step 1, and establish the Kriging model y(X) through the existing input-output data, that is, the obtained performance function data set. Repeat step 6 until the U function converges. Go back to step 4; *

[0058] Further, the convergence condition described in step 6 is min(U(X R )) > 2.

[0059] 7. Based on the final Kriging surrogate model y(X), the gradient information of the function is calculated using the approximate numerical differentiation method, and the reliability index is solved by the conjugate first-order reliability method. The formula for the approximate numerical differentiation method in step 7 is as follows:

[0060]

[0061] where y(·) is the obtained Kriging surrogate model; y′(x) is the derivative of the random variable x; and Δx = 0.001 is the approximate differential offset.

[0062] The specific steps of the conjugate first-order reliability method in step 7 are as follows:

[0063] 7.1. For a specific structural reliability analysis problem, determine the performance function G, the mean μ x and the standard deviation σ x of the random variable x, where the performance function G = y(X) is the Kriging surrogate model, and the random variable x represents the specific value of the random variable X in the solution process of the conjugate first-order reliability method.

[0064] 7.2. Initialization: k = 0, d k = 0, x k = μ x , ε = 1×10 -6 . Here, k is used to record the number of iterations, ||·|| is the norm of the matrix, λ k represents the step size of the k-th iteration, and x k represents the value of the random variable at the k-th iteration.

[0065] 7.3. Equivalent normal transformation u k = T(x k ), where the equivalent normal transformation is as follows:

[0066] F X (x) = Φ(u)

[0067]

[0068]

[0069]

[0070] where T represents the equivalent normal transformation; F X (x) is the cumulative distribution function of the random variable X; Φ(·) is the standard normal cumulative distribution function; is the equivalent standard deviation; is the standard normal probability density function; Φ -1 (·) is the standard normal inverse cumulative distribution function; is the equivalent mean value; u is the standardized random variable;

[0071] 7.4. Calculate the conjugate gradient vector: Wherein, represents the gradient of the performance function; here represents the transpose of the matrix;

[0072] In general engineering applications, the calculation of the function gradient belongs to the existing general technology, and the present invention will not elaborate.

[0073] 7.5. Calculate the conjugate unit vector

[0074] 7.6. Calculate the new iteration point:

[0075] 7.7. Inverse transformation of equivalent normalization x k = T -1 (u k ), wherein, T -1 represents the inverse transformation of equivalent normalization,

[0076] 7.8. Judgment: Whether the convergence condition is satisfied: If the condition is satisfied, continue to the next step; otherwise, k = k + 1, update the finite step size Return to step 7.3, wherein, ε represents the constant for judging convergence, ε = 1×10 -6 ;

[0077] 7.9. After satisfying the convergence condition in step 7.8, the reliability index calculation formula is:

[0078]

[0079] Wherein, is the reliability index; is the function gradient; T is the transpose of the vector; is the conjugate unit vector. The entire calculation process ends.

[0080] Those skilled in the art should know that in the structural reliability analysis, the relationship between the reliability index β and the failure probability Pf is: Pf = Φ(-β). Φ() represents the cumulative distribution function of the standard normal distribution. The two are completely equivalent. In related research, generally, either one can be obtained.

[0081] Example 1

[0082] This Example 1 verifies this method through a two-dimensional structural reliability analysis case, and this example is a typical example of highly nonlinear problems in structural reliability analysis.

[0083] 1. Analyze the composition, function, and working conditions of the structure based on information such as the specification, design standard, expert opinion, and historical data of the reliability index to be evaluated, determine the failure mode of the structure and the corresponding performance function G(X), and obtain the random variable X affecting the performance function of the structure and its distribution information;

[0084] The performance function in this embodiment is expressed as:

[0085]

[0086] where X = (x1, x2), and the dimension n of the structure performance function is 2;

[0087] The distribution information of the random variable is as follows: x1 and x2 are independent of each other. x1 follows a lognormal distribution with a mean of 5 and a standard deviation of 1, and x2 follows a Gumbel extreme value distribution with a mean of 10 and a standard deviation of 10. Its statistical information is shown in Table 1.

[0088] Table 1 Statistical information of random variables in this case

[0089] Random variable Mean value Standard deviation Distribution function <![CDATA[x1]]> 5 1 Log-normal distribution <![CDATA[x2]]> 10 10 Gumbel distribution

[0090] 2. Determine the initial sample size according to the dimension n = 2 of the random variable: m = 5 * n = 10, and obtain its initial sample space X using the uniform sampling method according to the distribution function of the random variable in step 1 m , and obtain the corresponding system response G(X m ) according to the performance function in step 1, as shown in Table 2.

[0091] Table 2 Initial sample input space, performance function output

[0092] Number <![CDATA[x1]]> <![CDATA[x2]]> G(X) 1 13.65 53.06 23299.24 2 1.22 16.15 100.64 3 20.14 11.50 1615.50 4 16.50 209.26 60569.30 5 9.51 38.87 1522.87 6 7.08 10.15 1 5659.78 7 19.20 100.33 50172.94 8 8.20 168.71 235385.42 9 10.14 -4.95 84111.78 10 17.77 124.34 54023.47

[0093] Establish an initial Kriging surrogate model y(X) based on the existing input-output data, that is, the initial sample space X m and the corresponding system response G(X m ).

[0094] Obtain candidate sample points X U using the uniform sampling method according to the distribution function of the random variable in step 1, where the sample size N U = 1000.

[0095] For the above uniform sampling method, its sampling formula is:

[0096] X = F -1 (p)+[F -1 (1 - p)-F -1(p)] * rand

[0097] Among them, F -1 (·) represents the inverse function of the cumulative distribution function of the random variable X; p = 1×10 -6 is the boundary probability of the interval; rand is a random number between 0 and 1.

[0098] In this embodiment, the boundaries of uniform sampling are x1 ∈ [1.0775, 22.3091] and x2 ∈ [-21.5801, 256.8503].

[0099] 3. According to the sample input with a sample size of N U in step 2, obtain the mean μ y (X U ) and variance

[0100] 4. Substitute the mean μ y (X U ) and variance obtained in step 3 into the U learning function, and determine whether the convergence condition is satisfied: If the convergence condition is not satisfied, find the sample point X U corresponding to the maximum value of the U learning function U(X * ), calculate the system output G(X * ) according to the performance function in step 1, establish a Kriging model through the existing input-output data, and jump to step 5; if the convergence condition is satisfied, then jump to step 7;

[0101] Among them, the U learning function is:

[0102]

[0103] Furthermore, the convergence condition in step 4 is min(U(X U )) > 2.

[0104] 5. As Figure 2 shown, according to the sample point X * corresponding to the maximum value of the U learning function obtained in step 4, centered on this sample point, use the uniform sampling method according to the distribution function of the random variable in step 1 to resample the sample space X R with a sample size of N R = 10000;

[0105] For the resampling strategy, the sampling formula is:

[0106]

[0107]

[0108] X′ = Lower + (Upper - Lower) * rand

[0109] where X * is the sample point of the maximum value of the U learning function; X′ is the resampled sample point; Upper and Lower are the upper and lower boundaries of resampling.

[0110] 6. Based on the sample X in step 5 R obtain the mean μ y (X R ) and variance According to the mean μ y (X R ) and variance calculate the U learning function U(X R ), and determine whether U(X R ) converges. If it does not converge, find the sample point X * of the maximum value of the U function, calculate the system output G(X * ) according to the performance function in step 1, and establish the Kriging model y(X) through the existing input-output data, that is, the obtained performance function data set. As shown in Table 3, repeat step 6 until the U function converges, and return to step 4;

[0111] Furthermore, the convergence condition in step 6 is min(U(X R )) > 2. In this embodiment, when finally converging, min(U) = 20.98.

[0112] Table 3 Uniform sampling and resampling process

[0113]

[0114] 7. According to the final Kriging surrogate model y(X), use the approximate numerical differentiation method to calculate the gradient information of the function, and solve the reliability index through the conjugate first-order reliability method. The formula of the approximate numerical differentiation method in step 7 is:

[0115]

[0116] where y(·) is the obtained Kriging surrogate model; y′(x) is the derivative of the random variable x; Δx = 0.001 is the approximate differential offset.

[0117] As Figure 4 (a) shows, the specific steps of the conjugate first-order reliability method in step 7 are as follows:

[0118] 7.1. For a specific structural reliability analysis problem, determine the performance function G, the mean μ x of the random variable x x, where the functional function \(G = y(X)\) is a Kriging surrogate model.

[0119] 7.2. Initialization: \(k = 0\), d k = 0, \(x\) k = \(\mu\) x , \(\varepsilon = 1\times10\) -6 . Here, \(k\) is used to record the number of iterations, and \(\|\cdot\|\) is the norm of the matrix.

[0120] 7.3. Equivalent normal transformation \(u\) k = \(T(x\) k ), where the equivalent normal transformation is:

[0121] F X (x)=\(\varPhi(u)\)

[0122]

[0123]

[0124]

[0125] where \(F\) X (x) is the cumulative distribution function of the random variable \(X\); \(\varPhi(\cdot)\) is the standard normal cumulative distribution function; is the equivalent standard deviation; is the standard normal probability density function; \(\varPhi\) -1 (\cdot) is the standard normal inverse cumulative distribution function; is the equivalent mean; \(u\) is the standardized random variable;

[0126] 7.4. Calculate the conjugate gradient vector: where represents the gradient of the functional function; here represents the transpose of the matrix;

[0127] In general engineering applications, the calculation of the function gradient belongs to the existing general technology, and the present invention will not elaborate.

[0128] 7.5. Calculate the conjugate unit vector

[0129] 7.6. Calculate the new iteration point:

[0130] 7.7. Equivalent normal inverse transformation \(x\) k = \(T\) -1 (u k ), where \(T\) -1 represents the equivalent normal inverse transformation,

[0131] 7.8. Judgment: Check whether the convergence condition is met: If the condition is met, proceed to the next step; otherwise, k = k + 1 and update the finite step size Go back to step 7.3, where ε represents the constant for judging convergence, ε = 1×10 -6 ;

[0132] 7.9. After meeting the convergence condition in step 7.8, the reliability index calculation formula is:

[0133]

[0134] where, is the reliability index; is the function gradient; T is the transpose of the vector; is the conjugate unit vector. The entire calculation process ends.

[0135] In this embodiment, when the conjugate first - order reliability method converges, the obtained design point is: x = [2.6475, 0.9315].

[0136] As shown in Table 4, the final result of this case takes the Monte Carlo simulation (MCS) with a sample size of 1×10 7 as a reference, and its reliability index is 3.5613. Among them, "-" indicates no such item or the method fails; the reliability index of the method proposed in the present invention is 3.2587. In addition, by comparing the differences in the "number of times of calling the surrogate model" and the "number of times of calling the system response" among the active Kriging Monte Carlo method, the first - order reliability method, and the conjugate first - order reliability method, where the active Kriging method and the first - order reliability method are common methods and will not be elaborated here, and the conjugate first - order reliability method is an analytical method without considering the surrogate model. The results show that the method proposed in the present invention can efficiently and accurately calculate the reliability index of the structure in the embodiment. Although the relative error between the proposed method and the Monte Carlo simulation is large, the accuracy of the present invention fully meets the engineering requirements. Furthermore, the method proposed in the present invention has a lower number of times of calling the surrogate model and the system response than other methods, showing the advantage of this method in terms of efficiency.

[0137] Table 4 Comparison of Results in Example 1

[0138]

[0139] Example 2

[0140] To further demonstrate the effectiveness of the method proposed in the present invention, a common engineering system is proposed as an example to elaborate on the method proposed in the present invention in detail.

[0141] Example 2 is a roof truss structure. The roof truss is as Figure 5 shown. The materials of the bottom and tension-bearing members are steel, and the top and compression-bearing members are reinforced with cement. It is assumed that a uniformly distributed load is applied to the roof truss and evenly applied to the roof.

[0142] 1. Analyze the composition, function, and working conditions of the structure based on information such as the structure's specification, design standards, expert opinions, and historical data for the reliability index to be evaluated. Determine the failure mode of the structure and the corresponding performance function G(X), and obtain the random variables X affecting the performance function of the structure and their distribution information;

[0143] In this Example 2, the failure mode is that the displacement at point C exceeds the predetermined value. Therefore, the performance function of the structure is expressed as:

[0144]

[0145] where X = (q, l, A s , A c , E s , E c ). q is the load on the roof truss; l is the length of the truss; A s and A c represent the cross-sectional areas of reinforced concrete and steel respectively; E s and E c represent the elastic moduli of reinforced concrete and steel respectively. The dimension n of the structure performance function is 6; and the distribution information of the random variables is shown in Table 5, and each random variable is independent of each other.

[0146] Table 5 Statistical Characteristics of Random Variables of Roof Truss Structure

[0147]

[0148] 2. Determine the initial sample size according to the dimension n = 6 of the random variables: m = 5 * n = 30, and use the uniform sampling method to obtain its initial sample space X m according to the distribution function of the random variables in step 1, and obtain the corresponding system response G(X m ) according to the performance function in step 1. Part of the initial samples are shown in Table 6.

[0149] Table 6 Initial Sample Input Space and Performance Function Output

[0150]

[0151] Based on the existing input-output data, that is, the initial sample space X m and the corresponding system response G(X m ), establish the initial Kriging surrogate model y(X).

[0152] Obtain the candidate sample point X using the uniform sampling method according to the distribution function of the random variable in step 1 U , where the sample size N U = 1000.

[0153] 3. According to the sample input with a sample size of N U in step 2, obtain the mean μ y (X U ) and variance

[0154] 4. Substitute the mean μ y (X U ) and variance obtained in step 3 into the U learning function to determine whether the convergence condition is satisfied: If the convergence condition is not satisfied, find the sample point X U at the maximum value of the U learning function U(X * ), calculate the system output G(X * ) according to the performance function in step 1, establish a Kriging model through the existing input-output data, and jump to step 5; if the convergence condition is satisfied, jump to step 7;

[0155] Among them, the U learning function is:

[0156]

[0157] Furthermore, the convergence condition in step 4 is min(U(X U )) > 2.

[0158] 5. As Figure 2 shown, according to the sample point X * at the maximum value of the U learning function obtained in step 4, centered on this sample point, use the uniform sampling method according to the distribution function of the random variable in step 1 to resample the sample space X with a sample size of N R = 1000 R ;

[0159] For the resampling strategy, the sampling formula is:

[0160]

[0161]

[0162] X′ = Lower + (Upper - Lower) * rand

[0163] Among them, X *The sample point for the maximum value of the U learning function; X′ is the resampled sample point; Upper and Lower are the upper and lower boundaries of resampling.

[0164] 6. According to the sample X in step 5 R Obtain the mean μ of the Kriging model y(X) y (X R ) and variance According to the mean μ y (X R ) and variance Calculate the U learning function U(X R ), and determine whether U(X R ) converges. If it does not converge, find the sample point X of the maximum value of the U function * , calculate the system output G(X * ) according to the performance function in step 1, and establish the Kriging model y(X) through the existing input-output data, that is, the obtained performance function data set. As shown in Table 7, repeat step 6 until the U function converges, and return to step 4;

[0165] Furthermore, the convergence condition described in step 6 is min(U(X R )) > 2. In this embodiment, when finally converging, min(U) = 2.03.

[0166] Table 7 Changes in the U function

[0167]

[0168] 7. According to the final Kriging surrogate model y(X), use the approximate numerical differentiation method to calculate the gradient information of the function, and solve the reliability index through the conjugate first-order reliability method. The formula for the approximate numerical differentiation method in step 7 is:

[0169]

[0170] Among them, y(-) is the obtained Kriging surrogate model; y′(x) is the derivative of the random variable x; Δx = 0.001 is the approximate differential offset.

[0171] As Figure 4 (a) shows, the specific steps of the conjugate first-order reliability method in step 7 are:

[0172] 7.1. For a specific structural reliability analysis problem, determine the performance function G, the mean μ x and standard deviation σ x of the random variable x, where the performance function G = y(X) is the Kriging surrogate model.

[0173] 7.2. Initialization: k = 0, dk = 0, x k = μ x , ε = 1 × 10 -6 . Among them, k is used to record the number of iterations, and ||·|| is the norm of the matrix.

[0174] 7.3. Equivalent normal transformation u k = T(x k ), where the equivalent normal transformation is:

[0175] F X (x) = Φ(u)

[0176]

[0177]

[0178]

[0179] Among them, F X (x) is the cumulative distribution function of the random variable X; Φ(·) is the standard normal cumulative distribution function; is the equivalent standard deviation; is the standard normal probability density function; Φ -1 (·) is the standard normal inverse cumulative distribution function; is the equivalent mean; u is the standardized random variable;

[0180] 7.4. Calculate the conjugate gradient vector: Among them, represents the gradient of the performance function; here represents the transpose of the matrix;

[0181] In general engineering applications, the calculation of the function gradient belongs to the existing general technology, and the present invention will not elaborate.

[0182] 7.5. Calculate the conjugate unit vector

[0183] 7.6. Calculate the new iteration point:

[0184] 7.7. Equivalent normal inverse transformation x k = T -1 (u k ), where T -1 represents the equivalent normal inverse transformation,

[0185] 7.8. Judgment: Whether the convergence condition is satisfied: If the condition is met, proceed to the next step; otherwise, k = k + 1 and update the finite step size. Return to step 7.3, where ε represents the constant for judging convergence, ε = 1×10 -6 ;

[0186] 7.9. After meeting the convergence condition in step 7.8, the reliability index calculation formula is:

[0187]

[0188] where is the reliability index; is the function gradient; T is the transpose of the vector; is the conjugate unit vector. The entire calculation process ends.

[0189] In this embodiment, when the conjugate first-order reliability method converges, the obtained design point is: x = [2.32×10 4 , 12.09, 8.74×10 -4 , 0.032, 8.97×10 10 , 1.91×10 10 .

[0190] As shown in Table 8, the final result of this case is referenced by the Monte Carlo simulation (MCS) with a sample size of 1×10 7 , and its reliability index is 3.8112. Among them, "-" indicates no such item or the method fails; the reliability index of the method proposed in the present invention is 3.9127. In addition, by comparing the differences in the "number of calls to the surrogate model" and "number of calls to the system response" among the active Kriging Monte Carlo method, the first-order reliability method, and the conjugate first-order reliability method, where the active Kriging method and the first-order reliability method are general methods and will not be elaborated here, and the conjugate first-order reliability method is an analytical method that does not consider the surrogate model. The results show that the method proposed in the present invention can efficiently and accurately calculate the reliability index of the structure in the embodiment, and the accuracy of the present invention fully meets the engineering requirements. Furthermore, the method proposed in the present invention has a lower number of calls to the surrogate model and the system response than other methods, showing the efficiency advantage of this method.

[0191] Table 8 Comparison of Results in Example 2

[0192]

[0193] Those of ordinary skill in the art will realize that the embodiments described herein are provided to assist the reader in understanding the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Various modifications and variations can be made to the present invention for those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A conjugate first-order reliability analysis method coupled with an active Kriging algorithm, characterized in that Including: S1. Analyze the composition, function, and working conditions of the structure of the reliability index to be evaluated, determine the failure mode of the structure and the corresponding performance function, and obtain the random variables affecting the performance function of the structure and their distribution information; S2. According to the performance function, random variables, and their distribution information determined in step S1, establish a Kriging surrogate model y(X); According to the distribution information of the random variables in step S1, use the uniform sampling method to obtain candidate sample points; then go to step S3; S3. Use the current candidate sample points as the input of the structure of the reliability index to be evaluated, and obtain the mean and variance of the Kriging surrogate model y(X); S4. Substitute the current mean and variance into the U learning function. If the convergence condition is not satisfied, find the sample point X U that maximizes the U learning function U(X * ). Calculate the structural output G(X * ) of the reliability index to be evaluated according to the function function in step S1. Establish a Kriging model through the existing input-output data, and jump to step S5; if the convergence condition is satisfied, jump to step S6; S5. Using the sample point X of the maximum value of the U learning function obtained in step S4 * as the center, according to the distribution information of the random variable in step S1, a new candidate sample space is resampled by using the uniform sampling method; go to step S3; S6. According to the final Kriging surrogate model y(X), use the approximate numerical differentiation method to calculate the gradient information of the function, and solve the reliability index by the conjugate first-order reliability method.

2. A conjugate first-order reliability analysis method of a coupled active Kriging algorithm according to claim 1, characterized in that The process of establishing the initial Kriging surrogate model y(X) described in step S2 includes the following steps: Determine the initial sample size according to the dimension of the random variable in step S1, and obtain its initial sample space X using the uniform sampling method based on the initial sample size and the distribution information of the random variable in step S1 m , based on the initial sample space X m , obtain the corresponding system response G(X m ) according to the performance function in step S1; Take the initial sample space X m as the input of the structure for which the reliability index is to be evaluated, and X m the corresponding system response G(X m ) as the output of the structure for which the reliability index is to be evaluated, and establish the Kriging surrogate model y(X).

3. A conjugate first-order reliability analysis method using the coupled active Kriging algorithm according to claim 2, characterized in that The sampling formula corresponding to step S2 is: X = F -1 (p) + [F -1 (1 - p) - F -1 (p)] * rand Among them, F -1 (·) represents the inverse function of the cumulative distribution function of the random variable X; p is the boundary probability of the interval; rand is a random number between 0 and 1.

4. A conjugate first-order reliability analysis method of a coupled active Kriging algorithm according to claim 3, characterized in that The convergence condition in step S4 is min(U(X U )) > 2.

5. A conjugate first-order reliability analysis method using a coupled active Kriging algorithm according to claim 4, characterized in that The sampling formula corresponding to the resampling in step S5 is: X′ = Lower+(Upper - Lower)*rand Among them, X * is the sample point of the maximum value of the U learning function; X′ is the resampled sample point; Upper and Lower are the upper and lower boundaries of resampling.

6. The conjugate first-order reliability analysis method of a coupled active Kriging algorithm according to claim 5, characterized in that The expression of the U learning function is: where, μ y (X) represents the mean of the Kriging surrogate model y(X), and σ y (X) represents the variance of the Kriging surrogate model y(X).

7. A conjugate first-order reliability analysis method using the coupled active Kriging algorithm according to claim 6, characterized in that, Step S6 specifically includes the following sub-steps: S61. Determine the function G of the structure of the reliability index to be evaluated and the mean value μ and standard deviation σ of the random variable x, where the function G = y(X) is a Kriging surrogate model; x and the standard deviation σ x , where the function G = y(X) is a Kriging surrogate model; S62. Initialization: k = 0, d k = 0, x k = μ x , ε = 1×10 -6 ; where k is used to record the number of iterations, represents the gradient of the objective function, ||·|| is the norm of the matrix; S63, equivalent normalizing transformation u k = T(x k ), where the equivalent normalizing transformation is as follows: F X f(x) = Φ(u) where, F X (x) is the cumulative distribution function of the random variable X; Φ(·) is the standard normal cumulative distribution function; is the equivalent standard deviation; is the standard normal probability density function; Φ -1 (·) is the standard normal inverse cumulative distribution function; is the equivalent mean; u is the standardized random variable; S64. Calculate the conjugate gradient vector: where represents the gradient of the performance function; here represents the transpose of the matrix; S65. Calculate the conjugate unit vector S66. Calculate a new iteration point: S67, equivalent normal inverse transformation x k = T -1 (u k ), where T -1 represents the equivalent normal inverse transformation, S68. Judgment: Whether the convergence condition is satisfied: If the condition is satisfied, proceed to the next step; otherwise, k = k + 1 and update the finite step size. Go back to step S63, where ε represents the constant for judging convergence. S69. After meeting the convergence condition of step S68, the reliability index calculation formula is: Among them, is the reliability index; is the function gradient; T is the transpose of the vector; is the conjugate unit vector.

Images