Reliability analysis method based on direct optimization and two-sample point updating

Abstract

The application discloses a reliability analysis method based on Direct optimization and double sample point updating. The application is based on a Direct optimization algorithm, sets an optimization target, carries out preliminary sampling on samples, then respectively in a safe domain and an unsafe domain, based on an adaptive selection rule of Euclidean distance, constantly selects two sample points to update a sample set by using an adaptive strategy, and uses a support vector regression to fit an approximate function function, uses an inner product kernel function to replace nonlinear mapping to a high-dimensional space, enhances nonlinear adaptability of a response surface method, and carries out Monte Carlo simulation on the basis to analyze structural reliability. The application has good universality in structural reliability analysis, can adapt to various nonlinear problems, expands the applicable range of the Direct optimization algorithm and the support vector machine machine learning method in the field of structural reliability analysis, and has important theoretical and engineering significance.

Description

Technical Field

[0001] This invention relates to the field of structural reliability analysis technology, and in particular to structural reliability analysis using response surface methodology combined with Monte Carlo simulation, specifically to a reliability analysis method based on Direct optimization and dual-sample point update. Background Technology

[0002] Reliability analysis of structures or products in fields such as civil engineering, mechanical engineering, and aerospace reasonably considers the uncertainties in engineering projects. It is widely accepted by engineering professionals and is an important means of developing engineering structure or product design theory. Stochastic structural or product reliability analysis mainly analyzes the influence of objective factors stemming from loads, material properties, and the manufacturing process of the structure or product. It is of great significance for safety assessment in engineering practice, safe operation of structures or products, and improving key influencing factors to enhance safety reserves.

[0003] In fields such as civil engineering, mechanical engineering, and aerospace, the functional functions characterizing the normal operating capability or critical safety of large and complex structures or products are often highly nonlinear and implicitly expressed. In such cases, reliability analysis of structures or products using either the classical first-order second-moment method or the Monte Carlo method is difficult or inefficient, either lacking accuracy or being extremely time-consuming. This is especially true when large-scale numerical methods such as the finite element method are needed to perform extensive analysis of the structural and product response, making it difficult to meet the efficiency and accuracy requirements of reliability analysis in engineering practice.

[0004] Monte Carlo random sampling and quasi-Monte Carlo sampling methods, while producing samples with high randomness or good uniformity, are not necessarily efficient sample construction methods that significantly impact the accuracy of failure probability. Response surface methodology, based on a small number of representative structural response analyses, constructs substitute functions to approximate the true performance functions for reliability analysis of structures or products in civil engineering, mechanical engineering, and aerospace fields. Especially when combined with Monte Carlo simulation methods, it avoids extensive structural response analysis while ensuring high accuracy in reliability analysis, greatly improving the efficiency of reliability analysis and gaining increasing attention and application in engineering practice. Summary of the Invention

[0005] The purpose of this invention is to address the aforementioned deficiencies in the prior art by providing a reliability analysis method based on Direct optimization and dual-sample point updates. This method is highly universal and applicable to the structural reliability analysis of various nonlinear performance functions. It uses the Direct optimization algorithm combined with a learning function to select suitable small samples and uses support vector regression to fit an approximate function. The inner product kernel function replaces the nonlinear mapping to a high-dimensional space, enhancing the nonlinear adaptability of the response surface method. Based on this, Monte Carlo simulation is performed to conduct structural reliability analysis, improving the accuracy of the response surface method using support vector machine regression in calculating structural reliability indices. This method is an extension of existing structural reliability methods.

[0006] The objective of this invention can be achieved by adopting the following technical solutions:

[0007] A reliability analysis method based on Direct optimization and dual-sample point update, characterized in that the reliability analysis method includes the following steps:

[0008] S1, specifying the product structure of the domain to be analyzed, the function g(x) reflecting the normal working capability or critical state of safe operation of the structure or product in the domain to be analyzed, and the random variable x = (x1, x2, ..., x...). n ) and its characteristic parameters, where n is the number of elements in the random variable x, x1, x2, ..., xn. n These are the components of the random variable x, and the fields to be analyzed include civil engineering, mechanical and electronic engineering, and aerospace.

[0009] S2. Let the random variable x = (x1, x2, ..., x...) n Transform the variable into a standard normal variable z = (z1, z2, ..., z) using the Rosenblatt transform. n ), z1, z2, ..., z n These are the components of the standard normal variable z;

[0010] S3. Set the loop variable iter = 1;

[0011] S4. Generate 2n+1 initial sample points using the Direct optimization algorithm, calculate the structural response corresponding to the sample based on the function g(x), establish a sample training set, and establish a support vector machine regression function. Using f(z) as the objective function, another iteration is performed using the Direct optimization algorithm. The k samples with smaller f(z) values ​​generated in the iteration are added to the previous samples. The true structural response corresponding to each sample is calculated, and a new training set is established. The sample set consisting of 2n+k+1 samples is denoted as Z0. The support vector regression response surface function is then updated.

[0012] S5. Let iter = iter + 1. Randomly generate N candidate sample points using the Monte Carlo method. After an adaptive point selection strategy, select one candidate sample point from the safe region and one from the unsafe region, and calculate the corresponding true structural response to add it to the sample set Z. 0 In the middle, the support vector regression response surface function is updated again.

[0013] S6. Response surface function based on support vector regression Calculate the probability of structural failure;

[0014] S7. Verify whether the structural failure probability converges. If it converges, obtain the structural failure probability and reliability index β = Φ. -1 (P f ), where Φ -1 (·) is the inverse function of the standard normal distribution function; otherwise, return to step S5 to continue execution.

[0015] Furthermore, the value of k in step S4 is selected as follows:

[0016] When n ≤ 10, k = n; when n > 10, k = 10.

[0017] Furthermore, in step S2, it is assumed that the joint cumulative distribution function of vector x is F. x (x), the Rosenblatt transform is calculated according to the following expression.

[0018]

[0019] Where Φ(·) and Φ -1 (·) represents the standard normal probability distribution function and its inverse function, respectively; the Rosenblatt transform can be found in the monograph "Zhang Ming. Structural Reliability Analysis: Methods and Procedures. Science Press, 2009."

[0020] Furthermore, in step S4, the objective function f(z) is defined as:

[0021]

[0022] in, Let p(z) be the support vector regression response surface function, and p(z) be the joint probability density function. w represents the standard normal variable z = (z1, z2, ..., zn) n The w-th component in f) zw (z w Let z = (z1, z2, ..., zn) be a standard normal variable. n The density function of the w-th standard random variable in ), i.e.

[0023] Furthermore, the adaptive point selection strategy process in step S5 is as follows:

[0024] S51, using support vector regression response surface function N candidate sample points are randomly generated, and these N candidate sample points are divided into N1 sample points in a safe region and N2 sample points in an unsafe region. The safe region corresponds to... Insecure domain correspondence

[0025] S52. Suppose N3 is the smaller of N1 and N2, let N3 = min(N1, N2), and define... The absolute value is used as the criterion for selecting sample points, and N3 / 2 absolute values ​​are taken from the safe region and the unsafe region respectively. Smaller candidate sample points constitute the candidate sample set. and

[0026] S53. Calculate the candidate sample sets respectively. and The Euclidean distances between N3 / 2 candidate sample points are used to select the candidate sample point with the largest minimum Euclidean distance and add it to the training set.

[0027]

[0028] Where min represents finding the minimum value, and max represents finding the maximum value. This means finding the minimum value of the function with respect to the variable z, and then finding the maximum value. 0 Let Z be any point in the point set Z1, and ||·|| be the Euclidean distance. or

[0029] Furthermore, the support vector machine regression model It has the following form:

[0030]

[0031] Where l is the number of sample support vectors, i is the subscript or superscript variable, and z (i) Let K(z,z) represent the support vector corresponding to the i-th standard random variable z generated from the samples. When choosing the Gaussian kernel function, K(z,z) (i) )=exp(-γ||zz (i) ||) 2 γ is the kernel parameter, α i , are the first and second Lagrange multipliers of the aforementioned support vector machine model, respectively, and b is the hyperplane parameter. These parameters are obtained by solving the optimization algorithm in the support vector machine machine learning method.

[0032] The present invention has the following advantages and effects compared with the prior art:

[0033] (1) This invention introduces the Direct optimization algorithm combined with an adaptive point selection strategy. It uses Euclidean distance to find suitable small samples and captures the sample points that have failed with a large contribution to participate in the training of support vector machine regression. This can achieve satisfactory reliability analysis accuracy and avoid the instability of analysis results.

[0034] (2) Meanwhile, the training samples are fitted with support vector regression to approximate the function function, and the inner product kernel function is used to replace the nonlinear mapping to the high-dimensional space to enhance the nonlinear adaptability of the response surface method. On this basis, Monte Carlo simulation is performed to perform structural reliability analysis, which ensures the accuracy of the solution and enhances the applicability of the reliability method. Attached Figure Description

[0035] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0036] Figure 1 This is a flowchart of a reliability analysis method based on Direct optimization and dual-sample point update provided by the present invention;

[0037] Figure 2 This is a schematic diagram of the ten-bar truss structure in Embodiment 2 of the present invention. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0039] Example 1

[0040] This embodiment 1 further illustrates the invention using an application example containing six random variables. A reliability analysis method based on Direct optimization and two-sample point update includes the following steps:

[0041] S1. Specify the structure to be analyzed. Taking the failure probability estimation of a three-variable function as an example, the corresponding function is:

[0042] g=(πx1x2 2 cos(x3) 2 )-1.65(x6 / 0.66+x5 / (0.41x4)) / 10 6

[0043] Table 1. Distribution of random variables in Example 1

[0044] variable Distribution pattern mean Standard deviation <![CDATA[x1]]> normal 3500 175 <![CDATA[x2]]> normal <![CDATA[2.5×10 -3 ]]> <![CDATA[1.25×10 -4 ]]> <![CDATA[x3]]> normal 0.524 <![CDATA[1.048×10 -2 ]]> <![CDATA[x4]]> normal 0.9 <![CDATA[2.25×10 -2 ]]> <![CDATA[x5]]> Log-normal 5000 250 <![CDATA[x6]]> Log-normal 6000 300

[0045] S2. Transform random variables into standard normal variables using transformations such as Rosenblatt;

[0046] S3. Set the loop variable iter = 1;

[0047] S4. Generate 2n+1 initial sample points using the Direct optimization algorithm, calculate the structural response corresponding to the sample based on the function g(x), establish a sample training set, and establish a support vector machine regression function. Using f(z) as the objective function, another iteration is performed using the Direct optimization algorithm. The k samples with smaller f(z) values ​​generated in the iteration are added to the previous samples. The true structural response corresponding to each sample is calculated, and a new training set is established. The sample set consisting of 2n+k+1 samples is denoted as Z. 0 Update the support vector regression response surface function

[0048] S5. Let iter = iter + 1. Randomly generate N candidate sample points using the Monte Carlo method. After an adaptive point selection strategy, select one candidate sample point from the safe region and one from the unsafe region, and calculate the corresponding true structural response to add it to the sample set Z. 0 In the middle, the support vector regression response surface function is updated again.

[0049] S6. Response surface function based on support vector regression Calculate the probability of structural failure;

[0050] S7. Verify whether the structural failure probability converges. If it converges, obtain the structural failure probability and reliability index β = Φ. -1 (P f ), where Φ -1 (·) is the inverse function of the standard normal distribution function; otherwise, return to step S5 to continue execution.

[0051] Table 2 compares the failure probabilities and their relative errors calculated by the reliability analysis method disclosed in Example 1 with those calculated by other methods. As can be seen from Table 2, the failure probability calculated by the reliability analysis method based on Direct optimization and dual sample point update of the present invention has a small relative error and can meet the actual needs of engineering.

[0052] Table 2. Comparison of failure probabilities and their relative errors calculated by various methods in Example 1

[0053]

[0054] Example 2

[0055] This embodiment 2 further illustrates the invention using an application example containing 5 random variables. A reliability analysis method based on Direct optimization and two-sample point update includes the following steps:

[0056] S1. Specify the structure to be analyzed. A simplified calculation diagram of the ten-bar truss structure is shown below. Figure 2 The lengths of the horizontal and vertical bars are constants, 9m, and the random variable is the cross-sectional area A of each unit bar. i The cross-sectional area of ​​the vertical members is A1, the cross-sectional area of ​​the horizontal members is A2, and the cross-sectional area of ​​the diagonal members is A3. The elastic modulus of all members is E, and the concentrated load is P. The maximum deformation is controlled by the horizontal displacement D(x) at the right apex, and the maximum allowable displacement is specified to be no more than 0.1m. The function of the structure is: g(x) = 0.1 - D(x). The five random variables are the areas A1, A2, A3, the elastic modulus E, and the concentrated load P. Their statistical parameters are shown in Table 3.

[0057] Table 3. Distribution of random variables in Example 2

[0058] variable Distribution pattern mean Standard deviation unit <![CDATA[A1]]> normal <![CDATA[2×10 -2 ]]> <![CDATA[1×10 -3 ]]> <![CDATA[m 2 ]]> <![CDATA[A2]]> normal <![CDATA[2×10 -3 ]]> <![CDATA[1×10 -4 ]]> <![CDATA[m 2 ]]> <![CDATA[A3]]> normal <![CDATA[5.5×10 -3 ]]> <![CDATA[2.5×10 -4 ]]> <![CDATA[m 2 ]]> P logarithm <![CDATA[3×10 5 ]]> <![CDATA[3×10 4 ]]> N E Extreme Value Type I <![CDATA[7.1×10 4 ]]> <![CDATA[3.55×10 3 ]]> Mpa

[0059] S2. Transform random variables into standard normal variables using transformations such as Rosenblatt;

[0060] S3. Set the loop variable iter = 1;

[0061] S4. Generate 2n+1 initial sample points using the Direct optimization algorithm, calculate the structural response corresponding to the sample based on the function g(x), establish a sample training set, and establish a support vector machine regression function. Using f(z) as the objective function, another iteration is performed using the Direct optimization algorithm. The k samples with smaller f(z) values ​​generated in the iteration are added to the previous samples. The true structural response corresponding to each sample is calculated, and a new training set is established. The sample set consisting of 2n+k+1 samples is denoted as Z.0 Update the support vector regression response surface function

[0062] S5. Let iter = iter + 1. Randomly generate N candidate sample points using the Monte Carlo method. After an adaptive point selection strategy, select one candidate sample point from the safe region and one from the unsafe region, and calculate the corresponding true structural response to add it to the sample set Z. 0 In the middle, the support vector regression response surface function is updated again.

[0063] S6. Response surface function based on support vector regression Calculate the probability of structural failure and reliability indicators;

[0064] S7. Check if the structural failure probability converges. If it converges, obtain the structural failure probability and reliability index. Otherwise, return to step S5 to continue execution.

[0065] Table 4 compares the failure probabilities and their relative errors calculated by the reliability analysis method disclosed in Example 2 with those calculated by other methods. As can be seen from Table 4, the failure probability calculated by the reliability analysis method based on Direct optimization and dual sample point update of the present invention has a small relative error and can meet the actual needs of engineering.

[0066] Table 4. Comparison of failure probabilities and their relative errors calculated by various methods in Example 2

[0067]

[0068] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A reliability analysis method based on Direct optimization and two-sample point updating, characterized in that, The reliability analysis method comprises the following steps: S1, specifying the product structure of the domain to be analyzed, the function g(x) reflecting the normal working capability or critical state of safe operation of the structure or product in the domain to be analyzed, and the random variable x = (x1, x2, ..., x...). n ) and its characteristic parameters, where n is the number of elements in the random variable x, x1, x2, ..., xn. n These are the components of the random variable x, and the fields to be analyzed include civil engineering, mechanical and electronic engineering, and aerospace. S2, transform the random variable x = (x1, x2,..., x n ) into a standard normal variable z = (z1, z2,..., z n ) by the Rosenblatt transformation, z1, z2,..., z n are the components of the standard normal variable z; S3, setting a loop variable iter=1; S4, generating 2n+1 initial sample points by Direct optimization algorithm, calculating the corresponding structural response of the sample according to the function g(x), establishing the sample training set, and establishing the support vector machine regression function Taking f(z) as the optimization objective function and performing another iteration through Direct optimization algorithm, adding the k samples with smaller f(z) values to the previous samples, calculating the corresponding real structural response of the samples, establishing a new sample training set, and recording the sample set composed of 2n+k+1 samples as Z 0 , updating the support vector regression response surface function S5, let iter = iter + 1, N candidate sample points are randomly generated by Monte Carlo method, and one candidate sample point is selected from the safe domain and the unsafe domain respectively through the adaptive sampling strategy, and the corresponding true structure response is calculated and added to the sample set Z 0 In the middle, the support vector regression response surface function is updated again S6、based on support vector regression response surface function computing a structural failure probability; S7, check if the structure failure probability converges, if it converges, get the structure failure probability and reliability index β = Φ -1 (P f ), wherein Φ -1 (·) is the inverse function of the standard normal distribution function, otherwise, return to step S5 for continuous execution.

2. The method of reliability analysis based on Direct optimization and two sample point updating according to claim 1, characterized in that, The value of k in the step S4 is selected as follows: When n≤10, k=n; when n>10, k=10.

3. The method of claim 1, wherein, In the step S4, the optimization objective function f(z) is defined as: where is the support vector regression response surface function, p(z) is the joint probability density function, w denotes the w-th component of the standard normal variable z = (z1, z2,..., z n ), is the density function of the w-th standard random variable in the standard normal variable z = (z1, z2,..., z n ), 4. The method of claim 1, wherein, In the step S5, the adaptive sampling strategy process is as follows: S51, generating N candidate sample points randomly N candidate sample points are generated randomly, and the N candidate sample points are divided into N1 sample points in the safe domain and N2 sample points in the unsafe domain, the safe domain corresponds to the unsafe domain corresponds to S52, assuming N3 is the smaller of N1 and N2, let N3 = min(N1, N2), and define the absolute value of the difference between the two as the condition for selecting a sample point, take N3 / 2 sample points from the safe domain and N3 / 2 sample points from the unsafe domain that have the smallest absolute values of the difference and S53, calculate the candidate sample set respectively and Euclidean distance of N3 / 2 candidate sample points in the middle, select the candidate sample point with the maximum minimum Euclidean distance to add to the training set, that is: where min denotes the minimum and max denotes the maximum, denotes the maximum of the minimum of the function over the variable z, z 0 for any point in the point set Z1, ||·|| is the Euclidean distance, or 5. The method of claim 1, wherein, The support vector regression response surface function has the form: wherein, l is the number of sample support vectors, i is a subscript or superscript variable, z (i) represents the support vector corresponding to the ith standard random variable generated from the sample, when the Gaussian kernel function is selected, K(z, z (i) ) = exp(-γ||z-z (i) ||) 2 , γ is the kernel parameter, α i 、 are the first and second Lagrange multipliers of the above support vector machine model respectively, and b is the hyperplane parameter.

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