Fast convergence reliability algorithm based on improved AK-MCS model

Abstract

The invention discloses a fast convergence reliability algorithm based on an improved AK-MCS model, mainly relates to the technical field of reliability analysis, and comprises a weight clustering algorithm, a multi-trend selection method and a sample number control method adopted for constructing the AK-MCS model, and the three algorithms can be combined for use or can be used independently; according to the method, the number of times of experimental point calculation can be greatly reduced while the model precision is ensured, and the failure rate calculation efficiency is improved.

Description

Technical Field

[0001] The invention mainly relates to the technical field of reliability analysis, and in particular to a fast convergence reliability algorithm based on an improved AK-MCS model. Background Art

[0002] In the field of engineering reliability analysis, evaluating the failure rate of events usually requires a large amount of scenario analysis calculations, so the computing power requirements are very high. One way to solve this problem is to establish a proxy model, in which the original computationally expensive model is approximated by a simpler proxy model to improve the efficiency of reliability calculation. The AK-MCS model is an active learning proxy model that combines Kriging and Monte Carlo operations and has been used in reliability analysis. However, in practical applications, the efficiency and accuracy of the AK-MCS model prediction are greatly affected by the size of the sampling set and the Kriging trend. Especially for high-dimensional, highly nonlinear, and low failure rate problems, the model is sensitive to the initial parameter selection and the convergence is unstable. In addition, the AK-MCS model uses a single-point addition method, which cannot achieve parallel computing, thus greatly reducing the efficiency of model learning. In view of the above problems, in order to improve the contribution rate of the addition and the number of word additions, a fast convergence reliability algorithm based on the improved AK-MCS model is proposed. Summary of the invention

[0003] 1. Technical issues to be resolved

[0004] In view of the above-mentioned problems, the present invention needs to provide a fast convergence reliability algorithm based on the improved AK-MCS model, which can integrate three algorithms: weight clustering algorithm, multi-trend selection method and sample number control method into the AK-MCS model to achieve fast and stable convergence of the model, thereby improving the calculation accuracy and the convergence speed.

[0005] (II) Technical solution

[0006] In response to the above technical problems, the present invention provides a fast convergence reliability algorithm based on an improved AK-MCS model, including three algorithms: a weighted clustering algorithm, a multi-trend selection method, and a sample number control method used to construct the AK-MCS model. The three algorithms can be used in combination or individually.

[0007] Furthermore, the weighted clustering algorithm is used to select the experimental points for the next iteration, and the process is:

[0008] Step 1: Use K-means clustering algorithm to divide the samples in MSC sampling set D into n clusters;

[0009] S21: Initialization of MSC sampling set D samples;

[0010] S22: Randomly select n sample points in the MSC sampling set D as the center of the cluster;

[0011] S23: Using Euclidean distance as a metric, assign the sample points in the MSC sampling set D to the nearest cluster center;

[0012] S24: For each class, recalculate the mean of the samples within the class as the center of the class;

[0013] S25: Repeat S23 and S24 until convergence, and the clustering process ends;

[0014] Step 2: Use the weight algorithm to determine the center of each cluster and add the experimental point set;

[0015] S26: Randomly select any sample point in each cluster in step 1 as the center of the cluster;

[0016] S27: Calculate the weighted distance from each sample point in the cluster to the center. The weighted distance is in the following form:

[0017]

[0018] Among them, n k is the number of samples in the kth cluster, d euclidean,i is the Euclidean distance from each sample point to the center, α is the uncertainty relaxation factor, U cc is the regularized U function value of the cluster center; the U function is in the following form:

[0019]

[0020] in, is the Kriging predicted mean of the experimental point x, is the Kriging prediction standard deviation of the experimental point x;

[0021] S28: Select the cluster with the minimum weight distance d w The sample points are added to the experimental point set.

[0022] Furthermore, the multi-trend selection method uses four trend functions, namely constant, linear, quadratic and chaotic polynomial, to construct the Kriging model; the four trend functions β T f(x) has the following form:

[0023] Constant: β T f(x)=β0

[0024] Linear:

[0025] Quadratic:

[0026] Chaotic polynomial:

[0027] Where n is the system dimension, β is the regression coefficient, is a multivariate polynomial, are the coefficients of the multivariate polynomial.

[0028] Furthermore, the sample number control method determines the sample number of the MSC sampling set D in the next cycle step, and the process is:

[0029] Step 1: Calculate the target number of samples for the next cycle using the following formula:

[0030]

[0031] Where j is the number of cycle steps, To predict the failure rate, for The coefficient of variation of

[0032] Step 2: Determine the number of samples for the next cycle by the following judgment:

[0033] If δ·n target,j >n j :Increase the number of samples in MSC sampling set D to n j+1 =δ·n target,j ;

[0034] If δ·n target,j <n j / 2: MSC sampling set D remains unchanged;

[0035] If j / 2<δ·n target,j <n j :Reselect MSC sampling set D, the number of samples is n j+1 =δ·n target,j ;

[0036] Among them, j is the number of loop steps and δ is the sample number relaxation factor.

[0037] Furthermore, in the second step, the sample number relaxation factor δ can be set within the range of 0.5<δ≤2.0 according to requirements, and the default value is 1.2.

[0038] Furthermore, the process of using the three algorithms simultaneously is as follows:

[0039] Step 1: Use Monte-Carlo sampling method to generate MSC sampling set D, the number of samples in the sampling set is n1;

[0040] Step 2: Initialize i=1, j=1, Determine the number of points added per time;

[0041] Step 3: Use Latin hypercube sampling to generate the initial DoE experimental set x = {x 1 ,…,x m}, the number of samples is m;

[0042] Step 4: Calculate the response value g(x) of the DoE experiment set x = {g 1 ,…,g m};

[0043] Step 5: Use the multi-trend selection method to establish the Kriging model;

[0044] Step 6: Use the Kriging model established in step 5 to calculate the response value and prediction failure rate error value of MSC sampling set D respectively Select the failure rate error value The smallest Kriging model is used as the optimal model for the current calculation;

[0045] Step 7: Determine the convergence condition. If Then proceed to step 8; if Then use the weighted clustering algorithm to select the optimal point and add it to the DoE experimental set x, and return to step 4;

[0046] Step 8: If Then proceed to step 9; if Update And use the weighted clustering algorithm to select the optimal point and add it to the DoE experimental set x, and return to step 4;

[0047] Step 9: Calculation And make a convergence judgment, if Then it ends and outputs the failure rate like Then the sample number control method is used to update the sample number of the MSC sampling set D, and return to step 6;

[0048] Where i is the number of iteration steps, j is the number of loop steps, is the convergence threshold of the i-th iteration step, is the initial convergence threshold, is the final convergence threshold, is the predicted failure rate error value, CoV thr is the coefficient of variation threshold.

[0049] Furthermore, the Monte-Carlo sampling method in the first step can be subdivided into variants of the Monte-Carlo sampling method such as Latin hypercube sampling, importance sampling, and subset simulation according to the different problem dimensions and the number of sample sets.

[0050] Furthermore, in the seventh and eighth steps, the convergence threshold is gradually reduced during the iteration process until it is no greater than the final convergence threshold; the final convergence threshold According to the needs The default value is 5%; the initial convergence threshold in the second step According to the needs The coefficient of variation threshold in the ninth step can be set within the range of 0 <CoV thr The value range is ≤10%, and the default value is 5%.

[0051] Compared with the prior art, the invention has the following beneficial effects: 1. The invention integrates the weighted clustering algorithm, the multi-trend selection method and the sample number control method into the construction of the AK-MCS model to achieve rapid and stable convergence of the model; 2. A weighted clustering algorithm is used to realize the addition of multiple experimental points on the basis of considering the spatial distribution of the response surface and the prediction error value of the failure rate, so as to improve the contribution of the experimental points to the convergence of the model and make full use of the parallel capability of the computer; 3. A multi-trend selection method is used to compare the prediction error values ​​of the failure rate of the AK-MCS model constructed with different trends, so as to reduce the risk of unstable model convergence caused by inaccurate Kriging trend selection during model construction; 4. A sample number control method is used to update the MSC sample set and the number of samples required for each step in an iterative manner, so as to reduce the risk of inefficient model convergence caused by inaccurate initial sample set number; 5. The FC AK-MCS algorithm is suitable for various reliability analysis and calculation, especially for highly nonlinear, high-dimensional and low failure rate problems, and can greatly reduce the number of experimental point calculations while ensuring the accuracy of the model, thereby improving the efficiency of failure rate calculation; the three algorithms can be used in combination or separately as needed. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is the algorithm flow chart of the present invention.

[0053] Figure 2 It is a function schematic diagram of the present invention.

[0054] Figure 3 It is the experimental point distribution diagram of the present invention.

[0055] Figure 4 It is a convergence process diagram of the present invention. DETAILED DESCRIPTION

[0056] The present invention will be further described below in conjunction with specific embodiments. The present invention is explained by means of the illustrative embodiments and descriptions of the present invention, but is not intended to be a limitation of the present invention.

[0057] Example: Figure 1-Figure 4 A fast convergence reliability algorithm based on an improved AK-MCS model is shown; the principle, steps, and process of a fast convergence reliability algorithm based on an improved AK-MCS model of the present invention are described and introduced in detail below in conjunction with the accompanying drawings and embodiments, so that those skilled in the art can better understand the technical ideas and connotations of the present invention. The technical solution of the present invention is not limited to the specific contents described in the embodiments; the process is as follows Figure 1 .

[0058] Here, the failure rate calculation of a two-dimensional four-branch function is used as an example to illustrate the algorithm flow and function, but the algorithm is not limited to this. The schematic diagram of the four-branch function G(x1,x2) is shown in Figure 2 , which has the following form:

[0059]

[0060] The two dimensions (x1, x2) of the function both obey the standard normal distribution; the problem is to solve the failure rate P of the two-dimensional four-branch function when the variables (x1, x2) ~ N(0,1) f The failure rate is defined as correspond Figure 2 Each of the four branches represents a failure mode, and the part enclosed by the four branches is the failure part.

[0061] Step 1: Use Latin hypercube sampling to generate MSC sampling set D, the number of samples in the sampling set is 10 6 ;

[0062] Step 2: Initialize i=1, j=1, Determine the single point addition number to be 6;

[0063] Step 3: Use Latin hypercube sampling to generate the initial DoE experimental set x = {x 1 ,…,x 12}, the number of samples is 12;

[0064] Step 4: Calculate the four-branch function response value g(x) corresponding to the DoE experiment set x =

[0065] {g 1 ,…,g 12};

[0066] Step 5: Use the multi-trend selection method to establish the Kriging model;

[0067] The multi-trend selection method uses four trend functions, namely constant, linear, quadratic and chaotic polynomial, to construct the Kriging model.

[0068] Step 6: Use the Kriging model established in step 5 to calculate the response value and prediction failure rate error value of MSC sampling set D respectively Select the failure rate error value The smallest Kriging model is used as the optimal model for the current calculation;

[0069] Step 7: Determine the convergence condition. If Then proceed to step 8; if Then use the weighted clustering algorithm to select the 6 optimal points and add them to the DoE experimental set x, and return to the fourth step.

[0070] The weighted clustering algorithm is used to select the experimental points for the next iteration. The process is:

[0071] The K-means clustering algorithm is used to divide the samples in the MSC sampling set D into 6 clusters;

[0072] S21: Initialization of MSC sampling set D samples;

[0073] S22: Randomly select 6 sample points in the MSC sampling set D as the center of the cluster;

[0074] S23: Using Euclidean distance as a metric, assign the sample points in the MSC sampling set D to the nearest cluster center;

[0075] S24: For each class, recalculate the mean of the samples within the class as the center of the class;

[0076] S25: Repeat S23 and S24 until convergence, and the clustering process ends.

[0077] The weight algorithm is used to determine the center of each cluster and added to the DoE experiment set x;

[0078] S26: Randomly select any sample point in each cluster in step 1 as the center of the cluster;

[0079] S27: Calculate the weighted distance from each sample point in the cluster to the center;

[0080] S28: Select the minimum weight distance in each cluster The sample points are added to the DoE experimental set x, a total of 6.

[0081] Step 8: If Then proceed to step 9; if Update And use the weighted clustering algorithm to select the optimal point and add it to the DoE experimental set x, and return to step 4;

[0082] Step 9: Calculation And make a convergence judgment, if Then it ends and outputs the failure rate like Then use the sample number control method to update the sample number of the MSC sampling set D and return to step 6.

[0083] The sample number control method determines the sample number of the MSC sampling set D in the next cycle step. The process is:

[0084] The target number of samples for the next cycle is calculated using the following formula:

[0085]

[0086] The number of samples for the next cycle step is determined by the following judgment, and δ = 1.2 is selected:

[0087] If 1.2n target,j >n j :Increase the number of samples in MSC sampling set D to n j+1 =1.2n target,j ;

[0088] If 1.2n target,j <n j / 2: MSC sampling set D remains unchanged;

[0089] If j / 2<1.2n target,j <n j :Reselect MSC sampling set D, the number of samples is n j+1 =1.2n target,j .

[0090] The algorithm iterates 12 times, converges after adding 72 experimental points, and outputs the failure rate The deviation from the standard value is 0.67%, and the convergence curve is as follows Figure 3 ; The experimental point set in the iterative process is as follows Figure 4 , it can be seen that the experimental points are mostly concentrated near the failure boundary of the function, thus effectively predicting the failure rate.

[0091] Any matters not described in the present invention are applicable to the prior art.

Claims

1. A fast convergence reliability algorithm based on an improved AK-MCS model, characterized in that: It includes three algorithms used to construct the AK-MCS model: weight clustering algorithm, multi-trend selection method and sample number control method. The three algorithms can be used in combination or separately.

2. According to claim 1, a fast convergence reliability algorithm based on an improved AK-MCS model is characterized in that: The weighted clustering algorithm is used to select the experimental points for the next iteration, and the process is: Step 1: Use K-means clustering algorithm to divide the samples in MSC sampling set D into n clusters; S21: Initialization of MSC sampling set D samples; S22: Randomly select n sample points in the MSC sampling set D as the center of the cluster; S23: Using Euclidean distance as a metric, assign the sample points in the MSC sampling set D to the nearest cluster center; S24: For each class, recalculate the mean of the samples within the class as the center of the class; S25: Repeat S23 and S24 until convergence, and the clustering process ends; Step 2: Use the weight algorithm to determine the center of each cluster and add the experimental point set; S26: Randomly select any sample point in each cluster in step 1 as the center of the cluster; S27: Calculate the weighted distance from each sample point in the cluster to the center. The weighted distance is in the following form: Among them, n k is the number of samples in the kth cluster, d euclidean,i is the Euclidean distance from each sample point to the center, α is the uncertainty relaxation factor, U cc is the regularized U function value of the cluster center; the U function is in the following form: in, is the Kriging predicted mean of the experimental point x, is the Kriging prediction standard deviation of the experimental point x; S28: Select the cluster with the minimum weight distance d w The sample points are added to the experimental point set.

3. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 1 is characterized in that: The multi-trend selection method uses four trend functions, namely constant, linear, quadratic and chaotic polynomial, to construct the Kriging model; the four trend functions β T f(x) has the following form: Constant: β T f(x)=β0 Linear: Quadratic: Chaotic polynomial: Where n is the system dimension, β is the regression coefficient, is a multivariate polynomial, are the coefficients of the multivariate polynomial.

4. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 1 is characterized in that: The sample number control method determines the sample number of the MSC sampling set D in the next cycle step, and the process is: Step 1: Calculate the target number of samples for the next cycle using the following formula: Where j is the number of loop steps, To predict the failure rate, for The coefficient of variation of Step 2: Determine the number of samples for the next cycle by the following judgment: If δ·n target，j >n j :Increase the number of samples in MSC sampling set D to n j+1 =δ·n target，j ; If δ·n target，j <n j / 2: MSC sampling set D remains unchanged; If j / 2<δ·n target，j <n j :Reselect MSC sampling set D, the number of samples is n j+1 =δ·n target，j ; Among them, j is the number of loop steps and δ is the sample number relaxation factor.

5. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 4 is characterized in that: In the second step, the sample number relaxation factor δ can be set within the range of 0.5<δ≤2.0 according to requirements, and the default value is 1.

2.

6. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 1 is characterized in that: The process of using the three algorithms simultaneously is as follows: Step 1: Use Monte-Carlo sampling method to generate MSC sampling set D, the number of samples in the sampling set is n1; Step 2: Initialization Determine the number of points added per time; Step 3: Use Latin hypercube sampling to generate the initial DoE experiment set x = {x 1 , ..., x m }, the number of samples is m; Step 4: Calculate the response value g(x) of the DoE experiment set x = {g 1 , ..., g m }; Step 5: Use the multi-trend selection method to establish the Kriging model; Step 6: Use the Kriging model established in step 5 to calculate the response value and prediction failure rate error value of MSC sampling set D respectively Select the failure rate error value The smallest Kriging model is used as the optimal model for the current calculation; Step 7: Determine the convergence condition. If Then proceed to step 8; if Then use the weighted clustering algorithm to select the optimal point and add it to the DoE experimental set x, and return to step 4; Step 8: If Then proceed to step 9; if Update And use the weighted clustering algorithm to select the optimal point and add it to the DoE experimental set x, and return to step 4; Step 9: Calculation And make a convergence judgment, if Then it ends and outputs the failure rate like Then the sample number control method is used to update the sample number of the MSC sampling set D, and return to step 6; Where i is the number of iteration steps, j is the number of loop steps, is the convergence threshold of the i-th iteration step, is the initial convergence threshold, is the final convergence threshold, is the predicted failure rate error value, CoV thr is the coefficient of variation threshold.

7. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 6 is characterized in that: The Monte-Carlo sampling method in the first step can be subdivided into variants of the Monte-Carlo sampling method, such as Latin hypercube sampling, importance sampling, and subset simulation, according to the different problem dimensions and the number of sample sets.

8. The fast convergence reliability algorithm based on the improved AK-MCS model according to claim 6 is characterized in that: In the seventh and eighth steps, the convergence threshold is gradually reduced during the iteration process until it is no greater than the final convergence threshold; the final convergence threshold According to the needs The default value is 5%; the initial convergence threshold in the second step According to the needs The coefficient of variation threshold in the ninth step can be set within the range of 0 <CoV thr The value range is ≤10%, and the default value is 5%.

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