Adaptive PC-Kriging reliability analysis method and system based on active learning

Abstract

The invention discloses an adaptive PC-Kriging reliability analysis method and system based on active learning, and the method comprises the steps: obtaining uniformly distributed first candidate samples from a pre-generated MC sample pool through employing a weight clustering method, and constructing an initial PC-Kriging model; selecting samples by using an interval reduction method to construct a new sample pool, and dividing the new sample pool into two sub-sample pools of a security domain and an invalidation domain; aiming at the two sub-sample pools, adopting the weight clustering method again, and respectively selecting uniformly distributed second candidate samples from the two sub-sample pools; constructing a crossing point by using the second candidate samples distributed in the failure domain and the security domain; taking the constructed crossing point as a newly added experimental point, and carrying out iterative updating on the PC-Kriging model; and judging convergence and outputting a result. According to the technical scheme, through interval reduction of the focusing key area and model updating based on the crossing points, the limit state surface can be more accurately approached, the number of experimental points is reduced, the prediction precision is improved, and the calculation resource utilization is more efficient.

Description

Technical Field

[0001] The present invention relates to the technical field of reliability analysis of mechanical systems, and particularly to an adaptive PC-Kriging reliability analysis method and system based on active learning. Background Art

[0002] With the increasing complexity of the equipment mechanism of construction machinery, the increasing requirements for the diversity of the working environment, and the increasing requirements for the accuracy and stability of the equipment, higher requirements are put forward for the reliability of engineering equipment. Since mechanical systems need to operate in complex and changeable service environments, involving many uncertain factors such as random excitation, material properties, and geometric dimensions. The above factors are one of the main reasons for the performance fluctuations and fault failures of mechanical systems. In this context, using advanced theories and methods to evaluate and analyze the mechanism reliability of mechanical equipment has become a key way to ensure product quality and performance.

[0003] In the applicant's prior application CN 118862647A, an RBDO method based on the PC-Kriging model and quantile evaluation was proposed. By constructing a unified surrogate model and introducing quantile judgment to constrain feasibility, the number of calls to the original model was reduced. However, this method may lead to deviation of the optimization result due to the cumulative error of quantile sign judgment in high-nonlinear problems, and its "MP+EI" combined sampling strategy is prone to cause the ill-conditioning of the correlation matrix when dealing with near sample points, reducing the prediction stability of the model. In addition, its global surrogate model enrichment strategy depends on a large number of space-filling samples in the initial stage, resulting in waste of computing resources and difficulty in meeting the rapid convergence requirements of high-dimensional complex problems.

[0004] The AK-MCS-K method proposed in the paper "Reliability Analysis of Complex Mechanical Structures Based on Active Learning (Journal of Northeastern University: Natural Science Edition, 2020, Vol. 41, Cao Runan, etc.)" combines k-means clustering and parallel computing, and reduces the simulation time by selecting multiple sample points in each iteration. Although its computing efficiency is significantly improved, there are still the following limitations: First, the selection of initial sample points depends on Latin hypercube sampling, and the probability density distribution characteristics are not fully combined, resulting in insufficient sample coverage in low-probability regions and affecting the generalization ability of the model; Second, the sample screening mechanism near the limit state surface is relatively rough, and the key areas cannot be accurately located, resulting in redundant iteration times; Finally, there is no adaptive interval reduction mechanism when dynamically adjusting the sample pool, and it is impossible to efficiently focus on the boundary area between the failure domain and the safety domain, affecting the convergence speed and accuracy.

[0005] The core problems commonly faced by the above-mentioned existing technologies are as follows: the uneven distribution of the initial samples leads to limited generalization ability of the model, the inaccurate approximation of the limit state surface causes low convergence efficiency, and the inflexible dynamic adjustment strategy results in waste of computing resources. Especially when dealing with the reliability analysis of mechanical structures with high-dimensional and non-linear implicit functional functions, these problems are further amplified, making it difficult to meet the dual requirements of accuracy and efficiency in practical engineering. Summary of the Invention

[0006] Object of the Invention: In order to overcome the deficiencies in the existing technologies, the present invention provides an active learning-based adaptive PC-Kriging reliability analysis method and system, aiming to optimize the initial sample distribution through weight clustering, dynamically focus on key regions through interval reduction, and construct an accurate approximation of the limit state surface in combination with the crossing points, so as to solve the deficiencies of the existing technologies and achieve the unity of high efficiency and accuracy in the reliability analysis of complex mechanical structures.

[0007] Technical Solution: To achieve the above object, the active learning-based adaptive PC-Kriging reliability analysis method of the present invention includes the following steps:

[0008] Using the weight clustering method to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool, and accordingly constructing an initial PC-Kriging model;

[0009] Using the interval reduction method, selecting samples with predicted values of the limit state function in a specific interval as the new sample pool; according to the positive and negative of the predicted values, dividing the new sample pool into two sub-sample pools: a safety domain and a failure domain;

[0010] For the two sub-sample pools, using the weight clustering method to respectively select uniformly distributed second candidate samples from them;

[0011] Using the second candidate samples distributed in the failure domain and the safety domain to construct crossing points;

[0012] Taking the constructed crossing points as new experimental points to update the PC-Kriging model;

[0013] According to the predicted values of the current PC-Kriging model, dynamically adjust the upper and lower thresholds of the sample pool, and output the result when the termination iteration condition is met.

[0014] Furthermore, the step of using the weight clustering method to obtain uniformly distributed first candidate samples from the pre-generated MC sample pool and directly using the first candidate sample points as experimental points to construct an initial PC-Kriging model includes:

[0015] Constructing weight coefficients based on sample probabilities, and using the weighted sampling method to obtain a uniformly distributed weight sample set;

[0016] Based on the weighted sample set, K-means is used to further select representative uniform samples.

[0017] Furthermore, the interval reduction method is based on an interval reduction function, and the interval reduction function is:

[0018]

[0019] where and are the upper and lower thresholds of the predicted value of the limit state function in the j-th iteration, respectively; is the predicted value of the surrogate model for the sample x; s mc is the sample set; β represents the reduction rate, which is a function of τ, specifically:

[0020]

[0021] In the formula, β min represents the minimum value of the reduction rate; β max represents the maximum value of the reduction rate; κ determines the change speed of the function; τ represents the sample rejection rate, which is the ratio of the number of rejected samples to the total number of samples in the sample pool after the previous iteration; τ c is the function center position parameter.

[0022] Furthermore, constructing the crossing points by using the second candidate samples distributed in the failure domain and the safety domain includes:

[0023] Based on the source of the second candidate samples, the second candidate samples are divided into a safety domain sample subset and a failure domain sample subset, and any sample point in the safety domain sample subset is connected to any sample point in the failure domain sample subset to form a crossing line;

[0024] Discretize multiple points on each of the crossing lines, and use the KO learning function and the WKO learning function to screen out the crossing points near the limit state surface;

[0025] The KO learning function is expressed as:

[0026]

[0027] where P KO (·) is the probability that the sample point is near the limit state surface, is the expected response function, and ε is the error of the expected region; is a pair of sample points The k-th discrete point on the crossing line between; Φ(·) represents the cumulative distribution function of the standard normal distribution; is the standard deviation; is the mathematical expectation of the normal distribution;

[0028] The WKO learning function is expressed as:

[0029]

[0030] where WKO(·) represents the probability of the weighted Kriging occurring; W(·) represents the probability weight of the discrete points; Ρ ij is the discrete point set; is a pair of sample points is the crossing point on the crossing line between them.

[0031] Furthermore, the improved WKO learning function is defined as:

[0032]

[0033] The distance condition from the initial sample points is:

[0034]

[0035] The finally obtained crossing point x new is:

[0036]

[0037] Furthermore, the termination condition of the iteration is expressed as:

[0038]

[0039] where is the predicted value of the failure probability at the i-th iteration, and ε cv is the critical value.

[0040] An adaptive PC-Kriging reliability analysis system based on active learning, the system includes:

[0041] A model construction module, which is used to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool by using the weight clustering method, and directly use the first candidate sample points as experimental points to construct an initial PC-Kriging model;

[0042] A first selection module, which is used to use the interval reduction method to select samples with predicted values of the limit state function in a specific interval as a new sample pool; according to the positive and negative of the predicted values, the new sample pool is divided into two sub-sample pools, a safe domain and a failure domain;

[0043] A second selection module, which is used to, for the two sub-sample pools, again use the weight clustering method to respectively select uniformly distributed second candidate samples;

[0044] A crossover point construction module, which is used to construct crossover points by using second candidate samples distributed in the failure domain and the safety domain;

[0045] An iterative update module, which is used to take the constructed crossover points as new experimental points and iteratively update the PC-Kriging model;

[0046] A discrimination and output module, which is used to dynamically adjust the upper and lower thresholds of the sample pool according to the predicted values of the current PC-Kriging model, gradually focus on the sample interval near the limit state surface, and terminate the iteration and output the result when the relative error of the predicted failure probability values in two adjacent iterations is less than a preset critical value.

[0047] Beneficial effects: The adaptive PC-Kriging reliability analysis method and system based on active learning of the present invention have the following beneficial effects:

[0048] (1) Compared with the prior art mentioned in the background art, the technical solution of the present invention has obvious advantages. It focuses on the key area through interval reduction, updates the model based on the crossover point, can approximate the limit state surface more accurately, reduces the number of experimental points and improves the prediction accuracy. Moreover, it is more efficient in using computing resources and does not need to call the real performance function in the intermediate process. In terms of dynamic adjustment, it can flexibly optimize sample selection and model update according to the model prediction value. When facing the reliability analysis of complex and changeable mechanical structures, it can better balance the computing efficiency and accuracy, be more adaptable to the actual engineering needs, and ensure the accuracy and reliability of the analysis results.

[0049] (2) When selecting samples, by constructing weight coefficients based on sample probabilities, the deficiencies of using traditional joint probability distributions as weights are avoided. Small weights can be assigned to samples with large probability densities, so that the obtained weighted sample set is evenly distributed. Through normalization processing and random number eigenvalue screening, it is ensured that the weighted samples are also distributed in the low-probability area. Then, K-means is used to select uniform samples from the weighted sample set, making the distribution of experimental points of the initial PC-Kriging model better, effectively improving the generalization ability of the model and reducing the deviation.

[0050] (3) Based on the interval reduction function, the upper and lower thresholds of the limit state function in the MC sample pool can be predicted by the PC-Kriging model, and these thresholds are used to screen out the sample points close to the LSS from the sample pool as the sample pool for the next iterative update of the PC-Kriging model. The above algorithm can accelerate the convergence speed of the function in the initial stage of iteration, and as the number of iterations increases, the number of samples in the sample pool decreases exponentially without causing the premature termination of model update.

[0051] (4) By dividing the sample subset to form a traversal line, the intersection area with the possible limit state surface can be accurately located. Points are discretely generated on the traversal line, and the KO and WKO learning functions are used to screen the traversal points, so that the points with the highest occurrence probability of the response value on the limit state surface can be found from the discrete point set. The WKO function considering the probability weight of the sample points comprehensively considers the position probability of the discrete points and the sample space distribution, and the screened traversal points are more representative, which can effectively improve the approximation accuracy of the PC-Kriging model near the limit state surface. Description of the Drawings

[0052] Figure 1 It is a schematic flow diagram of the adaptive PC-Kriging reliability analysis method based on active learning;

[0053] Figure 2 It is a distribution diagram of the initial uniformly sampled points;

[0054] Figure 3 It is the reduced interval after the first iteration and the corresponding uniformly sampled points, traversal points and newly added experimental points;

[0055] Figure 4 It is the reduced interval after the second iteration and the corresponding uniformly sampled points, traversal points and newly added experimental points;

[0056] Figure 5 It is the reduced interval after the third iteration and the corresponding uniformly sampled points, traversal points and newly added experimental points;

[0057] Figure 6 It is the reduced interval after the fourth iteration and the corresponding uniformly sampled points, traversal points and newly added experimental points;

[0058] Figure 7 It is the reduced interval after the fifth iteration and the corresponding uniformly sampled points, traversal points and newly added experimental points;

[0059] Figure 8 It is a schematic diagram of the KO function;

[0060] Figure 9 It is a schematic diagram of the system composition of the adaptive PC-Kriging reliability analysis system based on active learning. Detailed Implementation Manner

[0061] The present invention will be further described below with reference to the accompanying drawings.

[0062] As Figure 1 shown in the adaptive PC-Kriging reliability analysis method based on active learning, the method includes the following steps S101-S106:

[0063] Step S101, obtain a uniformly distributed first candidate sample from a pre-generated MC sample pool by using a weighted clustering method, directly use the first candidate sample points as experimental points, and construct an initial PC-Kriging model;

[0064] Step S102, use an interval reduction method to select samples whose predicted values of the limit state function are within a specific interval as a new sample pool; according to the positive or negative of the predicted values, divide the new sample pool into two sub-sample pools: a safe domain and a failure domain;

[0065] Step S103, for the two sub-sample pools, use the weighted clustering method again to respectively select uniformly distributed second candidate samples from them;

[0066] Step S104, use the second candidate samples distributed in the failure domain and the safe domain to construct crossing points;

[0067] Step S105, use the constructed crossing points as new experimental points to iteratively update the PC-Kriging model;

[0068] Step S106, according to the predicted values of the current PC-Kriging model, dynamically adjust the upper and lower thresholds of the sample pool, gradually focus on the sample interval near the limit state surface. When the relative error of the predicted failure probability values of two adjacent iterations is less than a preset critical value, terminate the iteration and output the result.

[0069] Compared with the prior art mentioned in the background art, the technical solution of the present invention has obvious advantages. It focuses on the key area through interval reduction, updates the model based on the crossing points, can more accurately approximate the limit state surface, reduce the number of experimental points and improve the prediction accuracy. Moreover, it is more efficient in using computing resources and does not need to call the real performance function in the intermediate process. In terms of dynamic adjustment, it can flexibly optimize sample selection and model update according to the model predicted values. When facing the reliability analysis of complex and changeable mechanical structures, it can better balance the computing efficiency and accuracy, be more adaptable to the actual engineering requirements, and ensure the accuracy and reliability of the analysis results.

[0070] In order to make the constructed PC-Kriging model have better generalization ability and lower deviation, it is expected that the obtained crossing points can be widely distributed along the limit state surface. Traditional MC samples are not uniform in the variable space. If candidate samples are drawn from the sample pool with equal probability, the selected candidate points will inevitably gather in the area with larger probability density, which will cause the crossing points to gather in the local area, while the area with lower probability density cannot be obtained. To solve this problem, the present invention proposes a K-Weight Clustering method, which includes (that is, the above step S101 includes) the following steps S201 - step S202:

[0071] Step S201, constructing weight coefficients based on sample probabilities, and using weighted sampling methods to obtain uniformly distributed weighted sample sets;

[0072] In order to obtain evenly distributed candidate points, it is necessary to assign a smaller weight coefficient to samples with larger probability density. Since the evaluation of the joint probability distribution at a certain point may not represent the probability of the point within the specified area, and ignores the influence of the shape of the joint probability distribution around the research point, the joint probability distribution cannot be directly used as the weight. Therefore, this step specifically includes:

[0073] -Define the probability of occurrence of a sample point as the probability of occurrence of a sample point in a neighborhood where the boundary of each input random variable is set to 10% of the standard deviation;

[0074] -The weight coefficient W of the i-th sample (i) It is expressed as:

[0075]

[0076] in,

[0077] P(x) is the sample x=[x1,x2,...,x n ]The probability of occurrence, x (i) represents the i-th sample; is the standard deviation of x; f(x j ) represents the random variable x j The probability distribution function of represents the random variable x j Cumulative distribution function of x1,x2,...,x n Independent of each other; α represents the weight smoothing adjustment parameter. The main purpose of setting this parameter is to prevent extremely high or low weights due to differences in probability distribution functions and to increase the probability of each region being equally sampled. The smaller the value of α, the greater the relative importance of low-weight samples. On the contrary, the larger the value, the greater the probability of extracting high-weight samples. In the method proposed by the invention, its value is set to 0.5-1;

[0078] -Normalize the weight coefficients and express them as:

[0079]

[0080] -Assign a random number u to each sample (i) =rand(0,1), and calculate the eigenvalue of each sample Finally, select M samples with the largest eigenvalues as the result of weighted sampling; it should be noted that due to the subsequent reduction of the interval, it may cause multi-regionality of the sample interval. Therefore, in order to ensure that the weighted samples are also distributed in the low-probability region, the value of M cannot be too small. In the present invention, the value of M is taken as 1% of the MC sample size;

[0081] Step S202: Based on the weighted sample set, use K-means to further select representative uniform samples.

[0082] In the above steps S201 - S202, by constructing the weight coefficient with the sample probability, the deficiency of using the traditional joint probability distribution as the weight is avoided. Small weights can be assigned to samples with large probability densities, so that the obtained weighted sample set is evenly distributed. After normalization processing and random number eigenvalue screening, it is ensured that the weighted samples are also distributed in the low-probability region. Then, use K-means to select uniform samples from the weighted sample set, making the distribution of experimental points of the initial PC-Kriging model better, effectively improving the generalization ability of the model and reducing the bias.

[0083] The reliability prediction accuracy based on the surrogate model mainly depends on the prediction performance in the important region near the LSS (limit state surface where the limit state function is equal to zero). Therefore, the key to active learning is to accurately locate the sampling points on or near the LSS.

[0084] The interval reduction method in the above step S102 is based on an interval reduction function, and the interval reduction function is:

[0085]

[0086] Where, and are the upper and lower thresholds of the predicted value of the limit state function in the j-th iteration respectively;

[0087]

[0088] In the formula, β min represents the minimum value of the reduction rate. In this embodiment, β min takes a value of 0.4; β max represents the maximum value of the reduction rate. In this embodiment, β max takes a value of 0.7; κ determines the change speed of the function. In this embodiment, κ takes a value of 20; τ represents the sample rejection rate, which is the ratio of the number of rejected samples to the total sample pool after the previous iteration; τ c is the function center position parameter. In this embodiment, τ c takes a value of 0.5.

[0089] Based on this function, the upper and lower thresholds of the limit state function in the MC sample pool can be predicted through the PC-Kriging model, and these thresholds are used to screen out the sample points close to the LSS from the sample pool as the sample pool for the next iterative update of the PC-Kriging model. The above algorithm can accelerate the convergence rate of the function at the initial stage of iteration, and as the number of iterations increases, the number of samples in the sample pool decreases at an exponential rate without causing the premature termination of model update. Figures 2 to 7 Schematic diagram of the reduction of the example, where Figure 2 is the distribution diagram of the initial uniform sampling points, Figures 3 to 7 is the distribution diagram of the reduced interval and the corresponding uniform sampling points, crossing points and newly added experimental points after multiple rounds of iteration. In each figure, the purple points are the sampling points, the black dotted line is the crossing line, and the black wavy line is the limit state curve.

[0090] Through the interval reduction strategy, the above method can gradually filter out the samples far from the limit state surface and gradually focus on the sample interval near the LSS. This not only avoids the computational consumption of samples in the non-sensitive area, but also maintains the high performance of the surrogate model in predicting the LSS even without taking other optimization measures. During the iterative process, the continuous change of the sample interval not only increases the diversity of candidate points, but also improves the quality of crossing points. At the same time, as the number of iterations increases, the candidate points and crossing points will approach the true LSS faster, thus improving the convergence rate of the model.

[0091] In the above method, first, a small number of initial training samples are selected from the MC sample pool using the uniform sampling strategy to construct the initial PC-Kriging predictor After j rounds of iteration, assume that the sample set after (j - 1) times of region reduction is According to the current PC-Kriging predictor can be further divided into into and which correspond to two sub-sample pools with positive and negative predicted values respectively. Subsequently, using the uniform sampling strategy, n S and n F sample points are selected from the two sub-sample pools respectively, denoted as and

[0092] Constructing the crossing points by using the second candidate samples distributed in the failure domain and the safety domain in step S104 described above includes the following steps S301 - S304:

[0093] Step S301: Based on the source of the second candidate samples, divide the second candidate samples into a safe domain sample subset and a failure domain sample subset, and connect any sample point in the safe domain sample subset with any sample point in the failure domain sample subset to form a crossing line; that is, for any pair of sample points The line connecting the two points is the crossing line, and there is at least one intersection between the crossing line and the limit state surface (or limit state curve), and this intersection is the ideal crossing point;

[0094] Step S302: Discretize and generate multiple points on each of the crossing lines, and use the KO (Kriging Occurrence) learning function and the WKO (Weighted Kriging Occurrence) learning function to screen out the crossing points near the limit state surface;

[0095] In this step, a discrete point set Ρ between two points is generated by linear interpolation ij , assuming that we discretize the crossing line into m points, then the k-th discrete point can be expressed as:

[0096]

[0097] After obtaining the discrete points on the crossing line, the most crucial thing is how to screen out the crossing points from the discrete points. It should be noted that during the algorithm iteration process, since the PC-Kriging model cannot fully fit the limit state surface, the searched crossing points are not necessarily the intersection points of the limit state surface, but the points in the discrete point set whose corresponding response values have the highest occurrence probability on the limit state surface. The KO function introduced in the present invention is defined as the probability that the response of point X (i.e., ) appears in the expected region, that is, the area of the probability density function (PDF) of the response of point X within the range corresponding to the expected region. As Figure 8 shown, the KO function is the shaded area under the distribution of each point from point A to point E, and a candidate point should be selected from it to train the current Kriging predictor For example, the mean value of point A is outside the expected region, and its KO value is zero, which means it will not be considered as the next candidate point. The mean values of point B and point D are close to the expected region and have positive KO values, but only a small part of their distributions is considered, so the chance of becoming the next candidate point is small. The mean values of point C and point E are within the expected region, and the chance of becoming the next candidate point is large. The error of the expected region (i.e., ε) is a function of the standard deviation (i.e., ), and is usually selected as The latter results in the expansion of the expected region of point E compared to point C with a smaller standard deviation and a sharper distribution.

[0098] The mathematical definition of the KO function at any point X is represented by the Kriging predictor The corresponding response appears in the desired region The probability, and its formula is as follows:

[0099]

[0100] where f(·) represents the probability distribution function of the Kriging predictor;

[0101] Therefore, the KO learning function is represented as:

[0102]

[0103] where P KO (·) is the likelihood that the sample point is near the limit state surface, is the expected response function, and ε is the error of the desired region; is a pair of sample points The k-th discrete point on the crossing line between them; Φ(·) represents the cumulative distribution function of the standard normal distribution; is the standard deviation; is the mathematical expectation of the normal distribution;

[0104] In the first embodiment, the WKO learning function is represented as:

[0105]

[0106] where WKO(·) represents the probability of the weighted Kriging occurrence, which is a comprehensive measure of the likelihood that the discrete point is near the limit state surface, considering the probability weight of the sample point itself and its Kriging occurrence probability; when screening the crossing points, the larger the value of WKO(·), the higher the probability that the discrete point is both near the limit state surface and has a suitable probability weight in the entire sample space), so the possibility and importance of this discrete point as a crossing point are higher; W(·) represents the probability weight of the discrete point; Ρ ij is the set of discrete points; is a pair of sample points The crossing point on the crossing line between them.

[0107] By performing the WKO learning function on the discrete points of each crossing line, we can easily obtain n S ×n F crossing points, denoted as

[0108] In the above method, by dividing the sample subset to form a traversing line, the intersection area with the possible limit state surface can be accurately located. Points are discretely generated on the traversing line, and the KO and WKO learning functions are used to screen the traversing points, so as to find out the points with the highest occurrence probability of the response value on the limit state surface from the discrete point set. The WKO function considering the probability weight of the sample points comprehensively considers the position probability of the discrete points and the sample space distribution, and the screened traversing points are more representative, which can effectively improve the approximation accuracy of the PC-Kriging model near the limit state surface.

[0109] In a preferred embodiment, the improved WKO learning function is defined as:

[0110]

[0111] The distance condition from the initial sample points is:

[0112]

[0113] The finally obtained traversing point x new is:

[0114]

[0115] Compared with the previous embodiment, in this embodiment, the WKO learning function is improved, which can avoid the sample points being too close.

[0116] The termination condition of the iteration in step S106 is expressed as:

[0117]

[0118] where, is the predicted failure probability value of the i-th iteration, and ε cv is the critical value.

[0119] In the above embodiments, i, j, k, etc. are all local variables.

[0120] Generally, when the error fluctuation of the failure probability is less than 0.005, it is considered that the calculation of the failure probability is relatively stable. However, according to the different requirements of the accuracy and efficiency of the reliability problems studied, setting strict convergence criteria will lead to non-convergence of the model or excessive calculation amount. In order to balance the calculation efficiency and calculation accuracy, the critical value of the relative error is usually in the range of 10 -5 ≤ε cv ≤10 -2 range.

[0121] The present invention also provides an active learning-based adaptive PC-Kriging reliability analysis system, and the system includes:

[0122] The model construction module 401 is configured to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool by using a weight clustering method, directly use the first candidate sample points as experimental points, and construct an initial PC-Kriging model;

[0123] The first selection module 402 is configured to use an interval reduction method to select samples with predicted values of the limit state function in a specific interval as a new sample pool; according to the positive and negative of the predicted values, the new sample pool is divided into two sub-sample pools, namely a safety domain and a failure domain;

[0124] The second selection module 403 is configured to, for the two sub-sample pools, again use the weight clustering method to respectively select uniformly distributed second candidate samples therefrom;

[0125] The crossing point construction module 404 is configured to construct crossing points by using the second candidate samples distributed in the failure domain and the safety domain;

[0126] The iterative update module 405 is configured to use the constructed crossing points as new experimental points to iteratively update the PC-Kriging model;

[0127] The discrimination output module 406 is configured to dynamically adjust the upper and lower thresholds of the sample pool according to the predicted values of the current PC-Kriging model, gradually focus on the sample interval near the limit state surface, and terminate the iteration and output the result when the relative error of the predicted failure probability values of two adjacent iterations is less than a preset critical value.

[0128] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention.

Claims

1. An adaptive PC-Kriging reliability analysis method based on active learning, characterized in that The method includes: Using the weighted clustering method to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool, and accordingly constructing an initial PC-Kriging model; Using the interval reduction method, selecting samples with predicted values of the limit state function in a specific interval as a new sample pool; according to the positive and negative of the predicted values, dividing the new sample pool into two sub-sample pools, namely the safety domain and the failure domain; For the two sub-sample pools, using the weighted clustering method to respectively select uniformly distributed second candidate samples from them; Using the second candidate samples distributed in the failure domain and the safety domain to construct crossing points; Taking the constructed crossing points as new experimental points to update the PC-Kriging model; According to the predicted values of the current PC-Kriging model, dynamically adjusting the upper and lower thresholds of the sample pool, and outputting the result when the termination iteration condition is met.

2. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, wherein The step of using the weighted clustering method to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool and directly using the first candidate sample points as experimental points to construct an initial PC-Kriging model includes: Constructing weight coefficients based on sample probabilities, and using the weighted sampling method to obtain a uniformly distributed weight sample set; Using K-means to further select representative uniform samples on the basis of the weight sample set.

3. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, characterized in that The interval reduction method is based on an interval reduction function, and the interval reduction function is: Among them, and are the upper and lower thresholds of the predicted value of the limit state function at the j-th iteration respectively; is the predicted value of the surrogate model for the sample x; s mc is the sample set; β represents the reduction rate, which is a function of τ, specifically: In the formula, β min represents the minimum value of the reduction rate; β max represents the maximum value of the reduction rate; κ determines the change speed of the function; τ represents the rejection rate of the samples, which is the ratio of the number of rejected samples to the total number of samples in the sample pool after the previous iteration; τ c is the function center position parameter.

4. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 3, wherein The step of using the second candidate samples distributed in the failure domain and the safety domain to construct crossing points includes: Based on the source of the second candidate samples, dividing the second candidate samples into a safety domain sample subset and a failure domain sample subset, connecting any sample points in the safety domain sample subset with any sample points in the failure domain sample subset to form a crossing line; Discretely generating multiple points on each crossing line, and using the KO learning function and the WKO learning function to screen out the crossing points near the limit state surface; The KO learning function is expressed as: Among them, P KO (·) represents the probability that the sample point is near the limit state surface, is the expected response function, and ε is the error of the expected region; is a pair of sample points The k-th discrete point on the crossing line between; Φ(·) represents the cumulative distribution function of the standard normal distribution; is the standard deviation; is the mathematical expectation of the normal distribution; The WKO learning function is expressed as: Among them, WKO(·) represents the probability of the weighted Kriging; W(·) represents the probability weight of the discrete points; Ρ ij is the discrete point set; is a pair of sample points and is the intersection point on the intersection line between them.

5. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 4, wherein The improved WKO learning function is defined as: The distance condition from the initial sample points is: The finally obtained penetration point x new is as follows:

6. The adaptive PC-Kriging reliability analysis method based on active learning according to claim 1, characterized in that The termination iteration condition is expressed as: Among them, is the predicted failure probability value for the i-th iteration, and ε cv is the critical value.

7. An adaptive PC-Kriging reliability analysis system based on active learning, characterized in that The system includes: A model construction module, which is used to use the weighted clustering method to obtain uniformly distributed first candidate samples from a pre-generated MC sample pool, and directly use the first candidate sample points as experimental points to construct an initial PC-Kriging model; A first selection module, which is used to use the interval reduction method to select samples with predicted values of the limit state function in a specific interval as a new sample pool; according to the positive and negative of the predicted values, dividing the new sample pool into two sub-sample pools, namely the safety domain and the failure domain; A second selection module, which is used to, for the two sub-sample pools, use the weighted clustering method again to respectively select uniformly distributed second candidate samples from them; A crossing point construction module, which is used to use the second candidate samples distributed in the failure domain and the safety domain to construct crossing points; An iterative update module, which is used to take the constructed crossing points as new experimental points to iteratively update the PC-Kriging model; The discrimination output module is used to dynamically adjust the upper and lower thresholds of the sample pool according to the predicted values of the current PC-Kriging model, gradually focus on the sample interval near the limit state surface, and terminate the iteration and output the result when the relative error of the predicted failure probability values in two adjacent iterations is less than the preset critical value.

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