Multi-failure domain structure reliability optimization method and system based on adaptive clustering

Abstract

The invention discloses a multi-failure domain structure reliability optimization method and system based on adaptive clustering, and the method comprises the steps: carrying out the first-stage training of a Kriging agent model through an active learning method based on engineering structure data, and constructing a first-stage Kriging agent model; performing failure domain identification on the first-stage Kriging agent model based on an adaptive clustering algorithm, and constructing an important sampling density; based on the important sampling density, performing second-stage training on the first-stage Kriging agent model through an active learning method, and outputting a second-stage Kriging agent model; and performing failure probability calculation on the engineering structure based on the second-stage Kriging agent model to obtain a reliability calculation result of the engineering structure. According to the method, the potential sub-failure domain of the engineering structure can be accurately identified, so that the calculation efficiency of the failure probability of the engineering structure is improved. The multi-failure domain structure reliability optimization method and system based on adaptive clustering can be widely applied to the technical field of structure reliability optimization.

Description

Technical Field

[0001] The present invention relates to the technical field of structural reliability optimization, and particularly to a multi-failure domain structural reliability optimization method and system based on adaptive clustering. Background Art

[0002] Structural reliability analysis can quantify the uncertainties existing in engineering structures and the loads they bear, which is of great significance for accurately evaluating structural safety, preventing structural failures, and ensuring life and property safety. Structural reliability is a quantitative index of structural reliability. Structural reliability is usually characterized by the failure probability, and the main goal of structural reliability calculation is to obtain the failure probability of the structure. Although great progress has been made in the theory and calculation methods of structural reliability analysis at the present stage, the calculation of the failure probability is still very difficult. First, the structural performance function is usually expressed by models such as finite element models, and the computational cost of a single call is very large. Second, engineering structures usually have a large safety margin, making structural failure a rare event. Therefore, the calculation of structural reliability usually requires a large number of simulations (thus a large number of calls to the structural performance function), significantly increasing the required computational cost. The computational cost required for structural reliability analysis is controlled by the number of calls to the structural performance function. The basic problem of structural reliability analysis is to develop efficient calculation methods so as to obtain a sufficiently accurate failure probability at the cost of as few calls to the structural performance function as possible.

[0003] The existing structural reliability calculation methods mainly include approximate analytical methods, numerical simulation methods, surrogate model methods, etc. The approximate analytical methods mainly include the first-order reliability method (FORM) and the second-order reliability method (SORM). These methods have high calculation efficiency, but have large errors for strongly nonlinear problems. The numerical simulation methods mainly include Monte Carlo simulation and various improved methods, such as the importance sampling method (IS), the subset simulation method (SS), the line sampling method (LS), and the directional sampling method (DS), etc. However, due to the need to extract a large number of samples for simulation, the calculation amount is too large. The surrogate model method uses a surrogate model that is easy to calculate to replace the true structural performance function, thus significantly reducing the calculation amount. Common surrogate models include the response surface method (RSM), artificial neural network (ANN), support vector machine (SVM), and Kriging model, etc. The main disadvantage of the surrogate model method is that it cannot quantify the approximation error brought by the surrogate model, resulting in a biased estimate of the failure probability. The active learning method combines the advantages of high accuracy of the numerical simulation method and high calculation efficiency of the surrogate model method. Its core lies in using active learning to train the surrogate model, significantly reducing the number of calls to the structural performance function required for training the surrogate model, that is, significantly reducing the calculation amount required for training. However, since the failure probability of large and complex engineering structures is usually small, the sample pool for the active learning method to train the surrogate model is too large, resulting in too high a calculation amount for the training of the surrogate model itself. In addition, there are usually multiple failure domains in engineering structures, and how to accurately identify and quantify the contribution of each failure domain to the failure probability has not been well solved. Summary of the Invention

[0004] In order to solve the above technical problems, the object of the present invention is to provide a multi-failure domain structural reliability optimization method and system based on adaptive clustering, which can accurately identify potential sub-failure domains of engineering structures, and further improve the calculation efficiency of the failure probability of engineering structures.

[0005] The first technical solution adopted by the present invention is: a multi-failure domain structural reliability optimization method based on adaptive clustering, comprising the following steps: Based on engineering structure data, the Kriging surrogate model is trained in the first stage by the active learning method to construct the first-stage Kriging surrogate model; Based on the adaptive clustering algorithm, the failure domain of the first-stage Kriging surrogate model is identified to construct the importance sampling density; Based on the importance sampling density, the first-stage Kriging surrogate model is trained in the second stage by the active learning method, and the second-stage Kriging surrogate model is output; Calculate the failure probability of the engineering structure based on the second-stage Kriging surrogate model to obtain the calculation result of the engineering structure reliability.

[0006] Furthermore, the step of constructing the first-stage Kriging surrogate model by training the Kriging surrogate model in the first stage through the active learning method based on the engineering structure data specifically includes: Obtain the engineering structure data and generate an initial experimental design and an initial sample pool within the preset standard deviation range of the standard normal space based on the Latin hypercube sampling method. The initial experimental design includes the corresponding structural responses. Construct a Kriging surrogate model through a preset software toolbox based on the initial experimental design. Based on the initial sample pool, identify and obtain the first optimal learning sample through the active learning function, and calculate the structural response of the first optimal learning sample through the structural function. Expand the structural response of the first optimal learning sample to the initial experimental design, and iteratively update the Kriging surrogate model based on the expanded experimental design until the preset stop condition is met, and output the first-stage Kriging surrogate model.

[0007] Furthermore, the expression of the active learning function is specifically as follows: ; In the above formula, represents the first optimal learning sample, represents the active learning function, represents the initial sample pool, represents the sample in the initial experimental design, and respectively represent the mean and standard deviation of the Kriging surrogate model at the sample points.

[0008] Furthermore, the expression of the preset stop condition is specifically as follows: ; In the above formula, represents the stop condition index based on the stability of the failure domain, represents the stop condition index based on the change of the sample response sign in two adjacent iterations, , represent the critical thresholds, represents the number of failure samples in the th round of active learning iteration, represents the number of failure samples in the th round of active learning iteration, represents the sample size. denotes the Kriging surrogate model obtained from the th round of active learning training, denotes the Kriging surrogate model obtained from the th round of active learning training, denotes the th sample.

[0009] Further, the step of identifying the failure domain of the Kriging surrogate model in the first stage based on the adaptive clustering algorithm and constructing the importance sampling density specifically includes: Generating samples within the failure domain corresponding to the Kriging surrogate model in the first stage through the Markov chain Monte Carlo method; Performing partition clustering on the samples within the failure domain through the adaptive clustering algorithm to obtain the best clustering result; Fitting the best clustering result through the Gaussian mixture density function to obtain the importance sampling density.

[0010] Further, the step of performing partition clustering on the samples within the failure domain through the adaptive clustering algorithm to obtain the best clustering result specifically includes: Determining the range of candidate cluster numbers; Performing K-means clustering on the samples within the failure domain based on the range of candidate cluster numbers to obtain the preliminary clustering result; Calculating the DI value for the preliminary clustering result to obtain the Dunn validity index; Selecting the DI value corresponding to the maximum Dunn validity index as the optimal number of clusters; Obtaining the clustering result corresponding to the optimal number of clusters as the basis for partitioning the potential failure domain, and partitioning the samples within the failure domain to obtain the best clustering result.

[0011] Further, the expression of the Dunn validity index is as follows: ; In the above formula, represents the DI value, represents the range of candidate cluster numbers, , represent the clustering results, represents the clustering and 's set distance, represents the diameter of the clustering result .

[0012] Further, the step of performing the second-stage training on the first-stage Kriging surrogate model through an active learning method based on the importance sampling density and outputting the second-stage Kriging surrogate model specifically includes: Combining the first-stage Kriging surrogate model with the initial experimental design to construct the second-stage initial Kriging surrogate model; Generating importance sampling samples based on the importance sampling density, and calculating the responses corresponding to the importance sampling samples based on the second-stage initial Kriging surrogate model to obtain the second optimal learning samples; Judging the second-stage initial Kriging surrogate model according to the preset stopping condition; If the second-stage initial Kriging surrogate model does not meet the preset stopping condition, calculate the structural responses corresponding to the second optimal learning samples, and expand the second optimal learning samples and the corresponding structural responses to the initial experimental design to update the second-stage initial Kriging surrogate model; If the second-stage initial Kriging surrogate model meets the preset stopping condition, calculate the failure probability based on the current second-stage initial Kriging surrogate model to obtain the coefficient of variation; If the coefficient of variation does not meet the preset target coefficient of variation, generate another group of importance sampling samples based on the importance sampling density, expand the existing importance sampling sample pool, and train the second-stage initial Kriging surrogate model through the active learning method based on the expanded importance sampling sample pool until the second-stage initial Kriging surrogate model meets the preset stopping condition, calculate the failure probability and the corresponding coefficient of variation, and re-judge whether it meets the preset target coefficient of variation; Until the coefficient of variation meets the preset target coefficient of variation, output the second-stage Kriging surrogate model.

[0013] Further, the calculation expression of the coefficient of variation is specifically as follows: ; In the above formula, represents the importance sampling estimate of the failure probability, represents the coefficient of variation of the failure probability estimate, represents the variance of the failure probability estimate.

[0014] The second technical solution adopted by the present invention is: a multi-failure domain structural reliability optimization system based on adaptive clustering, including: The first module is used to perform the first-stage training on the Kriging surrogate model through an active learning method based on engineering structure data to construct the first-stage Kriging surrogate model; The second module is used to identify the failure domain of the first-stage Kriging surrogate model based on the adaptive clustering algorithm and construct the importance sampling density; The third module is used to perform the second-stage training on the first-stage Kriging surrogate model by the active learning method based on the importance sampling density and output the second-stage Kriging surrogate model; The fourth module is used to calculate the failure probability of the engineering structure based on the second-stage Kriging surrogate model and obtain the calculation result of the engineering structure reliability.

[0015] The beneficial effects of the method and system of the present invention are as follows: Through the engineering structure data, the present invention performs the first-stage training on the Kriging surrogate model by the active learning method to construct the first-stage Kriging surrogate model, enabling the model to better approximate the main shape of the failure domain. Further, based on the adaptive clustering algorithm, the failure domain of the first-stage Kriging surrogate model is identified to construct the importance sampling density. The adaptive clustering algorithm is used to automatically identify multiple potential sub-failure domains, and then the GMM is constructed as the importance sampling density function. Furthermore, based on the importance sampling density, the second-stage training is performed on the first-stage Kriging surrogate model by the active learning method to output the second-stage Kriging surrogate model. A large number of importance sampling samples are generated based on the constructed GMM importance sampling density function. Combining the AK-IS, the second stage uses the active learning method to train the Kriging model to efficiently calculate the failure probability. Finally, the failure probability of the engineering structure is calculated based on the second-stage Kriging surrogate model to obtain the calculation result of the engineering structure reliability, which can accurately identify the potential sub-failure domains of the engineering structure and thus improve the calculation efficiency of the failure probability of the engineering structure. Description of the Drawings

[0016] Figure 1 is the flowchart of the steps of the multi-failure domain structure reliability optimization method based on adaptive clustering of the present invention; Figure 2 is the structural block diagram of the multi-failure domain structure reliability optimization system based on adaptive clustering of the present invention; Figure 3 is the schematic diagram of the steps of the active learning method for multi-failure domain structure reliability calculation based on adaptive clustering provided by the specific embodiment of the present invention; Figure 4 is the result schematic diagram of the first specific embodiment of the present invention; Figure 5 is the result schematic diagram of the second specific embodiment of the present invention. Detailed Embodiments

[0017] The following further elaborates on the present invention in conjunction with the accompanying drawings and specific embodiments. For the step numbers in the following embodiments, they are only set for the convenience of elaboration and explanation, and no limitation is imposed on the order between steps. The execution order of each step in the embodiments can be adaptively adjusted according to the understanding of those skilled in the art.

[0018] First of all, it should be noted that engineering structures such as super high-rise buildings, large-span building structures, long-span bridges, reservoir dams, nuclear power plants and other large and complex structures have the characteristics of high investment, large volume, complex structure, and key social service functions. These engineering structures are faced with various random and uncertain factors during the design, construction and operation and maintenance processes, including but not limited to aspects such as material mechanical properties, geometric parameters, loads, etc. Under the influence of these random factors, the structural response also has strong randomness. Structural reliability analysis is an effective means to analyze and evaluate the safety and reliability of engineering structures under the influence of various random factors.

[0019] The embodiments of the present invention are mainly carried out based on the active learning surrogate model importance sampling method (AK-IS). AK-IS has a high solution efficiency for small failure probability problems. However, it depends on the FORM search design point, which limits its application in multi-failure domain problems. Therefore, the embodiments of the present invention propose an active learning method for calculating the structural reliability of multi-failure domains based on adaptive clustering (AC-AK-IS). This method uses adaptive clustering to automatically identify multiple potential failure domains, thus providing an effective means for the structural reliability analysis of multi-failure domains.

[0020] Among them, the main goal of structural reliability calculation is to calculate the failure probability, which can be expressed as: ; In the above formula, is the joint Gaussian probability density function of the basic random variables . Let be the structural performance function, then is the failure domain, is the indicator function of the failure domain. If , then , otherwise .

[0021] Therefore, the basic idea of the AC-AK-IS embodiment of the present invention is to first train the first-stage Kriging surrogate model using the active learning method, then use the adaptive clustering algorithm to automatically identify multiple potential failure domains, construct the Gaussian mixture density (GMM) based on the identified failure domains, use this GMM as the importance sampling density, and finally extract a large number of importance sampling samples as the sample pool and call the active learning method to train the second-stage Kriging surrogate model. Based on the second-stage Kriging surrogate model, the failure probability can be quickly calculated.

[0022] Referring to Figure 1 With Figure 3 , the present invention provides a multi-failure domain structural reliability optimization method based on adaptive clustering, and the method includes the following steps: S100. Based on the engineering structure data, perform the first-stage training on the Kriging surrogate model through the active learning method to construct the first-stage Kriging surrogate model; Based on the sample pool generated by Latin hypercube sampling (LHS), use the active learning method to train the first-stage Kriging model to capture the global behavior of the structural performance function, facilitating the subsequent automatic identification of multiple potential failure domains by adaptive clustering.

[0023] Specifically, obtain the engineering structure data and generate the initial experimental design and the initial sample pool within the preset standard deviation range in the standard normal space based on the Latin hypercube sampling method. The initial experimental design includes the corresponding structural responses; based on the initial experimental design, construct the Kriging surrogate model through the preset software toolbox; based on the initial sample pool, identify and obtain the first best learning sample through the active learning function, and calculate the structural response of the first best learning sample through the structural performance function; expand the structural response of the first best learning sample to the initial experimental design, and iteratively update the Kriging surrogate model based on the expanded experimental design until the preset stop condition is met, and output the first-stage Kriging surrogate model.

[0024] In this embodiment, first use LHS to generate the initial experimental design (DoE) including samples within the times standard deviation range in the standard normal space, including inputting the initial DoE: and the corresponding structural responses: , where [[ID=((26))]] , where times the standard deviation range of is the truncation coefficient. Construct the first-stage Kriging model based on the initial experimental design, denoted as . Then in the same region ( A large number of samples are generated by LHS within [number of standard deviations], denoted as the initial sample pool. , where is the sample size. The active learning method is used to obtain the best learning samples . Calculate the response of the structural performance function at . And expand it into the DoE. Further update the Kriging model based on the expanded DoE. Repeat the above steps until the stopping condition is met. The stopping condition can be expressed as: ; ; The first equation in the above formula depicts the stability of the failure domain and can be further written as: ; In the above formula, is the number of failure samples in the th round of active learning iteration. The second stopping condition depicts the proportion of the change in the response sign of the LHS sample pool in adjacent two iterations, and can be written as: ; In the above formula, is the Kriging model obtained from the th round of active learning training, represents the exclusive OR operation, and both of the two critical thresholds and are taken as 0.001. When the stopping conditions are met in two consecutive iterations, it is considered that the accuracy of the trained Kriging model meets the requirements, and the active learning process stops.

[0025] In this embodiment, the LHS samples have better spatial coverage, so only a small number of samples are needed to train a Kriging model with high accuracy. In addition, active learning makes full use of the current information to obtain the next training sample, significantly reducing the sample size required for Kriging model training and further improving the computational efficiency.

[0026] In some specific embodiments, LHS is used to generate the initial DoE in the standard normal space: and the initial sample pool: . Calculate the corresponding structural response.

[0027] Train the first-stage Kriging model : Based on the current DoE, use the MATLAB toolbox DACE to construct the Kriging model . Use the active learning function to identify the best learning samples from it , the expression of the active learning function is as follows: ; In the above formula, is the active learning function, expressed as: ; In the above formula, and are the mean and standard deviation of the Kriging model at the sample points, respectively. Check whether the stopping condition is satisfied (both critical thresholds and are 0.001). If the stopping condition is satisfied, the Kriging model is considered accurate enough. Otherwise, calculate the structural response corresponding to the optimal learning sample , and expand to the DoE, and repeat the steps to train the Kriging model in the first stage until the stopping condition is satisfied.

[0028] S200. Identify the failure domain of the Kriging surrogate model in the first stage based on the adaptive clustering algorithm, and construct the importance sampling density; Based on the Kriging model in the first stage, use Markov chain Monte Carlo (MCMC) to generate samples within the corresponding failure domain. Then use the adaptive clustering algorithm to automatically identify the failure domain. Finally, fit the Gaussian mixture density for the identified failure domain and use it as the importance sampling density for AK-IS.

[0029] Specifically, generate samples within the failure domain of the Kriging surrogate model in the first stage through the Markov chain Monte Carlo method; perform partition clustering on the samples within the failure domain through the adaptive clustering algorithm to obtain the best clustering result; fit the best clustering result through the Gaussian mixture density function to obtain the importance sampling density.

[0030] More specifically, determine the range of candidate clustering numbers; based on the range of candidate clustering numbers, perform K-means clustering on the samples within the failure domain to obtain the preliminary clustering result; calculate the DI value for the preliminary clustering result to obtain the Dunn validity index; select the DI value corresponding to the maximum Dunn validity index as the optimal clustering number; obtain the clustering result corresponding to the optimal clustering number and use it as the basis for partitioning the potential failure domain, and partition the samples within the failure domain to obtain the best clustering result.

[0031] In this embodiment, first, use MCMC to generate samples within the failure domain.

[0032] In some specific embodiments, The corresponding failure domain is denoted as . Samples within are generated using MCMC and denoted as , where asymptotically follows a normal distribution truncated from above in the failure domain : . In the formula, is the indicator function of the failure domain . MCMC is implemented using a component-based Metropolis–Hastings algorithm.

[0033] The seed of MCMC is the sample with a negative response in the current DoE. Starting from the seed, multiple Markov chains are generated simultaneously to enhance the exploration of the failure domain. The proposal distribution of MCMC is taken as a uniform distribution, with the center of the distribution being the current sample point and the distribution interval being twice the standard deviation ( ). In addition, a burn-in period and thinning technique are used to reduce the correlation of MCMC samples. Only the Kriging model is called to calculate the structural response during the sample generation stage of MCMC, so the computational cost is very small.

[0034] Secondly, identify the failure domain.

[0035] In some specific embodiments, an adaptive clustering algorithm is used to divide the samples within the failure domain into several groups. It can be considered that each group of samples obtained by clustering represents a potential failure domain. The adaptive clustering algorithm mainly uses the K-means clustering and Dunn clustering validity index.

[0036] First, determine the candidate interval for the number of clusters as , where is the larger value between the number of clusters corresponding to the first occurrence of an empty cluster when performing K-means clustering on and in ascending order. Secondly, use K-means clustering to divide into clusters, where . Calculate the Dunn index corresponding to each . The optimal clustering is the for which the Dunn index takes the maximum value, as shown in the following formula: ; In the above formula, is the optimal number of clusters, and is the Dunn index corresponding to clusters.

[0037] Given clusters, each cluster denoted as , the Dunn index is: ; In the above formula, is the diameter of cluster , written as: ; In the above formula, is the center of cluster , is the and Euclidean distance, is the sample size of cluster , is the set distance between cluster and , denoted as: ; The optimal number of clusters is used as the number of potential failure domains, and the corresponding clusters can be used as representatives of each failure domain.

[0038] Furthermore, the GMM is used to construct the importance sampling density.

[0039] In some specific embodiments, based on the optimal clustering of the sample family within the failure domain , the GMM can be fitted and used as the importance sampling density. Let be the sample family of the optimal clustering, then the GMM can be expressed as: ; In the above formula, is the normalized weight, proportional to the sample size within , satisfying , and are the mean and covariance matrix of the -th Gaussian distribution fitted to the sample family respectively.

[0040] Finally, using the GMM as the importance sampling density, combined with the first-stage Kriging model and the initial DoE, the second stage of AK-IS can be carried out to efficiently calculate the failure probability.

[0041] In summary, MCMC is used to generate failure samples within the surrogate model failure domain, and the MCMC algorithm is used to generate samples within the surrogate model failure domain , denoted as . The proposal distribution of MCMC is a centered on the current sample.Uniform distribution on the interval. The burn-in period length and slice length of MCMC are taken as and .

[0042] Adaptive clustering, using the adaptive clustering algorithm to obtain the optimal clustering of the sample family , that is, is divided into clusters, denoted as .

[0043] Construct the importance sampling density, and construct the GMM based on the clustering result . This GMM is used as the importance sampling density in the second stage of AK-IS.

[0044] S300. Based on the importance sampling density, the second-stage training of the first-stage Kriging surrogate model is carried out by the active learning method, and the second-stage Kriging surrogate model is output; Specifically, combining the first-stage Kriging surrogate model with the initial experimental design, construct the second-stage initial Kriging surrogate model; generate importance sampling samples based on the importance sampling density, and calculate the responses corresponding to the importance sampling samples based on the second-stage initial Kriging surrogate model to obtain the second-best learning samples; judge the second-stage initial Kriging surrogate model according to the preset stop condition; if the second-stage initial Kriging surrogate model does not meet the preset stop condition, calculate the structural responses corresponding to the second-best learning samples, and expand the second-best learning samples and the corresponding structural responses to the initial experimental design to update the second-stage initial Kriging surrogate model; if the second-stage initial Kriging surrogate model meets the preset stop condition, calculate the failure probability based on the current second-stage initial Kriging surrogate model to obtain the coefficient of variation; if the coefficient of variation does not meet the preset target coefficient of variation, generate another group of importance sampling samples based on the importance sampling density, expand the existing importance sampling sample pool, and train the second-stage initial Kriging surrogate model by the active learning method based on the expanded importance sampling sample pool until the second-stage initial Kriging surrogate model meets the preset stop condition, calculate the failure probability and the corresponding coefficient of variation, and re-judge whether it meets the preset target coefficient of variation; until the coefficient of variation meets the preset target coefficient of variation, output the second-stage Kriging surrogate model.

[0045] In the embodiment of the present invention, construct the second-stage initial Kriging model, and respectively use the first-stage Kriging model and DoE as the initial Kriging model and DoE of this stage. Denote the second-stage Kriging model as .

[0046] Search for the best learning samples , and use the obtained GMM as the importance sampling density to generate importance sampling samples . Call the current Kriging model to calculate the responses of all samples within, and denote the corresponding mean and standard deviation as and respectively. Determine the best learning samples using the same learning function . It should be noted that the candidate sample pool is , and the surrogate model is .

[0047] Judge whether it converges. Based on the current surrogate model and the importance sampling sample pool , judge whether the preset stop condition is satisfied. Note that the two critical thresholds and are both 0.0001. If the stop condition is satisfied, jump to the step of expanding the DoE based on the best learning samples . Otherwise, the method jumps to the step of calculating the coefficient of variation of the failure probability estimate and continues the active learning

[0048] Expand the DoE based on the best learning samples , calculate the structural response corresponding to the best learning samples , and expand into the current DoE. The method jumps to the step of constructing the initial Kriging model in the second stage and updates the Kriging model .

[0049] Calculate the coefficient of variation of the failure probability estimate. Use the current Kriging model to calculate the failure probability and its coefficient of variation as shown in the following formula ; In the above formula is the importance sampling estimate of the failure probability, and its expression is ; is the indicator function corresponding to the failure domain of the surrogate model in the second stage , and

[0050] [[ID=7i]] is the variance of the failure probability estimator, and its expression is ; Check whether the following condition is satisfied , where is a pre-specified target coefficient of variation. If the following condition is satisfied , it is considered that the estimated value of the failure probability is accurate enough, and the method stops. Otherwise, generate another set of importance sampling samples, expand it to , and the method jumps to the step of searching for the best learning samples , and start a new round of active learning until the surrogate model meets the target accuracy.

[0051] S400. Calculate the failure probability of the engineering structure based on the second-stage Kriging surrogate model to obtain the calculation result of the reliability of the engineering structure.

[0052] In summary, the method proposed in the embodiment of the present invention uses the active learning method to train the Kriging surrogate model for approximating the true structural performance function. The sample size required for training the surrogate model (and the corresponding number of actual engineering structure analyses) is significantly reduced, and the computational efficiency of the surrogate model itself is relatively high. Based on the trained Kriging surrogate model, the active learning method and the importance sampling method are used to calculate the structural failure probability, further improving the computational efficiency of the reliability analysis. Finally, the Kriging surrogate model of the structural performance function and the failure probability are obtained, which can provide technical support for the design, optimization, safety assessment, etc. of engineering structures.

[0053] Furthermore, in combination with Attached Drawing Figure 4 and Attached Drawing Figure 5 , the embodiment of the present invention is described. First, parameter descriptions are given. For the first-stage Kriging model, the truncation coefficient of the LHS sampling region is taken as , that is, the sampling region is . The initial DoE and the capacity of the LHS sample pool are and respectively. The threshold of the stopping condition is 0.001, that is . For the adaptive clustering stage, the capacity of the MCMC sample family is , and the burn-in period length and the slice length are and respectively. For the AK-IS second stage, the capacity of the importance sampling sample family is , and the threshold of the stopping condition is 0.0001, that is . The target coefficient of variation for estimating the failure probability is taken as . The number of calls of the structural performance function , the failure probability estimation , the coefficient of variation Cov, and the relative error are used to compare the performance of the proposed AC-AK-IS with the existing methods. Specific Embodiment 1: This embodiment is a series system with four failure domains, and the structural performance function is as follows: ; In the above formula, are all standard normal random variables.

[0055] The typical process is as Figure 4 shown. As shown in (a) of Figure 4 : In the first stage, the newly added DoE of the Kriging model for active learning mainly focuses on the vicinity of the limit state surface. As shown in (b) of Figure 4 : The samples within the failure domain of the surrogate model generated by MCMC and the optimal clustering results indicate that the adaptive clustering algorithm can effectively identify sub-failure domains. As shown in (c) of Figure 4 : The GMM constructed based on the identified sub-failure domains is a multimodal probability density function. As shown in (d) of Figure 4 : The importance sampling sample pool and the finally trained Kriging model.

[0056] The calculation results are shown in Table 1. It can be seen that the performance of the proposed AC-AK-IS method exceeds that of the simulation methods (MCS, DS, and IS), Meta-IS, and MetaAK-IS2, and a similar or higher-precision failure probability estimate can be obtained with a smaller number of calls to the structural performance function Although the of AC-AK-IS is slightly larger than that of AK-SS and ALK-KDE-IS, the coefficient of variation Cov of the failure probability estimate is smaller, meaning that the accuracy of the failure probability estimate given by it is higher. Since AK-MCS+U and AK-MCS+EFF do not give Cov, the performance of AC-AK-IS cannot be compared with these two methods. The overall performance of AC-AK-IS is slightly lower than that of Meta-IS-AK, but it should be noted that AC-AK-IS can explicitly identify sub-failure domains, providing an effective way to quantify the contribution of each sub-failure domain to the overall failure.

[0057] Table 1 Reliability calculation results of the first specific embodiment

[0058] Specific embodiment two: This embodiment is a system with two failure domains, and the structural performance function is as follows: ; In the above formula, are independent standard normal random variables. The parameters take 3, 4, and 5 respectively, and the corresponding order of magnitude of the failure probability drops from to .

[0059] Figure 5 Give the typical calculation process at Figure 5 In (a), it is the first-stage Kriging model in Figure 5 In (b), it shows that the adaptive clustering algorithm can accurately identify two sub-failure domains Figure 5 In (c), it is the fitted GMM probability density function Figure 5 In (d), it is the importance sampling samples and the finally trained Kriging model

[0060] As shown in Table 2, the reliability calculation results are given. It can be seen that the performance of the proposed AC-AK-IS significantly exceeds that of MCS, Au&Beck, and Meta-IS, and the number of function calls required is significantly reduced. Although the required by AC-AK-IS is close to that of MetaAK-IS2 and ALK-KDE-IS, the coefficient of variation of failure probability Cov of AC-AK-IS is smaller, indicating higher calculation accuracy. In addition, as the order of magnitude of the failure probability decreases, the required by AC-AK-IS does not increase significantly, demonstrating the robustness to the change of the order of magnitude of the failure probability.

[0061] Table 2 Reliability calculation results of the second specific embodiment

[0062] In summary, in the embodiment of the present invention, first, the first-stage active learning Kriging model is trained based on the LHS sample pool, and this model can better approximate the main shape of the failure domain. Secondly, the adaptive clustering algorithm is used to automatically identify multiple potential sub-failure domains, and then the GMM is constructed as the importance sampling density function. Finally, a large number of importance sampling samples are generated based on the constructed GMM importance sampling density function, and the Kriging model is trained by the active learning method in the second stage of AK-IS to efficiently calculate the failure probability. The proposed method AC-AK-IS combines the importance sampling method and the active learning Kriging model based on adaptive clustering, and can provide an effective means for the structural reliability analysis with the characteristics of multiple failure domains and small failure probability.

[0063] Combined with specific embodiments, it shows that: (1) the proposed method can accurately identify potential sub-failure domains; (2) compared with the existing methods, it can generally improve the calculation efficiency; (3) it can provide an effective means for quantifying the contribution of each sub-failure domain to the structural failure.

[0064] Refer to Figure 2 , the multi-failure domain structural reliability optimization system based on adaptive clustering includes: The first module 201 is used to perform the first-stage training on the Kriging surrogate model through an active learning method based on engineering structure data, and construct the first-stage Kriging surrogate model; The second module 202 is used to identify the failure domain of the first-stage Kriging surrogate model based on an adaptive clustering algorithm, and construct an importance sampling density; The third module 203 is used to perform the second-stage training on the first-stage Kriging surrogate model through an active learning method based on the importance sampling density, and output the second-stage Kriging surrogate model; The fourth module 204 is used to calculate the failure probability of the engineering structure based on the second-stage Kriging surrogate model, and obtain the calculation result of the engineering structure reliability.

[0065] The content in the above method embodiments is applicable to the present system embodiment. The functions specifically implemented by the present system embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those in the above method embodiments.

[0066] The above has specifically described the preferred embodiments of the present invention. However, the present invention is not limited to the above embodiments. Those skilled in the art can make various equivalent deformations or substitutions without departing from the spirit of the present invention, and these equivalent deformations or substitutions are all included in the scope defined by the claims of this application.

Claims

1. An adaptive clustering-based multi-failure domain structural reliability optimization method, characterized in that It includes the following steps: Based on the engineering structure data, the Kriging surrogate model is trained in the first stage by the active learning method to construct the first-stage Kriging surrogate model; Based on the adaptive clustering algorithm, the failure domain of the first-stage Kriging surrogate model is identified to construct the importance sampling density; Based on the importance sampling density, the first-stage Kriging surrogate model is trained in the second stage by the active learning method to output the second-stage Kriging surrogate model; Based on the second-stage Kriging surrogate model, the failure probability of the engineering structure is calculated to obtain the calculation result of the engineering structure reliability; 2. The method for optimizing the structural reliability of multiple failure domains based on adaptive clustering according to claim 1, wherein The step of "Based on the engineering structure data, the Kriging surrogate model is trained in the first stage by the active learning method to construct the first-stage Kriging surrogate model" specifically includes: Obtain the engineering structure data and generate the initial experimental design and the initial sample pool within the preset standard deviation range of the standard normal space based on the Latin hypercube sampling method. The initial experimental design includes the corresponding structural responses; Based on the initial experimental design, construct the Kriging surrogate model through the preset software toolbox; Based on the initial sample pool, identify and obtain the first optimal learning sample through the active learning function, and calculate the structural response of the first optimal learning sample through the structural function; Expand the structural response of the first optimal learning sample to the initial experimental design, and iteratively update the Kriging surrogate model based on the expanded experimental design until the preset stop condition is met, and output the first-stage Kriging surrogate model.

3. The multi-failure domain structural reliability optimization method based on adaptive clustering according to claim 2, characterized in that The expression of the active learning function is specifically as follows: ； In the above formula, represents the first optimal learning sample, represents the active learning function, represents the initial sample pool, represents the sample in the initial experimental design, and respectively represent the mean and standard deviation of the Kriging surrogate model at the sample points.

4. The reliability optimization method of multi-failure domain structure based on adaptive clustering according to claim 3, wherein The expression of the preset stop condition is specifically as follows: ； In the above formula, represents the stopping condition index based on the stability of the failure domain, represents the stopping condition index based on the change in the sign of the sample response in two adjacent iterations, , represents the critical threshold, represents the number of failed samples in the th round of active learning iteration, represents the number of failed samples in the th round of active learning iteration, represents the sample size, represents the Kriging surrogate model obtained from the th round of active learning training, represents the Kriging surrogate model obtained from the th round of active learning training, represents the th sample.

5. The method for optimizing the structural reliability of multiple failure domains based on adaptive clustering according to claim 4, wherein The step of "Based on the adaptive clustering algorithm, the failure domain of the first-stage Kriging surrogate model is identified to construct the importance sampling density" specifically includes: Generate samples within the failure domain corresponding to the first-stage Kriging surrogate model through the Markov chain Monte Carlo method; Perform partition clustering on the samples within the failure domain through the adaptive clustering algorithm to obtain the best clustering result; Fit the best clustering result through the Gaussian mixture density function to obtain the importance sampling density.

6. The reliability optimization method for multi-failure domain structures based on adaptive clustering according to claim 5, wherein The step of "Perform partition clustering on the samples within the failure domain through the adaptive clustering algorithm to obtain the best clustering result" specifically includes: Determine the range of candidate clustering numbers; Based on the range of candidate clustering numbers, perform K-means clustering on the samples within the failure domain to obtain the preliminary clustering result; Calculate the DI value for the preliminary clustering result to obtain the Dunn validity index; Select the DI value corresponding to the maximum Dunn validity index as the optimal clustering number; Obtain the clustering result corresponding to the optimal clustering number as the basis for dividing the potential failure domain, and divide the samples within the failure domain to obtain the best clustering result.

7. The method for optimizing the structural reliability of multiple failure domains based on adaptive clustering according to claim 6, wherein, The expression of the Dunn validity index is as follows: ； In the above formula, represents the DI value, represents the range of candidate cluster numbers, , represents the clustering result, represents clustering and the set distance of, represents the clustering result the diameter of.

8. The reliability optimization method for multi-failure domain structures based on adaptive clustering according to claim 7, wherein The step of performing the second-stage training on the first-stage Kriging surrogate model by an active learning method based on the importance sampling density and outputting the second-stage Kriging surrogate model specifically includes: Combining the first-stage Kriging surrogate model with the initial experimental design to construct the second-stage initial Kriging surrogate model; Generating importance sampling samples based on the importance sampling density and calculating the responses corresponding to the importance sampling samples based on the second-stage initial Kriging surrogate model to obtain the second-best learning samples; Judging the second-stage initial Kriging surrogate model according to the preset stopping condition; If the second-stage initial Kriging surrogate model does not meet the preset stopping condition, calculating the structural response corresponding to the second-best learning samples, expanding the second-best learning samples and the corresponding structural responses to the initial experimental design, and updating the second-stage initial Kriging surrogate model; If the second-stage initial Kriging surrogate model meets the preset stopping condition, calculating the failure probability based on the current second-stage initial Kriging surrogate model to obtain the coefficient of variation; If the coefficient of variation does not meet the preset target coefficient of variation, generating another group of importance sampling samples based on the importance sampling density, expanding the existing importance sampling sample pool, and training the second-stage initial Kriging surrogate model by an active learning method based on the expanded importance sampling sample pool until the second-stage initial Kriging surrogate model meets the preset stopping condition, calculating the failure probability and the corresponding coefficient of variation, and re-judging whether it meets the preset target coefficient of variation; Until the coefficient of variation meets the preset target coefficient of variation, outputting the second-stage Kriging surrogate model.

9. The method for optimizing the structural reliability of multiple failure domains based on adaptive clustering according to claim 8, wherein The specific calculation expression of the coefficient of variation is as follows: ； In the above formula, represents the importance sampling estimate of the failure probability, represents the coefficient of variation of the failure probability estimate, represents the variance of the failure probability estimate.

10. The multi-failure domain structural reliability optimization system based on adaptive clustering is characterized in that Including the following modules: The first module is used to perform the first-stage training on the Kriging surrogate model by an active learning method based on engineering structure data and construct the first-stage Kriging surrogate model; The second module is used to identify the failure domain of the first-stage Kriging surrogate model based on the adaptive clustering algorithm and construct the importance sampling density; The third module is used to perform the second-stage training on the first-stage Kriging surrogate model by an active learning method based on the importance sampling density and output the second-stage Kriging surrogate model; The fourth module is used to calculate the failure probability of the engineering structure based on the second-stage Kriging surrogate model to obtain the calculation result of the engineering structure reliability.

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