Structural reliability analysis method based on Kriging model and truncation importance sampling

Abstract

The invention belongs to the technical field of mechanical structure reliability analysis, and particularly relates to a structure reliability analysis method based on a Kriging model and truncation importance sampling, which comprises the following steps: S1, analyzing a mechanical structure, identifying key factors and establishing a performance function; s2, estimating a probability distribution form and parameters of an input variable; s3, sampling to generate a multi-dimensional random sample, calculating a response by using a performance function, and forming a training sample set; s4, optimizing related function parameters of the Kriging model by using the training sample set, and constructing an initial Kriging model; s5, using a Kriging model to evaluate MPP, forming a TIS density function, and extracting candidate sample points; s6, selecting a candidate sample point which enables the learning function value to be minimum as a newly added training sample point, and continuing to optimize the Kriging model; if not, returning to S5, and if so, turning to S7; s7, calculating a failure probability and a variable coefficient according to the prediction response of the Kriging model with local convergence; and if not, returning to S5. According to the method, the calculation efficiency of reliability analysis of the low-failure-probability complex structure can be improved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of mechanical structure reliability analysis, and particularly relates to a structure reliability analysis method based on a Kriging model and a truncated importance sampling. BACKGROUND

[0002] There are various uncertain factors in the whole life cycle of mechanical parts and equipment products, the dispersion of material performance and organization distribution in the material preparation stage, the process error in the processing, manufacturing and assembling stage, and the randomness of external load conditions and human operation and use in the service stage. These uncertain factors fluctuate about 5% to 10% of the design value, which leads to the instability of product performance in use, which causes great safety hazards to the service of mechanical equipment. Therefore, at the beginning of design, it is necessary to analyze the reliability of mechanical structure to prevent the loss of mechanical function, increase the planned maintenance cost, cause chain economic loss or accident.

[0003] The function function g(x) of the mechanical structure is usually implicit, and analyzing its reliability requires large-scale finite element calculation or testing, which is costly. Therefore, many scholars have conducted research on it for many years, and the mechanical structure reliability analysis method is mainly divided into: approximate analytical method, digital simulation method, and proxy model method. The approximate analytical method mainly includes the first order reliability method (FORM) and the second order reliability method (SORM), etc. This method linearizes or quadratic approximates the implicit structure function function at the mean point or the design point (the maximum point of the joint probability density function in the failure domain) through Taylor expansion, and then calculates the failure probability, but ignores the high-order terms, which is difficult to converge in high nonlinear problems and lacks accuracy. The digital simulation method mainly refers to the Monte Carlo method (MCS) and the corresponding improved method. This method is based on the law of large numbers, and the failure probability is approximated by the exceeding frequency of the limit state. In order to ensure the accuracy and convergence of the calculation results, a large number of samples and a large number of function function calculations (large-scale finite element calculations) are required, resulting in very low computational efficiency of reliability analysis in actual engineering or even impossible to complete, especially for problems with very low failure probability. The core of the proxy model method is to establish a high-precision proxy model to simulate the true structure function function, and based on this, the numerical simulation technology is used to evaluate the failure probability, thereby significantly reducing the number of calls for computationally intensive simulations such as finite element analysis, and greatly improving the calculation efficiency. However, it has limitations: the training samples are usually uniformly distributed in the input space, resulting in insufficient sample density near the failure boundary, which may affect the accuracy of failure probability estimation; the proxy model construction and reliability analysis process are independent of each other, and the model built is difficult to ensure the accuracy of subsequent failure probability evaluation.

[0004] To overcome the above defects, researchers have proposed advanced methods such as active learning Kriging model combined with Monte Carlo simulation (AK-MCS). Such methods take advantage of the Kriging model, which can not only predict the response value but also provide the prediction uncertainty. By designing a learning function (such as U function), the most influential candidate sample points on the failure probability estimation are actively identified, and the real function function is called for "point filling" training to realize the adaptive optimization of the model in the key area. However, existing active learning methods still face severe challenges when dealing with low failure probability complex mechanical structure problems: on the one hand, due to the extremely small proportion of the failure domain in high-dimensional space, the traditional Monte Carlo sampling is sparse in the failure area, resulting in low efficiency of the learning process; on the other hand, if the global sampling strategy is still used, a large amount of computing resources is wasted in the safe area with minimal contribution to the failure probability, making it difficult to fully improve the modeling accuracy near the limit state in a limited iteration, resulting in a large deviation in the final failure probability estimation.

[0005] Therefore, how to improve the computational efficiency of low failure probability complex structure reliability analysis while ensuring high precision has become a problem to be solved. SUMMARY

[0006] In view of the above deficiencies of the prior art, the present application provides a structure reliability analysis method based on Kriging model and truncated importance sampling, which can improve the computational efficiency of low failure probability complex structure reliability analysis while ensuring high precision.

[0007] To solve the above technical problems, the present application adopts the following technical solutions:

[0008] The structure reliability analysis method based on Kriging model and truncated importance sampling comprises the following steps:

[0009] S1, analyze the mechanical structure, determine its key failure mode and failure mechanism; identify the key factors of failure behavior as the input variables of reliability analysis; and establish the function function of the mechanical structure combined with the key failure mode;

[0010] S2, statistics the relevant information data of the input variables, estimate and verify the probability distribution form and parameters of the input variables;

[0011] S3, sample to generate multi-dimensional random samples according to the input variable distribution obtained in S2, and calculate the response of each sample point using the function function to form a training sample set;

[0012] S4, use the training sample set to optimize the correlation function parameters θ of the Kriging model, and construct an initial Kriging model;

[0013] S5, evaluate the most probable point of failure (MPP) in the failure domain using the current Kriging model, construct an importance sampling density function based on the MPP, and truncate the importance sampling density function in the vicinity of the limit state to form a truncated importance sampling (TIS) density function; extract N TIS candidate sample points based on the TIS density function;

[0014] S6, predict the response of each candidate sample point using the current Kriging model, and calculate the learning function value through the learning function; select the candidate sample point that minimizes the learning function value as a new training sample point, and calculate its response using the function function to obtain new training data; continue to optimize the correlation function parameter theta of the Kriging model using the new training data; and determine whether the Kriging model converges, if not, return to S5, if converges, go to S7;

[0015] S7, predict the response of the candidate sample point of S5 using the converged Kriging model, and calculate the failure probability and coefficient of variation of the mechanical structure based on the prediction result; if the coefficient of variation meets the requirements, the result is determined to be converged; if not, return to S5 to generate new candidate sample points for continued iteration.

[0016] Compared with the prior art, the present application has the following beneficial effects:

[0017] 1. With the help of truncated importance sampling (TIS) mechanism, the utilization efficiency of candidate samples and the convergence speed of learning process are greatly improved. The existing AK-MCS method usually selects candidate points in the global sample pool, resulting in a large number of candidate points in the safe region far from the limit state, which contributes little to the failure probability estimation, causing waste of computing resources. In S5, the method innovatively combines the most probable point of failure (MPP) to construct an importance sampling density, and truncates it in the vicinity of the limit state to form a TIS density function. The candidate sample points generated by this method are naturally concentrated in the potential failure region, greatly improving the information value of the selected samples in each model update, making the learning process more efficient, and significantly speeding up the convergence of the Kriging model to the limit state.

[0018] 2. Realize the double convergence control of failure probability evaluation process, and consider the calculation efficiency and result reliability. The method sets double judgment mechanism of model convergence and result convergence in S6 and S7 respectively. S6 ensures that the Kriging model has sufficient accuracy in the key area, and S7 monitors the statistical stability of the failure probability estimation through the coefficient of variation. This phased iteration strategy avoids redundant calculation caused by blindly increasing samples, and prevents result deviation caused by insufficient model convergence. Compared with the traditional method which only relies on fixed sample size or single convergence criterion, the method is more robust and adaptive, and can obtain reliable failure probability estimation in fewer times.

[0019] 3. Through the active learning mechanism, the model precision is self-adaptively improved, and the approximation ability of the surrogate model in the key area is significantly enhanced. Unlike the traditional surrogate model method which uniformly distributes points in the input space and lacks a feedback mechanism, the method introduces an active learning strategy based on a learning function (such as the U function) in S6. It can use the prediction uncertainty information provided by the Kriging model to intelligently identify candidate sample points near the limit state boundary that are least accurate in prediction or have the greatest impact on failure probability. By supplementing real response data at these key locations and continuously optimizing model parameters, focused modeling of the performance function in the vicinity of the failure domain is achieved, effectively overcoming the precision deficiency problem caused by the sparsity of boundary samples in traditional methods.

[0020] 4. The precision and efficiency in high nonlinearity problems are effectively balanced, and the applicability of the surrogate model method in complex engineering scenarios is expanded. For highly nonlinear, multi-peak, or small failure probability complex structure problems, the approximate analytic method is not accurate enough, the Monte Carlo method is too high in computational cost, and the traditional surrogate model is difficult to handle. The method deeply integrates the TIS-guided sampling strategy and the active learning mechanism, retains the good fitting ability of the Kriging model for nonlinear response, and greatly reduces the number of calls to the high-cost simulation model through intelligent sampling, making it possible to perform high-precision reliability evaluation of complex structures under limited computational budget.

[0021] In summary, the method can improve the computational efficiency of reliability analysis of complex structures with low failure probability under the premise of ensuring high precision.

[0022] Preferably, in S4, the whale optimization algorithm WOA is used to optimize the correlation function parameters θ of the Kriging model; the optimization objective of WOA is to minimize the fitness function based on θ, and the optimization problem model is expressed using the following formula:

[0023]

[0024] In the formula, according to the basic theory of the Kriging model, φ(θ) is the fitness function of the correlation parameter θ in the Kriging model; R is the m x m correlation matrix of any two sample data points; is the process variance, which is a function of θ; n is the dimension of the sample; θ k is the k-dimensional component of the correlation parameter vector.

[0025] Such a setting, the correlation function parameter theta of Kriging model is based on the sample, using whale optimization algorithm instead of matlab dace toolbox pattern search method, not dependent on the initial solution for global optimization, can more effectively search parameter space, improve the model precision. Whale optimization algorithm (WOA) is a kind of optimization algorithm based on swarm intelligence, which is inspired by the hunting behavior of humpback whales. Compared with traditional gradient descent and other local optimization methods, WOA has stronger global search ability. In the optimization of the correlation function parameter theta of Kriging model, WOA can more effectively explore the parameter space and avoid falling into local optimal solution. This global search ability helps to find better parameter configuration, thus improving the prediction performance of Kriging model.

[0026] Preferably, in S7, if the coefficient of variation of failure probability estimation is lower than the preset threshold, it is determined that the Kriging model converges.

[0027] Preferably, in S7, the calculation formula of failure probability is:

[0028]

[0029] The calculation formula of coefficient of variation is:

[0030]

[0031] In the formula, I F (x) is the failure domain indicator function, which takes the value of 1 when the sample point is located in the failure domain, otherwise 0; I β (x) is the truncation indicator function; φ n (x) is the original probability density function, is the importance sampling density function.

[0032] Such a setting, 1. Provides accurate failure probability estimation, enhances the accuracy of reliability analysis. The calculation formula of failure probability combines the failure domain indicator function I F (x) and the truncation indicator function I β (x), which can accurately identify the sample points located in the failure domain, and through the importance sampling density function) Weighted processing is performed on these sample points. Compared with the traditional Monte Carlo simulation method, this method can provide more accurate failure probability estimation with less sample size, significantly improving the accuracy of reliability analysis.

[0033] 2. Evaluate the statistical stability of the results to ensure the reliability of the analysis process. The formula for calculating the coefficient of variation is used to evaluate the statistical variability of the failure probability estimate. By calculating the coefficient of variation, the uncertainty of the failure probability estimate can be quantified, thereby judging the stability and reliability of the analysis results. When the coefficient of variation meets the preset requirements, it indicates that the failure probability estimate has high precision and stability, otherwise further increase the sample size or optimize the model parameters are needed. This explicit evaluation of the stability of the results enhances the reliability of the entire analysis process.

[0034] Preferably, in S5, the whale optimization algorithm is used to evaluate the most probable failure point MPP in the failure domain.

[0035] Such settings, using whale optimization algorithm to evaluate the most probable failure point MPP, not only overcome the limitations of traditional gradient method in nonlinear, implicit problems, but also through its global search ability, no gradient characteristics and swarm intelligence advantages, significantly improve the accuracy, robustness and engineering applicability of MPP identification.

[0036] Preferably, in S5, when truncating in the vicinity of the limit state, the truncated region is defined as: the inside of the β-sphere contains part or all of the safety zone, and the sample points in the β-sphere are considered completely safe and do not need to be evaluated by the limit state function; The other regions outside the β-sphere include all failure zones and part of the safety zone, and the sample points outside the β-sphere may be in the failure domain or the safety domain, and need to be further judged by the function function.

[0037] Preferably, in S5, in the process of forming the truncated importance sampling TIS density function, the objective function of MPP search is:

[0038]

[0039] And use the following formula as the constraint:

[0040]

[0041] In the formula, u is a random variable vector in the standard normal space; And The mean and standard deviation of the random variable x i , respectively; is the limit state function constructed by the Kriging model.

[0042] Such setting can improve the accuracy and efficiency of MPP search. In reliability analysis, MPP is the key point connecting the design space and the standard normal space, and its accuracy directly affects the effectiveness of the subsequent TIS strategy. Through the minimization of the objective function in the method, the nearest failure boundary point to the origin, i.e. MPP, can be ensured to be found. This method can effectively avoid the deviation of MPP positioning caused by improper selection of initial points or local extreme values, thereby improving the accuracy of MPP search. At the same time, the limit state function constructed by the Kriging model As a constraint condition, it can quickly converge to the true MPP in complex nonlinear problems, improving the search efficiency.

[0043] Preferably, the learning function is a U learning function; the definition of the U learning function is as follows:

[0044]

[0045] In the formula, U(x i ) is the value of the learning function; respectively, the mean and standard deviation of the predicted response value of the Kriging model;

[0046] A new sample is added according to the following formula:

[0047]

[0048] In the formula, S is a candidate sample set composed of N TIS sample points;

[0049] The stopping criterion is:

[0050]

[0051] Such setting can improve the prediction accuracy and generalization ability of the Kriging model. The U learning function combines the mean and standard deviation of the predicted response value of the Kriging model, and can consider the prediction uncertainty and deviation of the model at a certain point. By selecting the new sample point that minimizes the U learning function value, the area with high prediction uncertainty of the model can be effectively filled, thereby improving the overall prediction accuracy and generalization ability of the Kriging model. Compared with random sampling or uniform sampling methods, the sample selection strategy based on the U learning function can more efficiently utilize limited sample resources and improve model performance.

[0052] Preferably, the expression of the Kriging model is as follows:

[0053] ξ(x)=ψ(x) T β+z(x);

[0054] In the formula, ξ(x) is a predicted value function; ψ(x)T β is the regression part of the model, ψ(x) is the base function of the model, β is the regression coefficient; z(x) is a Gaussian random process with mean 0, and the covariance is cov(z(x (i) ),z(x (j) ))=σ 2 R(x (i) ,x (j) ,θ), wherein R(x (i) ,x (j) ,θ) represents the spatial correlation function of any two sample points , θ represents the corresponding correlation parameter vector, and the superscript S represents the dimension.

[0055] Such settings, Kriging model as a powerful proxy modeling tool, in the mechanical structure reliability analysis shows excellent performance. It can not only provide accurate prediction results and global approximation ability, but also effectively quantify uncertainty and support efficient optimization.

[0056] Preferably, in S3, a multi-dimensional random sample is generated by Latin hypercube sampling.

[0057] Such settings, compared with the traditional simple random sampling, Latin hypercube sampling through stratified sampling mechanism, ensure that each input variable in its distribution range is evenly divided in the interval of exactly one sample. This strategy effectively avoids the sample aggregation or void phenomenon that may occur in random sampling, significantly improves the uniformity and coverage of the initial training sample in the high-dimensional input space. BRIEF DESCRIPTION OF DRAWINGS

[0058] In order to make the purpose, technical scheme and advantages of the application clearer, the application will be further described in detail below in combination with the drawings, wherein:

[0059] Figure 1 is a flowchart of the present method;

[0060] Figure 2 is a schematic diagram of example 2 in example two. DETAILED DESCRIPTION

[0061] The following will be further described in detail through specific embodiments:

[0062] Example one

[0063] As shown in Figure 1 , the present embodiment discloses a structure reliability analysis method based on Kriging model and truncated importance sampling, comprising the following steps:

[0064] S1, analyze the mechanical structure, determine its key failure mode and failure mechanism; identify the key factors of failure behavior as the input variables of reliability analysis; combine the key failure mode to establish the function function of the mechanical structure;

[0065] S2, estimate and test the probability distribution form and parameters of the input variables by statistical information data of the input variables;

[0066] S3, according to the input variable distribution obtained in S2, sample generation multi-dimensional random sample, and use the function function to calculate the response of each sample point to form the training sample set.

[0067] In specific implementation, in S3, Latin hypercube sampling is used to generate multi-dimensional random samples. Compared with traditional simple random sampling, Latin hypercube sampling ensures that each input variable is evenly divided in the interval by stratified sampling mechanism. This strategy effectively avoids the sample aggregation or hollow phenomenon that may occur in random sampling, and significantly improves the uniformity and coverage of the initial training sample in high-dimensional input space.

[0068] S4, use the training sample set to optimize the correlation function parameters θ of Kriging model, and construct the initial Kriging model;

[0069] The expression of Kriging model is:

[0070] ξ(x)=ψ(x) T β+z(x);

[0071] In the formula, ξ(x) is the prediction value function; ψ(x) T β is the regression part of the model, ψ(x) is the basic function of the model, β is the regression coefficient; z(x) is a Gaussian random process with mean 0, and its covariance is cov(z(x (i) ),z(x (j) ))=σ 2 R(x (i) ,x (j) ,θ), where R(x (i) ,x (j) ,θ) represents the spatial correlation function of any two sample points , θ represents the corresponding correlation parameter vector, and the superscript S represents the dimension.

[0072] Kriging model as a powerful proxy modeling tool, shows excellent performance in mechanical structure reliability analysis. It not only can provide accurate prediction results and global approximation ability, but also can effectively quantify uncertainty and support efficient optimization.

[0073] In implementation, the whale optimization algorithm (WOA) is used to optimize the correlation function parameters θ of the Kriging model. The optimization objective of WOA is to minimize the fitness function based on θ, and the optimization problem model is expressed using the following formula:

[0074]

[0075] In the formula, according to the basic theory of the Kriging model, φ(θ) is the fitness function of the correlation parameter θ in the Kriging model; R is the m×m correlation matrix of any two sample data points; σ2 is the process variance, which is a function of θ; n is the dimension of the sample; θ k is the k-dimensional component of the correlation parameter vector.

[0076] The correlation function parameters θ of the Kriging model are based on samples, and the whale optimization algorithm is used instead of the pattern search method of the dace toolbox in matlab for global optimization without relying on initial solutions. It can more effectively search the parameter space and improve the model accuracy. The whale optimization algorithm (WOA) is a swarm intelligence-based optimization algorithm, which is inspired by the hunting behavior of humpback whales. Compared with traditional local optimization methods such as gradient descent, WOA has stronger global search ability. When optimizing the correlation function parameters θ of the Kriging model, WOA can more effectively explore the parameter space and avoid falling into local optimal solutions. This global search ability helps to find better parameter configurations, thereby improving the prediction performance of the Kriging model.

[0077] S5, using the current Kriging model to evaluate the most likely failure point MPP in the failure domain, constructing the importance sampling density function based on MPP, and truncating the region near the limit state to form the truncated importance sampling (TIS) density function; based on the TIS density function, N TIS candidate sample points are extracted.

[0078] Wherein, when truncating the region near the limit state, the truncated region is defined as: the internal part of the β-sphere contains part or all of the safety zone, and the sample points in the β-sphere are considered completely safe and do not need to be evaluated by the limit state function; the other regions outside the β-sphere include all failure zones and part of the safety zone, and the sample points outside the β-sphere may be in the failure domain or the safety domain, which need to be further judged by the function function.

[0079] In specific implementation, the whale optimization algorithm is used to evaluate the most probable point of failure (MPP) in the failure domain. The whale optimization algorithm is used to evaluate the most probable point of failure (MPP) in the failure domain. The whale optimization algorithm not only overcomes the limitations of traditional gradient methods in nonlinear and implicit problems, but also significantly improves the accuracy, robustness and engineering applicability of MPP identification through its global search ability, gradient-free characteristics and swarm intelligence advantages.

[0080] In the process of forming the truncated importance sampling (TIS) density function, the objective function of the MPP search is:

[0081]

[0082] And use the following formula as a constraint:

[0083]

[0084] In the formula, u is a random variable vector in the standard normal space; and are the mean and standard deviation of the random variable x i , respectively; is the limit state function constructed by the Kriging model.

[0085] In this way, the accuracy and efficiency of MPP search can be improved. In reliability analysis, MPP is the key point connecting the design space and the standard normal space, and its accuracy directly affects the effectiveness of subsequent TIS strategies. By minimizing the objective function in this method, the nearest failure boundary point to the origin, i.e. MPP, can be ensured. This method can effectively avoid the deviation of MPP positioning caused by improper initial point selection or local extremum, thereby improving the accuracy of MPP search. At the same time, the limit state function constructed by the Kriging model as a constraint condition can quickly converge to the true MPP in complex nonlinear problems, improving the search efficiency.

[0086] S6, use the current Kriging model to predict the response of each candidate sample point, and calculate the learning function value through the learning function; select the candidate sample point that minimizes the learning function value as the new training sample point, and calculate its response using the function function to obtain new training data; continue to optimize the correlation function parameters θ of the Kriging model using the new training data; and determine whether the Kriging model converges, if not, return to S5, if yes, go to S7;

[0087] Wherein, the learning function is a U learning function; the definition of the U learning function is:

[0088]

[0089] In the formula, U(xi ) is a learning function value; are the mean and standard deviation of the Kriging model predicted response value, respectively;

[0090] A new sample is added according to the following formula:

[0091]

[0092] where S is a candidate sample set composed of N TIS candidate sample points;

[0093] The stopping criterion is:

[0094]

[0095] In this way, the prediction accuracy and generalization ability of the Kriging model can be improved. The U learning function combines the mean and standard deviation of the Kriging model predicted response value, and can comprehensively consider the prediction uncertainty and bias of the model at a certain point. By selecting a new sample point that minimizes the U learning function value, the area with high prediction uncertainty of the model can be effectively filled, thereby improving the overall prediction accuracy and generalization ability of the Kriging model. Compared with random sampling or uniform sampling methods, the sample selection strategy based on the U learning function can more efficiently utilize limited sample resources and improve model performance.

[0096] S7, the candidate sample points of S5 are predicted with the converged Kriging model, and the failure probability and coefficient of variation of the mechanical structure are calculated based on the prediction results; if the coefficient of variation meets the requirements, the result is determined to be converged; if it is not converged, return to S5 to generate new candidate sample points for continued iteration.

[0097] If the coefficient of variation of the failure probability estimate is lower than a preset threshold, it is determined that the Kriging model is converged. In specific implementation, the preset threshold is 0.05.

[0098] The calculation formula of the failure probability is:

[0099]

[0100] The calculation formula of the coefficient of variation is:

[0101]

[0102] where I F (x) is a failure domain indicator function, taking a value of 1 when the sample point is located in the failure domain, and 0 otherwise; I β (x) is a truncated indicator function; φ n (x) is the original probability density function, is the importance sampling density function.

[0103] The calculation formula of failure probability combines the failure domain indicator function I F (x)I and the truncated indicator function I β (x), which can accurately identify sample points located in the failure domain and pass through the importance sampling density function) These sample points are weighted. Compared with the traditional Monte Carlo simulation method, this method can provide more accurate failure probability estimation with less sample size, significantly improving the accuracy of reliability analysis. The calculation formula of the coefficient of variation is used to evaluate the statistical variability of the failure probability estimation. By calculating the coefficient of variation, the uncertainty of the failure probability estimation can be quantified, so as to judge the stability and credibility of the analysis results. When the coefficient of variation meets the preset requirements, it indicates that the failure probability estimation has high precision and stability, otherwise it needs to further increase the sample size or optimize the model parameters. This explicit evaluation of the stability of the results enhances the reliability of the entire analysis process.

[0104] The existing AK-MCS method usually selects candidate points in the global sample pool, resulting in a large number of candidate points located in the safe region far from the limit state, which contributes little to the failure probability estimation, causing waste of computing resources. This method innovatively constructs the importance sampling density in S5 combined with the most probable failure point (MPP), and performs truncation near the limit state to form the TIS density function. The candidate sample points generated by this method are naturally concentrated in the potential failure region, greatly improving the information value of the selected samples in each model update, making the learning process more efficient, and significantly speeding up the convergence of the Kriging model to the limit state. In addition, this method sets up a double judgment mechanism of model convergence and result convergence in S6 and S7 respectively. S6 ensures that the Kriging model has sufficient accuracy in the key area, and S7 monitors the statistical stability of the failure probability estimation through the coefficient of variation. This phased iteration strategy avoids redundant calculation caused by blindly increasing samples, and also prevents result deviation caused by insufficient model convergence. Compared with the traditional method which only relies on fixed sample size or single convergence criterion, this method is more robust and adaptive, and can obtain reliable failure probability estimation with fewer iterations.

[0105] And, unlike the traditional surrogate model method, which lacks a feedback mechanism and is uniformly distributed in the input space, the method introduces an active learning strategy based on a learning function (such as the U function) in S6, which can intelligently identify candidate sample points near the limit state boundary that are most inaccurate in prediction or have the greatest impact on failure probability using the prediction uncertainty information provided by the Kriging model. By supplementing real response data at these key locations and continuously optimizing model parameters, focused modeling of the performance function in the vicinity of the failure domain is achieved, effectively overcoming the precision problem caused by the sparsity of boundary samples in traditional methods. In addition, for highly nonlinear, multi-peak, or small failure probability complex structure problems, the approximate analytic method lacks precision, the Monte Carlo method has high computational cost, and the traditional surrogate model is difficult to handle. The method combines the sampling strategy guided by TIS with the deep integration of the active learning mechanism, which not only retains the good fitting ability of the Kriging model for nonlinear response, but also significantly reduces the number of calls to the high-cost simulation model through intelligent sampling, making it possible to perform high-precision reliability analysis on complex structures under limited computational budget.

[0106] The method can improve the computational efficiency of reliability analysis of complex structures with low failure probability while ensuring high precision.

[0107] Embodiment Two

[0108] To better understand the method, the following illustrative description is provided.

[0109] Example 1

[0110] Example 1 further illustrates the present application with an example of a series system consisting of four failure sub-regions, which is a typical example in the field of structural reliability analysis.

[0111] (1) Determine the failure mode of the structure according to the specification and the corresponding performance function g(x), and obtain the parameter distribution information of the variable x;

[0112] In Example 1, the performance function of the structure is represented as:

[0113]

[0114] where x1 and x2 are independent standard normal variables.

[0115] (2) Use matlab to extract 15 initial sample points by Latin hypercube sampling method, calculate the sample response values using the performance function g(x), and form the training sample set;

[0116] (3) According to the obtained training set, the correlation function parameter θ of the Kriging model is optimized by matlab combined with whale optimization algorithm (WOA), the population size is set to 25, and when the maximum iteration number is equal to 100, the global optimal correlation parameter is obtained. An initial WOA-Kriging surrogate model is established, and the correlation function is selected as a Gaussian function.

[0117] (4) The most probable point (MPP) is evaluated. The most probable point (MPP) in the failure domain is searched by WOA algorithm through matlab. The probability density function of TIS is constructed with the MPP obtained in the foregoing as the center, and N TIS sample points are generated to cover the potential failure region. The response values of these sample points are evaluated by the WOA-Kriging model.

[0118] (5) Active learning and model updating are performed by matlab: the learning function of each sample point is calculated, the optimal sample is selected according to the adding point criterion of U learning function, the response value of the new sample is calculated by the function function g(x), the training data set is added, the Kriging model parameters are re-optimized by WOA, and the surrogate model is updated. The learning and updating process continues until the stop criterion is met.

[0119] (6) The failure probability and its coefficient of variation are evaluated. Based on the WOA-Kriging model updated by the active learning method and the truncated importance sampling sample, statistical methods are used for reliability analysis. The failure probability and its coefficient of variation are estimated, and if the coefficient of variation meets the requirements, the result is determined to be convergent. Otherwise, return to step (4) to generate new TIS sample points to enrich the sample set. The unbiased estimate of the failure probability is output. When the threshold value is met, the algorithm terminates, and the unbiased estimate of the failure probability is output.

[0120] Table 1 Comparison of the results of example 1 between the method of the present application and the typical method

[0121] Analysis method Required sample size Failure probability / % Relative error / % to Monte Carlo simulation MCS 1 x 10 6 ]]> 2.220 x 10 -3 ]]> — AK-MCS-U 15+81 2.235 x 10 -3 ]]> 0.67 AK-MCS-EFF 15+101 2.159 x 10 -3 ]] 2.7 The method of the invention 15+72 2.219 x 10 -3 ]] 0.05

[0122] According to the results in table 1, compared with the Monte Carlo simulation results, the structure reliability analysis method based on active learning Kriging model, truncated importance sampling and whale optimization algorithm (WOA) proposed in the present application can efficiently and accurately estimate the failure probability of the structure. Compared with the typical method, the relative error of the Monte Carlo simulation is better than that of the typical method. Furthermore, the sample size required by the method proposed in the present application is less than that of the typical method, indicating that the method of the present application is efficient.

[0123] Example 2

[0124] To further show the effectiveness of the method proposed in the application in the small failure probability problem of complex variables, a common engineering system is taken as Example 2 to illustrate the method proposed in the application in detail.

[0125] Example 2 is a nonlinear undamped single degree of freedom system subjected to a rectangular load pulse. As shown in Figure 2 .

[0126] (1) According to the description, the failure mode of the structure and the corresponding function function g(x) are determined, and the parameter distribution information of the variables is obtained:

[0127] Table 2 shows the parameter distribution table of the input variables of Example 2

[0128] Random variable Distribution Mean Standard deviation m Normal distribution 1 0.05 k1 Normal distribution 1 0.1 k2 Normal distribution 0.1 0.01 R Normal distribution 0.5 0.05 t1 Normal distribution 1 0.2 F1 Normal distribution 0.6 0.2

[0129] In this example 2, the function function of the structure is represented as:

[0130]

[0131] Where x1 and x2 are independent standard normal variables.

[0132] (2) 15 initial sample points are extracted by using the Latin hypercube sampling method through matlab, and the sample response values are calculated by using the function function g(x) to form a training sample set;

[0133] (3) According to the obtained training set, the correlation function parameters θ of the Kriging model are optimized by matlab combined with the whale optimization algorithm (WOA), the population size is set to 25, and when the maximum iteration number is equal to 100, the global optimal correlation parameters are obtained. The initial WOA-Kriging surrogate model is established, and the correlation function is selected as the Gaussian function.

[0134] (4) Evaluate the most probable failure point (MPP). The most probable failure point (MPP) in the failure domain is searched by using matlab through the WOA algorithm. The probability density function of TIS is constructed with the MPP obtained as the center, and N TIS sample points are generated to cover the potential failure region. The response values of these sample points are evaluated by the WOA-Kriging model.

[0135] (5) Active learning and model updating are performed by using matlab: the learning function of each alternative sample point is calculated, the optimal sample is selected according to the adding point criterion of U learning function, the response value of the new sample is calculated by using the function function g(x), the training data set is added, the Kriging model parameters are re-optimized by using WOA, and the surrogate model is updated. The learning and updating process continues until the stopping criterion is met.

[0136] (6) Estimate the failure probability and its coefficient of variation. Based on the updated WOA-Kriging model and truncated importance sampling samples, the reliability analysis is carried out by using statistical methods. Estimate the failure probability and its coefficient of variation. If the coefficient of variation meets the requirements, the result converges. Otherwise, return to step (4) to generate new TIS sample points to enrich the sample set. Output the unbiased estimate of the failure probability. When the threshold is met, the algorithm terminates and outputs the unbiased estimate of the failure probability.

[0137] Table 3 Comparison of the results of the method of the present application and the typical method in Example 2

[0138] Analysis method Required sample size Failure probability / % Relative error / % to Monte Carlo simulation MCS 1 x 10 6 ]]> 5.46 x 10 -4 ]]> — PAK-Bn 20+106 5.50 x 10 -4 ]]> 0.733 AK-MCS-U 15+133 5.53 x 10 -4 ]] 1.3 AK-IS 15+205 5.32 x 10 -4 ]]> 2.56 Improved AK-MCS 15+60 5.35 x 10 -4 ]]> 2.0

[0139] According to the results in Table 3, it can be known that compared with the Monte Carlo simulation results, the structural reliability analysis method based on active learning Kriging model, truncated importance sampling and whale optimization algorithm (WOA) can efficiently and accurately estimate the failure probability of the structure. Compared with the typical method, although the relative error of the Monte Carlo simulation is not as good as that of the typical method, the accuracy of the present application also fully meets the engineering needs. Further, the sample size required by the method of the present application is much smaller than that of the typical method and the advanced method in foreign periodicals, indicating the efficiency of the method of the present application.

[0140] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application and not to limit the technical solutions. Those skilled in the art should understand that those who modify or equivalently replace the technical solutions of the present application without departing from the purpose and scope of the technical solutions should be covered in the scope of the claims of the present application.

Claims

1. A structural reliability analysis method based on the Kriging model and truncated importance sampling, characterized in that, Includes the following steps: S1. Analyze the mechanical structure to determine its key failure modes and failure mechanisms; Identify the key factors of failure behavior and use them as input variables for reliability analysis; Establish the function functions of the mechanical structure by combining key failure modes; S2. Collect relevant information data of the input variables, estimate and test the probability distribution form and parameters of the input variables; S3. Based on the input variable distribution obtained in S2, sample and generate multidimensional random samples, and use the function to calculate the response of each sample point to form a training sample set. S4. Using the training sample set, optimize the relevant function parameters θ of the Kriging model and construct the initial Kriging model; S5. Use the current Kriging model to evaluate the most probable failure point (MPP) within the failure domain. Construct an importance sampling density function based on the MPP, and truncate it in the region near the limit state to form a truncated importance sampling density function (TIS). Extract N based on the TIS density function. TIS 1 candidate sample point; S6. Predict the response of each candidate sample point using the current Kriging model, and calculate the learning function value using the learning function; select the candidate sample point that minimizes the learning function value as the new training sample point, and calculate its response using the function to obtain the new training data; use the new training data to continue to optimize the relevant function parameters θ of the Kriging model; and determine whether the Kriging model has converged. If it has not converged, return to S5; if it has converged, go to S7. S7. For the candidate sample points of S5, predict the response using a converged Kriging model, and calculate the failure probability and coefficient of variation of the mechanical structure based on the prediction results. If the coefficient of variation meets the requirements, the result is considered convergent. If convergence is not achieved, return to S5 to generate new candidate sample points and continue iterating.

2. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S4, the Whale Optimization Algorithm (WOA) is used to optimize the relevant function parameters θ of the Kriging model. The optimization objective of WOA is to minimize the fitness function based on θ, and the optimization problem model is expressed by the following formula: In the formula, according to the basic theory of the Kriging model, φ(θ) is the fitness function of the relevant parameter θ in the Kriging model; R is the m×m correlation matrix of any two sample data points; Let be the process variance, a function of θ; n be the dimension of the sample; θ k represents the k-dimensional components of the relevant parameter vector.

3. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S7, if the coefficient of variation of the failure probability estimate is lower than the preset threshold, the Kriging model is determined to be converged.

4. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 3, characterized in that: In S7, the formula for calculating the failure probability is: The formula for calculating the coefficient of variation is: In the formula, I F (x) is the failure domain indicator function, which takes a value of 1 when the sample point is located in the failure domain, and 0 otherwise; β (x) is the truncation indicator function; φ n (x) is the original probability density function. It is the importance sampling density function.

5. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S5, the whale optimization algorithm is used to evaluate the most likely failure point (MPP) within the failure domain.

6. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S5, when truncation is performed in the region near the limit state, the resulting truncation region is defined as follows: the interior of the β-sphere contains part or all of the safe region, and sample points inside the β-sphere are considered completely safe and do not require evaluation by calling the limit state function; other regions outside the β-sphere include all failure regions and part of the safe region, and sample points outside the β-sphere may be located in the failure domain or the safe domain, and their state needs to be further determined by the function.

7. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S5, during the formation of the truncated importance sampling (TIS) density function, the objective function of the MPP search is: And use the following formula as a constraint: In the formula, u is a vector of random variables in the standard normal space; and These are random variables x i The mean and standard deviation; The limit state function constructed for the Kriging model.

8. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S6, the learning function is the U-learning function; the definition of the U-learning function is: In the formula, U(x) i () represents the value of the learning function; These are the mean and standard deviation of the response values ​​predicted by the Kriging model, respectively. Add new samples according to the following formula: In the formula, S is N TIS A candidate sample set consisting of 10 candidate sample points; The stopping criteria are:

9. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: The expression for the Kriging model is: ξ(x)=ψ(x) T β+z(x); In the formula, ξ(x) is the predicted value function; ψ(x) T β represents the regression component of the model, ψ(x) is the basic function of the model, and β is the regression coefficient; z(x) is a Gaussian random process with a mean of 0 and a covariance of cov(z(x)). (i) ),z(x (j) ))=σ 2 R(x (i) ,x (j) ,θ), where R(x (i) ,x (j) ,θ) represent any two sample points The spatial correlation function, where θ represents the corresponding correlation parameter vector, and the superscript S represents the dimension.

10. The structural reliability analysis method based on the Kriging model and truncated importance sampling as described in claim 1, characterized in that: In S3, multidimensional random samples are generated using Latin hypercube sampling.

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