Engineering structure reliability analysis method based on extended kriging surrogate model

Abstract

This invention discloses a method for reliability analysis of engineering structures based on an extended Kriging surrogate model, comprising: using prior random variables obtained from a prior dataset of the object to be analyzed and prior probability distribution parameters that change accordingly as prior extended input variables; constructing a prior initial Kriging surrogate model for the object to be analyzed based on a prior initial experimental design set; expanding the prior initial experimental design set by obtaining the best sample from a prior candidate sample set and the corresponding true response value; updating the initial Kriging surrogate model based on the expanded prior experimental design set to obtain the prior failure probability of the object to be analyzed; and calculating the prior reliability index of the object to be analyzed based on the prior failure probability. This embodiment provides a method for reliability analysis of engineering structures based on an extended Kriging surrogate model, applicable to scenarios where probability distribution parameters change.

Description

Technical Field

[0001] This invention relates to the field of reliability analysis technology, and specifically to a reliability analysis method for engineering structures based on an extended Kriging surrogate model. Background Technology

[0002] Due to the non-homogeneity of materials, the complexity of the environment, and the randomness of loads, engineering structures face numerous uncertainties during design, construction, and use. These uncertainties are transmitted to the structural response, thereby affecting structural performance. However, with the development of reliability technology, structural performance can be rationally analyzed and evaluated. Reliability analysis methods can be mainly divided into four categories: methods based on the most probable failure point, sampling-based methods, statistical moment-based methods, and surrogate model-based methods.

[0003] Methods based on the most likely failure point, such as first-order and second-order reliability methods, approximate the solution by performing linear or quadratic Taylor expansions of the function at the most likely failure point. These methods are computationally efficient, but their accuracy is difficult to guarantee for highly nonlinear function problems. Monte Carlo simulation is a typical sampling-based method that directly generates samples based on the distribution of random variables. This method is simple, accurate, and robust, and is often used as a reference solution to verify the effectiveness of other methods. However, its drawback is its high computational cost, which can be prohibitive in some complex engineering problems. To reduce computational costs, some improved methods have been proposed, such as importance sampling, subset simulation, and line sampling. Although these methods reduce computational load to some extent, they are still insufficient for handling complex problems.

[0004] To address these issues, researchers have proposed various methods based on statistical moments, such as Pearson and Johnson systems, generalized λ-distributions, and maximum entropy methods. The core idea of ​​these methods is to reconstruct the probability density function based on the statistical moments of the function of performance, thus avoiding the need for derivatives and the most likely failure point. However, the calculation of statistical moments typically involves multidimensional integrals that cannot be directly solved, and the accuracy of calculating higher-order moments remains problematic for highly nonlinear function of performance. To achieve a good balance between efficiency and accuracy, surrogate model-based methods have received widespread attention in the field of reliability analysis. The computationally expensive finite element model can be replaced by a computationally inexpensive "black box" surrogate model. Currently, various surrogate models have been developed for structural reliability analysis, including response surface methodology, polynomial chaotic expansion, support vector machines, radial basis functions, neural networks, and Kriging models. Among these, the Kriging model is the most popular due to its unique advantage of simultaneously providing predicted values ​​and predicted variances of the response.

[0005] In the field of reliability research, structural parameters often exhibit uncertainty and are typically modeled using random variables. The distribution type (e.g., normal or log-normal) and parameters (e.g., mean and standard deviation) of these random variables are determined through statistical analysis of measurement data obtained from field or laboratory tests. For specific data, the distribution type and related parameters of the random variables can be determined. Subsequently, the aforementioned reliability analysis methods can be used to analyze the structural reliability. However, in the initial stages, due to time or cost constraints, the amount of data is often severely insufficient. As subsequent data is expanded and updated, the distribution parameters of the random variables will change. This parameter change alters the probability distribution of the function, thus affecting reliability. Therefore, once the distribution parameters change, reliability needs to be reassessed. This leads to underutilization of the previous stage of reliability analysis, resulting in wasted resources and computational redundancy. Summary of the Invention

[0006] In view of this, embodiments of the present invention provide a reliability analysis method for engineering structures based on an extended Kriging surrogate model, in order to solve the problem that when using the Kriging model for reliability analysis in the prior art, once the distributed parameters change, the reliability needs to be reassessed, resulting in insufficient utilization of the reliability analysis in the previous stage, causing resource waste and computational redundancy.

[0007] This invention provides a method for reliability analysis of engineering structures based on an extended Kriging surrogate model, comprising:

[0008] Obtain the prior dataset of the object to be analyzed; the object to be analyzed is a civil or mechanical engineering structure.

[0009] The prior random variables obtained based on the prior dataset and the prior probability distribution parameters that change accordingly with the prior random variables are used as prior extended input variables;

[0010] Sampling is performed on the prior extended input variables to obtain the prior initial experimental design set and the prior candidate sample set, respectively;

[0011] Construct a prior initial kriging proxy model about the object to be analyzed based on a prior initial experimental design set;

[0012] The prior initial experimental design set is expanded by obtaining the best sample and the corresponding true response value from the prior candidate sample set, and the expansion is stopped after the iteration termination condition is met.

[0013] The initial Kriging surrogate model is updated based on the expanded prior experimental design set to obtain the prior failure probability of the object to be analyzed calculated by the Kriging surrogate model under the prior probability distribution parameters.

[0014] Based on the prior failure probability, the prior reliability index of the object to be analyzed is calculated.

[0015] Optionally, when the probability distribution parameters of the object to be analyzed change, the method further includes:

[0016] Obtain the posterior dataset of the object to be analyzed;

[0017] The prior experimental design set is used as the posterior initial experimental design set. Based on the posterior dataset, the posterior Kriging surrogate model is obtained, and the posterior failure probability of the object to be analyzed is calculated by the Kriging surrogate model under the posterior probability distribution parameters.

[0018] The posterior reliability index of the object to be analyzed is obtained based on the posterior failure probability calculation.

[0019] Optionally, the prior experimental design set is used as the posterior initial experimental design set. Based on the posterior dataset, a posterior Kriging surrogate model is obtained, and the failure probability under the posterior probability distribution parameters is calculated, including:

[0020] Based on the posterior dataset, obtain the posterior random variable and the posterior probability distribution parameters corresponding to the posterior random variable as the posterior extended input variable;

[0021] Sampling is performed on the posterior extended input variables to obtain a posterior candidate sample set;

[0022] Construct a posterior initial Kriging surrogate model based on a posterior initial experimental design set;

[0023] The posterior initial experimental design set is expanded by obtaining the best sample from the posterior candidate sample set and the corresponding true response value, and the iteration stops when the termination condition is met.

[0024] The posterior kriging surrogate model is updated based on the expanded posterior experimental design set, and the failure probability under the posterior probability distribution parameters is calculated.

[0025] Optionally, the prior extended input variables are sampled to obtain the prior initial experimental design set and the prior candidate sample set, including:

[0026] Based on the probability distribution type of the prior extended input variables, the Latin hypercube sampling method is used to generate several prior initial samples and calculate the corresponding true response values.

[0027] Several prior initial samples and their corresponding real response values ​​are used as the prior initial experimental design set.

[0028] Optionally, sampling the prior extended input variables to obtain the prior initial experimental design set and the prior candidate sample set, further includes:

[0029] Based on the probability distribution type of the prior extended input variables, Monte Carlo random sampling is used to generate several prior random samples as a prior candidate sample set.

[0030] The number of prior initial samples is much smaller than the number of prior random samples.

[0031] Optionally, the prior initial experimental design set is expanded by obtaining the best sample from the prior candidate sample set and the corresponding true response value, and the iteration stops when the termination condition is met, including:

[0032] The best sample is obtained from the prior candidate sample set by learning the U function;

[0033] The criteria for selecting the best sample include:

[0034] ;

[0035] ;

[0036] in, Let N be the prior candidate sample set, and N be the number of samples. The learning function U(*) is used to evaluate the uncertainty of the samples, and the sample corresponding to the minimum value of U(*) is the best sample. This represents the predicted response value obtained using the Kriging surrogate model; and These represent the mean and standard deviation of the predicted response value, respectively.

[0037] Optionally, the iteration termination condition is:

[0038] Prior candidate sample set satisfies .

[0039] Optionally, constructing a prior initial Kriging surrogate model based on a prior initial experimental design set includes:

[0040] Generate M sets of prior initial samples Calculate its corresponding true response value ;

[0041] Define the prior initial experimental design set as The function is called M times in the current step.

[0042] Optionally, it also includes:

[0043] For each additional set of optimal samples and corresponding true response values ​​added to the initial experimental design set, the number of function calls in the current step increases by 1.

[0044] Optionally, the total number of function calls under the prior probability distribution parameters is N. call=M+M0; The total number of function calls under the posterior probability distribution parameters is N. call =M+M0+M1; where M is the number of prior initial samples, M0 is the number of samples added to update the prior initial Kriging surrogate model, and M1 is the number of samples added to update the posterior initial Kriging surrogate model.

[0045] The beneficial effects of this invention are:

[0046] 1. This invention provides a reliability analysis method for engineering structures based on an extended Kriging surrogate model, applicable to scenarios where probability distribution parameters change. Prior random variables and their corresponding changing prior probability distribution parameters are used as prior extended input variables. Based on this, a Kriging surrogate model is constructed and updated to derive the prior failure probability.

[0047] The final Kriging surrogate model under the prior parameters can be directly used as the initial surrogate model under the posterior parameters. There is no need to rebuild the initial Kriging model under the change of probability distribution parameters, which reduces the number of initial samples required to build the initial surrogate model under the posterior parameters and can significantly improve computational efficiency.

[0048] 2. The final experimental design set under the prior parameters is used for analysis and calculation under the posterior parameters, making full use of the data resources under the prior parameters. This makes the samples in the experimental design set under the posterior parameters more diverse, which can effectively reduce the number of samples required to update the surrogate model under the posterior parameters, thereby further improving the computational efficiency.

[0049] 3. This embodiment provides a method for engineering structure reliability analysis based on an extended Kriging surrogate model. The extended input variables not only include random variables but also cover changing probability distribution parameters. Before and after parameter changes, only one Kriging surrogate model needs to be constructed and updated, making it simpler and more efficient to implement than traditional methods. Attached Figure Description

[0050] The features and advantages of the invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the invention in any way. In the drawings:

[0051] Figure 1 A flowchart of an engineering structure reliability analysis method based on an extended Kriging proxy model is shown in an embodiment of the present invention.

[0052] Figure 2 This illustration shows the impact of changes in probability distribution parameters on reliability analysis of an engineering structure reliability analysis method based on an extended Kriging surrogate model in an embodiment of the present invention.

[0053] Figure 3 A comparison diagram of an extended Kriging proxy model and the original Kriging proxy model in an embodiment of the present invention is shown;

[0054] Figure 4 This illustration shows a step diagram of a method for reliability analysis of engineering structures based on an extended Kriging surrogate model under varying probability distribution parameters, according to an embodiment of the present invention.

[0055] Figure 5 This figure shows a comparison of the prior and posterior probability density functions of a random variable X1 in an embodiment of the present invention;

[0056] Figure 6 This diagram illustrates the sample addition process based on an extended Kriging proxy model under prior parameters in an embodiment of the present invention.

[0057] Figure 7 This illustration shows a schematic diagram of the sample addition process based on an extended Kriging proxy model under posterior parameters in an embodiment of the present invention;

[0058] Figure 8 This diagram illustrates the sample addition process based on the original Kriging proxy model under prior parameters in an embodiment of the present invention.

[0059] Figure 9 This illustration shows a process for adding samples based on the original Kriging proxy model under posterior parameters in an embodiment of the present invention.

[0060] Figure 10 The diagram shows the overall convergence curves of the calculated failure probability using the Monte Carlo simulation method, the original Kriging surrogate model, and the extended Kriging surrogate model.

[0061] Figure 11 The image shows a magnified portion of the convergence curves for calculating the failure probability of an engineering structure using the Monte Carlo simulation method, the original Kriging surrogate model, and the extended Kriging surrogate model.

[0062] Figure 12 A schematic diagram of a roof truss structure according to an embodiment of the present invention is shown;

[0063] Figure 13 The convergence curves of calculating the failure probability of the roof truss structure using the Monte Carlo simulation method, the original Kriging surrogate model, and the extended Kriging surrogate model are shown. Detailed Implementation

[0064] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0065] like Figure 1 As shown, this embodiment of the invention provides a method for reliability analysis of engineering structures based on an extended Kriging surrogate model, including:

[0066] Step S10: Obtain the prior dataset of the object to be analyzed.

[0067] In this embodiment, a priori dataset is obtained based on field measured data or experimental data. The priori dataset is used to determine the probability distribution type and distribution parameters of the random variable.

[0068] Step S20: The prior random variables obtained based on the prior dataset and the prior probability distribution parameters that change accordingly with the prior random variables are used as prior extended input variables.

[0069] In this embodiment, the random variable is determined based on the prior dataset S in step S10. The type of prior probability distribution and the parameters of the prior probability distribution. Besides the random variable X, what are the probability distribution parameters that will change? This also includes the extended input variable U=[X, P], which has a dimension of n. u =n x +n p n x and n p Let X and P be the dimensions of X and P, respectively.

[0070] Step S30: Sample the prior extended input variables to obtain the prior initial experimental design set and the prior candidate sample set.

[0071] In this embodiment, based on the probability distribution type of the prior extended input variable U, the Latin hypercube sampling method is used to generate M sets of prior initial samples. Calculate its corresponding true response value Define the prior initial experimental design set as follows: The subscript pr represents data related to the prior probability distribution.

[0072] Based on the probability distribution type of the prior extended input variables, Monte Carlo random sampling is used to generate several random samples as a prior candidate sample set. N is the number of samples.

[0073] The number of prior initial samples M is much smaller than the number of random samples N. In a specific embodiment, the number of prior initial samples is 10 to 15, and the number of random samples N is 10. 5 ~10 6 indivual.

[0074] Step S40: Construct a prior initial Kriging surrogate model based on the prior initial experimental design set.

[0075] In this embodiment, based on the M sets of prior initial samples generated by the Latin hypercube sampling method in step S30, the number of function calls N in the current step of the prior initial Kriging surrogate model is... call =M.

[0076] Step S50: Expand the prior initial experimental design set by obtaining the best sample from the prior candidate sample set and the corresponding true response value, and stop after the iteration termination condition is met.

[0077] The best sample is obtained from the prior candidate sample set by learning a function;

[0078] The criteria for selecting the best sample include:

[0079] ;

[0080] ;

[0081] in, Let N be the prior candidate sample set, and N be the number of samples. The learning function U(*) is used to evaluate the uncertainty of the samples, and the sample corresponding to the minimum value of U(*) is the best sample. This represents the predicted response value obtained using the Kriging proxy model. and These represent the mean and standard deviation of the predicted response value, respectively.

[0082] When the value of U(u) is very small, it means Approaching the limiting state G=0, or The value of U(u) is very large, meaning that the uncertainty of sample u is very high. Therefore, the sample u corresponding to the minimum value of U(u) is the optimal sample u*.

[0083] Calculate the true response value G*=G(u*) corresponding to the optimal sample u*, expand the experimental design set D using {u*, G*}, and the number of function calls in the current step is N. call =N call +1. Before proceeding to the next iteration, remove the current best sample from the prior candidate sample set. At this point, the number of samples in the prior candidate sample set is N = N - 1.

[0084] If for the prior candidate sample set , If it holds, the algorithm ends, obtaining the final Kriging surrogate model, and calculating the failure probability. Otherwise, continue to expand the experimental design set D.

[0085] Step S60: Update the initial Kriging surrogate model based on the expanded prior experimental design set, and calculate the prior failure probability under the prior probability distribution parameters.

[0086] In this embodiment, the formula for calculating the failure probability is:

[0087] ;

[0088] ;

[0089] where N is the capacity of the candidate samples, represents the predicted response value of the i-th candidate sample u i obtained using the Kriging model.

[0090] Step S70: Obtain the prior reliability index of the object to be analyzed based on the prior failure probability.

[0091] In this embodiment, the relationship between the reliability index and the failure probability is:

[0092] ; <​​​​​​​​​​​​​​​​​​​In this embodiment, the posterior dataset is obtained by adding measured or experimental data to the prior dataset. Compared with the prior dataset, the data has changed, and the distribution parameters of the random variables have also changed accordingly.

[0099] Step S90: Use the prior experimental design set as the posterior initial experimental design set, obtain the posterior Kriging surrogate model based on the posterior dataset, and obtain the posterior failure probability of the object to be analyzed calculated by the Kriging surrogate model under the posterior probability distribution parameters.

[0100] Step S100: Calculate the posterior reliability index of the object to be analyzed based on the posterior failure probability.

[0101] For the posterior probability distribution parameters, the following steps are included:

[0102] Step S91: Obtain the posterior random variable and the posterior probability distribution parameters corresponding to the posterior random variable as posterior extended input variables based on the posterior dataset.

[0103] In this embodiment, based on the expanded posterior dataset, the posterior probability distribution type and posterior probability distribution parameters of the random variable are redefined.

[0104] Step S92: Sample the posterior extended input variables to obtain the posterior candidate sample set.

[0105] In this embodiment, based on the probability distribution type of the posterior extended input variable, a number of random samples are generated as candidate samples using the Monte Carlo random sampling method.

[0106] Several random samples are generated using the Monte Carlo random sampling method as the posterior candidate sample set. N represents the sample size. In a specific embodiment, the number of posterior candidate samples is the same as the number of prior candidate samples in the prior distribution parameter section. In a specific embodiment, the number of prior candidate samples and the number of posterior candidate samples are set according to actual needs. The subscript po represents data related to the posterior probability distribution.

[0107] Step S93: Construct a posterior initial Kriging surrogate model based on the posterior initial experimental design set.

[0108] In this embodiment, the final experimental design set D under the prior parameters can be directly used as the initial experimental design set D under the posterior parameters, without the need to redefine the initial experimental design set D. The number of function calls in the current step is N. call =M+M0.

[0109] Step S94: Expand the initial posterior experimental design set by obtaining the best sample from the posterior candidate sample set and the corresponding true response value, and stop the iteration when the termination condition is met.

[0110] In this embodiment, the specific steps are as described in step S50.

[0111] Step S95: Based on the expanded posterior experimental design set, update the posterior Kriging surrogate model and calculate the failure probability under the posterior probability distribution parameters.

[0112] In this embodiment, the formula for calculating the failure probability is the same as in step S60. The difference lies in the total number of function calls under the posterior probability distribution parameter being N. call =M+M0+M1, where M1 is the number of samples added to the Kriging model to update the posterior parameters.

[0113] This embodiment provides a reliability analysis method based on the extended Kriging surrogate model, which is applied to scenarios where probability distribution parameters change. The final Kriging surrogate model under the prior parameters can be directly used as the initial surrogate model under the posterior parameters, without the need to rebuild the initial Kriging model under the change of probability distribution parameters. This reduces the number of initial samples required to build the initial surrogate model under the posterior parameters and can significantly improve computational efficiency.

[0114] The final experimental design set under the prior parameters is used for analysis and calculation under the posterior parameters, making full use of the data resources under the prior parameters. This makes the samples in the experimental design set under the posterior parameters more diverse, which can effectively reduce the number of samples required to update the surrogate model under the posterior parameters, thereby further improving computational efficiency.

[0115] This embodiment provides a reliability analysis method based on an extended Kriging surrogate model. The extended input variables not only include random variables but also cover changing probability distribution parameters. Before and after parameter changes, only one Kriging surrogate model needs to be built and updated, making it simpler and more efficient to implement than traditional methods.

[0116] As the dataset expands, the distribution parameters of the random variables will change. These parameter changes will alter the probability distribution of the function, thus affecting reliability. Figure 2 As shown. To address the above situation, this embodiment proposes a reliability analysis method based on an extended Kriging surrogate model under varying probability distribution parameters. This method uses random variables and varying probability distribution parameters as extended input variables to construct an extended Kriging surrogate model, as shown... Figure 3 As shown.

[0117] When parameters change, a reliability analysis method based on an extended Kriging proxy model, as provided in this embodiment, is used for analysis. The specific steps are as follows: Figure 4As shown, the final surrogate model under the prior parameters can be directly used as the initial surrogate model under the posterior parameters. In this embodiment, there is no need to reconstruct the Kriging model under the change of probability distribution parameters, which can significantly improve computational efficiency.

[0118] Example 1: Four-branch series system.

[0119] The function of a four-branch series system is defined as follows:

[0120] .

[0121] in, They are normally distributed random variables, and Table 1 shows the corresponding probability distributions.

[0122] Table 1. Probability distribution of random variables in a series system

[0123] ;

[0124] It can be seen that the mean and standard deviation of the random variable X1 are the probability distribution parameters P=[μ, Then the extended input variables can be represented as U=[X, P]=[X1, X2, μ, ... Its dimension is n u =n x +n p =2+2=4. As the dataset expands, the distribution parameters of the random variable will change, ultimately leading to a change in the probability distribution.

[0125] Table 2. Estimation results of probability distribution parameters

[0126] ;

[0127] Based on datasets of different sizes, the maximum likelihood estimation method was used to determine the values ​​of the distribution parameters, and the estimation results are shown in Table 2.

[0128] Different dataset sizes lead to different distribution parameters, ultimately resulting in different probability distributions. The prior and posterior probability density functions of a random variable X1 are as follows: Figure 5 As shown in Table 2 and Figure 5 It can be seen that the prior parameter P=[0.1231, 1.3377] is estimated based on 100 data points. As the dataset expands to 200, the posterior parameter becomes P=[0.0252, 1.1784].

[0129] To address the reliability calculation problem under this parameter variation, a comparative analysis is conducted on the Monte Carlo simulation method, the method based on the original Kriging surrogate model, and the method based on the extended Kriging surrogate model presented in this embodiment. The results of the different methods are compared, for example... Figures 5-11 As shown in Table 3.

[0130] Table 3 Failure probability results for different methods

[0131] ;

[0132] exist Figures 6 to 9 In the diagram, black triangles represent the initial samples, while red squares and blue circles represent additional samples in the reliability analysis method based on the extended Kriging surrogate model and the method based on the original Kriging surrogate model, respectively. The solid black line represents the true function G(X) = 0, and the dashed red and blue lines represent the predicted functions of the proposed method and the method based on the original Kriging surrogate model, respectively. =0.

[0133] Monte Carlo simulation is often used as a benchmark method due to its high accuracy and robustness. First, N=10 prior parameters for X are generated. 6 A random sample is generated, and the corresponding true response value is calculated. Then, the proportion of failed samples to the total number of samples is counted, which is the failure probability. If the distribution parameters of the random variable change, the samples in the Monte Carlo simulation method need to be regenerated, and the reliability needs to be reassessed. For the posterior parameters, an N=10 ... 6 A set of random samples is used, and then the failure probability is calculated. Therefore, the total number of function calls for this method under varying probability distribution parameters is N. call =N+N=2×10 6 .

[0134] The method based on the original Kriging surrogate model first generates M=12 initial samples of X to construct an initial surrogate model under the prior parameters. Then, it updates the Kriging model using an additional N0=82 samples. Afterward, the failure probability is calculated using the updated Kriging model. Once the distribution parameters of the random variable change, all samples need to be regenerated, the Kriging model needs to be rebuilt, and the reliability needs to be reassessed. For the posterior parameters, the number of samples required to construct and update the Kriging model are M=12 and N1=50, respectively. Therefore, the total number of function calls for this method under changing probability distribution parameters is N. call =(M+N0)+(M+N1)=12+82+12+50=156.

[0135] In the reliability analysis method provided in this embodiment, in addition to the random variable X=[X1, X2], the probability distribution parameters of the change... , They are also included, together forming the extended input variable U=[X, P]=[X1, X2, , Its dimension is n u =n x +n p =2+2=4.

[0136] First, M=12 initial samples of U are generated to construct an initial surrogate model under the prior parameters. Then, an additional M0=54 samples are used to update the Kriging model. Afterward, the failure probability is calculated using the final Kriging surrogate model. Unlike Monte Carlo simulation methods and methods based on the original Kriging surrogate model, the method proposed in this embodiment does not require regenerating samples or rebuilding the Kriging model. This method only needs to construct one Kriging model, while methods based on the original Kriging surrogate model require constructing two Kriging models. Therefore, for the posterior parameters, the number of samples required to update the Kriging model is M1=44. Finally, the total number of function calls under the change in probability distribution parameters is N. call =M+M0+M1=12+54+44=110.

[0137] from Figure 10 and Figure 11 As can be seen, the engineering structure reliability analysis method based on the extended Kriging surrogate model provided in this embodiment performs poorly in the initial stage when directly applying the final surrogate model under prior parameters to the reliability assessment under posterior parameters, resulting in a low failure probability value P. f =1 shows a significant bias. However, after updating the Kriging proxy model with a small number of sample points, the model's accuracy improved rapidly.

[0138] Using the Monte Carlo simulation results as a benchmark, Table 3 shows that both the method based on the original Kriging proxy model and the method proposed in this embodiment exhibit high computational accuracy, with errors less than 5% in all cases. However, the method based on the extended Kriging proxy model proposed in this embodiment has limitations in the number of function calls N. call This demonstrates higher computational efficiency. Specifically, the computational cost of this embodiment is significantly lower than that of the method based on the original Kriging proxy model, being only about two-thirds of it.

[0139] Example 2: Roof truss structure.

[0140] like Figure 12 The roof truss structure shown has a concrete upper chord and compression members, while the steel lower chord and tension members are made of steel. The structure is subjected to a uniformly distributed vertical load q, which can be converted into three nodal loads, each with a magnitude of P = ql / 4.

[0141] The function is expressed as follows:

[0142] ;

[0143] Table 4. Probability distribution of random variables in truss structures

[0144] ;

[0145] in, This is the allowable displacement of the roof truss. This is the maximum vertical displacement. X=[q, l, A] C E C A S E S ] is a normally distributed random variable, and its specific probability distribution is shown in Table 4. It can be seen that the mean and standard deviation of random variables q and l are the probability distribution parameters of their variation. , , , Then the extended input variables can be represented as U=[X, P]=[q, l, A] C E C A S E S , , , , Its dimension is n u =n x +n p =6+4=10. Assume the changes in the distribution parameters are as shown in Table 5.

[0146] Table 5. Changes in probability distribution parameters

[0147] ;

[0148] As shown in Table 5, the prior parameter is P=[30000, 2000, 10, 0.10], and the posterior parameter becomes P=[20000, 1400, 12, 0.12]. Reliability is evaluated using Monte Carlo simulation, a method based on the original Kriging surrogate model, and the method based on the extended Kriging surrogate model provided in this embodiment. In the Monte Carlo simulation method, reliability calculations under both prior and posterior parameters require generating N=10... 6 For a random sample of group X, the total number of function calls under varying probability distribution parameters is N. call =N+N=2×10 6 Failure probability P under prior parameters f =2.321×10 -2 The failure probability P under the posterior parameterf =9.343×10 -3 It serves as a benchmark for results obtained by other methods. The results of these methods are as follows: Figure 13 As shown in Table 6.

[0149] Table 6 Failure probability results for different methods

[0150] ;

[0151] Based on the probability distribution type of X, the method based on the original Kriging surrogate model first uses M=12 initial samples and N0=292 additional samples to construct and update the Kriging surrogate model under the prior parameters. Then, it uses M=12 samples and N1=227 samples to construct and update another Kriging surrogate model under the posterior parameters. Therefore, the total number of function calls for this method under varying probability distribution parameters is N. call =(M+N0)+(M+N1)=12+292+12+227=543.

[0152] Unlike the two methods mentioned above, the reliability analysis method provided in this embodiment expands the input variables to U=[X, P]=[q, l, A]. C E C A S E S , , , , Its dimension is n u =n x +n p =6+4=10. The reliability analysis method proposed in this embodiment only requires the construction of one Kriging surrogate model. Specifically, based on the probability distribution type of U, the Kriging surrogate model under the prior parameters is first constructed and updated using M=12 initial samples and M0=198 additional samples. Subsequently, the final surrogate model under the prior parameters is directly used as the initial surrogate model under the posterior parameters, and updated using an additional M1=176 additional samples. Finally, the total number of function calls under the change of probability distribution parameters is N. call =M+M0+M1=12+198+176 =386.

[0153] Depend on Figure 13 As shown in Table 6, the engineering structure reliability analysis method based on the extended Kriging proxy model proposed in this embodiment has similar reliability calculation accuracy to other methods, but with the highest computational efficiency. Compared with the method based on the original Kriging proxy model, the method provided in this embodiment significantly reduces the number of function calls, to only two-thirds of the former.

[0154] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A reliability analysis method for engineering structures based on an extended Kriging surrogate model, characterized in that, include: Obtain the prior dataset of the object to be analyzed; the object to be analyzed is a civil or mechanical engineering structure. The prior random variables obtained based on the prior dataset and the prior probability distribution parameters that change corresponding to the prior random variables are used as prior extended input variables; The prior extended input variables are sampled to obtain the prior initial experimental design set and the prior candidate sample set, respectively; Construct a prior initial Kriging proxy model for the object to be analyzed based on the prior initial experimental design set; The prior initial experimental design set is expanded by obtaining the best sample and the corresponding true response value from the prior candidate sample set, and the expansion stops when the iteration termination condition is met. The initial Kriging surrogate model is updated based on the expanded prior experimental design set to obtain the prior failure probability of the object to be analyzed calculated by the Kriging surrogate model under the prior probability distribution parameters. Based on the prior failure probability, the prior reliability index of the object to be analyzed is calculated.

2. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 1, characterized in that, When the probability distribution parameters of the object to be analyzed change, the method further includes: Obtain the posterior dataset of the object to be analyzed; The prior experimental design set is used as the posterior initial experimental design set. Based on the posterior dataset, a posterior Kriging surrogate model is obtained, and the posterior failure probability of the object to be analyzed is calculated by the Kriging surrogate model under the posterior probability distribution parameters. The posterior reliability index of the object to be analyzed is calculated based on the posterior failure probability.

3. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 2, characterized in that, Using the prior experimental design set as the posterior initial experimental design set, a posterior Kriging surrogate model is obtained based on the posterior dataset. The failure probability under the posterior probability distribution parameters is calculated, including: Based on the posterior dataset, obtain posterior random variables and posterior probability distribution parameters corresponding to the posterior random variables as posterior extended input variables; The posterior extended input variables are sampled to obtain a posterior candidate sample set; Construct a posterior initial Kriging proxy model based on the aforementioned posterior initial experimental design set; The posterior initial experimental design set is expanded by obtaining the best sample and the corresponding true response value from the posterior candidate sample set, and the iteration stops when the termination condition is met. The posterior kriging surrogate model is updated based on the expanded posterior experimental design set, and the failure probability under the posterior probability distribution parameters is calculated.

4. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 3, characterized in that, The prior extended input variables are sampled to obtain the prior initial experimental design set and the prior candidate sample set, respectively, including: Based on the probability distribution type of the prior extended input variables, several prior initial samples are generated using the Latin hypercube sampling method, and the corresponding true response values ​​are calculated. The prior initial samples and their corresponding real response values ​​are used as the prior initial experimental design set.

5. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 4, characterized in that, Sampling the prior extended input variables to obtain the prior initial experimental design set and the prior candidate sample set, respectively, also includes: Based on the probability distribution type of the prior extended input variables, a number of prior random samples are generated using the Monte Carlo random sampling method as the prior candidate sample set. The number of prior initial samples is much smaller than the number of prior random samples.

6. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 5, characterized in that, The prior initial experimental design set is expanded by obtaining the best sample and the corresponding true response value from the prior candidate sample set, and the iteration stops when the termination condition is met, including: pass U The learning function obtains the best sample from the prior candidate sample set; The criteria for selecting the best sample include: ； ； in, The prior candidate sample set. N Number of samples; learning function U (*) is used to assess the uncertainty of a sample. U The sample corresponding to the minimum value of (*) is the optimal sample; This represents the predicted response value obtained using the Kriging surrogate model; and These represent the mean and standard deviation of the predicted response value, respectively.

7. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 6, characterized in that, The iteration termination condition is: The prior candidate sample set satisfies .

8. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 4, characterized in that, Constructing a prior initial Kriging proxy model based on the aforementioned prior initial experimental design set includes: generate M The prior initial samples described in the group Calculate its corresponding true response value ; The prior initial experimental design set is defined as follows: The number of function calls in the current step is M Second-rate.

9. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 1, characterized in that, Also includes: For each additional set of optimal samples and corresponding true response values ​​added to the prior initial experimental design set, the number of function calls in the current step increases by 1.

10. The engineering structure reliability analysis method based on the extended Kriging surrogate model according to claim 4, characterized in that, The total number of function calls under the prior probability distribution parameters is N call = M + M 0; The total number of function calls under the posterior probability distribution parameters is N call = M + M 0+ M 1; in, M The number of the prior initial samples. M 0 is the number of samples added to update the prior initial Kriging proxy model; M 1 is the number of samples added to update the posterior initial Kriging proxy model.

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