Adaptive structure reliability analysis method based on Co-kriging proxy model

Abstract

The invention relates to the technical field of structural reliability analysis, in particular to an adaptive structural reliability analysis method based on a Co-kriging proxy model, which comprises the following steps: determining a structural failure mode, acquiring corresponding high-fidelity and low-fidelity performance functions and inputting distribution information of variables; a high-fidelity sample set and a low-fidelity sample set are obtained through sampling; obtaining responses corresponding to the two fidelity samples, and constructing a Co-kriging agent model; carrying out structural reliability analysis by utilizing a Monte Carlo simulation method, and calculating a failure probability of each iteration; judging whether a convergence condition is met; adding new sample points into the training set by using a learning function, constructing a Co-kriging proxy model, and obtaining a final failure probability; according to the method, the Co-kriging proxy model is combined with the learning function MPO (x, m), multi-precision sample data can be fused, and on the premise that it is ensured that the failure probability estimation result has high precision, the calculation process is optimized, unnecessary calculation resource consumption is effectively reduced, and the overall calculation cost is remarkably reduced.

Description

Technical Field

[0001] The present invention relates to the technical field of structural reliability analysis, and in particular to an adaptive structural reliability analysis method based on a Co-kriging proxy model. Background Art

[0002] The Monte Carlo method is widely used in engineering design, optimization, and uncertainty analysis due to its simplicity and universal applicability. While it can provide highly accurate solutions to structural failure probabilities, convergence is often required for low-probability events, often requiring millions or even more simulation runs. This computational cost is prohibitive for computationally intensive models like computational fluid dynamics and finite element analysis, exceeding acceptable engineering costs. This limitation is particularly exacerbated in high-dimensional uncertainty analysis scenarios by the "curse of dimensionality," severely limiting the engineering applicability of traditional Monte Carlo methods.

[0003] To address the challenges of data collection, time-consuming optimization, and the inherent difficulty of data collection during optimization, surrogate model methods have emerged and are becoming a key branch of multidisciplinary optimization and a key technology. Surrogate model methods are used to replace the true model in costly or complex computations. By fitting existing data, a relatively simple model is constructed to approximate the true computational model, significantly reducing computational costs, time, and resource consumption. Common surrogate models include polynomial response surface models (RSMs), Kriging models, and radial basis functions (RBFs).

[0004] The Kriging surrogate model, based on a Gaussian process, not only predicts output values but also derives the prediction variance of sample points, thereby reflecting the uncertainty of the predicted values. This characteristic makes it naturally suited for the active learning framework used in reliability analysis. By constructing learning functions (such as the EI function in Efficient Global Optimization and the U learning function in AK-MCS), the candidate sample points that contribute most to the failure probability are dynamically identified, enabling adaptive updates of the surrogate model.

[0005] The co-kriging surrogate model is a method based on the Kriging surrogate model. This method can integrate multiple precision models and jointly model high-precision model data with low-precision model data by constructing a multi-level covariance function. Its core is to use low-precision models to capture global trends and correct local deviations with high-precision data, thereby significantly reducing the number of calls to high-cost, high-precision models while maintaining accuracy. The co-kriging surrogate model method has greater advantages than the single-precision kriging model when the system involves simulation models of different precision levels (such as a high-precision 3D finite element model and a simplified 2D analytical model) or multi-source heterogeneous data (experimental data, simulation data, expert experience, etc.).

[0006] Like the traditional Kriging method, the Co-kriging method also provides a measure of predictive uncertainty. When performing adaptive structural reliability analysis using the Co-kriging model, new training points are obtained through a learning function to update the sample set, thereby constructing a new Co-kriging model and continuously improving the accuracy of the resulting surrogate model. In this process, a suitable learning function plays a crucial role in achieving efficient and high-precision estimation of structural failure probabilities. New sample points obtained based on the learning function significantly impact the efficiency and computational accuracy of the Co-kriging construction. Therefore, a novel learning function combined with the Co-kriging model is of great significance in the field of adaptive structural reliability. Summary of the Invention

[0007] In order to overcome the problems raised in the above background technology, the present invention proposes an adaptive structural reliability analysis method based on a Co-kriging surrogate model.

[0008] The technical solution of the present invention is: an adaptive structural reliability analysis method based on a Co-kriging surrogate model, comprising the following steps:

[0009] S11: Determine the failure mode of the structure and the corresponding high-fidelity function g HF (x) and the low-fidelity function g LF (x), and according to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l ;

[0010] S12: The two fidelity training sets n obtained in step S11 h and n l, obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration

[0011] S13: Use the Monte Carlo simulation method to perform structural reliability analysis on the Co-kriging surrogate model constructed in step S12 and calculate the failure probability of the kth iteration

[0012] S14: Determine whether the failure probability of the kth iteration meets the convergence condition. If so, go to step S16. If not, update the number of iterations: k=k+1, and proceed to the next step.

[0013] S15: Determine the new training point x using the MPO(x,m) learning function new , and add the new training point to the training set, and return to step S12;

[0014] S16: Finally obtain the failure probability of the structure

[0015] As a preference, according to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l Specifically including:

[0016] S21: Based on the distribution information of the structural input variable x, the Latin hypercube method is used to extract candidate points as the Monte Carlo candidate sample set n MCS ;

[0017] S22: extract N h and N l Initial sample points are used as the initial high-fidelity sample set n h and the initial low-fidelity sample set n l , let the initial number of iterations k = 1.

[0018] As a preference, according to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l When , the input variable x is expressed as:

[0019] x=(x (1) ,x (2) ,x (3) ,x (b) );

[0020] Where b represents the variable dimension.

[0021] As an example, in the two fidelity training sets n obtained in step S11, h and n l , obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration When , the predicted mean of the Co-kriging multi-fidelity surrogate model is:

[0022] μ ξ (x)=ρμl(x)+μ δ (x);

[0023] Among them, ρ is the scaling factor, which is obtained by solving the optimization problem, μ l (x) is the predicted mean of the low-fidelity kriging surrogate model, μ δ (x) is the predicted mean of the residual function.

[0024] As an example, in the two fidelity training sets n obtained in step S11, h and n l , obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration When , the residual function of the Co-kriging surrogate model is expressed as:

[0025] y δ =y h -ρy l (x h );

[0026] Among them, y h is the high-fidelity response value, y l (x h ) is x h Sample response values in the low-fidelity model.

[0027] As a preference, the structural reliability analysis of the Co-kriging proxy model constructed in step S12 is performed using the Monte Carlo simulation method, and the failure probability of the kth iteration is calculated. When Calculated using the following formula:

[0028]

[0029] Among them, n MCS represents the number of samples in the Monte Carlo sample set, and I(·) is the characteristic function, which satisfies hour, on the contrary,

[0030] Preferably, when judging whether the failure probability of the k-th iteration meets the convergence condition, the convergence condition is:

[0031]

[0032] Among them, n MCS represents the number of samples in the Monte Carlo sample set, μ ξ (x i ) and σ ξ (x i ) represent the Co-kriging proxy model The predicted mean and predicted standard deviation of the i-th sample point are obtained.

[0033] As a preference, the new training point x is determined by using the MPO(x,m) learning function. new When , the learning function MPO(x,m) is expressed as:

[0034]

[0035] Among them, μ ξ (x) and σ ξ (x) represent the predicted mean and predicted standard deviation of the Co-kriging proxy model, σ l (x) represents the predicted standard deviation of the kriging surrogate model constructed only from low-fidelity samples, P represents the cost coefficient, and P = C HF / C LF , C HF and C LF They represent the time cost of obtaining high-fidelity sample and low-fidelity sample responses respectively, m represents the accuracy level, m is 1, which means the obtained sample belongs to the low-fidelity sample set, and m is 2, which means the obtained sample belongs to the high-fidelity sample set.

[0036] As a preference, the new training point x is determined by using the MPO(x,m) learning function. new When , the prediction standard deviation of the Co-kriging proxy model is calculated by the following formula:

[0037]

[0038] Among them, σ l (x) is the standard deviation of the low-fidelity kriging surrogate model prediction, σ δ (x) is the predicted standard deviation of the residual function.

[0039] As a preference, the new training point x is determined by using the MPO(x,m) learning function. new When, through the formula:

[0040] Identify new training points.

[0041] Beneficial effects of the present invention:

[0042] 1. This paper uses a co-kriging surrogate model combined with a learning function for adaptive reliability analysis. By effectively leveraging the correlations between multiple fidelity variables, the co-kriging method reduces the reliance on complex system models in sample response acquisition. This multi-scale, multi-dimensional information fusion significantly reduces the number of high-fidelity, high-precision samples required compared to traditional kriging methods, thereby accelerating the analysis process and reducing computational costs.

[0043] 2. This paper introduces a new learning function that leverages the Co-kriging surrogate model's ability to predict uncertainty measures. By integrating the influence of the predicted mean and prediction variance into the learning function, the paper prioritizes sample points with the largest change in misjudgment error, achieving a more accurate sample point selection strategy. Furthermore, the learning function considers the cost coefficient, balancing these multiple factors to select the most appropriate new sample points for inclusion in the training set. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a flow chart of the adaptive structural reliability analysis method combining the learning function and the Co-kriging model proposed in the present invention;

[0045] Figure 2 1 is a schematic diagram comparing the final sampling results of the method of the present invention and the typical method in Example 1 of the present invention and the proxy model with the real limit state function of the structure;

[0046] Figure 3 1 is a graph showing the convergence of failure probability predictions by the method of the present invention and a typical method in Example 1 of the present invention;

[0047] Figure 4 is a schematic diagram of a nonlinear oscillation system in Example 2 of the present invention;

[0048] Figure 5 1 is a graph showing the convergence of failure probability predictions by the method of the present invention and a typical method in Example 2 of the present invention. DETAILED DESCRIPTION

[0049] The present invention will be further described below with reference to the accompanying drawings and examples.

[0050] Example 1

[0051] See also Figure 1-Figure 3In this embodiment 1, a four-branch function with four failure modes is used to further illustrate the present invention. This example is a common engineering example in the field of reliability analysis.

[0052] Step 1: Determine the failure mode of the structure and the corresponding high-fidelity function g HF (x) and the low-fidelity function g LF (x), according to the distribution information of the structural input variable x, the Latin hypercube method is used to extract N candidate points as the Monte Carlo candidate sample set n MCS , and extract N h and N l Initial sample points are used as the initial high-fidelity sample set n h and the initial low-fidelity sample set n l , the initial number of iterations k = 1;

[0053] In this embodiment, the structural high-fidelity function g HF (x) and the low-fidelity function g LF (x) is as follows:

[0054]

[0055]

[0056] Where x=(x (1) ,x (2) ),variable x (1) and x (2) All of them obey the standard normal distribution, and the Latin hypercube method is used to first extract 10 5 The candidate points form the Monte Carlo sample set n MCS , and again extract 12 candidate points as the initial low-fidelity sample set n l and 3 candidate points as the initial high-fidelity sample set n h ;

[0057] Step 2: The two fidelity training sets n obtained in step 1 h and n l , through the function g HF (x) and g LF (x) Obtain the corresponding high-fidelity response and low-fidelity response, and construct the Co-kriging agent model of the kth iteration

[0058] The predicted mean of the Co-kriging multi-fidelity surrogate model can be expressed as μ ξ (x)=ρμ l (x)+μ δ(x), ρ is the scaling factor, which is obtained by solving the optimization problem, μ l (x) is the predicted mean of the low-fidelity kriging surrogate model, μ δ (x) is the predicted mean of the residual function;

[0059] Step 3: Use the Monte Carlo simulation method to perform structural reliability analysis on the Co-kriging surrogate model constructed in step 2 and calculate the failure probability of the first iteration

[0060] Step 4: Determine whether the failure probability of the kth iteration meets the convergence condition. If so, go to step S16. If not, update the number of iterations: k = k + 1, and proceed to the next step.

[0061] The convergence condition is expressed as

[0062] Among them, n MCS represents the number of samples in the Monte Carlo sample set, and I(·) is the characteristic function, which satisfies hour, on the contrary,

[0063] Step 5: Use the MPO(x,m) learning function to determine the new training point x new , and add the new training point to the training set, and go back to step 2;

[0064] The learning function is expressed as:

[0065]

[0066] Among them, μ ξ (x) and σ ξ (x) represent the predicted mean and predicted standard deviation of the Co-kriging proxy model, σ l (x) represents the predicted standard deviation of the kriging surrogate model constructed only from low-fidelity samples, P represents the cost coefficient, and P = C HF / C LF , C HF and C LF Represent the time cost of obtaining high-fidelity sample and low-fidelity sample responses respectively, m represents the accuracy level, m is 1, indicating that the obtained sample belongs to the low-fidelity sample set, and m is 2, indicating that the obtained sample belongs to the high-fidelity sample set;

[0067] pass Identify new training points.

[0068] In engineering examples, high-fidelity sample responses require high precision and incur high costs in terms of model construction, computing resources, and experimental verification. Low-fidelity sample responses are simplified and relatively inexpensive. The cost difference between the two is significant, usually ranging from dozens to hundreds of times. In this example, P = 15.

[0069] Step 6: Finally obtain the failure probability of the structure

[0070] Figure 2 The comparison between the Co-kriging proxy model obtained by the method of the present invention and the actual performance function is given. Figure 3 The variation curves between the failure probability and the sample acquisition cost during the iteration process of the method of the present invention and the typical method are given. Among them, the typical method is the U learning function method, which is not described here. The cost coefficient P is 15, so the low-fidelity sample acquisition cost is recorded as 1 and the high-fidelity sample acquisition cost is recorded as 15.

[0071] The specific calculation results of this example are shown in Table 1:

[0072] Table 1 Comparison of the results of the method of the present invention and the typical method in Example 1

[0073]

[0074]

[0075] The results in Table 1 show that, compared to Monte Carlo simulation results, the adaptive structural reliability analysis method proposed in this paper, based on the Co-kriging surrogate model, can efficiently and accurately estimate the failure probability of a structure. Compared with typical methods, this method significantly reduces the number of high-fidelity samples required and increases the number of low-fidelity samples. This significantly reduces the total cost of sample acquisition, while also reducing the relative error compared to the Monte Carlo method.

[0076] Example 2

[0077] See also Figure 4-Figure 5 This embodiment 2 further illustrates the present invention by combining a typical 6-dimensional failure problem. Figure 4 The nonlinear oscillator shown is widely used in the validation of reliability analysis methods.

[0078] Step 1: Determine the failure mode of the structure and the corresponding high-fidelity function g HF (x) and the low-fidelity function g LF (x), according to the distribution information of the structural input variable x, the Latin hypercube method is used to extract N candidate points as the Monte Carlo candidate sample set n MCS , and extract Nh and N l Initial sample points are used as the initial high-fidelity sample set n h and the initial low-fidelity sample set n l , let the initial iteration number k = 1;

[0079] In this embodiment, the structural high-fidelity function g HF (x) and the low-fidelity function g LF (x) is as follows:

[0080]

[0081] in, Random variable x = (c1, c2, r, F1, m, t1), c1 and c2 are the elastic stiffness of the spring, r is the resistance of the oscillator, F1 is the force on the oscillator, m is the mass of the trolley, and t1 is the time the oscillator is subjected to the force.

[0082] Table 2 Distribution of random variables in Example 2

[0083] random variable mean Standard deviation Distribution type <![CDATA[c1]]> 1 0.1 normal distribution <![CDATA[c2]]> 0.1 0.01 normal distribution r 0.5 0.05 normal distribution <![CDATA[F1]]> 1 0.2 normal distribution m 1 0.05 normal distribution <![CDATA[t1]]> 1 0.2 normal distribution

[0084] Use the Latin hypercube method to first extract 10 5 candidate points as the Monte Carlo sample set n MCS , and again extract 12 candidate points as the initial low-fidelity sample set n l and 4 candidate points as the initial high-fidelity sample set n h ;

[0085] Step 2: The two fidelity training sets n obtained from step 1 h and n l , through the function g HF (x) and g LF (x) Obtain the corresponding high-fidelity response and low-fidelity response, and construct the Co-kriging agent model of the kth iteration

[0086] The predicted mean of the Co-kriging multi-fidelity surrogate model can be expressed as μ ξ (x)=ρμ l (x)+μ δ (x), ρ is the scaling factor, which is obtained by solving the optimization problem, μ l (x) is the predicted mean of the low-fidelity kriging surrogate model, and the predicted variance is expressed as σ l (x) is the standard deviation of the low-fidelity kriging surrogate model prediction, σ δ (x) is the predicted standard deviation of the residual function;

[0087] Step 3: Use the Monte Carlo simulation method to perform structural reliability analysis on the Co-kriging surrogate model constructed in step 2 and calculate the failure probability of the kth iteration

[0088] Step 4: Determine whether the convergence condition is met. If so, go to step 6. If not, update the number of iterations: k = k + 1 and proceed to the next step.

[0089] The convergence condition is expressed as

[0090] Among them, n MCS represents the number of samples in the Monte Carlo sample set, μ ξ (x i ) and σ ξ (x i ) represent the Co-kriging proxy model The predicted mean and predicted standard deviation of the i-th sample point are obtained;

[0091] Step 5: Use the learning function to determine the new training point x new , and add the new training point to the training set, and go back to step 2;

[0092] The learning function is expressed as:

[0093]

[0094] Among them, μ ξ (x) and σ ξ (x) represent the predicted mean and predicted standard deviation of the Co-kriging proxy model, σ l (x) represents the predicted standard deviation of the kriging surrogate model constructed only from low-fidelity samples, P represents the cost coefficient, and P = C HF / C LF , C HF and C LF They represent the time cost of obtaining high-fidelity sample and low-fidelity sample responses respectively, m represents the accuracy level, m is 1, which means the obtained sample belongs to the low-fidelity sample set, and m is 2, which means the obtained sample belongs to the high-fidelity sample set.

[0095] pass Identify new training points.

[0096] In this example, P = 25;

[0097] Step 6: Finally obtain the failure probability of the structure

[0098] Figure 2The comparison between the Co-kriging proxy model obtained by the method of the present invention and the actual performance function is given. Figure 3 The variation curves between the failure probability and the sample acquisition cost during the iteration process of the method of the present invention and the typical method are given. Among them, the typical method is the U learning function method, and the cost coefficient P is 25, so the low-fidelity sample acquisition cost is recorded as 1 and the high-fidelity sample acquisition cost is recorded as 25.

[0099] The specific calculation results of this example are shown in Table 3.

[0100] Table 3 Comparison of the results of the method of the present invention and the typical method in Example 2

[0101]

[0102] The results in Table 3 show that, compared to Monte Carlo simulation results, the proposed adaptive structural reliability analysis method based on the Co-kriging surrogate model can efficiently and accurately estimate the failure probability of a structure. While the relative error with the Monte Carlo method is slightly higher than that of typical methods, the prediction accuracy of the proposed method fully meets engineering requirements. Furthermore, the required number of high-fidelity samples is significantly reduced, while a certain number of low-fidelity samples are added, significantly reducing the total cost of the required samples.

[0103] The embodiments of the present invention are described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the scope of knowledge of those skilled in the art without departing from the spirit of the present invention.

Claims

1. An adaptive structural reliability analysis method based on a Co-kriging surrogate model; characterized by: The following steps are included: S11: Determine the failure mode of the structure and the corresponding high-fidelity function g HF (x) and the low-fidelity function g LF (x), and according to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l ; S12: The two fidelity training sets n obtained in step S11 h and n l , obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration S13: Use the Monte Carlo simulation method to perform structural reliability analysis on the Co-kriging surrogate model constructed in step S12 and calculate the failure probability of the kth iteration S14: Determine whether the failure probability of the kth iteration meets the convergence condition. If so, go to step S16. If not, update the number of iterations: k=k+1, and proceed to the next step. S15: Determine the new training point x using the MPO(x,m) learning function new , and add the new training point to the training set, and return to step S12; S16: Finally obtain the failure probability of the structure 2. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 1, characterized in that: According to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l Specifically including: S21: Based on the distribution information of the structural input variable x, the Latin hypercube method is used to extract candidate points as the Monte Carlo candidate sample set n MCS ; S22: extract N h and N l Initial sample points are used as the initial high-fidelity sample set n h and the initial low-fidelity sample set n l , let the initial number of iterations k = 1.

3. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 2, characterized in that: According to the high-fidelity function g HF (x) and the low-fidelity function g LF (x) Obtain the initial high-fidelity sample set n h and the initial low-fidelity sample set n l When , the input variable x is expressed as: x=(x (1 ,x (2 ,x (3 ,x (b ); Where b represents the variable dimension.

4. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 3 is characterized in that: In the two fidelity training sets n obtained in step S11 h and n l , obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration When , the predicted mean of the Co-kriging multi-fidelity surrogate model is: m ξ (x)=rm l (x)+μ δ (x); Among them, ρ is the scaling factor, which is obtained by solving the optimization problem, μ l (x) is the predicted mean of the low-fidelity kriging surrogate model, μ δ (x) is the predicted mean of the residual function.

5. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 4 is characterized in that: In the two fidelity training sets n obtained in step S11 h and n l , obtain the corresponding high-fidelity response and low-fidelity response through the functional function, and construct the Co-kriging agent model of the kth iteration When , the residual function of the Co-kriging surrogate model is expressed as: y δ =y h -ρy l (x h ); Among them, y h is the high-fidelity response value, y l (x h ) is x h Sample response values in the low-fidelity model.

6. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 5, characterized in that: The structural reliability analysis of the Co-kriging surrogate model constructed in step S12 is performed using the Monte Carlo simulation method, and the failure probability of the kth iteration is calculated. When Calculated using the following formula: Among them, n MCS represents the number of samples in the Monte Carlo sample set, and I(·) is the characteristic function, which satisfies hour, on the contrary, 7. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 6, characterized in that: When judging whether the failure probability of the kth iteration meets the convergence condition, the convergence condition is: Among them, n MCS represents the number of samples in the Monte Carlo sample set, μ ξ (x i ) and σ ξ (x i ) represent the Co-kriging proxy model The predicted mean and predicted standard deviation of the i-th sample point are obtained.

8. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 7, characterized in that: When using the MPO(x,m) learning function to determine the new training point x new When , the learning function MPO(x,m) is expressed as: Among them, μ ξ (x) and σ ξ (x) represents the predicted mean and predicted standard deviation of the Co-kriging proxy model, σ l (x) represents the predicted standard deviation of the kriging surrogate model constructed only from low-fidelity samples, P represents the cost coefficient, and P = C HF / C LF , C HF and C LF They represent the time cost of obtaining high-fidelity sample and low-fidelity sample responses respectively, m represents the accuracy level, m is 1, which means the obtained sample belongs to the low-fidelity sample set, and m is 2, which means the obtained sample belongs to the high-fidelity sample set.

9. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 8, characterized in that: When using the MPO(x,m) learning function to determine the new training point x new When , the prediction standard deviation of the Co-kriging proxy model is calculated by the following formula: Among them, σ l (x) is the standard deviation of the low-fidelity kriging surrogate model prediction, σ δ (x) is the predicted standard deviation of the residual function.

10. The adaptive structural reliability analysis method based on the Co-kriging surrogate model according to claim 9, characterized in that: When using the MPO(x,m) learning function to determine the new training point x new When, through the formula: Identify new training points.

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