Reliability analysis method of complex structure based on RBF model under extremely small failure probability

Abstract

The application discloses a complex structure active learning reliability analysis method based on an RBF model with a very small failure probability, proposes a combination of a radial basis function (RBF) model and active learning, uses the RBF model as a proxy model to approximate a real function function, and uses an important sampling technique to improve the sampling efficiency near a failure area. By combining a multi-modal optimization technique, the method can identify and use multiple most probable failure points (MPPs), thereby more accurately estimating a failure probability. The AL-RBF-IS method proposed in the application can significantly reduce the number of required training points and calculation time while ensuring accuracy, and performs well when dealing with small failure probability problems, and is more convenient when analyzing bogie frame static strength reliability.

Description

Technical Field

[0001] The present invention relates to the technical field of active learning radial basis function (RBF), and in particular to an active learning reliability analysis method for complex structures based on an RBF model under extremely low failure probability. Background Art

[0002] The bogie frame is an important component of locomotives and the main load-bearing component of the running gear of rail vehicles. The safety and reliability of the bogie frame directly affect the safety of rail vehicles. Therefore, accurate assessment of the bogie frame reliability is of great significance to ensuring the safety of railway transportation. Most domestic scholars' research on bogie frames focuses on static strength, fatigue life, and fatigue strength analysis. In this type of analysis, basic parameters are regarded as fixed values, and the calculated results are also deterministic. However, in engineering practice, the structural strength of the bogie frame is often affected by many uncertain factors, such as material properties, geometric parameters, and loads. Therefore, combining reliability analysis theory with the finite element method can more appropriately and reasonably evaluate the static performance of the bogie frame during the design stage.

[0003] To date, structural reliability analysis methods can be roughly divided into three categories. The first category is analytical methods, represented by the first-order reliability method (FORM) and the second-order reliability method (SORM). These methods approximate the limit state surface (LSS) using a Taylor expansion at the point closest to the origin (MPP) in standard normal space. The second category includes digital sampling methods, such as Monte Carlo simulation (MCS), subset simulation (SS method, and importance sampling (IS) method). The third category is reliability methods based on surrogate models. Common surrogate models include the response surface method (RSM), polynomial chaos expansion (PCE) model, neural network model, Kriging model, and radial basis function (RBF) model. In recent years, researchers have continuously deepened their research on the AL-KRG model, thereby improving its efficiency and enriching its application scenarios. These studies mainly fall into the following aspects: ① Point addition criteria. By constructing more reasonable point addition criteria, more optimal training points can be selected. ② Demand-oriented convergence criteria.

[0004] In the realm of approximate models, no single surrogate model consistently outperforms the others. Therefore, in addition to the Kriging model, other types of surrogate models have also been extensively studied in structural reliability analysis. Some studies have shown that RBF models outperform Kriging models in areas with high nonlinearity. Considering that the sample size required for importance sampling for low-probability failures is far smaller than that of traditional Monte Carlo methods, this paper proposes an active learning method based on radial basis functions (RBFs) for bogie frame static strength reliability analysis. SUMMARY

[0005] In order to solve the problems in the prior art, the purpose of the present application is to provide a complex structure active learning reliability analysis method based on RBF model with extremely small failure probability, which obtains the uncertain parameters of the RBF model by using the Jackknife cross-validation method, obtains the limit state surface (LSS) of the agent according to the uncertain parameters of the RBF model, and obtains the MPP by using the evolutionary multi-objective optimization multi-modal optimization (EMO-MMO) algorithm, and establishes the auxiliary probability density function (iPDF) by considering the weight of the MPP; finally, the RBF model is made accurate enough by constantly adding new training points according to the learning function, and the structure failure probability is solved by using the RBF model reaching the convergence threshold.

[0006] In order to achieve the above purpose, the technical scheme adopted by the present application is as follows: a complex structure active learning reliability analysis method based on RBF model with extremely small failure probability, comprising the following steps:

[0007] Step 1: converting random variables into standard normal space, generating a small number of initial training points by using the Latin hypercube sampling method LHS, calculating the true value of the function function, and forming an initial experimental design DoE;

[0008] Step 2: training the initial RBF model according to the initial experimental design DoE;

[0009] Step 3: obtaining the proxy limit state surface LSS according to the prediction information of the current RBF model;

[0010] Step 4: obtaining a plurality of points MPP closest to the origin in the standard normal space according to the multi-modal multi-objective optimization algorithm EMO-MMO;

[0011] Step 5: calculating the weight of the point MPP closest to the origin in the standard normal space and establishing the auxiliary probability density function iPDF, generating N IS important samples;

[0012] Step 6: judging whether the RBF model reaches the preset convergence threshold;

[0013] Step 7: when the RBF model does not reach the preset convergence threshold, obtaining the optimal training point by using the active learning function;

[0014] Step 8: adding the optimal training point to the initial experimental design DoE, updating the RBF model, and predicting the failure probability by using the RBF model reaching the convergence criterion.

[0015] As a further improvement of the present application, the step 1 is specifically as follows:

[0016] Let the random variable be x=[x1,…,x k ], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design DoE (u (i) ,G(u (i) ))(i=1,…,k).

[0017] As a further improvement of the present invention, in step 2, the performance function G(u) is expressed by the RBF model as follows:

[0018]

[0019] in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; That is the radial basis function; r u,i =||Uu (i) || represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function;

[0020] Substituting the initial experimental design DoE into (1) yields:

[0021]

[0022] Its matrix form is:

[0023] G=Φβ (3)

[0024] Among them, G=[(G(u (1) ),…,G(u k )] T ; Φ is the RBF matrix of the training point position; then the weight coefficient vector β is:

[0025] β=Φ -1 G(4)

[0026] The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1); the shape parameter c of the radial basis function has a great influence on the accuracy of the RBF model and usually requires cross-validation technology to be corrected.

[0027] As a further improvement of the present invention, the step 3 is specifically as follows:

[0028] The prediction mean and variance of the current RBF model are obtained using the cross-validation method; the current DoE is (u (i) ,G(u (i)))(i=1,…,k), if a training point (u (i) ,G(u (i) )), then the RBF model becomes:

[0029] G (-i) =Φ (-i) β (-i) (5)

[0030] Among them, G (-i) means G remove G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector; remove a training point (u (i) ,G(u (i) ), the predicted value of the RBF model at the unknown point is:

[0031]

[0032] in, By deletion The i-th column element is obtained; by using the block inversion operation, the overall time complexity is reduced to o(m 3 ), as shown below:

[0033]

[0034] After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u are defined according to the Jackknife method:

[0035]

[0036] and

[0037] The limit state plane (LSS) divides the random space into failure zone and safety zone. The uncertainty of the prediction makes it impossible for the RBF model to distinguish the failure zone from the safety zone. Considering the uncertainty, the region with a certain probability or confidence level of becoming the failure zone is expressed as:

[0038]

[0039] Among them, α is a constant that controls the confidence level; the area where the positive and negative signs of the performance function are likely to be incorrectly predicted is:

[0040]

[0041] use and The set of ensures that no potential failure area is missed, which is expressed as:

[0042]

[0043] Proxy failure area The control function is:

[0044]

[0045] This is the proxy limit state plane LSS.

[0046] As a further improvement of the present invention, the step 4 is specifically as follows:

[0047] In the standard normal space, the optimal solution to the constrained optimization problem is MPP, which is expressed as:

[0048]

[0049] in, is the prescribed search area;

[0050] Equation (14) is solved based on the multimodal multi-objective optimization algorithm EMO-MMO.

[0051] As a further improvement of the present invention, the step 5 is specifically as follows:

[0052] use represents the MPPs obtained using the multimodal multi-objective optimization algorithm EMO-MMO. The weight of each MPP is defined as:

[0053]

[0054] in, is the cumulative function of the standard normal distribution The function value calculated at , the auxiliary probability density function i PDF is expressed as:

[0055]

[0056] in:

[0057]

[0058] Generate N according to the importance sampling method IS Obedience important samples.

[0059] As a further improvement of the present invention, in step 6, the convergence condition is defined as:

[0060]

[0061] in, and respectively are rationality measure and belief measure of system failure region; γ is convergence threshold; and The calculation formula of is:

[0062]

[0063] Where, I F (u) is a failure indicator function; φ(u) is the probability density function PDF of standard normal space.

[0064] As a further improvement of the present application, the step 7 is specifically as follows:

[0065] The active learning function is:

[0066]

[0067] The variance of |G(u)| is:

[0068]

[0069] Considering the expectation and variance simultaneously, the coefficient of variation CV is obtained as the learning standard:

[0070]

[0071] Equation (23) is named as CV |G| (u) (24) function, the maximum point of CV |G| (u) means that the point is close to the agent LSS, that is, the value of E(|G(u)|) is small, or has a large prediction uncertainty, that is, the value of V(|G(u)|) is large, or both; then the best training point of each iteration is:

[0072] u*=argmaxCV |G| (u) (24).

[0073] As a further improvement of the present application, in step 8, the failure probability is estimated as:

[0074]

[0075] The coefficient of variation is represented as:

[0076]

[0077] The variance of the failure probability is represented as:

[0078]

[0079] The application also provides an application of the complex structure active learning reliability analysis method based on the RBF model under a minimum failure probability.

[0080] The application has the following advantages:

[0081] 1. The RBF model is combined with active learning in the application, and the important sampling technology is used to improve the sampling efficiency near the failure region. By combining the multi-modal optimization technology, the method can identify and use multiple most probable failure points (MPPs), so that the failure probability is more accurately estimated.

[0082] 2. The application takes the coefficient of variation (CV) of the function |G(u)| as the learning standard, which can significantly reduce the number of required training points and calculation time while ensuring accuracy, and improves the calculation efficiency of the bogie frame static strength reliability analysis problem. BRIEF DESCRIPTION OF DRAWINGS

[0083] Figure 1 The flowchart of the embodiment of the application is shown in the figure;

[0084] Figure 2 The figure shows the area with a high probability of failure in the embodiment of the application and the area with a positive and negative sign that is incorrectly predicted ;

[0085] Figure 3 The figure shows the 23-bar truss structure in the embodiment of the application;

[0086] Figure 4 The figure shows the finite element model of the bogie frame in the embodiment of the application. DETAILED DESCRIPTION

[0087] The embodiments of the application will be described in detail below with reference to the accompanying drawings.

[0088] EMBODIMENT

[0089] This embodiment takes the reliability analysis of the bogie frame static strength as an example to illustrate the embodiment, as shown in the figure, a bogie frame static strength reliability analysis method based on active learning radial basis function (RBF), comprising: Figure 1

[0090] Step S1: Convert the random variable to a standard normal space, generate a small number of initial training points by using the Latin hypercube sampling (LHS) method, calculate the true value of the function, and form an initial experimental design (DoE);

[0091] The random variable is x = [x1, …, x k ​], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design (DoE) (u (i) ,G(u (i) ))(i=1,…,k).

[0092] Step S2: train the initial RBF model according to the initial DoE;

[0093] The performance function G(u) is expressed by the RBF model as:

[0094]

[0095] in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; This is the radial basis function. u,i =||uu (i) || represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function.

[0096] Substituting the initial DoE into equation (1) yields:

[0097]

[0098] Its matrix form is:

[0099] G=Φβ (3)

[0100] Among them, G=[G(u (1) ),…,G(u k )] T ; Φ is the RBF matrix at the training point. Then the coefficient vector β is:

[0101] β=Φ -1 G(4)

[0102] The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1). The shape parameter c has a great influence on the accuracy of the RBF model and usually requires cross-validation technology to correct it.

[0103] Step S3: Obtain the proxy limit state plane LSS based on the current RBF model prediction information;

[0104] The prediction mean and variance of the current RBF model can be obtained using the cross-validation method. The current DoE is for:

[0105] G(-i) =Φ (-i) β (-i) (5)

[0106] Among them, G (-i) means G remove G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector. Remove a training point (u (i) ,G(u (i) )), the predicted value of the RBF model at the unknown point is:

[0107]

[0108] in, By deletion The i-th column element of is obtained. In formula (6), there is Φ (-i) The computational complexity of the inverse solution is o(m 3 The Jackknife method requires m inverse operations to estimate the model's predicted mean and standard deviation. Therefore, the total time complexity is o(m 4 By using block inversion operations, the overall time complexity can be reduced to o(m 3 ), as shown below.

[0109]

[0110] After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u can be defined according to the Jackknife method:

[0111]

[0112] and

[0113] like Figure 2 As shown in Figure 2, the limit state plane LSS can divide the random space into failure zone and safe zone. However, the uncertainty of the prediction makes it impossible for the RBF model to distinguish the failure zone from the safe zone. Considering the uncertainty, the area with a greater probability or confidence to become the failure zone is expressed as:

[0114]

[0115] Where α is a constant that controls the confidence level. If α is set to 1.96, the confidence level will be 95%. The areas where the sign of the function is more likely to be incorrectly predicted are:

[0116]

[0117] Using and the set of

[0118]

[0119] proxy failure region The control function of

[0120]

[0121] is the proxy LSS.

[0122] Step S4: Obtain multiple MPPs according to the multi-modal multi-objective optimization (EMO-MMO) algorithm.

[0123] In the standard normal space, the optimal solution of the constrained optimization problem is the MPP, denoted as:

[0124]

[0125] wherein, is the specified search region; in actual applications, G(u) is likely to be a multi-modal function, and thus is also multi-modal.

[0126] In addition, the LSS obtained from the initial RBF model is very inaccurate, and thus although G(u) is unimodal, it is likely that is also unimodal, and thus is multi-modal, and thus formula (14) can have multiple global optimal solutions. Even if formula (14) has only one global optimal solution, the contribution of some local optimal solutions to the failure probability integral cannot be directly ignored. If the iPDF is constructed only around one MPP, the error of the biased estimate will become very large. Therefore, searching for all MPPs on the proxy LSS is meaningful for improving the accuracy of model estimation. Many optimization problems in actual engineering belong to multi-modal optimization (MMO), that is, multiple optimal solutions are found in one run. The multi-modal optimization (EMO-MMO) algorithm based on evolutionary multi-objective optimization is used to solve formula (14).

[0127] Step S5: Calculate the MPP weights and establish the iPDF to generate N IS important samples;

[0128] The EMO-MMO algorithm can find multiple solutions of mixed global optimal solutions, local optimal solutions, and even inferior solutions. This requires establishing an iPDF that can consider the weight of each solution, so as to avoid wasting training points in unimportant failure regions.

[0129] use represents the MPPs obtained using the EMO-MMO algorithm, and the weight of each MPP is defined as:

[0130]

[0131] in, is the cumulative function of the standard normal distribution The function value calculated at , then iPDF can be expressed as:

[0132]

[0133] in:

[0134]

[0135] Generate N according to the importance sampling method IS Obedience important samples.

[0136] Step S6: Determine whether the RBF model is accurate enough and reaches a preset convergence threshold;

[0137] The convergence condition is defined as:

[0138]

[0139] in, and are the rationality measure and belief measure of the system failure area respectively; γ = 0.95 is the convergence threshold specified in this paper. and The calculation formula is:

[0140]

[0141] Among them, I F (u) is the failure indicator function; φ(u) is the probability density function PDF of the standard normal space.

[0142] Step S7: When the RBF model is not less than the preset convergence threshold, the active learning function is used to obtain the optimal training point;

[0143] The active learning function is as follows:

[0144]

[0145] The variance of |G(u)| is:

[0146]

[0147] Taking both expectation and variance into account, we get the coefficient of variation (CV) as the learning criterion:

[0148]

[0149] Formula (23) is named CV |G| Function, CV |G| The maximum value of means that the point is very close to the agent LSS (E(|G(u)|) has a small value) or has a large prediction uncertainty (V(|G(u)|) has a large value), or both. Therefore, the optimal training point for each iteration is:

[0150] u * =argmaxCV |G| (u) (24)

[0151] Step S8: Add the optimal training point to the DoE, update the RBF model, and use the RBF model that reaches the convergence criterion to predict the failure probability.

[0152] The failure probability can be estimated as:

[0153]

[0154] The coefficient of variation can be expressed as:

[0155]

[0156] The variance of the failure probability can be expressed as:

[0157]

[0158] The 23-bar truss structure is used as a test case to verify the above steps. Figure 3 As shown in Figure 1, the structure is subjected to six vertical forces. There are 10 independent random variables in this case, which are the elastic modulus (E1 and E2) and cross-sectional area (A1 and A2) of the horizontal and diagonal bars and the six vertical forces. The specific distribution is shown in Table 1. The performance function is defined as:

[0159] G(u)=160-Δ(u) (28)

[0160] where Δ is the deflection at the midpoint of the truss base and u is a vector of random variables.

[0161] Table 1

[0162]

[0163] Table 2 shows the calculation results of different methods. It can be seen that the AL-RBF-IS method proposed in this embodiment requires the least number of function calls, which reflects the advantages of the AL-RBF-IS method.

[0164] Table 2

[0165]

[0166] The reliability analysis method of this application is verified using a bogie frame static strength reliability analysis case.

[0167] The bogie frame finite element model was established in HyperMesh software, using COMBIN14 elements to simulate the primary and secondary suspensions; BEAM188 beam elements to simulate components such as wheelsets, swing arm positioning, and axle boxes; and MASS21 mass elements to simulate the mass of components such as motors and gearboxes. The bogie frame is primarily meshed using Shell181 shell elements, with Solid185 solid elements used for some complex components. The total number of nodes in the frame finite element model is 178,622, and the total number of elements is 771,466. The specific finite element model is as follows: Figure 4 shown.

[0168] This application selects the worst abnormal load condition as the reliability analysis condition, considering the influence of nine uncertain factors such as load, plate thickness, and material properties. The main parameters of the framework are shown in Table 3. The functional function of the framework is defined as:

[0169] G(u)=σ s -σ(u) max (29)

[0170] Among them, σ s is the yield strength of the material, σ max is the equivalent stress on the frame under abnormal load conditions, u=(u1,u2,…,u n ) represents the input random variable, and G(u)≤0 indicates structural failure.

[0171] Table 3

[0172]

[0173] The random input variable settings of the frame are shown in Table 4, and the calculation results are shown in Table 5. The reliability analysis of the bogie frame belongs to the problem of small failure probability. The AL-RBF-IS method is used for calculation. Only 44 real functional functions are called, with 12 initial sample points. 32 new sample points are added during the learning process. The failure probability is solved as P f =3.40×10 -7 , the coefficient of variation is CoV=3.49%.

[0174] Table 4

[0175]

[0176] Table 5

[0177]

[0178] The above case results show that the AL-RBF-IS method proposed in the embodiment can well solve the bogie frame static strength reliability problem. As can be seen from the results, even in the face of extremely small failure probability problem, the AL-RBF-IS method can obtain accurate results, and has certain advantages compared with other algorithms. In solving implicit problems in practical engineering, the AL-RBF-IS method also has excellent performance.

[0179] The above-described embodiments only express the specific implementation of the present application, which is described in detail and is not considered as a limitation to the scope of the patent of the present application. It should be pointed out that for ordinary skilled in the art, without departing from the concept of the present application, a number of modifications and improvements can be made, which are within the scope of protection of the present application.

Claims

1. A complex structure active learning reliability analysis method based on RBF model under minimal failure probability, characterized by: Used to analyze the static strength of the bogie frame, including the following steps: Step 1: Convert the random variable into the standard normal space, use the Latin hypercube sampling method (LHS) to generate a small number of initial training points, calculate the true value of the performance function, and form the initial experimental design (DoE). The step 1 is specifically as follows: Let the random variable be x=[x1,…,x k ], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design DoE (u (i) ,G(u (i) )), i=1,…,k; A finite element model of the bogie frame was established in HyperMesh software, using COMBIN14 elements to simulate the primary and secondary suspensions; BEAM188 beam elements to simulate the wheelset, swing arm positioning, and axlebox components; and MASS21 mass elements to simulate the mass of the motor and gearbox components. The bogie frame was meshed using Shell181 shell elements, and some complex components were meshed using Solid185 solid elements. Selecting the worst abnormal load condition as the reliability analysis condition, considering the influence of uncertainty factors such as load, plate thickness, and material properties, the performance function of the bogie frame is defined as: G(u)=σ s -σ(u) max Among them, σ s is the yield strength of the material, σ max is the equivalent stress on the frame under abnormal load conditions, u=(u1,u2,…,u k ) represents the input random variable, G(u)≤0 indicates structural failure; Step 2: Train the initial RBF model based on the initial experimental design DoE; Step 3: Obtain the proxy limit state plane LSS based on the current RBF model prediction information; Step 4: Obtain the point MPP closest to the origin in multiple standard normal spaces according to the multimodal multi-objective optimization algorithm EMO-MMO; Step 5: Calculate the weight of the point MPP closest to the origin in the standard normal space and establish the auxiliary probability density function iPDF to generate N IS An important sample; Step 6: Determine whether the RBF model has reached the preset convergence threshold; Step 7: When the RBF model is not less than the preset convergence threshold, the active learning function is used to obtain the optimal training point; Step 8: Add the optimal training points to the initial experimental design DoE, update the RBF model, and use the RBF model that reaches the convergence criterion to predict the failure probability.

2. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 1 is characterized in that: In step 2, the performance function G(u) is expressed using the RBF model as: in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; That is the radial basis function; r u,i =‖uu (i) ‖ represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function; Substituting the initial experimental design DoE into (1) yields: Its matrix form is: G=Φβ (3) Among them, G=[G(u (1) ),…,G(u k )] T ; Φ is the RBF matrix of the training point position; then the weight coefficient vector β is: β=Φ -1 G (4) The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1); the shape parameter c of the radial basis function has a great influence on the accuracy of the RBF model and usually requires cross-validation technology to be corrected.

3. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 2 is characterized in that: The step 3 is specifically as follows: The prediction mean and variance of the current RBF model are obtained using the cross-validation method; the current DoE is (u (i) ,G(u (i) )), i=1,…,k, if a training point (u (i) ,G(u (i) )), then the RBF model becomes: G (-i) =Φ (-i) b (-i) (5) Among them, G (-i) means G remove G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector; remove a training point (u (i) ,G(u (i) )), the predicted value of the RBF model at the unknown point is: in, By deletion The i-th column element is obtained; by using the block inversion operation, the overall time complexity is reduced to O(m 3 ), as shown below: After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u are defined according to the Jackknife method: and The limit state plane (LSS) divides the random space into failure zone and safety zone. The uncertainty of the prediction makes it impossible for the RBF model to distinguish the failure zone from the safety zone. Considering the uncertainty, the region with a certain probability or confidence level of becoming the failure zone is expressed as: Among them, α is a constant that controls the confidence level; the area where the positive and negative signs of the performance function are likely to be incorrectly predicted is: use and The set of ensures that no potential failure area is missed, which is expressed as: Proxy failure area The control function is: This is the proxy limit state plane LSS.

4. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 3 is characterized in that: The step 4 is specifically as follows: In the standard normal space, the optimal solution to the constrained optimization problem is MPP, which is expressed as: in, is the prescribed search area; Equation (14) is solved based on the multimodal multi-objective optimization algorithm EMO-MMO.

5. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 4 is characterized in that: The step 5 is specifically as follows: use represents the MPPs obtained using the multimodal multi-objective optimization algorithm EMO-MMO. The weight of each MPP is defined as: in, is the cumulative function of the standard normal distribution The function value calculated at , the auxiliary probability density function iPDF is expressed as: in: Generate N according to the importance sampling method IS Obedience important samples.

6. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 5 is characterized in that: In step 6, the convergence condition is defined as: in, and are the rationality measure and belief measure of the system failure area respectively; γ is the convergence threshold; and The calculation formula is: Among them, I F (u) is the failure indicator function; φ(u) is the probability density function PDF of the standard normal space.

7. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 6 is characterized in that: The step 7 is specifically as follows: The active learning function is: The variance of |G(u)| is: Considering both expectation and variance, we get the coefficient of variation CV as the learning criterion: Formula (23) is named CV |G| Function, CV |G| The maximum value of means that the point is close to the agent LSS, that is, the value of E(|G(u)|) is small, or has a large prediction uncertainty, that is, the value of V(|G(u)|) is large, or both; then the optimal training point for each iteration is: u*=argmax CV |G| (u) (24)。 8. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 7 is characterized in that: In step 8, the failure probability is estimated as: The coefficient of variation is expressed as: The variance of the failure probability is expressed as:

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