MRBF and iCE-m fused composite material structure high-dimension reliability analysis method

Abstract

The invention discloses a high-dimension reliability analysis method for a composite material structure fusing MRBF and iCE-m, and the method comprises the steps: initializing a training sample set of a high-dimension small-failure probability composite material structure, calculating a true value of a target function, and forming an experimental design DoE; based on the current DoE, training an RBF full model, and defining a quasi-optimal iPDF; an important sample obeying the quasi-optimal iPDF is generated; constructing an active learning function, and obtaining an optimal training point; and S4, according to the optimal training point obtained in the step S4, determining a stop criterion of active learning, and outputting the failure probability of the to-be-analyzed high-dimension small-failure-probability composite material structure. According to the method, the MRBF and iCE-m are fused, the reliability of the high-dimension small-failure-probability problem is analyzed, a new learning function is provided according to prediction information of the MRBF model, a limit state plane in the reliability analysis problem is better approximated, and therefore the analysis efficiency of reliability analysis of the high-dimension small-failure-probability composite material structure is greatly improved.

Description

Technical Field

[0001] The present invention relates to the technical field of high-dimensional small failure probability composite materials, and more specifically to a high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m. Background Art

[0002] Composite structures have remarkable properties, including high mechanical strength, stiffness, corrosion resistance, thermal insulation, and low weight ratio. These structures can be tailored to specific application requirements to optimize structural performance and are widely used in industries such as aerospace, automotive, construction, and shipbuilding. However, their structural performance is often subject to a certain degree of uncertainty due to the complexity of the manufacturing process and the inherent variability of material parameters.

[0003] The performance of complex composite structures is usually affected by multiple uncertain parameters, especially material parameters, number of layers, thickness of each layer, and angle of each layer. These uncertain factors may lead to uncertainty in the mechanical properties of composite structures. Therefore, it is of great significance to comprehensively consider the uncertainty of these parameters and perform reliability analysis on the uncertainty of each input parameter.

[0004] Reliability analysis of composite structures is a typical high-dimensional problem. Most traditional reliability analysis methods require multiple calculations of the functional function of the structure. In actual engineering, the calculation of the functional function often requires the use of commercial software such as finite element, multibody dynamics, and computational fluid dynamics, which are time-consuming. Therefore, studying a set of efficient and high-precision reliability analysis methods has always been the goal pursued by scholars at home and abroad. At present, the commonly used structural reliability analysis calculation methods are mainly divided into three categories: approximate analytical method, sampling simulation method, and surrogate model method. The first-order reliability method (FORM) and the second-order reliability method (SORM) are the most basic approximate methods, which make first-order or second-order approximations to the function; the Monte Carlo simulation method and the importance sampling method are classic sampling simulation methods, which can obtain more accurate results through a large number of function calculations. Commonly used surrogate models include polynomial response surface (PRS), Kriging, radial basis function (RBF), artificial neural network (ANN), support vector regression (SVR), etc. These methods first use surrogate models to approximate the function, and then use sampling simulation methods to solve the approximate failure probability.

[0005] Unlike Kriging models, PCE, SVR, neural networks and other proxy models that are prone to dimensionality disasters, the RBF model does not require too much training time to obtain high accuracy. This means that the RBF model does not need to rely on feature screening or mapping dimensionality reduction, and can complete model training directly in the high-dimensional source space with high accuracy. Moreover, with the help of cross-validation technology, the prediction variance of the proxy model can also be obtained, thereby completing active learning. Related research has been confirmed in the field of low-dimensional reliability analysis. However, in the context of high-dimensional reliability analysis, the computational time consumed to obtain the predicted mean and variance of the RBF model increases significantly, and the computational time of active learning itself (excluding the computational time of the functional function) will exceed the acceptable range.

[0006] The MRBF model uses matrix operations to complete the prediction of massive candidate points, which greatly reduces the prediction time of the RBF model and makes active learning possible. However, this method is not suitable for problems with extremely small failure probabilities. -5 ), a large number of MCS samples are required to ensure the accuracy of the estimation. -5 ~10 -10 The magnitude of the function is often 4×10 7 ~4×10 12 Only with a large number of simulation samples, MCS can provide an estimation result with a coefficient of variation lower than 5%. Based on the ALR method of MCS, each iteration needs to predict the response of such a large number of MCS samples, and the calculation time is unacceptable. Summary of the invention

[0007] The technical problem to be solved by the present invention is to provide a high-dimensional reliability analysis method for composite material structures that integrates MRBF and iCE-m, introduces the dimensionality reduction idea of ​​iCE-m and MRBF fusion, builds a bridge for collaboration between the two, and helps select the optimal training point.

[0008] In order to solve the above technical problems, the technical solutions adopted by the present invention are as follows.

[0009] A high-dimensional reliability analysis method for composite structures integrating MRBF and iCE-m includes the following steps:

[0010] S1. Initialize the training sample set of high-dimensional composite structure with small failure probability, and calculate the true value of the objective function to form the experimental design DoE;

[0011] S2. Based on the current DoE, train the RBF full model and define the quasi-optimal iPDF;

[0012] S3. Generate important samples that obey the quasi-optimal iPDF;

[0013] S4. Construct an active learning function to obtain the optimal training point;

[0014] S5. According to the optimal training point obtained in step S4, a stop criterion for active learning is determined, and the failure probability of the high-dimensional small failure probability composite material structure to be analyzed is output.

[0015] To further optimize the technical solution, in step S1, the high-dimensional small failure probability composite material training sample set is initialized using the Latin hypercube sampling method, the training sample set is extracted in the original high-dimensional variable space, and the true response value of the objective function is calculated to form the experimental design DoE (x (i) ,g(x (i) ))(i=1,…,m).

[0016] To further optimize the technical solution, in step S2, the RBF full model is trained and the method for defining the quasi-optimal iPDF is:

[0017] According to the given DoE(x (i) ,g(x (i) ))(i=1,…,m), the RBF model approximates the objective function as a linear combination of m RBFs, that is:

[0018]

[0019] Where: ||xx (i) || represents any point x and x (i) The spatial distance is usually the Euclidean distance. is the radial basis function; c is the shape parameter of the radial basis function; β = [β1,…,β m ] T is the coefficient vector of the basis function;

[0020] The unknown coefficients can be obtained as

[0021] β=Φ -1 g

[0022] Where Φ is the RBF matrix of the training point, g = [g(x (1) ),…,g(x (m) )] T ;

[0023] The expression for obtaining the optimal shape parameter of the full RBF model is as follows:

[0024]

[0025] Among them, β i is the i-th element of β, is Φ -1The i-th diagonal element of ;

[0026] The prediction mean and prediction variance of the RBF model are obtained by using LOOCV technology or Jackknife method as the mean and variance of the prediction values ​​of m RBF sub-models, that is,

[0027]

[0028]

[0029] in, is the predicted value of the ith sub-model, assuming Subject to the mean μ g (x), the standard deviation is Normal distribution of

[0030] The matrix formed by the predicted values ​​of all sub-models is The matrix formed by the RBF values ​​between the training points and the prediction points is F X , It can be expressed in matrix form, that is,

[0031]

[0032] in, and

[0033]

[0034] F X It can be expressed as

[0035]

[0036] Where D is the Euclidean distance matrix between all training points and prediction points.

[0037] To further optimize the technical solution, in step S3, in the importance sampling method, the optimal iPDF is:

[0038]

[0039] Where f(x) is the probability density function, P f represents the true failure probability;

[0040] The basic idea of ​​iCE-IS is to use a probability density function h(x,q) with parameter q to approximate h * (x), q by minimizing h(x,q) and h * (x) is determined by the KL difference between

[0041]

[0042] In the case of small failure probability problems, it is usually difficult to directly obtain a large number of failure samples, and it is also difficult to directly approximate h * (x), therefore, iCE-IS approximates h in a hierarchical manner. * (x), define the following iPDF series in

[0043]

[0044] Among them, P t is a normalizing constant, σ t represents the smoothness parameter, and ∞>σ1>…>σ T >0; as σ t increase, Close to h * (x);

[0045] iCE-IS from approximate Start with h(x,q t ) to approximate The importance sampling method was used to calculate h(x,q t )and The KL difference between Make all intermediate samples participate in q t The estimation results are better.

[0046] For Gaussian density, the parameter to be estimated is q t =(μ t ,Σ t ), where μ t ∈R D is the mean of the Gaussian density, Σ t ∈R D×D is the covariance matrix of the Gaussian density,

[0047]

[0048]

[0049] in, in

[0050] For high-dimensional problems, the above covariance matrix estimate will suddenly collapse to 0, so a mapping dimensionality reduction strategy is proposed to estimate the covariance matrix in a one-dimensional subspace:

[0051] Map random samples into a low-dimensional subspace

[0052] y (i) =R T x(i)

[0053] in, The projection direction is defined;

[0054] Compute the variance of sample points in a low-dimensional subspace:

[0055]

[0056] Estimate the covariance matrix of samples in the high-dimensional original space:

[0057] Σ t =(v-1)RR T +I D

[0058] In this way, the covariance matrix is ​​estimated only in the direction of the mean of the multivariate Gaussian distribution;

[0059] To fuse iCE-m with the RBF model, we should first define the optimal iPDF as the target distribution of iCE-m, so as to extract “high-quality” candidate samples;

[0060] Considering the uncertainty of the prediction results, is the area near the limit state plane, It is the failure area with a high confidence level; set α=1.96, and the confidence level is 95%. represents a potential failure domain;

[0061] Based on this, the following quasi-optimal iPDF is proposed

[0062]

[0063] Among them, P is the normalization coefficient, g Sur (x) is the potential failure plane,

[0064] g Sur (x) = μ g (x)-ασ g (x)

[0065] H * (x) is the target distribution, and the generated samples will be mostly located in In order to meet the above needs; in order to obtain obedience An important sample of iPDF series is defined as

[0066]

[0067] Then we can perform iCE-m, and the samples in the last layer are those that obey H * (x) is an important sample.

[0068] To further optimize the technical solution, in step S4, the method for obtaining the optimal training point is:

[0069] Since g(x) is random, we need to find g 2 The mathematical expectation of (x),

[0070]

[0071] In order to keep the training points dispersed, g 2 Variance of (x):

[0072]

[0073] In order to balance the two objectives, namely, minimizing E(g 2 (x)), maximize Var(g 2 (x)), using g 2 The coefficient of variation of (x) is used as a learning function:

[0074]

[0075] The optimal training point is chosen as

[0076] x*=argmax Cov(g 2 (x)).

[0077] To further optimize the technical solution, in step S5,

[0078] The actual failure probability is The failure probability predicted by the RBF model is The relative error of importance sampling is

[0079]

[0080] Among them, I WSP (x i ) and I F (x i ) are indicator functions, which represent x i Whether the sign of the position function is mispredicted or not i Is it a failed sample? The relative error takes into account the weight W(x i ), for ε IS Perform interval estimation and use the upper bound of the confidence interval As the convergence condition of the relative error of failure probability;

[0081] It is also necessary to evaluate the ability of the sample set to estimate the true iPDF and monitor the sample set’s Estimation capability of

[0082]

[0083] when and When both are less than the given threshold, the active learning process is stopped and the failure probability is calculated.

[0084] Due to the adoption of the above technical scheme, the technical progress achieved by the present invention is as follows.

[0085] The present invention provides a high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m. It proposes integrating MRBF and iCE-m to analyze the reliability of high-dimensional small failure probability problems. According to the prediction information of the MRBF model, a new learning function is proposed to better approximate the limit state plane in the reliability analysis problem, thereby greatly improving the analysis efficiency of the reliability analysis of high-dimensional small failure probability composite material structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 is a logic flow chart of the present invention;

[0087] Figure 2 It is a schematic diagram of the structure of the composite material reinforced plate of the present invention;

[0088] Figure 3 is a curve diagram showing the failure probability and the estimated relative error of the present invention;

[0089] Figure 4 The convergence process of active learning and the distribution diagram of DoE. DETAILED DESCRIPTION

[0090] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0091] A high-dimensional reliability analysis method for composite material structures that integrates MRBF and iCE-m is a high-dimensional reliability analysis method for composite material structures that integrates a matrix operation radial basis function (MRBF) model and cross entropy importance sampling based on mean dimensionality reduction (iCE-m). The logical process is as follows Figure 1 As shown, it includes the following steps:

[0092] S1. Initialize the training sample set of high-dimensional small failure probability composite structures, and calculate the true value of the objective function to form the experimental design DoE.

[0093] The Latin hypercube sampling method is used to initialize the training sample set of high-dimensional composite materials with small failure probability. The training sample set is extracted in the original high-dimensional variable space, and the true response value of the objective function is calculated to form the experimental design DoE (x (i) ,g(x (i)))(i=1,…,m).

[0094] S2. Based on the current DoE, train the full RBF model and define the quasi-optimal iPDF.

[0095] The method for training the full RBF model and defining the quasi-optimal iPDF is:

[0096] According to the given DoE(x (i) ,g(x (i) ))(i=1,…,m), the RBF model approximates the objective function as a linear combination of m RBFs, that is:

[0097]

[0098] Where: ||xx (i) || represents any point x and x (i) The spatial distance is usually the Euclidean distance. is the radial basis function; c is the shape parameter of the radial basis function; β = [β1,…,β m ] T is the coefficient vector of the basis function;

[0099] The unknown coefficients can be obtained as

[0100] β=Φ -1 g

[0101] Where Φ is the RBF matrix of the training point, g = [g(x (1) ),…,g(x (m) )] T ;

[0102] The expression for obtaining the optimal shape parameter of the full RBF model is as follows:

[0103]

[0104] Among them, β i is the i-th element of β, is Φ -1 The i-th diagonal element of ;

[0105] The prediction mean and prediction variance of the RBF model are obtained by using LOOCV technology or Jackknife method as the mean and variance of the prediction values ​​of m RBF sub-models, that is,

[0106]

[0107]

[0108] in, is the predicted value of the ith sub-model, assuming Subject to the mean μ g (x), the standard deviation is The normal distribution of .

[0109] The matrix formed by the predicted values ​​of all sub-models is The matrix formed by the RBF values ​​between the training points and the prediction points is F X , It can be expressed in matrix form, that is,

[0110]

[0111] Where B = [β1,…,β m ],and

[0112]

[0113] F X It can be expressed as

[0114]

[0115] Where D is the Euclidean distance matrix between all training points and prediction points.

[0116] S3. Generate important samples that obey the quasi-optimal iPDF.

[0117] In the importance sampling method, the optimal iPDF is:

[0118]

[0119] Where f(x) is the probability density function, P f represents the true failure probability;

[0120] The basic idea of ​​iCE-IS is to use a probability density function h(x,q) with parameter q to approximate h * (x), q by minimizing h(x,q) and h * (x) is determined by the KL difference between

[0121]

[0122] In the case of small failure probability problems, it is usually difficult to directly obtain a large number of failure samples, and it is also difficult to directly approximate h * (x), therefore, iCE-IS approximates h in a hierarchical manner. * (x), define the following iPDF series in

[0123]

[0124] Among them, P tis a normalizing constant, σ t represents the smoothness parameter, and ∞>σ1>…>σ T >0; as σ t increase, Close to h * (x);

[0125] iCE-IS from approximate Start with h(x,q t ) to approximate The importance sampling method was used to calculate h(x,q t )and The KL difference between Make all intermediate samples participate in q t The estimation results are better.

[0126] For Gaussian density, the parameter to be estimated is q t =(μ t ,Σ t ), where μ t ∈R D is the mean of the Gaussian density, Σ t ∈R D×D is the covariance matrix of the Gaussian density,

[0127]

[0128]

[0129] in, in

[0130] For high-dimensional problems, the above covariance matrix estimate will suddenly collapse to 0, so a mapping dimensionality reduction strategy is proposed to estimate the covariance matrix in a one-dimensional subspace:

[0131] Map random samples into a low-dimensional subspace

[0132] y (i) =R T x (i)

[0133] in, The projection direction is defined;

[0134] Compute the variance of sample points in a low-dimensional subspace:

[0135]

[0136] Estimate the covariance matrix of samples in the high-dimensional original space:

[0137] Σ t =(v-1)RR T +I D

[0138] In this way, the covariance matrix is ​​estimated only in the direction of the mean of the multivariate Gaussian distribution;

[0139] To fuse iCE-m with the RBF model, we should first define the optimal iPDF as the target distribution of iCE-m, so as to extract “high-quality” candidate samples;

[0140] Considering the uncertainty of the prediction results, is the area near the limit state plane, It is the failure area with a high confidence level; set α=1.96, and the confidence level is 95%. represents a potential failure domain;

[0141] Based on this, the following quasi-optimal iPDF is proposed

[0142]

[0143] Among them, P is the normalization coefficient, g Sur (x) is the potential failure plane,

[0144] g Sur (x) = μ g (x)-ασ g (x)

[0145] H * (x) is the target distribution, and the generated samples will be mostly located in In order to meet the above needs; in order to obtain obedience An important sample of iPDF series is defined as

[0146]

[0147] Then we can perform iCE-m, and the samples in the last layer are those that obey H * (x) is an important sample.

[0148] S4. Build an active learning function to obtain the optimal training point.

[0149] The method to obtain the optimal training point is:

[0150] Since g(x) is random, we need to find g 2 The mathematical expectation of (x),

[0151]

[0152] In order to keep the training points dispersed, g 2 Variance of (x):

[0153]

[0154] In order to balance the two objectives, namely, minimizing E(g 2 (x)), maximize Var(g 2 (x)), using g 2 The coefficient of variation of (x) is used as a learning function:

[0155]

[0156] The optimal training point is chosen as

[0157] x*=argmax Cov(g 2 (x)).

[0158] S5. According to the optimal training point obtained in step S4, a stop criterion for active learning is determined, and the failure probability of the high-dimensional small failure probability composite material structure to be analyzed is output.

[0159] The actual failure probability is The failure probability predicted by the RBF model is The relative error of importance sampling is

[0160]

[0161] Among them, I WSP (x i ) and I F (x i ) are indicator functions, which represent x i Whether the sign of the position function is mispredicted or not i Is it a failed sample? The relative error takes into account the weight W(x i ), for ε IS Perform interval estimation and use the upper bound of the confidence interval As the convergence condition of the relative error of failure probability;

[0162] It is also necessary to evaluate the ability of the sample set to estimate the true iPDF and monitor the sample set’s Estimation capability of

[0163]

[0164] when and When both are less than the given threshold, the active learning process is stopped and the failure probability is calculated.

[0165] An explicit nonlinear function is introduced as a test to verify the above steps. The dimension D is set to 100 and 200, where:

[0166]

[0167] The experimental results are shown in Table 1:

[0168] Table 1 Comparison of results of different methods

[0169]

[0170] It can be seen that the number of training points required by the AL-MRBF-iCE-m method proposed in the present invention is less than that of other methods. AL-MRBF-MCS and AL-MRBF-iCE do not require dimensionality reduction or feature screening, and directly identify the optimal training points in the high-dimensional space. Therefore, the number of required function calculations is greatly reduced. AL-MRBF-MCS needs to convert 1×10 7 MCS sample points are used as prediction samples, while AL-MRBF-iCE uses iCE-m samples as prediction samples. Therefore, the calculation time of AL-MRBF-iCE is significantly lower than that of AL-MRBF-MCS.

[0171] The following is an example of a high-dimensional small failure probability composite structure reliability analysis problem:

[0172] like Figure 2 The figure shows a composite reinforced plate with skin width W = 610 mm and length L = 865 mm. The rib spacing S = 17.8 mm. The single layer thickness of the composite laminate is 0.125 mm. The material parameters are: elastic modulus E1 in the first direction, elastic modulus E2 in the second direction, Poisson's ratio μ, shear modulus G 12 ,G 13 and G 23 The skin consists of 16 layers with ply angles θ1 to θ 16 . The reinforcement cap is laid with 50 layers and the laying angle is θ 17 ~θ 66 . 22 layers of stiffener web, layer angle θ 67 ~θ 88 . 22 layers of stiffener flange, layer angle θ 89 ~θ 110. Four sets of stiffeners are bonded to the skin using BSL322 film adhesive with an adhesive layer thickness of 0.1 mm, an elastic modulus of 1000 GPa, and a shear modulus of 1000 GPa. The random variables in this case are 6 material parameters and 110 ply angle parameters. The distribution type and distribution parameters of the random variables are shown in Table 2. The finite element model of the composite stiffened panel was established using ABAQUS, in which the laminate was simulated using shell elements, the adhesive layer was simulated using cohesive elements, and the adhesive layer and the shell were bound using Tie connections. A unit load was applied along the laminate stiffener direction (Z axis), and a linear buckling analysis was performed.

[0173] Table 2 Random variable distribution parameters

[0174]

[0175] In the presence of random parameters, there is a certain dispersion in the buckling load that the panel can withstand. According to the design requirements, this value should be greater than 90kN, so the performance function is defined as

[0176] g(x)=λ(x)F0-90

[0177] Among them, F0 = 10.32 kN is the total load applied to the structure, and λ(x) is the first-order buckling factor of the structure.

[0178] The proposed method is used to solve the failure probability, and the calculation results are shown in Table 3. The present invention uses 100 initial training points and calls the finite element software 100 times as the initial DoE of AL-MRBF-iCE. 61 new training points are added, and the finite element software is called 61 times. The proposed method has completed convergence. The total cost of active learning is 4577 seconds, of which about 4336 seconds are used for finite element model calculation and only 241 seconds are used for the algorithm itself. Figure 3 As shown in Figure 2, most of the added training points are located near the limit state surface. The variation curves of failure probability and estimated relative error during the iteration process are shown in Figure 2. Figure 3 shown.

[0179] Table 3 Calculation results of composite stiffened plate

[0180]

[0181] In order to verify the accuracy of the proposed method, in the variable space [μ j -6σ j ,μ j +6σ j], 1000 samples were extracted, 700 of which were used as training points to approximate the global MRBF model, and 300 were used as verification points. The accuracy of the MRBF model was verified, and its determination coefficient R2 = 0.9998 was obtained. Based on the global MRBF model, the failure probability obtained by iCE-m is 2.91×10 -6 This shows that the calculation results of AL-MRBF-iCE have high accuracy. The convergence process of active learning is similar to the distribution of DoE. Figure 4 As shown in Figure 2, it can be seen that the proposed method can robustly complete active learning, and the added training points are concentrated near the limit state.

[0182] The above example results show that the AL-MRBF-iCE-m model proposed in this application can well evaluate the reliability problem of high-dimensional small failure probability composite material structures. In terms of computational cost, the AL-MRBF-iCE-m model proposed in this application requires less than 200 sample points, and the calculated failure probability is highly accurate, which is generally considered to be computationally feasible in engineering practice. Compared with existing methods, it solves the difficult problem of reliability analysis of high-dimensional small probability composite material structures, greatly reduces the amount of calculation, and improves computational efficiency.

Claims

1. A high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m, characterized by: It includes the following steps: S1. Initialize the training sample set of high-dimensional composite structure with small failure probability, and calculate the true value of the objective function to form the experimental design DoE; S2. Based on the current DoE, train the RBF full model and define the quasi-optimal iPDF; S3. Generate important samples that obey the quasi-optimal iPDF; S4. Construct an active learning function to obtain the optimal training point; S5. According to the optimal training point obtained in step S4, a stop criterion for active learning is determined, and the failure probability of the high-dimensional small failure probability composite material structure to be analyzed is output.

2. According to claim 1, a high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m is characterized by: In step S1, the high-dimensional small failure probability composite material training sample set is initialized by using the Latin hypercube sampling method to extract the training sample set in the original high-dimensional variable space, and calculate the true response value of the objective function to form the experimental design DoE (x (i) ,g(x (i) ))(i=1,…,m).

3. The high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m according to claim 1, characterized in that: In step S2, the RBF full model is trained and the method of defining the quasi-optimal iPDF is: According to the given DoE(x (i) ,g(x (i) ))(i=1,…,m), the RBF model approximates the objective function as a linear combination of m RBFs, that is: Where: ||xx (i) || represents any point x and x (i) The spatial distance is usually the Euclidean distance. is the radial basis function; c is the shape parameter of the radial basis function; β = [β1,…,β m ] T is the coefficient vector of the basis function; The unknown coefficients can be obtained as β=Φ -1 g Among them, Φ is the RBF matrix of the training point, g=[g ( x (1) ) ,…,g ( x (m) ) ] T ; The expression for obtaining the optimal shape parameter of the full RBF model is as follows: Among them, β i is the i-th element of β, is Φ -1 The i-th diagonal element of ; The prediction mean and prediction variance of the RBF model are obtained by using LOOCV technology or Jackknife method as the mean and variance of the prediction values ​​of m RBF sub-models, that is, in, is the predicted value of the ith sub-model, assuming Subject to the mean μ g (x), the standard deviation is Normal distribution of The matrix formed by the predicted values ​​of all sub-models is The matrix formed by the RBF values ​​between the training points and the prediction points is F X , It can be expressed in matrix form, that is, Where B = [β1,…,β m ],and F X It can be expressed as Where D is the Euclidean distance matrix between all training points and prediction points.

4. The high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m according to claim 1, characterized in that: In step S3, in the importance sampling method, the optimal iPDF is: Where f(x) is the probability density function, P f represents the true failure probability; The basic idea of ​​iCE-IS is to use a probability density function h(x,q) with a parameter q to approximate h * (x), q by minimizing h(x,q) and h * (x) is determined by the KL difference between In the case of small failure probability problems, it is usually difficult to directly obtain a large number of failure samples, and it is also difficult to directly approximate h * (x), therefore, iCE-IS approximates h in a hierarchical manner. * (x), define the following iPDF series in Among them, P t is a normalizing constant, σ t represents the smoothness parameter, and ∞>σ1>…>σ T >0; as σ t increase, Close to h * (x); iCE-IS from approximate Start with h(x,q t ) to approximate The importance sampling method was used to calculate h(x,q t )and The KL difference between Make all intermediate samples participate in q t The estimation results are better. For Gaussian density, the parameter to be estimated is q t =(μ t ,Σ t ), where μ t ∈R D is the mean of the Gaussian density, Σ t ∈R D ×D is the covariance matrix of the Gaussian density, in, in For high-dimensional problems, the above covariance matrix estimate will suddenly collapse to 0, so a mapping dimensionality reduction strategy is proposed to estimate the covariance matrix in a one-dimensional subspace: Map random samples into a low-dimensional subspace y (i) =R T x (i) in, The projection direction is defined; Compute the variance of sample points in a low-dimensional subspace: Estimate the covariance matrix of samples in the high-dimensional original space: S t =(v-1)RR T +I D In this way, the covariance matrix is ​​estimated only in the direction of the mean of the multivariate Gaussian distribution; To fuse iCE-m with the RBF model, we should first define the optimal iPDF as the target distribution of iCE-m, so as to extract "high-quality" candidate samples; Considering the uncertainty of the prediction results, is the area near the limit state plane, It is a failure region with a large confidence level; set α = 1.96, the confidence level is 95%, represents a potential failure domain; Based on this, the following quasi-optimal iPDF is proposed Among them, P is the normalization coefficient, g Sur (x) is the potential failure plane, g Sur (x)=μ g (x)-as g (x) H * (x) is the target distribution, and the generated samples will be mostly located in To meet the above needs; To gain obedience An important sample of iPDF series is defined as Then we can perform iCE-m, and the samples in the last layer are those that obey H * (x) is an important sample.

5. The high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m according to claim 1, characterized in that: In step S4, the method for obtaining the optimal training point is: Since g(x) is random, we need to find g 2 The mathematical expectation of (x), In order to keep the training points dispersed, g 2 Variance of (x): In order to balance the two objectives, namely, minimizing E(g 2 (x)), maximize Var(g 2 (x)), using g 2 The coefficient of variation of (x) is used as a learning function: The optimal training point is chosen as x*=argmax Cov(g 2 (x)).

6. The high-dimensional reliability analysis method for composite material structures integrating MRBF and iCE-m according to claim 1, characterized in that: In the step S5, The actual failure probability is The failure probability predicted by the RBF model is The relative error of importance sampling is Among them, I WSP (x i ) and I F (x i ) are indicator functions, which respectively represent x i Whether the sign of the position function is mispredicted or not i Is it a failed sample? The relative error takes into account the weight W(x i ), for ε IS Perform interval estimation and use the upper bound of the confidence interval As the convergence condition of the relative error of failure probability; It is also necessary to evaluate the ability of the sample set to estimate the true iPDF and monitor the sample set’s Estimation capability of when and When both are less than the given threshold, the active learning process is stopped and the failure probability is calculated.

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