A High-Dimensional Composite Material Structure Reliability Analysis Method Based on Active Learning Model

Abstract

The present invention discloses a high-dimensional composite material structure reliability analysis method based on an active learning model, proposes to use the Euclidean distance matrix operation method to calculate the RBF matrix, calculate the RBF sub-model prediction value through matrix operation, calculate the prediction mean and prediction variance according to the jackknife method, use the learning function to calculate the coefficient of variation, construct an active learning RBF model, and combine the Monte Carlo simulation MCS with the active learning RBF model to evaluate the failure probability of the high-dimensional composite material structure system. The present invention has the advantages of simple logic, easy implementation, high precision, short calculation time, and is suitable for the reliability analysis of high-dimensional composite material structures. The present invention provides a powerful tool for reliability analysis of high-dimensional composite material structures.

Description

Technical Field

[0001] The present invention relates to the field of composite material structure analysis, and in particular to a high-dimensional composite material structure reliability analysis method based on an active learning model. Background Art

[0002] Compared with traditional metal materials, fiber reinforced composite laminates (hereinafter referred to as composite materials) have a more complex manufacturing process, so their material properties, layer thickness, and layer direction are more dispersed. In addition, during the operation of mechanical equipment using composite materials, there is a large uncertainty in the load borne by the structure, the working environment temperature and humidity. Therefore, the 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 time-consuming commercial software such as finite element, multi-body dynamics, and computational fluid dynamics. Therefore, the reliability analysis of high-dimensional composite structures has always been a research hotspot for scholars at home and abroad.

[0003] The active learning reliability analysis (ALR) method based on the surrogate model can actively select training points in the area of ​​interest and actively terminate the learning process, thereby obtaining high-precision calculation results with as few functional function calculations as possible. ALR provides a practical solution to structural reliability analysis problems in medium and low dimensions.

[0004] However, when dealing with high-dimensional problems, the hyperparameters of most proxy models, such as the Kriging model, SVR model, PCE and neural network model, increase significantly, and the number of training points required increases significantly. At this time, the computational time for training the proxy model will increase significantly, and may even reach or exceed the time for calling the numerical simulation model. Therefore, ALR currently encounters the "dimensionality disaster" when dealing with high-dimensional composite material structure reliability analysis.

[0005] Unlike proxy models such as the Kriging model, PCE, SVR, and neural networks that are prone to the "curse of dimensionality", the RBF model can complete model training directly in the high-dimensional source space without requiring too much training time, and has high accuracy. In fact, in the field of structural optimization design, the RBF model has demonstrated excellent high-dimensional function approximation performance. However, in the field of high-dimensional reliability analysis, the RBF-based active learning method is difficult to apply due to the cross-validation operation mechanism. At present, the active learning method based on the RBF model in the field of reliability analysis still remains in the medium and low-dimensional field, and the dimension of the problem analyzed basically does not exceed 30.

[0006] In the current ALR field, when dealing with conventional failure problems (non-minimum failure probability problems), MCS sample points are usually used as candidate points. The sample points extracted by MCS can cover all failure areas without any prior information. The MCS-based ALR method has the inherent advantages of being robust and efficient. Under this framework, each time the AL-RBF model adds a point, the RBF model needs to be trained on a large number of MCS samples (10 5 ~10 6 In a high-dimensional context, the computation time required to obtain the predicted mean and variance of the RBF model increases significantly, and the computation time of active learning itself (excluding the computation time of the functional function) will exceed the acceptable range.

[0007] In view of this, the present invention aims to propose an active learning method based on the RBF model for reliability analysis of high-dimensional composite materials. In the face of large-scale data calculations, implementing matrix operations of the algorithm is an excellent way to reduce calculation time. Matrix operations can make full use of the parallel instruction set for matrix operations in modern computer CPUs to carry out parallel operations; matrix operations can make full use of the highly optimized linear algebra basic library in high-level programming languages ​​to reduce memory access delays. To this end, the present application proposes a new matrix operation method for Euclidean distance and RBF sub-model prediction values, which significantly shortens the calculation time of the RBF model for response prediction of high-dimensional and massive MCS samples of composite materials structures during active learning. Summary of the invention

[0008] In response to the above problems, the present invention provides a reliability analysis method for high-dimensional composite materials based on an active learning model. Through the matrix operation method of Euclidean distance and RBF sub-model prediction values, the calculation time of the RBF model for response prediction of high-dimensional and massive MCS samples of composite materials structures in the active learning process can be significantly shortened.

[0009] The technical solution adopted by the present invention is as follows:

[0010] A high-dimensional composite material structure reliability analysis method based on an active learning model includes the following steps:

[0011] S1. Prepare a training sample set of high-dimensional composite structures and calculate the true value of the objective function to form an experimental design (DoE);

[0012] S2, generate candidate sample points, estimate failure probability and prediction error;

[0013] S3, based on the current DoE, train the full RBF model;

[0014] S4, obtaining the optimal shape parameters of the full RBF model;

[0015] S5, calculating the RBF matrix;

[0016] S6, obtaining the predicted values ​​of all RBF sub-models;

[0017] S7, calculating the predicted mean and predicted variance;

[0018] S8, construct an active learning function to obtain the optimal training point;

[0019] S9. According to the optimal training point obtained in step S8, a stop criterion for active learning is determined, and the failure probability of the high-dimensional composite material structure to be analyzed is output.

[0020] In step S1, the high-dimensional composite material training sample set is extracted in the original high-dimensional variable space using the Latin hypercube sampling method, and the true response value of the objective function is calculated to form an experimental design DoE (x (i) ,g(x (i) ))(i=1,…,m).

[0021] In step S2, a large number of candidate points are generated through MCS to estimate the failure probability and prediction error.

[0022] In step S3, the expression of the RBF model is:

[0023]

[0024] Among them, ||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. (i) The RBF function between

[0025]

[0026] The vector formed by the RBF function between point x and all training points is

[0027]

[0028] g=Φβ

[0029] Where Φ is the RBF matrix of the training point, expressed as:

[0030]

[0031] where d i,j =||x (i)-x (j) ||, g = [g(x (1) ),…,g(x (m) )] T .

[0032] In step S4, the optimal shape parameter expression of the full RBF model is obtained as follows:

[0033]

[0034] Among them, β i is the i-th element of β, is Φ -1 The i-th diagonal element of .

[0035] In step S5, the RBF matrix is ​​calculated using the Euclidean distance matrix operation method. N ] and training point X T =[x (1) ,…,x (m) ] is called the Euclidean distance matrix, denoted by D:

[0036]

[0037] If D is known, the RBF matrix F X It can be easily obtained by matrix operation:

[0038]

[0039] The Euclidean distance matrix can be completed using matrix operations:

[0040]

[0041]

[0042]

[0043] D can be expressed in the following matrix form:

[0044]

[0045] in, Represents element-wise square root. s X and T The matrix operation form is:

[0046]

[0047]

[0048] in is the Hadamard product. So, find the RBF matrix F X All formulas of are expressed in the form of matrix operations.

[0049] In step S6, the predicted values ​​of all RBF sub-models are calculated using a matrix operation method. N If we want to obtain prediction information of m sub-models for N unknown points x1,…,x N The predicted value of .

[0050] If the i-th training point is removed from the DoE, the radial basis function model is:

[0051] Φ (-i) β (-i) =g (-i)

[0052] Among them, g (-i) =[g(x (1) ),…,g(x (i-1) ),g(x (i+1) ),…,g(x (m) )] T is to remove g(x (i) )The vector obtained. Φ (-i) is the training point [x (1) ,…,x (i-1) ,x (i+1) ,…,x (m) ] T The RBF matrix is ​​formed by removing the i-th row and i-th column elements of Φ.

[0053] Among them, (Φ (-i) ) -1 Expressed as:

[0054]

[0055] in, is the matrix Φ -1 The m-1-order square matrix after removing the elements in the i-th row and i-th column; is the matrix Φ -1 The m-1 row vector after removing the i-th element from the i-th column vector; is Φ -1 The i-th row and i-th column element of .

[0056] Thus, we get β (-i) expression:

[0057]

[0058] In order to facilitate matrix operations, a “zero padding strategy” is proposed to uniformly express the prediction results of the RBF sub-model as follows:

[0059]

[0060] in,

[0061]

[0062] The matrix composed of these predicted values ​​is recorded as

[0063]

[0064] In this way, m sub-models are used for N unknown points X = [x1,…,x N The prediction result of ] can be expressed in the following matrix form:

[0065]

[0066] Where B = [β1,…,β m ]. In this way, the predicted values ​​of m RBF sub-models of x are obtained.

[0067] In step S7, the predicted mean and predicted variance are calculated according to the Jackknife method, and the predicted mean and standard deviation of x are defined as:

[0068]

[0069]

[0070] and

[0071] In step S8, the active learning function is as follows:

[0072] You can choose a variety of learning functions such as U learning function, EFF learning function, CV |g| Function, etc. The optimal training point in each iteration is x*.

[0073] In step S9, the stopping criterion judgment rule of active learning is as follows:

[0074] The relative error of failure probability prediction is:

[0075]

[0076] in, is the predicted failure probability, P f is the “true” failure probability derived from MCS; N WSP is the number of samples whose signs of g(x) are incorrectly predicted; N Fis the number of failed samples. N F and N WSP It obeys normal distribution, that is

[0077]

[0078]

[0079] Obtain The confidence interval of To conservatively estimate ε. The stopping criterion is expressed as:

[0080]

[0081] If the above formula is satisfied for five consecutive iterations, the learning process is terminated and the failure probability of the high-dimensional composite structure to be analyzed is output. If it is not satisfied, the optimal training point is added into DoE and the RBF model is retrained until the conditions are met.

[0082] The beneficial effects of the present invention are:

[0083] 1. Calculating the RBF matrix in the form of matrix operations can improve the training speed of the RBF model.

[0084] 2. Carrying out sub-model prediction in the form of matrix operations can greatly reduce the sub-model prediction time.

[0085] 3. Combine the RBF model with active learning and extend it to the field of reliability analysis of high-dimensional composite materials. The model is suitable for reliability analysis of high-dimensional composite materials.

[0086] 4. The present invention can improve the fitting speed of the RBF model, combine the RBF model with active learning and expand it to high-dimensional problems, quantify the influence of random variables and related parameters on the reliability of high-dimensional composite material structures, and effectively improve the computational efficiency of high-dimensional composite material structure reliability analysis problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 is a logic flow chart of the present invention;

[0088] Figure 2 This is a schematic diagram of the composite reinforced plate structure of Case 2;

[0089] Figure 3 For the AL-MRBF-MCS learning process;

[0090] Figure 4 Validation of the MRBF model. DETAILED DESCRIPTION

[0091] The embodiments of the present invention are further described below in conjunction with the accompanying drawings.

[0092] Example:

[0093] like Figure 1 As shown, a high-dimensional composite material structure reliability analysis method based on an active learning model includes the following steps:

[0094] S1. Prepare a training sample set of high-dimensional composite structures and calculate the true value of the objective function to form an experimental design (DoE);

[0095] Latin hypercube sampling is used to extract random samples in the original high-dimensional variable space and calculate the true value of the objective function. (i) (i=1,…,m) is the m-dimensional input variable, g(x (i) ) is the objective function, and the current DoE(x (i) ,g(x (i) ))(i=1,…,m);

[0096] S2, generate candidate sample points, estimate failure probability and prediction error;

[0097] Generate a large number of candidate points through MCS to estimate the failure probability and prediction error, and select additional training points from them;

[0098] S3, based on the current DoE, train the full RBF model;

[0099] Substituting into the DoE, we get the linear equation system:

[0100] g=Φβ

[0101] Where Φ is the RBF matrix of the training point, expressed as:

[0102]

[0103] where d i,j =||x (i) -x (j )||,g=[g(x (1) ),…,g(x (m) )] T ;

[0104] S4, obtaining the optimal shape parameters of the full RBF model;

[0105]

[0106] Among them, β i is the i-th element of β, is Φ -1 The i-th diagonal element of ;

[0107] S5, calculating the RBF matrix;

[0108] The Euclidean distance matrix operation method is used to calculate the RBF matrix. Matrix operations can make full use of the parallel instruction set for matrix operations in modern computer CPUs to carry out parallel operations; matrix operations can make full use of the highly optimized linear algebra basic library in high-level programming languages ​​to reduce memory access delays. This makes the matrix operation speed of large-scale data sets significantly faster than the traditional element-level operation speed under the same time complexity. Therefore, the Euclidean distance matrix operation method is used to calculate the RBF matrix.

[0109] All prediction points X=[x1,…,x N ] and training point X T =[x (1) ,…,x (m) ] is called the Euclidean distance matrix, denoted by D:

[0110]

[0111] The Euclidean distance matrix can be completed using matrix operations.

[0112]

[0113]

[0114] And considering

[0115]

[0116] D can be expressed in the following matrix form:

[0117]

[0118] in, Represents element-wise square root. s X and T The matrix operation form is:

[0119]

[0120]

[0121] in is the Hadamard product. So, find the RBF matrix F X All formulas of are expressed in the form of matrix operations.

[0122] If D is known, the RBF matrix F X It can be easily obtained by matrix operation:

[0123]

[0124] Where c is the shape parameter of the radial basis function.

[0125] There are many types of radial basis functions, the common ones are Gaussian radial basis function (GS-RBF), thin plate spline function (TH-RBF), multi-quadratic function (MQ-RBF), etc. See Table 1 for details.

[0126] Table 1 Common radial basis functions

[0127]

[0128] S6, obtaining the predicted values ​​of all RBF sub-models;

[0129] To obtain N unknown points x1,…,x N If we want to obtain prediction information of m sub-models for N unknown points x1,…,x N If the training points are removed from the DoE, the radial basis function model is:

[0130] Φ (-i) β (-i) =g (-i)

[0131] Among them, g (-i) =[g(x (1) ),…,g(x (i-1) ),g(x (i+1) ),…,g(x (m) )] T is to remove g(x (i) )The vector obtained. Φ (-i) is the training point [x (1) ,…,x (i-1) ,x (i+1) ,…,x (m) ] T The RBF matrix is ​​formed by removing the i-th row and i-th column of Φ.

[0132] After removing a training point, the predicted value of the RBF model at the unknown point x is for:

[0133]

[0134] Among them, (Φ (-i) ) -1 Expressed as:

[0135]

[0136] in, is the matrix Φ -1 The m-1-order square matrix after removing the elements in the i-th row and i-th column; is the matrix Φ -1 The m-1 row vector after removing the i-th element from the i-th column vector; is Φ -1 The i-th row and i-th column elements of . In this way, we get β (-i) expression:

[0137]

[0138] In order to facilitate matrix operations, a “zero padding strategy” is proposed to uniformly express the prediction results of the RBF sub-model as follows:

[0139]

[0140] in,

[0141]

[0142] The matrix composed of these predicted values ​​is recorded as

[0143]

[0144] In this way, m sub-models are used for N unknown points X = [x1,…,x N The prediction result of ] can be expressed in the following matrix form:

[0145]

[0146] Where B = [β1,…,β m ]. In this way, the predicted values ​​of m RBF sub-models of x are obtained.

[0147] S7, calculating the predicted mean and predicted variance;

[0148] Following Jackknife's method, define the predicted mean and variance of x:

[0149]

[0150]

[0151] and

[0152] S8, construct an active learning function to obtain the optimal training point;

[0153] You can choose a variety of learning functions such as U learning function, EFF learning function, CV g | function, etc. The optimal training point in each iteration is x*.

[0154] S9. According to the optimal training point obtained in step S8, a stop criterion for active learning is determined, and the failure probability of the high-dimensional composite material structure to be analyzed is output.

[0155] Calculate the convergence criterion. If it is satisfied, output the failure probability. Otherwise, add the optimal training point to DoE and retrain the RBF model.

[0156] The relative error of failure probability prediction is:

[0157]

[0158] in, is the predicted failure probability, P f is the “true” failure probability derived from MCS; N WSP is the number of samples whose signs of g(x) are incorrectly predicted; N F is the number of failed samples. N F and N WSP It obeys normal distribution. but:

[0159]

[0160]

[0161] Obtain The confidence interval of To conservatively estimate ε. The stopping criterion is expressed as:

[0162]

[0163] If the above equation is satisfied for 5 consecutive iterations, the learning process is terminated. Otherwise, the optimal training point is added to DoE to retrain the RBF model.

[0164] At the same time, an explicit nonlinear function is introduced as a test to verify the above steps. The dimension D is set to 40, 50 and 100, where:

[0165]

[0166] The experimental results are shown in Table 2. It can be seen that the number of training points required by the AL-MRBF-MCS method proposed in the present invention is less than that of other methods. When D=100, the reason why AL-MRBF-MCS requires so much CPU time is that MRBF needs to train 10 7 However, the calculation time is only 50 minutes, which is acceptable considering the complex finite element analysis.

[0167] Table 2 Comparison of results of different methods

[0168]

[0169] The following is an example of a high-dimensional composite material structure reliability analysis problem:

[0170] like Figure 2 The figure shows a composite stiffened panel with skin width W = 610 mm and length L = 865 mm. The spacing between stiffeners S = 17.8 mm. The thickness of a single layer of the composite laminate is 0.125 mm, and the material parameters are: elastic modulus E1 in the first direction, elastic modulus E2 in the second direction, Poisson's ratio μ, shear modulus G12, G13 and G23. The skin contains 16 layers of plies with ply angles θ1 to θ16. The stiffener cap has 50 layers with ply angles θ17 to θ66. The stiffener web has 22 layers with ply angles θ67 to θ88. The stiffener flange has 22 layers with ply angles θ89 to θ110. Four groups 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 3. The finite element model of the composite stiffened plate was established using ABAQUS, where 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.

[0171] Table 3 Random variable distribution parameters

[0172]

[0173] 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 96kN, so the performance function is defined as

[0174] g(x)=λ(x)F0-96

[0175] 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.

[0176] The failure probability is solved using the proposed method. The calculation results are shown in Table 4. The initial number of samples of AL-MRBF-MCS is 100. After 52 learning iterations, 52 training points are added and converged. Figure 3 As shown in , most of the added training points are located near the limit state surface. The change curves of failure probability and estimated relative error during the iteration process are shown in Figure 3 shown.

[0177] In order to verify the accuracy of the proposed method, in the variable space [μ j -5σ j ,μ j +5σ j ] 500 groups of samples were selected as training points to obtain the global approximate MRBF model. In order to verify the accuracy of the model, another 300 groups of test samples were generated. The predicted values ​​of the global MRBF model at the test samples are shown in Figure 4 As shown. Determination coefficient R 2 It reaches 0.9996. This shows that the global MRBF model is accurate enough to obtain accurate failure probability. The failure probability prediction value of the global MRBF model is 0.00287, while the calculation result of the proposed method is 0.00285. The proposed method only mobilizes 152 finite element calculations to obtain highly accurate prediction results.

[0178] Table 4 Calculation results of composite stiffened plate

[0179]

[0180] The above example results show that the AL-MRBF-MCS model proposed in this application can well evaluate the reliability of high-dimensional composite materials. In terms of computational cost, the AL-MRBF-MCS 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 problem of reliability analysis of high-dimensional composite materials, greatly reduces the amount of calculation, and improves computational efficiency.

[0181] The above embodiments are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any changes that adopt the design principles of the present invention and are made through non-creative labor on this basis should fall within the protection scope of the present invention.

Claims

1. A high-dimensional composite material structure reliability analysis method based on an active learning model, characterized in that: The steps include: S1. Prepare a training sample set of high-dimensional composite structures and calculate the true value of the objective function to form an experimental design (DoE); S2, generate candidate sample points, estimate failure probability and prediction error; S3, based on the current DoE, train the full RBF model; S4, obtaining the optimal shape parameters of the full RBF model; S5, calculating the RBF matrix; S6, obtaining the predicted values ​​of all RBF sub-models; S7, calculating the predicted mean and predicted variance; S8, construct an active learning function to obtain the optimal training point; S9, determining a stop criterion for active learning according to the optimal training point obtained in step S8, and outputting a failure probability of the high-dimensional composite material structure to be analyzed; In step S5, the RBF matrix is ​​calculated using the Euclidean distance matrix operation method; all prediction points X = [x1, ..., x N ] and training point X T =[x (1) ,…,x (m) ] is called the Euclidean distance matrix; The step S6, using a matrix operation method to calculate the predicted values ​​of all RBF sub-models; To obtain N unknown points x1,…,x N If we want to obtain prediction information of m sub-models for N unknown points x1,…,x N The predicted value of If the i-th training point is removed from the DoE, the radial basis function model is: F (-i) b (-i) =g (-i) Among them, g (-i) =[g ( x (1) ) ,…,g ( x (i-1) ) ,g ( x (i+1) ) ,…,g ( x (m) ) ] T is to remove g from g ( x (i) ) The resulting vector, Φ (-i) is the training point [x (1) ,…,x (i-1) ,x (i+1) ,…,x (m) ] T The RBF matrix is ​​obtained by removing the i-th row and i-th column elements of Φ; Among them, (Φ (-i) ) -1 Expressed as: in, is the matrix Φ -1 The m-1-order square matrix after removing the elements in the i-th row and i-th column; is the matrix Φ -1 The m-1 row vector after removing the i-th element from the i-th column vector; is Φ -1 The i-th row and i-th column element of ; thus, we get β (-i) expression: In order to facilitate matrix operations, a "zero padding strategy" is proposed to uniformly express the prediction results of the RBF sub-model as follows: in, The matrix composed of these predicted values ​​is recorded as In this way, m sub-models are used for N unknown points X = [x1,…,x N The prediction result of ] is expressed in the following matrix form: Where B = [β1,…,β m ], thus obtaining the predicted values ​​of m RBF sub-models of x.

2. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S1, the high-dimensional composite material training sample set is extracted in the original high-dimensional variable space using the Latin hypercube sampling method, and the true response value of the objective function is calculated to form an experimental design DoE (x (i) ,g(x (i) ))(i=1,…,m).

3. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S2, a large number of candidate points are generated through MCS to estimate the failure probability and prediction error.

4. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S3, the expression of the RBF model is: Among them, ||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 point x and any training point x (i) The RBF function between is: The vector formed by the RBF function between point x and all training points is: g=Φβ Where Φ is the RBF matrix of the training point, expressed as: where d i,j =||x (i) -x (j) ||, g = [g(x (1) ),…,g(x (m) )] T .

5. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S4, the optimal shape parameter expression of the full RBF model is obtained as follows: Among them, β i is the i-th element of β, is Φ -1 The i-th diagonal element of .

6. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S5, the RBF matrix is ​​calculated using the Euclidean distance matrix operation method; all prediction points X = [x1, ..., x N ] and training point X T =[x (1) ,…,x (m) ] is called the Euclidean distance matrix, denoted by D: If D is known, the RBF matrix F X It can be easily obtained by matrix operation: The Euclidean distance matrix can be completed using matrix operations, let: D can be expressed in the following matrix form: in, Represents element-wise square root. s X and T The matrix operation form is: in is the Hadamard product.

7. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S7, the predicted mean and predicted variance are calculated according to the Jackknife method, and the predicted mean and standard deviation of x are defined as: and 8. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S8, the active learning function is as follows: Choose from a variety of learning functions, including U learning function, EFF learning function, CV |g| Function, the optimal training point in each iteration is x*.

9. The high-dimensional composite material structure reliability analysis method based on active learning model according to claim 1 is characterized in that: In step S9, the stopping criterion judgment rule of active learning is as follows: The relative error of failure probability prediction is: in, is the predicted failure probability, P f is the "real" failure probability derived from MCS; N WSP is the number of samples whose signs of g(x) are incorrectly predicted; N F is the number of failed samples, N F and N WSP It obeys normal distribution, that is Obtain The confidence interval of To conservatively estimate ε ， The stopping criterion is expressed as: If the above formula is satisfied for five consecutive iterations, the learning process is terminated and the failure probability of the high-dimensional composite structure to be analyzed is output. If it is not satisfied, the optimal training point is added into DoE and the RBF model is retrained until the conditions are met.

Images