Mechanical structure reliability evaluation method based on hybrid R-Vine Copula model

Abstract

The invention discloses a mechanical structure reliability evaluation method based on a hybrid R-Vine Copula model, and the method comprises the steps: firstly, selecting an optimal R-Vine tree type according to sample data by employing an AIC criterion and the basic definition of an R-Vine tree type, employing a hybrid Pair-Copula function for the first layer of R-Vine tree type to improve the sample adaptability, employing Clayton, Gumbel and Frank Copula functions to construct a hybrid Copula function, and employing a Pair-Copula function to construct a Copula model; calculating a weight coefficient and a dependent parameter of the mixed Pair-Copula function by using an expectation maximization algorithm; a subsequent tree type is constructed by adopting a single Pair-Copula function, so that the optimal balance between the calculation efficiency and the calculation precision is achieved, and the optimal construction based on the mixed R-Vine Copula model is completed; and finally, solving the reliability index and the failure probability of the structure based on a mixed R-Vine Copula model in combination with an improved first-order second-order moment method.

Description

Technical Field

[0001] The present invention relates to the field of mechanical structure reliability assessment, and particularly to a mechanical structure reliability assessment method based on a hybrid R-Vine Copula model. Background Art

[0002] For high-precision and high-tech products, the structural reliability of complex equipment is crucial. Especially for extreme manufacturing, such as extremely large structural weights, extremely small structural dimensions, and extremely high product operating speeds, etc., such structures are often related to the major national life and property safety. Therefore, their reliability and safety cannot be underestimated.

[0003] Regarding the design of mechanical structures, the existing design methods have the following problems:

[0004] 1. In the design stage of construction machinery structures, uncertain factors are inevitable and cannot be ignored. In the design and manufacturing stage of product structures, ignoring reliability analysis related to uncertainties will have a huge impact on the safe and reliable operation of the structures, misestimate the expected life of the product structures, leaving potential safety hazards. Excessive redundant design will cause a large amount of waste of financial and human resources, especially in the case of a huge market entity in our country.

[0005] 2. Regarding the reliability design method of mechanical structures, currently, the structural reliability analysis based on Copula functions mainly uses multivariate Copula functions and R-Vine Copula models to describe the correlation problems among multivariate data. However, there are certain limitations. For example, the selection of the R-Vine Copula model is fixed and single Copula models are not sufficient to accurately fit the sample data model, and the solution of relevant parameters is inaccurate. For the situation of coexistence of high-dimensional complex multi-source correlations, it is unable to cope effectively. Therefore, it is urgent to study a mathematical correlation model that can handle the coexistence of high-dimensional complex multivariate correlations, so as to be able to describe variable correlations more flexibly, efficiently, and accurately, improve sample adaptability, which is of great significance for improving the accuracy of multivariate structural reliability analysis results. Summary of the Invention

[0006] In order to overcome the above problems, the present invention proposes a mechanical structure reliability assessment method based on a hybrid R-Vine Copula model.

[0007] The technical solution adopted by the present invention to solve its technical problems is: a mechanical structure reliability assessment method based on a hybrid R-Vine Copula model, including the following steps:

[0008] Step 1: Analyze the random variables in the mechanical structure design process, and establish the limit state function g(X) and failure probability p for mechanical structure reliability assessmentf :

[0009] g(X)=g(x1,…,x d ) Formula (1)

[0010] P f =∫…∫ g(X)≤0 f(x1,…,x d )dx1…dx d Formula (2)

[0011] Wherein, f(x1,…,x d ) represents the joint probability density function of the d-dimensional design variable X=(x1,…,x d );

[0012] Step 2: Based on the collected variable sample point set, use the maximum likelihood estimation method to obtain the parameter estimation values of all marginal Pair-Copula modules of the first-layer tree T1, and then calculate the AIC value. The Pair-Copula module with the smallest AIC value for each marginal is the optimal Pair-Copula module for that marginal:

[0013]

[0014] Wherein, c represents the pair-copula function, θ represents the parameter of the copula, and k represents the number of parameters of the Copula function;

[0015] Step 3: Construct the first-layer tree of the optimal R-Vine tree in the form of a mixed Pair-Copula function, and calculate the weight coefficient and the dependence parameter by using the expectation-maximization algorithm and Formulas (4) and (5);

[0016]

[0017]

[0018] Step 4: Construct all the margins of the j-th layer tree T j , j = 2,..., d - 1, convert the variable sample point set to the conditional variables corresponding to all the margins of the tree T j by using the h function described in Formula (6), and again use the maximum likelihood estimation method to obtain the parameter estimation values of all the marginal Pair-Copula modules of the T j -th layer tree, and then calculate the AIC value until the selection of the optimal Pair-Copula modules for all the margins of the tree T j is completed;

[0019]

[0020] Step 5: Repeat the operation in Step 4 until the optimal Pair-Copula module in tree T is selected; d-1 in;

[0021] Step 6: Construct a joint probability density function based on the optimal Pair-Copula module and the optimal hybrid R-Vine model of the set of the first-layer hybrid Pair-Copula;

[0022] Step 7: Based on the optimal hybrid R-Vine tree type obtained by the point-by-point optimization method, transform the initial sample X s into an independent standard normal vector I s :

[0023]

[0024] where Φ represents the distribution function of the standard normal function;

[0025] Step 8: Solve the reliability index β and the failure probability p by using a first-order second-moment method based on the hybrid R-Vine Copula model f :

[0026]

[0027] P f = Φ(-β) Formula (9)

[0028] wherein, represents the norm of, and the performance function corresponds to G(l).

[0029] Preferably, in Step 3, the first-layer tree type is constructed by using a hybrid Pair-Copula module, the hybrid Copula function is constructed by using Clayton, Gumbel, and Frank Copula functions, and the maximum expectation algorithm is used to solve the weight coefficient and the dependence parameter of the hybrid Copula function.

[0030] Preferably, a first-order second-moment method based on the hybrid R-Vine Copula model in Step 8 adopts an improved HL-RF algorithm to solve the reliability index β and the failure probability p f .

[0031] The beneficial effects of the present invention are as follows:

[0032] 1. Aiming at the first point proposed in the background art, the present invention adopts a point-by-point optimization strategy of the R-Vine Copula tree type to model the uncertain design variables of the mechanical structure, thereby considering the influence of the uncertainty-related variables on the design result.

[0033] 2. Regarding the second point raised in the background art, the present invention proposes a mechanical structure reliability analysis method based on a hybrid R-Vine Copula model. First, according to the sample data, the optimal R-Vine tree is selected using the AIC criterion and the basic definition of the R-Vine tree. For the first-layer tree, i.e., the binary variables, a hybrid Copula function is used to describe them. The hybrid Copula function is constructed using three functions: Clayton, Gumbel, and Frank Copula. Selecting these three types of Copula functions to construct the hybrid Copula can describe the complex correlations of variables, especially the situation where multiple correlations coexist. Secondly, a single Copula form is used for all other R-Vine tree layers to complete the construction of an optimal hybrid R-Vine Copula model. Finally, a hybrid R-Vine Copula model is combined with an improved first-order second-moment method to solve the structural reliability index and failure probability.

[0034] Note: The above designs are not in any particular order, and each one makes the present invention distinct and significantly advanced compared to the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 is a flowchart of a mechanical structure reliability assessment method based on a hybrid R-Vine Copula model of the present invention

[0036] Figure 2 is a schematic diagram of the model variable positions of a tubular cantilever beam in a specific embodiment

[0037] Figure 3 is the optimal tree and tree matrix of the tree-by-tree optimization of a tubular cantilever beam in a specific embodiment

[0038] In the figure, the reference numerals are as follows:

[0039] 1. External force F1 (N) 2. External force F2 (N) 3. External force P (N) 4. Torque T (N·m) 5. Wall thickness t (mm) 6. Pipe diameter d (mm) 7. Angle θ1 8. Angle θ2 9. Length L2 (mm) DETAILED DESCRIPTION OF THE EMBODIMENTS

[0040] The general method of the present invention will be described below with reference to the accompanying drawings:

[0041] As Figure 1 shown, a mechanical structure reliability assessment method based on a hybrid R-Vine Copula model includes the following steps:

[0042] Step 1: Analyze the random variables in the mechanical structure design process, and establish the limit state function g(X) and failure probability p for mechanical structure reliability assessment f :

[0043] g(X) = g(x1, …, x d ) Formula (1)

[0044] P f = ∫…∫ g(X)≤0 f(x1, …, x d )dx1…dx d Formula (2)

[0045] Wherein, f(x1, …, x d ) represents the joint probability density function of the d-dimensional design variable X = (x1, …, x d );

[0046] Step 2: Based on the collected variable sample point set, use the maximum likelihood estimation method to obtain the parameter estimation values of all marginal Pair-Copula modules of the first-layer tree T1, and then calculate the AIC value. The Pair-Copula module with the smallest AIC value for each margin is the optimal Pair-Copula module for that margin:

[0047]

[0048] Wherein, c represents the pair-copula function, θ represents the parameter of the copula, and k represents the number of parameters of the Copula function;

[0049] Step 3: Construct the first-layer tree of the optimal R-Vine tree in the form of a mixed Pair-Copula function. Through the expectation-maximization algorithm, use Formulas (4) and (5) to calculate the weight coefficient and the dependence parameter respectively;

[0050]

[0051]

[0052] Step 4: Construct all margins of the jth-layer tree T j , j = 2, …, d - 1. Convert the variable sample point set to the conditional variables corresponding to all margins of the tree T through the h function described by Formula (6). Again, use the maximum likelihood estimation method to obtain the parameter estimation values of all marginal Pair-Copula modules of the T j layer tree, and then calculate the AIC value until the selection of the optimal Pair-Copula module for all margins of the tree T j is completed; j Select the best of all marginal optimal Pair-Copula modules;

[0053]

[0054] Step 5: Repeat the operation of Step 4 until the tree T is selected.d-1 Optimal Pair-Copula module in

[0055] Step 6: Construct the joint probability density function based on the optimal hybrid R-Vine model of the optimal Pair-Copula module and the first-layer hybrid Pair-Copula;

[0056] Step 7: Based on the optimal hybrid R-Vine tree obtained by the point-by-point optimization method, transform the initial sample X s into the independent standard normal vector I s :

[0057]

[0058] where Φ represents the distribution function of the standard normal function;

[0059] Step 8: Use a first-order second-moment method based on the hybrid R-Vine Copula model to solve the reliability index β and the failure probability p f :

[0060]

[0061] P f = Φ(-β) Formula (9)

[0062] where, represents the norm of, and the performance function corresponds to G(l).

[0063] To further elaborate on the present invention in more detail, a specific embodiment is hereinafter combined to illustrate the solution of the present invention. This embodiment takes the reliability design of a tubular cantilever beam structure as an example, and is implemented on the premise of the technical solution of the present invention, and gives the detailed implementation manner and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0064] As Figure 2 shown, it is a schematic diagram of the model of the tubular cantilever beam structure targeted by the method of the present invention. Implement according to Figure 1 the process shown, a method for mechanical structure reliability assessment based on a hybrid R-Vine Copula model, and its specific steps are as follows:

[0065] Step 1: Analyze the uncertainty variables of the tubular cantilever beam structure. Now, the wall thickness t (mm), the pipe diameter d (mm), the angle θ1, the angle θ2, and the length L2 (mm) are sequentially used as five-dimensional random variables X = (X1, X2, X3, X4, X5), and its limit state function:

[0066] g(t, d, θ1, θ2, L1, L2, F1, F2, P, T) = σ max -σ s Equation (10)

[0067] Table 1 Distribution Parameters of Random Variables of Tubular Cantilever Beam

[0068]

[0069] Step 2: Based on the variable sample point set collected from the tubular cantilever beam structure, use the maximum likelihood estimation method to obtain the parameter estimates of all marginal Pair-Copula modules of the first-layer tree T1. Subsequently, calculate the AIC value. The Pair-Copula module with the smallest AIC value for each marginal is the optimal Pair-Copula module for that marginal; for the five-dimensional problem of the above tubular cantilever beam structure, the first-layer tree has marginal edges. Therefore, perform AIC selection for each marginal. The alternative Copulas are as follows: Gaussian, t, Clayton, Gumbel, Frank, Joe Copula and their corresponding rotated types. The Copula with the smallest calculated AIC value is used as the optimal Copula; according to the AIC values of each marginal shown in Table 2, the first-layer margins are selected in ascending order of AIC as follows: 34, 45, 12, 13;

[0070]

[0071] Table 2 Optimal Copula Functions and Their AIC Values for All Marginal Combinations of the First-Layer Tree

[0072]

[0073] Step 3: Construct the first-layer tree of the optimal R-Vine tree in the form of a mixed Pair-Copula function. Through the expectation-maximization algorithm, use Equation (12) and Equation (13) to calculate the weight coefficient and the dependence parameter respectively;

[0074]

[0075]

[0076] Table 3 Weight Coefficients and Dependence Parameters of Mixed Copulas

[0077]

[0078] Step 4: Construct all marginal edges of the j-th layer tree T j , j = 2, …, d - 1, and convert the variable sample point set to the tree T through the h function described by Equation (14)j For the conditional variables corresponding to all edges, the maximum likelihood estimation method is used again to obtain the parameter estimation values of all edge Pair-Copula modules of the $T$-th layer tree. Subsequently, the AIC value is calculated until the optimal Pair-Copula module selection for all edges of tree $T$ is completed; j j

[0079]

[0080] Step 5: Repeat the operation of Step 4 until the optimal Pair-Copula module in tree $T$ is selected; d-1

[0081] For the second-layer tree, the following edge combinations exist for the 4 nodes described in Step 2: 23|1, 14|3, 35|4. According to the judgment method in Step 4, only 3 edges are required for the second layer, and there are only 3 edges available for selection in the second layer, which conforms to the definition of the R-Vine tree type. The number of available edges in the second-layer tree type is equal to the number of required edges. Therefore, this tree type meets the requirements and is uniquely determined, and there is no need to select points one by one for subsequent tree layers; After selecting the best one layer by layer and point by point, the finally selected tree type is configured as shown in the appendix; Figure 3

[0082] Step 6: Based on the optimal Pair-Copula module and the set of optimal mixed R-Vine models of the first-layer mixed Pair-Copula, construct the joint probability density function. The joint probability density function of the optimal tree type is shown in formula (15);

[0083]

[0084] Step 7: Based on the optimal mixed R-Vine tree type selected point by point, transform the initial sample $X$ s into an independent standard normal vector $I$ s :

[0085]

[0086] where $\Phi$ represents the distribution function of the standard normal function;

[0087] Step 8: Use a first-order second-moment method based on the mixed R-Vine Copula model to solve the reliability index $\beta$ and the failure probability $p$ f :

[0088]

[0089] $P$ f $=\Phi(-\beta)$ Formula (18)

[0090] where,​​​​ denote the norm of, and the corresponding performance function is G(l).

[0091] In this embodiment, according to a structural reliability analysis method based on a hybrid R-Vine Copula model, after selecting the optimal R-Vine tree type, the weight coefficients and dependence parameters of the hybrid Copula, the first-order second-moment method based on the hybrid R-Vine Copula model (MRVC-FORM), the first-order second-moment method based on the non-hybrid R-Vine Copula model (RVC-FORM), and the reliability solution method based on the Monte Carlo simulation method (MCS) are calculated respectively. Finally, the reliability results are shown in Table 4. The results show that the average errors of the reliability index and failure probability of MRVC-FORM are only 0.64% and 1.87% respectively, while the average errors of the reliability index and failure probability of RVC-FORM without using a hybrid R-Vine Copula model reach 14.04% and 34.39%. This shows that using a hybrid R-Vine Copula model can effectively improve the accuracy of the reliability analysis results.

[0092] Table 4 Reliability analysis results of the tubular cantilever beam structure under different thresholds and their average errors compared with the Monte Carlo simulation

[0093]

[0094] The above detailed description is a specific description of the feasible embodiments of the present invention. This embodiment is not intended to limit the patent scope of the present invention. Any equivalent implementation or modification without departing from the present invention shall be included in the patent scope of this case.

Claims

1. A mechanical structure reliability assessment method based on a hybrid R-Vine Copula model, characterized in that: The steps include: Step 1: Analyze the random variables in the mechanical structure design process and establish the limit state function g(X) and failure probability p for mechanical structure reliability evaluation f : g(X)=g(x1,…,x d ) Formula (1) P f =∫…∫ g(X)≤0 f(x1,…,x d )dx1…dx d Formula (2) In the formula, f(x1,…,x d ) represents the d-dimensional design variable X=(x1,…,x d )’s joint probability density function; Step 2: Based on the collected variable sample point set, use the maximum likelihood estimation method to obtain the parameter estimates of all edge Pair-Copula modules of the first layer tree T1, and then calculate the AIC value. The Pair-Copula module with the smallest AIC value on each edge is the optimal Pair-Copula module for that edge: Where c represents the pair-copula function, θ represents the copula parameter, and k represents the number of parameters of the Copula function; Step 3: The first layer of the optimal R-Vine tree is constructed in the form of a hybrid Pair-Copula function, and the weight coefficient and dependent parameter are calculated using the maximum expectation algorithm using formula (4) and formula (5); Step 4: Construct the j-th level tree T j , j = 2, ..., d-1, and transform the variable sample point set into the tree T through the h function described by formula (6). j The conditional variables corresponding to all edges are estimated again using the maximum likelihood estimation method to obtain the Tth j The parameter estimates of all edge Pair-Copula modules of the layer tree are then calculated, and the AIC value is then calculated until the tree T is completed. j The optimal Pair-Copula module of all edges is selected; Step 5: Repeat step 4 until tree T is selected. d-1 The optimal Pair-Copula module in; Step 6: Construct a joint probability density function based on the optimal hybrid R-Vine model of the optimal Pair-Copula module and the first layer of hybrid Pair-Copula; Step 7: Based on the optimal hybrid R-Vine tree obtained by the point-by-point optimization method, the initial sample X s Transformed into independent standard normal vector I s : Where Φ represents the distribution function of the standard normal function; Step 8: Use a first-order second moment method based on the hybrid R-Vine Copula model to solve the reliability index β and failure probability p f : P f = Φ(-β) Equation (9) In the formula, express The norm of , the corresponding functional function is G(l).

2. The mechanical structure reliability assessment method based on the hybrid R-Vine Copula model according to claim 1 is characterized in that: In the step 3, the first layer tree type is constructed using a hybrid Pair-Copula module, the Clayton, Gumbel, and Frank Copula functions are used to construct a hybrid Copula function, and the maximum expectation algorithm is used to solve the weight coefficient and dependent parameters of the hybrid Copula function.

3. The mechanical structure reliability assessment method based on the hybrid R-Vine Copula model according to claim 1 is characterized in that: In step 8, a first-order second moment method based on a hybrid R-Vine Copula model uses an improved HL-RF algorithm to solve the reliability index β and the failure probability p f .

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