A Copula-based global sensitivity analysis method for aircraft engine disks

Abstract

This invention discloses a Copula-based global sensitivity analysis method for aero-engine rotor disks. First, finite element simulation is performed on the aero-engine rotor disk structure to obtain the Mises stress distribution, enabling the division of hazardous areas, determination of failure modes, and identification of uncertain input parameters. Next, a set of independent samples following a standard uniform distribution is generated based on the dimension of the input parameters. Conditional samples of this set are generated using the Vine-Copula function, and inversely transformed according to the input parameter distribution. The transformed samples are used as new input variables. Finally, the input samples are divided into several equal-probability spaces, and the mean, variance, expectation, and variance of the conditional probabilities within the subspaces are calculated. This allows for the determination and ranking of the importance of input parameters in each hazardous area of ​​the rotor disk structure. This invention overcomes the problem of traditional methods failing to consider the correlation between input variables, filling a gap in the field of global sensitivity analysis for rotor disks.

Description

Technical Field

[0001] This invention relates to the field of aero-engines, and in particular to a method for global sensitivity analysis of engine disks that takes into account correlation. Background Technology

[0002] As a critical component of aero-engines, the turbine disk plays a vital role in the safe fixing and assembly of turbine blades. Operating under high temperature and pressure, the disk is subjected to enormous centrifugal forces and thermal stresses, potentially leading to failure. Furthermore, the prolonged operation and mode transitions of the engine impose significant cyclic loading on the disk, resulting in various failure mechanisms throughout its service life. Due to the complexity and uncertainties of the disk's crack propagation mechanisms, structural reliability analysis of the disk is of great significance for improving the availability, safety, and reliability of aero-engines. For the disk, fracture failure represents one of the most challenging and severe failures that can occur during its operation.

[0003] The structure of an aero-engine rotor disc contains numerous input variables, and these variables exhibit significant uncertainty. This uncertainty can have a substantial impact on structural design, leading to problems such as inaccurate hazardous area delineation, large errors in uncertainty calculation parameters, and discrepancies between importance measurements and actual conditions. Furthermore, the degree of influence of each input variable on structural safety varies. Therefore, prioritizing critical parameters is crucial in structural design and subsequent optimization. Consequently, global sensitivity analysis of the engine rotor is essential. Current engineering techniques often assume that the individual random variables of an engine rotor structure are independent. However, in practical applications, different input random variables are typically correlated, meaning they influence each other. Therefore, global sensitivity analysis must consider the correlation between input variables. Moreover, common global sensitivity analysis methods often require large sample sizes and computational loads to ensure accuracy. For implicit structures like aero-engine rotors, this approach significantly reduces computational efficiency. Therefore, a more efficient global sensitivity analysis method that considers correlation is urgently needed for global sensitivity analysis of aero-engine rotors. Summary of the Invention

[0004] To overcome the shortcomings of existing methods and technologies and solve the problem of prioritizing the importance of input variables for aero-engine rotor disks, this invention proposes a Copula-based global sensitivity analysis method for aero-engine rotor disks. This method can effectively divide the hazardous areas of the engine rotor disk, consider the correlations between various input variables of the engine rotor disk, and prioritize their importance. At the same time, this method improves the computational efficiency of general global sensitivity analysis methods.

[0005] The calculation scheme of this invention is as follows:

[0006] A Copula-based global sensitivity analysis method for aero-engine rotor disks includes the following steps:

[0007] Step 1: Based on the actual working conditions of the aero-engine rotor disk, a deterministic analysis is performed using finite element simulation to extract the Mises stress results of the disk and obtain the stress distribution characteristics of the disk under centrifugal load. Based on the stress distribution and combined with the cross-sectional view of the simulated disk, the critical parts of the disk structure are identified and critical areas are delineated. According to the stress analysis results, four critical areas are delineated: the inner surface of the disk center, the end face of the disk center hole, the disk surface area, and the disk subsurface area.

[0008] Step 2: Based on the Paris-Erdogan crack propagation model and stress-strength interference theory, conduct preliminary work for uncertainty analysis of the engine disk, establish failure modes and limit state functions, and clarify the input random variables and parameters of the engine disk;

[0009] Step 3: Based on the spatial segmentation method, establish a variance global sensitivity expression that considers the correlation between each input variable, calculate the importance of each input random variable in Step 2 and sort them. This method includes three stages: the limit state function fitting stage, the sample generation stage, and the spatial segmentation method to solve the Sobol index.

[0010] 1) Limit state function fitting stage

[0011] ① For each hazardous area defined in step 2, a non-repeating input random variable is extracted based on the sample distribution information using a random sampling method (GLPM-PSS method), and the output response value (Mises stress) corresponding to each input sample is calculated through finite element simulation. 100 sets of input and output samples are calculated as the sample pool.

[0012] ② Select several samples from the sample pool as initial samples, and use the remaining samples as the updated sample pool; construct a Kriging surrogate model using the initial sample points, and continuously update the surrogate model using the U learning function until the surrogate model converges;

[0013] ③The established proxy model is used as an equivalent replacement for the limit state function of the roulette structure.

[0014] 2) Sample generation stage

[0015] ① A set of samples following a standard uniform distribution is drawn using a random sampling method. The Vine Copula function is used to perform correlation processing on the drawn uniformly distributed samples. After processing, the input variables of the samples are not independent of each other but have a certain correlation.

[0016] ② The processed samples are transformed inversely to generate input random variables that take into account correlation and follow the actual distribution of the roulette structure. The generated input random variables will be used as input samples for the global sensitivity analysis of the roulette.

[0017] 3) Solving the Sobol index using the spatial partitioning method

[0018] ① Based on the total variance formula and the total expectation formula, two expressions for the global variance sensitivity index are derived: as well as

[0019] ② Assume random variable X i The value range of is (b1, b2), which is divided into s consecutive non-overlapping subintervals A. k =[a k-1 ,a k ), (k=1,…,s);

[0020] ③ Transform the two indicators in ① into a spatial partitioning form using the expectation and variance formulas and the integral mean value theorem: as well as In the formula For X i Falling into interval A k =[a k-1 ,a k The probability of ).

[0021] ④ Calculate the response value, mean, and variance of the sample generated by the transformation in 2), and divide the sample into s non-overlapping continuous sub-intervals according to ③;

[0022] ⑤ Calculate the probability that the sample falls into each sub-interval and the conditional variance within each sub-interval.

[0023] Step 4: Substitute the relevant parameters calculated in step ③ into the transformed Sobol index for solution to obtain the importance ranking of each input variable of the aero-engine disk structure.

[0024] Beneficial effects

[0025] 1. This invention proposes a global sensitivity analysis method for aero-engine rotor disks, used to quantify the importance of various input variables of the aero-engine rotor disk. Currently, there is a significant gap in global sensitivity analysis of engine rotor disks. The method disclosed in this invention is applied to the field of aero-engine rotor disks. By dividing the rotor disk into dangerous regions to find the most likely points of structural failure, the Copula-spatial segmentation method is used to rank the importance of the rotor disk structural input variables, filling a gap in this field.

[0026] 2. This invention employs Copula and spatial partitioning to solve for the variance sensitivity Sobol index, significantly improving computational accuracy and efficiency compared to existing global sensitivity analysis methods. ① It considers the correlation between input variables. Existing methods generally treat input variables as independent when performing global sensitivity analysis on engineering structures. This invention quantifies the correlation between input variables based on correlation theory and generates the samples required for global sensitivity analysis using VineCopula theory, greatly improving the accuracy of global sensitivity analysis compared to general methods. ② When calculating the global sensitivity Sobol index, this invention uses spatial partitioning to perform an equivalent transformation and solve for the Sobol index. The computational cost of solving the variance global sensitivity index using spatial partitioning depends only on the number of samples N and is independent of the sample dimension n. This significantly reduces computational costs and improves computational efficiency compared to other global sensitivity analysis methods. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below.

[0028] Figure 1 This is a schematic diagram of an aircraft engine disc structure according to an embodiment of the present invention;

[0029] Figure 2 This is a finite element stress cloud diagram of an aero-engine wheel disk according to an embodiment of the present invention;

[0030] Figure 3 This is a schematic diagram illustrating the dangerous zone division of a roulette wheel according to an embodiment of the present invention;

[0031] Figure 4 The result of calculating the global sensitivity index of hazardous area 1 according to an embodiment of the present invention;

[0032] Figure 5 The result of calculating the global sensitivity index of hazardous area 2 according to an embodiment of the present invention;

[0033] Figure 6 The global sensitivity index calculation result of hazardous area 3 in one embodiment of the present invention;

[0034] Figure 7 This is the calculation result of the global sensitivity index of the dangerous area 4 in one embodiment of the present invention;

[0035] Figure 8 This refers to the rate of change of variance of each input variable in hazardous areas 1-4 according to an embodiment of the present invention.

[0036] Figure 9 This is a schematic diagram illustrating the solution of the copula function according to an embodiment of the present invention. Detailed Implementation

[0037] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of the present invention.

[0038] This invention discloses a Copula-based global sensitivity analysis method for aero-engine rotor disks. First, finite element simulation is performed on the aero-engine rotor disk structure. The resulting Mises stress distribution is used to divide the disk into hazardous regions. Failure modes of the rotor structure are determined based on crack propagation theory, and uncertain input parameters are determined according to the rotor's geometry and material properties. Next, a set of independent samples following a standard uniform distribution is generated based on the dimension of the input parameters. Conditional samples of this set are generated using the Vine-Copula function and inversely transformed according to the input parameter distribution. The transformed samples are then used as new input variables. The input samples are further divided into several equal-probability spaces, and the mean, variance, expectation, and variance of the conditional probabilities within each subspace are calculated. This allows for the determination and ranking of the importance of the input parameters in each hazardous region of the rotor structure.

[0039] Step 1: For Figure 1 Considering the aero-engine disk structure shown, a deterministic analysis of the disk structure should be performed before conducting a global sensitivity analysis. First, the surface area of ​​the engine disk is meshed. The purpose of meshing is to clarify the stress distribution of the disk under external load conditions and to delineate critical areas. After finite element simulation calculations, the results are as follows... Figure 2 The stress distribution of the wheel structure is shown, and then it is divided into the following categories based on the stress level: Figure 3 Dangerous areas shown:

[0040] ① Consider using FGH96 alloy as the machining material for the wheel structure. Its material properties are shown in Table 1.

[0041] Table 1 Material Properties

[0042]

[0043] ② In the finite element model of the wheel, a centrifugal force of 38000 rad / min is applied to the wheel, and a uniform temperature field of 500 is applied. The axial displacement at the center hole of the wheel is constrained for stress analysis.

[0044] ③ Based on the finite element simulation results of the roulette wheel, the cross-sectional view of the roulette wheel is divided as follows: Figure 3 The danger zones are shown. The roulette wheel is divided into five zones: Zone 1 represents the inner surface of the central hole, where the cracks are generally semi-elliptical; Zone 2 represents the end face of the central hole, where the cracks are quarter-circular corner cracks; Zone 3 represents the surface area of ​​the wheel, where the cracks are semi-elliptical shallow cracks; Zone 4 represents the subsurface area of ​​the wheel, where the cracks are near-edge elliptical cracks; and Zone 5 represents the interior of the wheel, where the cracks are elliptical. Based on the finite element stress analysis of the roulette wheel model, the areas with higher stress are more dangerous, indicating that the danger zones are mainly concentrated near the central hole of the wheel.

[0045] ④ The element stresses near the center hole of the disk are relatively high in regions 1, 2, 3 and 4, so they are identified as the dangerous areas of the disk.

[0046] Step 2: Perform crack propagation analysis on the wheel to clarify its failure conditions and criteria, and select the types and parameters of input variables for global sensitivity analysis:

[0047] ① The Paris-Erdogan model is used to establish the crack propagation life model for the wheel disk, and its expression is as follows:

[0048]

[0049] In the formula, N is the number of load cycles; a is the size of the crack propagation under N cycles; ΔK is the range of stress intensity factor; and C and m are material parameters.

[0050] ② The crack propagation life of the disk can be obtained by direct integration, i.e.

[0051]

[0052] Where a0 is the initial crack size, a c This represents the critical crack size.

[0053] ③ Based on the crack propagation principle, a function is constructed using the disk life interference model. That is, if the actual crack propagation life of the disk during operation does not exceed the specified lifespan, the disk is considered to have failed. Its expression is:

[0054] G = N0 - N c

[0055] In the formula N c The specified lifespan.

[0056] ④ Based on the finite element analysis results of the wheel, the maximum stress value in each critical area can be determined. For i = 1, 2, 3, 4, considering the randomness of the load and materials, and based on experience taking the coefficient of variation of the load as 0.1, the distribution forms and distribution parameters of each random variable required to calculate the failure probability of the hazardous area are shown in Table 2:

[0057] Table 2 Input Random Variable Parameters

[0058]

[0059]

[0060] ⑤ An adaptive Kriging surrogate model with a roulette structure was established using actual sample data. The initial sample points were 16, the sample pool had 150 samples, and the U learning function was used for point addition training.

[0061] Step 3: Establish a spatial segmentation global sensitivity index that considers the correlation of input variables, calculate the importance of each input random variable in Step 2, and rank them:

[0062] ① Generate a set of mutually independent vectors (q1, q2, ..., q) that follow a standard uniform distribution. n ).

[0063] ② Construct Vine Copula trees and select the optimal Copula function for each variable in the T1 tree based on partially known sample data and the AIC criterion, and calculate the parameters of its Copula function; similarly, calculate the Copula parameters in other Vine Copula trees. It should be noted that in the sensitivity analysis, four input variables were selected, and the Copula modeling should be performed separately for C and m, C and a0, C and σ. max m and a0, m and σ max a0 and σ max Calculating the parameter θ pairwise, that is, calculating the parameter between any two different input variables, such as... Figure 9 As shown, the parameters to be obtained are the copula functions of each level of the tree, and the solution is obtained by solving layer by layer downwards.

[0064] The Copula function expression between the input variables, constructed based on the known engine disk data, is as follows:

[0065]

[0066] In the formula, c(·) is the Copula density function, and f(x) is the density function of the Copula. i ) is x i The marginal probability density function.

[0067] ③ Let u1 = q1, and suppose the input sample considering correlation is x = (x1, x2, ... x n Then you can get make Where C 12 (·) is a Copula function between the first and second dimension input variables, thus yielding u2 = h -1 (q2,u1) and By analogy, we can obtain a sample of input random variables (x1, x2, ... x) that takes correlation into account. n ).

[0068] ④ Calculate the (x1, x2, ... x) in ③ using the adaptive Kriging surrogate model constructed in step 2. n The corresponding response value Y = {Y1, Y2, ..., Y} N Meanwhile, the mean value corresponding to the response value is calculated. and variance Used for solving the sobol index in subsequent step ⑧.

[0069] ⑤ To transform the SOBO index into an expression that is easy to spatially partition, consider defining it in R. n The model function in space is Y = g(X), and the input variables are X = (X1, X2, ..., X...). n Therefore, the Sobol exponent can be decomposed into:

[0070]

[0071] According to the total variance formula, the above equation can be further expressed as:

[0072]

[0073] Substituting the total expectation formula into the above equation, we get:

[0074]

[0075] ⑥ Spatial segmentation process of the first indicator: The input variable X... i The sample space is divided into s non-overlapping continuous sub-intervals A.k =[a k-1 ,a k ), 1≤k≤s, for the above equation, according to the definition of mathematical expectation and the properties of definite integrals, we can obtain:

[0076]

[0077] By the mean value theorem for integrals, we can obtain that in each subinterval A k =[a k-1 ,a k There exists a point ε. k ∈A k ={a k-1 ≤x i <a k}, so that:

[0078]

[0079] In the formula For X i Falling into interval A k =[a k-1 ,a k The probability of ). When At that time, there is A k →ε k Then we have:

[0080]

[0081] This leads to the first form of the spatial partitioning Sobol index:

[0082]

[0083] ⑦ The spatial segmentation process of the second indicator: For the Sobol indicator in ⑤, the key to its calculation lies in E(E 2 (Y∣X i The solution to )) can be obtained based on the definition of expectation and the mean value theorem for integrals:

[0084]

[0085] In the formula ε k ∈A k ={a k-1 ≤x i <a k} is A k Hit a point, For X i Falling into [a k-1 ,a k The probability of ) can then be transformed into:

[0086]

[0087] This leads to the second form of the spatial partitioning Sobol index:

[0088]

[0089] ⑧ The formula for calculating the solution terms required by the expression after spatial partitioning is given:

[0090] For the first form of the spatial partitioning algorithm, the calculation... For Algorithm 2, calculate

[0091] Step 4: Substitute the relevant data from the engine wheel disc into steps 1-3 to calculate the importance ranking of the wheel disc input variables. The calculation results are as follows: Figures 4-7 As shown in the figure, for danger zone 1, the importance ranking of each input random variable is a0 > σ. 1max >lgC>>m; For hazardous area 2, the importance ranking is: a0>σ 2max >lgC>m; For hazardous area 3, the importance ranking is σ 3max >a0>lgC>>m; For hazardous area 4, the importance ranking is: a0>σ 4max >lgC>>m. For the global sensitivity index calculated by this method, an RMSE check was performed using a simplified MCS method, and the check results are shown in Table 3:

[0092] Table 3. RMSE Verification of Calculation Results for Each Hazardous Area (Compared to Simplified MCS)

[0093]

[0094] Note: Algorithm 1 and Algorithm 2 in the table are the first and second forms of the Sobol index derived in steps ⑥ and ⑦ above, respectively.

[0095] As shown in Table 3, the RMSE index of the method proposed in this invention is very small compared with the simplified MCS, which is within an acceptable range, indicating that the calculation results are accurate. Furthermore, the proposed method achieves an 83.3% reduction in computational load, significantly improving efficiency. In addition, the accuracy of the global sensitivity calculation results considering correlation can be verified by using the controlled variable method. That is, by keeping other variables constant and only changing the interval of a certain variable, the influence of changes in each variable on the system variance can be explored. Figure 8 As shown. By Figure 8It can be seen that in each danger zone, the input variables that have a greater impact on variance fluctuations are the variables with larger sensitivity indices calculated by the proposed method. This shows that the global sensitivity analysis performed by the proposed method after considering correlation is accurate and in line with reality.

[0096] For the input variables to be analyzed, where some data are known and roughly follow a distribution, a large number of randomly sampled data following a distribution is needed for subsequent solution when some data is unknown. This invention utilizes the Copula-based spatial segmentation global sensitivity analysis method to transform these samples into samples that follow an actual distribution. This transformed sample considers both correlation and actual distribution. The transformed samples are used as input samples for the aero-engine wheel structure. A surrogate model is used to estimate its response value under these input samples, and two spatial segmentation methods are used to solve for the Sobol index corresponding to each input variable. The importance of each input variable in the wheel is then ranked. The analytical method disclosed in this invention overcomes the problem of traditional methods failing to consider the correlation of input variables, achieving efficient and accurate solution for global sensitivity analysis of the wheel structure, thus filling a gap in the field of global sensitivity analysis for wheel structures.

[0097] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A Copula-based global sensitivity analysis method for aero-engine rotor disks, characterized in that, Applied to aircraft engine discs, the process includes the following steps: Step 1: Apply finite element simulation to perform stress analysis on the aero-engine disk to obtain the stress distribution characteristics of the disk under centrifugal load; based on the stress distribution and combined with the cross-sectional view of the disk simulation component, identify the dangerous parts of the disk structure and delineate the dangerous areas. Step 2: Determine the failure mode and limit state function of the wheel structure based on the Paris-Erdogan crack propagation model and stress-strength interference theory, and determine the types and distribution of uncertainty input parameters based on the wheel geometry and material properties. Step 3: Perform global sensitivity analysis on the aero-engine rotor disk using the Copula-based spatial segmentation global sensitivity analysis method. Calculate and rank the importance of the uncertainty input parameters mentioned in Step 2. This method includes three stages: limit state function fitting, sample generation, and solving the Sobol index using the spatial segmentation method. Specifically... Step 3.1, Limit State Function Fitting Stage: Using a finite number of known data, the limit state function G of the roulette structure is fitted through the Kriging surrogate model for subsequent sample response value calculation. Step 3.2, Sample Generation Stage: Generate a set of samples that follow a standard uniform distribution. Use the Vine Copula model to perform correlation processing on the samples. Then, use the inverse transformation of the processed samples to generate input random variables that take correlation into account and follow the actual distribution of the roulette structure. Step 3.3, the stage of solving the Sobol index using the spatial partitioning method, firstly, the Sobol index of the global sensitivity analysis method is transformed into a probability P using the expectation, variance formula, and integral mean value theorem. k and conditional variance The form of spatial division; Step 3.4: Calculate the response values ​​of the input random variables described in Step 3.2 based on the Kriging surrogate model obtained in Step 3.1, and calculate the mean of the response values. The variance is calculated, and the sample is divided into s non-overlapping continuous sub-intervals; the probability P of the sample falling into each sub-interval is calculated. k and the conditional variance within each subinterval Step 4, use the P calculated in step 3.4 k , Substitute these values ​​into step 3.3 to solve the problem and obtain the importance ranking of each input variable for the aero-engine disk structure.

2. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 1, characterized in that, In step 2, the limit state function of the aero-engine rotor disk is expressed as: G=N0-N c In the formula N c The crack propagation life of disk N0 is defined as the specified lifespan. According to the crack propagation model, N0 is represented as: In the formula, a0 is the initial crack size, a c ΔK represents the critical crack size, ΔK represents the stress intensity factor range, and C and m are material parameters.

3. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 1, characterized in that, In step 2, the uncertainty input parameters include: material parameters C and m, initial crack size a0, and maximum stress σ in each critical region of the disc. max The parameter distributions are all normally distributed.

4. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 1, characterized in that, The specific steps of the limit state function fitting stage are as follows: Step 3.1.1: Based on each danger zone defined in Step 2, sample non-repeating engine wheel disc input-output data to establish a sample pool for the wheel disc Kriging proxy model; Step 3.1.2: Select initial sample points to establish an initial Kriging surrogate model, and use the U learning function for adaptive learning until convergence; Step 3.1.3: Replace the limit state function of the roulette structure with the established surrogate model for subsequent global sensitivity analysis.

5. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 1, characterized in that, The sample generation stage specifically includes: Step 3.2.1: Generate samples (q1, q2, ... q) that follow a standard uniform distribution through random sampling. n ); Step 3.2.2: Construct a Vine Copula tree and select the optimal Copula function for each variable in the T1 tree based on some known sample data and the AIC criterion, and calculate the parameters of its Copula function; similarly, calculate the Copula parameters in other Vine Copula trees; Step 3.2.3: Based on the constructed Copula function and its parameters, process the independent samples (q1, q2, ... q) generated in Step 3.2.

1. n Transform the sample into one that considers correlation (x1, x2, ... x) n ).

6. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 5, characterized in that, In step 3.2.2, the Copula function is represented as: In the formula, c(·) is the Copula density function, and f(x) is the density function of the Copula. i ) is x i The marginal probability density function.

7. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 5, characterized in that, In step 3.2.3, the sample x = (x1, x2, ... x) is considered for correlation. n The conversion method is as follows: Let u1 = q1, then make C 12 (·) is a Copula function between the first and second dimension input variables, yielding u2 = h -1 (q2,u1) and ...and so on, to obtain the input random variable samples (x1, x2, ... x...) that consider correlation. n ).

8. The Copula-based global sensitivity analysis method for aero-engine rotor disks according to claim 1, characterized in that, The specific steps in solving the Sobol index using the spatial partitioning method are as follows: Step 3.3.1: Based on the total variance formula and the total variance formula, perform an equivalent transformation on the Sobol index of variance sensitivity. The transformation yields two forms: as well as Step 3.3.2, divide it into s consecutive non-overlapping sub-intervals A k =[a k-1 ,a k ), (k=1,...,s), and according to the definition of expectation and the mean value theorem for integrals, the transformed Sobol index is converted into a spatial partitioning form, that is: as well as In the formula For X i Falling into interval A k =[a k-1 ,a k The probability of ).

9. The method for solving the Sobol index according to claim 8, characterized in that: In step 3.4, the mean of the response values ​​is... The probability that a sample falls into each subinterval and the conditional variance within each subinterval The operator P that incorporates the two Sobol index expressions Si k , as well as In this process, the Sobol index of each input variable is obtained.

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