Model global sensitivity evaluation method based on moment integral

Abstract

The invention belongs to the technical field of electric digital data processing, and provides a moment integral-based model global sensitivity evaluation method, a model is a damping oscillator model or a wing surrounding fluid velocity model, and the method comprises the following steps: constructing a damping oscillator or wing surrounding fluid univariate quadrature expression; expanding the univariate quadrature expression into a multivariate quadrature expression based on the tensor product; performing model evaluation based on a multivariable quadrature expression to obtain a function response; calculating a mean value and a total variance of the model response; and calculating first N-order global sensitivity factors of the damping oscillator model or the wing surrounding fluid velocity model, and determining key parameters, which influence position uncertainty and velocity uncertainty, in the first sensitivity parameters, or determining key parameters, which influence steady-state maximum velocity, in the second sensitivity parameters. The method can be applied to uncertainty analysis and evaluation of various engineering scenes, and model global sensitivity evaluation is realized by performing probability analysis and combining motion simulation and the like.

Description

Technical Field

[0001] This invention belongs to the field of electrical digital data processing technology, and in particular, it is a method for evaluating the global sensitivity of a model based on moment integral. Background Technology

[0002] The Sobol index, a key indicator for ranking uncertain variables and reducing computational model complexity under uncertainty, has been widely applied across multiple disciplines. Its advantage lies in treating the model as a black box system, requiring only the model output for evaluation. The practicality of the Sobol index has been demonstrated in numerous interdisciplinary studies, and it has a theoretical connection to more general analysis of variance techniques such as ANOVA decomposition. By calculating the Sobol index, the contribution of input variables to the variability of model output can be quantified, effectively revealing the main effects of each input variable and their interaction effects, providing strong support for further adjustments to model structure or decision-making processes.

[0003] Currently, this index is mainly calculated using Monte Carlo (MC) methods. However, traditional MC methods suffer from sampling uncertainty and require large sample sizes to ensure convergence. Furthermore, since the estimation error of MC methods is inversely proportional to the square root of the sample size, calculating low Sobol exponents is particularly difficult. Quasi-Monte Carlo methods, using low-discrepancy sequences, can significantly improve the convergence rate, but this is affected by the problem dimension. Furthermore, scholars have proposed various improvement schemes to enhance the accuracy and robustness of Sobol exponent estimation: Homma and Saltelli established a MC estimator for the total effect exponent; Archer et al. developed a bootstrap strategy to obtain the confidence region of the Sobol exponent; Sobol and Levitan introduced variance reduction multipliers, equivalent to weighted uniform sampling, to reduce estimation volatility; Sobol and Myshetskaya proposed two techniques to improve the accuracy of small exponents: mean reduction and associated sampling; Owen improved the convergence performance of the MC estimator for small exponents; the dynamic adaptive variance algorithm proposed by Azzini's team performed well in estimating univariate main effects and group effects; Damblin and Ghione developed an adaptive method based on Latin hypercube sampling; Kucherenko and Song verified the advantages of the double cyclic reordering (DLR) method in first-order exponent estimation. Existing improvement methods mostly focus on first-order exponents and generally suffer from problems such as complex sample operations and limited applicability to higher-order exponents. Even the optimal balance scheme proposed by Azzini may still produce negative or fluctuating results when estimating unknown and uniformly small Sobol exponents. Besides Monte Carlo (MC) methods, sensitivity analysis techniques based on surrogate models such as multinomial chaotic expansion, Kriging models, response surface methodology, and machine learning have also been developed. However, except for multinomial chaotic models where variance can be directly calculated, other surrogate models still require evaluation using MC estimators, and their accuracy heavily depends on the design of high-dimensional space sampling strategies. This poses a significant challenge for computationally intensive black-box models such as the finite element method. Therefore, to ensure the non-negativity of the results while avoiding surrogate model construction errors and sampling optimization problems, it is urgent and necessary to seek a global sensitivity evaluation method based on moment integrals that can significantly improve computational efficiency while maintaining accuracy. Summary of the Invention

[0004] This invention addresses the shortcomings of existing technologies by proposing a global sensitivity evaluation method for models based on moment integrals. This method includes constructing Hankel and Jacobi matrices based on origin moments to obtain univariate quadrature expressions; extending the univariate quadrature expressions to multivariate quadrature expressions based on tensor products; evaluating the model based on the multivariate quadrature expressions to obtain the function response; calculating the mean and total variance of the model response; and calculating the first N global sensitivity factors of the model, i.e., the first N Sobol exponents. This invention effectively reduces sampling errors and fluctuations in small exponent estimation, avoids surrogate model construction errors and sampling optimization problems, ensures non-negativity of results, achieves simultaneous calculation of Sobol exponents of all orders, significantly improves computational efficiency, and can be used for global sensitivity evaluation of models in multiple engineering fields.

[0005] In a first aspect, the present invention provides a method for evaluating the global sensitivity of a model based on moment integrals, wherein the model is a damped oscillator model, and the method includes the following steps: S1. Constructing the single-variable quadrature expression for the damped oscillator: Construct a damped oscillator model under external load, determine the random variable of the first sensitivity parameter, and define the first probability density function as follows. Before calculation First-order origin moment Construct the first matrix Second matrix ; Calculate the second matrix eigenvalues With feature vectors Thus, the single-variable integral expression for the damped oscillator is obtained; S2. Constructing the multivariate quadrature expression for the damped oscillator: Based on the tensor product, the univariate quadrature expression for the damped oscillator is extended to a multivariate quadrature expression. Let the i-th dimension be... The dimension of the node product expression is... Multivariable quadrature expression for damped oscillator for: ; in, Represents the i-th dimension The expression for the product of nodes, i = 1, 2, ..., d; S3. Evaluate the damped oscillator model based on the multivariate quadrature expression of the damped oscillator and obtain the position-time response. and speed time response Calculate the location-time response of each group of nodes. and speed time response ,in Indicates the first The first group node One component; S4. Calculate the position-time response of the damped oscillator model using the standard ordinary differential solver. and speed time response The mean and total variance; S5. Calculate the first N global sensitivity factors of the damped oscillator model and determine the key parameters affecting position uncertainty and velocity uncertainty in the first sensitivity parameters: The first N global sensitivity factors of the damped oscillator model include the first global sensitivity factor, the second global sensitivity factor, the third global sensitivity factor, ... and the Nth global sensitivity factor of the damped oscillator model.

[0006] Preferably, the damped oscillator model under external load in step S1 is represented as follows: (1) in, Indicates the damping coefficient; Indicates stiffness; Indicates the amplitude of the external load; Indicates frequency; and These represent the initial position and initial velocity, respectively; the first sensitivity parameter has multiple parameters, including... Furthermore, by using scaling and translation transformations, the first sensitivity parameter is transformed to conform to a standard uniform distribution. A random variable.

[0007] Preferably, for the first matrix in step S1 Decomposition yields: ; in, Represents an upper triangular matrix; forward First-order origin moment for: ; Where E represents expectation; Let j represent a set of randomly generated random variables; j represents the order and ; Indicates the range of the random variable.

[0008] Preferably, in step S1, the first matrix Second matrix They are represented as follows: ; ; in, Let j represent the j-th main variable, and express it as... ; Let j represent the j-th variable, and express it as... .

[0009] Preferably, the total number of nodes in the multivariate product expression in step S2 is... for: ; in, This indicates a multiplication operation.

[0010] Preferably, in step S4, each dimension of the damped oscillator model adopts... Use the dot product expression to calculate the position-time response of the damped oscillator model. and speed time response mean and total variance : ; ; ; .

[0011] Secondly, the present invention provides a method for evaluating the global sensitivity of a model based on moment integrals, wherein the model is a fluid velocity model around an airfoil, and the method includes the following steps: S1. Constructing the univariate quadrature expression for the fluid around the wing: Construct a velocity model for the fluid around the wing, determine the random variable of the second sensitivity parameter, and define the second probability density function as follows. Before calculation Second-order original moment Construct the Hankel and Jacobi matrices and calculate the eigenvalues ​​and eigenvectors to obtain the univariate quadrature expression for the fluid around the wing; the second sensitivity parameters include the angle of attack, the inlet flow centerline velocity, and the Reynolds number; S2. Constructing a multivariate quadrature expression for the fluid around the wing: Based on the tensor product, the univariate quadrature expression for the fluid around the wing is extended into a multivariate quadrature expression for the fluid around the wing; S3. Evaluate the fluid velocity model around the wing based on the multivariate quadrature expression of the fluid around the wing, obtain the fluid velocity time response, and calculate the fluid velocity time response of each group of nodes. S4. Mean and total variance of the fluid velocity time response of the fluid velocity model around the computer wing based on the lattice Boltzmann method; S5. The first N Sobol exponents of the fluid velocity model around the wing are used to determine the key parameters affecting the steady-state maximum velocity in the second sensitivity parameter: The first N Sobol exponents of the fluid velocity model around the wing include the first Sobol exponent of the fluid velocity around the wing, the second Sobol exponent of the fluid velocity model around the wing, the third Sobol exponent of the fluid velocity around the wing, ... and the Nth Sobol exponent of the fluid velocity around the wing.

[0012] Thirdly, the present invention provides a method for evaluating the global sensitivity of a model based on moment integrals, which includes the following steps: S1. Constructing the single-variable quadrature expression: Define the probability density function of the random variable as follows: Before calculation First-order origin moment Construct the Hankel matrix and Jacobi matrix, calculate the eigenvalues ​​and eigenvectors, and obtain the univariate quadrature expression; S2. Constructing a multivariable quadrature expression: Extending the univariate quadrature expression into a multivariate quadrature expression based on the tensor product; S3. Evaluate the model based on the multivariate quadrature expression and obtain the function response: Let the model response function be... Calculate the function response of each group of nodes in the multivariate quadrature expression. ; S4. Calculate the mean and total variance of the model response; S5. Calculate the first three Sobol indices of the model: The first three Sobol indices of the model include the first Sobol index. Second-order Sobol exponent and the third-order Sobol exponent .

[0013] Preferably, in step S5, the first-order Sobol exponent is first determined. Second-order Sobol exponent and the third-order Sobol exponent Then replace 1, 2, and 3 in the result with m, n, and q respectively to obtain , The first constant, the second constant, and the third constant are respectively, and satisfy the following conditions: .

[0014] Compared with the prior art, the technical effects of the present invention are as follows: 1. The global sensitivity evaluation method for models based on moment integrals proposed in this invention constructs a univariate quadrature expression based on the origin moment, and then constructs a multivariate quadrature expression based on tensor product extension. This can effectively avoid sampling errors and fluctuations in small exponent estimation, and ensure the non-negativity of the results. At the same time, the proposed method can effectively avoid the problems of traditional surrogate model construction errors and sampling optimization, and achieve convergence of Sobol exponent results.

[0015] 2. The global sensitivity evaluation method for models based on moment integrals proposed in this invention adopts deterministic nodes and weights in the standard hypercube space, which can realize the simultaneous calculation of all-order Sobol exponents, including mean effect, first / second order and total effect. It significantly improves the computational efficiency while ensuring accuracy, and has stronger practicality and applicability.

[0016] 3. The model global sensitivity evaluation method based on moment integral proposed in this invention can be applied to uncertainty analysis and evaluation in various engineering scenarios. For example, it can be applied to the global sensitivity evaluation of damped oscillator models or the global sensitivity evaluation of fluid velocity models around airfoils. By performing probability analysis and combining it with physics engine, motion simulation, rotation simulation, rolling simulation and collision simulation, the global sensitivity evaluation of the model can be achieved. Attached Figure Description

[0017] Other features, objects, and advantages of this application will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings.

[0018] Figure 1 This is a flowchart of the model global sensitivity evaluation method based on moment integral of the present invention; Figure 2 This is a curve showing the position and velocity changes of the damped oscillator in the first 20 seconds in a specific embodiment of the present invention; Figure 3 This is a first-order Sobol exponential curve of random parameters with respect to the position 20 seconds prior in a specific embodiment of the present invention; Figure 4 This is a first-order Sobol exponential curve of random parameters with respect to the velocity for the first 20 seconds in a specific embodiment of the present invention; Figure 5 This is a second-order Sobol exponential curve of random parameters with respect to the position 20 seconds prior in a specific embodiment of the present invention; Figure 6 This is a second-order Sobol exponential curve of random parameters with respect to the velocity for the first 20 seconds in a specific embodiment of the present invention; Figure 7 This is a first-order Sobol exponential curve of the random parameter position response for the first 20 seconds in a specific embodiment of the present invention; Figure 8 This is a second-order Sobol exponential curve of the random parameter position response in a specific embodiment of the present invention for the first 20 seconds; Figure 9 This is a first-order Sobol exponential curve of the velocity response of random parameters in the first 20 seconds of a specific embodiment of the present invention; Figure 10 This is a second-order Sobol exponential curve of the velocity response of random parameters in the first 20 seconds of a specific embodiment of the present invention; Figure 11 This is an evaluation curve of the first-order Sobol exponential model of the method proposed in a specific embodiment of the present invention; Figure 12 This is an evaluation curve of the first-order Sobol exponential model of the MC method in a specific embodiment of the present invention; Figure 13 This is an evaluation curve of the second-order Sobol exponential model of the method proposed in a specific embodiment of the present invention; Figure 14 This is an evaluation curve of the second-order Sobol exponential model of the MC method in a specific embodiment of the present invention; Figure 15 This is a schematic diagram of a space truss structure in a specific embodiment of the present invention; Figure 16 This is a schematic diagram of the fluid structure around the wing in a specific embodiment of the present invention; Figure 17 This is a specific embodiment of the present invention, showing the dynamic response of airflow at different time steps calculated with different parameters; Figure 18 This is a curve showing the maximum value of the fluid velocity around the wing versus time in a specific embodiment of the present invention; Figure 19 This is a curve showing the change of the Sobol exponent with the number of function evaluations in a specific embodiment of the present invention. Detailed Implementation

[0019] The present application will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other. The present application will now be described in detail with reference to the accompanying drawings and embodiments.

[0020] Figure 1 illustrates the global sensitivity evaluation method based on moment integrals of the present invention, such as... Figure 1 As shown, the method includes the following steps: S1. Constructing the single-variable quadrature expression: Define the probability density function of the random variable as follows: Before calculation First-order origin moment Construct the Hankel matrix and Jacobi matrix, and calculate the eigenvalues ​​and eigenvectors to obtain the univariate quadrature expression.

[0021] S2. Constructing a multivariate quadrature expression: Based on the tensor product, extend the univariate quadrature expression into a multivariate quadrature expression.

[0022] S3. Evaluate the model based on the multivariate quadrature expression and obtain the function response: Let the model response function be... Calculate the function response of each group of nodes in the multivariate quadrature expression. .

[0023] S4. Calculate the mean and total variance of the model response.

[0024] S5. Calculate the first three Sobol indices of the model: The first three Sobol indices of the model include the first Sobol index. Second-order Sobol exponent and the third-order Sobol exponent In step S5, the first Sobol exponent is first determined. Second-order Sobol exponent and the third-order Sobol exponent Then replace 1, 2, and 3 in the result with m, n, and q respectively to obtain , The first constant, the second constant, and the third constant are respectively, and satisfy the following conditions: .

[0025] The method for evaluating the global sensitivity of a model based on moment integrals according to the present invention will be described in detail below with reference to specific embodiments, which includes the following steps: S1. Construct a single-variable product expression, that is, determine the product rule; S11. Define the probability density function of a random variable as follows: Calculate its predecessor First-order origin moment for: ; Where E represents expectation; Let j represent a set of randomly generated random variables; j represents the order and ; Indicates the range of the random variable.

[0026] S12, Construct the first matrix The first matrix is ​​a Hankel matrix, that is: .

[0027] right Perform Cholesky decomposition: ; in, Describes an upper triangular matrix and is: ; in, Indicates the first Line number Column variables.

[0028] S13, Construct the second matrix The second matrix is ​​a Jacobi matrix, that is: ; in, Let j represent the j-th main variable, and express it as... ; Let j represent the j-th variable, and express it as... .

[0029] S14, Calculation eigenvalues With feature vectors This yields the single-variable product expression; eigenvectors ,in, Represents the k-th eigenvector; the univariate quadrature expression includes integration nodes. and the corresponding integral weights And satisfy and .

[0030] For uniform random variables ,That The first-order origin moment is: .

[0031] The corresponding Hankel matrix is: .

[0032] The Jacobi matrix is: .

[0033] The calculated univariate quadrature expressions for the first ten nodes are shown in Table 1.

[0034] Table 1 For the remaining distribution functions are random variables By making transformations Convert it to Substitute the data from Table 1 into the calculation.

[0035] S2. Constructing a multivariate quadrature expression: Based on the tensor product, the univariate quadrature expression is extended into a multivariate quadrature expression. Let the i-th dimension be... The dimension of the node product expression is... Multivariable quadrature expression for: ; in, Represents the i-th dimension Nodal integration expression.

[0036] Total number of nodes in a multivariate product expression for: ; in, This indicates a multiplication operation.

[0037] In one specific embodiment, if all dimensions adopt The expression for the product of nodes is given, in which case the dimension is... Multivariable quadrature expression and the total number of its nodes They are respectively: .

[0038] .

[0039] S3. Evaluate the model based on the multivariate quadrature expression and obtain the function response: Let the model response function be... Calculate the function response of each group of nodes in the multivariate quadrature expression. ,in In a multivariate product expression, the first... The first group node Each component.

[0040] S4. Calculate the mean and total variance of the model response: (This is the process of calculating the mean and total variance of the model response function.) The input parameter distribution is transformed into a uniform random variable. All dimensions adopt The expression for the product of nodes is used to calculate its mean. and total variance for: .

[0041] .

[0042] S5. Calculate the first three Sobol indices of the model: The first three Sobol indices of the model include the first Sobol index. Second-order Sobol exponent and the third-order Sobol exponent . .

[0043] .

[0044] .

[0045] in, express Except for the first All other variables besides the one variable, express Except for the first All other variables besides the one variable, express Except for the first All other variables besides the one variable.

[0046] Other Sobol exponents Need to Replace the corresponding subscript with the corresponding subscript. That's all. The first, second, and third constants are respectively, and satisfy the following conditions: .

[0047] An important improvement of this invention lies in: steps S4 and S5 All calculations have been completed in step S3, meaning the proposed method can calculate all Sobol exponents simultaneously without additional computation, with a total computational cost of [missing information]. .

[0048] In a specific application, this invention provides a method for evaluating the global sensitivity of a damped oscillator model based on moment integrals, which includes the following steps: S1. Constructing the single-variable quadrature expression for the damped oscillator: Construct a damped oscillator model under external load and determine the random variable of the first sensitivity parameter. Its first probability density function is defined as , where random variables Variables include damping coefficient, stiffness, amplitude of external load, initial position and initial velocity of the oscillator, etc. Represents random variables The probability of taking the value of a real number x; before calculation First-order origin moment Construct the first matrix Second matrix ; Calculate the second matrix eigenvalues With feature vectors Thus, the single-variable integral expression for the damped oscillator is obtained. The damped oscillator model under external load is expressed as: (1) in, Indicates the damping coefficient; Indicates stiffness; Indicates the amplitude of the external load; Indicates frequency; and These represent the initial position and initial velocity, respectively; the first sensitivity parameter has multiple parameters, including... Furthermore, by using scaling and translation transformations, the first sensitivity parameter is transformed to conform to a standard uniform distribution. A random variable.

[0049] For the first matrix Decomposition yields: ; in, Represents an upper triangular matrix; forward First-order origin moment for: ; Where E represents expectation; Let j represent the first sensitivity parameter raised to the power of j, where the first sensitivity parameter may include the damping coefficient, stiffness, amplitude of the external load, initial position of the oscillator, and initial velocity; j represents the order and ; Indicates the range of the random variable.

[0050] First matrix Second matrix They are represented as follows: ; ; in, Let j represent the j-th main variable, and express it as... ; Let j represent the j-th variable, and express it as... .

[0051] S2. Constructing the multivariate quadrature expression for the damped oscillator: Based on the tensor product, the univariate quadrature expression for the damped oscillator is extended to a multivariate quadrature expression. Let the i-th dimension be... The nodal quadrature expression, for example, for a damped oscillator model under external load, where the first dimension of the sensitivity parameter represents the damping coefficient, and for this random variable, the sample size is designed as follows: The damping coefficient sample points and their corresponding weights are denoted as . The second dimension of the sensitivity parameter represents stiffness, and the designed sample size is... The stiffness sample points and their corresponding weights are denoted as... ; and so on, the sample size is obtained as External load amplitude sample points and corresponding weights Sample size is Sample points of the initial position of the oscillator and their corresponding weights and sample size is Sample points of the initial velocity of the oscillator Then the dimension is The multivariate integral expression for the damped oscillator, i.e., the generated global sample points of the damped oscillator and their corresponding weights. for: ; in, Represents the i-th dimension The nodal quadrature expression, i = 1, 2, ..., d. Here, the total number of nodes in the multivariate quadrature expression is the global sample size of the generated damped oscillator. for: ; in, This indicates a multiplication operation.

[0052] In a preferred embodiment of the present invention, for the damped oscillator model under external load in S1, the value of d is 5.

[0053] S3. Evaluate the damped oscillator model based on the multivariate quadrature expression of the damped oscillator and obtain the position-time response. and speed time response Calculate the location-time response of each group of nodes. and speed time response ,in Indicates the first The first group node Each component.

[0054] S4. Calculate the position-time response of the damped oscillator model using the standard ordinary differential solver. and speed time response The mean and total variance. All dimensions of the damped oscillator model adopt... Use the dot product expression to calculate the position-time response of the damped oscillator model. and speed time response mean and total variance : ; ; ; .

[0055] S5. Calculate the first N global sensitivity factors of the damped oscillator model to determine the key parameters affecting position and velocity uncertainties among the first sensitivity parameters: the first N global sensitivity factors of the damped oscillator model include the first-order global sensitivity factor, the second-order global sensitivity factor, the third-order global sensitivity factor, ... and the Nth-order global sensitivity factor of the damped oscillator model. In a preferred embodiment of the present invention, step S5 specifically involves: calculating the first N Sobol exponents of the damped oscillator model to determine the key parameters affecting position and velocity uncertainties among the first sensitivity parameters: the first N Sobol exponents of the damped oscillator model include the first-order Sobol exponent, the second-order Sobol exponent, the third-order Sobol exponent, ... and the Nth-order Sobol exponent of the damped oscillator model. In a specific embodiment, a damped oscillator under external load is used as an example for model global sensitivity measurement.

[0056] To examine the performance of the proposed method in calculating the sensitivity index of non-monotonic time-varying processes, a damped linear oscillator described by the following expression was investigated: (1) in, Indicates the damping coefficient; Indicates stiffness; Indicates the amplitude of the external load; Indicates frequency; and These represent the initial position and initial velocity, respectively. Parameters in a damped oscillator that consider the influence of random factors are called random parameters. These parameters are affected by the manufacturing process of the material, microscopic defects, measurement accuracy, and environmental randomness, making it difficult to express them with deterministic values. Therefore, random parameters are embedded as random variables into the model. The random parameters of the damped oscillator model in expression (1) include... The distribution of as a random variable is shown in Table 2.

[0057] Table 2 The position and velocity changes of the damped linear oscillator in the first 20 seconds were calculated. For simplicity, the frequency parameter was set to 1. Equation (1) was solved using a standard ordinary differential solver, and the mean values ​​of position and velocity over time were obtained as follows: Figure 2 As shown.

[0058] By applying scaling and translation transformations, the random parameters are transformed into a standard uniform distribution. The random variables are embedded into the model, and then the global sensitivity factor, i.e., the Sobol exponent, is obtained using the aforementioned method. A 2-node univariate product expression is then used ( The Sobol index at each time point can be obtained through a 32-order function evaluation. Figure 3 and Figure 4 The first-order Sobol exponents of the five random parameters in Table 2 are given. Figure 3 This is a diagram illustrating how the location changes over time. Figure 4 This is a diagram illustrating the change in speed over time.

[0059] The first-order Sobol exponent describes the contribution of a single random parameter to the overall response of a damped oscillator model. Indicates the first in Table 2 The contribution of each random parameter to the overall response of the damped oscillator model. Figure 3 As can be seen from this, the first-order Sobol exponent is... to It exhibits highly nonlinear temporal variation. The main contributing indicators over the time span are... and They correspond to and The results show that the initial position It dominates initially, then its Sobol exponent declines rapidly because the uncertainty of the position variable at the initial moment depends only on the initial position. For Figure 4 The velocity-related first-order Sobol exponent shown is primarily contributed by... It corresponds to the initial velocity. . The term is 1 at the beginning, which is consistent with the previously mentioned fact that at the initial time... The initial velocity determines the uncertainty at that point in time.

[0060] The second-order Sobol exponent describes the contribution of the interaction between two random parameters to the overall response of the damped oscillator model. Indicates the first in Table 2 The and the first The contribution of the interaction of individual random parameters to the overall response of the damped oscillator model. Figure 5 and Figure 6 Schematic diagrams showing the second-order Sobol exponential results for position and velocity as a function of time are displayed. For the position response, and (Right now The interaction of ) in Figure 3 A noticeable bulge appears after about 15 seconds, which is related to Figure 3 The observations are consistent with those in the study, namely and This is the main factor causing positional uncertainty. For the velocity response, the contribution of the interaction term is almost negligible. Figure 6 The maximum Sobol exponent is 0.017. For the third-order Sobol exponent, which is the contribution of the interaction of the three random parameters to the overall response of the damped oscillator model, the value of its maximum Sobol exponent is less than... The difference is negligible, so the results for the third-order Sobol exponent with respect to time are omitted.

[0061] To verify the accuracy of the proposed method, the results are compared with those of the MC method. The sample size for each dimension of the MC estimator is chosen to be... For first-order Sobol exponent estimation, the MC method requires a total of The secondary model evaluation requires a total of [number] steps for estimating the second-order Sobol exponent. The results of the two methods in the second model evaluation are as follows: Figures 7-10 As shown.

[0062] Figure 7 and Figure 8 The results are the first-order and second-order Sobol exponents of the position response, respectively. Figure 9 and Figure 10 The figures show the first and second order Sobol exponents of the velocity response, respectively. As can be seen from the figures, the proposed method achieves results almost identical to MC using only 32 model evaluations. To further verify that the proposed method converges using 32 model evaluations, three nodes are used for each dimension (…). ) and 4 nodes ( The calculations were performed, with a total of 243 and 1024 model evaluations respectively. The results are as follows: Figure 11 As shown in the figure. The result is convergent. Meanwhile, for the MC method, the single-dimensional sample sizes are compared as follows: and The result is as follows Figure 12As shown, where At the 14th second and A significant deviation can be observed at the 18th second, indicating that... The MC method does not converge for the first-order Sobol exponent.

[0063] When engineering problems have small Sobol exponents, the convergence problem of the MC method faces significant challenges, but the proposed method is not limited by this situation. The second-order Sobol exponent in this case is representative of this scenario. Figure 13 Given At that time, the proposed method evaluated the second-order Sobol exponent of the displacement response. It can be seen from this that even for [the following conditions], [the method is effective]. arrive Even with a very small range of Sobol exponents, convergence can be guaranteed with 32 evaluations. For the MC method, Figure 14 China points out The evaluation results of this model are non-convergent, in particular, Figures 7-10 The study points out that the MC method requires 300 million evaluations to converge.

[0064] In a specific example of the present invention, the spatial truss problem will be used as an example for detailed explanation.

[0065] use Figure 15 The proposed method is demonstrated in a spatial truss structure to be applicable to real-world structures. All members are made of aluminum of the same diameter. The numbers in parentheses represent nodes. Figure 15 The various intersection points of the members of the mid-space truss structure. Nodes (1-6) are subjected to random loads. The function of the suffix, in which Represents the node number. The direction of force at node (7-10) represents the force direction; node (7-10) is constrained by the ground. The random variables are the Young's modulus of the material and... The power of time See Table 3 for details.

[0066] Table 3 The goal is to evaluate the effect of each variable on node 1. The contribution of directional displacement variability. A finite element model of the space truss structure is established and solved.

[0067] The Sobol exponents of the 13 random variables in the table above were calculated using both the proposed method and the McLeod method. For the proposed method, a 2-node univariate integral expression was used for each dimension, requiring a total of 8192 evaluations. For the McLeod method, the sample size for each dimension was chosen as [value missing]. A total of 28 is needed. Second evaluation. Used. The results of the MC method were used as the true values ​​to calculate the relative error between the two methods, and the results are shown in Table 4. For all first-order Sobol exponents, the proposed method produces a maximum relative error of 0.56%, which is better than the MC estimation method, and the number of function evaluations required is only a fraction of that of the MC estimation method, less than 1 / 30. It is worth noting that when using the proposed method, second-order and third-order Sobol exponents do not require additional function evaluations, while the number of additional function evaluations required by the MC method exceeds the number required for first-order calculations.

[0068] Table 4 In another specific embodiment of the present invention, the model is a fluid velocity model around the wing or a fluid velocity field model around the wing, which includes the following steps: S1. Constructing the univariate quadrature expression for the fluid around the wing: Construct a velocity model for the fluid around the wing, determine the random variable of the second sensitivity parameter, and define the second probability density function as follows. Before calculation Second-order original moment Construct the Hankel and Jacobi matrices and calculate the eigenvalues ​​and eigenvectors to obtain the univariate quadrature expression for the fluid around the wing; the second sensitivity parameters include the angle of attack, the inlet flow centerline velocity, and the Reynolds number.

[0069] S2. Construct a multivariate quadrature expression for the fluid around the wing: Based on the tensor product, extend the univariate quadrature expression for the fluid around the wing into a multivariate quadrature expression for the fluid around the wing.

[0070] S3. Evaluate the fluid velocity model around the wing based on the multivariate integral expression of the fluid around the wing, obtain the fluid velocity time response, and calculate the fluid velocity time response for each group of nodes.

[0071] S4. Mean and total variance of the fluid velocity time response of the fluid velocity model around the computer wing based on the lattice Boltzmann method.

[0072] S5. The first N Sobol exponents of the fluid velocity model around the wing are used to determine the key parameters affecting the steady-state maximum velocity in the second sensitivity parameter: The first N Sobol exponents of the fluid velocity model around the wing include the first Sobol exponent of the fluid velocity around the wing, the second Sobol exponent of the fluid velocity model around the wing, the third Sobol exponent of the fluid velocity around the wing, ..., the Nth Sobol exponent of the fluid velocity around the wing.

[0073] In this embodiment, the fluid velocity problem around an airfoil is taken as an example. A computationally demanding time-varying fluid problem is introduced to demonstrate the unique advantages of the proposed method.

[0074] The computational cost of current MC-based methods in this field is too high. A NACA 6412 wing with a nominal angle of attack of 3° is placed in a channel, such as... Figure 16 As shown. The velocity profile of the inlet fluid is a quadratic curve, with a nominal velocity of 0.2 at the centerline (dimensionless units are used for demonstration), and a velocity of zero at the channel boundary, i.e., Poiseuille flow.

[0075] The nominal Reynolds number is defined as 20. The model response is the velocity of the fluid around the wing, requiring sensitivity analysis of three variables: angle of attack, inlet centerline velocity, and Reynolds number. The parameters of these three random variables are shown in Table 5; the nominal parameters and their ranges are for demonstration purposes only.

[0076] Table 5 Using a two-dimensional grid The dynamic response of the fluid was computed using the lattice Boltzmann method (LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator. A total of [number missing] were evaluated. At each time step, the airflow around the wing stabilizes after approximately 6,000 time steps. Figure 17 A series of snapshots are shown at different time steps (i.e., 0.2, 20, and 3) calculated using the nominal values ​​of the three parameters in Table 4. Figure 18 The maximum fluid velocity around the wing is shown to vary over time, stabilizing after 6000 time steps. The magnitude of the steady-state maximum velocity around the wing is extracted from the last time step. Clearly, as the three parameters fluctuate around their nominal values, the magnitude of the steady-state maximum velocity will change due to the propagation of uncertainties in the time-varying computational model.

[0077] Clearly, for such a computational model, element-wise calculations of the mesh in nine directions are required at each time step; therefore, the computational cost is extremely high, making the MC-based method unsuitable, while the time-varying characteristics do not allow for robust and efficient surrogate model-based methods. Using the proposed method, a two-point moment quadrature expression is employed for each dimension, i.e., two inlet flow centerline velocity samples, two Reynolds number samples, and two angle-of-attack samples are designed to form eight sets of global random parameter samples, and eight model evaluations are performed. Table 6 lists the evaluation results of the Sobol exponent. Since there is no true solution to this complex problem, further evaluations were conducted to verify the convergence of the results. The Sobol index results are obtained by using 3, 4, and 5 random parameter samples for each random parameter, such as the inlet flow centerline velocity, Reynolds number, and angle of attack, respectively. As can be seen from the table, using... The Sobol exponent obtained by this method is convergent. Figure 19 The results are shown to change with the number of function evaluations, clearly demonstrating the convergence of the results.

[0078] Table 6 This invention proposes a global sensitivity evaluation method for models based on moment integrals. This method can construct a univariate quadrature expression based on the origin moments, and then construct a multivariate quadrature expression based on tensor product extension. This effectively avoids sampling errors and fluctuations in small exponent estimation, ensuring the non-negativity of the results. At the same time, the proposed method can effectively avoid the problems of traditional surrogate model construction errors and sampling optimization, and achieve convergence of Sobol exponent results. The proposed method uses deterministic nodes and weights in the standard hypercubic space, which can realize the simultaneous calculation of all-order Sobol exponents, including mean effect, first-order, second-order and total effect. It significantly improves the computational efficiency while ensuring accuracy, and has stronger practicality and applicability.

[0079] Finally, it should be noted that the above embodiments are for illustration only and not for limiting the technical solutions of the present invention. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the present invention without departing from the spirit and scope of the present invention. Any modifications or partial substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for evaluating the global sensitivity of a model based on moment integrals, characterized in that, The model is a damped oscillator model, which includes the following steps: S1. Constructing the single-variable quadrature expression for the damped oscillator: Construct a damped oscillator model under external load, determine the random variable of the first sensitivity parameter, and define the first probability density function as follows. Before calculation First-order origin moment Construct the first matrix Second matrix ; Calculate the second matrix eigenvalues With feature vectors Thus, the single-variable integral expression for the damped oscillator is obtained; S2. Constructing the multivariate quadrature expression for the damped oscillator: Based on the tensor product, the univariate quadrature expression for the damped oscillator is extended to a multivariate quadrature expression. Let the i-th dimension be... The dimension of the node product expression is... Multivariable quadrature expression for damped oscillator for: ； in, Represents the i-th dimension The expression for the product of nodes, i = 1, 2, ..., d; S3. Evaluate the damped oscillator model based on the multivariate quadrature expression of the damped oscillator and obtain the position-time response. and speed time response Calculate the location-time response of each group of nodes. and speed time response ,in Indicates the first The first group node One component; S4. Calculate the position-time response of the damped oscillator model using the standard ordinary differential solver. and speed time response The mean and total variance; S5. Calculate the first N global sensitivity factors of the damped oscillator model and determine the key parameters affecting position uncertainty and velocity uncertainty in the first sensitivity parameters: The first N global sensitivity factors of the damped oscillator model include the first global sensitivity factor, the second global sensitivity factor, the third global sensitivity factor, ... and the Nth global sensitivity factor of the damped oscillator model.

2. The model global sensitivity evaluation method based on moment integrals according to claim 1, characterized in that, The damped oscillator model under external load in step S1 is represented as follows: （1）； in, Indicates the damping coefficient; Indicates stiffness; Indicates the amplitude of the external load; Indicates frequency; and These represent the initial position and initial velocity, respectively; the first sensitivity parameter has multiple parameters, including... Furthermore, by using scaling and translation transformations, the first sensitivity parameter is transformed to conform to a standard uniform distribution. A random variable.

3. The model global sensitivity evaluation method based on moment integrals according to claim 1, characterized in that, For the first matrix in step S1 Decomposition yields: ； in, Represents an upper triangular matrix; forward First-order origin moment for: ； Where E represents expectation; Let j represent a set of randomly generated random variables; j represents the order and ; Indicates the range of the random variable.

4. The model global sensitivity evaluation method based on moment integrals according to claim 1, characterized in that, The first matrix in step S1 Second matrix They are represented as follows: ； ； in, Let j represent the j-th main variable, and express it as... ; Let j represent the j-th variable, and express it as... .

5. The model global sensitivity evaluation method based on moment integrals according to claim 1, characterized in that, The total number of nodes in the multivariate product expression in step S2 for: ； in, This indicates a multiplication operation.

6. The model global sensitivity evaluation method based on moment integrals according to claim 1, characterized in that, In step S4, all dimensions of the damped oscillator model are adopted. Use the dot product expression to calculate the position-time response of the damped oscillator model. and speed time response mean and total variance : ； ； ； 。 7. A method for evaluating the global sensitivity of a model based on moment integrals, characterized in that, The model is a fluid velocity model around the wing, which includes the following steps: S1. Constructing the univariate quadrature expression for the fluid around the wing: Construct a velocity model for the fluid around the wing, determine the random variable of the second sensitivity parameter, and define the second probability density function as follows. Before calculation Second-order original moment Construct the Hankel and Jacobi matrices and calculate the eigenvalues ​​and eigenvectors to obtain the univariate quadrature expression for the fluid around the wing; the second sensitivity parameters include the angle of attack, the inlet flow centerline velocity, and the Reynolds number; S2. Constructing a multivariate quadrature expression for the fluid around the wing: Based on the tensor product, the univariate quadrature expression for the fluid around the wing is extended into a multivariate quadrature expression for the fluid around the wing; S3. Evaluate the fluid velocity model around the wing based on the multivariate quadrature expression of the fluid around the wing, obtain the fluid velocity time response, and calculate the fluid velocity time response of each group of nodes. S4. Mean and total variance of the fluid velocity time response of the fluid velocity model around the computer wing based on the lattice Boltzmann method; S5. The first N Sobol exponents of the fluid velocity model around the wing are used to determine the key parameters affecting the steady-state maximum velocity in the second sensitivity parameter: The first N Sobol exponents of the fluid velocity model around the wing include the first Sobol exponent of the fluid velocity around the wing, the second Sobol exponent of the fluid velocity model around the wing, the third Sobol exponent of the fluid velocity around the wing, ... and the Nth Sobol exponent of the fluid velocity around the wing.

8. A method for evaluating the global sensitivity of a model based on moment integrals, characterized in that, It includes the following steps: S1. Constructing the single-variable quadrature expression: Define the probability density function of the random variable as follows: Before calculation First-order origin moment Construct the Hankel matrix and Jacobi matrix, calculate the eigenvalues ​​and eigenvectors, and obtain the univariate quadrature expression; S2. Constructing a multivariable quadrature expression: Extending the univariate quadrature expression into a multivariate quadrature expression based on the tensor product; S3. Evaluate the model based on the multivariate quadrature expression and obtain the function response: Let the model response function be... Calculate the function response of each group of nodes in the multivariate quadrature expression. ; S4. Calculate the mean and total variance of the model response; S5. Calculate the first three Sobol indices of the model: The first three Sobol indices of the model include the first Sobol index. Second-order Sobol exponent and the third-order Sobol exponent .

9. The model global sensitivity evaluation method based on moment integrals according to claim 8, characterized in that, In step S5, the first-order Sobol exponent is first determined. Second-order Sobol exponent and the third-order Sobol exponent Then replace 1, 2, and 3 in the result with m, n, and q respectively to obtain , The first constant, the second constant, and the third constant are respectively, and satisfy the following conditions: .

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