Random collocation method for solving global sensitivity of augmented failure probability of turbine blade structure

Abstract

The invention provides a random point collocation method for solving global sensitivity of an augmented failure probability of a turbine blade structure, and belongs to the technical field of reliability design. Comprising the following steps: constructing an expected density function of an n-dimensional random input variable of a turbine blade structure to obtain an n-dimensional expected density function; calculating an unconditional augmentation failure probability of the turbine blade structure based on the n-dimensional expected density function and the performance function; a Gauss-Hermite integral quadrature node of the n-dimensional expected density function is determined; constructing an expected density function of (n-1)-dimensional random input variables except the i-th dimension of the turbine blade structure to obtain an (n-1)-dimensional expected density function except the i-th dimension; based on the (n-1)-dimensional expected density function and the conditional performance function, the conditional augmentation failure probability of the turbine blade structure corresponding to the Gauss-Hermite integral quadrature node is calculated; and determining the augmented failure probability global sensitivity of the turbine blade structure. According to the invention, the calculation efficiency of the global sensitivity of the augmentation failure probability of the turbine blade structure can be improved.

Description

Technical Field

[0001] This disclosure relates to the field of reliability design technology, and in particular to a stochastic collocation method for solving the global sensitivity of augmented failure probability of turbine blade structures. Background Technology

[0002] Currently, considering the uncertainties of input variables and their distribution parameters, the global sensitivity of augmented failure probability can reasonably measure the influence of each random input on the augmented failure probability of turbine blade structures. However, due to the complexity of turbine blade structures and loads, the computational cost of solving the global sensitivity of augmented failure probability using simulation-based methods is too high and time-consuming. The results obtained using surrogate models are random, and the construction and updating of surrogate models are also very time-consuming. Therefore, existing technologies lack a high-precision and efficient method for solving the global sensitivity of augmented failure probability for turbine blade structures.

[0003] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0004] This disclosure provides a stochastic collocation method for solving the global sensitivity of augmented failure probability of turbine blade structures, thereby at least partially solving the problem of the lack of high-precision and efficient global sensitivity methods for solving the augmented failure probability of turbine blade structures in the prior art.

[0005] Other features and advantages of this disclosure will become apparent from the following detailed description, or may be learned in part from practice of this disclosure.

[0006] According to one aspect of this disclosure, a stochastic collocation method is provided for solving the global sensitivity of the augmented failure probability of a turbine blade structure, comprising: constructing the expected density function of the n-dimensional random input variables of the turbine blade structure to obtain the n-dimensional expected density function; calculating the unconditional augmented failure probability of the turbine blade structure based on the n-dimensional expected density function and the function; determining five Gauss-Hermite integral quadrature nodes under a one-dimensional standard normal distribution, and transforming them into five Gauss-Hermite integral quadrature nodes under the i-th dimension expected density function through an equal probability transformation; wherein, 1≤i≤n, and n is a positive integer; constructing the expected density function of the turbine blade structure in n-1 dimensions other than the i-th dimension. The expected density function is used to obtain the (n-1)-dimensional expected density function excluding the i-th dimension; based on the (n-1)-dimensional expected density function excluding the i-th dimension and the conditional function when the i-th input variable is fixed under the i-th expected density function, the conditional augmented failure probability of the turbine blade structure corresponding to the k-th Gauss-Hermite integral quadrature node is calculated; where 1≤k≤5; according to the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node, the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th input variable is determined.

[0007] In an exemplary embodiment of this disclosure, the step of calculating the unconditional augmented failure probability of the turbine blade structure based on the n-dimensional expected density function and the function includes: obtaining n-dimensional standard random placement points corresponding to the n-dimensional standard normal distribution density function and the weights corresponding to the n-dimensional standard random placement points, and converting the n-dimensional standard random placement points into n-dimensional random placement points corresponding to the n-dimensional expected density function; calculating the function function value corresponding to each n-dimensional random placement point according to the n-dimensional random placement points corresponding to the n-dimensional expected density function and the weights corresponding to each random placement point; performing a transformation on the function function and calculating the first four statistical moments of the transformed function function; performing a transformation on the function function and calculating the first four statistical moments of the transformed function function; and calculating the unconditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the function function.

[0008] In one exemplary embodiment of this disclosure, the expected density function of the n-dimensional random input variable for:

[0009]

[0010] in, Represents the n-dimensional expectation density function. Describes the i-th dimension input variable x i The expected density function, Describes the i-th dimension input variable x i The distribution parameters, x represents i by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

[0011] In one exemplary embodiment of this disclosure, the transformation of the function includes: performing a Box-Cox transformation on the function to determine the transformed function; wherein, performing the Box-Cox transformation on the function includes: performing a Box-Cox transformation on the function using the following transformation formula:

[0012]

[0013] Where g(x) represents the function, This represents the transformed function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

[0014] In one exemplary embodiment of this disclosure, calculating the first four statistical moments of the transformed function includes: calculating the first four statistical moments of the function using the following formula:

[0015]

[0016] in, Representing the function The mean, standard deviation, skewness, and kurtosis estimates, x (z) Let ω represent the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. (z) Represents a random allocation point x (z) The corresponding weight, N c This represents the number of n-dimensional randomized points corresponding to the n-dimensional expected density function.

[0017] The step of calculating the unconditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the function includes: determining the fourth-order reliability index estimate of the function based on the first four statistical moments of the function, and calculating the unconditional augmented failure probability estimate of the turbine blade structure based on the fourth-order reliability index estimate of the function; the fourth-order reliability index estimate and the unconditional augmented failure probability estimate of the function are expressed as follows:

[0018]

[0019] in, Represents function The estimated value of the fourth-order reliability index, Let Φ(·) represent the unconditionally augmented failure probability estimate, and let Φ(·) represent the cumulative distribution function of the standard normal distribution.

[0020] In one exemplary embodiment of this disclosure, the step of calculating the conditional augmentation failure probability of the turbine blade structure at the Gauss-Hermite integral quadrature node based on the (n-1)-dimensional expectation density function excluding the i-th dimension and the conditional function when the i-th input variable is fixed under the i-th dimension expectation density function includes: obtaining the (n-1)-dimensional standard random placement points corresponding to the (n-1)-dimensional standard normal distribution density function and the weights corresponding to the standard random placement points, and converting the (n-1)-dimensional standard random placement points into the (n-1)-dimensional expectation density function excluding the i-th dimension. The density function corresponds to n-1 dimensional random placement points; based on the n-1 dimensional random placement points corresponding to the n-1 dimensional expected density function (excluding the i-th dimension) and the weights corresponding to each random placement point, the conditional function value is calculated for each of the n-1 dimensional random placement points when the i-th input variable is fixed at the k-th Gauss-Hermite integral quadrature node under the i-th dimensional expected density function; the conditional function is transformed, and the first four statistical moments of the transformed conditional function are calculated; the conditional augmentation failure probability of the turbine blade structure is calculated based on the first four statistical moments of the conditional function.

[0021] In one exemplary embodiment of this disclosure, the n-1 dimensional random input variable The expected density function is:

[0022]

[0023] in, Describes the (n-1)-dimensional expectation density function. Describes the j-th dimension input variable x j The expected density function, Describes the j-th dimension input variable x j The distribution parameters, x represents j by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

[0024] In one exemplary embodiment of this disclosure, the transformation of the conditional function includes: performing a Box-Cox transformation on the conditional function to determine the transformed conditional function; wherein, performing the Box-Cox transformation on the conditional function includes: transforming the conditional function using the following transformation formula. Perform Box-Cox transformation:

[0025]

[0026] in, x represents i Fixed in Conditional function at time, This represents the transformed conditional function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

[0027] In one exemplary embodiment of this disclosure, calculating the first four statistical moments of the transformed conditional function includes: calculating the first four statistical moments of the conditional function using the following formula:

[0028]

[0029] in, Representing conditional function Estimates of the mean, standard deviation, skewness, and kurtosis. This represents the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. Represents a random configuration point The corresponding weight, N c ′ represents the number of n-1 dimensional randomized points.

[0030] The step of calculating the conditionally augmented failure probability of the turbine blade structure based on the first four statistical moments of the conditional function includes: determining the fourth-order reliability index estimate of the conditional function based on the first four statistical moments of the conditional function, and calculating the conditionally augmented failure probability estimate of the turbine blade structure based on the fourth-order reliability index estimate of the conditional function; the fourth-order reliability index estimate and the conditionally augmented failure probability estimate of the conditional function are expressed as follows:

[0031]

[0032] in, x represents i Fixed in Conditional function at time, Represents a conditional function The estimated value of the fourth-order reliability index, Φ(·) represents the estimated probability of conditional augmentation failure, and Φ(·) represents the cumulative distribution function of the standard normal distribution.

[0033] In an exemplary embodiment of this disclosure, determining the global sensitivity of the augmented failure probability of the turbine blade structure based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node includes: determining the global sensitivity of the augmented failure probability of the turbine blade structure using the following formula based on the estimated value of the unconditional augmented failure probability and the estimated value of the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node:

[0034]

[0035] in, This represents an estimate of the global sensitivity to the augmented failure probability. This represents the estimate of the unconditionally augmented failure probability. ξ represents the estimated probability of conditional augmentation failure. (k) This represents the Gauss-Hermite integral quadrature node. The corresponding weights.

[0036] The exemplary embodiments disclosed herein have the following beneficial effects:

[0037] Construct the expected density function of the n-dimensional random input variables of the turbine blade structure to obtain the n-dimensional expected density function; calculate the unconditional augmented failure probability of the turbine blade structure based on the n-dimensional expected density function and the function of performance; determine 5 Gauss-Hermite integral quadrature nodes under the one-dimensional standard normal distribution, and transform them into 5 Gauss-Hermite integral quadrature nodes under the i-th dimension expected density function through equal probability transformation; where 1≤i≤n, and n is a positive integer; construct the expected density function of the (n-1)-dimensional random input variables of the turbine blade structure other than the i-th dimension to obtain the (n-1)-dimensional expected density function other than the i-th dimension. The expected density function is used; based on the (n-1)-dimensional expected density functions excluding the i-th dimension and the conditional function when the i-th input variable is fixed under the i-th expected density function, the conditional augmented failure probability of the turbine blade structure corresponding to the k-th Gauss-Hermite integral quadrature node is calculated; where 1≤k≤5; based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node, the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th input variable is determined. This exemplary embodiment proposes a new method for solving the global sensitivity of the augmented failure probability of turbine blades, which can provide reference and guidance for the reliability optimization design of turbine blade structures, thereby saving a lot of economic and time costs. Specifically, the random collocation method provided in this exemplary embodiment can calculate the augmented failure probability function value of any distribution parameter of any input variable in one go. In the solution process, as long as the dimension of the input variable is determined, the computational cost can be determined. This solution method can significantly improve the solution efficiency of the global sensitivity of the augmented failure probability of turbine blades.

[0038] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit this disclosure. Attached Figure Description

[0039] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure. It is obvious that the drawings described below are merely some embodiments of this disclosure, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0040] Figure 1 This schematically illustrates a flowchart of a stochastic collocation method for solving the global sensitivity of augmented failure probability of a turbine blade structure in this exemplary embodiment.

[0041] Figure 2This schematic diagram illustrates a turbine blade structure in an exemplary embodiment of the present invention.

[0042] Figure 3 The schematic diagram illustrates a sub-flowchart of a stochastic collocation method for solving the global sensitivity of augmented failure probability of a turbine blade structure in this exemplary embodiment.

[0043] Figure 4 This schematically illustrates another sub-flowchart of a stochastic collocation method for solving the global sensitivity of augmented failure probability of a turbine blade structure in this exemplary embodiment.

[0044] Figure 5 A schematic diagram is shown showing the global sensitivity solution for calculating the augmented failure probability of a turbine blade structure based on the method of this exemplary embodiment. Detailed Implementation

[0045] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that this disclosure will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0046] The exemplary embodiments of this disclosure first provide a stochastic collocation method for solving the global sensitivity of the augmented failure probability of a turbine blade structure.

[0047] The following is in conjunction with the appendix Figure 1 The exemplary embodiments will be further described as follows: Figure 1 As shown, the stochastic collocation method for solving the global sensitivity of the augmented failure probability of turbine blade structures may include the following steps S110 to S160:

[0048] Step S110: Construct the expected density function of the n-dimensional random input variables of the turbine blade structure to obtain the n-dimensional expected density function.

[0049] Turbine blades are crucial components in aero-engines, gas turbines, and other similar equipment. Mounted on a rotor, they convert the kinetic and thermal energy of high-temperature, high-pressure gas into mechanical energy through high-speed rotation, thereby driving the turbine. Various uncertainties arise during the analysis, simulation, or optimization of turbine blade structures. These uncertainties can include material properties, geometric dimensions, and operating conditions. This exemplary embodiment can first construct the expected density function of the n-dimensional random input variables of the turbine blade structure to obtain an n-dimensional expected density function, which is then used for subsequent failure probability calculations.

[0050] In step S110, the turbine blade structure described above can be as follows: Figure 2 As shown, in this exemplary embodiment, the number of input variables can be seven, i.e., n=7, specifically representing yield strength x1, Young's modulus x2, coefficient of thermal expansion of nickel-based alloys x3, Poisson's ratio x4, blade pressure side load x5, blade suction side load x6, and thermal conductivity x7. It should be noted that this exemplary embodiment is merely illustrative; in practical applications, the number of input variables is not limited to seven. For example, the number of input variables can also be eight, nine, etc., specifically including shear modulus, density, etc. This disclosure does not specifically limit this.

[0051] In this exemplary embodiment, Table 1 shows the distribution form and distribution parameters of each input variable:

[0052] Table 1

[0053]

[0054] Step S120: Calculate the unconditional augmented failure probability of the turbine blade structure based on the n-dimensional expected density function and the function.

[0055] The function in question refers to the function used to calculate the unconditional augmentation failure probability of a turbine blade structure. For example, based on the input variables shown in Table 1, the following function can be constructed:

[0056] y = g(x) = g(x1, x2, ..., x n ),n=7 (1)

[0057] This exemplary embodiment can calculate the augmented failure probability of a turbine blade under the influence of n-dimensional random input variables using an n-dimensional expectation density function and a function function without fixing any conditions.

[0058] Specifically, in an exemplary embodiment, such as Figure 3 As shown, step S120 above may include the following steps:

[0059] Step S310: Obtain the n-dimensional standard random allocation points corresponding to the n-dimensional standard normal distribution density function and the weights corresponding to the n-dimensional standard random allocation points, and convert the n-dimensional standard random allocation points into n-dimensional random allocation points corresponding to the n-dimensional expected density function.

[0060] Step S320: Based on the n-dimensional random allocation points corresponding to the n-dimensional expected density function and the weights corresponding to each random allocation point, calculate the function value corresponding to each n-dimensional random allocation point through the function.

[0061] Step S330: Transform the function and calculate the first four statistical moments of the transformed function.

[0062] Step S340: Calculate the unconditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the function.

[0063] Here, an n-dimensional standard random placement point refers to a random point generated according to a standard normal distribution in an n-dimensional space. The corresponding weight is a measure of the importance of each random placement point in a certain calculation. This exemplary embodiment can use a sparse grid method to generate n-dimensional standard random placement points and their corresponding weights under an n-dimensional standard normal distribution. Then, the standard random placement points are transformed into random placement points of other types of distributions. For example, random placement points of an n-dimensional expected density function can be determined through an equal probability transformation, and the corresponding function value can be calculated using a function of performance.

[0064] Furthermore, the function can be transformed, and the first four statistical moments after the transformation can be calculated. Transformation methods can include Box-Cox transformation, etc. Finally, the unconditional augmented failure probability of the turbine blade structure can be calculated based on the first four statistical moments of the function. Specifically, this can be done by first calculating the fourth-order reliability index estimate based on the first four statistical moments, and then calculating the unconditional augmented failure probability of the turbine blade structure based on the fourth-order reliability index estimate.

[0065] In an exemplary embodiment, the expected density function of the above-mentioned n-dimensional random input variables for:

[0066]

[0067] in, Represents the n-dimensional expectation density function. Describes the i-th dimension input variable x i The expected density function, Describes the i-th dimension input variable x i The distribution parameters, x represents i by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

[0068] In an exemplary embodiment, step S310 can use standard random locus points under an n-dimensional standard normal distribution generated by the sparse grid method. (z = 1, 2, ..., N) c ), whose corresponding weight is ω (z) .

[0069] In an exemplary embodiment, step S320 can be performed by an equal probability transformation. Determine the desired density function The random allocation point is S = {x} (z) z = 1, 2, ..., N c}, and its corresponding function value is G={g(x (z) ), z = 1, 2, ..., N c}

[0070] In an exemplary embodiment, the above-described transformation process of the function may include:

[0071] Perform a Box-Cox transformation on the function to determine the transformed function;

[0072] The Box-Cox transformation of the function includes:

[0073] The Box-Cox transformation of the function is performed using the following transformation formula:

[0074]

[0075] Where g(x) represents the function, This represents the transformed function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

[0076] The Box-Cox transformation is a commonly used transformation method in statistics. Its main purpose is to convert data or functions that do not conform to a normal distribution into a form that approximates a normal distribution by adjusting certain parameters. This exemplary embodiment can perform a Box-Cox transformation on the function g(x) using the formula described above.

[0077] In this exemplary embodiment, a genetic algorithm can be used first to determine the parameters that need to be determined in the Box-Cox transformation. Then, based on the Box-Cox transformation parameters, the transformed function is determined. The genetic algorithm is an optimization algorithm that simulates the process of biological evolution. It finds the optimal solution to the problem by simulating biological evolution mechanisms such as natural selection, heredity, and mutation. After determining the optimal Box-Cox transformation parameters using the genetic algorithm, the original function can be substituted into the Box-Cox transformation formula for transformation, thereby obtaining the transformed function.

[0078] Specifically, in an exemplary embodiment, the optimization model for determining the Box-Cox transformation parameter τ can be expressed as:

[0079]

[0080] in and It can be represented as:

[0081]

[0082]

[0083] The optimization model can be solved using a genetic algorithm, where L represents a function of the transformation parameter τ, and the transformation parameter τ can be solved by minimizing the distance L. Representing the function Mean, standard deviation, skewness, kurtosis, N c ω represents the number of n-dimensional randomized points. (z) Represents a random allocation point x (z) The corresponding weights.

[0084] In formula (3), Including the transformation parameter τ, in formula (4), L includes implicit... of and Therefore, this exemplary embodiment can minimize the skewness-kurtosis pair of the transformed function (i.e., and The transformation parameter τ is determined by the distance L between the skewness-kurtosis pair (i.e., 0 and 3) of the normal distribution.

[0085] In an exemplary embodiment, the calculation of the first four statistical moments of the transformed function may include:

[0086] The first four statistical moments of the function are calculated using the following formula:

[0087]

[0088] in, Representing the function The mean, standard deviation, skewness, and kurtosis estimates, x (z) Let ω represent the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. (z) Represents a random allocation point x (z) The corresponding weight, N c This represents the number of n-dimensional randomized points corresponding to the n-dimensional expected density function.

[0089] In an exemplary embodiment, the calculation of the unconditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the function may include:

[0090] The fourth reliability index estimate of the function is determined based on the first four statistical moments of the function, and the unconditional augmented failure probability estimate of the turbine blade structure is calculated based on the fourth reliability index estimate.

[0091] The fourth-order reliability index estimate and the unconditional augmented failure probability estimate are expressed as follows:

[0092]

[0093] in, Represents function The estimated value of the fourth-order reliability index, Let Φ(·) represent the unconditionally augmented failure probability estimate, and let Φ(·) represent the cumulative distribution function of the standard normal distribution.

[0094] In this exemplary embodiment, after calculating the first four statistical moments, the fourth reliability index estimate can be calculated using formula (7), and then the fourth reliability index estimate can be substituted into formula (8) to determine the unconditional augmented failure probability value.

[0095] Step S130: Determine 5 Gauss-Hermite integral quadrature nodes under the one-dimensional standard normal distribution, and transform them into 5 Gauss-Hermite integral quadrature nodes under the i-th dimension expectation density function through equal probability transformation; where 1≤i≤n, and n is a positive integer.

[0096] In this exemplary embodiment, five Gauss-Hermite integral quadrature nodes under a one-dimensional standard normal distribution can be determined. (k = 1, 2, ..., 5), and the corresponding weights are Then, through equal probability transformation, it is converted into 5 Gauss-Hermite integral quadrature nodes under the i-th dimension expectation density function, such as 5 Gauss-Hermite integral quadrature nodes under the 1-dimensional expectation density function, 5 Gauss-Hermite integral quadrature nodes under the 2-dimensional expectation density function, and so on, for a total of 5n Gauss-Hermite integral quadrature nodes. Furthermore, the conditional augmentation failure probability corresponding to each Gauss-Hermite integral quadrature node can be calculated separately.

[0097] Step S140: Construct the expected density function of the (n-1)-dimensional random input variables of the turbine blade structure, excluding the i-th dimension, to obtain the (n-1)-dimensional expected density function excluding the i-th dimension.

[0098] In this exemplary embodiment, the expected density function of the (n-1)-dimensional random input variables of the turbine blade structure, excluding the i-th dimension, can also be constructed to obtain the (n-1)-dimensional expected density function, excluding the i-th dimension, where i can be any dimension from 1 to n, and n-1 dimensions refer to the remaining dimensions of n dimensions excluding the i-th dimension.

[0099] Step S150: Based on the (n-1)-dimensional expected density function excluding the i-th dimension and the conditional function of the i-th input variable fixed under the i-th expected density function at the k-th Gauss-Hermite integral quadrature node, calculate the conditional augmented failure probability of the turbine blade structure corresponding to the k-th Gauss-Hermite integral quadrature node; where 1≤k≤5.

[0100] Here, the conditional function refers to the conditional function used when calculating the conditional augmented failure probability of the turbine blade structure. It differs from the function corresponding to the random placement points of the n-dimensional expectation density function. In the conditional function corresponding to the random placement points of the (n-1)-dimensional expectation density function, the i-th dimension is already fixed, thus effectively reducing the dimension by one. This exemplary embodiment can fix the i-th dimension data at different Gauss-Hermite integral quadrature nodes and calculate the conditional augmented failure probability corresponding to each Gauss-Hermite integral quadrature node.

[0101] In one exemplary embodiment, such as Figure 4 As shown, the above-mentioned conditional augmented failure probability of the turbine blade structure at the Gauss-Hermite integral quadrature node, based on the (n-1)-dimensional expected density function excluding the i-th dimension and the conditional function when the i-th input variable is fixed under the i-th dimension expected density function, may include the following steps:

[0102] Step S410: Obtain the n-1-dimensional standard random allocation points and the weights corresponding to the n-1-dimensional standard normal distribution density function, and convert the n-1-dimensional standard random allocation points into n-1-dimensional random allocation points corresponding to the n-1-dimensional expected density function excluding the i-th dimension.

[0103] Step S420: Based on the n-1-dimensional random placement points corresponding to the n-1-dimensional expected density function (excluding the i-th dimension) and the weights corresponding to each random placement point, calculate the conditional function value of each n-1-dimensional random placement point when the i-th dimension input variable is fixed at the k-th Gauss-Hermite integral quadrature node under the i-th dimension expected density function through the conditional function.

[0104] Step S430: Transform the conditional function and calculate the first four statistical moments of the transformed conditional function.

[0105] Step S440: Calculate the conditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the conditional function.

[0106] Here, the n-1 dimensional standard random placement point refers to the random placement point corresponding to the n-1 dimensional standard normal distribution density function. This exemplary embodiment can use the sparse grid method to generate the standard random placement points and their corresponding weights under the n-1 dimensional standard normal distribution, and determine the random placement points of the n-1 dimensional expected density function through equal probability transformation, and then use the conditional function to calculate the corresponding function value.

[0107] Then, by transforming the conditional function (CDF) using methods such as the Box-Cox transform, the transformed CDF is determined, and the first four statistical moments of the CDF are calculated. Finally, the conditional augmented failure probability of the turbine blade structure is calculated based on the first four statistical moments. Specifically, this can be achieved by first calculating the fourth-order reliability index estimate based on the first four statistical moments, and then calculating the conditional augmented failure probability of the turbine blade structure based on the fourth-order reliability index estimate. It should be noted that the first four statistical moments of the CDF are used to calculate the unconditional augmented failure probability, while the first four statistical moments of the CDF are used to calculate the conditional augmented failure probability.

[0108] Similar to the calculation of the failure probability of unconditional augmentation, the difference is that the failure probability of unconditional augmentation is a single result, while the failure probability of conditional augmentation is multiple results. The number of results is related to the number of nodes in the Gauss-Hermite integral.

[0109] In one exemplary embodiment, the above-mentioned n-1 dimensional random input variables The expected density function is:

[0110]

[0111] in, Describes the (n-1)-dimensional expectation density function. Describes the j-th dimension input variable x j The expected density function, Describes the j-th dimension input variable x j The distribution parameters, x represents j by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

[0112] In an exemplary embodiment, step S410 can use standard random locus points under an n-1 dimensional standard normal distribution generated by the sparse grid method. Its corresponding weight is

[0113] In an exemplary embodiment, step S420 can be performed by an equal probability transformation. Determine the desired density function The random allocation point is Its corresponding function value is z = 1, 2, ..., N′ c ,k=1,2,…,5}.

[0114] In an exemplary embodiment, the above-described transformation of the conditional function may include:

[0115] Perform a Box-Cox transformation on the conditional function to determine the transformed conditional function;

[0116] The Box-Cox transformation of the conditional function can include:

[0117] The conditional function is transformed using the following formula. Perform Box-Cox transformation:

[0118]

[0119] in, x represents i Fixed in Conditional function at time, This represents the transformed conditional function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

[0120] In this exemplary embodiment, a genetic algorithm can be used first to determine the parameters that need to be determined in the Box-Cox transformation. Then, based on the Box-Cox transformation parameters, the transformed conditional function can be determined. For example, a genetic algorithm can be used to determine the optimal Box-Cox transformation parameters, and the original conditional function can be substituted into the Box-Cox transformation formula for transformation to obtain the transformed conditional function.

[0121] In an exemplary embodiment, the optimization model for determining the Box-Cox transformation parameter τ can be expressed as:

[0122]

[0123] in, and It can be represented as:

[0124]

[0125] The optimization model can be solved using a genetic algorithm, where L represents a function of the transformation parameter τ, and the transformation parameter τ can be solved by minimizing the distance L. Representing conditional function Mean, standard deviation, skewness, kurtosis, N c ′ represents the number of n-dimensional randomized points. Represents a random configuration point The corresponding weights.

[0126] In an exemplary embodiment, calculating the first four statistical moments of the transformed conditional function may include:

[0127] The first four statistical moments of the conditional function are calculated using the following formula:

[0128]

[0129]

[0130] in, Representing conditional function Estimates of the mean, standard deviation, skewness, and kurtosis. This represents the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. Represents a random configuration point The corresponding weight, N c ′ represents the number of n-1 dimensional randomized points.

[0131] In an exemplary embodiment, the calculation of the conditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the conditional function may include:

[0132] The fourth reliability index estimate of the conditional function is determined based on the first four statistical moments of the conditional function, and the conditional augmented failure probability estimate of the turbine blade structure is calculated based on the fourth reliability index estimate.

[0133] The fourth-order reliability index estimate and the conditionally augmented failure probability estimate of the conditional function are expressed as follows:

[0134]

[0135] in, x represents i Fixed in Conditional function at time, Represents a conditional function The estimated value of the fourth-order reliability index, Φ(·) represents the estimated probability of conditional augmentation failure, and Φ(·) represents the cumulative distribution function of the standard normal distribution.

[0136] In this exemplary embodiment, after calculating the first four statistical moments, the fourth reliability index estimate can be calculated using formula (14), and then the fourth reliability index estimate can be substituted into formula (15) to determine the condition augmented failure probability estimate.

[0137] Step S160: Based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral integration node, determine the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th dimension input variable.

[0138] After determining the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node, the global sensitivity of the augmented failure probability of the turbine blade structure can be calculated. For example, the global sensitivity of the augmented failure probability of the turbine blade structure can be determined by weighting the unconditional augmented failure probability and the conditional augmented failure probability. In this exemplary embodiment, there can be one unconditional augmented failure probability and multiple conditional augmented failure probabilities. For example, the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th dimension input variable can be calculated based on one unconditional augmented failure probability and five conditional augmented failure probabilities. During the calculation, the weight corresponding to each Gauss-Hermite integral quadrature node should also be considered.

[0139] In an exemplary embodiment, step S160 may include:

[0140] Based on the unconditional augmented failure probability estimate and the conditional augmented failure probability estimate of the turbine blade structure corresponding to each Gauss-Hermite integral node, the global sensitivity of the augmented failure probability of the turbine blade structure is determined by the following formula:

[0141]

[0142] in, This represents the global sensitivity of the augmented failure probability of the turbine blade structure. This represents the estimate of the unconditionally augmented failure probability. ξ represents the estimated probability of conditional augmentation failure. (k) express The corresponding weights.

[0143] Figure 5 A schematic diagram is shown showing the global sensitivity solution for calculating the augmented failure probability of a turbine blade structure based on the method of this exemplary embodiment.

[0144] Based on the above description, in this exemplary embodiment, the expected density function of the n-dimensional random input variables of the turbine blade structure is constructed to obtain the n-dimensional expected density function; the unconditional augmented failure probability of the turbine blade structure is calculated based on the n-dimensional expected density function and the function; five Gauss-Hermite integral quadrature nodes under the one-dimensional standard normal distribution are determined, and they are transformed into five Gauss-Hermite integral quadrature nodes under the i-th dimension expected density function through equal probability transformation; where 1≤i≤n, and n is a positive integer; the expected density function of the (n-1)-dimensional random input variables of the turbine blade structure other than the i-th dimension is constructed to obtain the expected density function other than the i-th dimension. The following methods are proposed: 1) Obtain the (n-1)-dimensional expected density function excluding the i-th dimension; 2) Calculate the conditional augmented failure probability of the turbine blade structure at the k-th Gauss-Hermite integral quadrature node when the i-th input variable is fixed under the i-th expected density function; where 1 ≤ k ≤ 5; 3) Determine the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th input variable based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral quadrature node. This exemplary embodiment proposes a novel method for solving the global sensitivity of the augmented failure probability of turbine blades, which can provide reference and guidance for the reliability optimization design of turbine blade structures, thereby saving significant economic and time costs. Specifically, the random collocation method provided in this exemplary embodiment can calculate the augmented failure probability function value of any distribution parameter of any input variable in one go. In the solution process, as long as the dimension of the input variable is determined, the computational cost can be determined. This solution method can significantly improve the solution efficiency of the global sensitivity of the augmented failure probability of turbine blades.

[0145] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the claims.

[0146] It should be understood that this disclosure is not limited to the precise structures described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this disclosure is defined only by the appended claims.

Claims

1. A stochastic collocation method for solving the global sensitivity of augmented failure probability of turbine blade structures, characterized in that, include: Construct the expected density function of the n-dimensional random input variables of the turbine blade structure to obtain the n-dimensional expected density function; The unconditional augmented failure probability of the turbine blade structure is calculated based on the n-dimensional expected density function and the function. Five Gauss-Hermite integral quadrature nodes under a one-dimensional standard normal distribution are identified, and they are transformed into five Gauss-Hermite integral quadrature nodes under the i-th dimension expectation density function through an equal probability transformation; where 1≤i≤n, and n is a positive integer; Construct the expected density function of the (n-1)-dimensional random input variables other than the i-th dimension of the turbine blade structure to obtain the (n-1)-dimensional expected density function other than the i-th dimension. Based on the (n-1)-dimensional expected density function excluding the i-th dimension and the conditional function of the k-th Gauss-Hermite integral quadrature node with the i-th input variable fixed under the i-th expected density function, the conditional augmented failure probability of the turbine blade structure corresponding to the k-th Gauss-Hermite integral quadrature node is calculated; where 1≤k≤5; Based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral node, determine the global sensitivity of the augmented failure probability of the turbine blade structure corresponding to the i-th input variable.

2. The method according to claim 1, characterized in that, The calculation of the unconditional augmented failure probability of the turbine blade structure based on the n-dimensional expected density function and the function of performance includes: Obtain the n-dimensional standard random placement points corresponding to the n-dimensional standard normal distribution density function and the weights corresponding to the n-dimensional standard random placement points, and convert the n-dimensional standard random placement points into n-dimensional random placement points corresponding to the n-dimensional expected density function; Based on the n-dimensional random placement points corresponding to the n-dimensional expected density function and the weights corresponding to each random placement point, the function function value corresponding to each of the n-dimensional random placement points is calculated through the function function. The function is transformed, and the first four statistical moments of the transformed function are calculated. The unconditional augmented failure probability of the turbine blade structure is calculated based on the first four statistical moments of the function.

3. The method according to claim 2, characterized in that, The expected density function of the n-dimensional random input variables for: in, Represents the n-dimensional expectation density function. Describes the i-th dimension input variable x i The expected density function, Describes the i-th dimension input variable x i The distribution parameters, x represents i by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

4. The method according to claim 2, characterized in that, The transformation process of the function includes: Perform a Box-Cox transformation on the function to determine the transformed function; The Box-Cox transformation of the function includes: The Box-Cox transformation of the function is performed using the following transformation formula: Where g(x) represents the function, This represents the transformed function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

5. The method according to claim 2, characterized in that, The calculation of the first four statistical moments of the transformed function includes: The first four statistical moments of the function are calculated using the following formula: in, Representing the function The mean, standard deviation, skewness, and kurtosis estimates, x (z) Let ω represent the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. (z) Represents a random allocation point x (z) The corresponding weight, N c This represents the number of n-dimensional randomized points corresponding to the n-dimensional expected density function. The calculation of the unconditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the function includes: The fourth-order reliability index estimate of the function is determined based on the first four statistical moments of the function, and the unconditional augmented failure probability estimate of the turbine blade structure is calculated based on the fourth-order reliability index estimate of the function. The fourth-order reliability index estimate and the unconditional augmented failure probability estimate of the function are expressed as follows: in, Represents function The estimated value of the fourth-order reliability index, Let Φ(·) represent the unconditionally augmented failure probability estimate, and let Φ(·) represent the cumulative distribution function of the standard normal distribution.

6. The method according to claim 1, characterized in that, The conditional augmented failure probability of the turbine blade structure at the Gauss-Hermite integral quadrature node is calculated based on the (n-1)-dimensional expected density function excluding the i-th dimension and the conditional function when the i-th input variable is fixed under the i-th dimension expected density function. This includes: Obtain the n-1-dimensional standard random placement points corresponding to the n-1-dimensional standard normal distribution density function and the weights corresponding to the standard random placement points, and convert the standard random placement points into n-1-dimensional random placement points corresponding to the n-1-dimensional expected density function excluding the i-th dimension; Based on the n-1-dimensional random placement points corresponding to the n-1-dimensional expected density function (excluding the i-th dimension) and the weights corresponding to each random placement point, the conditional function value of each n-1-dimensional random placement point is calculated using the conditional function when the i-th input variable is fixed at the k-th Gauss-Hermite integral quadrature node under the i-th expected density function. The conditional function is transformed, and the first four statistical moments of the transformed conditional function are calculated. The conditional augmented failure probability of the turbine blade structure is calculated based on the first four statistical moments of the conditional function.

7. The method according to claim 6, characterized in that, The n-1 dimensional random input variable The expected density function is: in, Describes the (n-1)-dimensional expectation density function. Describes the j-th dimension input variable x j The expected density function, Describes the j-th dimension input variable x j The distribution parameters, x represents j by Let be the joint probability density function of the distribution parameters. express The joint probability density function.

8. The method according to claim 6, characterized in that, The transformation process of the conditional function includes: Perform a Box-Cox transformation on the conditional function to determine the transformed conditional function; The Box-Cox transformation of the conditional function includes: The conditional function is transformed using the following formula. Perform Box-Cox transformation: in, x represents i Fixed in Conditional function at time, This represents the transformed conditional function, and τ represents the parameters that need to be determined in the Box-Cox transformation.

9. The method according to claim 6, characterized in that, The calculation of the first four statistical moments of the conditional function after transformation includes: The first four statistical moments of the conditional function are calculated using the following formula: in, Representing conditional function Estimates of the mean, standard deviation, skewness, and kurtosis. This represents the n-dimensional random locus of points corresponding to the n-dimensional expectation density function. Represents a random allocation point The corresponding weight, N c ′ represents the number of n-1 dimensional randomized points. The calculation of the conditional augmented failure probability of the turbine blade structure based on the first four statistical moments of the conditional function includes: The fourth-order reliability index estimate of the conditional function is determined based on the first four statistical moments of the conditional function, and the conditional augmented failure probability estimate of the turbine blade structure is calculated based on the fourth-order reliability index estimate of the function. The fourth-order reliability index estimate and the conditionally augmented failure probability estimate of the conditional function are expressed as follows: in, x represents i Fixed in Conditional function at time, Represents a conditional function The estimated value of the fourth-order reliability index, Φ(·) represents the estimated probability of conditional augmentation failure, and Φ(·) represents the cumulative distribution function of the standard normal distribution.

10. The method according to claim 1, characterized in that, The step of determining the global sensitivity of the augmented failure probability of the turbine blade structure based on the unconditional augmented failure probability and the conditional augmented failure probability of the turbine blade structure corresponding to each Gauss-Hermite integral node includes: Based on the unconditional augmented failure probability estimate and the conditional augmented failure probability estimate of the turbine blade structure corresponding to each Gauss-Hermite integral node, the global sensitivity of the augmented failure probability of the turbine blade structure is determined by the following formula: in, This represents an estimate of the global sensitivity to the augmented failure probability. This represents the estimate of the unconditionally augmented failure probability. ξ represents the estimated probability of conditional augmentation failure. (k) This represents the Gauss-Hermite integral quadrature node. The corresponding weights.

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