A method for quantifying aerodynamic uncertainty of airfoil shape and application thereof

Abstract

The application of a kind of airfoil aerodynamic uncertainty quantification method and application belongs to compressor technical field;The specific steps of the method are as follows: the geometric profile of the blade is parameterized designed;The probability density function of the uncertainty input parameter is calculated;The orthogonal basis function of the data-driven proxy model is solved;The undetermined coefficient of the data-driven proxy model is solved;The uncertainty quantification of aerodynamic performance parameters.The aerodynamic uncertainty quantification of the coupling influence of the compressor airfoil installation angle error and profile error is obtained by the application, the first three order statistical moments of the aerodynamic coefficient obtained by the application are basically consistent with the Monte Carlo method, but the training set sample number required by the application is about 0.006 times of the Monte Carlo method.At the same time, the statistical histogram of the aerodynamic coefficient obtained by the application and the Monte Carlo method is almost coincident, so the application method has high aerodynamic uncertainty quantification precision.

Description

A method for quantifying aerodynamic uncertainties of airfoil and its application Technical Field

[0001] This invention belongs to the field of compressor technology, specifically relating to a method for quantifying the aerodynamic uncertainty of airfoil and its application. Background Technology

[0002] In the design process of any engineering system, uncertainties in design parameters are always present and cannot be completely eliminated. In the field of aerodynamics, uncertainties such as manufacturing deviations cause the aerodynamic shape of the system to deviate from the design shape, significantly impacting performance robustness. Advanced compressor blades not only require high performance but also high performance robustness under the influence of uncertainties. Therefore, conducting statistical research on the aerodynamic uncertainties of compressor blades and achieving accurate quantification is of great significance for the robust optimization design of aerodynamic shapes.

[0003] The analytical method widely used in the quantification of aerodynamic uncertainties in compressor blades is the Generalized Non-Intrusive Polynomial Chaos (GNIPC) method. GNIPC integrates numerical simulation tools and can quickly output responses for given input parameters, making it relatively convenient to implement. The core idea of ​​GNIPC is to project the uncertain output in the problem onto a random space composed of a set of optimal orthogonal bases containing standard random input variables. In uncertainty analysis, the probability distribution of the standard random input needs to match the optimal orthogonal base, as shown in the literature (D. Xiu, GE Karniadakis. Modelling uncertainty in flow simulations via generalized polynomial chaos[J]. Journal of Computational and Physics, 2003, 187(1):137-167). When the probability distribution of the random input does not match the optimal orthogonal base, GNIPC has low convergence. Therefore, GNIPC is limited to solving problems where uncertain input variables follow standard probability distributions, such as uniform distributions and Gaussian distributions. However, most current domestic and international studies on the quantification of aerodynamic uncertainties in compressor blades make subjective assumptions about the probability distribution of input variables, assuming that they follow a certain standard probability distribution. In actual engineering, the data for uncertain inputs are usually limited, causing the probability distribution of uncertain inputs to deviate significantly from the standard probability distribution. This leads to large errors in the uncertainty quantification results of GNIPC.

[0004] To avoid fitting errors in quantification results caused by subjective assumptions about the distribution of input parameters, existing technologies apply data-driven, non-embedded multinomial chaos to aerodynamic uncertainty quantification research. This technique constructs the corresponding optimal orthogonal basis using the statistical moments of random input variables to propagate probabilistic information, thus eliminating the need for subjective assumptions about the probability distribution of random inputs. When multiple random input parameters are involved, the optimal orthogonal basis corresponding to the multidimensional random input is obtained by performing a tensor product on the optimal orthogonal basis corresponding to all one-dimensional random inputs. In this case, the orthogonality of the optimal orthogonal basis corresponding to the multidimensional random input holds only when the one-dimensional random inputs are independent. Therefore, DNIPC is limited to solving the uncertainty propagation problem where multidimensional random input variables satisfy the independence assumption. In practical engineering, compressor blades have high degrees of bending, twisting, and sweeping, making them difficult to manufacture, resulting in numerous manufacturing deviation parameters, and the statistical information of these error parameters is dependent on each other. Therefore, in order to accurately assess the robustness of compressor blade performance and thus guide aerodynamic design, it is necessary to develop a new data-driven method for quantifying aerodynamic uncertainties in compressor blade profiles to address the problem of uncertainty propagation from non-standard and non-independent random input parameters. Summary of the Invention

[0005] The technical problem to be solved:

[0006] To overcome the shortcomings of existing technologies, this invention provides a method for quantifying aerodynamic uncertainties in airfoil configurations. This method uses the joint probability density function of measurement data to propagate probabilistic information from random inputs, solving the problem of uncertainty propagation from non-independent and non-standard random inputs in existing aerodynamic uncertainty quantification techniques. It also avoids errors in the uncertainty quantification results caused by assumptions about the distribution and dependencies of input parameters. Using this method, the installation angle error and profile error of compressor blades can be eliminated.

[0007] The technical solution of this invention is: a method for quantifying the aerodynamic uncertainty of airfoils, the specific steps of which are as follows:

[0008] Step 1: Parametrically design the geometry of the blades;

[0009] Step 2: Calculate the probability density function of the uncertain input parameters;

[0010] Based on n finite measurement data, the D-dimensional uncertainty input parameters x = (x1, x2, ..., xn) are calculated using kernel density estimation techniques. D The probability density function of ).

[0011] Step 3: Solve for the orthogonal basis functions of the data-driven proxy model;

[0012] S3.1, using a one-dimensional random variable x∈x=(x1, x2, ..., x...D Define a simple polynomial sequence {v} i}:

[0013] v i (x)=x i-1

[0014] S3.2, Create a D-dimensional random variable x = (x1, x2, ..., xn) D The corresponding multivariate polynomial sequence {v} i The formula is as follows:

[0015]

[0016] In the formula, the subscript i represents the multiple index i = (i1, i2, ..., i...) using a hierarchical reverse lexicographical sort. D The amount of )

[0017] S3.3, let Let v\v1 denote the vector of v with its first component v1 = 1 removed, and use a decorrelation algorithm to calculate the orthogonal basis function vector. in That is, orthogonal basis functions; the specific formulas are as follows:

[0018] Φ=(L -1 (v\v1-E[v\v1]))∪Φ1

[0019] In the formula, Φ1=1 represents the first component of the orthogonal basis function vector Φ, and matrix L is solved using the improved Cholesky decomposition, as shown in the following formula:

[0020]

[0021] In the formula, Δ is a diagonal stable matrix, and the matrix element Cov(v) i v j ) is defined as:

[0022] Cov(v i v j )=∫(v i -E[v i ])(v j -E[v j ])ρ joi (x)dx

[0023] The above calculations were performed using Clenshaw-Curtis numerical integration.

[0024] Step 4: Solve for the undetermined coefficients of the data-driven proxy model;

[0025] The formula for calculating the undetermined coefficients is as follows:

[0026]

[0027] In the formula, ψ is defined as

[0028]

[0029] Step 5: Quantification of the uncertainty of aerodynamic performance parameters;

[0030] Using the orthogonality of the orthogonal basis functions obtained in step 3 and the undetermined coefficients obtained in step 4, the mean μ(X) and standard deviation σ(X) of the aerodynamic performance parameter X are calculated using the following formulas:

[0031] μ(X)=a1

[0032]

[0033] The skewness δ(X) of the aerodynamic performance parameter is obtained by statistically analyzing the predicted values ​​of the surrogate model, and is defined as follows:

[0034]

[0035] In the formula, N represents the predicted value of the proxy model. s Represents the joint probability density function ρ joi The number of Monte Carlo random samplings for (x).

[0036] A further technical solution of the present invention is as follows: In step 1, firstly, the mid-arc line of the designed blade is parameterized using a Bezier curve; then, the suction and pressure surface profile control points of the designed blade are constructed by superimposing the thickness of the mid-arc line, and the suction and pressure surface profiles of the designed blade are obtained by connecting the control points through a B-Spline curve; finally, the suction and pressure surface profiles are connected using the leading and trailing edges of a circle to obtain the complete parameterized profile of the designed blade.

[0037] A further technical solution of the present invention is: in step 2, the uncertainty input parameters include the marginal probability density function ρ(x) and the two-dimensional joint probability density function ρ joi (x i The joint probability density function ρ of dimension xx) and D dimension joi (x), the specific calculation formula is as follows:

[0038]

[0039]

[0040]

[0041] In the formula, h b , and Let K, K″, and K represent the bandwidth calculated according to Silverman's rule. D These represent one-dimensional, two-dimensional, and D-dimensional Gaussian kernel functions, respectively.

[0042] A further technical solution of the present invention is: in step 3, the domain of the uncertain input parameter is divided into 100 uniformly spaced sub-intervals, and each interval is integrated using 101 Clenshaw-Curtis nodes.

[0043] A further technical solution of the present invention is: in step 4, the aerodynamic performance parameter values... The steps are as follows: First, let p = 4, and with uncertainty, input x = (x1, x2, ..., x...). D On the probability space of N, N is obtained by sampling using Halton pseudo-random sequences. t =2N p Leaf sample Then for these N t Numerical simulations of the flow field were performed on samples to obtain N. t Aerodynamic performance parameters of individual blade samples

[0044] A further technical solution of the present invention is: in step 5, the recommended number of Monte Carlo random samplings is 10,000.

[0045] An application of a blade aerodynamic uncertainty quantification method is disclosed, which quantifies the uncertainty of the total pressure loss coefficient and static pressure coefficient of compressor blades, thereby eliminating the installation angle error and profile error of compressor blades.

[0046] Beneficial effects

[0047] The advantages of this invention are as follows: compared to traditional uncertainty quantification methods that require assumptions that the probability distributions of uncertain inputs are standard and independent, this invention does not require assumptions about the distribution and independence of random input parameters, but rather propagates statistical information based on the joint probability density function of the input parameters.

[0048] The decorrelation algorithm developed in step 3 of this invention does not need to make assumptions about the distribution and independence of uncertain input parameters when creating the basis function of the surrogate model. Instead, it propagates statistical information based on the joint probability density function of the input parameters. Therefore, it can avoid the error caused by the assumption of the distribution and dependence of the input parameters to the uncertainty quantification result.

[0049] This invention quantifies the aerodynamic uncertainty caused by the coupled effects of compressor blade installation angle error and profile error. The first three statistical moments of the aerodynamic coefficients obtained by this invention are basically consistent with those obtained by the Monte Carlo method, but the number of training samples required by this invention is approximately 0.006 times that of the Monte Carlo method. Furthermore, the statistical histograms of the aerodynamic coefficients obtained by this invention and the Monte Carlo method almost overlap, thus this invention has higher accuracy in quantifying aerodynamic uncertainty. Attached Figure Description

[0050] Figure 1 is a technical roadmap of the method of the present invention.

[0051] Figure 2 shows the design geometry of an embodiment of the present invention.

[0052] Figure 3 shows the parameterization scheme of the geometry in an embodiment of the present invention.

[0053] Figure 4 shows the statistical histogram of the measured installation angle error and profile error.

[0054] Figure 5 shows the marginal and joint probability density distributions of installation angle error and profile error.

[0055] Figure 6 shows the orthogonal basis functions based on the installation angle error and profile error data.

[0056] Figure 7 shows the convergence curve of the Monte Carlo method.

[0057] Figure 8 is a statistical histogram of the total pressure loss coefficient obtained by the Monte Carlo method and the method of the present invention at an angle of attack of 2.5°.

[0058] Figure 9 shows the statistical histograms of the static pressure coefficients obtained by the Monte Carlo method and the method of this invention at an angle of attack of 2.5°. Detailed Implementation

[0059] The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the invention, and should not be construed as limiting the invention.

[0060] This embodiment specifically presents a data-driven method for quantifying the aerodynamic uncertainty of compressor blades. It uses the joint probability density function of the measured data to propagate the probability information of random inputs, solving the problem of uncertainty propagation of non-independent and non-standard random inputs in the existing aerodynamic uncertainty quantification technology. It also avoids the errors in uncertainty quantification results caused by assumptions about the distribution form and dependencies of input parameters.

[0061] The solution to the technical problem is as follows: The mid-arc line is parameterized using Bezier curves, and a geometric line is constructed by superimposing thicknesses on the mid-arc line and using B-Spline curves. Based on limited measurement data, the probability density function of the random input parameters is determined using kernel density estimation, and the basis functions of the surrogate model are generated using a decorrelation algorithm. Training samples are obtained using Halton pseudo-random sequences in the probability space of the random input. The aerodynamic performance parameter values ​​of the training samples are obtained through numerical simulation, and the surrogate model is trained using a least-squares regression algorithm. Finally, the trained surrogate model is used to quantify aerodynamic uncertainties.

[0062] The above technical solution quantifies the aerodynamic uncertainty caused by the coupled effects of compressor blade installation angle error and profile error. The first three statistical moments of the aerodynamic coefficients obtained are basically consistent with those obtained by the Monte Carlo method; however, the number of training set samples required in this embodiment is approximately 0.006 times that of the Monte Carlo method. Furthermore, the statistical histograms of the aerodynamic coefficients obtained in this embodiment and the Monte Carlo method almost overlap, therefore, the method in this embodiment has higher accuracy in quantifying aerodynamic uncertainty.

[0063] Example:

[0064] The technical solution will be described in detail below with reference to the accompanying drawings:

[0065] This embodiment presents a data-driven method for quantifying the aerodynamic uncertainties of compressor blade profiles, as shown in Figure 1. Installation angle error and profile error are the two most common types of errors in compressor blade manufacturing. This embodiment quantifies the uncertainties of the total pressure loss coefficient and static pressure coefficient of a controllable diffusion compressor blade profile, focusing on the installation angle error and profile error. The invention is further described in conjunction with the accompanying drawings. The design geometry of the blade profile is shown in Figure 2.

[0066] Step 1: Parametric design of blade geometry:

[0067] The Bezier curve is used to parameterize the mid-curve of the designed blade. Then, the thickness of the mid-curve is superimposed to construct the control points of the suction and pressure surface profiles of the designed blade. The control points are connected by B-Spline curves to obtain the suction and pressure surface profiles of the designed blade. Finally, the leading and trailing edges of the circles are used to connect the suction and pressure surface profiles to obtain the complete parameterized profile of the designed blade. Based on this, the blade model with geometric changes is completed.

[0068] As shown in Figure 3, the mid-arc line in this embodiment is fitted by a Bezier curve controlled by 13 control points, and the thickness distribution is described by 16 symmetrical thicknesses H along the mid-arc line. The airfoil with installation angle error θ and profile error e can be described by equations (1)-(3):

[0069] γ actual =γ+θ (1)

[0070] R i-actual =R i +ei∈{LE,TE} (2)

[0071] h i-actual =h i +e for i=1, 2,..., 16 (3)

[0072] In the formula, γ represents the installation angle, and R LE and R TE These represent the leading edge radius and trailing edge radius, respectively, with the subscript actual indicating the airfoil with machining errors.

[0073] Step 2: Calculate the probability density function of the uncertain input parameters:

[0074] Based on n finite measurement data, the D-dimensional uncertainty input parameters x = (x1, x2, ..., xn) are calculated using kernel density estimation techniques. D The probability density function of ) . Uncertainty input parameters include the marginal probability density function ρ(x) and the two-dimensional joint probability density function ρ joi (x i x j ) and D-dimensional joint probability density function ρ joi (x), the specific calculation formula is as follows:

[0075]

[0076]

[0077]

[0078] In the formula, h b , and Let K, K″, and K represent the bandwidth calculated according to Silverman's rule. D These represent one-dimensional, two-dimensional, and D-dimensional Gaussian kernel functions, respectively.

[0079] After measuring and statistically analyzing the blade profile errors of 100 actual machined blades, the statistical histograms of the installation angle error θ and profile error e are shown in Figure 4. Based on these 100 sets of measurement data, the probability density functions ρ(θ), ρ(e), and ρe were calculated using kernel density estimation techniques. joi (θ, e) is shown in Figure 5. It can be seen that the probability density distributions of the installation angle error and the profile error are not standard distributions, and there is a correlation between the two, that is, they are not independent of each other.

[0080] Step 3: Solve for the basis functions of the data-driven proxy model:

[0081] For a stochastic physical model X(x, ξ), its output can be expressed as a linear combination of p-order orthogonal basis functions on the physical random space ξ:

[0082]

[0083] In the formula a j Φ represents the undetermined coefficient. j (x) represents a D-dimensional uncertain input x = (x1, x2, ..., xn) D The corresponding orthogonal basis functions; N p The number of orthogonal basis functions is defined as follows:

[0084]

[0085] The core of the data-driven agent model is to solve for orthogonal basis functions based on the joint probability density function of uncertain inputs.

[0086] First, using a one-dimensional random variable x∈x=(x1, x2, ..., x D Define a simple polynomial sequence {v} i}:

[0087] v i (x)=x i-1 (9)

[0088] Subsequently, a D-dimensional random variable x = (x1, x2, ..., xn) D The corresponding multivariate polynomial sequence {v} i} can be created by the tensor quadrature operation shown in equation (10):

[0089]

[0090] In the formula, the subscript i represents the multiple index i = (i1, i2, ..., i...) using graded reverse lexicographical ordering. D The amount of ).

[0091] Finally, set Let v\v1 denote the vector of v with its first component v1 = 1 removed, and use a decorrelation algorithm to calculate the orthogonal basis function vector. in That is, orthogonal basis functions.

[0092] The decorrelation algorithm can be expressed by equation (11):

[0093] Φ=(L -1 (v\v1-E[v\v1]))∪Φ1(11)

[0094] In the formula, Φ1=1 represents the first component of the orthogonal basis function vector Φ, and matrix L can be obtained by Cholesky decomposition of the covariance matrix Cov(v\v1). Since the standard Cholesky decomposition is prone to rounding errors, an improved Cholesky decomposition is used to solve matrix L, which can be expressed by equation (12):

[0095]

[0096] In the formula, Δ is a diagonal stable matrix, and the matrix element Cov(v) i v j ) is defined as

[0097] Cov(v i v j )=∫(v i -E[v i ])(v j -E[v j ])ρ joi (x)dx (13)

[0098] Equation (13) is calculated using Clenshaw-Curtis numerical integration. The domain of the uncertain input parameter is divided into 100 uniformly spaced sub-intervals, and each interval is integrated using 101 Clenshaw-Curtis nodes.

[0099] Taking the mean and covariance of equation (11) yields the following results.

[0100] E[Φ\Φ1]=L-1 (E[v\v1]-E[v\v1])=0 (14)

[0101] Cov(Φ\Φ1)=L -1 Cov(v\v1)L -T =L -1 LL T L -T =I (15)

[0102] In the formula, I represents the identity matrix. According to equations (14)-(15), the vector inner product can be obtained.

[0103] <Φ i Φ j >=E[Φ i Φ j ]=E[Φ i Φ j ]-E[Φ i ]E[Φ j ]=Cov(Φ i Φ j )=γ ij ,i≠1,j≠1 (16)

[0104] In the formula, γ ij Let Kronecker function be denoted as Kronecker function. From equation (16), it can be seen that for an uncertain input x = (x1, x2, ..., xn) following an arbitrary probability distribution... D The orthogonal basis function vectors Φ\Φ1 obtained by solving equation (11) satisfy orthogonality. Since Φ1=1, the complete orthogonal basis function vector Φ also satisfies orthogonality, that is... They satisfy orthogonality. Furthermore, the orthogonal basis functions obtained by solving equation (11) are not obtained by performing tensor products on the orthogonal basis functions corresponding to all one-dimensional random inputs. Therefore, the solved orthogonal basis functions can be used to propagate the uncertainty of non-independent random input parameters.

[0105] In this embodiment, the installation angle error and the profile error are uncertain input variables. Based on the probability density distribution shown in Figure 5, the first 5 orthogonal basis functions calculated by Equation (11) are shown in Figure 6.

[0106] Step 4: Solve for the undetermined coefficients of the data-driven proxy model:

[0107] Given an uncertain input x = (x1, x2, ..., x...), D On the probability space of N, N is obtained by sampling using Halton pseudo-random sequences. t =2N p Leaf sample Then for these N tNumerical simulations of the flow field were performed on samples to obtain N. t Aerodynamic performance parameters of individual blade samples Finally, the least squares regression method is used to solve for the undetermined coefficient a in equation (7). j :

[0108]

[0109] In the formula, ψ is defined as

[0110]

[0111] Based on research experience, a linear combination of p = 4th order orthogonal basis functions is sufficient to quantify the uncertainty problem of compressor internal flow. t =2N p Training with leaf samples can make the undetermined coefficients a j The solution converges. Therefore, for this embodiment, when the joint probability density function is ρ joi On the probability space of (θ, e), obtained by sampling from Halton pseudo-random sequences. Thirty blade samples were used to solve for the undetermined coefficients. The geometry corresponding to the thirty blade samples was constructed by equations (1)-(3). The flow field of each blade sample was numerically simulated, and the calculation results were post-processed to obtain the total pressure loss coefficient and static pressure coefficient corresponding to each blade sample. Finally, the results of the total pressure loss coefficient and static pressure coefficient were substituted into equation (17) to calculate the undetermined coefficients based on the data-driven surrogate model.

[0112] Using the average error ε avr and root mean square error ε ms To test the accuracy of the surrogate model, the following definition is used:

[0113]

[0114]

[0115] In the formula f and These represent the calculated value from the numerical simulation and the predicted value from the surrogate model, respectively; M represents the number of test set samples, which is set to 5000 in this embodiment.

[0116] Table 1 shows the average and root mean square errors of the total pressure loss coefficient and static pressure coefficient of the airfoil at 0° and 2.5° angles of attack. As can be seen from the table, the accuracy of the method of this invention reaches 10%. -5 The magnitude of the value indicates high model prediction accuracy. Therefore, p=4 and N are used. t =2N pIt is reasonable to create a proxy model.

[0117] Step 5: Quantification of the uncertainty of aerodynamic performance parameters:

[0118] Using the orthogonality of the orthogonal basis functions obtained in step three, the mean μ(X) and standard deviation σ(X) of the aerodynamic performance parameter X can be directly calculated by equations (21) and (22):

[0119] μ(X)=a1(21)

[0120]

[0121] The skewness δ(X) of the aerodynamic performance parameters can be obtained by statistically analyzing the predicted values ​​of the surrogate model, and is defined as follows:

[0122]

[0123] In the formula N s Represents the joint probability density function ρ joi The recommended value for the number of Monte Carlo random samplings for (x) is 10000.

[0124] To verify the accuracy of aerodynamic uncertainty quantification using the method of this invention, Table 2 compares the first three statistical moments of the aerodynamic coefficients obtained by the Monte Carlo method and the method of this invention at a 2.5° angle of attack. The convergence curve of the Monte Carlo method is shown in Figure 7. Table 2 shows that the first three statistical moments obtained by the method of this invention are basically consistent with those obtained by the Monte Carlo method, but the number of training samples required by the method of this invention is approximately 0.006 times that of the Monte Carlo method. Furthermore, Figures 8 and 9 compare the statistical histograms of the aerodynamic coefficients obtained by the Monte Carlo method and the method of this invention at a 2.5° angle of attack. It can be seen that the statistical histograms of the aerodynamic coefficients obtained by the two methods almost overlap, which further illustrates the high accuracy of aerodynamic uncertainty quantification using the method of this invention.

[0125] Table 1 shows the average and root mean square errors of the aerodynamic coefficients at 0° and 2.5° angles of attack.

[0126] Angle of attack (°) Aerodynamic coefficient Average error Root mean square error 0 Total pressure loss coefficient 2.0E-0.5 7.0E-0.5 0 Static pressure coefficient 1.7E-0.5 1.9E-0.5 2.5 Total pressure loss coefficient 2.0E-0.5 3.0E-0.5 2.5 Static pressure coefficient 4.0E-0.5 5.0E-0.5 surface

[0127] Table 2 shows the statistical moments of aerodynamic coefficients obtained by different uncertainty quantification methods at an angle of attack of 2.5°.

[0128]

[0129]

[0130] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for quantifying aerodynamic uncertainties of airfoil shapes, characterized in that... The specific steps are as follows: Step 1: Parametrically design the blade's geometry; Step 2: Calculate the probability density function of the uncertain input parameters; based on Using a limited set of measurement data, kernel density estimation techniques are employed to calculate... 3D uncertainty input parameters The probability density function; Step 3: Solve for the orthogonal basis functions of the data-driven surrogate model; S3.1, using one-dimensional random variables Define a sequence of simple polynomials ： S3.2, Create 3D random variables The corresponding multivariate polynomial sequence The formula is as follows: In the formula, the subscript This indicates a multi-index using a hierarchical reverse lexicographical sort. The components; S3.3, let ,make express Remove the first component The vectors are used, and the orthogonal basis function vectors are calculated using a decorrelation algorithm. ,in That is, orthogonal basis functions; the specific formulas are as follows: In the formula, Represents orthogonal basis function vectors The first component, matrix The improved Cholesky decomposition is used for the solution, as shown in the following formula: In the formula, For a diagonally stable matrix, the matrix elements are... Defined as: The above calculations are performed using Clenshaw-Curtis numerical integration; Step 4: Solve for the undetermined coefficients of the data-driven proxy model; the formula for calculating the undetermined coefficients is as follows: In the formula, Defined as Step 5: Quantification of uncertainty in aerodynamic performance parameters; using the orthogonality of the orthogonal basis functions obtained in Step 3 and the undetermined coefficients obtained in Step 4, calculate the aerodynamic performance parameters. mean and standard deviation The formula is: The skewness of the aerodynamic performance parameters The values ​​predicted by the surrogate model are statistically analyzed and defined as follows: In the formula, This represents the predicted value from the proxy model. Represented by joint probability density function The number of Monte Carlo random samplings.

2. The method for quantifying aerodynamic uncertainties of airfoils according to claim 1, characterized in that: In step 1, firstly, the mid-curve of the designed blade is parameterized using a Bezier curve; then, the suction and pressure surface profile control points of the designed blade are constructed by superimposing the thickness of the mid-curve; the suction and pressure surface profiles of the designed blade are obtained by connecting the control points using a B-Spline curve; finally, the suction and pressure surface profiles are connected using the leading and trailing edges of a circle to obtain the complete parameterized profile of the designed blade.

3. The method for quantifying aerodynamic uncertainties of airfoils according to claim 2, characterized in that: In step 2, the uncertainty input parameters include the marginal probability density function. Two-dimensional joint probability density function and Joint probability density function The specific calculation formula is as follows: (1) (2) (3) In the formula, 、 and This represents the bandwidth calculated according to Silverman's rule. 、 and Representing one-dimensional, two-dimensional, and respectively Gaussian kernel function.

4. The method for quantifying aerodynamic uncertainties of airfoil according to claim 3, characterized in that: In step 3, the domain of the uncertain input parameters is divided into 100 evenly spaced sub-intervals, and each interval is integrated using 101 Clenshaw-Curtis nodes.

5. The method for quantifying aerodynamic uncertainties of airfoil according to claim 3, characterized in that: In step 4, the aerodynamic performance parameter values The steps to obtain the result are as follows: First, let... In the face of uncertain input On the probability space, sampling using Halton pseudo-random sequences is used to obtain... Leaf sample Then regarding this Numerical simulations of the flow field were performed on each sample to obtain... Aerodynamic performance parameters of individual blade samples 。 6. The method for quantifying aerodynamic uncertainties of airfoils according to claim 1, characterized in that: In step 5, the number of Monte Carlo random samplings is 10,000.

7. The method for quantifying aerodynamic uncertainties of airfoils according to any one of claims 1-6, characterized in that: The method quantifies the uncertainties of the total pressure loss coefficient and static pressure coefficient of the compressor blades, thereby eliminating the installation angle error and profile error of the compressor blades.

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