CN112613236A - Method and system for processing wing profile parameters - Google Patents
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Abstract
本发明公开一种翼型参数处理方法及系统，包括：首先，确定翼型的上翼面与下翼面的形状，将所述翼型的上翼面与下翼面的形状映射为类函数和型函数；其次，基于所述类函数和型函数，使用拉丁超立方算法生成控制翼型前缘形状与翼型后缘形状的样本点；然后，根据所述控制翼型前缘形状与翼型后缘形状的样本点，使用径向基神经网络模型建立均方根误差的代理模型；最后，使用遗传算法对所述均方根误差的代理模型进行优化求解，得到翼型参数化模型的前缘基函数和尾缘基函数。将LEM CST方法和改进的Hicks‑Henne型函数方法进行有效地结合，从而对翼型进行高精度地拟合，同时减少了翼型参数化设计变量的数目，提高了翼型气动优化的效率。
The invention discloses an airfoil parameter processing method and system, comprising: first, determining the shapes of the upper airfoil surface and the lower airfoil surface of the airfoil, and mapping the shapes of the upper airfoil surface and the lower airfoil surface of the airfoil into a class function and shape function; secondly, based on the class function and shape function, use the Latin hypercube algorithm to generate sample points that control the shape of the leading edge of the airfoil and the shape of the trailing edge of the airfoil; then, according to the shape of the leading edge of the airfoil and the shape of the wing The sample points of the shape of the trailing edge of the airfoil are used to establish a surrogate model of the root mean square error by using the radial basis neural network model; finally, the surrogate model of the root mean square error is optimized and solved by using the genetic algorithm, and the Leading edge basis functions and trailing edge basis functions. The LEM CST method and the improved Hicks-Henne type function method are effectively combined to fit the airfoil with high precision, while reducing the number of parametric design variables of the airfoil and improving the efficiency of aerodynamic optimization of the airfoil.
Description
Technical Field
The invention belongs to the technical field of aerodynamic design of an airfoil profile, and particularly relates to an airfoil profile parameter processing method and system.
Background
The parameterization of the shape of the wing profile is to use mathematical equations to express the profile of the wing profile, which is an important part in the aerodynamic optimization of the wing profile. In the airfoil shape parameterization, different design variables can produce different airfoils, so the shape parameterization can directly influence the design space of the airfoil. And the more accurate the profile of the airfoil, the more accurate the aerodynamic design of the airfoil. Therefore, in the aerodynamic optimization design of the wing profile, the selection of a proper shape parameterization method is very important.
There are many references describing airfoil parameterization methods, such as the lattice node method, the B-spline method, the Hicks-Henne type function method, and the Parametric Section (PARSEC), in Castonguy P, Nadarjah S.Effect of shape parameterization on aerodynamic shape optimization [ A ].45th AIAA Aerospace Sciences evaluation and optimization [ C ]. Reno, Nevada,2007: 59; references Sripawadkul V, Padulo M, Guenov M.A compliance of air foil shape parameter optimization [ A ].13th AIAA/ISSMO multiple characterization optimization concept [ C ]. Fort Worth, Texas,2010:9050. Fragson curve equation, Hicks-Henne type function equation, B spline curve equation, PARSEC equation, and shape-like function transformation equation are aligned according to simplicity, intuitiveness, orthogonality, completeness, and absence. References Masters D A, Taylor N J, Rendall T, et al, geometric comprison of Aerofoil Parameterization Methods [ J ]. AIAA Journal,2017,55(5):1575-1589. class/Shape function transformation method, B-spline curve method, Hicks-Henne type function method, radial basis principal element method, Bessel surface equation, singular value decomposition modal method and PARSEC method were analyzed and the efficiency of these Parameterization Methods was tested.
The class/shape function transformation parameterization method (CST) is improved by reference to Kulfan B M.modification of CST air foil reconstruction method, http:// www.brendakulfan.com/docs/CST8.pdf, and a wing parameterization method for leading edge modification (LEM CST) is provided. When the LEM CST method is used for fitting the large-camber and complex airfoil profile, the fitting precision near the trailing edge of the airfoil profile is low, and the adjusting capability of the leading edge of the airfoil profile is limited. The references Liang X, Meng G, Tong S, et al, Rapid design and optimization of air foil based on improved genetic algorithm [ J ], ActaAerodynamica Sinica,2016,31(6):803-812. an improved Hicks-Henne type function method is proposed, which can improve the control capability of the airfoil trailing edge and expand the design space of the airfoil, but the accuracy of the airfoil fitting is poor, and more design variables are needed to reach the standard of typical experimental wind tunnel fitting.
Disclosure of Invention
The invention aims to provide a method and a system for processing wing section parameters, which solve the problem of reducing the number of wing section parametric design variables to perform high-precision fitting.
In view of the above, the present invention provides a method for processing airfoil parameters, which includes:
firstly, determining the shapes of an upper wing surface and a lower wing surface of an airfoil profile, and mapping the shapes of the upper wing surface and the lower wing surface of the airfoil profile into a class function and a type function;
secondly, based on the class function and the type function, generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm;
then, according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil profile, a radial basis function neural network model is used for establishing a proxy model of the root mean square error;
and finally, carrying out optimization solution on the proxy model of the root mean square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
Further, the shapes of the upper airfoil surface and the lower airfoil surface of the airfoil profile are mapped into a class function and a type function, and control parameters for controlling the slope and the attenuation of the basis function are increased for indication.
Further, generating sample points controlling the airfoil leading edge shape and the airfoil trailing edge shape using a latin hypercube algorithm, comprising: and acquiring coordinates of an upper wing surface and a lower wing surface of the airfoil.
Further, the proxy model of the root mean square error is optimized and solved by using a genetic algorithm, and the method comprises the following steps: the minimum root mean square error was solved using the genetic algorithm toolkit of MATLAB.
Another object of the present invention is to provide a system for processing airfoil parameters, comprising:
the mapping module is used for determining the shapes of the upper wing surface and the lower wing surface of the wing profile and mapping the shapes of the upper wing surface and the lower wing surface of the wing profile into a class function and a type function;
the generating module is used for generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm based on the class function and the type function;
the building module is used for building a proxy model of the root mean square error by using a radial basis function neural network model according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil;
and the calculating module is used for carrying out optimization solving on the proxy model of the root-mean-square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
Further, the mapping module is instructed to increase the control parameter for controlling the slope and the attenuation of the basis function.
Further, the generating module comprises an obtaining sub-module for obtaining coordinates of an upper airfoil surface and a lower airfoil surface of the airfoil.
Further, the solution module uses a genetic algorithm toolset of MATLAB to solve for the minimum root mean square error.
The invention achieves the following significant beneficial effects:
the realization is simple, include: firstly, determining the shapes of an upper wing surface and a lower wing surface of an airfoil profile, and mapping the shapes of the upper wing surface and the lower wing surface of the airfoil profile into a class function and a type function; secondly, based on the class function and the type function, generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm; then, according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil profile, a radial basis function neural network model is used for establishing a proxy model of the root mean square error; and finally, carrying out optimization solution on the proxy model of the root mean square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model. The LEM CST method and the improved Hicks-Henne type function method are effectively combined, so that the wing profile is fitted with high precision, the number of wing profile parametric design variables is reduced, and the efficiency of wing profile pneumatic optimization is improved.
Drawings
FIG. 1 is a flow chart of a method of processing airfoil parameters in accordance with the present invention;
FIG. 2 is a schematic illustration of p and β and RMSE for the S1223 airfoil upper surface;
FIG. 3 is a schematic representation of p and β and RMSE for the S1223 airfoil lower airfoil surface;
FIG. 4 is a schematic diagram of an embodiment of the calculation of the control parameters p and β for the NEW CST method;
FIG. 5 is a comparison of the fit of the original CST, LEM CST and NEW CST methods to the S1223 airfoil profile;
FIG. 6 is a schematic diagram comparing the fit of the original CST, LEM CST and NEW CST methods to FX63_137 airfoils;
FIG. 7 is a comparison of the fit of the raw CST, LEM CST and NEW CST methods to an E216 airfoil.
Detailed Description
The advantages and features of the present invention will become more fully apparent from the following description and appended claims, taken in conjunction with the accompanying drawings and detailed description of specific embodiments of the invention. It is to be noted that the drawings are in a very simplified form and are not to scale, which is intended merely for convenience and clarity in describing embodiments of the invention.
It should be noted that, for clarity of description of the present invention, various embodiments are specifically described to further illustrate different implementations of the present invention, wherein the embodiments are illustrative and not exhaustive. In addition, for simplicity of description, the contents mentioned in the previous embodiments are often omitted in the following embodiments, and therefore, the contents not mentioned in the following embodiments may be referred to the previous embodiments accordingly.
While the invention is amenable to various modifications and alternative forms, specifics thereof have been shown by way of example in the drawings and will be described in detail. It should be understood that the inventors do not intend to limit the invention to the particular embodiments described, but intend to protect all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the claims. The same meta-module part number may be used throughout the drawings to represent the same or similar parts.
Referring to fig. 1 to 3, a method for processing airfoil parameters according to the present invention includes:
step S101, determining the shapes of an upper wing surface and a lower wing surface of an airfoil profile, and mapping the shapes of the upper wing surface and the lower wing surface of the airfoil profile into a class function and a type function;
step S102, based on the class function and the type function, generating sample points for controlling the shape of the leading edge and the shape of the trailing edge of the airfoil profile by using a Latin hypercube algorithm;
step S103, establishing a proxy model of the root mean square error by using a radial basis function neural network model according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil;
and S104, carrying out optimization solution on the proxy model of the root-mean-square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
In one embodiment, the shapes of the upper and lower airfoil surfaces of the airfoil are mapped to a class function and a type function, and the indication is performed by increasing a control parameter for controlling the slope and attenuation of the basis function.
In one embodiment, generating sample points controlling the airfoil leading edge shape and the airfoil trailing edge shape using a latin hypercube algorithm comprises: and acquiring coordinates of an upper wing surface and a lower wing surface of the airfoil.
In one embodiment, the proxy model of root mean square error is optimized for solution using a genetic algorithm, comprising: the minimum root mean square error was solved using the genetic algorithm toolkit of MATLAB.
Another object of the present invention is to provide a system for processing airfoil parameters, comprising:
the mapping module is used for determining the shapes of the upper wing surface and the lower wing surface of the wing profile and mapping the shapes of the upper wing surface and the lower wing surface of the wing profile into a class function and a type function;
the generating module is used for generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm based on the class function and the type function;
the building module is used for building a proxy model of the root mean square error by using a radial basis function neural network model according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil;
and the calculating module is used for carrying out optimization solving on the proxy model of the root-mean-square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
In one embodiment, the mapping module is instructed to increase the control parameter for controlling the slope and attenuation of the basis function.
In one embodiment, the generation module includes an acquisition submodule for acquiring coordinates of an upper airfoil surface and a lower airfoil surface of the airfoil.
In one embodiment, the solution module uses the genetic algorithm toolset of MATLAB to solve for the minimum root mean square error.
As a specific embodiment, the method is suitable for high-precision fitting of the two-dimensional airfoil profile, and the expression of the model is as follows:
in the formula: the 1 st part is an original airfoil parametric equation part, the 2 nd part is an airfoil Trailing Edge Modification (TEM) equation, and the 3 rd part is an airfoil Leading Edge Modification (LEM) equation. A. thenAnd An+1Is a weight coefficient, and is related to the trailing edge angle of the airfoil profile and the vertical position of the trailing edge.
The airfoil profile is represented using a class function and a type function:
ζ(ψ)＝C(ψ)S(ψ)+ψζT
defining function C (psi) ═ psia(1-ψ)bA and b are class parameters, and for most round-nose tip-tail airfoils, a is 0.5, and b is 1; function of type
Expressed by a Bernstein polynomial.
The front edge modification parameterization method adds 1 additional basis function in an airfoil parameterization equation for improving the expression capacity of the CST method near the airfoil front edge, and the LEM CST expression is as follows:
the improved Hicks-Henne type function is adopted, the design space near the tail edge of the airfoil profile is expanded, and the expression is as follows:
where α and β are control parameters for increasing basis functions, α can control the slope of the basis function, and β can control the decay of the basis function.
Referring to fig. 4 to 7, a novel airfoil profile parameterization method of the present invention includes:
the LEM CST method adds 1 additional basis function in the wing profile parametric equation and is used for improving the expression capacity of the CST method near the leading edge of the wing profile, and the formula of the LEM CST method is as follows:
the improved Hicks-Henne type function method expands the design space near the trailing edge of the airfoil profile, and the specific formula is as follows:
where α and β are control parameters for increasing basis functions, α can control the slope of the basis function, and β can control the decay of the basis function.
The essence of the airfoil parameterization is that different basis functions, which can produce different fitting errors, are added together to approximate the original airfoil. The invention combines the new leading edge modification basis function and the trailing edge basis function together to parameterize the airfoil profile. Introducing a parameter p to control the leading edge of the airfoil profile to improve the LEM CST parameterization method, wherein different p can generate different leading edge base functions, so that the fitting capacity of a new leading edge modification base function is stronger than that of LEM CST, and the parameter q can control the amplitude of the leading edge base function; a trailing edge modification function (TEM) is introduced into the NEW CST method, where α and β are control parameters of the trailing edge basis function, where α can control the magnitude of the basis function. Different beta can generate different basis functions, so the trailing edge modification function can improve the fitting accuracy of the trailing edge. The NEW CST parameterized equation is:
the above formula can be written as:
weight coefficient A in the formulanAnd An+1In relation to the trailing edge angle of the airfoil and the vertical position of the trailing edge.
To determine the p and β values of the NEW CST equation, the present invention calculates the NEW CST method using Root Mean Square Error (RMSE) as a standard statistical measure, the RMSE calculation formula being:
in the formula, zi,original(psi) coordinates of the initial airfoil profile, zi,fitting(psi) represents the coordinates of the fitted airfoil profile, and n is the number of coordinate points of the upper and lower airfoil profiles.
The relation between control parameters p and beta of the NEW CST method and RMSE is complex and nonlinear, in order to obtain the approximate relation between p and beta and RMSE and further obtain the leading edge basis function and the trailing edge basis function of the NEW CST method, the invention provides a method adopting a proxy model, adopting a Latin hypercube algorithm (LHD) to sample p and beta, utilizing sampling points to establish a radial basis neural network model of p and beta and RMSE, and utilizing a genetic algorithm tool box of MATLAB to solve the minimum RMSE, wherein the specific optimization process is shown in figure 4.
As a specific embodiment, the novel wing profile parameterization method provided by the invention uses S1223 wing profiles to analyze the relation between p and beta and RMSE in the parameterization method. When the number of NEW CST process control parameters is equal to the LEM CST process, both 15, the RMSE value of the upper airfoil surface of S1223 in the LEM CST process is 1.3637X 10-4And the RMSE value of the lower airfoil surface of S1223 is 7.7261X 10-5. When 0 is present<p<At 0.2, the effect of different β values (β ═ 20, β ═ 50, β ═ 100) of the NEW CST method on the upper and lower airfoil surface RMSE was analyzed; when 0 is present<β<At 300, the influence of different p values (p is 0.015, p is 0.02, and p is 0.035) of the NEW CST method on the upper airfoil RMSE was analyzed; when 0 is present<β<At 100, the effect of different p values (p 0.015, 0.02, 0.025, 0.08, 0.1) of the NEW CST method on the lower airfoil RMSE was analyzed, and the results shown in fig. 2 and 3 were obtained.
In FIG. 2, when the p-value of the airfoil surface at S1223 was varied from 0 to 0.04, the RMSE had 1 trough and the minimum RMSE was 6.736X 10-5(ii) a When the p value of the upper airfoil surface of S1223 is changed from 0.04 to 0.2, the minimum RMSE of the upper airfoil surface is 6.7465 x 10-5(ii) a When the beta value of the airfoil surface on S1223 was varied from 0 to 300, the RMSE had 1 trough and the minimum RMSE was 4.1581X 10-5(p ═ 0.015,. beta.180). In FIG. 3, when the p-value of the airfoil at S1223 was varied from 0 to 0.2, there were two troughs in RMSE, which were 3.5394X 10, respectively-5(p ═ 0.015,. beta.20) and 3.0859 × 10-5(p ═ 0.08, β ═ 20); when 0 is present<β<At 100, there is a minimum RMSE for the lower airfoil surface of S1223.
As shown in fig. 4, a calculation flow of the control parameters p and β of the NEW CST method is established, which includes: and adopting a Latin hypercube algorithm to sample p and beta, establishing a radial basis function neural network model of the sampling points p and beta and RMSE, solving the minimum RMSE of the proxy model by using a genetic algorithm, and finally obtaining control parameters p and beta of a novel parameterization method. In this case, 500 sample points are generated by the LHD method, the RMSE values of the sample points are calculated, and the sample points and the RMSE values are approximated by using the RBF model. Setting parameters of a genetic algorithm: the population size is 10, the population quantity is 10, the total evolution generation number is 10, the cross probability is 0.95, the variation probability is 0.01, the population migration probability is 0.3, and the interval generation number of population migration is 4. Obtaining a parameterized equation of the upper wing surface and the lower wing surface of the S1223 wing profile through optimization:
to compare the fit errors of the NEW CST method, the original CST method, and the LEM CST method, demonstrating the superiority of the NEW airfoil parameterization method, the present example selects the high camber high lift S1223, FX63 — 137, and E216 airfoils, as shown in fig. 5, 6, and 7.
In fig. 5: for S1223 wing profile, the fitting precision of the CST method using 26 control parameters can not meet the requirement of a typical wind tunnel experiment, the LEM CST method can meet the fitting requirement by 15 control parameters in total, the NEW CST method can meet the error precision requirement of the S1223 wing profile by only 8 control parameters, if the control parameters of the NEW CST method are also 15, the NEW CST parameterization method is more accurate than the LEM CST method, and the error of the front edge and the tail edge of the S1223 wing profile is smaller than that of the LEM CST method.
In fig. 6: for the FX63 — 137 airfoil, to meet the wind tunnel experimental error requirements, the number of control parameters of the CST method is 16, the number of control parameters of the LEM CST parameterization method is 10, and the number of control parameters of the NEW CST parameterization method is 9; when the number of the control parameters of the NEW CST method is 10, the NEW CST parameterization method has higher fitting accuracy than the LEM CST method, and fitting errors of the leading edge and the trailing edge of the airfoil profile are smaller than the LEM CST method.
In fig. 7: for the E216 airfoil profile, the fitting precision of the CST method using 23 control parameters can meet the requirement of a typical wind tunnel experiment, the LEM CST method needs 14 control parameters in total to meet the fitting requirement, and the NEW CST method only needs 8 control parameters to meet the error precision requirement of the S1223 airfoil profile; when the control parameter of the NEW CST is 11, it can achieve the LEM CST method fitting accuracy with the control parameter of 14.
The invention achieves the following significant beneficial effects:
the realization is simple, include: firstly, determining the shapes of an upper wing surface and a lower wing surface of an airfoil profile, and mapping the shapes of the upper wing surface and the lower wing surface of the airfoil profile into a class function and a type function; secondly, based on the class function and the type function, generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm; then, according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil profile, a radial basis function neural network model is used for establishing a proxy model of the root mean square error; and finally, carrying out optimization solution on the proxy model of the root mean square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model. The LEM CST method and the improved Hicks-Henne type function method are effectively combined, so that the wing profile is fitted with high precision, the number of wing profile parametric design variables is reduced, and the efficiency of wing profile pneumatic optimization is improved.
Any other suitable modifications can be made according to the technical scheme and the conception of the invention. All such alternatives, modifications and improvements as would be obvious to one skilled in the art are intended to be included within the scope of the invention as defined by the appended claims.
Claims (8)
1. An airfoil parameter processing method, comprising:
firstly, determining the shapes of an upper wing surface and a lower wing surface of an airfoil profile, and mapping the shapes of the upper wing surface and the lower wing surface of the airfoil profile into a class function and a type function;
secondly, based on the class function and the type function, generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm;
then, according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil profile, a radial basis function neural network model is used for establishing a proxy model of the root mean square error;
and finally, carrying out optimization solution on the proxy model of the root mean square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
2. The method of claim 1, wherein the shape of the upper and lower airfoil surfaces of the airfoil is mapped to a class function and a type function, and the indication is performed by increasing a control parameter for controlling the slope and attenuation of the basis function.
3. The method of claim 1, wherein generating sample points controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge using a latin hypercube algorithm comprises: and acquiring coordinates of an upper wing surface and a lower wing surface of the airfoil.
4. The airfoil parameter processing method according to claim 1, wherein the proxy model of root mean square error is optimized for solution using a genetic algorithm, comprising: the minimum root mean square error was solved using the genetic algorithm toolkit of MATLAB.
5. An airfoil parameter processing system, comprising:
the mapping module is used for determining the shapes of the upper wing surface and the lower wing surface of the wing profile and mapping the shapes of the upper wing surface and the lower wing surface of the wing profile into a class function and a type function;
the generating module is used for generating sample points for controlling the shape of the airfoil leading edge and the shape of the airfoil trailing edge by using a Latin hypercube algorithm based on the class function and the type function;
the building module is used for building a proxy model of the root mean square error by using a radial basis function neural network model according to the sample points for controlling the shape of the front edge and the shape of the rear edge of the airfoil;
and the calculating module is used for carrying out optimization solving on the proxy model of the root-mean-square error by using a genetic algorithm to obtain a leading edge basis function and a trailing edge basis function of the wing profile parameterized model.
6. An airfoil parameter processing system according to claim 5, wherein the mapping module instructs with increasing control parameters for controlling slope and attenuation of the basis functions.
7. An airfoil parameter processing system according to claim 5, wherein the generation module includes an acquisition submodule for acquiring coordinates of an upper airfoil surface and a lower airfoil surface of an airfoil.
8. An airfoil parameter processing system according to claim 5, wherein the solution module uses a genetic algorithm toolkit of MATLAB to solve for the minimum root mean square error.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113656895A (en) * | 2021-08-20 | 2021-11-16 | 中国空气动力研究与发展中心高速空气动力研究所 | Improved Parsec parameterization method for describing supercritical airfoil |
| CN117113571A (en) * | 2023-08-24 | 2023-11-24 | 江苏大学 | A CST parameterization method for airfoil after rotation and translation transformation |
| CN118468453A (en) * | 2024-07-10 | 2024-08-09 | 西安羚控电子科技有限公司 | A method and device for parametric design of airfoil |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113656895A (en) * | 2021-08-20 | 2021-11-16 | 中国空气动力研究与发展中心高速空气动力研究所 | Improved Parsec parameterization method for describing supercritical airfoil |
| CN117113571A (en) * | 2023-08-24 | 2023-11-24 | 江苏大学 | A CST parameterization method for airfoil after rotation and translation transformation |
| CN118468453A (en) * | 2024-07-10 | 2024-08-09 | 西安羚控电子科技有限公司 | A method and device for parametric design of airfoil |
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