CN117669264A - Wind turbine airfoil parameterization method based on improved NURBS - Google Patents

Wind turbine airfoil parameterization method based on improved NURBS Download PDF

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CN117669264A
CN117669264A CN202311851884.7A CN202311851884A CN117669264A CN 117669264 A CN117669264 A CN 117669264A CN 202311851884 A CN202311851884 A CN 202311851884A CN 117669264 A CN117669264 A CN 117669264A
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airfoil
nurbs
curve
point
parameterization
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马剑龙
苏宏杰
张晓涛
王琰锴
赵明博
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Inner Mongolia University of Technology
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Abstract

The invention discloses a wind turbine airfoil parameterization method based on an improved NURBS, which comprises the steps of preprocessing coordinate points of an original airfoil to obtain an upper surface fitting curve and a lower surface fitting curve; and separating the upper surface and the lower surface of the airfoil, optimizing the fitting curve of the upper surface and the fitting curve of the lower surface by using NURBS curve representation, obtaining control points and weight values, and performing airfoil parameterization. The sequence quadratic programming algorithm based on the gradient takes the minimized mean error and the minimized maximum error as objective functions, takes control points and weight values as design variables, optimizes the control points and the weight values, and obtains parameterized optimal results. The method improves the fitting precision of the wing profile, keeps good smoothness, increases the solving speed and the solving quality, and reduces the error.

Description

Wind turbine airfoil parameterization method based on improved NURBS
Technical Field
The invention belongs to the technical field of wind turbine design, and particularly relates to a wind turbine airfoil parameterization method based on improved NURBS.
Background
Airfoil parameterization is the basis of numerical calculation and is one of important contents of the current airfoil optimization design research. The existing airfoil parameterization method mainly comprises the following parameterization methods: hicks-Henne shape function, PARSEC feature parameter, CST parameterization, B-spline parameterization, and spline parameterization. The parameterization method of the Hicks-Henne profile function method is characterized in that the camber and thickness of the airfoil are changed as parameters, and then the airfoil is overlapped with the original airfoil, so that the profile of the airfoil is controlled. The method has very strong control on the appearance, but the representation of the trailing edge is not smooth enough, and the wing profile represented by the method is not smooth when the description parameter difference is relatively large, so the method is suitable for optimizing the appearance on the reference wing profile and is not suitable for describing a large design space. PARSEC (Parametric Section), the airfoil profile is smooth, wave is not generated generally, and the robustness is good, but the control capability on the airfoil profile is poor, so that the method is not suitable for fine design. A parameterization method of CST, an airfoil parameterization method proposed by the United states Boeing company describes the appearance of an airfoil through a class function and a shape function, the basis function of the shape function is Bernstein polynomial, and the coefficient of the basis function is a design variable of the CST airfoil parameterization optimization method. The CST method is used for improving the control of the appearance by increasing the times of Bernstein polynomials, lane and the like propose that the CST parameterization method is less in design parameters than a Bezier curve by selecting a proper class function, the control of the appearance shape is very strong, a large design space can be described, the CST method has less parameters and higher precision, waves and protruding points generally cannot occur, the robustness on the expression of different airfoils is poor, the control capability on different airfoils is different, the local repair capability is not achieved, and particularly, the effect on the optimal design of the supercritical airfoils is quite unsatisfactory. Spline parameterization generally refers to a method for representing an airfoil curve by using a Bezier curve, a B spline curve or a non-uniform rational B spline (NURBS) curve, which is widely used in CAD, has strong flexibility, can perform local control and smoothness treatment on the generated airfoil curve, and can have a phenomenon of unsmooth optimization result of the B spline parameterization, and the piecewise rational B spline curve is connected end to represent the parameterization method of the supercritical airfoil. The method can solve the problem of non-smooth phenomenon of the optimization result, can reduce the number of the optimization design variables, and can accurately represent the common supercritical airfoil profile, but the operation process is very troublesome, the problem of splicing is considered, and the calculation time of the optimization method is longer.
In view of the foregoing, there is a need for an improved NURBS-based wind turbine airfoil parameterization method.
Disclosure of Invention
In order to solve the technical problems, the invention provides a wind turbine airfoil parameterization method based on an improved NURBS, which improves the fitting precision of airfoils, keeps good smoothness, increases the solving speed and the solving quality and reduces errors.
To achieve the above object, the present invention provides a wind turbine airfoil parameterization method based on improved NURBS, comprising:
preprocessing coordinate points of an original airfoil profile to obtain an upper surface fitting curve and a lower surface fitting curve;
and separating the upper surface and the lower surface of the airfoil, optimizing the fitting curve of the upper surface and the fitting curve of the lower surface by using NURBS curve representation, obtaining control points and weight values, and performing airfoil parameterization.
The sequence quadratic programming algorithm based on the gradient takes the minimized mean error and the minimized maximum error as objective functions, takes control points and weight values as design variables, optimizes the control points and the weight values, and obtains parameterized optimal results.
Alternatively, the method of separating the upper and lower surfaces of the airfoil using NURBS curves is:
wherein P is i To control the point coordinates, ω i For the respective weight value, N i,p Is a p-th order b spline basis function, A (u) is the position of a point on the curve, R i，p (mu) is mobile base connection mu E [0,1 ]]Is a piecewise rational function of (a).
Optionally, preprocessing the coordinate points of the airfoil, and obtaining the upper surface fitting curve and the lower surface fitting curve includes: preprocessing the coordinate points of the wing profile, respectively fixing the first coordinate point and the last coordinate point as a first control point and a last control point at the front edge and the tail edge of the wing profile, respectively setting a plurality of control points along each curve by adopting a uniform parameter method, and optimizing the plurality of control points to obtain the upper surface fitting curve and the lower surface fitting curve.
Optionally, the method for obtaining the control point and the weight value by fitting the curve to the upper surface and the curve to the lower surface includes:
the calculation node vector is calculated as follows,
wherein U is a node vector value, P i Coordinates of control points;
the back-calculation control vertex is calculated as follows,
the problem is converted into a least squares problem by constructing a system of linear equations of the form:
(N T ·N)·crtP＝N T ·r
wherein N is T N represents transposition of N, N represents a node vector coefficient matrix, r represents the difference between a data point and a head-to-tail control point, and crtP is a control point matrix;
the r matrix is calculated by traversing the data points as follows:
r i ＝d i -d 1 ·N(u i )-d m ·N(u i )
wherein d i Representing the coordinates of the ith data point, u i Parameter value representing the ith data point, N (u i ) Expressed in the parameter value u i Values of spline basis function at d m Representing coordinates of an mth data point;
the N matrix is calculated by traversing the data points as follows:
N ij ＝N(u j,i )
wherein N is ij Expressed in the parameter value u j The value of the ith component of the spline basis function at, u j,i Representing an ith row and jth column node vector;
the control point matrix ctrP is obtained by solving a linear equation set, and the formula is as follows:
crtP＝(N T ·N) -1 ·N T ·r。
optionally, in the gradient-based sequence quadratic programming algorithm, taking the minimized mean error and the minimized maximum error as objective functions, the method for obtaining the minimized mean error and the minimized maximum error is as follows:
wherein ε mea Is the mean error, d i C is the chord length of the airfoil, n is the number of each airfoil point and epsilon is the distance between the original airfoil and the target curve projection max Is the maximum error.
Optionally, the objective function is expressed as: f (X) =2ε mea +ε max
Wherein ε mea Is the mean error epsilon max For maximum error, X is a design variable.
Alternatively, provided thatThe design variables are expressed as: x= { X 1 ,y 1 ,ω 1 ,...,x n ,y n ,ω n }
Wherein x is n Is the x coordinate value, y of the airfoil control point n For the y-coordinate value, ω, of the airfoil control point n Is the corresponding weight vector.
The invention has the technical effects that: the invention discloses a wind turbine airfoil parameterization method based on an improved NURBS, which is a new free deformation parameterization method, the airfoil parameterization is carried out by using the method without limiting the number of design variables, the initial appearance is not required to be fitted, the operation is easy, the error is minimum as an objective function, the control point and the weight vector are optimized, and the fitting error is greatly reduced; the NURBS airfoil parameterization method is improved, the generated airfoil profile is smooth, and the local concave-convex phenomenon can not occur.
Drawings
The accompanying drawings, which are included to provide a further understanding of the application, illustrate and explain the application and are not to be construed as limiting the application. In the drawings:
FIG. 1 is a flow diagram of a method for parameterizing a wind turbine airfoil based on an improved NURBS in accordance with an embodiment of the present invention;
FIG. 2 is an example of an airfoil NURBS curve fitting result of the present invention, wherein (a) is an upper surface fitting result and (b) a lower surface fitting result;
FIG. 3 is an airfoil NURBS curve that has resulted in minimum fitting error control point coordinates in accordance with an embodiment of the present invention.
Detailed Description
It should be noted that, in the case of no conflict, the embodiments and features in the embodiments may be combined with each other. The present application will be described in detail below with reference to the accompanying drawings in conjunction with embodiments.
It should be noted that the steps illustrated in the flowcharts of the figures may be performed in a computer system such as a set of computer executable instructions, and that although a logical order is illustrated in the flowcharts, in some cases the steps illustrated or described may be performed in an order other than that illustrated herein.
As shown in fig. 1, in this embodiment, a wind turbine airfoil parameterization method based on improved NURBS is provided, including:
preprocessing the coordinate points of the wing profile to obtain an upper surface fitting curve and a lower surface fitting curve;
separating the upper surface and the lower surface of the airfoil, and carrying out airfoil parameterization by using NURBS curve representation;
optimizing the upper surface fitting curve and the lower surface fitting curve to obtain control points and weight values;
the sequence quadratic programming algorithm based on the gradient takes the minimized mean error and the minimized maximum error as objective functions, takes control points and weight values as design variables, optimizes the control points and weight of parameters, and obtains parameterized results.
NURBS airfoil representation
One NURBS curve is expressed as:
wherein P is i To control the point coordinates, ω i For the respective weight value, N i,p A (u) is the position of a point on the curve, which is a b spline basis function. The basis function is obtained by a node vector, which represents the break point of the function, in the form of:
the use of NURBS parameterization ensures curvature continuity of the airfoil geometric representation. The higher the order of the basis function, the higher the order of the curvature continuity. Three-degree basis functions are considered here for airfoil parameterization; this corresponds to a polynomial curve with cubic terms in NURBS representation.
Back-computing control points, giving a set of airfoil data points, namely model value points, generating NURBS curves passing through the model value points, namely back-computing the curves, (1) calculating node vectors; (2) back-calculating the control vertex.
Calculating node vectors by passing a p-th NURBS curve through a given set of airfoil data points (model value points) Q i = (i=0, 1,) n, guarantee that the end point of the curve coincides with the type value point, guarantee Q i Node u in the domain defined by the sequential and construction curves i+p (i=0, 1,) corresponding to n, a parameterization process is typically required for the model value points to determine the model value point Q i Parameter value u of (2) i+p (i=0, 1., n.). Having n+1 type value points Q i Will be represented by n+3 control points P i And weight factors and node vector U definitions. The invention uses the accumulated chord length parameterization method, shows the distribution condition of data points according to the chord lengths, and can obtain a curve with better smoothness, and the parameterization method meets the following conditions:
obtaining a node vector:
U＝[u 0 ,u 1 ,…u n+k+3 ]。
the inverse calculation of the control vertex is performed, and the control point of the curve is calculated by using a least square method, which is an optimization method for fitting the relationship between the data points and the model. In NURBS curve fitting, it is desirable to adjust the position of the control points so that the curve fits as closely as possible to the given data. The goal of (a) is to find a control point matrix ctrP that minimizes the error between the data points and the NURBS curve. The problem is converted into a least squares problem by constructing a system of linear equations of the form:
(N T ·N)·crtP＝N T ·r
wherein N is T Representing a transpose of N, r representing the data point and end-to-end controlDifferences between points.
The calculation of the r matrix is done by traversing the data points as follows:
r i ＝d i -d 1 ·N(u i )-d m ·N(u i )
wherein d i Representing the coordinates of the ith data point, u i Parameter value representing the ith data point, N (u i ) Expressed in the parameter value u i The value of the spline basis function at.
The calculation of the N matrix is also done by traversing the data points as follows:
N ij ＝N(u j,i )
wherein N is ij Expressed in the parameter value u j Spline basis function at i The value of the individual components.
By solving the linear equation set, the control point matrix ctrP is obtained as follows:
crtP＝(N T ·N) -1 ·N T ·r
and calculating a curve control point according to the formula, wherein the weight values are all 1.
In order to obtain more accurate airfoil geometry parameters, instead of parameterizing the entire airfoil together, as in the parameterized DU25 airfoil of FIG. 2, the upper and lower surfaces of the airfoil are separated by NURBS curves, respectively, the DU25 airfoil having a maximum relative thickness of 25% with the first and last control points fixed at the leading and trailing edges of the airfoil. The parameterization method selects an accumulated chord length parameter method, 7 control points are distributed along each curve respectively, and the 7 control points are selected according to the fitting effect. And optimizing with fewer variables in the later-period optimized wing profile conveniently according to the fitting result error and the number of the control points.
Optimizing NURBS curves:
the traditional NURBS curve parameterization only considers the coordinates of the control points as design variables, and the weight values are fixed to parameterize the airfoil curves, and the error of curve parameterization is larger because the weight values of the control points are not changed. Therefore, in order to reduce the error of NURBS curve parameterization fitting and improve the parameterization precision, the invention introducesInto the weight parameter omega i As parameterized variables. In order to obtain the appropriate control point weights and distributions, the parameterization problem needs to be optimized. The representation error between the original airfoil and the fitted geometry can be used to average epsilon mea And a maximum value epsilon max To show that parameterized control variables and weights are optimized by taking minimized mean error and minimized maximum error as objective functions and coordinates and weight values of control points as design variables.
The maximum error and the average error are used as objective functions, which are typical multi-objective optimization problems, and the multi-objective optimization problems are converted into single-objective problems by using a weighting method.
Epsilon in mea Is the mean error, d i C is the chord length of the airfoil, n is the number of each airfoil point and epsilon is the distance between the original airfoil and the target curve projection max Is the maximum error.
The objective function is:
F(X)＝2ε mea +ε max
the variables are:
X＝{x 1 ,y 1 ,ω 1 ,...,x n ,y n ,ω n }
and the obtained control points and the corresponding weights form a final NURBS airfoil profile fitting curve, and the parameterized result is output.
Clearly, the F (x) objective function is a typical nonlinear optimization problem, and the design variables are optimized by using a gradient-based sequence quadratic programming algorithm (Sequential quadratic programming, abbreviated as SQP) algorithm, and the main advantages of the SQP algorithm are local super-linear convergence and global convergence.
As shown in FIG. 3, the control point coordinates with the least fitting error have been obtained, providing the basis for the next step in optimization of the airfoil. A comparison of DU21 NURBS shape and original airfoil is shown in table 1.
TABLE 1
Error after optimization, lower surface average error 3.0785e -4 Upper surface average error 8.7009e -4 。
After optimization, the error is greatly reduced, the fitting precision is improved, and a foundation is laid for the follow-up optimization of the aerofoil aerodynamic profile.
The invention discloses a wind turbine airfoil parameterization method based on an improved NURBS, which is a free deformation parameterization method, wherein the airfoil parameterization is performed by using the method without limiting the number of design variables, and the method does not need to fit an initial appearance, is easy to operate, optimizes control points and weight vectors by taking the minimum error as an objective function, and greatly reduces the fitting error; the NURBS airfoil parameterization method is improved, the generated airfoil profile is smooth, and the local concave-convex phenomenon can not occur.
The foregoing is merely a preferred embodiment of the present application, but the scope of the present application is not limited thereto, and any changes or substitutions easily conceivable by those skilled in the art within the technical scope of the present application should be covered in the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.

Claims (7)

1. A method of parameterizing a wind turbine airfoil based on an improved NURBS, comprising:
preprocessing coordinate points of an original airfoil profile to obtain an upper surface fitting curve and a lower surface fitting curve;
and separating the upper surface and the lower surface of the airfoil, optimizing the fitting curve of the upper surface and the fitting curve of the lower surface by using NURBS curve representation, obtaining control points and weight values, and performing airfoil parameterization.
The sequence quadratic programming algorithm based on the gradient takes the minimized mean error and the minimized maximum error as objective functions, takes control points and weight values as design variables, optimizes the control points and the weight values, and obtains parameterized optimal results.
2. The improved NURBS-based wind turbine airfoil parameterization method of claim 1,
the method for separating the upper surface and the lower surface of the airfoil by using NURBS curves comprises the following steps:
wherein P is i To control the point coordinates, ω i For the respective weight value, N i,p Is a p-th order b spline basis function, A (u) is the position of a point on the curve, R i,p (u) is mobile base connection mu E [0,1 ]]Is a piecewise rational function of (a).
3. The improved NURBS-based wind turbine airfoil parameterization method of claim 1,
preprocessing the coordinate points of the airfoil, and acquiring an upper surface fitting curve and a lower surface fitting curve, wherein the process comprises the following steps: preprocessing the coordinate points of the wing profile, respectively fixing the first coordinate point and the last coordinate point as a first control point and a last control point at the front edge and the tail edge of the wing profile, respectively setting a plurality of control points along each curve by adopting a uniform parameter method, and optimizing the plurality of control points to obtain the upper surface fitting curve and the lower surface fitting curve.
4. The improved NURBS-based wind turbine airfoil parameterization method of claim 1,
fitting a curve to the upper surface and a curve to the lower surface, and obtaining control points and weight values comprises the following steps:
the calculation node vector is calculated as follows,
U＝[u 0 ,u 1 ,…u n+k+3 ]
wherein U is a node vector value, P i Coordinates of control points;
the back-calculation control vertex is calculated as follows,
the problem is converted into a least squares problem by constructing a system of linear equations of the form:
(N T ·N)·crtP＝N T ·r
wherein N is T N represents transposition of N, N represents a node vector coefficient matrix, r represents the difference between a data point and a head-to-tail control point, and crtP is a control point matrix;
the r matrix is calculated by traversing the data points as follows:
r i ＝d i -d 1 ·N(u i )-d m ·N(u i )
wherein d i Representing the coordinates of the ith data point, u i Parameter value representing the ith data point, N (u i ) Expressed in the parameter value u i Values of spline basis function at d m Representing coordinates of an mth data point;
the N matrix is calculated by traversing the data points as follows:
N ij ＝N(u j’i )
wherein N is ij Expressed in the parameter value u j The value of the ith component of the spline basis function at, u j’i Representing an ith row and jth column node vector;
the control point matrix ctrP is obtained by solving a linear equation set, and the formula is as follows:
crtP＝(N T ·N) -1 ·N T ·r。
5. the improved NURBS-based wind turbine airfoil parameterization method of claim 1,
the method for acquiring the minimum mean error and the minimum maximum error in the gradient-based sequence quadratic programming algorithm by taking the minimum mean error and the minimum maximum error as objective functions comprises the following steps:
wherein ε mea Is the mean error, d i C is the chord length of the airfoil, n is the number of each airfoil point and epsilon is the distance between the original airfoil and the target curve projection max Is the maximum error.
6. The improved NURBS-based wind turbine airfoil parameterization method of claim 1,
the objective function is expressed as: f (X) =2ε mea +ε max
Wherein ε mea Is the mean error epsilon max For maximum error, X is a design variable.
7. The improved NURBS-based wind turbine airfoil parameterization method of claim 1,
the design variable tableThe method is shown as follows: x= { X 1 ,y 1 ,ω 1 ,...,x n ,y n ,ω n }
Wherein x is n Is the x coordinate value, y of the airfoil control point n For the y-coordinate value, ω, of the airfoil control point n Is the corresponding weight vector.
CN202311851884.7A 2023-12-29 2023-12-29 Wind turbine airfoil parameterization method based on improved NURBS Pending CN117669264A (en)

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