CN117669264A - A parameterization method for wind turbine airfoil based on improved NURBS - Google Patents

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

A parameterization method for wind turbine airfoil based on improved NURBS

Technical field

The invention belongs to the technical field of wind turbine design, and in particular relates to a wind turbine airfoil parameterization method based on improved NURBS.

Background technique

Airfoil parameterization is the basis of numerical calculation and one of the important contents of current airfoil optimization design research. The current airfoil parameterization methods mainly include the following parameterization methods: Hicks-Henne shape function method, PARSEC characteristic parameter method, CST parameterization method, B-spline parameterization, and spline parameterization method. The parametric method of the Hicks-Henne shape function method uses the changes in camber and thickness of the airfoil as parameters, and then superimposes them with the original airfoil to control the shape of the airfoil. This method is very powerful in controlling the shape, but its representation of the trailing edge is not smooth enough. When the difference in description parameters is relatively large, the airfoil represented is not smooth. Therefore, this method is suitable for optimizing the shape on the benchmark airfoil. Not suitable for describing larger design spaces. PARSEC (Parametric Section) airfoil parameterization method. In this method, the description parameters of the airfoil have very clear meanings. The generated airfoil has a smooth shape, generally does not appear wavy, and has good robustness. , but the ability to control the airfoil shape is relatively poor, so it is not suitable for refined design. CST parameterization method, an airfoil parameterization method proposed by the American Boeing Company, describes the shape of the airfoil through category functions and shape functions. The basis function of the shape function is the Bernstein polynomial, and the coefficient of the basis function is the CST airfoil. Design variables for parametric optimization methods. By increasing the degree of Bernstein polynomials to improve the control of the shape of the CST method, Lane et al. proposed that by selecting an appropriate category function, the CST parametric method has fewer design parameters than the Bézier curve, has very strong control over the shape of the shape, and has the ability to describe the shape of the shape. With a large design space, the CST method uses fewer parameters and has higher accuracy. Generally, there will be no waves or bumps. It has poor robustness in expressing different airfoils, has different control capabilities for different airfoils, and has different control capabilities. It has local repair capabilities, especially the effect on the optimization design of supercritical airfoils is very unsatisfactory. The spline parameterization method usually refers to the method of using Bézier curves, B-spline curves or non-uniform rational B-splines (NURBS) curves to represent airfoil curves. It is widely used in CAD and is highly flexible. The generated airfoil curve can be locally controlled and smoothed. However, the optimization results of the B-spline parametric method may not be smooth. The piecewise rational Bézier curves are connected end to end to represent the parameters of the supercritical airfoil. ization method. This method can solve the uneven phenomenon of optimization results, and at the same time, it can reduce the number of optimization design variables and can more accurately represent common supercritical airfoils. However, the operation process is very troublesome and the splicing problem must be taken into consideration. Optimization methods also take longer to calculate.

In summary, it is urgent to propose a wind turbine airfoil parameterization method based on improved NURBS.

Contents of the invention

In order to solve the above technical problems, the present invention proposes a wind turbine airfoil parameterization method based on improved NURBS to improve the airfoil fitting accuracy while maintaining good smoothness, increasing the speed of solution and quality of the solution, and reducing errors.

In order to achieve the above objectives, the present invention provides a wind turbine airfoil parameterization method based on improved NURBS, including:

Preprocess the coordinate points of the original airfoil to obtain the upper surface fitting curve and the lower surface fitting curve;

The upper surface and lower surface of the separated airfoil are represented by NURBS curves to optimize the upper surface fitting curve and the lower surface fitting curve, obtain control points and weight values, and parameterize the airfoil.

The gradient-based sequential quadratic programming algorithm takes minimizing the mean error and minimizing the maximum error as the objective function, and uses the control points and weight values as the design variables. It optimizes the control points and weight values to obtain the parameterized optimal results.

Optionally, the method of using NURBS curves to represent the upper and lower surfaces of the separated airfoil is:

Among them, P _i is the coordinate of the control point, ω _i is the respective weight value, N _{i, p} is the p-th degree b-spline basis function, A(u) is the position of a certain point on the curve, R _{i, p} (μ) is Moving basis connects piecewise rational functions of μ∈[0,1].

Optionally, the process of 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 airfoil, and converting the first coordinate point into and the last coordinate point are respectively fixed on the leading edge and trailing edge of the airfoil as the first control point and the last control point. A number of control points are set along each curve using the uniform parameter method, and the control points are Optimize to obtain the upper surface fitting curve and the lower surface fitting curve.

Optionally, the method of obtaining control points and weight values for the upper surface fitting curve and the lower surface fitting curve includes:

The calculation node vector is calculated as follows,

Among them, U is the node vector value, _Pi is the control point coordinate;

The inverse control vertex calculation is as follows,

The problem is transformed into a least squares problem by constructing a system of linear equations in the following form:

( ^NT ·N)·crtP＝ ^NT ·r

Among them, N ^T represents the transpose of N, N represents the node vector coefficient matrix, r represents the difference between the data point and the first and last control points, and crtP is the control point matrix;

Calculate the r matrix by traversing the data points, the formula is as follows:

r _i =d _i -d ₁ ·N(u _i )-d _m ·N(u _i )

Among them, _di represents the coordinates of the i-th data point, u _i represents the parameter value of the i-th data point, N(u _i ) represents the value of the spline basis function at parameter value u _i , and d _m represents the m-th coordinates of data points;

Calculate the N matrix by traversing the data points, the formula is as follows:

N _ij =N(u _j,i )

Among them, N _ij represents the value of the i-th component of the spline basis function at parameter value u _j , u _j,i represents the node vector of the i-th row and j-th column;

The control point matrix ctrP is obtained by solving the linear equations, and the formula is as follows:

crtP=( ^NT ·N) ^-1 · ^NT ·r.

Optionally, in the gradient-based sequential quadratic programming algorithm, minimizing the mean error and minimizing the maximum error are the objective functions. The method for obtaining the minimized mean error and the minimized maximum error is:

Among them, ε _mea is the mean error, _di is the distance between the original airfoil and the target curve projection, C is the chord length of the airfoil, n is the number of each airfoil point, and ε _max is the maximum error.

Optionally, the objective function is expressed as: F(X)=2ε _mea +ε _max

Among them, ε _mea is the mean error, ε _max is the maximum error, and X is the design variable.

Optionally, the design variables are expressed as: X={x ₁ ,y ₁ ,ω ₁ ,...,x _n ,y _n ,ω _n }

Among them, x _n is the x coordinate value of the airfoil control point, y _n is the y coordinate value of the airfoil control point, and ω _n is the corresponding weight vector.

Technical effects of the present invention: The present invention discloses a wind turbine airfoil parameterization method based on improved NURBS. NURBS is a new free deformation parameterization method. Using this method to perform airfoil parameterization does not limit the number of design variables. , and does not need to fit the initial shape, and is easy to operate. This invention takes the minimum error as the objective function, optimizes the control points and weight vectors, and greatly reduces the fitting error; improves the NURBS airfoil parameterization method, and generates airfoils The appearance is smooth and there will be no local unevenness.

Description of drawings

The drawings that form a part of this application are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an improper limitation of this application. In the attached picture:

Figure 1 is a schematic flow chart of a wind turbine airfoil parameterization method based on improved NURBS according to an embodiment of the present invention;

Figure 2 shows the NURBS curve fitting results of the airfoil according to the embodiment of the present invention, where (a) is the upper surface fitting result, (b) the lower surface fitting result;

Figure 3 shows the airfoil NURBS curve of the control point coordinates with the smallest fitting error obtained according to the embodiment of the present invention.

Detailed ways

It should be noted that, as long as there is no conflict, the embodiments and features in the embodiments of this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and embodiments.

It should be noted that the steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and, although a logical sequence is shown in the flowchart, in some cases, The steps shown or described may be performed in a different order than here.

As shown in Figure 1, this embodiment provides a wind turbine airfoil parameterization method based on improved NURBS, including:

Preprocess the coordinate points of the airfoil to obtain the upper surface fitting curve and the lower surface fitting curve;

The upper and lower surfaces of the separated airfoil are represented by NURBS curves for airfoil parameterization;

Optimize the upper surface fitting curve and the lower surface fitting curve to obtain control points and weight values;

The gradient-based sequential quadratic programming algorithm takes minimizing the mean error and minimizing the maximum error as the objective function, and uses the control points and weight values as the design variables to optimize the control points and weights of the parameters to obtain parameterized results.

NURBS airfoil representation

A NURBS curve is expressed as:

Among them, P _i is the coordinate of the control point, ω _i is the respective weight value, N _{i, p} is the p-th degree b-spline basis function, and A(u) is the position of a certain point on the curve. The basis function is obtained through a node vector, which represents the breakpoint of the function, and its form is:

Use NURBS parameterization to ensure curvature continuity of the airfoil geometric representation. The higher the order of the basis function, the higher the order of curvature continuity. Here cubic basis functions are considered for the airfoil parameterization; this corresponds to a polynomial curve with cubic terms in the NURBS representation.

Inverse calculation of control points, given a set of airfoil data points, that is, type value points, generates a NURBS curve passing through these type value points, which is called the inverse calculation of the curve, (1) Calculate the node vector; (2) Inverse calculation of the control vertex .

Calculate the node vector. If you want a NURBS curve of degree p to pass through a given set of airfoil data points (type value points) Q _i = (i = 0,1,...,n), ensure that the first part of the curve The end points coincide with the type value points, ensuring that Q _i corresponds to the nodes u _i+p (i=0,1,...,n) in the definition domain of the construction curve in turn. It is usually necessary to parameterize the type value points. , to determine the parameter value u _i+p (i=0,1,...,n) of the type value point _Qi . A NURBS curve of degree p with n+1 type value points Q _i will be defined by n+3 control points _Pi and weight factors and node vectors U. This invention uses the cumulative chord length parameterization method to show the distribution of data points according to individual chord lengths, and can obtain curves with better smoothness. This parameterization method satisfies:

Get the node vector:

U=[u ₀ ,u ₁ ,…u _n+k+3 ].

The inverse calculation of the control vertices calculates the control points of the curve using the least squares method, an optimization method used to fit the relationship between data points and the model. In NURBS curve fitting, it is hoped that by adjusting the positions of control points, the curve can fit the given data as closely as possible. The goal is to find a control point matrix ctrP that minimizes the error between the data points and the NURBS curve. Convert the problem into a least squares problem by constructing a system of linear equations of the following form:

( ^NT ·N)·crtP＝ ^NT ·r

Among them, N ^T represents the transpose of N, and r represents the difference between the data point and the first and last control points.

The calculation of the r matrix is completed by traversing the data points, and the formula is as follows:

r _i =d _i -d ₁ ·N(u _i )-d _m ·N(u _i )

Among them, _di represents the coordinates of the i-th data point, u _i represents the parameter value of the i-th data point, and N(u _i ) represents the value of the spline basis function at the parameter value u _i .

The calculation of the N matrix is also completed by traversing the data points. The formula is as follows:

N _ij =N(u _j,i )

Among them, N _ij represents the value of the _i- th component of the spline basis function at parameter value u _j .

By solving the system of linear equations, the control point matrix ctrP is obtained: the formula is as follows:

crtP=(N ^T ·N) ^-1 ·N ^T ·r

The curve control points are calculated according to the above formula, and the weight values are all 1.

In order to obtain more accurate airfoil geometric parameters, it can be achieved by separating the upper surface and lower surface of the airfoil and using NURBS curves respectively, instead of parameterizing the entire airfoil together. As shown in Figure 2, the parameterized DU25 airfoil, DU25 airfoil The maximum relative thickness is 25%, and the first and last control points are fixed at the leading and trailing edges of the airfoil. The parameterization method chooses the accumulated chord length parameter method, and 7 control points are distributed along each curve. The 7 control points are selected based on the fitting effect. According to the fitting result error and reducing the number of control points, it is convenient to optimize the airfoil with fewer variables in the later stage.

Optimize NURBS curves:

Traditional NURBS curve parameterization only considers the coordinates of control points as design variables and fixes the weight values to parameterize the airfoil curve. Since the weights of the control points do not change, the curve parameterization error is large. Therefore, in order to reduce the error of parameterized fitting of NURBS curves and improve the parameterization accuracy, the weight parameter ω _i is introduced as a parameterization variable in the invention. In order to obtain appropriate control point weights and distributions, the parameterized problem needs to be optimized. The representation error between the original airfoil and the fitted geometry can be expressed by the mean ε _mea and the maximum value ε _max . By minimizing the mean error and minimizing the maximum error as the objective function, the coordinates and weight values of the control points are Design variables to optimize parameterized control variables and weights.

Taking the maximum error and the average error as the objective function is a typical multi-objective optimization problem. In the present invention, the weight method is used to convert the multi-objective optimization problem into a single-objective problem.

In the formula, ε _mea is the mean error, di _is the distance between the original airfoil and the target curve projection, C is the chord length of the airfoil, n is the number of each airfoil point, and ε _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 }

The obtained control points and corresponding weights form the final NURBS airfoil fitting curve, and the parameterized result is output.

Obviously, the F(x) objective function is a typical nonlinear optimization problem. The gradient-based Sequential quadratic programming (SQP) algorithm is used to optimize the design variables. The main advantage of the SQP algorithm is that it has local super Linear convergence and global convergence.

As shown in Figure 3, the control point coordinates with the smallest fitting error have been obtained, providing a basis for the next step of airfoil optimization. A comparison of the DU21 NURBS shape and the original airfoil is shown in Table 1.

Table 1

After optimization, the average error of the lower surface is 3.0785e ^-4 and the average error of the upper surface is 8.7009e ^-4 .

After optimization, the error was greatly reduced, the fitting accuracy was improved, and it laid the foundation for subsequent optimization of the airfoil aerodynamic shape.

The invention discloses a wind turbine airfoil parameterization method based on improved NURBS. NURBS is a free deformation parameterization method. Using this method to perform airfoil parameterization does not limit the number of design variables and does not require initial fitting. appearance, easy to operate. This invention takes the minimum error as the objective function to optimize the control points and weight vectors, greatly reducing the fitting error; the NURBS airfoil parameterization method is improved, and the generated airfoil has a smooth appearance and no localized Concave-convex phenomenon.

The above are only preferred specific implementations of the present application, but the protection scope of the present application is not limited thereto. Any person familiar with the technical field can easily think of changes or substitutions within the technical scope disclosed in the present application. All are covered by the protection scope of this application. Therefore, the protection scope of this application should 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.