CN105574221A - Improved CST (Class Function/Shape Function Transformation) airfoil profile parametric method - Google Patents

Abstract

The invention discloses an improved CST airfoil parameterization method. First, the airfoil profile is preprocessed, and then the Ratio(x) curve of the airfoil is calculated, and then the node and order of the B-spline are determined, and the basis function As the shape function of the improved CST airfoil parameterization method, the linear sum of the airfoil is used to fit the Ratio(x) curve of the airfoil to obtain the value of the design variable, and then the fitted airfoil and the maximum fitting error are calculated. If the maximum If the fitting error meets the accuracy requirement, the parameterization ends, otherwise, the order of the B-spline is increased and the B-spline nodes are adjusted, and the above process is repeated until the parameterization ends.

Description

An Improved CST Airfoil Parameterization Method

technical field

The invention belongs to the field of aircraft design, in particular to airfoil parameterization method

Background technique

The airfoil parameterization method describes the airfoil as a form of design variable, and its ability to control the airfoil shape has an important impact on the final design result of the airfoil. The CST (Classfunction/ShapefunctionTransformation) parameterization method is a new type of parameterization method proposed by BrendaM.Kulfan of Boeing Company in the United States in 2006. , 2006, AIAA-2006-6948) and "AUniversalParametricGeometryRepresentationMethod-"CST"" (BrendaM.Kulfan, 45thAIAAAerospaceSciencesMeetingandExhibit, 2007, AIAA-2007-62). The parameterization method uses the combination of class function (Class function) and shape function (Shape function) to describe the geometric shape of two-dimensional curve. When parametrizing the airfoil, the class function of the CST parametric method is fixed, and its shape function is the linear sum of a set of Bernstein polynomials, and the coefficients of each Bernstein polynomial are the design variables of the CST airfoil parametric method. The geometric shape of the airfoil can be adjusted by changing the value of the design variable. The mathematical expression of the CST airfoil parameterization method is shown in formula ①:

$\mathrm{the\; y}\left(x\right)=C_{1.0}^{0.5}\left(x\right)\&Center\; Dot;\underset{i=0}{\overset{\mathrm{no}}{\mathrm{\Σ}}}{A}_i\·B_{i,\mathrm{no}}\left(x\right)+x\·{\mathrm{the\; y}}_{\mathrm{tail}},0\≤x\≤1$ ① Among them, y(x) is the shape of the airfoil; x is the chord coordinate of the airfoil, for class functions, is the shape function, A _i is the design variable, B _i,n (x) is a set of n-degree Bernstein polynomials, i=0...n, y _tail is the vertical coordinate of the trailing edge point of the airfoil. Transform formula ① to get:

$\underset{i=0}{\overset{\mathrm{no}}{\mathrm{\Σ}}}{A}_i\&Center\; Dot;B_{i,\mathrm{no}}\left(x\right)=\frac{(\mathrm{the\; y}\left(x\right)-x\·{\mathrm{the\; y}}_{\mathrm{tail}})}{C_{1.0}^{0.5}\left(x\right)}=\mathrm{Ratio}\left(x\right)$ ②

It can be seen from formula ② that the ability of the CST airfoil parameterization method to control the airfoil shape is directly related to the fitting ability of the shape function to the airfoil Ratio(x) curve. The shape function of the CST airfoil parameterization method is the linear sum of a set of Bernstein polynomials. Bernstein polynomials are a commonly used spline curve in the field of mathematics. See the literature "Shape Analysis in Computer-Aided Design and Manufacturing" (NicholasM. Takashi Maekama, Feng Jieqing, Ye Xiuzi translation, Machinery Industry Press, 2005: 6 ~ 12). Bernstein polynomials have mathematical properties such as non-negativity, unit division, symmetry, recursion, and ascending order, and their ability to fit strongly nonlinear curves is limited. However, the Ratio(x) curve of the airfoil has strong nonlinear characteristics near the leading edge, which leads to the disadvantage of poor control ability of the airfoil shape in the CST airfoil parameterization method.

Contents of the invention

The purpose of the present invention is: aiming at the problem that the CST airfoil parameterization method has a poor ability to control the airfoil shape, an improved CST airfoil parameterization method with precise airfoil shape control ability is proposed to meet the requirements of modern aircraft shapes. design requirements.

The technical scheme of the present invention is: a kind of parameterization method of improving CST airfoil, it is characterized in that, comprises the following steps:

(1) Use the formula ③ to preprocess the airfoil shape coordinate point (x, y(x)), x∈[0,1], so that the preprocessed airfoil ordinate Y(x) is at the trailing edge x=1 at 0,

Y(x)=y(x)-x y _tail ③wherein, y(x) is the real airfoil shape; x is the chord coordinate of the airfoil, and y _tail is the longitudinal coordinate of the trailing edge point of the airfoil;

(2) Using ④ to divide Y(x) by the class function of the CST airfoil parameterization method Obtain the fitting curve Ratio(x) of the CST airfoil parameterization method shape function;

$\mathrm{Ratio}\left(x\right)=\frac{Y\left(x\right)}{C_{1.0}^{0.5}\left(x\right)}=\frac{Y\left(x\right)}{\sqrt{x}(1-x)}$ ④

(3) Determine the node and order of B-spline, when the order of B-spline is m, its node has In the form of , that is, with m 0s and m 1s as boundary nodes and internal nodes containing 0.01, the linear sum of the B-spline basis functions N _i,k (x) As the shape function of the improved CST airfoil parameterization method, the coefficient A _i of the B-spline basis function is the design variable of the improved CST airfoil parameterization method;

(4) Utilize the shape function of the improved CST airfoil parameterization method in the step (3) to fit the Ratio (x) curve obtained in the step (2), obtain the value of the design variable A _i ;

(5) Calculate the fitting airfoil y'(x) of the improved CST airfoil parameterization method by using formula ⑤;

${\mathrm{the\; y}}^{\′}\left(x\right)=C_{1.0}^{0.5}\left(x\right)\&Center\; Dot;\underset{i=0}{\overset{\mathrm{no}}{\mathrm{\Σ}}}{A}_i\·N_{i,k}\left(x\right)+x\&Center\; Dot;{\mathrm{the\; y}}_{\mathrm{tail}}$ ⑤

(6) Using formula ⑥ to calculate the maximum fitting error Error of the improved CST airfoil parameterization method;

Error＝max|y(x)-y′(x)|, x∈[0,1]⑥

(7) If Error≤0.0007, it indicates that the current shape function can make the improved CST airfoil parameterization method accurately control the airfoil shape, and the parameterization is over; if Error>0.0007, it indicates that the current shape function cannot make the improved CST airfoil The parametric method accurately controls the shape of the airfoil, returns to step (3), increases the order of the B-spline, adjusts the B-spline nodes, and repeats the above process until Error≤0.0007.

Further, the B-spline in step (3) only uses 0.01 as the internal node, and the rest of the nodes are 0 and 1, that is, when the B-spline order is m, its node form is

Further, the B-spline in step (3) contains internal nodes other than 0.01, that is, when the B-spline order is m, its nodes have In the form of , where node set 1 and node set 2 are not all empty.

The advantages of the present invention are:

1. By introducing the node form as The m-order B-spline, and the linear sum of the B-spline basis functions are used as the shape function of the improved CST airfoil parameterization method, which greatly improves the fitting ability of the airfoil Ratio(x) curve, and solves the problem of the original CST The airfoil parameterization method is poor in controlling the airfoil shape.

2. When the B-spline only uses 0.01 as the internal node, that is, the node form is When , the airfoil profile generated by the improved CST airfoil parameterization method has smooth transition and no local concave-convex phenomenon, which is suitable for the optimal design of aircraft shape.

3. When the B-spline contains internal nodes other than 0.01, the node form is And when node set 1 and node set 2 are not all empty, the improved CST airfoil parameterization method has the ability to control the local shape, which is convenient for designers to locally modify the airfoil shape based on their own experience.

Description of drawings

Figure 1 is the algorithm flow of the improved CST airfoil parameterization method.

Figures 2-18 are examples of the process of parameterizing the NASAC(2)-0414 airfoil using the improved CST airfoil parameterization method.

Example

1. Embodiment one:

The improved CST airfoil parameterization method is used to parameterize the NASASC(2)-0414 airfoil. Figure 2 shows the profile of the NASAC(2)-0414 airfoil. At this time, the node form is The m-order B-spline of , the spline only has 0.01 as internal nodes. Specific steps are as follows:

(1) The shape of the NASASC(2)-0414 airfoil is preprocessed by formula ③, so that the vertical coordinates of the trailing edge points of the upper and lower airfoils are 0. Figure 3 is the Y(x) curve after pretreatment.

(2) Use formula ④ to calculate the Ratio(x) curves of the upper and lower airfoils. Figure 4 shows the Ratio(x) curves of the upper airfoil, and Figure 5 shows the Ratio(x) curves of the lower airfoil.

(3) Select the node as The 8th order B-spline, Fig. 6 is this B-spline basis function curve, the linear sum of this B-spline basis function is used as the shape function of improving CST airfoil parameterization method;

(4) Utilize the shape function to carry out least squares fitting to the Ratio(x) curves of the upper and lower airfoils respectively, and obtain the values of the upper and lower airfoil design variables: upper airfoil design variable Paraup=[0.1683,0.2322,0.0914,0.2986,0.0105 ,0.3530,0.1185,0.2301,0.2149], lower airfoil design variable Paralow＝[-0.1682,-0.2323,-0.0891,-0.3205,0.0325,-0.4352,-0.0340,-0.0725,0.2232];

(5) Substituting the design variables into formula ⑤ to calculate the fitted airfoil y'(x), and Fig. 7 is the fitted airfoil profile;

(6) Calculate the fitting error. Fig. 8 is the fitting error distribution of the upper and lower airfoils, and adopt ⑥ formula to calculate the maximum fitting error Error, and the calculation result is Error=0.0013;

(7) Since Error>0.0007, it indicates that the current shape function cannot make the improved CST airfoil parameterization method accurately control the airfoil shape, return to step (3), increase the order of B-spline, and adjust the B-spline nodes, Repeat step (3) to step (6) until Error≤0.0007.

For NASASC(2)-0414 airfoil, when Error≤0.0007, the nodes are The 13th order B-spline, Figure 9 is the B-spline basis function curve. Figure 10 shows the fitted airfoil obtained by taking the linear sum of the B-spline basis functions as the shape function, and Figure 11 shows the fitting error distribution of the upper and lower airfoils, and the maximum fitting error at this time is Error=0.00064. Upper airfoil design variable Paraup＝[0.0937,0.2425,0.0930,0.3677,-0.2425,0.8884,-0.7827,1.1969,-0.6384,0.7661,-0.0971,0.3257,0.1684,0.2340]; lower airfoil design variable Paralow＝[- 0.0931, -0.2429, -0.0900, -0.3805, 0.2602, -0.9209, 0.7873, -1.1198, 0.4523, -0.5224, 0.0651, -0.1145, 0.1135, 0.1947].

2. Embodiment two:

Still using the improved CST airfoil parameterization method to parameterize the NASASC(2)-0414 airfoil. At this time, the node form is For B-splines of order m, node set 1 and node set 2 are not all empty. Specific steps are as follows:

(1) Use formula ③ to preprocess the profile of the NASASC(2)-0414 airfoil, so that the vertical coordinates of the trailing edge points of the upper and lower airfoils are 0. Figure 3 is the Y(x) curve after pretreatment;

(2) Calculate the Ratio(x) curves of the upper and lower airfoils by using ④ formula, Fig. 4 is the Ratio(x) curve of the upper airfoil, and Fig. 5 is the Ratio(x) curve of the lower airfoil;

(3) Select the node as The 6th-order B-splines, Figure 12 is the B-spline basis function curve, the linear sum of the B-spline basis functions is used as the shape function of the improved CST airfoil parameterization method;

(4) Utilize the shape function to carry out least squares fitting to the Ratio(x) curves of the upper and lower airfoils respectively, and obtain the values of the upper and lower airfoil design variables: upper airfoil design variable Paraup=[0.1803,0.2429,0.2042,0.1911,0.1761 ,0.1761,0.1751,0.1813,0.1882,0.1978,0.2011,0.2044,0.2074,0.2133,0.2216,0.2251], lower airfoil design variable Paralow＝[-0.1802,-0.2430,-0.2041,-0.1910,-0.17 -0.1783,-0.1807,-0.1822,-0.1663,-0.1224,-0.0498,0.0331,0.1216,0.1673,0.2108];

(5) Substituting the design variables into formula ⑤ to calculate the fitted airfoil y′(x), and Fig. 13 is the fitted airfoil profile;

(6) Calculate the fitting error, Fig. 14 is the fitting error distribution, and adopt ⑥ formula to calculate the maximum fitting error Error, the calculation result is Error=0.0011;

(7) Since Error>0.0007, it indicates that the current shape function cannot make the improved CST airfoil parameterization method accurately control the airfoil shape, return to step (3), increase the order of B-spline, and keep the internal node set Adjust the B-spline nodes under different conditions, and repeat steps (3) to (6) until Error≤0.0007.

For NASASC(2)-0414 airfoil, when Error≤0.0007, the nodes are The 9th order B-spline, Figure 15 is the B-spline basis function curve. Figure 16 shows the fitted airfoil obtained by taking the linear sum of the B-spline basis functions as the shape function, and Figure 17 shows the corresponding fitting error distribution, and the maximum fitting error at this time is Error=0.00059. Design variable Paraup of upper airfoil＝[0.1308, 0.2507, 0.2101, 0.2045, 0.1794, 0.1827, 0.1707, 0.1781, 0.1751, 0.1857, 0.1920, 0.2027, 0.2001, 0.2062, 0.2061, 0.217349, 0.2] 2 of lower airfoil Design variable Paralow＝[-0.1307,-0.2508,-0.2101,-0.2047,-0.1791,-0.1863,-0.1706,-0.1839,-0.1745,-0.1860,-0.1789,-0.1533,-0.0995,-0.0383,0.02035, ,0.1423,0.1873,0.2025].

It can be seen from Figure 15 that the B-spline basis function has a local support property, that is, the basis function is only positive in the range of x∈(a,b), where 0<a<b<1, and in x∈[0,a ] and x∈[b,1] are all 0. The improved CST airfoil parameterization method using the linearity of the B-spline basis function and the shape function has the ability of local shape control. Figure 18 shows the deformation effect when the value of the fifth design parameter of the upper airfoil design variable Paraup is changed from 0.1794 to 0.15.

Claims (3)

1. An improved CST airfoil parameterization method, is characterized in that, comprises the following steps: Step 1: Use formula (1) to preprocess the airfoil shape coordinate point (x, y(x)), x∈[0,1], so that the preprocessed airfoil ordinate Y(x) is at the trailing edge x = 0 at 1, Y(x)=y(x)-x y _tail (1) Among them, y(x) is the real airfoil shape; x is the chord coordinate of the airfoil, and y _tail is the vertical coordinate of the trailing edge point of the airfoil; Step 2: Use formula (2) to divide Y(x) by the class function of the CST airfoil parameterization method Obtain the fitting curve Ratio(x) of the CST airfoil parameterization method shape function; $\mathrm{<span\; class="notranslate">\; Ratio</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 1.0</span>}}^{\mathrm{<span\; class="notranslate">\; 0.5</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}{\sqrt{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ Step 3: Determine the nodes and order of B-spline, when the order of B-spline is m, its nodes have In the form of , that is, with m 0s and m 1s as boundary nodes and internal nodes containing 0.01, the linear sum of the B-spline basis functions N _i,k (x) As the shape function of the improved CST airfoil parameterization method, the coefficient A _i of the B-spline basis function is the design variable of the improved CST airfoil parameterization method; Step 4: Use the shape function of the improved CST airfoil parameterization method in step 3 to fit the Ratio(x) curve obtained in step 2 to obtain the value of the design variable A _i ; Step 5: Calculate the fitting airfoil y'(x) of the improved CST airfoil parameterization method by formula (3); ${\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 1.0</span>}}^{\mathrm{<span\; class="notranslate">\; 0.5</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; tail</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ Step 6: Calculate the maximum fitting error Error of the improved CST airfoil parameterization method by formula (4); Error=maxy(x)-y'(x), x∈[0,1](4) Step 7: If Error≤0.0007, it indicates that the current shape function can make the improved CST airfoil parameterization method accurately control the airfoil shape, and the parameterization ends; if Error>0.0007, it indicates that the current shape function cannot make the improved CST airfoil The parametric method accurately controls the shape of the airfoil, returns to step 3, increases the order of the B-spline, adjusts the B-spline nodes, and repeats the above process until Error≤0.0007. 2. the improved CST airfoil parameterization method according to claim 1, is characterized in that, the B-spline in step 3 only takes 0.01 as internal node, and all the other nodes are 0 and 1, that is, when the B-spline order When it is m, its node form is 3. The improved CST airfoil parameterization method according to claim 1, characterized in that, the B-spline in step 3 includes internal nodes beyond 0.01, that is, when the B-spline order is m, its node form for Among them, node set 1 and node set 2 are not all empty.