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

Abstract

The invention discloses an improved CST (Class Function/Shape Function Transformation) airfoil profile parametric method, which comprises the following steps: firstly, preprocessing the outline of an airfoil profile; then, calculating the Ratio(x) curve of the airfoil profile; determining the node and the order of a B spline, taking the linear sum of the primary function of the B spline as a shape function of the improved CST airfoil profile parametric method; utilizing the shape function to fit the Ratio(x) curve of the airfoil profile to obtain a value of a design variable; then, calculating and fitting the airfoil profile and a maximum fitting error, if the maximum fitting error meets precision requirements, finishing parameterization, otherwise, increasing the order of the B spline, and regulating the node of the B spline; and repeating the above process until parameterization is finished.

Description

A kind of improvement CST aerofoil profile parametric method

Technical field

The invention belongs to field of flight vehicle design, particularly aerofoil profile parametric method

Background technology

Aerofoil profile is described as the form of design variable by aerofoil profile parametric method, and it exists material impact to the control ability of aerofoil profiles to the design result that aerofoil profile is final.CST (Classfunction/ShapefunctionTransformation) parametric method is a kind of new type of parametric method proposed in 2006 by the BrendaM.Kulfan of Boeing Co., see document " " Fundamental " ParametricGeometryRepresentationsforAircraftComponentSha pes " (BrendaM.Kulfan, JohnE.Bussoletti, 11thAIAA/ISSMOMultidisciplinaryAnalysisandOptimizationCo nference, 2006, and " AUniversalParametricGeometryRepresentationMethod-" CST " " (BrendaM.Kulfan AIAA-2006-6948), 45thAIAAAerospaceSciencesMeetingandExhibit, 2007, AIAA-2007-62).The method that this parametric method adopts class function (Classfunction) and shape function (Shapefunction) to combine describes two-dimensional curve geometric shape.When carrying out parametrization to aerofoil profile, the class function of CST parametric method immobilizes, its shape function be one group of Bernstein polynomial linear and, the polynomial coefficient of each Bernstein is the design variable of CST aerofoil profile parametric method.By changing the value of design variable and adjustable airfoil geometry shape, the mathematic(al) representation of CST aerofoil profile parametric method is such as formula shown in 1.:

$y\left(x\right)=C_{1.0}^{0.5}\left(x\right)\·\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{A}_i\·B_{i,n}\left(x\right)+x\·y_{\mathrm{tail}},0\≤x\≤1$ 1. wherein, y (x) is aerofoil profiles; X is the tangential coordinate of aerofoil profile, for class function, for shape function, A _ifor design variable, B _i,nx () is one group of n Bernstein polynomial expression, i=0 ... n, y _tailfor airfoil trailing edge point ordinate.1. formula is out of shape and can obtains:

$\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{A}_i\·B_{i,n}\left(x\right)=\frac{(y\left(x\right)-x\·y_{\mathrm{tail}})}{C_{1.0}^{0.5}\left(x\right)}=\mathrm{Ratio}\left(x\right)$ ②

From formula 2., CST aerofoil profile parametric method is directly related with the capability of fitting of its shape function to aerofoil profile Ratio (x) curve to the control ability of aerofoil profiles.The shape function of CST aerofoil profile parametric method be one group of Bernstein polynomial linear and, Bernstein polynomial expression is a kind of conventional SPL in art of mathematics, see document " contour analysis in computer aided design and manufacture " (NicholasM.Patrikalakis, TakashiMaekama work, Feng Jieqing, Ye Xiuzi translates, China Machine Press, 2005:6 ~ 12).Bernstein polynomial expression has nonnegativity, unit, symmetry, recursiveness and rises the mathematical property such as rank property, limited to the capability of fitting of strong nonlinearity curve.And aerofoil profile Ratio (x) curve exists strong nonlinear characteristic near leading edge, CST aerofoil profile parametric method is caused to exist the poor shortcoming of aerofoil profiles control ability.

Summary of the invention

The object of the invention is: for CST aerofoil profile parametric method to the poor problem of aerofoil profiles control ability, propose a kind of improvement CST aerofoil profile parametric method with accurate aerofoil profiles control ability, to meet the minute design demand of present generation aircraft profile.

Technical scheme of the present invention is: a kind of improvement CST aerofoil profile parametric method, is characterized in that, comprise the following steps:

(1) adopt 3. that formula is to aerofoil profiles coordinate points (x, y (x)), x ∈ [0,1] carries out pre-service, and after making pre-service, aerofoil profile ordinate Y (x) is 0 at trailing edge x=1 place,

Y (x)=y (x)-xy _tail3. wherein, y (x) is true aerofoil profiles; X is the tangential coordinate of aerofoil profile, y _tailfor airfoil trailing edge point ordinate;

(2) adopt 4. formula by the class function of Y (x) divided by CST aerofoil profile parametric method obtain matched curve Ratio (x) of CST aerofoil profile parametric 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 node and the exponent number of B-spline, when B-spline exponent number is m, its node has form, namely respectively with m 0 and m 1 for boundary node, and internal node comprises 0.01, by this B-spline basis function N _i,k(x) linear and as the shape function improving CST aerofoil profile parametric method, the coefficient A of B-spline basis function _ifor improving the design variable of CST aerofoil profile parametric method;

(4) utilize the shape function of the improvement CST aerofoil profile parametric method in step (3) to carry out matching to Ratio (x) curve obtained in step (2), obtain design variable A _ivalue;

(5) matching aerofoil profile y ' (x) of 5. formula computed improved CST aerofoil profile parametric method is adopted;

$y^{\′}\left(x\right)=C_{1.0}^{0.5}\left(x\right)\·\underset{i=0}{\overset{n}{\mathrm{\Σ}}}{A}_i\·N_{i,k}\left(x\right)+x\·y_{\mathrm{tail}}$ ⑤

(6) the maximum error of fitting Error of 6. formula computed improved CST aerofoil profile parametric method is adopted;

Error＝max|y(x)-y′(x)|,x∈[0,1]⑥

(7) if Error≤0.0007, show that current shape function can make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, parametrization terminates; If Error > 0.0007, then show that current shape function can not make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, return step (3), increase the exponent number of B-spline, adjustment B-spline node, and repeat said process, until Error≤0.0007.

Further, the B-spline in step (3) only with 0.01 for internal node, all the other nodes are 0 and 1, and namely when B-spline exponent number is m, its joint form is

Further, the B-spline in step (3) comprises the internal node outside 0.01, and namely when B-spline exponent number is m, its node has form, wherein set of node 1 and set of node 2 are not empty entirely.

Advantage of the present invention is:

1, by introducing joint form be m rank B-spline, and using the linear of this B-spline basis function with as the shape function improving CST aerofoil profile parametric method, significantly improve the capability of fitting to aerofoil profile Ratio (x) curve, solve former CST aerofoil profile parametric method to the poor problem of aerofoil profiles control ability.

2, when B-spline is only internal node with 0.01, namely joint form is time, the aerofoil profiles transition improving the generation of CST aerofoil profile parametric method is smooth, there will not be localized indentation to show especially and resembles, be applicable to Optimization of Aircraft Configuration Design problem.

3, when B-spline comprises the internal node outside 0.01, namely joint form is and when set of node 1 and set of node 2 are not empty entirely, improve CST aerofoil profile parametric method and there is local form's control ability, be convenient to designer, according to experience, local shape modifications carried out to aerofoil profiles.

Accompanying drawing explanation

Fig. 1 is the algorithm flow improving CST aerofoil profile parametric method.

Fig. 2-Figure 18 adopts improvement CST aerofoil profile parametric method to carry out parameterized process instance to NASASC (2)-0414 aerofoil profile.

Embodiment

1. embodiment one:

Adopt improvement CST aerofoil profile parametric method to carry out parametrization to NASASC (2)-0414 aerofoil profile, Fig. 2 is NASASC (2)-0414 aerofoil profiles.Now employing joint form is m rank B-spline, this batten only with 0.01 for internal node.Concrete steps are as follows:

(1) adopt 3. formula to carry out pre-service to NASASC (2)-0414 aerofoil profiles, making upper and lower airfoil trailing edge point ordinate be 0, Fig. 3 is pretreated Y (x) curve.

(2) adopt 4. formula to calculate Ratio (x) curve of upper lower aerofoil, Fig. 4 is top airfoil Ratio (x) curve, and Fig. 5 is lower aerofoil Ratio (x) curve.

(3) node is selected to be 8 rank B-spline, Fig. 6 is this B-spline basis function curve, using the linear of this B-spline basis function and the shape function as improvement CST aerofoil profile parametric method;

(4) utilize shape function to carry out least square fitting to Ratio (x) curve of upper lower aerofoil respectively, obtain the value of lower aerofoil design variable: top airfoil design variable Paraup=[0.1683,0.2322,0.0914,0.2986,0.0105,0.3530,0.1185,0.2301,0.2149], lower aerofoil design variable Paralow=[-0.1682 ,-0.2323 ,-0.0891,-0.3205,0.0325 ,-0.4352,-0.0340 ,-0.0725,0.2232];

(5) design variable is substituted into 5. formula digital simulation aerofoil profile y ' (x), Fig. 7 is matching aerofoil profiles;

(6) digital simulation error, Fig. 8 is the error of fitting distribution of upper lower aerofoil, and adopts 6. formula to calculate maximum error of fitting Error, and result of calculation is Error=0.0013;

(7) due to Error > 0.0007, show that current shape function can not make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, return step (3), increase the exponent number of B-spline, and adjust B-spline node, repeat step (3) to step (6), until Error≤0.0007.

For NASASC (2)-0414 aerofoil profile, during Error≤0.0007, employing to be node be 13 rank B-spline, Fig. 9 is this B-spline basis function curve.Figure 10 is linear with this B-spline basis function and the matching aerofoil profile that obtains for shape function, and Figure 11 is that the error of fitting of upper lower aerofoil distributes, maximum error of fitting Error=0.00064 now.Top 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 aerofoil 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 improvement CST aerofoil profile parametric method is adopted to carry out parametrization to NASASC (2)-0414 aerofoil profile.Now employing joint form is m rank B-spline, set of node 1 and set of node 2 are not empty entirely.Concrete steps are as follows:

(1) adopt 3. formula to carry out pre-service to NASASC (2)-0414 aerofoil profiles, making upper and lower airfoil trailing edge point ordinate be 0, Fig. 3 is pretreated Y (x) curve;

(2) adopt 4. formula to calculate Ratio (x) curve of upper lower aerofoil, Fig. 4 is top airfoil Ratio (x) curve, and Fig. 5 is lower aerofoil Ratio (x) curve;

(3) node is selected to be 6 rank B-spline, Figure 12 is this B-spline basis function curve, using the linear of this B-spline basis function and the shape function as improvement CST aerofoil profile parametric method;

(4) shape function is utilized to carry out least square fitting to Ratio (x) curve of upper lower aerofoil respectively, obtain the value of lower aerofoil design variable: top 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 aerofoil design variable Paralow=[-0.1802,-0.2430,-0.2041,-0.1910,-0.1782,-0.1781,-0.1783,-0.1807,-0.1822,-0.1663,-0.1224,-0.0498, 0.0331, 0.1216, 0.1673, 0.2108],

(5) design variable is substituted into 5. formula digital simulation aerofoil profile y ' (x), Figure 13 is matching aerofoil profiles;

(6) digital simulation error, Figure 14 is error of fitting distribution, and adopts 6. formula to calculate maximum error of fitting Error, and result of calculation is Error=0.0011;

(7) due to Error > 0.0007, show that current shape function can not make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, return step (3), increase the exponent number of B-spline, and keeping adjusting B-spline node under the constant condition of internal node collection, repeat step (3) to step (6), until Error≤0.0007.

For NASASC (2)-0414 aerofoil profile, during Error≤0.0007, employing to be node be 9 rank B-spline, Figure 15 is this B-spline basis function curve.Figure 16 is linear with this B-spline basis function and the matching aerofoil profile that obtains for shape function, and Figure 17 is that corresponding error of fitting distributes, maximum error of fitting Error=0.00059 now.Top airfoil design variable Paraup=[0.1308,0.2507,0.2101,0.2045,0.1794,0.1827,0.1707,0.1781,0.1751,0.1857,0.1920,0.2027,0.2001,0.2062,0.2061,0.2123,0.2171,0.2239,0.2243]; Lower aerofoil 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,-0.0383,0.0255,0.1039,0.1423,0.1873,0.2025].

As shown in Figure 15, this B-spline basis function has local support character, and namely basis function is only at x ∈ (a, b) in scope be on the occasion of, wherein 0<a<b<1, at x ∈ [0, a] and x ∈ [b, 1] scope in be 0.Linear and as shape function the improvement CST aerofoil profile parametric method of this B-spline basis function is adopted to have local form's control ability.Figure 18 is the deformation effect when value of the 5th of top airfoil design variable Paraup the design parameter is become 0.15 from 0.1794.

Claims (3)

1. improve a CST aerofoil profile parametric method, it is characterized in that, comprise the following steps:

Step 1: adopt (1) formula to aerofoil profiles coordinate points (x, y (x)), x ∈ [0,1] carries out pre-service, and after making pre-service, aerofoil profile ordinate Y (x) is 0 at trailing edge x=1 place,

Y(x)＝y(x)-x·y _tail(1)

Wherein, y (x) is true aerofoil profiles; X is the tangential coordinate of aerofoil profile, y _tailfor airfoil trailing edge point ordinate;

Step 2: adopt (2) formula by the class function of Y (x) divided by CST aerofoil profile parametric method obtain matched curve Ratio (x) of CST aerofoil profile parametric method shape function;

$\mathrm{Ratio}\left(x\right)=\frac{Y\left(x\right)}{C_{1.0}^{0.5}\left(x\right)}=\frac{Y\left(x\right)}{\sqrt{x}(1-x)}---\left(2\right)$

Step 3: node and the exponent number of determining B-spline, when B-spline exponent number is m, its node has form, namely respectively with m 0 and m 1 for boundary node, and internal node comprises 0.01, by this B-spline basis function N _i,k(x) linear and as the shape function improving CST aerofoil profile parametric method, the coefficient A of B-spline basis function _ifor improving the design variable of CST aerofoil profile parametric method;

Step 4: utilize in step 3 shape function improving CST aerofoil profile parametric method to carry out matching to Ratio (x) curve obtained in step 2, obtain design variable A _ivalue;

Step 5: matching aerofoil profile y ' (x) adopting (3) formula computed improved CST aerofoil profile parametric method;

$y^{\&prime;}\left(x\right)=C_{1.0}^{0.5}\left(x\right)\&CenterDot;\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{A}_i\&CenterDot;N_{i,k}\left(x\right)+x\&CenterDot;y_{\mathrm{tail}}---\left(3\right)$

Step 6: the maximum error of fitting Error adopting (4) formula computed improved CST aerofoil profile parametric method;

Error＝maxy(x)-y′(x),x∈[0,1](4)

Step 7: if Error≤0.0007, show that current shape function can make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, parametrization terminates; If Error > 0.0007, then show that current shape function can not make improvement CST aerofoil profile parametric method control aerofoil profiles accurately, return step 3, increase the exponent number of B-spline, adjustment B-spline node, and repeat said process, until Error≤0.0007.

2. improvement CST aerofoil profile parametric method according to claim 1, is characterized in that, the B-spline in step 3 only with 0.01 for internal node, all the other nodes are 0 and 1, and namely when B-spline exponent number is m, its joint form is

3. improvement CST aerofoil profile parametric method according to claim 1, it is characterized in that, the B-spline in step 3 comprises the internal node outside 0.01, and namely when B-spline exponent number is m, its joint form is wherein set of node 1 and set of node 2 are not empty entirely.