Bezier-Bernstein approximation for airfoil shape
 

Hello,

I'm interested if it is possible to approximate an airfoil shape with 2 or more Bezier curves of a moderate order (say 10-12). What I want to know is if it is possible that given a certain number of points (say 360 points) of an airfoil to fit through these a 12 order Bezier curve.

I've read an introductory chapter in Bezier curves but I didn't found any clear explanations of how you can approximate a given data set with a Bezier curve. The classical approach seems to be to use a Bezier curve of order N-1 where N is the number of your data points.

I would appreciate any insight in the subject.

Thanks,

Do

what is Shadows?
 

Hi all of you, I want to do meshing at the inetrface of two surfaces using gambit meshing. But at the interface i find error of Shadows when the geometry is exported to Fluent.

So, what is the meaning of Shadows? how to tackel the problems with these Shadows?


please reply.....

I am interested in the same question at present. I have been reading up on bsplines and nurbs. But the books never tell you how to get a curve to fit given data.

I guess you just have to do some least squares fit. Thats what I am going to try.

A new and better way to represent airfoil shapes has been proposed by Brenda Kulfan. Search for her name in aiaa papers and journals. This is especially interesting if you are going to do shape optimization.

Thanks Praveen,

least square approximation was what I have had in mind, basically you will determine by least squares the control points.

I've already implemented similar codes for Chebyshev polynomials and other orthogonal polynomials, so (in a first approx.) I will just replace the Chebyshev function with one that calculates a Bernestein polynomial! Sure in order to have a Bezier curve you need to interpolate exactly the first and the last point of your data set, so my code isn't exactly right, but it can be easily modified.

Shape optimization is what I'm trying to do:).

Do

It works! except the for the first and last points of course.

Do

I am assuming you are fitting two bezier curves, one for the upper and another for the lower curve. For each curve you can fix the first and last control points to be equal to the first and last point coordinates. Then determine the remaining control points by least squares. You can even put in information about the tangents at the LE/TE points into the bezier curve.

If you are using a high degree bezier curve, you should not directly evaluate the bernstein polynomials as that leads to loss of accuracy. A better way is to use the de Casteljau algorithm. In that case, you must do the least squares fitting by using an optimization method.

Yes, I use 2 Bezier curves of 12 order. I will check the accuracy of my Bernstein polynomials, but I suppose for the order 12 I will have no particular accuracy loss. Anyway I can check the error switching to quad precision versus double precision as I usually use in my code.

In case I will notice some degradation in accuracy I will use 4 Bezier curves of order 6, 2 for the upper side and 2 for the lower side of my airfoil.

Do

@Praveen

Just to be sure I'm not in the wrong direction, when you do shape optimization of an airfoil using a Bezier curve, the design variables are the control points of the Bezier curve. Correct ?

Thanks,

Do

Yes. You would fix the leading and trailing edge control points.

Thanks.

Do

Airfoil representation by bezier curves
 

Hi

I am trying to represent airfoil by using 2 bezier curves of order 12.
what i understand is i have to find control points of bezier curve by using the co-ordinates of the airfoil and doing least squares approximation.

My question is in typical least squares approx. we minimize vertical offset (i.e. we minimize {y - y(t)}^2 ). So if i do this i will find the y co-ordinates of my control points. Now, how do i find the x co-ordinates of my control points???

Can anyone please help me. :confused: