What you addressed is an issue for the assumption of first degree polynomial you use when a straigth line links two values. Why don't you think about a polynomial of higher degree? For example the cubic spline.

Filippo Maria Denaro Thanks very much for your reply. I did experiment with some higher order polynomials on excel, but a 6th order polynomial did not have a great fit. I was able to calculate values to plot a cubic spline which fit excellently, so thank you for your recommendation, however I lack the knowledge to extract the equation of this spline. The equation is obviously essential for finding the second derivative in my original question. Do you know how to find the equation of such a spline?

Thanks

Ed

Ed Frost

The second derivatives are exactly the values of the unknown in the nodes computed from the solution of the spline problem (a linear system with a tri-diagonal matrix). Therefore, once the coefficients are known, by definition the second derivative is a linear function between any couple of nodes (i,i+1).

This is a classical topic in the interpolation field, there are a lot of hystorical references for it in textobook of numerical analysis.

Just in Internet you can find an interactive form https://en.wikiversity.org/wiki/Cubic_Spline_Interpolation

If you have Maple, you can also use it in symbolic computation.

Ed Frost , what method/software did you use to fit the spline? If it's already calculated, then it should be possible to get the equation somehow.

On the other hand, you might not even need the splines to get what you need. I recommend you check the public domain aeronautical software collection, there is a Wavedrag by Area Rule Program:

http://www.pdas.com/wavedragdownload.html

They do a numerical estimation of the drag for arbitrary shapes, so essentially very similar to what you are doing and the algorithm should work (check papers for accuracy, they seem pretty good). Maybe you can take a look at their source code and see what they do after calculating the intersection areas.

hi frost,

If you can fit linear polynomial, you can again fit the non-linear polynomial function as piece -wise linear.

Say you are trying to fit

y=a0+_a1x+a2*x^2+a3*x^3+a4*x^4+a5*x^5+a6*x^6

then the equation could be written as

y=a0+_a1x(1+a2/a1*x)+a3*x^3(1+a4a3*x)+a5*x^5(1+a6/a5*x)

like wise i can reduce the complete equation interms of piece-wise linear polynomial and start solving the linear polynomial in the open interval.

Filippo Maria Denaro Thanks for the information, in the end I used a method of transforming the data points into a fourier series using matlab, and using those coefficients to compute the drag as I found others had done. I certainly appreciate your advice.

Srinvas Sharma Gangaraju Thanks for contributing advice, your post on the polynomial was a direction I considered, but as i mentioned I went with the Fourier series which I should have found before asking the question really!

I'm familiar with the wave drag program, it's a great resource particularly the references used for writing it. Unfortunately, all their work computes the areas in terms of (x,y) co-ordinates at each point, and does the maths accordingly. This doesn't fit my process unfortunately as I just have the areas from a Cad program read out. I'm working with quite complex geometries and when I tried to define it using the Wavedrag input instructions, I wasn't able to describe it in its entirety.

Thanks both for your help, I'm quite new to the community and I'm grateful for how helpful everyone has been.

Best regards

Ed

Thanks for the elaborate explanation about your work .

The problem seems to be interesting will try to work in this area.

All the very best frost