I wonder if you can interpolate the derivatives, instead of derivating the interpolation.

Thanks Gianluca for the suggestion!

I was trying to first take the route of finding an expression for the inverse, but I may have to look at the numerical inverse or just formulate a transformation from scratch to map interior points of an airfoil onto interior points of a circle.

The issue with the numerical inverse is that I plan to next find the deformation gradient matrix and its Jacobian from this map, which involves finding derivatives with respect to variables. That task would be "easier" if I could find an explicit expression ....

Thanks again,

Saliou

I made a few small modifications. I did not check which, if any, make a difference to the outcome.

(1) I made all numbers exact. This could be an issue, for example, in "symbolic" handling of square roots.

(2) I converted the equations to expressions.

(3) I did not define x and y to be functions of r and theta. I expect that this makes the reverse solving problem easier. In particular we can now solve for those rather than for the {r,theta} pair.

Ro = 2;
Ri = 1;
\[Rho] = 2;
\[Alpha] = -1/4;
\[Beta] = 0;
\[Lambda] = 7/4;

exprs = {-xd + (Ri*x)/(x^2 + y^2)^(1/2) + ((Ro - Ri)/Ro)*x, 
      -yd + (Ri*y)/(x^2 + y^2)^(1/2) + ((Ro - Ri)/Ro)*y};

Timing[solns = Simplify[Solve[exprs == 0, {x, y}]]]

(* Out[156]= {0.546003, {{x -> (
    2 (xd^3 + xd yd^2 - Sqrt[xd^2 (xd^2 + yd^2)]))/(xd^2 + yd^2), 
   y -> (2 yd (xd^3 + xd yd^2 - Sqrt[xd^2 (xd^2 + yd^2)]))/(
    xd (xd^2 + yd^2))}, {x -> (
    2 (xd^3 + xd yd^2 + Sqrt[xd^2 (xd^2 + yd^2)]))/(xd^2 + yd^2), 
   y -> (2 yd (xd^3 + xd yd^2 + Sqrt[xd^2 (xd^2 + yd^2)]))/(
    xd (xd^2 + yd^2))}}} *)

It might be overwhelming in terms of complexity to do a full symbolic solution of the original pair of equations. Specializing the parameters makes it viable. So you might want to consider having a function that just inverts when given specific values of those parameters.

Thanks Kay!

I will give it a try in the complex domain where it was first derived. However, the map in the complex form (w = f(z)) involved magnitude of z ( |z| ), which you can see in the algebraic form as sqrt (xd^2 + yd^2)... I was not sure would be easily inverted by Mathematica or any other tool.

But it is worth a try and thanks for the suggestion....

Regards,

Saliou

Thanks Kay!

This seems promising as it does appear to be able to solve in the complex form. I will look into the solution in more detail for correctness and also see if I can re-express it in Cartesian form, which is really the ideal form that I would need to work with.

I am more of a Matlab person but I am amazed by the power of Mathematica when it comes to symbolic calculations.....

Thanks again for your time,

Saliou

It would be more easy to test this if Mathematica code were provided.