I work currently on the calculation of magnetic resistance for a air-gap of an electrical machine. For this I am using conformal transformations. I am doing this on a 2D slice. As the machine is obviously round I first use a logarithmic transformation to convert it to a polygonal shape. After that I use a Schwarz-Christoffel transformation to map the polygon to a rectangle.
For the said rectangle I assume a homogeneous field distribution. A fine mesh is set to represent the homogeneous field distribution. This mesh is maped back to the real(physical) geometry. What I get is a very nice representation of the field and equipotential lines. I approximate every of the small quadrilaterals as rectangles and calculate the resistance of them. So I get the resistance. For the Schwarz-Christoffel Transformation I use the SC-Toolbox by Tobin A. Driscoll
Here you can see a simple example of what I'm trying to calculate: Larger image
My question is if the there is a more elegant way to get the resistance directly from the transformed (canonical) space, meaning the rectangle. I asked this question already on the math portal in more general terms, but I wasn't able to get an adequate answer
I am currently in the phase of writing documentation for my project, so I wanted to see if there is some deeper physical/mathematical explanation for it.