Understanding CoilSolver

[hbr]

Hi,

For my application I need to understand in detail what the solver is actually doing numerically when we call CoilSolver. I had a look at the manual and the code, which gave me some understanding, but I would like to ask if I got it correctly and also some details I might have missed, if that's alright. I'll lay out the recipe for the CoilSolver but please do let me know where I might have misunderstood what's actually being done. I have attached a diagram. For simplicity, let's consider a uniform scalar conductivity.

1. The problem is that the electric potential needs to be discontinuous in a closed coil. Thus, we pick two cross-sections (c1 and c2) on the coil which are approximately opposite to each other. We then create two "twin" coils, one which has a cross-sectional cut at c1, and the other coil has a cut at c2.

2. For each of these two twin coils, we introduce a tiny gap where the cut is, such that we can introduce boundary conditions where the potential is V_0 on one side and V_1 on the other. We then solve the Poisson equation for the electric potential within each of the twin coils.

3. Having the potential and the conductivity, we now calculate the current field for each twin coil.

4. Because of the tiny gaps introduced by the cuts, in each twin coil the current field will have a disturbance close to the cut, but the current will be correct far away from the cut. Thus, we stitch together the two solutions by taking from one twin coil the current far away from the cut and from the other coil the current far away from the cut (which together span the whole coil since the cuts were opposite to each other).

5. Finally, we integrate the current density over the entire coil and normalise it to the desired value of the total current.

CoilSolver.png

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There are some questions I would like to ask about this procedure, particularly about the criteria with which the two coils are put together in the end, but before I do, let me just make sure: is this how the CoilSolver works or did I mix things up? Thank you for the help!

[raback]

Yep, that's what it does.

A minor point is that the "cut" is not actually done in the mesh but just in the CRS matrix.

-Peter

[hbr]

Thank you! That's helpful. And what criterion do you use to "stitch" together the current from the two cut coils? Do you weigh them based on distance to the cut or something like that?