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MatthewGrim / plasma_physics

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Write basic Grad Shafranov solver using Deal II to solve Soloviev equilibria #32

Open MatthewGrim opened 5 years ago

MatthewGrim commented 5 years ago

This issue documents the use of Deal II to write a Grad Shafranov solver to solve basic MHD equilibria. This is just the first pass, using a rectangular grid, to solve Soloviev equilibria. Later issues will move to more advanced Deal II grids etc.

This issue will be complete when a Soloviev equilibria has been successfully computed.

MatthewGrim commented 5 years ago

Changed solution to be on a circular grid

I decided to change the solution to solve on a circular grid. This will make the solution much easier for the first pass, as circular plasma boundary conditions are simpler.

MatthewGrim commented 5 years ago

At this point, the solution is close to the exact solution, but clearly incorrect - the axis is not at the right point.

I have double checked the equation I have implemented and the boundary condition. They should both be correct. I am not sure if I have problems because I am using a rectangular mesh, going beyond the edge of the plasma, or there is something wrong with the way I have constructed my basis.

To check, I am going to jump straight to #34 and generate a circular grid that matches the plasma boundary expected for the domain. This should eliminate the possibility of the grid being a problem.

MatthewGrim commented 5 years ago

Now the solution is found correctly, what's left is to write out all equilibrium profiles, and use the interpolator routine to solve the grid, instead of the analytic function.

After this point, I can move onto solving non-linear problems.

MatthewGrim commented 5 years ago

Before I can move onto #33 and #35, I need to implement a Newton method within Deal II to find the axis of the plasma. This is what remains for this issue to be completed.