Posted by ADSW1243 on
URL: http://forum.featool.com/Vector-Poisson-Equation-for-a-3D-Current-Source-tp338.html

I thought given this is a separate issue, it would be worth a separate topic to make it easier to follow where I'm stuck.

In short, the aim of my model is to take a 3D current loop (initially with rectangular cross section - see my other post for a geometry reference) and model the resulting magnetic vector potential in 'all of space', where the containing box is assumed to be large enough that magnetic flux is completely confined within and Dirchlet zero conditions are appropriate for the outer faces. I also hope to calculate the related magnetic field over the same domain for further analysis.

For simplicity, the current density is assumed to have a constant vector magnitude throughout the coil, with vector components assigned to create a current flow around the 3D object. Something like this:



I'm fairly happy that this is possible within this toolbox, however after a significant number of hours effort I'm currently the limiting factor in making this work.

The simplified system of equation being studied is a vector Poisson equation for the magnetic vector potential due to a 3D current source. To implement this through the GUI I thought the inbuilt Poisson equation would be a good place to start to generate syntax giving (after removing the time derivative):

d_poi*(Ax_x + Ay_y + Az_z) = f_poi

From this, assuming mu = mu_o  = 4pi... throughout the domain:

d_poi = mu_o
Ax_x = d/dx Ax ect
f_poi = J.

This is where I come unstuck because I'm not sure how the div -D gradA is defined in this syntax to then convert that to
nabla A = - mu_o J;
d^2/dx^2 Ax = -mu_o * Jx ;
d^2/dy^2 Ay = -mu_o * Jy ;
d^2/dz^2 Az = -mu_o * Jz ;

I think a custom equation with 3 dependent variables is needed, but this scalar expression seems to have vector components too so I'm not actually certain. Also, changing the syntax/expression doesn't seem to alter the expression displayed in the GUI which is making this harder because I'm unsure how exactly the things I type are (or aren't) being interpreted.

I'm fairly certain this is a really simple task (though perhaps assuming the current flows maybe a little more difficult) but I'm struggling with the learning curve to setup something useful. Still working at this so will post an update if I make any progress - apologies for all the very basic questions and thanks for the help in advance.