I'm trying to model a bulk metal with anisotropic electrical conductivity such as graphite. I want to run a stationary current through the material and I am using the finite volume method to do this. My boundary conditions are fixed potentials at certain boundary positions of the simulated volume, just like electrical contacts. In a homogeneous isotropic medium, I simply use Poisson's equation (or Laplace to be more precise) to compute the potential and I use Ohm's law to derive the electric current from the potential.
Now on to an anisotropic medium: As far as I've understood, my conductivity becomes a tensor so that the current $j$ will be given as: $$j_i = \Sigma_{j} \sigma_{ij} E_j$$ This is where my confusion starts.
- Wouldn't it make more sense to compute the current from the electric displacement field $\vec{D}$?
- What happens to Poisson's equation? Do I also have to include an anisotropic permittivity in order to compute the electric field correctly? Or is this irrelevant for the electrical current? (It can't really be irrelevant since this would give me the same electric field for isotropic and anisotropic media)
UPDATE: Here is what I found out during further research and while running a few simulations on simple model systems:
Concerning 1.: Check the answer below.
Concerning 2.: The anisotropy needs to be included in Poisson's equation as well. Otherwise current is not conserved. In the case of graphite for example, the anisotropy coefficient for the conductivity can be applied to Poisson's equation as well. When the current is then derived from the gradient of the potential, I obtain the correct values.