Philip,
> In short, I am modeling the maxwell equations for the electric field and
> voltage scalar field. The equations that I am using are displayed below:
>
> div(*E*) = rho / epsilon where epsilon = epsilon_{0} * epsilon_{r} and
> rho is the charge density of the material.
>
> -grad(V) = *E*
> *
> *
> Using Dr. Bangerth's recommendation, I am solving for the scalar voltage
> field first then taking the gradient of my solution using the
> DataPostprocessorVector class. This has worked extremely well in my test
> program. For more information on that, see this post:
> https://groups.google.com/forum/#!topic/dealii/XIiPyMh0Jz4
> <https://groups.google.com/forum/#%21topic/dealii/XIiPyMh0Jz4>
>
> However, now that I am actually coding the solver of the simulation, I
> will need to expand on my test simulation to include modeling
> anisotropic materials.
>
> When the material is anisotropic, the epsilon value (the permittivity)
> of the material is represented by a tensor. To make things slightly
> simplified, I am only running 2D simulations.
>
> I am attempting to determine the best method on modeling these types of
> materials. One approach that I have considered is to still solve for the
> voltage scalar field. If I go with this approach, then I will end up
> simulating a Possion equation where f = rho / epsilon and epsilon is a
> tensor. My concern for this direction would be that the Laplacian
> operator results in a scalar value. So I am not sure how I would handle
> the tensor on the RHS. Unless Deal.II has some sort of provision for this.
Correct -- rho/epsilon doesn't make any sense.
The way to formulate this is to look at the mixed Laplace first. There,
if you start with the primal formulation
-div (K grad p) = f
and write it as a first order system, you arrive at the equation
K^{-1} \vec u - grad p = 0
div \vec u = -f
as is explained in step-20. If you have an anisotropic material
behavior, then K is not just a scalar, but in fact a matrix, and K^{-1}
is its inverse. For physical reasons, K will have to be a symmetric and
positive definite matrix, and consequently the same holds for K^{-1}.
Now how you implement something like this: This is actually already done
in step-20, where we describe K (or, rather, its inverse) as a dim x dim
tensor:
https://dealii.org/developer/doxygen/deal.II/step_20.html#Theinversepermeabilitytensor
This is then used in the matrix assembly here:
https://dealii.org/developer/doxygen/deal.II/step_20.html#MixedLaplaceProblemassemble_system
Best
WB
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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