Dear Philip,
Let me offer an alternative to what Wolfgang has suggested. I don’t think that
it is necessary to use a mixed formulation to introduce material anisotropy
into your problem. If you go back to the derivation in our previous discussion,
there was this line:
(6) div [\epsilon_{0} \epsilon_{r} e] = \rho^{f}
For tensor valued coefficient you could restate this in one of two ways. Either
(6a) div [ \epsilon_{r} e] = \rho^{f} / \epsilon_{0}
where \epsilon_{r} is a tensor of relative electric permittivity coefficients,
or
(6b) div [ \epsilon e] = \rho^{f}
where \epsilon is a tensor of electric permittivity coefficients.
The latter, (6b) is analogous to linear elasticity. In fact, we have a
code-gallery example
<https://github.com/dealii/code-gallery/tree/master/Linear_Elastic_Active_Skeletal_Muscle_Model>
(here’s a link to the theory pdf
<https://github.com/dealii/code-gallery/blob/master/Linear_Elastic_Active_Skeletal_Muscle_Model/doc/theory/theory-linear_elastic_active_muscle_model.pdf>)
that deals with anisotropic linear elasticity. I think that you would benefit
greatly by reading the literature on piezoelectric materials, which are
regularly modelled with a linear constitutive law.
Touching on one of your other questions, I would say that nowadays many people
would tend to use the electric scalar potential approach (solving for d with e
as the independent field) to modelling these materials under electrostatic
conditions, rather than the electric vector potential formulation (solving for
e with d as the independent field). I’m not saying that one is more correct
than the other - I’m just reporting what I perceive the trend in the literature
to be (I’d be happy to have someone offer a different opinion that me on this).
There is literally a mountain of literature dating back to the 1980’s on this
topic, so I leave it up to you to read into this more.
Best,
Jean-Paul
> On 02 Nov 2018, at 18:57, phillip mobley <phillipmobl...@gmail.com> wrote:
>
> Hello all,
>
> This is a follow up discussion to the Jean-Paul answer to the question "Is
> the approach for the electrostatic Bi-linear form correct?". The question
> (and answer) can be found at the following link:
>
> https://groups.google.com/forum/#!topic/dealii/i8P4JTwm7kQ
> <https://groups.google.com/forum/#!topic/dealii/i8P4JTwm7kQ>
>
> 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/#!topic/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.
>
> I have also been kicking around the idea of solving for the displacement
> vector D(i)= epsilon(i)(j) * E by substituting this equation into one of the
> equations above. Or at the very least, to use this equation as a constitutive
> relation to the equations above.
>
> A third approach that I haven't quite explored very much is solving for the
> polarization of the material. But I am not sure if this is a practical
> approach since I could unnecessarily complicate the problem.
>
> I wanted to post a discussion on this form to discuss what the best direction
> I should take to model anisotropic materials in Deal.ii? I have been looking
> at 3 different approaches and I would like to discuss which one of these 3
> directions is the better. Or if there might be others that I have not
> considered yet.
>
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