Hello Jean-Paul,
Thank you for the links and resources. I am going through them now. I am
not very familiar with linear elastic. I was able to better understand the
step-20 and how it applies to electrostatics because I was able to relate
the equations to their electrostatic equivalents.
I have been reading through you paper a couple of times now but I od not
see how it relates to anisotropic materials. I could be missing something
or misunderstanding some parts. From what I can tell from the paper, the
strong form of linear elasticity is:
∇· σ + b = 0 on Ω.
where σ is the stress tensor which is modeled by σ = σ m + σ f. The
isotropic linear constitutive law is given by: σm = C m : ε. Which is
basically Hooke's law. Now, hooke's law does have an analogue for the
polarization of a dielectric material when the material is exposed to an
electric field. I am not very sure if this is the direction that I need to
go in.
I have been looking up anisotropic materials for linear elasticity. I might
be missing something but I am having a hard time seeing how this relates to
the electrostatic case? Maybe I need to spend a little bit more time on
this topic. I do see that there is a similarity between div *[* *\epsilon*
*e*] = \rho^{f} and ∇· σ + b = 0 where σ = *\epsilon* *e *and b = -
\rho^{f}.
On Tuesday, November 6, 2018 at 3:51:43 AM UTC-5, Jean-Paul Pelteret wrote:
>
> 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 <phillip...@gmail.com
> <javascript:>> 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
>
> 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
>
> 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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