I almost forgot:
6) I am starting to get the feeling that the difference between the
isotropic and the anisotropic material is the stiffness matrix. This might
be a long shot but would it be possible for me to utilize the same weak
form as in the paper but change the code on the stiffness matrix? Or should
I re-derive the weak form with anisotropic materials in mind?
6a) As a sub note on this one, there would be a concern since the weak form
in the paper has a provision for "history variables". These variables also
appear before the weak form and are apart of the derivation. Now since I am
working on an electrostatic problem, I would not have these history
variables and thus, I am unable to utilize this weak form.
On Wednesday, November 7, 2018 at 9:58:33 PM UTC-5, phillip mobley wrote:
>
> Hello Dr. Bangerth,
>
> Thank you stating this!
>
> I was able to go through the paper from Jean-Paul more thoroughly a few
> times. I do have a few questions regarding the paper.
>
> 1) On equation 23, where do the subscripts ijkl come from? I understand
> that subscripts ij are from equation 5. But I do not see where the
> subscripts k and l factor into the problem.
>
> 2) Why is the shape function for the problem in the form as eqn. 23? Is
> this the basic definition of the shape function in deal.ii? Or is this
> specific to the problem?
>
> 3) How dos equation 27 reduce down from this:
>
>
> to this:
>
>
> Is this because of the Kronecker delta?
>
> 4) Also, I am not familiar with the abbreviation eps. Which math function
> does this relate to?
>
> 5) Since -div (C eps(u)) = b is equivalent to -div (K grad u) = , I am
> almost thinking I should do the Mixed Laplace approach for the problem?
> But, after reading the paper by Jean-Paul a few times, it would seem that
> the -div (C eps(u)) = b approach does not require Mixed Laplace.
>
>
>
> On Tuesday, November 6, 2018 at 11:15:32 PM UTC-5, Wolfgang Bangerth wrote:
>>
>>
>> > 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}.
>>
>> You have
>> sigma = C eps(u)
>> and
>> -div sigma = b
>> which together yields
>> -div (C eps(u)) = b
>> and is the equivalent of
>> -div (K grad u) = f
>>
>> The elastic material is anisotropic if the stress-strain tensor C
>> represents
>> an anisotropic material. In the elasticity case, it isn't just a dxd
>> matrix,
>> but a dxdxdxd (rank-4) tensor. So it looks more complicated, but it isn't
>> really.
>>
>> Best
>> W.
>>
>> --
>> ------------------------------------------------------------------------
>> Wolfgang Bangerth email: bang...@colostate.edu
>> www: http://www.math.colostate.edu/~bangerth/
>>
>>
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