Hello Dr. Bangerth,
Thank you for the reply to my discussion! I greatly I appreciate the help.
I did not realize that the flow equations are very similiar if not the same
to the electrostatic equations. I am still going through the material and
formulating a plan on how to code the solver. There are a few other
components that I would like to add to the data out put if possible.
>From what I have read so far, the pressure would be the voltage scalar
field and the velocity would be the electric field. The tutorial also
mentions a flux. In electrostatics, this is the Electric Displacement Field.
The equations for finding these are exactly in step-20. But the symbols
used are different.
I do have some questions regarding the material found in step-20 and with
finite element as a whole. I apologize if these are beginner questions but
there are many topics in finite elements that I am still learning.
1) Is Raviet-Thomas elements specific for mixed Laplace equations? I
understand that step-20 is a tutorial and for learning. Other then being a
good learning opportunity, is there any specific reason why RT elements
were chosen for step-20?
1b) From another perspective, If I am solving for the mixed Laplace
equation, do I need to setup the system to use the RT elements?
2) The weak derivation for step-20 has Dirichlet boundary values
incorporated into it. I believe the sentence is "Note how in this
formulation, Dirichlet boundary values of the original problem are
incorporated in the weak form." I am guessing that this is because of the p
= g on d(omega) term. Later down the road, I might need to implement
different B.C. such as a periodic, anti-periodic, or a mixed coefficient
B.C. If this is the case, would I need to re-derive the weak form? Or can I
utilize the one found in step-20? Or, would it be better for me to derive a
more general weak form that can be used with any B.C.?
3) From what I have read so far, I get the impression that the R.T elements
are also incorporated into the weak form. Basically this sentence here:
"Both xh and wh are from the space Xh=RT(k)×DQ(k), where RT(k) is itself a
space"? is this thought correct? If so, if I wanted to change the element
type later, would I need to re-derive the weak form?
4) The weak enforcement of the pressure boundary conditions, does this need
to be factored in even if the B.C are not Dirichlet?
On Monday, November 5, 2018 at 5:17:05 PM UTC-5, Wolfgang Bangerth wrote:
>
>
> 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
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
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