Dear Wolfgang, Thank you for your reply. It is a bit difficult to read formulas in text. So I have put a few questions I have in a pdf file. Formulas are better there. It is attached to this message. May I ask you to have a look at it?
Best,
John
On Friday, April 16, 2021 at 10:26:51 PM UTC+2 Wolfgang Bangerth wrote:
> On 4/16/21 11:36 AM, John Smith wrote:
> > However, the FE_Nedelec is different. It implements edge elements. They
> are
> > vector-based. That is, functions are represented by a superposition of
> > vector-valued shape functions:
> >
> > \vec{A} = \sum u_i vec{N}_i .
> >
> > Therefore, the output of the "value" method in “class CustomFunction :
> public
> > Function” must be vector-valued. Three components in a three-dimensional
> > space. Otherwise, there is no point in interpolation.
> >
> > This kind of vectorial approximations is a bread-and-butter topic in
> > magnetics. See, for example, equation (7) in:
> >
> > https://ieeexplore.ieee.org/document/497322
> > <
> https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F497322&data=04%7C01%7CWolfgang.Bangerth%40colostate.edu%7Cf7f578c8c6fc44890c3308d900fe30ef%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C1%7C637541914012798794%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=tJVhiTO272UFKlS0mWzTCrVp1VzPV3zbr2I16KxHkFI%3D&reserved=0
> >
> >
> > In short, the source, i.e the current vector potential, must be
> projected on
> > the space spanned by the vector-valued shape functions. Otherwise, the
> > simulation is numerically unstable. The last, from my experience, is
> > definitely true.
>
> I think you already found your solution, but just for clarity: When we use
> the
> term "interpolation", we typically ask for (scalar) coefficients U_i so
> that
>
> u_h(x_j) = sum U_i \phi_i(x_j) = g(x_j)
>
> where g is given and x_j are the node points. The problem is that for the
> Nedelec element, this is not always possible: Not all possible vectors
> g(x_j)
> can be represented. This makes sense because if you have N node points in
> 3d,
> then you have N scalar coefficients U_i, but you have 3N components of the
> values g(x_j).
>
> One way to deal with this is to associate a vector, let's say y_j with
> every
> node point x_j, and require that
>
> y_j \cdot u_h(x_j) = sum U_i y_j \cdot \phi_i(x_j) = y_j \cdot g(x_j)
>
> These y_j could, for example, be the tangential direction associated with
> the
> shape function phi_j. One *could* call this a variation of the term
> "interpolation", but it is not what VectorTools::interpolate() implements.
> It
> would probably not be terribly difficult to implement this kind of
> function,
> however!
>
> Best
> W.
>
>
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> www: http://www.math.colostate.edu/~bangerth/
>
>
--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see
https://groups.google.com/d/forum/dealii?hl=en
---
You received this message because you are subscribed to the Google Groups
"deal.II User Group" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To view this discussion on the web visit
https://groups.google.com/d/msgid/dealii/4149dd4c-baea-4bb7-9a5d-9d05a637d44fn%40googlegroups.com.
FE_Nedelec.pdf
Description: Adobe PDF document
