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.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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