Periodicity with relative phase

[HelgeGehring]

If I have a mesh like

and I want it to be periodic, i.e. left-right and top-bottom with the possibility to set a relative phase between the facets, how would I do that?

More precise: I want the DOFs at the left side to be the same as at the right side, but multiplied by something like i, which I'd like to define.

I'd guess I need for each side a facet basis and add a term, right? But I don't have a clue how that should look like :/ Would be awesome if you could give me some advise how to get started :)

[kinnala]

I'd start by trying to add a penalty term and then try to improve from there. I can try to make you an example.

[HelgeGehring]

Thanks for the new version!

And thanks for the examples! It seems to start working for me :)
If I understand it right, the penalty method adds a term which forces the DOFs to be equal (where a wrong choice of alpha would lead to problems (?)) and the Lagrange multipliers increase the matrix size by the number of DOFs which have to match and use the multipliers to force them to match.

"multi-point constraints" helped a lot for searching, thanks!
The Master Slave Method (see https://mashayekhi.iut.ac.ir/sites/mashayekhi.iut.ac.ir/files//files_course/lesson_16.pdf) would seem to be the ideal solution, right? It would be exact and would also decrease the size of the system. The implementation would probably be somehow similar to condense, where some DOFs are kicked out of the system and subsequently are derived from the solutions (?)

[kinnala]

If you come up with something like condense which deals with these multi-point constraints, I'd be happy to accept it as a PR to skfem.utils.

[kinnala]

If I understand your reference correctly:

• T and g specify the constraint and are thus given by the user.
• So we could have a call such as multipoint(K, f, T=T, g=g) which returns the modified system and can be directly passed to solve.

Now the user probably also wants to set boundary conditions by specifying D and I and x etc. I think these can be technically implemented by specific choices of T and g so perhaps we want to have "helper" functions for creating T and g for (a) boundary conditions (b) periodic conditions (c) possible other interesting scenarios.

[HelgeGehring]

That's exactly how I understand it! Seems like a more generalized form of condense.

I'm unsure what exactly the function signature of the helper functions should look like. In principal also a three or more point constraints would be doable like this. Also I'm thinking of mixing different conditions, i.e. two sides periodic and setting the other two sides to 0.

If we implement the "Nonhomogeneous MPCs", that should already cover (a) and (b).

Maybe something like

def constraint(Dx, x, *Ix): -> T,g

where Dx=(D,x) and D are the degrees of freedom to eliminate, x is the prefactor. Ix=(I,x) would be similar defined (and multiple Ix and also None can be put in) and x would be like in condense the x?
I know that the x in Dx would not be completely necessary, as it can be realized by the choice of Ix and g, but I'd guess that makes it way more convenient and better understandable.

Probably we would also want a function which can combine several constraints into one T,g (and checks their compatibility)?
This could either be done before creating T,g by combining Dx, x and Ix or after if there's a way to combine already created T,g efficiently?

[HelgeGehring]

see #1006

[iceneve]

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