I am interested in this too. Is there a way to invoke a periodic BC with a phase factor \exp(\Gamma l) ?
For example, in solving the Maxwell system on a 1-D periodic structure, we have two planes \partial\Omega_1, \partial\Omega_2 connected to each other via the relationship:
\mathbf{n}\times\mathbf{E}\vert_{\partial\Omega_1} \exp(-\Gamma l) = \mathbf{n}\times\mathbf{E}\vert_{\partial\Omega_2}
where l is the unit cell length and \Gamma is an eigenvalue that depends on the frequency of operation. One can find a dispersion relationship \Gamma (k).
I see two problems:
- The weighted periodic boundary condition.
- Properly setting up the eigenvalue problem so it is tractable using Fenics.
The first is most important, since a search algorithm could be used to find \Gamma (not the best option). Generally, for a 2D or 3D structure, the mode set would be reflected in the computation of \Gamma.