Thanks for the suggestion- I haven’t heard of that extension. I’ll make a feature request and in the meantime use a hacky solution like interpolating function values on the other boundary at each timestep and manually feeding them plus a constant in to the periodic boundary as Dirichlet.
Sure- I’m working on a multiscale time-dependent constitutive PDE. In its full multiscale glory, the PDE is
-\nabla \cdot(E^{\epsilon} \nabla u^{\epsilon} + \nu^{\epsilon} \partial t \nabla u^{\epsilon}) = f, x \in \Omega
u^{\epsilon} = 0 , x \in \partial \Omega
u^\epsilon(0) = u^{*} (Initial condition)
where y = \frac{x}{\epsilon} is the “fine scale” variable for \epsilon \ll 1. The idea is that the PDE parameters E^{\epsilon} and \nu^{\epsilon} have variation on scales that are too small to resolve on the full domain \Omega, but they are periodic in y. What we are actually interested in is a map from average strain \int_{\mathbb{T}^d} \nabla u^{\epsilon} \; dy to average stress \int_{\mathbb{T}^d} (E^{\epsilon} + \nu^{\epsilon}\partial t)\nabla u^{\epsilon} \; dy, where both of these are trajectories in time that no longer depend on the small scale variable. This is related to some key terms like “homogenization” and “effective properties.” If you’re unfamiliar with this, I recommend Chapter 12 of “Multiscale Methods” by Pavliotis and Stuart.
Long story short, the way to solve for the average stress given an average strain trajectory involves a PDE at each timestep that takes the following form (the problem is now posed on the microscale, so \epsilon notation is dropped and the forcing, which is assumed to be on a larger scale, vanishes)
-\nabla \cdot ((E + \nu \partial t)\nabla u) = 0 such that u(y + e_i) = u(y) + \overline{\epsilon}_i where \overline{\epsilon} is the average strain at that time. From this one can see that an attempt to separate the problem into one with a periodic boundary condition and one with a Dirichlet boundary condition would just result in a trivial zero solution for the periodic case and constant edges in the Dirichlet case.
I haven’t been able to explain in real detail here, but if you’re really interested, we have some related papers: links may be found here and here.