[Dune] Periodicity for unstructured grids

[Martin Nolte]

Hi Aleksejs,

there has been quite some discussion on periodic boundary conditions.
Currently, there are two approaches:

Approach 1:

Consider the flat torus T^n = R^n / Z^n. In this case, you can condider your
computational domain to be R^n itself and theoretically imagine a grid with
infinitely many elements such that for each (geometrical) element E and each i
in Z^n, E + i is also an element of the grid. Factoring out Z^n, you end up
with a grid for T^n.

Now, imagine you also have infinitely many processors and assign to each cube
[0,1]^n to one processor. This case can (theoretically) be handled by the
parallel interface. As each process only has a shifted copy of the grid, we
periodicity allows us to consider only one representative of the parallel
processes, i.e., 1 process (which then communicates with itself). I hope it is
clear how this can be extended to multiple "real" processors.

This approach is implemented in YaspGrid.

There is one little tweak to notice: As far as I know, the identified elements
are considered copies of each other. This might lead to problems with the
coordinate function, which is not periodic at all. I'm not sure whether this
question was thoroughly discussed, though.


Approach 2:

Let us consider the flat torus to be constructed by glueing together opposite
sides of the unit cube. In other words, we consider elements on the right
boundary to have an intersection with elements on the left boundary (and vice
versa). But that's it, the subentities are not identified.

In this case, parallelization is also clear. For each intersection, the
neighboring element should exist as a ghost.


A few years ago, there was quite some discussion on this, but people seem to
have lost interest. I'm not sure whether there is a "right and wrong".
Personally, I consider both approaches viable, depending on your goal.

Best,

Martin

PS: You can find some more information in the appendix of my phd thesis.


On 12/09/2016 11:52 AM, Aleksejs Fomins wrote:
> Dear Dune,
> 
> I need to implement periodic boundary conditions into dune-curvilineargrid now.
> 
> Would you be so kind to tell me if the current dune-grid interface envisions periodic boundaries and associated ghost elements, and how.
> 
> The questions of interest are: partiton type, behavior of intersection, behavior of DataHandle
> 
> Best regards,
> Aleksejs
> 
> 
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-- 
Dr. Martin Nolte <nolte at mathematik.uni-freiburg.de>

Universität Freiburg                                   phone: +49-761-203-5630
Abteilung für angewandte Mathematik                    fax:   +49-761-203-5632
Hermann-Herder-Straße 10
79104 Freiburg, Germany