Periodic Boundaries

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Periodic Boundaries

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FlexPDE supports periodic and antiperiodic boundary conditions in one, two or three dimensions.  

Periodicity in the X-Y Plane

Periodicity in a two-dimensional problem, or in the extrusion walls of a three-dimensional problem, is invoked by the PERIODIC or ANTIPERIODIC statement.  

The PERIODIC statement appears in the position of a boundary condition, but the syntax is slightly different, and the requirements and implications are more extensive.  

The syntax is:


PERIODIC ( X_mapping, Y_mapping )

ANTIPERIODIC ( X_mapping, Y_mapping )


The mapping expressions specify the arithmetic required to convert a point (X,Y) in the immediate boundary to a point (X',Y') on a remote boundary.  The mapping expressions must result in each point on the immediate boundary being mapped to a point on the remote boundary.  Segment endpoints must map to segment endpoints.  The transformation must be invertible; do not specify  constants as mapped coordinates, as this will create a singular transformation.

The periodic boundary statement terminates any boundary conditions in effect, and instead imposes equality of all variables on the two boundaries.  It is still possible to state a boundary condition on the remote boundary, but in most cases this would be inappropriate.

The periodic statement affects only the next following LINE or ARC path.  These paths may contain more than one segment, but the next appearing LINE or ARC statement terminates the periodic condition unless the periodic statement is repeated.

Periodicity in 1D

Periodicity in a one-dimensional problem is invoked by the POINT PERIODIC or POINT ANTIPERIODIC statement. All other aspects are similar to the description above for X-Y periodicity.  

Periodicity in the Z-Dimension

Periodicity In the extruded dimension is invoked by the modifier PERIODIC or ANTIPERIODIC before the EXTRUSION statement, for example,


In this case, the top and bottom extrusion surfaces are assumed to be conformable, and the values are forced equal (or sign-reversed) along these surfaces.

Periodicity for different variables

Periodicity can be mapped differently for each variable using the MAP qualifier. The syntax is :

PERIODIC ( X_mapping, Y_mapping )  MAP(variable)={1 or -1}

ANTIPERIODIC ( X_mapping, Y_mapping )  MAP(variable)={1 or -1}

The PERIODIC (or ANTIPERIODIC) statement presets the behavior of all variables. Variables then mapped to 1 will be periodic. Variables mapped to -1 will be antiperiodic. For example, assuming variables A, B, C, the following would make A and C periodic, and B antiperiodic :

PERIODIC(x-2,y) MAP(a) = 1 MAP(b) = -1

The MAP can also be used to exchange variables :

MAP(a) = b MAP(b) = a

Note :

Prior to version 7, each node in the finite element mesh could have at most one periodic image.  This meant that two-way or three-way periodicity at a single mesh node could not be supported.  Starting in version 7 multiple periodic images at one mesh node is supported.


Example Problems:

Samples | Usage | Periodicity | 1d_periodic.pde
Samples | Usage | Periodicity | 3d_antiperiodic.pde
Samples | Usage | Periodicity | 3d_periodic_exchange.pde
Samples | Usage | Periodicity | 3d_xperiodic.pde
Samples | Usage | Periodicity | 3d_zperiodic.pde
Samples | Usage | Periodicity | antiperiodic.pde
Samples | Usage | Periodicity | azimuthal_periodic.pde
Samples | Usage | Periodicity | periodic+time.pde
Samples | Usage | Periodicity | periodic.pde
Samples | Usage | Periodicity | periodic_contaminant.pde
Samples | Usage | Periodicity | periodic_map.pde
Samples | Usage | Periodicity | three-way_periodic.pde
Samples | Usage | Periodicity | two-way_periodic.pde
Samples | Usage | Periodicity | two-way_periodic_3d.pde