antiperiodic

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{  ANTIPERIODIC.PDE  

 

 This example shows the use of FlexPDE in applications with antiperiodic

  boundaries.

 

 The ANTIPERIODIC 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:

       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

 mapping 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 antiperiodic 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 antiperiodic 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.

 

}  

 

title 'ANTI-PERIODIC BOUNDARY TEST'  

 

Variables  

    u  

 

definitions  

   k = 0.1  

   h=0  

 

equations  

   u : div(K*grad(u)) + h = 0  

 

boundaries  

  Region 1  

    start(-1,-1)  

      value(u)=0   line to (1,-1)  

 

 

    { The following arc is required to be an antiperiodic image of an arc

           two units to its left. (This image boundary has not yet been defined.) }  

    antiperiodic(x-2,y) arc(center=-1,0) to (1.2,-0.2)  

    antiperiodic(x-2,y) line to (1.2,0.2)  

    antiperiodic(x-2,y) arc(center=-1,0) to (1,1)  

 

    value(u)=0 line to (-1,1)  

 

    { The following arc provides the required image boundary for the previous

           antiperiodic statement }  

    nobc(u)       { turn off the value BC }  

    arc(center= -3,0) to (-0.8,0.2) line to (-0.8,-0.2) arc(center=-3,0) to close  

 

  { an off-center heat source provides the asymmetric conditions to

       demonstrate the antiperiodicity of the solution }  

  region 2  h=10 k=10  

    start(1.2,-0.2) line to (1.2,0.2) to (1,0.2) to (1,-0.2) to close  

 

  region 3  h=-10 k=10  

    start(-0.6,-0.2) line to (-0.6,0.2) to (-0.8,0.2) to (-0.8,-0.2) to close  

 

monitors  

    grid(x,y)  

    contour(u)  

 

plots  

    grid(x,y)  

    contour(u)  

end