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{ 1D_PERIODIC.PDE  

 

 This example shows the use of FlexPDE in one-dimensional applications with periodic boundaries.

 

 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 in 1D is:

       POINT PERIODIC()

 The mapping expressions specify the arithmetic required to convert the immediate point

 (X) to a remote point (X').

 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 POINT_PERIODIC statement affects only the immediately preceding point.

 The next appearing LINE statement terminates the periodic condition unless the periodic

 statement is repeated.

 

 We will define a periodic segment (-1,1) that is conceptually repeated in + and -X.

 A heat source H2 is defined in a small subsection (X0,X1) of the line.

 Since there are no sidewalls to constrain the solution, we must program a

 heat sink H1 in the remaining parts of the domain, so that the heat generated by H2

 is exactly absorbed by H1.

 

}

 

title '1D PERIODIC BOUNDARY TEST'

 

coordinates cartesian1

 

variables

    u

 

definitions

   k

   h

   x0=0.5  

   x1=0.8  

   h2 = 1   ! the heat source in region 2 (energy per unit length)

   h1 = -(x1-x0)*h2/(2-x1+x0) ! this is the value of H1 which applied over (-1,X0) and (X1,1) will balance H2 applied over (X0,X1)

 

equations

   u : div(K*grad(u)) + h = 0

   

! since there are no fixed values in the domain, the system is ill-posed (solution is not unique).  So we add a constraint to make the solution unique.

constraints

   integral(u)=0

 

boundaries

  region 1

      k = 1e-3

      h = h1

      start(-1)

      line to (x0) to (x1) to (1)

 

    point periodic(x-2)

 

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

     demonstrate the periodicity of the solution }

  region 2  

       h=h2

       k=1e-4

      start(x0) line to (x1)

 

monitors

    grid(x)

    elevation(u)

 

plots

    grid(x)

    elevation(u)

    report(h1) report(h2)

    report(integral(h,1)) report(integral(h,2))

 

end