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3d_shell

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3d_shell

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{ 3D_SHELL.PDE

 

 This problem considers heatflow in a

 spherical shell.

 We solve a heatflow equation with

 fixed temperatures on inner and outer

 shell surfaces.

}

 

title '3D Test - Shell'

 

coordinates

   cartesian3

 

variables

   u

 

definitions

   k = 10                     { conductivity }

   heat =6*k                 { internal heat source }

   rad=sqrt(x^2+y^2)

   R1 = 1

   thick = staged(0.1,0.03,0.01)

   R2 = R1-thick

 

equations

   U: div(K*grad(u)) + heat   = 0

 

extrusion

  surface z =  -SPHERE ((0,0,0),R1)     { the bottom hemisphere }

  surface z =  -SPHERE ((0,0,0),R2)

  surface z = SPHERE ((0,0,0),R2)

  surface z = SPHERE ((0,0,0),R1)       { the top hemisphere }

 

boundaries

 

  surface 1 value(u) = 0     { fixed values on outer sphere surfaces }

  surface 4 value(u) = 0

 

  Region 1   { The outer boundary in the base projection }

      layer 1 k=0.1 mesh_spacing=10*thick { force resolution of shell curve }

      layer 2 k=0.1

      layer 3 k=0.1 mesh_spacing=10*thick

      start(R1,0)

      value(u) = 0           { Fixed value on outer vertical sides }

          arc(center=0,0) angle=180

      natural(u)=0 line to close

 

    Limited Region 2   { The inner cylinder shell boundary in the base projection }

      surface 2 value(u) = 1   { fixed values on inner sphere surfaces }

      surface 3 value(u) = 1

      layer 2 void           { empty center }

      start(R2,0)

      arc(center=0,0) angle=180

      nobc(u) line to close

 

monitors

    grid(x,y,z)

    grid(x,z) on y=0

    grid(rad,z) on x=y

    contour(u) on x=0         { YZ plane through diameter }

    contour(u) on y=0         { XZ plane through diameter }

    contour(u) on z=0         { XY plane through diameter }

    contour(u) on x=0.5       { YZ plane off center }

    contour(u) on y=0.5       { XZ plane off center }

 

definitions

    yp = 0.5

    R0 = (R1+R2)/2

    Rin = sqrt((R0-0.1)^2-yp^2)

    Rout = sqrt((R0+0.1)^2-yp^2)

    xin = Rin/sqrt(2)

    xout = Rout/sqrt(2)

 

plots

    grid(x,y,z)

    grid(x,z) on y=0

    grid(x,z) on y=yp

    contour(u) on x=0         as "Temp on YZ plane through diameter"

    contour(u) on y=0         as "Temp on XZ plane through diameter"

    contour(u) on z=0         as "Temp on XY plane through diameter"

    contour(u) on z=0.001         as "Temp on XY plane through diameter"

    contour(u) on x=0.5       as "Temp on YZ plane off center"

    contour(u) on y=0.5       as "Temp on XZ plane off center" report(Rin,Rout,xin,xout)

    contour(magnitude(grad(u))) on y=yp

                              zoom(xin,xin, xout-xin,xout-xin)

                              as "Flux on XZ plane off center"

                               report(yp)

 

end