3d_shells

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3d_shells

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{ 3D_SHELLS.PDE

 

 This problem demonstrates the construction

 of multiple nested spherical shells.

 We solve a heatflow equation with fixed

 temperatures on inner and outer

 shell surfaces.

}

 

title 'Nested 3D Shells'

 

coordinates

   cartesian3

 

variables

   u

 

definitions

   k = 10            

   heat =6*k        

   rad=sqrt(x^2+y^2)

   R1 = 1

   thick = 0.1  

   R2 = R1-thick

   R3 = R2-thick

   R4 = R3-thick

   R5 = R4-thick

 

equations

   U: div(K*grad(u)) + heat   = 0

 

 

extrusion

  surface 'SB1' z =  -SPHERE ((0,0,0),R1)     { the bottom hemisphere }

layer 'LB1'

  surface 'SB2' z =  -SPHERE ((0,0,0),R2)

layer 'LB2'

  surface 'SB3' z =  -SPHERE ((0,0,0),R3)

layer 'LB3'

  surface 'SB4' z =  -SPHERE ((0,0,0),R4)

layer 'LB4'

  surface 'SB5' z =  -SPHERE ((0,0,0),R5)

layer 'LB5'

  surface 'ST5' z = SPHERE ((0,0,0),R5)

layer 'LT4'

  surface 'ST4' z = SPHERE ((0,0,0),R4)

layer 'LT3'

  surface 'ST3' z = SPHERE ((0,0,0),R3)

layer 'LT2'

  surface 'ST2' z = SPHERE ((0,0,0),R2)

layer 'LT1'

  surface 'ST1' z = SPHERE ((0,0,0),R1)       { the top hemisphere }

 

boundaries

 

  surface 'SB1' value(u) = 0     { fixed values on outer sphere surfaces }

  surface 'ST1' value(u) = 0

 

  Region 1    

      layer 'LB1' k=1

      layer 'LT1' k=1

        start(R1,0)

      value(u) = 0            

          arc(center=0,0) angle=180

      natural(u)=0 line to close

 

    Limited Region 2    

      layer 'LB2' k=2

      layer 'LT2'  k=2

        ! include the region in all layers that must merge out:

        layer 'LB3' layer 'LB4' layer 'LB5' layer 'LT4' layer 'LT3'

      start(R2,0)

      arc(center=0,0) angle=180

      nobc(u) line to close

 

    Limited Region 3    

      layer 'LB3'  k=3

      layer 'LT3'  k=3

        ! include the region in all layers that must merge out:

        layer 'LB4' layer 'LB5' layer 'LT4'

      start(R3,0)

      arc(center=0,0) angle=180

      nobc(u) line to close

 

    Limited Region 4    

      layer 'LB4'  k=4

      layer 'LT4'  k=4

        ! include the region in all layers that must merge out:

        layer 'LB5'  

      start(R4,0)

      arc(center=0,0) angle=180

      nobc(u) line to close

 

    Limited Region 5    

      surface 'SB5' value(u) = 1   { fixed values on inner sphere surfaces }

      surface 'ST5' value(u) = 1

      layer 'LB5' void           { empty center }

      start(R5,0)

      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

    rp = sqrt(R2^2-yp^2)

    xp = rp/sqrt(2+thick)

plots

    grid(x,y,z)

    grid(x,z) on y=0

    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"

    contour(magnitude(grad(u))) on y=0.5 as "Flux on XZ plane off center"

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

                              zoom(xp,xp, thick*sqrt(2+thick),thick*sqrt(2+thick))      

                              as "Flux on XZ plane off center"

 

end