{
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
select aspect=8 { allow long thin cells in joint between hemispheres }
definitions
k = 10 { conductivity }
heat =6*k { internal heat source }
rad2 = x^2+y^2
rad=sqrt(rad2)
R1 = 1
thick = 0.01
R2 = R1-thick
zshell = sqrt(max(R2^2-rad2, 0))
equations
div(K*grad(u)) + heat = 0
extrusion
surface z = -sqrt(R1^2-rad2) { the bottom hemisphere }
surface z = -zshell
surface z = zshell
surface z = sqrt(R1^2-rad2) { 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 }
plots
grid(x,y,z)
grid(x,z) on y=0
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 z=0.001 { XY plane through diameter }
contour(u) on x=0.5 { YZ plane off center }
contour(u) on y=0.5 { XZ plane off center }
contour(magnitude(grad(u))) on y=0.5 { XZ plane off center }
end