Sample Problems > Usage > 3D_domains > 3d

{ 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

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

3d_shell

3d_shell