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{ 3D_BRICKS+TIME.PDE
This problem demonstrates the application of FlexPDE to time-dependent
three dimensional heat conduction. An assembly of bricks of differing
conductivities has a gaussian internal heat source, with all faces held
at zero temperature. After a time, the temperature reaches a stable
distribution.
This is the time-dependent analog of example problem 3D_BRICKS.PDE.
}
title 'time-dependent 3D heat conduction'
select
regrid=off { use fixed grid }
ngrid=5 { smaller grid for quicker run }
coordinates
cartesian3
variables
Tp(threshold=0.1) { the temperature variable, with approximate size }
definitions
long = 1
wide = 1
K { thermal conductivity -- values supplied later }
Q = 10*exp(-x^2-y^2-z^2) { thermal source }
tmax = 6 { plot range control }
initial values
Tp = 0.
equations
Tp : div(k*grad(Tp)) + Q = dt(Tp) { the heat equation }
extrusion z = -long,0,long { divide Z into two layers }
boundaries
Surface 1 value(Tp)=0 { fix bottom surface temp }
Surface 3 value(Tp)=0 { fix top surface temp }
Region 1 { define full domain boundary in base plane }
k = bylayer (1.0, 0.1) { bottom and top right brick }
start(-wide,-wide)
value(Tp) = 0 { fix all side temps }
line to (wide,-wide) { walk outer boundary in base plane }
to (wide,wide)
to (-wide,wide)
to close
Region 2 { overlay a second region in left half }
k = bylayer (0.2, 0.4) { bottom and top left brick }
start(-wide,-wide)
line to (0,-wide) { walk left half boundary in base plane }
to (0,wide)
to (-wide,wide)
to close
time 0 to 3 by 0.01 { establish time range and initial timestep }
monitors
for cycle=1
contour(Tp) on z=0 as "XY Temp" range=(0,tmax)
contour(Tp) on x=0 as "YZ Temp" range=(0,tmax)
contour(Tp) on y=0 as "XZ Temp" range=(0,tmax)
elevation(Tp) from (-wide,0,0) to (wide,0,0) as "X-Axis Temp" range=(0,tmax)
elevation(Tp) from (0,-wide,0) to (0,wide,0) as "Y-Axis Temp" range=(0,tmax)
elevation(Tp) from (0,0,-long) to (0,0,long) as "Z-Axis Temp" range=(0,tmax)
plots
for t = endtime
contour(Tp) on z=0 as "XY Temp" range=(0,tmax)
contour(Tp) on x=0 as "YZ Temp" range=(0,tmax)
contour(Tp) on y=0 as "XZ Temp" range=(0,tmax)
histories
history(Tp) at (wide/2,-wide/2,-long/2)
(wide/2,wide/2,-long/2)
(-wide/2,wide/2,-long/2)
(-wide/2,-wide/2,-long/2)
(wide/2,-wide/2,long/2)
(wide/2,wide/2,long/2)
(-wide/2,wide/2,long/2)
(0,0,0) range=(0,tmax)
end