|
<< Click to Display Table of Contents >> spacetime2 |
  
|
|
<< Click to Display Table of Contents >> spacetime2 |
|
{ SPACETIME2.PDE
This example is a modification of SPACETIME1.PDE, showing the solution of
one-dimensional transient heatflow with differing material properties,
cast as a boundary-value problem.
The time variable is represented by Y, and the temperature by u(x,y).
We specify two regions of differing conductivity, KX.
The initial Temperature is given as a truncated parabola along y=0.
We specify reflective boundary conditions in X (natural(u)=0) along
the sides x=0 and x=1.
The value of u is thus assigned everywhere on the boundary except
along the segment y=1, 0<x<1. Along that boundary, we use the
natural boundary condition,
natural(u) = 0,
since this corresponds to the application of no boundary sources.
}
title "1-D Transient Heatflow as a Boundary-Value Problem"
Variables
u { define U as the system variable }
definitions
kx { declare KX as a parameter, but leave the value for later }
Initial values
u = 0 { unimportant, since this problem is masquerading
as a linear boundary-value problem }
equations { define the heatflow equation }
U: dy(u) = dx(kx*dx(u))
boundaries
region 1
kx = 0.1 { conductivity = 0.1 in region 1 }
start(0,0)
value(u)=2.025-10*x^2 { define the temperature at t=0, x<=0.45 }
line to (0.45,0)
value(u) = 0 { force zero temperature for t=0, x>0.45 }
line to (1,0) to (1,1)
natural(u) = 0 { no flux across x=1 boundary }
line to (1,1)
natural(u) = 0 { no sources on t=1 boundary }
line to (0,1)
natural(u) = 0 { no flux across x=0 boundary }
line to close
region 2
kx = 0.01 { low conductivity in region 2 }
start(0.45,0) { lay region 2 over center strip of region 1 }
line to (0.55,0)
to (0.55,1)
to (0.45,1)
to close
monitors
contour(u)
plots
contour(u)
surface(u)
end