Spacetime2

{ SPACETIME2.PDE }

{ ***********************************************************

   This example is a modification of SPACETIME1, 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 }

    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