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Thread: steady state solution and element number

  1. #1
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    steady state solution and element number

    dear dr.sewell,
    I am using my pc version 0f pde2d to solve a simple steady state problem of heat 2d diffusion where conductivity is function of temperature and on the top of the layer there is a very thin layer with a very low conductivity.
    The problem is that using "adapt" and a growing number of elements I never found the same solution being the temperature field always increasing. May you help me? The pde2d
    file is already sent to you.
    thanks a lot

    costanzo federico

  2. #2
    Constanzo,
    Your nonlinear conductivity coefficient has a discontinuity at y=aH-10,
    I added an extra grid line:
    YGRID(NYGRID/2) = aH-10
    and now it seems to work fine (also increased NYGRID to 21, but that's not
    critical).

    The solution has a discontinuity in its derivatives along this line, so it is
    critical that no initial triangles, and thus no final triangles, straddle this
    interface. Remember that the approximate solution is very smooth
    (polynomial) everywhere except at the triangle boundaries, where it also
    has discontinuous first derivatives, so you get a MUCH better approximation
    of you match the discontinuity (in derivatives) of the approximate solution
    and the discontinuity of the true solution. You can imagine that it is hard
    to accurately approximate a solution with a discontinuous derivatives by
    a smooth polynomial.

    In fact it is so critical to avoid triangles which straddle such interfaces, that
    if you have a curved interface, PDE2D will follow the curved interface when
    refining the triangulation, even using isoparametric triangles (triangles with
    curved edges) along the discontinuity.

    With the new grid line, the solution is almost unchanged when you rerun
    with a different adaptive triangulation. This accurate solution has a temp
    of about 450 at the bottom, your previous runs produced solutions which
    changed each time you ran with a better adaptive triangulation, and
    appeared to be approaching this value from below, but very slowly.
    Even with an adaptive triangulation, the fact that triangles straddled the
    interface was a serious problem.

    Sincerely,

    Granville Sewell

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