sci.math.num-analysis

SUNA08, Discretization at Triangles
===================================

It was shown in the sci.math.research article "SUNA, 2-D Skew Upwind F.E.D."
how the equation for pure steady uniform convection in two dimensions can be
discretized for arbitrary curvilinear coordinates, applying a transformation
to the velocity components of the corresponding standard F.D. upwind scheme.

The analytical formulation reads as follows:

    u.dT/dx + v.dT/dy = 0                       ("d" = partial derivatives)

Where: (x,y) = global coordinates, (u,v) = velocity field, T = temperature.
At hand of the above equation, we want to do essentially the same thing, namely
discretize it for arbitrary curvilinear coordinates. But we will do it now in
a different manner.

As a first step, the terms  dT/d(x,y)  will be handled. In order to accomplish
their discretization, transformation to isoparametric local coordinates  (h,k)
at a Finite Element, being a general F.E. technique, seems to be advantageous.
Derivatives to the global coordinates must then be expressed in derivatives to
the local coordinates. Which is done as follows:

     dT/dh = dx/dh.dT/dx + dy/dh.dT/dy
     dT/dk = dx/dk.dT/dx + dy/dk.dT/dy

Solving for  dT/d(x,y)  by Cramer's rule, we find the general formulas:
                                                      ^^^^^^^
     dT/dx = (dy/dk.dT/dh - dy/dh.dT/dk)/J
     dT/dy = (dx/dh.dT/dk - dx/dk.dT/dh)/J

     where:  J = dx/dh.dy/dk - dx/dk.dy/dh

Let's see how such Finite Element discretization works out for the simplest
element in 2-D, a linear triangle:
                                               3 o
   x = x1 + (x2 - x1).h + (x3 - x1).k            | \
   y = y1 + (y2 - y1).h + (y3 - y1).k            |   \
   T = T1 + (T2 - T1).h + (T3 - T1).k            |     \
                                               1 o-------o 2
By differentiation and substitution:

J.dT/dx = (y3 - y1).(T2 - T1) - (y2 - y1).(T3 - T1)
J.dT/dy = (x2 - x1).(T3 - T1) - (x3 - x1).(T2 - T1)

Where:   J = (x2 - x1).(y3 - y1) - (x3 - x1).(y2 - y1)

Consequently:   J.[ u.dT/dx + v.dT/dy ] =

                [ + (y3 - y1).u - (x3 - x1).v ]/J . (T2 - T1) +
                [ - (y2 - y1).u + (x2 - x1).v ]/J . (T3 - T1) = 0

This is exactly the same expression as (5) in the abovementioned SUNA article,
apart from renumbering issues. Well, ... not entirely: information about (F.D.)
upwinding is _no_ longer available in the F.E. formulation.

To be continued ...
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* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
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