sci.math.numanalysis
SUNA08, Discretization at Triangles
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It was shown in the sci.math.research article "SUNA, 2D 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 2D, 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 oo 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 ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood