sci.math.research

SUNA06, 3-D Skew Upwind F.E.D.
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This article is part of the "Series on Unified Numerical Approximations".


          SKEW UPWIND FINITE ELEMENT DIFFERENCES (SUFED)

      FOR PURE STEADY UNIFORM CONVECTION IN THREE DIMENSIONS

                      PART I : GENERAL THEORY


Steady pure 3-D convection of heat is governed by the following equation:

    U.dT/dx + V.dT/dy + W.dT/dz = 0                                     (1)

Here (x,y,z) = coordinates; (U,V,W) = flow velocity; T = temperature.


1. Standard F.D. scheme

It is assumed for the moment that the coordinate system applies to a grid which
is typically Finite Difference: equidistant and regular (and unity).
Consider a piece of this grid around a mesh point P (sorry: NO drawing).
Six "points of the compass" are denoted: South, North, East, West, Top, Bottom.
Let there be a wind, blowing from North-East-Top. With other words, there is
pure convective heat transfer in the octant PENT. Then according to the law of
Upwind Differencing, the following equation holds:

    U.(Tp - Te) + V.(Tp - Tn) + W.(Tp - Tt) = 0                         (2)

    where:  U <= 0  ,  V <= 0  ,  W  <= 0 .

This scheme is generally accepted as a proper discretization for its analytical
idealization: equation (1).

However, there are no seven, but only four nodal values of  T  involved in this
upwind scheme. Why then employ a seven point star, if four nodes are sufficient?
Switching to the Finite Element viewpoint, we can abstract herefrom a linear
tetrahedron, which is at the same time the simplest 3-D element we can think of.
Instead of the F.D. compass-nomenclature, a F.E. local numbering should then
be adopted:  p = 0 , e = 1 , n = 2 , t = 3 . And the scheme (2) reads:

   U.(T0 - T1) + V.(T0 - T2) + W.(T0 - T3 ) = 0
                                                                        (3)
   where  U <= 0  ,  V <= 0  ,  W <= 0 .


2. Standard F.E. transformation

A unit tetrahedron can be transformed, according to standard F.E. techniques.
Then the base-vectors are transformed into:

 (1,0,0)  ->  (x1 - x0 , y1 - y0 , z1 - z0)
 (0,1,0)  ->  (x2 - x0 , y2 - y0 , z2 - z0)
 (0,0,1)  ->  (x3 - x0 , y3 - y0 , z3 - z0)

Which in turn induces a transformation of the velocities:

   u = (x1 - x0).U + (x2 - x0).V + (x3 - x0).W
   v = (y1 - y0).U + (y2 - y0).V + (y3 - y0).W                         (4)
   w = (z1 - z0).U + (z2 - z0).V + (z3 - z0).W

The inverse of this transformation is the one that's needed. Substitution of
the inverse transformation into (3) already yields the general SUFED scheme
in three dimensions, for pure steady (uniform) convection.

In fact, the the most important part of the 3-D theory is completed herewith:
it is known now how to perform upwind differencing on an arbitrary curvilinear
(Finite Element) mesh.

It is assumed in the next article that the velocity field does not vary across
an element, hence that it is uniform. This is one of the restrictions that has
to be removed later.

To be continued ...
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* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
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