sci.math.research

SUNA22, Staggered Triangles
===========================

A means for discretization of the ideal flow equations (the Labrujere problem)
was already suggested in "SUNA, Trace Weighting". Encountered in that poster
are expressions of the form:

   J.du/dx + J.dv/dy       ;      J.du/dy - J.dv/dx      ;      J = Jacobian

Integrating the ideal flow equations at a linear triangle precisely result in
these expressions, because the integrands are constants.
The other way around is converting them to line-integrals:       3
                                                                 o
   //                         /               /                /   \
  || (du/dx + dv/dy) dx.dy = O u.dy - v.dx = O dF          2 *       * 1
 //                    J    /               /              /           \
   //                         /               /          o ----- * ----- o
  || (dv/dx - du/dy) dx.dy = O u.dx + v.dy = O dP        1       3       2
 //                    J    /               /

Which reminds us of the SUNA poster "Stream Function & Flow Potential".
Here the stream function  F  and the flow potential  P  are concentrated in
nodes marked  with (o), while the flow velocities (u,v) are concentrated along
sides which are marked with (*). Numbering of the entities in the triangle is
in accordance with: same numbered side opposite to same numbered vertex (!).
The line-integrals are easily worked out:

     /
    O dF = (F2 - F1) + (F3 - F2) + (F1 - F3)
   /
         = (y2 - y1).u3 - (x2 - x1).v3             : the discretization for
         + (y3 - y2).u1 - (x3 - x2).v1
         + (y1 - y3).u2 - (x1 - x3).v2 = 0           du/dx + dv/dy = 0

     /
    O dP = (P2 - P1) + (P3 - P2) + (P1 - P3)
   /
         = (x2 - x1).u3 + (y2 - y1).v3             : the discretization for
         + (x3 - x2).u1 + (y3 - y2).v1
         + (x1 - x3).u2 + (y1 - y3).v2 = 0           dv/dx - dy/dy = 0

Here we are! Simple & straightforward. At first sight, anyway.

A small revolution has taken place, however. The above method could _only_ be
conceived _because_ we abandoned every adherence to Labrujere's finite element
method. The velocities are NO longer concentrated at the vertices of triangles,
to mention a detail. The idea of F.E. integration has no sense anymore in this
context, because we _have_ already evaluated F.D. line-integrals. What has been
left of the infamous Least Squares Finite Element Method is only the idea that
the whole system of F.D. equations must be squared and summed up. And even this
last procedure is questionable, according to many of us (?). Not to me, yet.

But wait! We almost forgot a minor detail: the number of equations must be made
equal to the number of unknowns. Our non-sophisticated method, remember? ;-)

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
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood