sci.math.research
SUNA28, Patched Quadrilaterals
==============================
So far, no single argument was found why linear triangles with velocities at
the vertices would not work. Yet we are still stucked with Labrujeres problem.
We can guess what's wrong by assuming an unusual topology for common triangles
which are to be assembled for a bigger mesh, as sketched on my ASCII notepad:
o  o  o For such a mesh of mutally _overlapping_ triangles, all
 \ /  \ /  nodes can be conceived as being of the type like (5) in
 X  X  the SUNA26 quadrilateral patch. This means that every
 / \  / \  nodal point (o) is accompanied with the two equations:
o  o  o
 \ /  \ /  2.u5 = u1 + u2 = u3 + u4
 X  X  2.v5 = v1 + v2 = v3 + v4
 / \  / \ 
o  o  o Due to the assumed topology, this is equivalent with
 \ /  \ /  a well known Finite Difference discretization of the
 X  X  analysis:
 / \  / \  (d/dx)^2 u = 0 ; (d/dy)^2 u = 0
o  o  o (d/dx)^2 v = 0 ; (d/dy)^2 v = 0
The only solution of the analysis is: u = A.x + B.y ; v = C.x + D.y .
Thus only velocity fields which are a LINEAR function of the global coordinates
are possible. This is too restrictive a condition here, of course.
To avoid misunderstanding, the above is NOT applicable to Labrujere's problem,
because _he_ just uses the common Finite Element topology, which is different.
The overlapping triangles suggest, however, that even the common F.E. topology
maybe is too restrictive in this case. Let's see what happens if we succeed in
weakening even the latter. Anyway, we _have_ adopted the requirement that there
must be as many equations as there are unknowns. And Finite Element topology
still gives rise to an overdetermined system of equations. A possible remedy is
simply to leave elements OUT, to create VOIDS in the mesh, thereby lessening in
quite a natural way the number of elements, hence the number of equations.
* * Just for convenience, let's surround each
/  \ /  \ of these quadrilaterals by yet another
/  \ /  \ quadrilateral:
/  \ /  \
*  *  *  *  * o  *  o = geometry
\  / \  /  /  \ 
\  / \  /  /  \ 
\  / \  /  /  \ 
* VOID * *  *  * = velocity
/  \ /  \  \  / 
/  \ /  \  \  / 
/  \ /  \  \  / 
*  *  *  *  * o  *  o
\  / \  /
\  / \  / But wait! This is a well known thing.
\  / \  / It reminds us of the other _working_
* * element that has been developed for this
problem: 10 unknowns, 6 equations ... Hmm.
It all sounds VERY much like "Staggered Quadrilaterals".
The velocity nodes are positioned at the middle of the sides of the geometry
quadrilateral. The central velocity node is the intersection of the diagonals
of the velocity quadrilateral. Elementary geometry reveals that the velocity
quadrilateral is always a paralellogram, and that the central velocity node is
at the (bary)centre of the geometry quadrilateral.
Especially the proof that the number of unknowns equals the number of equations
can be easily taken over from previous theory. But we are not satisfied, yet.
The two finite element matrices from SUNA27 are repeated:
 + o o  o +  o   +   + o o o o 
 o +  o  o o +  u   + +  o o o o  u
 o  + o + o o     + +  o o o o 
  o o + o  + o _____  +   + o o o o _____
 o  + o + o o    o o o o +   + 
 + o o  o +  o   o o o o  + +  
  o o + o  + o  v  o o o o  + +   v
 o +  o  o o +   o o o o +   + 
Cauchy Linear
Next, the following patch of _two_ reduced quadrilaterals is devised:
6 7 Contributions from the elements for u5 sum of:
o  o
  I :  u1 + u5  v2 + v3 : Cauchy
 II  + u1  u2  u3 + u5 : Linear
 1  II : + u5  u7 + v4  v6 : Cauchy
3 o  5  o 4 + u5  u4  u6 + u7 : Linear
 4 
 I  Contributions from the elements for v5 sum of:
 
o  o I : + u2  u3  v1 + v5 : Cauchy
1 2 + v1  v2  v3 + v5 : Linear
II :  u4 + u6 + v5  v7 : Cauchy
+ v5  v4  v6 + v7 : Linear
For u5 : 4.u5  u2  u3  u4  u6  v2 + v3 + v4  v6 = 0
For v5 : 4.v5  v2  v3  v4  v6 + u2  u3  u4 + u6 = 0
 
Laplacelike discretization (3) + (4) = (2) + (6) linearity
Anyway, these patched elements do _not_ seem so bad as reduced quadrilaterals:
there is no exclusive connectivity between even or odd numbers, or between just
one of the velocity components.
The only thing left now is to show that they actually give the _right_ answers
for ideal flow around a circular cylinder. Sigh! Why do there exist NO methods
for proving that a numerical scheme is always RIGHT, instead of proving that it
is not necessarily WRONG? Sigh!
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