sci.math.research

SUNA19, Least Squares F.E.M.
============================

Incompressible irrotational (ideal) flow of an inviscid fluid is described by:

               du/dx + dv/dy = 0
               dv/dx - du/dy = 0

Here "d" denotes partial differentiation; x,y = coordinates; u,v = velocity
components.

There does _not_ exist kind of "natural" variational principle for the above
differential equations. Conventional Finite Element Methods, however, depend
upon the existence of such principles. There must be _something_ to minimize
(or make "stationary").

Least Squares Finite Element Methods proceed by constructing an alternative
minimum principle: square the equations as-they-are (!), add them together,
integrate the sum of squares over the area of interest, minimize the result.
See also O.C. Zienkiewicz "The Finite Element Method" (1977) chapter 3.14.2.
In our case:

     //
    || [(du/dx + dv/dy)^2 + (du/dy - dv/dx)^2] dx.dy = minimum
   //

Simple as it sounds, but watch out! I spoiled VERY much time on getting this
method to work and, after five (yes!) years of research, I even had to give up.
Least Squares is the most perfidious (? Dutch: "verradelijke") Finite Element
Method which was ever invented.

L.S.FEM doesn't work for linear triangles, iff the method is applied to these
in a straightforward Finite Element manner. This is what *I* wish to call the
Labrujere problem (whether this author likes it or not ;-).
My purpose is to show in the sequel that the Labrujere problem can be "solved"
properly, however: by looking at it in a difference, pardon (-: different way.
It's due to the power of the Unification Principle (Re: SUNA, The Manifesto).
Without unification, there is no hope altogether. Take it or leave it ...

At first, the integral is splitted out over the elements in the mesh:

           //
   Sigma  || [ (du/dx + dv/dy)^2 + (du/dy - dv/dx)^2 ] dx.dy = minimum
     E   //
         E

Numerical integration, according to Zienkiewicz chapter 8.8, is applied:

   Sigma Sigma w.[ (du/dx + dv/dy)^2 + (du/dy - dv/dx)^2 ].J = minimum
     E     p

Here  J  is the Jacobian determinant of the coordinate transformation, which
is assumed to be positive;  w  are positive weighting factors; the expressions
 (du/dx + dv/dy)  and  (du/dy - dv/dx)  are discretized, according to previous
SUNA theory. What follows now is a small step for me, big leap for mankind (-:

   Sigma w.J.[ (du/dx + dv/dy)^2 + (du/dy - dv/dx)^2 ] = minimum
     i

Here (i) is now the _global_ index of an "integrating point" = E,p .

This merely says that summing over elements, together with their integrating
points, is _equivalent_ with summing over _all_ the integrating points in the
_whole_ domain of interest, in one sweep.
Integrating points can be looked upon as being more "basic" kind of elements.
Finite elements with more than one integrating point can be interpreted as a
superposition of "elementary" integrated elements, with only one integrating
point at each of them.

In order for L.S.FEM to work, the minimum required must be a small number,
preferrably approximating zero, as the size of the elements diminishes.
Thus maybe it would be not such a stupid idea to demand that this minimum
is merely _zero from the start_. But then the above "variational integral"
would have been equivalent to the following un-squared system of equations:

  du/dx + dv/dy = 0   and   dv/dx - du/dy = 0   : at each integrating point.

Let's go one more step further. It is realized that each "integrating point" in
the grid in fact does nothing else than contributing one (independent) equation
to a Finite Difference system of equations. Let's therefore at last replace the
notion of "an integrating point" simply by: "an F.D. equation". And that's it !

Conjecture (HdB)
----------------
|| Any workable Least Squares _Finite Element_ Method is entirely _equivalent_
|| with taking the square of a _Finite Difference_ system of equations.

Let's now look upon the Labrujere problem in the following way. In order for
the associated Finite Difference equations to have a solution, the number of
equations must _equal_ the number of unknowns. Otherwise, the system will be
overdetermined & it is doubtful if the Least Squares minimum approaches zero.
A simple count of triangles involved reveals that this is NOT the case here:
the number of elements outweights the number of nodal points by a factor 2.
This means that there are roughly twice as many "unsquared" F.D. equations
as there are unknowns. Now despite of any complicated kind of F.E. argument,
about higher order continuity and such, this surely is ... another problem.

Norrie and DeVries actually solved that problem. They first kept the triangles.
But in order to compensate for this, they had to introduce (_many_) additional
variables. (And they devised _quite_ a sophisticated argument for doing so.)
It is clearly seen now what kind of different approach may be feasable here.
Instead of increasing the number of unknowns, why not for example diminsh the
number of elements and/or the number of independent F.D. equations? Anyway,
our only care seems to be about matching the number of independent variables
and of independent equations. Quite an unsophisticated argument instead ;-).

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