sci.math.research

SUNA20, Trace Weighting
=======================

Or "SpoorWegen" (pun: "RailRoads" in Dutch :-) Ugh, perhaps weight tracing as
opposed to ray tracing ...

The Least Squares Finite Element "variational" integral can be modified a bit
as follows, according to Zienkiewicz chapter 3.14.2 equation (3.168):

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

Here  A  and  B  are "positive valued functions or constants", which may be
chosen in a convenient way. Quoting without permission from Zienkiewicz:
"Once again this weighting function could be chosen as to ensure a constant
ratio of terms contributed by various elements - although this has not yet
been put into practice." Well, _somebody_ has to be the first, eh?

The discretization, after numerical integration, now becomes:

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

Where (i) denotes the index of a non-squared F.D. scheme: see previous poster.

It is remarked here that for example the area of the elements is not relevant
anymore: it is "absorbed" into the constants A amd B, which are arbitrary. So
the constants  w  and  J  can be left out; appropriate weighting factors may
be chosen differently instead. Putting the idea of Zienkiewicz into practice,
this indeed may be used for optimizing the condition of the system's matrix.
Write as follows:

  Sigma [ w1.(J.du/dx + J.dv/dy)^2 + w2.(J.du/dy - J.dv/dx)^2 ] = minimum
    i

The two discretized Cauchy equations will correspond with two linear algebraic
F.D. equations, with six (3 x 2) unknown velocities  Wj  per element:

   (J.du/dx + J.dv/dy)  -->   Sigma  Dj.Wj
                                j

   (J.dv/dx - J.du/dy)  -->   Sigma  Oj.Wj
                                j

Now divide each F.D. scheme by its "length". Define:

   wD = SQRT( Sigma Dk^2 )   respectively   wO = SQRT( Sigma Ok^2 )
                k                                        k

The discretization then becomes:

    Sigma [ (Sigma Dj.Wj)^2 /wD^2 + (Sigma Oj.Wj)^2 /wO^2 ] = minimum
      i        j                       j

The minimum is obtainted by differentiating to the unknowns  Wj . This results
for each of the F.D. schemes in, say a 6 X 6 Finite Element matrix of the form:

 | F1.F1   F1.F2   F1.F3  ... |     Where  F  must be replaced by  D  or  O .
 | F2.F1   F2.F2   F2.F3  ... |     The symmetry of these matrices is obvious,
 | F3.F1   F3.F2   F3.F3  ... |     as well as their positive definiteness.
 |  ...     ...     ...   ... |

The _trace_ of such an element matrix is equal to  Sigma Fi^2  . Therefore
trace weighting can also be accomplished by dividing all coefficients of an
element matrix by the _trace_ of this matrix. Hence the name.

Little theorem: the trace of the whole system's matrix equals the number of
weighted elements involved. The proof of this is left as an exercise.

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