sci.math.research
SUNA18, The Labrujere Problem
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In February 1976 Dr. Th.E. Labrujere, at National Aerospace Laboratory NLR,
the Netherlands, wrote a Memorandum (WD76030) which is titled: De "Eindige
Elementen  Kleinste Kwadraten" Methode toegepast op de 2D Incompressibele
Stroming om een Cirkel Cylinder (: the Least Squares Finite Element Method
[L.S.FEM] applied to the 2D incompressible flow around a circular cylinder).
Really: incompressible and irrotational, ideal flow.
It was firmly established in this report that a straightforward application
of the Least Squares Method, using linear triangular Finite Elements, quite
unexpectedly, DOES NOT WORK!
In December 1976 the problem was "solved" by G. de Vries, T.E. Labrujere and
D.H. Norrie, at the mechanical Engineering Department of The University of
Calgary, Alberta, Canada. The result is written down in their Report no.86:
A Least Squares Finite Element Solution for Potential Flow. The abstract of
this report is quoted here without permission:
The leastsquares finite element method is formulated for the twodimensional,
irrotational flow of an incompressible, inviscid fluid. The continuity
requirements on the components of velocity are established and shown to be
more stringent than previously accepted. The development of the solution
procedure is therefore based on fifthorder [ WOW! ] trial functions for both
components of velocity. Using the leastsquares procedure, the flow past a
circular cylinder is calculated and the results shown to be in very close
agreement with the theoretical solution.
End of quotation. It turns out that the above "solution" is of theoretical
interest only, however. The apparent need for fifthorder trial functions
makes this method completely UNWORKABLE in practice. Worse, the method cannot
reasonably be generalized, in order to cope with less idealized situations.
In the end, 2D and 3D Navier Stokes equations at a curvilinear grid need
to be solved. So the starting point must be something _much_ more simple !
Especially the number of unknowns at each nodal point should not exeed the
absolute minimum: two. It is CLAIMED here that an alternative least squares
finite element solution, _having_ such desirable properties, is definitely
possible. As will be shown in subsequent SUNA posters.
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