sci.math.research SUNA18, The Labrujere Problem ============================= In February 1976 Dr. Th.E. Labrujere, at National Aerospace Laboratory NLR, the Netherlands, wrote a Memorandum (WD-76-030) 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 least-squares finite element method is formulated for the two-dimensional, 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 fifth-order [ WOW! ] trial functions for both components of velocity. Using the least-squares 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 fifth-order 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, 2-D and 3-D 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 * E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood