Finite Element Analysis of Fluid Flows
WASHINGTON UNIV SEATTLE DEPT OF AERONAUTICS AND ASTRONAUTICS
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The finite element method is applied to several simple cases of steady flow of a perfect, incompressible fluid. It is shown that the finite element representation accurately reflects the behavior of the classical flow equations. Finite elements form the basis for a versatile analysis procedure applicable to problems in several different fields. The earliest applications were to problems in structural mechanics. In recent years, nonstructural problems also have been treated by this method. The finite element method represents an approximate procedure for satisfying the problem in terms of its variational formulation. In structural mechanics this is generally accomplished by determining displacement fields based on satisfying the minimum potential energy theorem. Consequently, finite elements furnish a useful alternative scheme for applying the well-known Ritz method. For nonstructural problems, it is essential that the appropriate variational expressions be known beforehand. For the flow problems taken up in this paper, such expressions are well known. The governing matrix equation for the assemblage of elements is based on the properties derived for a single typical element. These properties, in turn, depend on assuming a mathematical form for the primary unknown of the problem and then satisfying the variational principle. For the elasticity problem, the unknowns are the displacements, while for the perfect incompressible fluid, either the velocity potential or the stream function may be used. Of great interest is that structural and nonstructural elements may often be identical in shape and may be represented by similar mathematical expressions. By way of illustration, the problems taken up in this paper were solved using the new ASTRA structural program developed at The Boeing Company. Finally, it should be pointed out that the major difference between the elasticity and fluid flow problems lies in the boundary conditions to be satisfied.
- Numerical Mathematics
- Theoretical Mathematics
- Fluid Mechanics