The Finite Element Method for Fluid Dynamics

The Finite Element Method for Fluid Dynamics: The present revision of The Finite Element Method was undertaken shortly before the passing in January 2009 of our close friend and co-author Olgierd C. (Olek) Zienkiewicz. His inspiration and guidance has been greatly missed in the intervening years. However, we hope that the essence of his writings is retained in the new work so that current and future scholars can continue to benefit from his insights and many contributions to the field of computational mechanics. The story of his life and works is summarized in International Journal for Numerical Methods in Engineering, 80, 2009, pp. 1–45.

In fluid dynamics, several difficulties arise. The first is that of dealing with incompressible or almost incompressible situations. These as we already know present special difficulties in formulation even in solids. The second difficulty is introduced by convection, which requires rather specialized treatment and stabilization.

Here, particularly in the field of compressible high-speed gas flow many alternative finite element approaches are possible and often different algorithms for different ranges of flow have been suggested. Although slow creeping flows may well be dealt with by procedures almost identical to those of solid mechanics, the high-speed range of the supersonic and hypersonic kind will require a very particular treatment.

In this text we shall use the so-called characteristic-based split (CBS) introduced several years ago by the authors. It turns out that this algorithm is applicable to all ranges of flow and indeed gives results which are at least equal to those of specialized methods.

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