Communicated by H. Altenbach and C. Wieners
A discrete-mechanical approach for computation of three-dimensional flows†
Version of Record online: 2 JUL 2013
Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume 93, Issue 12, pages 868–881, December 2013
How to Cite
Abali, B.E., Müller, W.H. and Georgievskii, D.V. (2013), A discrete-mechanical approach for computation of three-dimensional flows. Z. angew. Math. Mech., 93: 868–881. doi: 10.1002/zamm.201300080
- Issue online: 2 DEC 2013
- Version of Record online: 2 JUL 2013
- Manuscript Accepted: 24 MAY 2013
- Manuscript Received: 2 APR 2013
- Variational formulation;
- computational fluid dynamics;
- balance equations.
Dynamical processes modeled with a suitable set of differential equations can be approximately computed with finite element method. However, especially in fluid dynamics, numerical instabilities may occur. In order to circumvent such problems many techniques have been realized over the last four decades. Often the lack of stability is linked to numerical insufficiency, therefore the solution is sought by tuning functional spaces, cf.  or by adding stabilizing terms, cf. [18, 19, 41]. Stabilizing terms are needed and helpful, however, they involve coefficients to be found depending on the underlying problem. The purpose of this work is to propose another approach based on first principles producing similar terms so that the numerical approximation converges without using any other parameter but the material constants, which are measurable. The approach eliminates stability problems, first, by setting the primary variables in a fundamental way and, second, by using the balance equations in their global form in a discrete manner instead of a continuous one. Some applications at the end emphasize the usefulness of the lengthy calculations, which we develop in a straightforward and consistent way.