Alles im Fluss : Simulationstechniken der Strömungsmechanik in der Materials Chain

Dieser Beitrag handelt von numerischen Strömungssimulationen. Heute, 200 Jahre nach ihrer Formulierung, basieren die Simulationen auf Hochleistungscomputern auf den sogenannten Navier-Stokes-Gleichungen.
The numerical simulation of complex problems in fluid dynamics is becoming more and more important, not least due to the recent progresses in high-performance computing. As a consequence thereof, the development of robust and efficient numerical models and solvers is indispensable. Today’s object of research in fluid dynamics focuses on the description of the perfusion of complex structures based on the Navier- Stokes equations. Newtonian as well as non-Newtonian fluids are the object of ongoing research. A relatively new and promising method is based on the variational approach of the mixed least-squares finite element method (LSFEM) combined with efficient iterative solvers for a massive, parallel computer architecture. In the present article, a brief survey on the fundamentals of fluid dynamics is given first. In addition, the challenges of ongoing research are presented and a possible solution strategy in terms of the LSFEM is treated in detail. In the framework of the MERCUR project of Schröder, Turek and Schwarz, possible problem-solving approaches are demonstrated. We explain the leastsquares method and its principles in the context of statistics, as well as its transferability for constructing finite elements in the framework of computational fluid dynamics. Thereby, two different formulations are investigated in terms of their performance and efficiency. In benchmark problems the influence of different interpolation spaces for the different formulations with respect to convergence and precision is examined. Furthermore, a numerical example for non-Newtonian fluids is shown. The inherent a posteriori error estimator within the LSFEM is a vast advantage of the method and can be applied in adaptive mesh generation. The future challenges within the LSFEM are the simulation of geometrically complex structures and their interaction with passing/ surrounding fluid flows, i.e. fluidstructure- interaction (FSI), which arises in manufacturing engineering, for example. Along with the perfusion of (micro)structures, the fluid flow through elastic structures such as blood vessels in combination with stents are conceivable problems to solve. The extremely complex interaction conditions and the nonlinear coupling of both regimes (fluid and solid) are the challenges to meet. In addition, numerical problems including high Reynolds numbers due to high velocities and/or low viscosities demand a special treatment, which is not yet included in LSFEMs. These challenges describe the goals of ongoing research and the way to progress for least-squares finite element methods in combination with efficient, compatible solvers.
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