4 resultados para Numerical error
em ArchiMeD - Elektronische Publikationen der Universität Mainz - Alemanha
Resumo:
The thesis deals with numerical algorithms for fluid-structure interaction problems with application in blood flow modelling. It starts with a short introduction on the mathematical description of incompressible viscous flow with non-Newtonian viscosity and a moving linear viscoelastic structure. The mathematical model consists of the generalized Navier-Stokes equation used for the description of fluid flow and the generalized string model for structure movement. The arbitrary Lagrangian-Eulerian approach is used in order to take into account moving computational domain. A part of the thesis is devoted to the discussion on the non-Newtonian behaviour of shear-thinning fluids, which is in our case blood, and derivation of two non-Newtonian models frequently used in the blood flow modelling. Further we give a brief overview on recent fluid-structure interaction schemes with discussion about the difficulties arising in numerical modelling of blood flow. Our main contribution lies in numerical and experimental study of a new loosely-coupled partitioned scheme called the kinematic splitting fluid-structure interaction algorithm. We present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Here, we assume both, the nonlinearity in convective as well is diffusive term. We analyse the convergence of proposed numerical scheme for a simplified fluid model of the Oseen type. Moreover, we present series of experiments including numerical error analysis, comparison of hemodynamic parameters for the Newtonian and non-Newtonian fluids and comparison of several physiologically relevant computational geometries in terms of wall displacement and wall shear stress. Numerical analysis and extensive experimental study for several standard geometries confirm reliability and accuracy of the proposed kinematic splitting scheme in order to approximate fluid-structure interaction problems.
Resumo:
Allgemein erlaubt adaptive Gitterverfeinerung eine Steigerung der Effizienz numerischer Simulationen ohne dabei die Genauigkeit des Ergebnisses signifikant zu verschlechtern. Es ist jedoch noch nicht erforscht, in welchen Bereichen des Rechengebietes die räumliche Auflösung tatsächlich vergröbert werden kann, ohne die Genauigkeit des Ergebnisses signifikant zu beeinflussen. Diese Frage wird hier für ein konkretes Beispiel von trockener atmosphärischer Konvektion untersucht, nämlich der Simulation von warmen Luftblasen. Zu diesem Zweck wird ein neuartiges numerisches Modell entwickelt, das auf diese spezielle Anwendung ausgerichtet ist. Die kompressiblen Euler-Gleichungen werden mit einer unstetigen Galerkin Methode gelöst. Die Zeitintegration geschieht mit einer semi-implizite Methode und die dynamische Adaptivität verwendet raumfüllende Kurven mit Hilfe der Funktionsbibliothek AMATOS. Das numerische Modell wird validiert mit Hilfe einer Konvergenzstudie und fünf Standard-Testfällen. Eine Methode zum Vergleich der Genauigkeit von Simulationen mit verschiedenen Verfeinerungsgebieten wird eingeführt, die ohne das Vorhandensein einer exakten Lösung auskommt. Im Wesentlichen geschieht dies durch den Vergleich von Eigenschaften der Lösung, die stark von der verwendeten räumlichen Auflösung abhängen. Im Fall einer aufsteigenden Warmluftblase ist der zusätzliche numerische Fehler durch die Verwendung der Adaptivität kleiner als 1% des gesamten numerischen Fehlers, wenn die adaptive Simulation mehr als 50% der Elemente einer uniformen hoch-aufgelösten Simulation verwendet. Entsprechend ist die adaptive Simulation fast doppelt so schnell wie die uniforme Simulation.
Resumo:
Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.
Resumo:
Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.