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Level set methods for high-order unfitted discontinuous Galerkin schemes

Utz, Thomas (2018)
Level set methods for high-order unfitted discontinuous Galerkin schemes.
Technische Universität Darmstadt
Ph.D. Thesis, Primary publication

Abstract

This work presents three algorithms for the level set modeling of phase boundaries. The application of these algorithms are high-order extended discontinuous Galerkin methods for multiphase flow simulations. The first algorithm is a reinitialization method, which is based on solving an elliptic partial differential equation. The algorithm is high order accurate in global norms. This reinitialization technique can be applied to arbitrary problems by using a first-order solver as preconditioning. The second algorithm is a high-order accurate solver for extending quantities from the interface into the domain. This is especially helpful for using a so called extension velocity for cases, in which the velocity of the interface is not given by a global field. Like the reinitialization algorithm, the method relies on solving an elliptic partial differential equation. Based on the underlying level-set, this problem might be ill- posed. An extension by an artificial viscosity allows stable solutions even for these cases. The third algorithm is a coupling of these two algorithms to an upwind discretiza- tion of the level set transport equation using an implicit time stepping scheme. For sufficiently smooth problems, this coupling gives high order accuracy as well. Last, this coupled scheme is applied to the simulation of a rising bubble using an unfitted discontinuous Galerkin scheme, which shows good agreement with reference solutions from literature.

Item Type: Ph.D. Thesis
Erschienen: 2018
Creators: Utz, Thomas
Type of entry: Primary publication
Title: Level set methods for high-order unfitted discontinuous Galerkin schemes
Language: English
Referees: Oberlack, Prof. Martin ; Schäfer, Prof. Michael
Date: 27 August 2018
Place of Publication: Darmstadt
Refereed: 20 June 2018
URL / URN: https://tuprints.ulb.tu-darmstadt.de/7724
Abstract:

This work presents three algorithms for the level set modeling of phase boundaries. The application of these algorithms are high-order extended discontinuous Galerkin methods for multiphase flow simulations. The first algorithm is a reinitialization method, which is based on solving an elliptic partial differential equation. The algorithm is high order accurate in global norms. This reinitialization technique can be applied to arbitrary problems by using a first-order solver as preconditioning. The second algorithm is a high-order accurate solver for extending quantities from the interface into the domain. This is especially helpful for using a so called extension velocity for cases, in which the velocity of the interface is not given by a global field. Like the reinitialization algorithm, the method relies on solving an elliptic partial differential equation. Based on the underlying level-set, this problem might be ill- posed. An extension by an artificial viscosity allows stable solutions even for these cases. The third algorithm is a coupling of these two algorithms to an upwind discretiza- tion of the level set transport equation using an implicit time stepping scheme. For sufficiently smooth problems, this coupling gives high order accuracy as well. Last, this coupled scheme is applied to the simulation of a rising bubble using an unfitted discontinuous Galerkin scheme, which shows good agreement with reference solutions from literature.

Alternative Abstract:
Alternative abstract Language

Diese Arbeit beschreibt drei Algorithmen für die Modellierung von Phasengrenzen mittels der Level-Set Methode. Die Anwendung dafür ist die Simulation von Mehrphasenströmungen mittels einer discontinuous Galerkin Methode mit hoher Ansatzordnung und Erweiterung für geschnittene Zellen.

Der erste Algorithmus behandelt das sogenannte Reinitialisierungsproblem mittels der Lösung eines elliptischen Problems. Dieser Algorithmus zeigt Konvergenzverhalten hoher Ordnung in globalen Normen. Durch den Einsatz eines Präkonditionierers, basierend auf einem Verfahren erster Ordnung kann die Methode auf beliebige Problemstellungen angewendet werden.

Der zweite Algorithmus behandelt die Ausbreitung von Größen von der Phasengrenze in das Rechengebiet hinein. Diese Fragestellung ist besonders im Fall der sogenannten Extension Velocity relevant, bei der der Geschwindigkeitswert an der Phasengrenze nicht als globales Feld gegeben ist, das im gesamten Rechengebiet definiert ist. Wie der Algorithmus für Reinitialisierung basiert diese Methode auf der Lösung einer elliptischen Differentialgleichung. In Abhängigkeit des Level-Set Felds kann das Problem schlecht gestellt sein. Eine Erweiterung um eine sogenannte künstliche Viskosität stabilisiert den Löser auch für diese Fälle.

Der Dritte Algorithmus koppelt diese beiden Algorithmen mit einem Upwind-Fluss für die Level Set Transportgleichung unter Verwendung einer impliziten Zeitdiskretisierung. Für glatte Probleme zeigt dieser Algorithmus ebenfalls eine hohe Fehlerordnung. Abschließend wird dieses gekoppelte Schema auf die Simulation einer aufsteigenden Blase mittels einer nicht randangepassten discontinuous Galerkin Methode angewendet. Die Ergebnisse zeigen gute Übereinstimmung mit Referenzlösungen aus der Literatur.

German
Uncontrolled Keywords: BoSSS
URN: urn:nbn:de:tuda-tuprints-77240
Classification DDC: 500 Science and mathematics > 510 Mathematics
600 Technology, medicine, applied sciences > 620 Engineering and machine engineering
Divisions: 16 Department of Mechanical Engineering
16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
16 Department of Mechanical Engineering > Fluid Dynamics (fdy) > Mehrphasenströmung
16 Department of Mechanical Engineering > Fluid Dynamics (fdy) > Numerische Strömungssimulation
Date Deposited: 16 Sep 2018 19:55
Last Modified: 20 Nov 2018 13:14
PPN:
Referees: Oberlack, Prof. Martin ; Schäfer, Prof. Michael
Refereed / Verteidigung / mdl. Prüfung: 20 June 2018
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