Parallel multigrid method for adaptive finite elements with application to 3D flow problems [Elektronische Ressource] / vorgelegt von Thomas Richter

Sujets

• Mathematics

Informations

• Publié par: ruprecht-karls-universitat_heidelberg
• Publié le: 01 janvier 2005
• Nombre de lectures: 19
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Inaugural-Dissertation
zur Erlangung der Doktorwürde
der
Naturwissenschaftlich-Mathematischen Gesamtfakultät
der
Ruprecht-Karls-Universität
Heidelberg
vorgelegt von
Diplom-Mathematiker Thomas Richter
aus Marl
Tag der mündlichen Prüfung: 13.07.2005Parallel Multigrid Method for
Adaptive Finite Elements with
Application to 3D Flow Problems
05.05.2005
1. Gutachter: Prof. Dr. Rolf Rannacher
2. Gutachter: Prof. Dr. Peter BastianAbstract
Aim of this work is the examination of numerical methods for the solution of large systems
of PDE’s. The equations under consideration arise from chemically reacting flows.
A focal point is the analysis of a finite element discretization with stabilized finite elements
of degree two. Aspects of error estimation, solution techniques and mesh adaptivity are
discussedwithregard totheNavier-Stokes equations. Usingawellestablished Navier-Stokes
benchmark flow the discussed methods are verified.
To cope with the huge systems arising from reactive flow problems a parallel multigrid
method on locally refined meshes is presented.
Finally, we will perform a simulation of a methane burner in a complex three dimensional
geometry. We will use a detailed reaction mechanism with 39 chemical species.
Zusammenfassung
Gegenstand dieser Arbeit ist die Analyse von numerischen Verfahren zur Lösung von großen
Systemen partieller Differenzialgleichungen. Die betrachteten Gleichungen treten z.B. bei
der Simulation von reaktiven Strömungen auf.
Ein Schwerpunkt ist die Untersuchung einer stabilisierten Finite Elemente Diskretisierung
mit quadratischen Ansatzräumen. Anhand der Navier-Stokes Gleichungen werden Aspekte
wie das Lösen der Gleichungssysteme, Fehlerschätzung und Gitteradaption behandelt. Die
vorgestellten Verfahren werden an einem etablierten Navier-Stokes Benchmark verifiziert.
Bei der Simulation von reaktiven Strömungen vergrößert sich die Anzahl der Lösungskom-
ponenten um die Anzahl an chemischen Substanzen. Die implizit gekoppelte Lösung der
entsprechenden Gleichungen stellt hinsichtlich Rechen- und Zeitaufwand eine enorme An-
forderung an die Computer. Um eine Lösung überhaupt zu ermöglichen wird eine paralleles
Mehrgitterverfahren auf adaptiven Gittern vorgestellt.
Schließlich werden Simulationsrechnungen einer Methanflamme in einem Brenner mit kom-
plexer, dreidimensionaler Geometrie präsentiert. Die chemischen Reaktionen werden mit
einem detailierten Reaktionsmechanismus unter Berücksichtigung von 39 chemischen Sub-
stanzen modelliert.Contents
1. Introduction 1
1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. 3D Navier Stokes Benchmark Configuration . . . . . . . . . . . . . . . . . . . 3
1.3. 3D Methane Burner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Basic notations and finite element spaces 7
2.1. Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. Finite Element Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3. Finite Element Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3. FE discretization for 3D Navier-Stokes 15
3.1. Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2. Residual Based Stabilization Techniques . . . . . . . . . . . . . . . . . . . . . 16
3.3. Local Projection Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4. Stokes Stabilization on Anisotropic Meshes . . . . . . . . . . . . . . . . . . . 18
3.5. Quadratic Adaptive Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 26
3.5.1. Pressure Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5.2. Convection Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5.3. Implementational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.4. Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.5. LPS based on theQ / iso Q element . . . . . . . . . . . . . . . . . . 322 2
4. Error Estimation & Mesh Adaptation 35
4.1. Dual Weighted Residual Method . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2. Error Estimation with Q Elements . . . . . . . . . . . . . . . . . . . . . . . 372
4.3. Mesh Adaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4.1. Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . 47
5. Parallel Adaptive Finite Elements 51
5.1. Isoefficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2. Parallel Finite Elements Discretization . . . . . . . . . . . . . . . . . . . . . . 55
5.2.1. Distributing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.2. Implementation and Parallel Efficiency of the Matrix Vector Product . 58
5.2.3. Distributed Communication . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2.4. Hanging Nodes and Multigrid . . . . . . . . . . . . . . . . . . . . . . . 62
5.3. Parallel Multigrid Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1. Parallel Multigrid Smoother . . . . . . . . . . . . . . . . . . . . . . . . 69
iContents
5.3.2. Convergence Analysis for the Schwarz Iteration . . . . . . . . . . . . . 71
5.3.3. Convergence Analysis of the Multigrid Smoother . . . . . . . . . . . . 74
5.4. Implementational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5. Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6. Reactive Flows 87
6.1. Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2. Simplified Model for Chemically Reacting Flows . . . . . . . . . . . . . . . . 89
6.3. Finite Elements for Reactive Flows . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.1. Stabilization by Local Projections . . . . . . . . . . . . . . . . . . . . 92
6.3.2. Solution Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3.3. Homotopy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3.4. Advanced Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4. Numerical Study of a Methane Burner . . . . . . . . . . . . . . . . . . . . . . 96
6.4.1. 2D Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4.2. Numerical 3D Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A. Modeling of Chemical Reactions 111
B. Reaction Scheme for Methane Combustion 115
ii1. Introduction
1.1. Motivation
This work is devoted to the reliable solution of complex problems described by partial dif-
ferential equations. The problems under consideration originate from various processes of
nature specifically from flows and chemical reactions. Our focus here is the combination of
both. Thinking of chemical reactors or flames in burners, the reacting species – whether
fluid, gaseous, or a mixture of both – flow in some technical devices. In addition to basic
‘cold flow’ the reaction effects the flow by density variations of different species as well as
temperature gradients aroused by the reactions. In order to accurately solve such combined
problems, we have to treat the arising equations coupled simultaneously. Even with the use
of modern parallel computers, the sheer size of the resulting system overstresses available
capacity in terms of memory usage and computational time if we consider three dimensional
problems involving large reaction systems.
Hence, the usual approach for the numerical treatment of reactive flow problems is made
up of a two dimensional reduction of the geometry and a decoupling of the equations. This
decoupling is either done by splitting the equations into a flow and a reaction part and
iterating between the two or by some splitting scheme applied within the solution process.
However, if we like to apply adaptivity with reliable error control or if the problem is sub-
ject to some optimization, we need a coupled handling of the equations. Braack [Bra98]
proposed an adaptive finite element scheme for two dimensional chemically reacting flows.
The problem is treated fully coupled as a set of nonlinear equations. Error control and mesh
adaption is applied following the framework of the dual weighted residual method by Becker
& Rannacher [BR96], [BR01].
This work is dedicated to the extension of the already extensively analyzed adaptive finite
element method to the three dimensional case. Beside enhancing the finite element dis-
cretization, a crucial point is the parallelization of the solvers. Modern parallel computers
work with message passing protocols, in which the data is distributed to separate machines
by passing data packages through a network. Consequently this communication is decidedly
slow in comparison to local memory access. Communication between different machines has
to be limited to a minimum.
The parallelization of finite element methods is not new. Mainly two approaches are used:
one could split the computational domain into several parts and distribute local problems
to different processors, see e.g. Quarteroni & Valli [QV99] for these “domain decomposi-
tio