Iterative solvers in implicit time integration for compressible flows [Elektronische Ressource] / vorgelegt von Bernhard Pollul
200
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English
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2008
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200
pages
English
Ebook
2008
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Publié le
01 janvier 2008
Nombre de lectures
18
Langue
English
Poids de l'ouvrage
12 Mo
Publié le
01 janvier 2008
Nombre de lectures
18
Langue
English
Poids de l'ouvrage
12 Mo
Iterative Solvers in Implicit Time
Integration for Compressible Flows
Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften
der RWTH Aachen University zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker
Bernhard Pollul
aus Bonn
Berichter: Universit¨atsprofessor Dr. Arnold Reusken
Universit¨ atsprofessor Dr. Marek Behr
Tag der mundlic¨ hen Prufung:¨ 05. November 2008
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨Contents
1 Introduction 1
1.1 Motivation.................................... 1
1.2 The Adaptive Finite Volume Solver QUADFLOW ..... 2
1.3 Outline of the Main Results ...... 3
1.4 Overview of the Contents................. 7
2 Description of Motion for Inviscid Compressible Gas 11
2.1 The Euler Equations .............................. 1
2.2 Domains, Initial and Boundary Conditions ......... 13
2.3 First Example .............. 14
3 The QUADFLOW Solver 17
3.1 Features of QUADFLOW ........................... 17
3.2 Grid Generation............. 20
3.2.1 Nested Grid Hierarchy ..... 20
3.2.2 Parametric Meshes ................ 21
3.2.3 B-Spline Representation ............... 22
3.2.4 Multi-block Concept ...... 23
3.3 Flow Solver ........................ 23
3.3.1 Finite Volume Discretization.. 24
3.3.2 Upwind Methods ................... 25
3.3.3 Time Discretization................ 26
3.3.3.1 Implicit Time Integration Schemes .... 26
3.3.3.2 Selection of the Time Step Size ...... 28
3.3.4 Newton-Krylov Methods ........................ 29
3.4 Adaptation ............... 30
3.4.1 Local Multiscale Transformation 32
3.4.2 Thresholding ................... 33
3.4.3 Prediction and Grading ............... 34
3.4.4 Grid Adaptation ........ 35CONTENTS
3.4.5 Local Inverse Multiscale Transformation ............... 35
3.4.6 Error Analysis..................... 35
3.4.7 Nested Iteration......... 37
3.5 Remarks on Implementation ............... 37
3.5.1 Data Structures 37
3.5.2 Automatic Differentiation (AD) ........... 38
3.5.2.1 Implementation in QUADFLOW .............. 38
3.5.2.2 The Forward Mode of AD ......... 39
3.5.2.3 Matrix-free Computation of a Matrix-Vector Product ... 42
3.5.2.4 Available AD Tools...................... 43
3.5.3 Parallelization.......... 43
4 Introduction to Iterative Methods 47
4.1 Standard Iterative Methods .......................... 47
4.1.1 Consistency and Convergence . 47
4.1.2 Linear Iterative Methods .... 48
4.1.3 Jacobi and Gauss-Seidel......... 48
4.1.4 Convergence Results ................. 50
4.1.5 Arithmetic Costs ........ 51
4.2 Krylov Subspace Methods ................ 52
4.2.1 Method of Conjugate Gradients (CG)........ 52
4.2.2 Stabilized Bi-C Gradient Method (BiCGSTAB) ...... 5
4.3 Preconditioning ................................. 57
4.3.1 The Idea of Preconditioning .. 57
4.3.2 Preconditioning Krylov Methods .......... 58
4.3.3 Jacobi and Gauss-Seidel ............. 60
4.3.4 Incomplete Lower- Upper- Decomposition (ILU) .. 60
4.3.5 Sparse Approximate Inverse (SPAI) ......... 62
4.3.6 Other Preconditioners ......................... 62
5 Test Problems 65
5.1 Test Problem 1: Stationary Flow on Rectangular Domains ......... 65
5.1.1 Test Case 1A: Stationary 2D Euler with Constant Solution ..... 65
5.1.2 Test Case 1B: 2D Euler with Shock Reflection ...... 6
5.2 Test problem 2: Stationary Flow around NACA0012 Airfoil......... 67
5.3 Test problem 3: Flow around BAC 3-11/RES/30/21 Airfoil .. 68
5.4 Further test problems..................... 69
5.5 Numerical Results..................... 69CONTENTS
6 Point-Block Preconditioners 75
6.1 Methods ..................................... 75
6.1.1 Point-Block-Gauss-Seidel Method .......... 76
6.1.2 Point-Block-ILU(0) Method .. 76
6.1.3 Point-Block Sparse Approximate Inverse ............... 77
6.2 Numerical Experiments.................... 78
6.2.1 Arithmetic Costs ........ 79
6.2.2 Test Case 1A: Stationary 2D Euler with Constant Solution ..... 80
6.2.3 Test Case 1B: 2D Euler with Shock Reflection ...... 81
6.2.4 Test Problem 2: Stationary Flow around NACA0012 Airfoil .... 82
6.2.5 Test Problem 2 on Uniformly Refined Grids ............. 84
6.2.6 Test Problem 3: Stationary Flow around BAC 3-11 Airfoil ..... 86
6.3 Concluding Remarks ..................... 8
7 Renumbering Techniques 89
7.1 Methods .......................... 90
7.1.1 Construction of Weighted Directed Matrix GraphG(A)....... 90
ˆ7.1.2 Cons of Reduced Matrix GraphG ..... 91
ˆ7.1.3 Downwind Numbering based on (V,E) (Bey and Wittum) ..... 92
ˆ7.1.4 Down- and Upwind Numbering based on (V,E) (Hackbusch) .... 93
ˆ7.1.5 Weighted Reduced Graph Numbering based on (V,E,ω )...... 94
ˆ
|E
7.2 Numerical Experiments............................. 96
7.2.1 Test Problem 2: Stationary Flow around NACA0012 Airfoil .... 98
7.2.2 Test problem 3: Flow BAC 3-11 Airfoil .....103
7.2.3 Test Problem 4: Stationary Flow in an Oblique 3D-Channel ....107
7.3 Concluding Remarks ..............................108
8TimeIntegration 109
8.1 CFL Evolution Strategies ................10
8.1.1 Exponential Progression (EXP) ...........11
8.1.2 Switched Evolution Relaxation (SER)........111
8.1.3 Residual Difference Method (RDM) ..................12
8.2 Numerical Experiments.........13
8.2.1 Test Problem 2: Parameter Study on the CFL Control Parameters . 113
8.2.2 Test Problem 2: Results Mimicking a Non-Adaptive Scheme ....17
8.2.3 Test Problem 3: Parameter Study on the CFL Control Parameters . 120
8.3 Locally Optimal CFL Numbers ........................121
8.4 Sensitivity Analysis ...........126
8.5 Concluding Remarks ..........128CONTENTS
9 Matrix-free Methods for Second Order Jacobians 129
9.1 Computation of the Jacobian-Vector Product ................130
9.2 Implementation in QUADFLOW ..............131
9.3 Numerical Experiments.........132
9.3.1 Test Problem 2: Analysis of Newton’s Method ............133
9.3.2 Test Problem 2: The Impact of the CFL Number .137
9.3.3 Test Problems 2 and 3: Acceleration of Time Integration ......139
9.3.4 Test Problem 3: Discussion on the Preconditioner ..........142
9.3.5 Test Problem 5: Oscillating NACA00012 Airfoil ..14
9.3.6 Test Problem 6: 3D Swept Wing ..........147
9.3.7 Test Problem 7: Laminar Flow over a Flat Plate...........149
9.4 Concluding Remarks .....................152
10 Conclusions 153
Outlook .............................15
Acknowledgment 157
Dedication............................157
Picture Credits ................158
Publications within this Research ......158
Directories 159
List of Figures.....................................159
List of Tables161
Nomenclature 163
Bibliography 169
Index 187Chapter 1
Introduction
1.1 Motivation
In times of a more and more globalized world, the demand of new aircrafts is constantly
growing. Air traffic is currently increasing about 5% annually. The environmental impact
of aviation is playing a significant role in the actual discussion of the greenhouse effect.
Moreover, rising prices for kerosene and disturbances due to the noise emission are fre-
quently discussed. In order to limit the arising problems and accommodate the growth
forecasts, the market demands in particular airliners with higher capacities, less fuel con-
sumption, less noise production, and a faster decrease of the wake. For the designing of
new aircrafts, the use of computer simulations is gaining more importance from one aircraft
design to the next. More and more realistic models have become tractable both due to
the development of more efficient numerical methods and due to the increasing computer
power. While the state-of-the-art software packages cannot substitute years of simulations
in wind channels, designing a new aircraft completely in a digital environment will be pos-
sible in near future. This approach will fasten and cheapen the process of designing. Thus,
computational fluid dynamics (CFD) that was started in the 1960’s is still an up-to-date
research topic.
This thesis is part of the research within the Collaborative Research Center SFB 401
[181] “Modulation of flow and fluid-structure interaction at airplane wings” being concerned
with fundamental problems of high capacity aircrafts in transonic conditions [10, 11]. One
key issue of the SFB 401 is the development of a new adaptive finite volume flow solver
called QUADFLOW [35, 37, 38, 40, 41]. This thesis deals with iterative methods that arise
in most CFD simulations. These methods have been implemented in QUADFLOW. The
aim of the research is to improve the efficiency, the robustness, and the usability of the
simulations.
1CHAPTER 1. INTRODUCTION
1.2 The Adaptive Finite Volume Solver QUADFLOW
The results presented in this thesis are an outcome of a research project that is part of
the development of the QUADFLOW package [35, 37, 38, 40, 41]. This adaptive multi-
scale finite volume solver is designed for stationary and non-stationary compressible flow
computations.
Figure 1.1: The logo of the Collaborative Research Center SFB 401
In a realistic CFD-simulation an extremely high resolved solution is needed at least
i
