Parametric multi-block grid generation and application to adaptive flow simulations [Elektronische Ressource] / vorgelegt von Philipp Peter Lamby

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2007
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 14 Mo

Extrait

Parametric Multi Block Grid Generation
and Application to Adaptive Flow Simulations
Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften
der Rheinisch Westfal¨ ischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom Mathematiker
Philipp Peter Lamby
aus Bonn
Berichter: Universita¨tsprofessor Dr. rer. nat. W. Dahmen
Universita¨ssor Dr. Ing. J. Ballmann
¨ ¨Tag der mundlichen Prufung: 15.August 2007
Diese Dissertation ist auf den Internetseiten
der Hochschulbibliothek online verfugb¨ ar.Acknowledgments
The present thesis originated from my work at the Institut fur¨ Geometrie und Prak
tische Mathematik of the RWTH Aachen from 2000 2007. During this period I
held a position in the Collaborative Research Center SFB 401 ”Flow Modula
tion and Fluid Structure Interaction at Airplane Wings” funded by the Deutsche
Forschungsgemeinschaft.
For his guidance of my work I want to express my gratitude to Prof. Dr. Wolf
gang Dahmen. I also sincerely thank Prof. Dr. Josef Ballmann for taking on the
task of the second referee of this thesis. I am grateful to both of them for their
lectures that I could attend, their steady support of my studies, many helpful dis
cussions and for giving me the opportunity to work on this project.
I am deeply thankful to Priv. Doz. Dr. Siegfried Mu¨ller for many years of
fruitful cooperation and especially for his thorough proof reading of the first draft
of this thesis. I am also indebted to Dr. Karl Heinz Brakhage who introduced me
to the world of computer aided geometric design and contributed many ideas and
proposals to this work.
Finally I would like to express how much I enjoyed the friendship and the
help of my colleagues at the IGPM and at the Lehr- und Forschungsgebiet fur¨
Mechanik. Special thanks go to Dr. Frank Bramkamp with whom I spent so much
time on the QUADFLOW code of which he has developed the core part (and which
will play a prominent role in the following pages), Frank Knoben for his computer
support and for outwitting stubborn compilers, Dr. Michael Hesse for providing
data and grids during the early stages of this project, and Rodolphe Prignitz for
implementing and testing the code described in Section 8.4.3.Contents
1 Introduction 1
1.1 Grid Generation and Computer Aided Design . . . . . . . . . . . 1
1.2 Quadflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Some Remarks on Notation . . . . . . . . . . . . . . . . . . . . . 6
2 Fluid Structure Interaction 7
2.1 Aircraft Wing Flutter . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Reference Configurations . . . . . . . . . . . . . . . . . . . . . . 12
3 Basic Grid Generation Concepts 15
3.1 Motivation and Overview . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Discretizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Grids, Grid Cells and Grid Connectivity . . . . . . . . . . 18
3.2.2 Grid Properties . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 Nested Hierarchy of Grids . . . . . . . . . . . . . . . . . 21
3.3 Structured Grids and Coordinate Mappings . . . . . . . . . . . . 21
3.4 Multiblock Grids . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.1 Blockstructured Methods . . . . . . . . . . . . . . . . . . 24
3.4.2 Alternative Multiblock Techniques . . . . . . . . . . . . . 26
3.5 Transfinite Interpolation . . . . . . . . . . . . . . . . . . . . . . 26
3.5.1 Construction Principle . . . . . . . . . . . . . . . . . . . 27
3.5.2 2D Linear Blend . . . . . . . . . . . . . . . . . . . . . . 29
3.5.3 Discrete Linear Blend . . . . . . . . . . . . . . . . . . . 29
3.5.4 3D Linear Blend . . . . . . . . . . . . . . . . . . . . . . 30
3.5.5 Coons Patches . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Harmonic Grid Generation . . . . . . . . . . . . . . . . . . . . . 32
3.6.1 Fundamental Theory . . . . . . . . . . . . . . . . . . . . 33
3.6.2 Poisson Systems . . . . . . . . . . . . . . . . . . . . . . 34
3.6.3 Numerical Construction . . . . . . . . . . . . . . . . . . 35
iii CONTENTS
3.7 Stretching Methods . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 B Spline Grid Generation 41
4.1 B Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 B Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Definition and Basics Properties . . . . . . . . . . . . . . 44
4.2.2 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Knot Insertion . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.4 Bezi´ er Curves . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.5 Clamped Knot Vectors . . . . . . . . . . . . . . . . . . . 47
4.2.6 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Tensor Product B Splines . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Interpolation and Approximation . . . . . . . . . . . . . . . . . . 53
4.4.1 Curve interpolation . . . . . . . . . . . . . . . . . . . . . 53
4.4.2 Curve Approximation . . . . . . . . . . . . . . . . . . . 54
4.4.3 Tensor Product Interpolation . . . . . . . . . . . . . . . . 55
4.4.4 Tensor Product Approximation . . . . . . . . . . . . . . . 57
4.5 Curve and Surface Fairing . . . . . . . . . . . . . . . . . . . . . 60
4.5.1 The Necessity of Fairing . . . . . . . . . . . . . . . . . . 60
4.5.2 The Fairing Principle . . . . . . . . . . . . . . . . . . . . 61
4.5.3 Knot Removal - Knot Reinsertion Fairing . . . . . . . . . 62
4.5.4 Local Energy Fairing . . . . . . . . . . . . . . . . . . . . 63
4.5.5 The Fairing Algorithm . . . . . . . . . . . . . . . . . . . 65
4.6 B Spline Transfinite Interpolation . . . . . . . . . . . . . . . . . 65
5 A Multiblock Grid Manager 69
5.1 The Topological Grid Elements . . . . . . . . . . . . . . . . . . . 70
5.1.1 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.3 Vertices Embedded in Edges . . . . . . . . . . . . . . . . 71
5.1.4 Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.5 Vertices Embedded in Faces . . . . . . . . . . . . . . . . 73
5.1.6 Edges Embedded in Faces . . . . . . . . . . . . . . . . . 73
5.1.7 Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.1 The Grid Dependency Graph . . . . . . . . . . . . . . . . 76
5.2.2 General Remarks on the Implementation . . . . . . . . . 76
5.2.3 TVertexData . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.4 TEdgeData . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.5 TFaceData . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.6 TBlockData . . . . . . . . . . . . . . . . . . . . . . . . . 79CONTENTS iii
5.3 Usage of the Topology Manager . . . . . . . . . . . . . . . . . . 80
5.3.1 Grid Construction . . . . . . . . . . . . . . . . . . . . . 80
5.3.2 Grid Deformation . . . . . . . . . . . . . . . . . . . . . . 83
5.3.3 Changing the Topology . . . . . . . . . . . . . . . . . . . 85
5.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.1 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.2 Connectivity Evaluation . . . . . . . . . . . . . . . . . . 87
6 Grid Deformation 89
6.1 Algebraic Block Deformation . . . . . . . . . . . . . . . . . . . . 90
6.1.1 Transfinite Interpolation of the Displacements . . . . . . . 90
6.1.2 Angle Preserving Method . . . . . . . . . . . . . . . . . 91
6.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Deforming the Framework . . . . . . . . . . . . . . . . . . . . . 95
6.2.1 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Application to the Deformation Problem . . . . . . . . . . 98
6.2.3 Deformation of the Framework . . . . . . . . . . . . . . . 99
6.3 Example: High Lift Configuration . . . . . . . . . . . . . . . . . 100
7 Offset Curves and Surfaces 105
7.1 Evolution of Planar Curves . . . . . . . . . . . . . . . . . . . . . 106
7.2 Generation of Planar B Spline Offset Curves . . . . . . . . . . . . 108
7.2.1 A Remark On Grid Generation on the Line . . . . . . . . 113
27.2.2 ForcingC continuity at transition points . . . . . . . . . 115
7.3 Constructing the Transverse Lines . . . . . . . . . . . . . . . . . 116
7.4 Offset Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.4.1 A CAGD Model for an Airplane Wing . . . . . . . . . . 121
7.4.2 Reparameterization of the Wing Tip . . . . . . . . . . . . 122
7.4.3 An Offset Technique for Patched Surfaces . . . . . . . . . 125
8 Quadflow 131
8.1 Model Discretization for Finite Volume Schemes . . . . . . . . . 132
8.1.1 The ALE Form . . . . . . . . . . . . . . . . . . . . . . . 132
8.1.2 Model Discretization . . . . . . . . . . . . . . . . . . . . 133
8.1.3 The Geometric Conservation Laws . . . . . . . . . . . . . 135
8.2 The Quadflow Flow Solver . . . . . . . . . . . . . . . . . . . . . 137
8.2.1 Data Structures . . . . . . . . . . . . . . . . . . . . . . . 13