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ALE-type and fixed grid fluid-structure interaction involving the p-version of the finite element method [Elektronische Ressource] / Stefan Kollmannsberger

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Technische Universität MünchenLehrstuhl Computation in EngineeringALE-type and fixed grid fluid-structure interaction involving thep-version of the Finite Element MethodStefan KollmannsbergerVollständiger Abdruck der von der Fakultät für Bauingenieur- und Vermessungswesen derTechnischen Universität München zur Erlangung des akademischen Grades einesDoktor-Ingenieursgenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr.-Ing. K.-U. BletzingerPrüfer der Dissertation:1. Univ.-Prof. Dr.rer.nat. E. Rank2. Univ.-Prof. Dr.-Ing. habil. M. Krafczyk,Technische Universität Carolo-Wilhelmina zu BraunschweigDie Dissertation wurde am 24.09.2009 bei der Technischen Universität München eingereichtunddurchdieFakultätfürBauingenieur-undVermessungswesen am15.02.2010angenommen.para las pajaritasiiiAbstractThis treatise addresses the efficient, numerical simulation of the interaction between fluidsand structures. The discretization of the structure is either based on high-order hexahe-dral or quadrilateral elements (p-version). Both types of elements allow for an independentchoice of the polynomial degrees for the different local directions as well as for the differentcomponents of the cartesian displacement vectors, which provides scope for strictly two- orthree-dimensional structural discretizations. Three different approaches for the fluid are re-garded: the Finite Volume Method, the Spectral Element Method and the Lattice BoltzmannMethod.

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Technische Universität München
Lehrstuhl Computation in Engineering
ALE-type and fixed grid fluid-structure interaction involving the
p-version of the Finite Element Method
Stefan Kollmannsberger
Vollständiger Abdruck der von der Fakultät für Bauingenieur- und Vermessungswesen der
Technischen Universität München zur Erlangung des akademischen Grades eines
Doktor-Ingenieurs
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. K.-U. Bletzinger
Prüfer der Dissertation:
1. Univ.-Prof. Dr.rer.nat. E. Rank
2. Univ.-Prof. Dr.-Ing. habil. M. Krafczyk,
Technische Universität Carolo-Wilhelmina zu Braunschweig
Die Dissertation wurde am 24.09.2009 bei der Technischen Universität München eingereicht
unddurchdieFakultätfürBauingenieur-undVermessungswesen am15.02.2010angenommen.para las pajaritas
iiiAbstract
This treatise addresses the efficient, numerical simulation of the interaction between fluids
and structures. The discretization of the structure is either based on high-order hexahe-
dral or quadrilateral elements (p-version). Both types of elements allow for an independent
choice of the polynomial degrees for the different local directions as well as for the different
components of the cartesian displacement vectors, which provides scope for strictly two- or
three-dimensional structural discretizations. Three different approaches for the fluid are re-
garded: the Finite Volume Method, the Spectral Element Method and the Lattice Boltzmann
Method. Whereas the first two approaches discretize the ALE form of the incompressible
Navier-Stokes equations, the Lattice Boltzmann Method discretizes the Boltzmann equation
stemming fromthekinetic gastheorywitha fixed, hierarchical grid. Algorithmsaresuggested
forthecoupling ofthese discretizations tothestructuralp-version, which arethenverified and
validatedagainstbenchmarkexamples. Inthiscontext, thecoupling totheLatticeBoltzmann
Method turned out to be particularly favourable.
Zusammenfassung
Diese Arbeit befasst sich mit der effizienten, numerischen Simulation der gegenseitigen Wech-
selwirkung zwischen Fluiden und Strukturen. Die Diskretisierung der Struktur basiert auf
Hexaeder- oder auf Viereckselementen hoher Ordnung (p-Version). Beide Elementtypen er-
lauben eine anisotrope Wahl der Polynomgrade für die lokalen Richtungen und für die Kom-
ponenten des kartesischen Verschiebungsvektors. Dies ermöglicht effiziente, strikt zwei- oder
dreidimensionale Diskretisierungen. Für das Fluid werden unterschiedliche Ansätze verwen-
det: eine Finite Volumen Methode, eine Spektralelement Methode und eine Lattice Boltz-
mann Methode. Die ersten beiden Ansätze diskretisieren die ALE-Form der inkompressiblen
Navier-Stokes Gleichung; die Lattice Boltzmann Methode hingegen diskretisiert die aus der
Gaskinetik stammende Boltzmanngleichung mit einem Finite Differenzen Ansatz auf einem
ortsfesten, hierarchischen Gitter. Es werden Verfahren zur Kopplung dieser Ansätze an die
mit der p-Version diskretisierten Struktur vorgeschlagen und an Referenzbeispielen verifiziert
und validiert. Hierbei hat sich die Kopplung an die Lattice Boltzmann Methode als besonders
vorteilhaft herausgestellt.
vPreamble
This treatise emerged as part of my work at the Chair for Computation in Engineering in the
framework of the project Elemente hoher Ordung in der Fluid-Struktur Wechselwirkung from
fall 2005 to 2009. It was funded by the Deutsche Forschungsgemeinschaft.
I would now like to express my gratitude to everyone who has contributed to this thesis.
Special thanks go to the best boss and supervisor Prof. Dr.rer.nat. Ernst Rank. His enthusi-
asm to get involved creatively in many diverse topics was and still is most inspiring.
I would like to thank Prof. Dr.-Ing. habil. Manfred Krafczyk for accepting to act as a second
examiner. I am proud to have been guided by such a renowned expert in the field.
To Prof. Dr.-Ing. habil. Alexander Düster who co–initiated this project. His door was always
open and questions were never left unanswered. Thanks for the ongoing interest and support.
Alarge partof this thesis contains results achieved in close cooperation with Sebastian Geller.
One common goal and eternal debugging sessions made us understand each others way of
thinking. It was fun to tackle all these difficulties with you, Sebastian.
There are may others who have contributed to this work. Some, like Sebastian, also became
friendsalongtheway. IasonPapaioannou—Greekgenius—andChristianSorger—thesports
man. Both have contributed to this treatise in form of their own master thesis and assistant
work. Martin Schlaffer: thanks for being such a good partner in discussion and reflection.
ThankyouDominikScholz, mypredecessor inthisproject,fornotleaving amessbehindwhen
you left the chair and for introducing me to the topic. And thanks to Georg Sehlhorst for
criticalbutconstructivecommentsandbeingafunofficemateforthelargestpartofthisthesis.
I am very grateful to my parents who are always there when I need their support and formed
me as a person. Looks like their efforts went somewhere.
Thanks to Mirely who gave up her previous life for love, including a Fulbright scholarship and
a PhD position in the US. I still find it amazing that she did this to be with me in freaking
cold Germany. Without her and our daughter Abi, most of the things I do would be less than
half the fun.
Stefan Kollmannsberger,
Winter 2009/2010
viiContents
1 Introduction 1
2 Computational structural dynamics 4
2.1 Basic mechanics for large deflections . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 p-FEM for structural mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Preliminary discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Remarks on p-FEM for fluid-structure interaction . . . . . . . . . . . . 13
3 Computational fluid mechanics 14
3.1 Basic mechanics for incompressible fluids . . . . . . . . . . . . . . . . . . . . . 14
3.2 Discretization by the Finite Volume Method . . . . . . . . . . . . . . . . . . . 17
3.3 Discretization by the Spectral Element Method . . . . . . . . . . . . . . . . . 18
3.4 The Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 The Boltzmann equation and the Maxwell distribution . . . . . . . . . 22
3.4.2 Macroscopic values via moments of microscopic quantities . . . . . . . 24
3.4.3 Simplification of the collision operator: BGK-Modell . . . . . . . . . . 25
3.4.4 Discretization of the BGK-Modell . . . . . . . . . . . . . . . . . . . . . 25
3.4.5 The incompressible model . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.6 The multiple relaxation time model . . . . . . . . . . . . . . . . . . . . 28
4 Partitioned fluid-structure interaction 30
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Preliminary remarks on conservation of mass, momentum and energy . 31
4.2.2 Preliminary remarks on transfer of variables at the boundary . . . . . . 32
4.2.3 Traction transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.4 Velocity and displacement transfer . . . . . . . . . . . . . . . . . . . . 35
4.3 Algorithms for partitioned fluid-structure interaction . . . . . . . . . . . . . . 36
4.3.1 Explicit coupling schemes . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.2 Implicit coupling schemes . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.2.1 Block Gauss-Seidel Iteration . . . . . . . . . . . . . . . . . . . 40
4.3.2.2 Interface-GMRES. . . . . . . . . . . . . . . . . . . . . . . . . 41
ix5 Benchmarks 45
5.1 Numerical benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.2 Principle behaviour and values of comparison . . . . . . . . . . . . . . 46
5.2 Experimental benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Coupling to the Finite Volume Method 49
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Testing against Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2.1 A critical view on Interface-GMRES . . . . . . . . . . . . . . . . . . . 53
7 Coupling to the Spectral Element Method 60
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.1.1 Coupling algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.1.2 Transfer of tractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.1.3 Transfer of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2 Testing against Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2.1 Driven Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2.2 DFG Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8 Coupling to the Lattice Boltzmann Method 70
8.1 Recapitulation, Motivation and Questions to answer . . . . . . . . . . . . . . . 70
8.1.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.1.2 Boundary discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.2 Coupling algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2.1 Setup process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2.2 Core algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3 Fluid Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3.1 Kinetic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 76
8.3.2 Traction evaluation at the fluid boundary. . . . . . . . . . . . . . . . . 77
8.4 Solid Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.4.1 Force transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.4.1.1 Fluid loads for large structural displacements . . . . . . . . . 80
8.4.1.2 Composed integration . . . . . . . . . . . . . . . . . . . . . . 81
8.4.2 Displacement transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.5 Verification against numerical Benchmark . . . . . . . . . . . . . . . . . . . . 83
8.5.1 Sensitivity to selected discretization parameters . . . . . . . . . . . . . 85
8.5.2 Discussion of consistent traction evaluation in the structure . . . . . . . 89
8.5.3 Discussion of energy conservation at the interface . . . . . . . . . . . . 89
8.6 Validation against experimental Benchmark . . . . . . . . . . . . . . . . . . . 92
8.7 A three-dimensional Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9 Beyond Benchmarks 101
9.1 Very large structural deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.2 A free surface example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
9.3 An example of practical interest . . . . . . . . . . . . . . . . . . . . . . . . . . 102
x