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Spline-Based Methods for Aerothermoelastic Problems

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Spline-Based Methods for Aerothermoelastic Problems Spline-Based Methods for Aerothermoelastic Problems Spline-basierte Methoden für aerothermoelastische Probleme Von der Fakultät für Maschinenwesen der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften genehmigte Dissertation vorgelegt von Michaël Karl Petronella Make Berichter: Univ.-Prof. Marek Behr, Ph. D. Univ.-Prof. Dr.-Ing. Stefanie Elgeti Tag der mündlichen Prüfung: 29.03.2021 Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar. "The sail, the play of its pulse so like our lives so thin and yet so full of life, so noiseless when it labors hardest, noisy and impatient when least effective." Henry David Thoreau Abstract This thesis investigates the role of geometry representation in the numerical analysis of aerothermoelastic problems. Nowadays, numerical analysis on spline-based geometric objects is possible through isogeo- metric analysis (IGA) by utilizing the spline-basis for numerical analysis. Although IGA allows for the analysis of volumetric splines, generating such splines is not trivial. For the analysis of thin-walled elastic structures, this drawback can be circumvented by applying shell-theory. For most fluid problems, however, such a workaround does not exist. The NURBS-enhanced finite element method (NEFEM) solves this issue by requiring only the domain boundaries to be defined using splines. Both the NEFEM and IGA provide an exact geometric boundary representation for numerical analysis. In the current work, NEFEM and IGA are coupled to provide a spline-based coupling interface in the context of fluid-structure interaction (FSI). The coupling is done within a strongly coupled partitioned solver framework, which allows for Dirichlet-Neumann (DN) and Robin-Neumann (RN) coupling. Combining NEFEM and IGA leads to a geometrically compatible fluid-structure interface defined by a single common spline. This enables a consistent and conservative transfer of coupling data between the fluid and structural domains. Furthermore, the common spline interface enables the direct integration of coupling quantities on the fluid and structural domains using the spline-basis. The numerical performance of the spline-based solver framework is investigated through a set of example problems. For compressible and incompressible flow problems, not considering FSI, improved numerical accuracy is observed when the exact geometry is considered through the NEFEM. An extension of this investigation to FSI problems shows similar behavior. It is found that especially fully-enclosed Dirichlet-bounded problems can benefit from the accurate boundary representation provided by the proposed spline-based method. Furthermore, the given examples show that using a common spline-basis can improve the numerical stability of the employed spatial coupling procedures. This observation is especially relevant for thermal coupled problems, for which such instabilities could lead to the inability to obtain converged numerical solutions. i Kurzfassung In dieser Arbeit wird der Einfluss einer exakten Geometriedarstellung in der numerischen Analyse aerothermoelastischer Probleme untersucht. Eine solche Analyse wird durch isogeometric analysis (IGA) möglich, indem innerhalb des numerischen Verfahrens direkt auf die zur Geometriedarstellung verwende- ten Spline-Basis zurückgegriffen wird. Obwohl IGA grundsätzlich die Analyse mit volumetrische Splines ermöglicht, ist die Erzeugung solcher Splines nicht trivial. Zwar kann diese Problematik z.B. für die Untersuchung dünnwandiger elastischer Strukturen durch die Anwendung der Schalentheorie umgangen werden, aber insbesondere für Fluid-Probleme existiert eine solche Alternative nicht. NURBS-enhanced finite element method (NEFEM) löst dieses Problem, indem nur die Gebietsgrenzen durch Splines definiert werden müssen. Damit erlauben sowohl NEFEM als auch IGA eine exakte geometrische Randdarstellung innerhalb der numerische Analyse. In der vorliegenden Arbeit erfolgt die numerische Analyse mit Hilfe eines stark gekoppelten parti- tionierten Lösungsverfahrens, welches durch die Kopplung von Dirichlet-Neumann (DN) und Robin- Neumann (RN) entsteht. Dazu werden NEFEM und IGA verbunden, wodurch im Kontext von Fluid- Struktur Interaktion (FSI) eine identische, spline-basierte Schnittfläche geschaffen wird. Diese ermöglicht eine konsistente und konservative Übertragung von Kopplungsdaten zwischen dem Fluid- und dem Struk- turbereich. Darüber hinaus ermöglicht die gemeinsame Spline-Grenzfläche die direkte Integration von Kopplungsgrößen auf dem Fluid- und Strukturgebiet unter Verwendung der Spline-Basis. Die numerischen Eigenschaften des spline-basierten Lösungsverfahrens werden mit Hilfe von Beispiel- problemen untersucht. So wird schon in einer isolierten Betrachtung von kompressiblen und inkompressi- blen Strömungsprobleme eine verbesserte numerische Genauigkeit beobachtet, wenn die exakte Geometrie durch NEFEM berücksichtigt wird. Die Erweiterung dieser Untersuchung auf FSI-Probleme zeigt ein ähn- liches Ergebnis. Insbesondere für vollständig durch Dirichlet-Randbedingungen umschlossene Probleme können Vorteile der genauen Randdarstellung, die durch die vorgeschlagene spline-basierte Methode bereitgestellt wird, nachgewiesen werden. Darüber hinaus zeigen die präsentierten Beispiele, dass die Verwendung einer gemeinsamen Spline-Basis die numerische Stabilität der verwendeten räumlichen Kopplungsverfahren verbessern kann. Diese Beobachtung ist besonders für thermisch gekoppelte Pro- bleme relevant, bei denen solche Instabilitäten dazu führen können, dass keine konvergenten numerischen Lösungen berechnet werden können. iii Acknowledgment This thesis is realized during my time spent as a research assistant at the RWTH Aachen University in Germany. Thinking back at this period in my life brings back many positive memories. The people at CATS undoubtedly had a significant influence on this. The friendly and inclusive atmosphere was a perfect base for fruitful discussions on research-related topics and beyond. I have gained many skills and many friends. I want to thank everyone at CATS, past and present, for making this possible. Specifically, I want to express my gratitude towards my supervisors Marek Behr and Norbert Hosters. For your continuous encouragement and support. Without your efforts, this work would not have been possible. Thank you for giving me the opportunity and the freedom to develop myself on an academic and personal level. Furthermore, I would like to thank Norbert Hosters, Thomas Spenke, Max von Danwitz, Patrick Antony, Emre Ongut, and Manuel Brüderlin of the FSI research group for the many insightful technical discussions and for letting me be part of the joint effort in further developing our methods. It was a very valuable experience. I would also like to thank Linda Gesenhues, Norbert Hosters, Thomas Spenke, and Max Schüster for proofreading this thesis and their unconditional support during the final stage of writing this work. Your help was indispensable. Most notably, I thank my family for their love and support through these challenging years. I am grateful to have you in my life. Thank you. Vaals, April 2021 Michel Make v Contents List of Figures xi List of Tables xiii Abbreviations xv 1 Introduction 1 1.1 Numerical Methods and Fluid-Structure Interaction.................... 2 1.2 Motivation.......................................... 3 1.3 Outline ........................................... 5 2 Physical Background7 2.1 Fluid-Structure Interaction ................................. 7 2.1.1 Lagrangian and Eulerian Material Descriptions.................. 8 2.2 Compressible Flows..................................... 11 2.2.1 Constitutive Relations ............................... 12 2.2.2 Generalized Advection-Diffusion System..................... 12 2.2.3 Boundary and Initial Conditions.......................... 13 2.2.4 Choice of Variable Sets............................... 14 2.3 Incompressible Flows.................................... 15 2.4 Elastic Structures...................................... 17 2.4.1 Constitutive Models ................................ 17 2.5 Heat Transport in Solid Structures ............................. 18 2.5.1 Weakly-Coupled Thermal Strains ......................... 18 3 Concepts of The Finite Element Method 19 3.1 The Strong and Weak Form................................. 19 3.2 Discrete Weak Form .................................... 21 3.2.1 The Isoparametric Concept............................. 22 3.3 Time Discretization..................................... 23 3.3.1 The θ-Method ................................... 24 3.3.2 Space-Time Finite Element Methods........................ 25 vii 3.3.3 Generalized-α Method............................... 26 3.4 Stabilized Finite Element Formulations .......................... 28 3.5 Consistent Boundary Flux Method............................. 30 4 Spline-Based Methods 33 4.1 Non-Uniform Rational B-Splines (A Very Brief Introduction)............... 34 4.2 Isogeometric Analysis ................................... 36 4.3 NURBS-Enhanced Finite Elements............................. 38 4.3.1 Cartesian NEFEM ................................. 38 4.3.2 Non-Cartesian NEFEM............................... 40 4.3.3 On NEFEM and Space-Time Formulations .................... 44 4.3.4 On Numerical Integration and Non-Cartesian NEFEM.............. 45 4.3.5 Visualization of Spline-Based Solutions...................... 47 5 Numerical Methods 49
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