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 isogeometric 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.

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 verwendeten 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

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 5.1 FSI Solver Framework and Coupling Strategies...... 49 5.1.1 Spatial Coupling ...... 49 5.1.2 Temporal Coupling of Space-Time and FD Methods ...... 54 5.1.3 Iterative Solution Procedure and Field Partitioning ...... 55 5.2 Stabilized FEM for Compressible Flows...... 57 5.3 Stabilized FEM for Incompressible Flows...... 61 5.4 FEM for Elastodynamic Problems ...... 62 5.5 FEM for Heat Transport in Solids ...... 63

6 Application of Non-Cartesian Space-Time NEFEM to Compressible and Incompressible Flows 65 6.1 Application to 2D Compressible Flow Problems...... 65 6.1.1 Flow Around a 2D Cylinder ...... 65 6.1.2 Flow Around a 2D NACA0012 Wing Section...... 68 6.2 Application to 3D Incompressible Flow Problems...... 71 6.2.1 Steady Flow ...... 71 6.2.2 Unsteady Flow...... 73

7 Application of Spline-Based Methods to FSI Problems 75 7.1 Elastic Problems ...... 75 7.1.1 3D Cavity With Flexible Bottom...... 75 7.1.2 Inﬂation of a Fully Enclosed 3D Cylindrical Domain...... 78 7.2 Thermal Problems...... 82 7.2.1 Natural Convection through Heated Wall ...... 82 7.2.2 Thermal Expansion of a Heated 3D Supported Plate ...... 86

8 Summary and Outlook 93 8.1 Summary ...... 93 8.2 Outlook...... 95 viii Bibliography 97

List of Figures

1.1 Black-box solver framework...... 3 1.2 Various combinations of discretizations for an inﬂating balloon...... 5

2.1 Fluid and structural domain Ω¯ f and Ω¯ s, and their common interface Γfs...... 8

2.2 Lagrangian description with reference and current conﬁguration Ω¯ X and Ω¯ x...... 9 2.3 Arbitrary Lagrangian-Eulerian reference mesh in a ﬂuid...... 10

3.1 Triangular element in the physical space and the parametric space...... 22 3.2 Space and time discretization procedures...... 23 e 3.3 Space-time slab with element Qn...... 26 3.4 Computational domain of the two-dimensional Couette ﬂow problem...... 31 3.5 Dimensionless temperature proﬁle for P r = 100 and Ec = 1...... 32

4.1 Steps of the domain discretization of a NURBS-based CAD geometry for NEFEM simulations. 34 4.2 Cubic B-Spline basis...... 35 4.3 NURBS curve corresponding to the knot vector used in Figure 4.2...... 36 4.4 NURBS surface and its corresponding parametric coordinate system θ = (θ1, θ2)T ...... 36 4.5 Mapping from the physical space via the parametric space to the reference element...... 37 4.6 Possible common face or edge conﬁgurations of boundary elements...... 39 4.7 Integration point positioning within an NEFEM element along a curved NURBS surface. . . 39 4.8 Shape function φ(x) corresponding to the interior node of a non-Cartesian NEFEM element. 40 4.9 THT mapping from a reference tetrahedron to a tetrahedral for a face element...... 41 4.10 THT mapping from a reference tetrahedron to a tetrahedral for an edge element...... 43 4.11 Space-time slab for a two-dimensional non-Cartesian NEFEM element...... 45 4.12 Cuboid with a cylindrical 2π/3 segment cut-out...... 46

4.13 Relative error εrel between the numerical and exact volume V˜Ω and VΩ...... 47 4.14 Normalized pressure ﬁeld within the NEFEM mesh of Figure 4.14...... 48

5.1 FIE data transfer procedure...... 51 5.2 Time-level handling of spline-based space-time and semi-discrete meshes...... 54 5.3 Schematic of strong temporal coupling procedures...... 56

6.1 Computational domain and boundary conditions for a supersonic ﬂow around a 2D cylinder. 66

xi 6.2 Contour lines of the pressure coefficient for the supersonic cylinder problem...... 67 6.3 Pressure coefficient along the cylinder wall...... 68 6.4 Mesh convergence for supersonic flow around a cylinder...... 68 6.5 Domain and boundary conditions for the transonic flow around a NACA0012 wing section. . 69

6.6 Contour lines of Cp for the transonic flow around a NACA0012 wing section...... 70 6.7 Pressure coefficient along the wing section...... 71 6.8 Computational domain for the flow around a 3D cylinder problem...... 72 6.9 Mesh convergence for the steady flow around a 3D cylinder...... 73

6.10 Time history of the drag coefﬁcient CD of the 3D cylinder...... 74

7.1 Fluid and structural domain of the 3D cavity with ﬂexible bottom problem...... 76

7.2 Maximum bottom deﬂection Dm over time...... 77

7.3 The relative error εrel of the maximum deformation Dm...... 78 7.4 Three-dimensional inﬂatable circular domain enclosed by a thin-walled structure...... 79 7.5 Computational mesh with n = 28 linear tetrahedral elements in the circumferential direction. 80 7.6 The absolute error of the three-dimensional cylindrical domain radius...... 81

7.7 The absolute error εabs of the three-dimensional cylindrical domain radius...... 81 7.8 Problem setup for natural convection through 2D heated wall problem...... 82 7.9 Natural convection through a 2D heated wall...... 84 7.10 Thermal quantities at the fluid side of the fluid-structure interface Γfs,s...... 84 fs 7.11 Normal heat flux qn on the fluid side of the fluid-structure interface Γ ...... 85 7.12 3D heated cavity solution...... 86 7.13 Problem setup for the thermal expansion of a heated 3D supported plate problem...... 87 7.14 Cross-section of the thermal expansion of a heated 3D supported plate problem...... 87 7.15 Flow domain outline and heated plate...... 89 7.17 Temperature distribution at various time instances...... 90

7.18 Maximum plate deﬂection Dm over time...... 91 7.19 Relative error in plate deﬂections...... 91

xii List of Tables

3.1 Common choices of θ for the θ-scheme and their accuracy and stability characteristics. . . . 24 3.2 Parameters used for the 2D Couette flow problem...... 31 3.3 Normal heat flux at the top and bottom wall of the 2D Couette flow problem...... 32

4.1 Grids used for the volume computation comparison...... 46

6.1 Parameters used for the supersonic flow around a 2D cylinder...... 66 6.2 Meshes used for the flow around a 2D cylinder problem...... 67 6.3 Parameters used for the transonic flow around a NACA0012 wing section...... 69 6.4 Parameters used for the steady flow around a 3D cylinder benchmark case...... 72 6.5 Meshes used for the steady flow around a 3D cylinder problem...... 72 6.6 Parameters used for the unsteady flow around a 3D cylinder benchmark case...... 74

7.1 Parameters used for the 3D cavity with flexible bottom problem...... 76 7.2 Meshes used for the 3D cavity with flexible bottom problem...... 78 7.3 Parameters used for the 3D inflatable cylindrical domain problem...... 79 7.4 Parameters used for the natural convection through 2D heated wall problem...... 83 7.5 Parameters used for the thermally expanding supported plate...... 88

xiii

Abbreviations

AD Advection-Diffusion. ALE Arbitrary Lagrangian-Eulerian. CAD Computer-Aided Design. CBF Consistent Boundary Flux. CFD Computational Fluid Dynamics. CSD Computational Structural Dynamics. DN Dirichlet-Neumann. DSD/SST Deformable-Spatial-Domain / Stabilized Space-Time. FD Finite Difference. FE Finite Element. FEM Finite Element Method. FIE Finite Interpolation Elements. FSI Fluid-Structure Interaction. FV Finite Volume. GLS Galerkin / Least-Squares. IGA Isogeometric Analysis. NEFEM NURBS-Enhanced Finite Element Method. NURBS Non-Uniform Rational B-Splines. ODE Ordinary Differential Equation. PDE Partial Differential Equation. RN Robin-Neumann. SFEM Standard Finite Element Method. ST Space-Time. SUPG Streamline Upwind Petrov-Galerkin. THT Tetrahedron-Hexahedron-Tetrahedron. VIV Vortex-Induced Vibrations. VMS Variational Multiscale.

Chapter 1

Introduction

Whether it is a sailor trimming the sails of his yacht to keep heading or the sailor’s very own body pumping fresh ocean air into his lungs, the balancing act of fluid-structure interaction (FSI) is taking place everywhere around us. The synergy between the physics involved is truly magical. It even inspired artists in their work. With Strandbeesten, for example, the Dutch artist Theo Jansen devised a series of moving, organic looking, kinetic structures that can autonomously move around using solely wind power [1]. But the interest in FSI is not exclusive to the artistic world. For many years, engineers have been studying various forms of FSI. The goal was either to utilize FSI in their advantage in smart engineering designs or simply to understand the mechanisms underlying certain physical phenomena. And past experiences painfully show that gaining knowledge about the physics of FSI is extremely important for successful advancement in engineering technology. In that respect, a very well known and still studied example of FSI is that of the wind-induced dynamic behavior of the long-span suspension bridge over the Tacoma Narrows strait. The interaction between the wind loads and the deforming bridge resulted in excessive excitation of the bridge. That caused its eventual failure in 1940, the same year its construction was completed. The mechanism of this failure is a clear example of FSI which is believed to be caused by the wind-induced vortex shedding frequencies approaching the eigenfrequency of the bridge structure itself (see, e.g., the recent study in [2]). Since the Tacoma Narrows Bridge disaster, the general understanding of FSI greatly improved. The consideration of FSI is nowadays a standard procedure in the design process of many engineering systems (e.g., [3]). However, despite this improved understanding, FSI phenomena still occur unexpectedly from time to time. A beautiful example in recent history surprisingly involves another long-span suspension bridge. In 1996, only two months after the official opening, the Erasmus bridge in Rotterdam suffered excessive vibrations at unpredictable moments in time. This behavior occurred seemingly randomly, and the exact underlying mechanisms were poorly understood. After an in-depth study [4], it was found that the vortex shedding frequency of the vertical stay cables was altered by rainwater dripping down the cables. The consequential change of the cyclic wind load frequencies towards the eigenfrequencies of the stay cables resulted in unwanted vibrations of the entire bridge. Hence, on rainy days with just the right amount of wind, unwanted FSI started to occur. Luckily for the Erasmus bridge, and its daily passersby, modifications were made to the design of the bridge and the issue was resolved. The Erasmus bridge example nicely shows the strong need for analysis tools for FSI. In the past, such tools were rudimentary in complexity and accuracy. Typically, FSI problems have been analyzed using

1 analytical or semi-analytical methods. Despite being valuable, these tools are often limited in application and accuracy due to the strong assumptions under which they are valid. An example of such methods relevant to long-span suspension bridges is the work on analytic methods for vortex-induced vibrations (VIV) in, e.g., [5–7]. Although simple in nature, most such methods would not be able to predict the FSI observed with the Erasmus bridge. Hence, the need for more advanced methods arises. Numerical methods have gained popularity in recent years due to their ability to analyze problems of higher complexity and with continuously improving accuracy. With the development of modern-day numerical simulation methods and the continuously increasing computational power, simulating fluid and structural problems with a high level of detail has become standard in engineering. However, due to their complex, coupled, and typically non-linear nature, high-fidelity FSI simulations are still computationally very costly. Hence, the need for increasingly efficient and accurate numerical methods remains strong. In this thesis, an attempt to further this development is proposed by employing a spline-based solver framework specific for the analysis of FSI phenomena. More specifically the combination of NURBS- enhanced finite element method (NEFEM) and isogeometric analysis (IGA) for the fluid and structure problems within a partitioned solver framework is investigated. Before elaborating more on the motivation of this work in Section 1.2, it is imperative first to clarify some of the fundamental concepts in the context of numerical methods and FSI in the next section.

1.1 Numerical Methods and Fluid-Structure Interaction

Canonically speaking, a discrete FSI problem can be formulated as a fully coupled multi-ﬁeld problem:

F (f f , f s) = RHS, (1.1) where f f and f s represent the fluid and structure problem variables. RHS represents the right-hand side terms of the individual problems. The different approaches to solve F can be classified in two main categories, i.e., monolithic and partitioned solution methods [8]. Solving Equation (1.1) as a single problem is known as the monolithic approach [9–13]. Solving each individual sub-problem separately in an iterative manner is known as the partitioned or staggered approach [14–17]. Following the monolithic solution strategy, the complete discrete FSI problem is solved as a single system of equations. Generally speaking, this approach is more robust and, compared to the partitioned approach, allows for a larger time-step size [18]. Despite these advantages, the partitioned approach often remains the preferred choice. This is due to several reasons discussed next. Most fluid and structural solvers are developed completely separately, each with its own highly optimized and specialized features. Constructing a full monolithic discrete equation by integrating multiple solvers into a single software program is an intricate endeavor. Moreover, it typically results in complicated and hard-to-maintain software. A partitioned solution strategy, on the other hand, provides flexibility and modularity. In this case, only specific coupling data is required by the individual solvers, and the control and communication of this data between the individual solvers can be handled by a separate coupling module. An example of a partitioned solution strategy that includes a coupling module is depicted in Figure 1.1. In the context of a partitioned approach, the individual solvers are typically referred to as black-box solvers following the concept of minimal necessary data exchange to and from the coupling module.

2 Discrete nodal Black-Box Fluid Solver coupling data - Fluid - Temperature - Mesh deformation Coupling Module - Spatial coupling - Temporal coupling Discrete nodal - Relaxation Black-Box Structural Solver coupling data - Non-linear elastodynamics - Temperature

Figure 1.1: Black-box solver framework with a coupling module arranging the necessary communication.

In addition to the modularity argument, partitioned solver setups allow for different time-step sizes for the fluid and structure problems. This is useful, as typically both problems have a different stability constraint on the time-step size. Monolithic schemes on the other hand, require a single time-step for the complete system. Hence, the fluid and structure problems are solved using the smallest time-step required for stability of the sub-problems. This implicitly affects the computational efficiency of monolithic methods. A partitioned and strongly coupled solution strategy is used in the current work to exploit the modularity and flexibility. Here, the single-field solvers are coupled through a coupling module as shown in Figure 1.1. This module handles the exchange of interface data in accordance with the coupling conditions of the FSI problem at hand. The fluid and structural solvers are assumed to be black-box solvers. Hence, the coupling module does not have access to the internals of the individual solvers. The coupling module merely sends, receives, and manipulates the required nodal coupling data. The coupling module is also responsible for controlling the advancement in time of the individual solvers. Irrespective of the solution strategy used to achieve the highest possible computational efficiency, the employed solution strategy should allow for computational grids with non-matching interfaces. By doing so, an optimal mesh resolution for each of the sub-problems can be chosen. This often results in non-matching meshes at the coupling interface and requires additional measures for the proper transfer of discrete coupling data. Ideally, the applied method should be consistent and energy-conserving, meaning the transferred data is identical on both sides of the interface after the transfer.

1.2 Motivation

In many engineering design processes, exact geometries are created using widely available computer- aided design (CAD) tools. Most such tools use Non-Uniform Rational B-Splines (NURBS) to deﬁne curves, surfaces, and volumes. The choice for NURBS follows from the strong underlying mathematical foundation. This results in NURBS being easy to modify, manipulate, and extract geometric information from. Furthermore, the widespread use of NURBS makes it possible to use NURBS-based data formats between different software tools.

3 Many numerical methods in the field of computational fluid dynamics (CFD) and computational structural dynamics (CSD), however, rely on discrete approximations of the exact problem domains. In the context of the finite element method (FEM) a computational domain is first discretized by generating a partitioning of the domain using finite elements. In many real-world engineering problems the exact domain consists of complex shapes that cannot be represented using standard finite elements1. Hence, the discretization step introduces a geometric error with respect to the exact domain. Furthermore, the meshing process to obtain the finite element (FE) partitioning can be a tedious and time-consuming task.

The introduction of IGA[19] allows us to directly perform numerical analysis on exact NURBS- based geometric objects. This is achieved by utilizing the spline-bases of the NURBS within aFE-type formulation. With this approach, the partitioning step, which is time-consuming and introduces a geometric error, is omitted altogether. However, even though IGA directly uses NURBS-based geometries, most of the available CAD tools do not represent three-dimensional objects through volume splines. Instead, objects are represented by means of their external surfaces and curves using the corresponding NURBS only. The reason for this lies in the fact that it is not strictly necessary in deﬁning geometric objects. Moreover, generating volume splines of complex geometries is challenging and, in some cases, it is an impossible process [20, 21].

As most real-world problems involve volumetric domains, this can become problematic. While some structures can be modeled by shell-formulations with reduced spatial dimensions (see e.g., [22]), for flow problems there is no such alternative. With the recently developed NEFEM[23 –26], the difficulties that arise when using IGA for complex volumetric domains are avoided. NEFEM allows favorable geometric characteristics of NURBS to be utilized within a standard finite element method (SFEM). While IGA uses a NURBS basis to represent both geometry and numerical solution [27], NEFEM uses a NURBS n representation of the domain boundary only. Consequently, for problems in R sd (with nsd being the n −1 number of spatial dimensions), at most an R sd NURBS domain boundary surface is needed. The possibility to directly perform analysis on exact geometries via IGA and NEFEM opens several of opportunities for further research. In the context of FSI, the application of IGA and NEFEM to the structure and fluid problem, provides a geometrically exact spline-based fluid-structure interface. Having such a common spline representation for the fluid and structural problem allows for a direct transfer of coupling quantities while still permitting different refinement levels for the individual domain discretizations. An example in which conventional and spline-based methods are combined in various ways is depicted in Figure 1.2.

This concept has been previously studied by coupling a non-Cartesian variation of the space-time NEFEM formulation [26] with an IGA formulation for two-dimensional elastic problems involving incompressible flows [28, 29]. This study shows the benefits of spline-based methods for elastic fully enclosed Dirichlet-bounded problems. In the current work, the influence of exact geometries in the context of FSI is further investigated. By providing the necessary building blocks, the possible benefits of a geometrically exact spline-based FSI framework for three-dimensional, aerothermoelastic problems involving both compressible and incompressible flows are explored.

1In the current work, standard finite element method is referring to classic isoparametric finite element methods in conjunction with Lagrangian finite elements.

4 SFEM - SFEM F luid

SFEM - IGA

Structure

NEFEM - IGA

Figure 1.2: Various combinations of discretizations for an inﬂating balloon. Here standard SFEM, non- Cartesian NEFEM, and IGA are combined in various ways. Note that here the thin-walled structure of the balloon is represented by an IGA shell. The presented spline-based framework also allows for structures represented by volume splines. In that case, not the complete volume spline, but rather its surfaces are used for the non-Cartesian NEFEM formulation. In Chapter7, this concept is used in two- and three-dimensional problems.

1.3 Outline

The remainder of this thesis provides the building blocks and demonstrates the performance of a spline- based partitioned thermal FSI solver framework. The material presented in the individual chapters is structured as follows:

• Chapter2 focuses on the relevant underlying physics in FSI phenomena. Here, a set of common material description model, the necessary conditions to couple a ﬂuid and a structure, as well as the governing equations of the individual problems are discussed. This chapter is self contained in the sense that it covers all necessary models required to describe the physical aspects of the numerical examples given in Chapters6 and7.

• Since in this work all the numerical formulations areFE-based, Chapter3 formulates the mathematical foundation of the FEM. Using a simple model problem, topics such as the weak formulation, the isoparametric concept, and various time-discretization approaches, including space-time ﬁnite elements, are introduced.

5 • The geometrically exact spline-based solver framework proposed in this thesis uses NURBS-based geometry representations. In Chapter4 , the important concepts of NURBS are discussed and subsequently applied to an IGA formulation. The chapter concludes in Section 4.3 with the presentation of the newly proposed non-Cartesian space-time NEFEM and the corresponding necessary mappings.

• In Chapter5 the mathematical tools from Chapters3 and4 are applied to the governing equations discussed in Chapter2. The complete solution procedure for coupled aerothermoelastic FSI problems is given in this chapter. This also includes a discussion on consistency and energy conservation of various spatial coupling methods for non-matching interfaces.

• The theoretical and numerical frameworks covered in Chapters2 to5 are put to the test by solving a series of numerical examples. In Chapter6 , the focus is speciﬁcally on the numerical performance of the geometrically exact non-Cartesian space-time NEFEM. Here, 2D and 3D problems involving compressible and incompressible ﬂows are considered, and the non-Cartesian space-time NEFEM is compared with conventional FEM.

• In Chapter7 the proposed solver framework is tested in the context of FSI. For this, a series of thermal, elastic, and thermoelastically coupled problems is considered. Similar to that in Chapter6, a comparison between spline-based and conventional methods is made.

• The work is concluded in Chapter8 , where a summary of the observations from Chapters6 and7 and an outlook for further development of the methods and possible future research applications is given.

6 Chapter 2

Physical Background

FSI is characterized, as the term suggests, by the interaction between a fluid and a structure. Hence, when describing problems involving FSI, it seems natural to divide the problem into a fluid and structural portion. This partitioned character allows us to apply conventional models to each individual sub-problem and define the interaction exclusively at their common fluid-structure interface. This is possible under the loose assumption that the interaction takes place solely via the common interface. In this chapter, the general framework of thermal FSI problems is presented. First, a brief introduction to FSI is given in Section 2.1 by presenting the necessary coupling conditions, followed by a discussion on Lagrangian and Eulerian reference frames. The governing equations of the compressible and incompressible fluid flow problems are presented in Section 2.2 and 2.3. Subsequently, the models that describe thermoelastic behavior of solids are discussed in Section 2.4 and 2.5.

2.1 Fluid-Structure Interaction

As shown in Figure 2.1, FSI problems can be divided into two sub-problems with a common interface. Here, the individual problem domains and their common interface are defined as Ω¯ = Ω¯ f ∪ Ω¯ s and Γfs = Ω¯ f ∩Ω¯ s. The superscripts f and s refer to the fluid and structural part of the problem2, respectively. The governing equations of the fluid and structure are defined on Ω¯ f and Ω¯ s. The interaction between Ω¯ f and Ω¯ s takes place at their common interface Γfs. To consistently describe this interaction, a set of coupling conditions needs to be considered:

• Kinematic coupling conditions ensure the continuity of displacement, velocity, and acceleration across the interface:

f s fs d = d on Γt , t > 0, (2.1a) f s fs u = u on Γt , t > 0, (2.1b) f s fs a = a on Γt , t > 0, (2.1c)

where d, u, a represent the displacement, velocity, and acceleration at the interface. In the above, fs the subscript t indicates time variance of Γt in time t.

2Note that in the remainder of this thesis, superscripts f and s, indicating the ﬂuid or structure, are omitted when not strictly needed. E.g., while Sections 2.2 and 2.3 cover solely the ﬂuid problem, superscript f is dropped.

7 Ω¯ f

Γfs

Ω¯ s

Figure 2.1: Fluid and structural domain Ω¯ f and Ω¯ s, and their common ﬂuid-structure interface Γfs.

• Following Newton’s third law of motion, the dynamic coupling condition ensures that the stresses across the ﬂuid-structure interface are continuous:

f f s s fs T n = T n on Γt , t > 0, (2.2)

where Tf and Ts represent the Cauchy stress tensors of the ﬂuid and the structure, and n the fs outward unit vector at Γt .

• Finally, when thermal interaction between the ﬂuid and structure is considered, a set of thermal coupling conditions needs to be addressed:

f s fs Θ = Θ on Γt , t > 0, (2.3a) ∂Θf ∂Θs = on Γfs, t > 0, (2.3b) ∂n ∂n t where Θf and Θs represent the temperature of the ﬂuid and structure, respectively.

Satisfying the above conditions for continuous elastodynamic FSI problems ensures the conservation of fs mass, momentum, and energy across Γt [29]. Notice in Equations (2.1) to (2.3) that we allow for the sub-domains to deform in time by deﬁning the fs ¯ f ¯ s coupling conditions and Γt to be time-dependent. Consequently, sub-domains Ωt and Ωt also depend on time. In the remainder of this work, unless stated otherwise, it is assumed that both the ﬂuid and structural domains are allowed to deform even when no subscript t is written.

2.1.1 Lagrangian and Eulerian Material Descriptions

In the ﬁeld of FSI, problem domains often deform. This could result form elastic deformation, thermal expansion, or simply a free-surface enclosing the ﬂuid domain. Several approaches to describe the moving domains and the material contained within are discussed next.

8 X x

¯ Ω¯ X Ωx

Figure 2.2: Lagrangian description of motion with reference and current conﬁgurations Ω¯ X and Ω¯ x.

In continuum mechanics, two spatial domains are commonly used, i.e., the material domain Ω¯ X at time t = 0 with material particles X, and the spatial domain Ω¯ x containing the points x in space. A sketch of the domains is presented in Figure 2.2. The motion of a continuum can be described in various ways using the material and spatial domains. The Lagrangian description follows the material particles X and their motion. Consider a computational mesh on Ω¯ X attached to the material particles X. Deforming the domain from its initial state to a displaced and deformed conﬁguration Ω¯ x would yield a deformation of this mesh. When considering

Ω¯ X as being fixed, typically corresponding to a configuration at an initial time, it is referred to as total Lagrangian formulation. With the Lagrangian description, a computational mesh remains fixed to the same material points at every point in time. Hence, no convective effects are present, and the material derivative reduces to a time derivative only:

D ∂ D ∂ ≡ + (c · ∇) ⇒ = , (2.4) Dt ∂t Dt ∂t where c represents the convection velocity. The Lagrangian material description is typically used for problems where material deformations are limited. When large deformations and distortions are considered, the Lagrangian approach becomes impractical. A well-known example of such a case is vortices present in a fluid. Using a pure Lagrange- type approach for such problems is simply impossible, as it would lead to very distorted computational grids. When such large deformations do occur, an Eulerian-type description is more suitable instead. With the Eulerian description, not the movement of individual particles, but rather the physical quantities associated with the particles passing through a fixed region in space are considered. The particles are no longer associated with a fixed point in the domain. Thus, the Eulerian description model does not require a reference configuration of the material. Describing the movement of particles relative to a reference frame that is fixed in space yields the convective term in the total derivative in Equation (2.4).

9 Where c = ufluid represents the convective velocity of the fluid particles. However, in the context of numerical simulations, the introduction of convective terms can lead to numerical instabilities. Hence, when the Eulerian approach is applied, typically additional modifications of discretization schemes are needed to obtain stable numerical solutions, as discussed in the context of the FEM in Chapter3. It is evident that the Lagrangian and Eulerian description models each have their advantages and drawbacks when applied to certain problems. An additional description model can be used, which combines the Lagrangian and Eulerian models. Instead of fixing the reference frame in space or to material particles, the so-called arbitrary Lagrangian-Eulerian (ALE) approach uses another frame of reference. This frame of reference can move arbitrarily through space independent of the material particles. To explain the arbitrary movement of the reference frame, consider a computational mesh describing a fluid flow as depicted in Figure 2.3a. In this particular case, the mesh is fixed in space, resulting in an Eulerian-type description model. The observed convection velocity at the individual nodes of the mesh is equal to the local fluid velocity c = ufluid.

ufluid ufluid

umesh

(a) Undeformed (Eulerian) mesh in a fluid flow. (b) Deforming mesh in a fluid flow.

Figure 2.3: Arbitrary Lagrangian-Eulerian reference mesh in a ﬂuid.

When a moving computational mesh is assumed, such as presented in Figure 2.3b, the observed ﬂuid velocity at the individual mesh nodes is no longer equal to ufluid. In fact, the velocity of the moving mesh nodes needs to be considered. Consequently, the observed convective velocity at the nodes is given by c = ufluid − umesh. The ALE description model applied to the material derivative results in the following expression: D ∂ ≡ + ((u − u ) · ∇) . (2.5) Dt ∂t fluid mesh

Note, that the Eulerian and Lagrangian description models are both contained within the ALE description model. In case of the Eulerian description, the mesh is ﬁxed in space, or c = ufluid − 0. On the other hand, when a Lagrangian model is considered, the mesh is ﬁxed to the material particles. Hence, the observed velocity of the material at any of the mesh nodes is zero, or c = 0.

10 2.2 Compressible Flows

A set of governing equations is needed to describe a fluid in sub-domain Ωf of the FSI problem. In this work, both compressible and incompressible Newtonian fluids are considered on Ωf . The current chapter will address compressible fluid flow, while the governing equations for incompressible fluids are presented in Section 2.3. Fluid motion can be described through conservation laws for mass, momentum, and energy. For an arbitrary spatial material domain Ω with boundary Γ and outward unit normal vector n, conservation of mass can be written in conservative integral form as follows:

D Z ρ dΩ = 0, (2.6) Dt Ω

D with ρ and Dt being the ﬂuid density and material time derivative. Conservation of momentum follows directly from Newton’s second law of motion. This law states that the rate of change of momentum of Ω is directly proportional to the forces applied to Ω and Γ. In conservative integral form, this yields:

D Z I Z ρu dΩ − T · n dΓ − ρb dΩ = 0. (2.7) Dt Ω Γ Ω

The second and third integral in (2.7) represent the surface forces acting on Γ, and the body forces acting on Ω respectively. In this expression, T is the symmetric stress tensor according to Cauchy’s stress theorem (see Section 2.2.1), b is the body force per unit mass, and u the ﬂuid velocity vector. Conservation of energy, or the ﬁrst law of thermodynamics, states that the rate of change of total energy per unit mass E contained in Ω is equal to the work applied, and heat added to Ω. In conservative integral form, this yields:

D Z I I Z ρE dΩ − (T · u) · n dΓ + q · n dΓ − ρ (b · u + r) dΩ = 0, (2.8) Dt Ω Γ Γ Ω where q is the heat ﬂux vector and r a source term. The total energy itself consists of the internal and 1 2 kinetic energy, or, E = e + 2 ||u|| . Here, the second and third integral represent the amount of work applied to Γ and the net amount of heat added through Γ respectively. Equations (2.6) to (2.8) can be rewritten into differential form by applying Reynolds transport theorem, the divergence theorem, and subsequently using the fact that Ω can be chosen arbitrarily. This yields the following:

∂ρ + ∇ · (ρu) = 0 on Ω ∀ t ∈ (0,T ), (2.9a) ∂t ∂ρu + ∇ · (ρu u ) − ∇ · T − ρb = 0 on Ω ∀ t ∈ (0,T ), (2.9b) ∂t ∂ρE + ∇ · (ρEu ) − ∇ · (Tu) + ∇ · q − ρb · u − ρ r = 0 on Ω ∀ t ∈ (0,T ), (2.9c) ∂t

where T is the ﬁnal solution time. The three conservation laws presented describe the behavior of compressible ﬂuids in motion and are known as the Navier-Stokes equations . The unknown variables in these equations, ρ, ρu, and ρE are commonly referred to as the conserved quantities.

11 2.2.1 Constitutive Relations

The system of equations in (2.9) is closed using a Newtonian fluid model for the stress T, and Fourier’s law to define the heat flux q as a function of temperature Θ:

Ä ä T = −pI + µ ∇u + (∇u)T + λ(∇ · u)I, (2.10) q = −κ∇Θ, (2.11)

2 where p is the pressure, µ and λ the dynamic and bulk viscosity related by λ = − 3 µ, and κ the thermal conductivity. A temperature-dependent dynamic viscosity can be approximated via Sutherland’s law using the reference viscosity and temperature µref and Θref :

Ç å 3 Θ 2 Θref + S µ(Θ) = µref , (2.12) Θref Θ + S in which S is a temperature independent constant. Furthermore, assuming an ideal calorically perfect gas, the pressure p and internal energy e can be related to the temperature using:

R p = ρRΘ and e = Θ, (2.13) 1 − γ where R and γ represent the ideal gas constant and the ratio of the speciﬁc heat is given by γ = Cp/Cv.

2.2.2 Generalized Advection-Diffusion System

To conveniently solve the Navier-Stokes equations, (2.9) is rewritten in a generalized advection-diffusion T (AD) system using the set of unknown conservation variables U = (ρ, ρu1, ρu2, ρu3, ρE) . Furthermore, the assumptions and constitutive relations presented in Section 2.2.1 are applied. The resulting system can be written in the following short form:

adv diff U,t + Fi,i − Fi,i − S = 0 on Ω ∀ t ∈ (0,T ), (2.14) where Fadv , Fdiff , and S are the advective and diffusive flux terms and the vector of source terms, respectively:

      ρui 0 0        ρu u + p δ   τ   ρb   i 1 1i   1i   1  adv   diﬀ     F =  ρu u + p δ  , F =  τ  , S =  ρb  , (2.15) i  i 2 2i  i  2i   2         ρuiu3 + p δ3i   τ3i   ρb3        ρuiE + p ui τijuj − qi ρ (b · u + r) where τij are the components of the viscous stress tensor contained in Equation (2.10). Note that in Equation (2.14), the indices relate to the components along with the principal axis directions, the temporal and spatial derivatives are denoted by (·),t and (·),i, the Einstein summation convention is assumed for repeated indices, and δi refers to the Dirac delta function.

12 2.2.3 Boundary and Initial Conditions

To form a well-posed initial boundary value problem, the generalizedAD system (2.14) is supplemented with a set of boundary and initial conditions. In general, the initial condition needs to be defined such that it is physically consistent. The boundary conditions can be categorized into two classes: impermeable walls, which represent surfaces of the geometry, and artificial boundaries, e.g., to represent far-field conditions at the exterior boundaries of the computational domain.

Impermeable Walls

When considering inviscid fluid flows, a slip-condition at the impermeable walls can be imposed. This boundary condition states that the normal fluid velocity at the wall is equal to the normal velocity of the wall itself. Assuming a stationary boundary, this condition is stated as:

u · n = 0 on Γ ∀ t ∈ (0,T ). (2.16)

Viscous fluid flows, on the other hand, require a more stringent no-slip boundary condition. This condition states that the velocity of the fluid is equal to the velocity of the wall itself. Hence, when again assuming a stationary boundary, this condition states that:

u = 0 on Γ ∀ t ∈ (0,T ). (2.17)

Compressible ﬂuid ﬂows, in addition to the no-slip boundary condition, require a thermal boundary condition. In the case of isothermal walls, a Dirichlet condition can be imposed on the temperature:

Θ = Θwall on Γ ∀ t ∈ (0,T ), (2.18)

with Θwall being the wall temperature. Alternatively, to obtain an adiabatic wall, a homogeneous

Neumann condition is imposed by setting the normal heat ﬂux qn to zero:

qn = −κ∇Θ · n = 0 on Γ ∀ t ∈ (0,T ). (2.19)

Artiﬁcial Far-Field Boundaries

Defining artificial far-field boundary conditions to obtain a well-posed problem depends on local flow properties and is far from trivial. In general, the required flow conditions for well-posed problems can be categorized by the local Mach number M = ||un||/c where un and c represent the local normal fluid velocity and speed of sound, respectively. For supersonic flow (M ≥ 1):

T • Inﬂow boundaries, where un < 0 and |un| > c, all elements of U = (ρ, ρu1, ρu2, ρu3, ρE) are speciﬁed.

• Outﬂow boundaries, where un > 0 and |un| > c, no boundary condition needs to be imposed.

For subsonic flow where M < 1, the requirements on boundary conditions are more complicated. Again we distinguish between inflow and outflow conditions:

T • Inﬂow boundaries, where un < 0 and |un| < c, all but one element of U = (ρ, ρu1, ρu2, ρu3, ρE) is speciﬁed.

13 • Outﬂow boundaries, where un > 0 and |un| < c, one element of U is speciﬁed.

Which elements of U need to be imposed in case of subsonic ﬂow problems follows from a linearized Riemann analysis. In the current work, the boundary conditions are chosen such that a well-posed problem is obtained. An extensive review of boundary conditions for hyperbolic problems can be found, e.g., in [30–32].

Imposing Boundary Conditions

Both the impermeable and artiﬁcial boundary conditions lead to a set of Dirichlet and Neumann boundary conditions. In the framework of the generalizedAD system, these boundary conditions and initial condition are imposed as follows:

U = g on Γg ∀ t ∈ (0,T ), (2.20a) diﬀ n · F = h on Γh ∀ t ∈ (0,T ), (2.20b)

U(x, 0) = U0(x) on Ω0. (2.20c)

Here, g and h represent the Dirichlet and Neumann boundary conditions and U0 the initial condition at t = 0. Ω0 is the problem domain at time t = 0, and Γ = Γg ∪ Γh with Γg ∩ Γh = ∅. While the unknowns in the generalizedAD system in Equation (2.14) are solved for the conservation variables, it is not directly trivial how the above boundary conditions are imposed. This also holds for the coupling conditions introduced in Section 2.1 for FSI problems. Therefore, in the next section the generalizedAD system is modiﬁed to allow for different variable sets.

2.2.4 Choice of Variable Sets

As shown in Section 2.2.2, the Navier-Stokes equations are naturally written as a system of equations T with unknown solution vector U = (ρ, ρu1, ρu2, ρu3, ρE) . In this section, however, another choice of unknown solution variables is discussed. The material presented in this section is mostly based on the work in [33, 34]. Other than writing the generalizedAD system as a function of conservation variables U, Equa- tion (2.14) can also be rewritten as a function of other variable sets. In addition to conservation variables, primitive variables or entropy variables can be chosen. An extensive study on the choice of variables and how they affect the generalizedAD system can be found in [35–37]. In the current work, a convenient choice of variables known as the pressure-primitive T variable set defined by Y = (p, u1, u2, u3,T ) is used. This set consists of the primitive pressure, velocity, and temperature variables. The benefit of using Y is the simplicity in which boundary conditions of the pressure, velocity, and temperature can be imposed. This can be extremely useful in the context of fluid-structure interaction and the related coupling conditions discussed in Section 2.1. Furthermore, the formulation is well-defined in the incompressible limit, as the fluid’s density is not part of the variable set itself [36]. To obtain the pressure-primitive form of (2.14), it is convenient to rewrite the system in the reduced T energy form first using U˜ = (ρ, ρu1, ρu2, ρu3, ρe) . Here, the total energy per unit mass E is replaced by the internal energy per unit mass e.

14 The resulting reduced form of the generalizedAD system is now written as:

˜ adv\p p sp diﬀ U,t + Fi,i + Fi,i + F − Fi,i − S = 0 on Ω ∀ t ∈ (0,T ). (2.21)

adv p adv\p Here, the advective flux vectors are split into two parts: Fi = Fi + Fi , and additionally a stress power term Fsp is defined. These newly introduced terms are given by:         ρui 0 0 0          ρu u   p δ   0   τ   i 1   1i     1i  adv\p   p   sp   diff   F =  ρu u  , F =  p δ  , F =  0  , F =  τ  . i  i 2  i  2i    i  2i           ρuiu3   p δ3i   0   τ3i          ρuie 0 p ui,i − τi,jui,j −qi (2.22) The reduced energy form in Equation (2.21) can now be rewritten as a function of pressure-primitive variables using a change of variables: U˜ = U˜ (Y) [36]. For convenience, the following additional set of mappings is introduced:

∂U˜ ∂Fadv\p ∂Fp A = , Aadv\p = i , Ap = i . (2.23) 0 ∂Y i ∂Y i ∂Y For the stress power and diffusive ﬂux terms, we additionally deﬁne

sp sp diﬀ A Y,i = F , KijY,j = Fi . (2.24) The full matrix forms Equations (2.23) and (2.24) can be found in, e.g., in [34, 36]. Inserting the variable transformation U˜ = U˜ (Y) and mappings (2.23) and (2.24) into (2.21) yields the following residual form of the generalizedAD system as a function of Y:

adv\p p sp Res(Y) = A0Y,t + Ai + Ai + Ai Y,i − (KijY,j),i − S = 0 on Ω ∀ t ∈ (0,T ). (2.25) As already stated, this pressure-primitive form is especially suitable for problems involving ﬂuid-structure interaction phenomena. This follows from the fact that the quantities at the coupling interface, such as traction, heat ﬂux, velocity, and temperature, can easily be derived or set directly as boundary conditions.

2.3 Incompressible Flows

Although the compressible Navier-Stokes equations derived in the previous section are applicable for a wide range of Mach numbers, it is common to assume incompressibility when the Mach number approaches zero. Following the extensive discussion on the conditions for incompressible ﬂows in [32], for problems where M < 0.3, it is typically acceptable to assume incompressibility. In the incompressible limit (M → 0), the variation in ﬂuid density becomes negligibly small. In fact, ∂ρ it is generally accepted to assume that ∂t ≈ 0. Following this assumption, (2.9) can be reduced to the incompressible Navier-Stokes equations. The resulting model problem is given by the following system of equations:

∇ · u = 0 on Ω ∀t ∈ (0,T ), (2.26a) ∂u − ν∇2u + (u · ∇)u + ∇p − ρb = 0 on Ω ∀t ∈ (0,T ), (2.26b) ∂t ∂E ρ + ρ∇ · (Eu ) − ∇ · (Tu) + ∇ · q − ρb · u − ρr = 0 on Ω ∀t ∈ (0,T ), (2.26c) ∂t

15 Notice in (2.26), when neglecting viscous dissipation and assuming constant material properties, the energy equation becomes decoupled from the mass and momentum equation. Hence, the mass and momentum equation can now be solved separately from the temperature equation. Combining the mass and momentum equation with a set of initial and boundary conditions gives

∇ · u = 0 on Ω ∀t ∈ (0,T ), (2.27a) ∂u − ν∇2u + (u · ∇)u + ∇p − ρb = 0 on Ω ∀t ∈ (0,T ), (2.27b) ∂t

u = uD on ΓD ∀t ∈ (0,T ), (2.27c) T −p n + 2 ν n · (∇u + (∇u) ) = t on Γt ∀t ∈ (0,T ), (2.27d)

u(x, 0) = u0 on Ω0. (2.27e)

Here, ΓD and Γt represent the Dirichlet and Neumann portion of the boundary where the velocity uD or tractions t are imposed, respectively. Furthermore, when applying Fourier’s law, the energy equation (2.26c) can be written as a function of the temperature Θ. Combining the result with a set of initial and boundary conditions yields the following temperature problem: ∂Θ + u · ∇Θ − α∇2Θ − Φ = 0 on Ω ∀t ∈ (0,T ), (2.28a) ∂t

Θ = ΘD on ΓD ∀t ∈ (0,T ), (2.28b)

−n · α∇Θ = qn on Γqn ∀t ∈ (0,T ), (2.28c)

Θ(x, 0) = Θ0 on Ω0. (2.28d)

Here, α = κ/ρcp denotes the thermal diffusivity, Φ represents the viscous dissipation. Moreover, ΓD and

Γqn represent the boundary portions at which the Dirichlet temperature ΘD and normal heat ﬂux qn are imposed respectively. With a few modiﬁcations, (2.28) can be used to describe thermal behavior in solids, as is presented in Section 2.5.

Natural Convection using the Boussinesq Approximation

In most cases involving incompressible flows, it is justifiable to assume a decoupling of the energy equation from the mass and momentum equations. However, in some specific problems, this coupling is reintroduced. When simulating natural convection due to temperature-induced density variations, for example, the fluid density depends on the local temperature of the fluid. In the context of incompressible flows, such buoyancy effects can be approximated by adding a temperature-dependent body force b(Θ) per unit mass to the momentum equation. This gravity-based body force is derived based on the Boussinesq approximation [30, 38]:

b = (1 − β(Θ − Θref ))g, (2.29) where β is the thermal expansion coefﬁcient of the ﬂuid, Θref a reference temperature, and g the gravitational acceleration. Other cases that introduce a coupling between the mass, momentum equations and the temperature equation are temperature-dependent material properties such as the viscosity µ(Θ), or the consideration of viscous dissipation Φ(u).

16 2.4 Elastic Structures

As done to derive the derivation of the fundamental equations of ﬂuid motion, the elastodynamic response of structures can be described using Newton’s second law of motion. Using a structural domain Ω with boundary Γ, which we consider here from a Lagrangian point of view (see Section 2.1.1) the law can be expressed as:

d2d ρ = ∇ · T+ ρb on Ω, ∀ t ∈ (0,T ), (2.30) dt2 where d(x, t) denotes the material displacement and T the Cauchy stress tensor. As can be observed, Equation (2.30) describes the balance between the rate of change of momentum of the structure and the forces applied to it. Note that here we assume that the problem is isothermal. Combining Equation (2.30) with a set of initial and boundary condition yields the following model problem:

d2d Ä ä ρ = ∇ · SFT + b on Ω ∀ t ∈ (0,T ), (2.31a) dt2 0

d = dD on ΓD ∀ t ∈ (0,T ), (2.31b)

FS n0 = h on Γh ∀ t ∈ (0,T ), (2.31c)

d(x, 0) = d0(x) on Ω0. (2.31d)

Here, d0 is the initial displacement ﬁeld at t = 0, while dD and h represent the Dirichlet and Neumann boundary conditions on ΓD and Γh respectively. Furthermore, F represents the deformation gradient and S the Piola-Kirchoff stresses deﬁned in the next section. The superscript 0 in this equation refers to the reference state of the structure. Note that Equation (2.31) is already in closed form. This is achieved by applying constitutive models introduced in the next section.

2.4.1 Constitutive Models

In this work, plastic deformations are not considered while large elastic displacements are allowed. Hence, the structural continuum is considered to be homogeneous and isotropic via the linear elastic Saint Venant-Kirchoff material model. The Saint Venant-Kirchoff material model relates the second Piola-Kirchoff stresses S to the Green- Lagrange strains E using the linear stress-strain relation:

S = η tr(E) + 2ϕE. (2.32)

Here, η and ϕ are the Lamé parameters. Additionally, the Green-Lagrange strains are given by

1 î ó E = (F )T F − I , (2.33) 2 where I is the identity matrix. Since the Green-Lagrange strain deﬁnition is a nonlinear kinematic relation, Equation (2.31) is geometrically nonlinear, meaning that it allows for large displacements and rotations with only small strains [39, 40].

17 2.5 Heat Transport in Solid Structures

When we do not assume the structure to be isothermal, the thermal energy within the structure needs to be considered. This can be achieved by solving a heat equation similar to that for incompressible ﬂuids presented in Section 2.3. Heat transfer in a solid continuum, due to temperature differences and thermal loads, is described by the heat equation. Depending on the material type of the continuum, transport can take place through conduction. Note that for solid materials, convection plays no role. Furthermore, for simplicity, thermal heating through radiation is neglected in this work. n Using a domain Ω ∈ R sd with boundary Γ, the resulting equation that describes heat transfer in solid bodies by means of the temperature, Θ(x, t), can be stated as:

∂Θ ρ c − ∇ · (κ∇Θ) − G = 0 on Ω, ∀ t ∈ (0,T ). (2.34) p ∂t

Here, cp and κ are the specific heat and the thermal conductivity coefficient for the material under consideration. G denotes the thermal heat source per unit volume. Note that, analogous to the fluid equations, Fourier’s law given by Equation (2.11) is used to express the heat flux as a function of the temperature Θ. When isotropic materials and constant coefficients in space and time are assumed, Equation (2.34) can be rewritten and combined with a set of initial and boundary conditions. This yields the following model problem for heat transport in solids: ∂Θ G − α∇2Θ − = 0 on Ω, ∀ t ∈ (0,T ), (2.35a) ∂t ρcp

Θ = ΘD on ΓD, ∀ t ∈ (0,T ), (2.35b) ∂Θ α(n · ∇)Θ = α = q on Γ , ∀ t ∈ (0,T ), (2.35c) ∂n n qn

Θ(x, 0) = Θ0(x) on Ω0, (2.35d)

where α is the thermal diffusivity. The Neumann boundary condition qn defines the heat flux. When non-isothermal problems are considered, Equations (2.31) and (2.35) are coupled. In the current work, this coupling is achieved in a weak sense by imposing the temperature-dependent thermal strains onto Equation (2.31). These thermal strains are defined in next.

2.5.1 Weakly-Coupled Thermal Strains

If a thermo-mechanical problem is considered, the thermal stresses due to thermal expansion of the problem’s material need to be accounted for. In the current work, thermo-mechanical effects are treated by adding thermal strains to the elastodynamic problem. The temperature-induced thermal strain is deﬁned by:

EΘ = β(Θref − Θ), (2.36) where β is the thermal expansion coefﬁcient for the problem’s material and Θref is a zero strain reference temperature. In solving thermo-mechanical problems, Equation (2.36) is simply subtracted from the elastic strains. When a weak coupling for the structural problem is assumed, the added thermal strains result in an additional right-hand side term in Equation (2.31a).

18 Chapter 3

Concepts of The Finite Element Method

The model problems in Chapter2 are numerically solved using the FEM. Since its introduction to aerospace applications in the 1950s, FEM have evolved into a powerful tool within the field of computational engineering [14]. The popularity of FEM can be attributed to its versatility and the possibility of implementing the method in a generic fashion. The commonly used finite element methods are based upon the Galerkin formulation. Such formulations are very suitable for solving elliptic or parabolic partial differential equations (PDEs), as they lead to symmetric stiffness matrices. For such diffusion-type problems, the difference between the exact and finite element solution is minimized with respect to the energy norm [41]. Although FEM is the general choice for diffusion-type problems, difficulties arise when considering convection-dominated problems. In such cases, the standard Galerkin finite element formulation under- represents the amount of diffusion in the system, which can lead to spurious modes in the finite element solution. This phenomenon makes the use of widely studied stabilization techniques a necessity. In this chapter, the building blocks required to construct stabilized finite element formulations will be presented. For this, a simple generic model problem is used as an example. The techniques in this chapter are utilized in the remainder of this thesis.

3.1 The Strong and Weak Form

The necessary general ingredients of the FEM will be explained using a simple example problem. In this case we use the heat equation as given by (2.35) in Section 2.5. The simplicity of this scalar transport problem makes it very suitable for introducing all the important FEM building blocks. For convenience

Equation (2.35) is repeated below where; for simplicity, s = G/(ρcp).

∂Θ − α∇2Θ = s on Ω, ∀ t ∈ (0,T ), (3.1a) ∂t

Θ = ΘD on ΓD, ∀ t ∈ (0,T ), (3.1b)

α(n · ∇)Θ = qn on Γqn , ∀ t ∈ (0,T ), (3.1c)

Θ(x, 0) = Θ0(x) on Ω0. (3.1d)

19 The above set of equations represents the strong form of the model problem. This terminology reﬂects the strong smoothness requirements on Θ. That is, for Equation (3.1) the solution Θ must be C2 continuous on Ω. The ﬁrst step towards a discretizedFE formulation of (3.1) is to formulate its weak or variational form. This is achieved by multiplying (3.1a) with a weighting function w and integrating over the domain Ω: Z Å∂Θ ã Z w − α∇2Θ dΩ = w s dΩ. (3.2) Ω ∂t Ω Note that in this case, Θ still needs to be twice differentiable. However, the second derivatives of Θ do not need to be continuous but square-integrable only. Or Θ ∈ H2(Ω), where H2 is the second-order Sobolev space (see, e.g, [30]). Next, integration by parts is performed and the divergence theorem is applied to obtain:

Z Å∂Θ ã Z Z w − s dΩ + ∇w · (α∇Θ) dΩ = w (n · α∇Θ)dΓ. (3.3) Ω ∂t Ω Γ Equation (3.3) only contains ﬁrst-order spatial derivatives of both the weighting function w and the trial solution Θ. Hence, w and Θ and their spatial derivatives have to be square integrable, or, Θ, w ∈ H1. ∂Θ For the remainder of this section, let us assume ∂t = 0 for simplicity. In Section 3.3 the discussion on the FEM with respect to transient problems will be continued. All the requirements on the trial and weighting functions Θ and w are met when they are chosen from the following function spaces:

1 S = { Θ | Θ ∈ H (Ω), Θ = g on ΓD}, (3.4a) 1 V = { w | w ∈ H (Ω), w = 0 on ΓD}, (3.4b)

where ΓD ∪ Γqn = Γ. Note that the solution at the Dirichlet portion of the domain boundary is known. Hence, we can set w = 0 on ΓD. As a result, the boundary integral in Equation (3.3) is reduced to an integral over the Neumann portion of the domain boundary only. Incorporating this into the weak formulation yields: Z Z Z ∇w · (α∇Θ) dΩ = w s dΩ + w qn dΓ ∀w ∈ V. (3.5) Ω Ω Γqn The solution Θ of Equation (3.1) is, by construction, also a solution of (3.5). The existence and ∂Θ uniqueness of solution Θ ∈ S for steady problems ( ∂t = 0) can be shown using the Lax-Milgram lemma [42]. In addition to the full integral forms of the weak formulations, in the remainder of this work a short-hand notation is used for writing inner products. In short-hand notation, Equation (3.3) is written as:

a(w, Θ) = (w, s) + (w, h)Γqn , (3.6) with Z Z Z

a(w, Θ) = ∇w · (α∇Θ) dΩ, (w, s) = w s dΩ, (w, qn)Γqn = w qn dΓ. (3.7) Ω Ω Γqn Thus far, no discretization is applied. In the next section, Equation (3.7) will be discretized by replacing the functions spaces in Equation (3.4) by their ﬁnite counter-parts.

20 3.2 Discrete Weak Form

To solve Equation (3.7) numerically, it needs to be discretized using aFE mesh Ωh = T h(Ω). Trian- gulation Ωh and its boundary Γh with characteristic length h represent the approximation of Ω and Γ respectively. Subsequently, the trial and weighting functions are replaced by Θh and wh, which are now chosen from the ﬁnite dimensional function spaces Sh ⊂ S and Vh ⊂ V. Replacing the trial and weighing functions in (3.7) by their discrete counterpart Θh and wh gives us the following discrete ﬁnite element formulation: Find Θh ∈ Sh such that:

h h h h h h a(w , Θ ) = (w , s) + (w , h) h ∀w ∈ V . (3.8) Γqn

The above formulation is also known as Galerkin’s method, after Boris Galerkin, who is often credited as the inventor of the FEM. For Equation (3.8), we deﬁne the discrete solution Θh as a linear combination of shape functions Ni, with 1 ≤ i ≤ nn and nn being the number of nodes in the ﬁnite element mesh, as follows:

nin nn h X X Θ (x) = Ni(x)Θi + Ni(x)ΘD(xi). (3.9) i=1 i=nin+1

Here, nin denotes the number of internal nodes that are not on the Dirichlet boundary. Moreover, the arbitrary set of weighting functions is deﬁned as:

h h w ∈ V := span {Ni} . (3.10) i∈nin Inserting Equations (3.9) and (3.10) into (3.8) and assuming homogeneous Dirichlet and Neumann boundary conditions leads to a set of nin equations:

n Xin Z Z ∇N1(x) · α∇Ni(x) dΩΘi = N1(x) s dΩ, h h i=1 Ω Ω n Xin Z Z ∇N2(x) · α∇Ni(x) dΩΘi = N2(x) s dΩ, h h i=1 Ω Ω (3.11) . . n Xin Z Z ∇Nnin (x) · α∇Ni(x) dΩΘi = Nnin (x) s dΩ. h h i=1 Ω Ω This set of equations can be written as a linear system of equations:

KΘ = f, (3.12) where matrix K is the so-called stiffness matrix, Θ the vector of unknown temperatures at nodes i, and f the known terms such as possible source terms and non-zero boundary conditions. The above procedure to obtain a solvable linear system of equations applies to most diffusion-type problems. For non-linear PDEs such as the Navier-Stokes Equations, linearization using, e.g., Newton’s or a quasi-Newton method is necessary [43]. Furthermore, if the FEM is applied to convection-dominated problems, the standard Galerkin formulation leads to numerical instabilities. In this case, stabilization techniques are needed to obtain stable numerical results. More details on stabilization methods in the context ofFE are presented in Section 3.4.

21 3.2.1 The Isoparametric Concept

The system of equations (3.8) consists of a large number of integrals over the domain. In many FEM implementations the integral terms are evaluated element-wise, after which they are assembled into a global system matrix. This is possible due to the use of shape functions with compact support. Hence, the integral contributions are only non-zero on the element under consideration. To simplify the integral evaluations in the assembly process, the isoparametric concept is applied. This means that both the unknown solution and known geometry are defined using the same basis. By doing so, the element-wise integral terms can be mapped to a parametric reference element before the evaluation of the integrals themselves. By mapping all elements to a single reference element, numerical integration can be performed efficiently by an integration rule tailored specifically for the reference element. A linear polynomial triangular finite element, for example, is mapped onto a reference element with parametric coordinates (ξ, η) ∈ [0, 1] × [0, 1]. A graphic representation of this mapping is presented in Figure 3.1. y η

3 1.0

Ωe 2 =⇒

Ωref

1 3 1 0.0 x ξ 0.0 1.0

Figure 3.1: Triangular element Ωe in the physical space and its corresponding reference element Ωref in the parametric space.

The linear polynomial shape functions deﬁned on the reference element which correspond to the element node numbering in Figure 3.1 are given by:

N1(ξ, η) = ξ, N2(ξ, η) = η, N3(ξ, η) = 1 − ξ − η. (3.13) By applying the isoparametric concept, both the global coordinates x = (x, y)T and the solution Θ can be expressed as a function of the reference coordinates, by interpolation using the shape functions above:

nen h X x (ξ, η) = Ni(ξ, η) xi, (3.14) i=1 nen h X h Θ (ξ, η) = Ni(ξ, η)Θi , (3.15) i=1 with xi and Θi being the nodal coordinates and solution. nen represents the number of element nodes. ∂x From Equation (3.14), the Jacobian J = ∂ξ can be computed by applying the chain rule. Finally, using Je corresponding to element e, a change of variables can be applied to the element integrals. Or for an integral over element e, this yields: Z Z f(x) dx dy = f(x(ξ, η)) |detJe| dξ dη, (3.16) h Ωe Ωref

22 h where Ωe and Ωref represent a global element and the reference element, respectively. Note that the reference element is identical for all elements in the physical space, regardless of their shape or size. Additionally, the mapping information between global and reference elements is contained solely within J. Consequently, efﬁcient numerical integration procedures can be applied, such as Gauss quadrature, which, for standard Lagrangian ﬁnite elements and an adequate number of Gauss points, is exact.

3.3 Time Discretization

∂Θ To obtain a discrete formulation for unsteady problems ( ∂t 6= 0), Equation (3.3) needs to be discretized not only in space, but also in time. This can be achieved by the method of Rothe, method of lines, or directly using a space-time formulation, as depicted in Figure 3.2.

Temporal Discretization (FD) Continuous Coupled PDEs Problem in Space Method of Rothe

Spatial Space-Time Discretization Discretization (FE) (FE)

Method of Lines

st 1 Order in Time Discrete Coupled ODEs Problem

Figure 3.2: Discretization approaches in space and time. Via the method of lines and method of Rothe, the continuous problem is discretized in space and time usingFE and ﬁnite difference (FD) methods. Space-time formulations, on the other hand, discretize both space and time directly using ﬁnite elements.

Via semi-discretization [44, 45], the method of lines and method of Rothe lead to an intermediate semi-discrete form:

• With the method of lines, the continuous problem is ﬁrst discretized in space. This yields a semi- discrete system of ordinary differential equations (ODEs) in time. The semi-discrete form is then discretized in time to obtain the fully discrete form.

• For the method of Rothe, the order of discretizing in space and time is reversed, which leads to an intermediate semi-discrete system of PDEs in space before spatial discretization is applied.

The popularity of the FEM, driven by its ﬂexibility and strong mathematical foundation, led to the development of space-time (ST) ﬁnite element formulations (see, e.g., [46,47]). Such methods useFE

23 interpolation in both space and time. Hence, variational formulations are deﬁned and solved over the wholeST domain. One of the main advantages ofST formulations is that arbitrary domain deformations in time are naturally accounted for. This makesST methods very attractive for applications involving moving boundaries. The remainder of this section is dedicated to the application of the method of lines and theST ﬁnite element approach to model problem (3.1) and elastodynamic equation (2.31) as presented in Section 2.4. First aFD-based scheme is discussed in Section 3.3.1, followed by a discontinuous in timeST formulation in Section 3.3.2. Next, a generalized-α method is applied to the geometrically non-linear structural dynamics problem in Section 3.3.3.

3.3.1 The θ-Method

A very well-known family of finite difference time discretization methods is defined using the θ-method. This family of methods is often used for solving first-order PDEs. The θ-method is a single-step method which solves for the solution Θn+1 at time tn+1 using the previous solution Θn at time tn. The general form of the θ-method, using the incremental unknown ∆Θ = Θn+1 − Θn and time-step size

∆t = tn+1 − tn, is deﬁned as: ∆Θ − θ∆Θ = (Θ ) + O((1/2 − θ)∆t, ∆t2), (3.17) ∆t t n with θ ∈ [0, 1]. Note that (3.17) gives us ∆Θ, consequently, the solution at tn+1 follows from

Θn+1 = Θn + ∆Θ. As depicted in Table 3.1, the choice of θ leads to different numerical schemes with varying stability and convergence characteristics.

Table 3.1: Common choices of θ for the θ-scheme and their corresponding accuracy and stability characteristic.

θ Scheme Accuracy Stability θ = 0 Euler O(∆t) Conditional θ = 1/2 Crank-Nicolson O(∆t2) Unconditional θ = 2/3 Galerkin O(∆t) Unconditional θ = 1 Backward Euler O(∆t) Unconditional

To apply the θ-scheme to model problem (3.1), the weak form is obtained using the ﬁnite dimensional trial and weighting spaces. This directly gives us the weak form discretized in space: For any t ∈ [0,T ] h h ﬁnd Θ ∈ St such that: h h h h h h h h (w , Θ ) + a(w , Θ ) = (w , s) + (w , qn) h ∀w ∈ V . (3.18) t Γqn Note that Θ(x, t) is a function of space and time for unsteady problems. Consequently, the trial space of our problem is now time-dependent. The solution Θ lies in the trial space St written as:

1 St = { Θ | Θ(·, t) ∈ H (Ω), t ∈ [0,T ] and Θ(x, t) = g on ΓD}. (3.19) For simplicity, in the remainder of this section, the superscript h, indicating the use of ﬁnite- dimensional subspaces, is dropped. To apply the θ-scheme to (3.19), ﬁrst, Equation (3.17) is rewritten into its variational form: Å ∆Θã w, − θ (w, ∆Θ ) = (w, (Θ ) ) . (3.20) ∆t t t n

24 Next, we combine (3.18) and (3.20) for ∆Θt and (Θt)n (with ∆Θt = (Θt)n+1 − (Θt)n). This yields the following fully discrete form:

Å ∆Θã w, − θa (w, ∆Θ) = −a(w, Θ ) (3.21) ∆t n

+ (w, θ sn+1 + (1 − θ ) sn) + (w, θ (qn)n+1 + (1 − θ )(qn)n)Γqn . (3.22)

Here, weighted averaging is applied to the source term and the Neumann boundary condition s and qn, respectively. This formulation can now be solved for ∆Θ, which in turn gives us the new solution

Θ(tn+1). Note, to obtain Equation (3.22), the model problem is ﬁrst discretized in space followed by the time discretization. Hence, following Figure 3.2, this approach can be classiﬁed as a method of lines.

3.3.2 Space-Time Finite Element Methods

The earlyST formulations, e.g. proposed in [46, 47], employ continuous interpolation in time. A discontinuous-in-time approach is proposed in [48]. Rather than solving the completeST domain simultaneously, the problem can be reduced to smaller problems, which are then solved iteratively in time. In the context of fluid flow problems, the discontinuous-in-timeST formulation is supplemented with a characteristic streamline diffusion method in [49]. Further developments led to the deformable-spatial- domain / stabilized space-time (DSD/SST) method in [50]. This particular formulation resulted in further improved stability of the method and employed equal-order interpolation in space and time [50,51]. In the context of stableST finite element methods for compressible flows, significant advancements were made in [37, 52] using entropy variables. This was later extended for arbitrary variable sets in [35, 53]. With the work in [54], the concept ofST finite elements was further developed to support simplexST meshes. This allows for local time-step refinement, as was also demonstrated in [55] for thermally coupled two-phase flow simulations of mold filling processes. In the current work, however, we restrict ourselves to the flat ST DSD/SST formulation as proposed in [50, 51, 56, 57]. To obtain the finite element function spaces for the DSD/SST method, the time interval t ∈ [0,T ] is subdivided into sub intervals In = [tn, tn+1], with tn+1 = tn + ∆t. Time instances tn and tn+1 are chosen from an ordered series of time instances: t0 = 0 < t1 < . . . < tN = T .

With the assumption of deforming domains, the spatial domain and its boundary at tn are deﬁned as Ωn = Ω(tn) and Γn = Γ(tn). Furthermore, the surface Γ(t) as it traverses from Γn to Γn+1 in In is named Pn. 3 Using the above, a so-called time-slab Qn is deﬁned as the volume enclosed by Ωn, Ωn+1, and Pn.

A graphic representation of aST slab for a 2D spatial domain with triangulation T (Ωn) is presented in e 4 Figure 3.3. Here, Qn represents a triangularST element . TheFE discretization strategies presented in Sections 3.1 and 3.2 can also be applied toST finite elements. For this, let us first define the function spaces for the trial and weighting functions of a single time-slab:

1 (S)n = { Θ | Θ ∈ H (Qn), Θ = g on (Pn)D}, (3.23a) 1 (V)n = { w | w ∈ H (Qn), w = 0 on (Pn)D}. (3.23b)

3 Here, the term volume is used loosely for the domain enclosed in space and time by Ωn, Ωn+1, and Pn. 4 e While Qn in Figure 3.3 is a prism-type element, in this work theST elements are referred to as their space-only equivalent.

25 Ωn+1

t y

x e Ωn Qn

e Figure 3.3:ST slab withST element Qn. TheST slab is enclosed by spatial domains Ωn and Ωn+1 together with Pn.

Here, (Pn)D refers to the Dirichlet portion of Pn. The variational form of the model problem can now be obtained by multiplying its strong form (3.1) with the weighting function w, integrating over the time-slab

Qn, and applying integration by parts. This yields the following expression: Find Θ ∈ (S)n, such that

∀ w ∈ (V)n the following holds: Z Z Z + Ä + −ä w Θt dQ + ∇w · (α∇Θ) dQ+ wn Θn − Θn dΩ (3.24) Qn Qn Ωn Z Z = w s dQ + w qn dP. (3.25) Qn (Pn)qn This variational form makes use of the following limit and integral deﬁnitions:

± Θn = lim Θ(x, tn ± ε), (3.26a) ε→0 Z Z Z ... dQ = ... dΩdt, (3.26b) Qn In Ωn Z Z Z ... dP = ... dΓdt. (3.26c) (Pn)qn In (Γn)qn The third integral term in (3.25) is known as the jump-term. This term added to the variational form to ± weakly enforce continuity over the interface between consecutive time slabs. Here, Θn is the solution at tn defined on time-slab Qn−1 or Qn. Note that since the variational form in (3.25) is defined on theST slab Qn, the DSD/SST formulation implicitly accounts for arbitrary domain deformations. This makes its use for the analysis of problems involving FSI phenomena a natural choice. Similar to the derivation in Section 3.2, the discrete variational form of (3.25) can be obtained Ä ä Ä ä using finite dimensional sub spaces Sh ⊂ (S) and Vh ⊂ (V) . The resulting discrete form is n n n n unconditionally stable and has an accuracy of O(∆t3) in time [37].

3.3.3 Generalized-α Method

When considering time integration for linear structural dynamics, the focus is typically on the order of accuracy. This is because unconditional stability is easily satisﬁed for most algorithms. For non-linear

26 structural dynamics, on the other hand, the main point of concern often is numerical stability. The reason is that many methods that are unconditionally stable for linear problems potentially become unstable in the non-linear case. In [58], a generalized-α method is proposed. This method includes commonly used numerical dissipative and non-dissipative integration algorithms as special cases (i.e., Newmark’s method [59], the Hilber-α method [60], and the Bossak-α method [61]). The generalized-α method is applied to non-linear structural dynamics in [62], where the optimal algorithmic parameters are determined to obtain a time integration with low numerical dissipation for low-frequency modes, and at the same time high numerical dissipation for high-frequency modes. Furthermore, rather than the mid-point rule, a weighting average of the internal forces at the beginning and end of a time step is used. For linear problems, the generalized-α method is unconditionally stable, second-order accurate, and maintains an optimal balance of numerical dissipation in the high- and low-frequency range. Although originally designed for linear problems, the generalized-α scheme is successfully used in the context of non-linear problems in [63]. The concept of the generalized-α method and the various schemes it contains make it a popular choice for time integration of structural dynamics problems. In the current work, the generalized-α method as described in [63] is used. A brief summary of the derivation is given next. Starting with the equation of motion presented in Section 2.4 and rewritten in matrix form:

¨ Mdn+1−αm + Nn+1−αf (d) = Rn+1−αf . (3.27)

Herein, M is the mass matrix, N the nonlinear stiffness matrix, and R the vector of external and body forces. The vectors d and d¨ represent the displacement and its second-order temporal derivative, i.e., acceleration. Hence, Md¨ and N(d) denote the inertial and internal forces of the dynamic system. Physical damping effects are not taken into account in this work. Consequently, no damping terms occur in (3.27). For the interested reader, a derivation of the generalized-α method including damping terms can be found in the original publication [63]. The subscripts in (3.27) denote a weighted average between the values at the beginning and at the end of time interval [tn, tn+1]. For the displacement and acceleration vectors, as well as the external force vector, this weighted average is deﬁned as follows:

dn+1−αf = (1 − αf ) dn+1 + αf dn (3.28a) ¨ ¨ ¨ dn+1−αm = (1 − αm) dn+1 + αmdn, (3.28b)

Rn+1−αf = (1 − αf ) Rn+1 + αf Rn. (3.28c)

Notice here the distinction between the acceleration, displacement, and forcing terms by using different weighting coefﬁcients αm and αf . The mass matrix in Equation (3.27) is assumed constant in time. The internal forces, on the other hand, depend non-linearly on the displacement. Consequently, as suggested in [63], the time-discrete internal force vector Nn+1−αf can be deﬁned as:

Nn+1−α = (1 − αf ) Nn+1 + αf Nn, f (3.29) = (1 − αf ) N(dn+1) + αf N(dn).

27 Alternatively, the time-discrete displacement (3.28a) can be inserted directly into N:

Nn+1−αf = N(dn+1−αf ). (3.30)

Under the assumption of linear acceleration, and after deﬁning all the weighted average expressions for the individual terms of Equation (3.27), the so-called Newmark relations are used to reduce the number of unknowns [59, 64, 65]. These relations are deﬁned as follows for the displacement and velocity:

î ó d˙ n+1 = d˙ n + ∆t γd¨n+1 + (1 − γ)d¨n , (3.31a) 1 î ó d = d + ∆td˙ + ∆t2 2βd¨ + (1 − 2β)d¨ . (3.31b) n+1 n n 2 n+1 n In these expressions, the parameters β and γ control the numerical characteristics of the algorithm such as accuracy, stability, and numerical damping [59, 65]. Next, by solving Equation (3.31b) for d¨n and substituting the result in (3.31a) we obtain:

1 î ó ï 1 ò d¨ = d − d − ∆td˙ − − 1 d¨ . (3.32) n+1 β∆t2 n+1 n n 2β n

Inserting this result into the time-discrete acceleration (3.28b) yields: 1 − α 1 − α 1 − α − 2β d¨ = m (d − d ) − m d˙ − m d¨ . (3.33) n+1−αm β∆t2 n+1 n β∆t n 2β n

With this, an expression for the weighted average of the acceleration is obtained and naturally, the time-discrete equation of motion follows: Å1 − α 1 − α 1 − α − 2β ã M m (d − d ) − m d˙ − m d¨ + N = R . (3.34) β∆t2 n+1 n β∆t n 2β n n+1−αf n+1−αf

The choice of parameters in Equation (3.34) leads to different time integration schemes, each with their own numerical characteristics. The generalized-α method is obtained when choosing the following parameter values:

2ρ∞ − 1 ρ∞ αm = , αf = , ρ∞ − 1 ρ∞ − 1 1 1 β = (1 − α + α )2, γ = − α + α , 4 m f 2 m f

with ρ∞ being the spectral radius of the method used. The spectral radius can be seen as a measure of numerical dissipation, where a smaller spectral radius leads to an increase in numerical dissipation.

A detailed discussion on ρ∞ can be found in [58]. An overview of other parameter choices and the corresponding integration schemes they result in is provided, e.g., in [63, 66]. After discretization in space and time, what remains to be done before solving Equation (3.34), is the linearization of the non-linear terms. In the current work, the powerful and well-known Newton-Raphson method is applied. A detailed derivation of the Newton-Raphson method can be found, e.g., in [67].

3.4 Stabilized Finite Element Formulations

As already stated in Section 3.4, when considering convection-dominated ﬂow problems, the use of a FEM is not straight forward. The now present convection operators in the Galerkin formulation

28 are non-symmetric. As a result, the best approximation property in the energy norm is lost, and the GalerkinFE solution can suffer from node-to-node oscillations when convection dominated problems are considered. These spurious modes can only be removed through mesh refinement or by introducing a form of stabilization. For obvious reasons, refining the computational mesh for large and high-Reynolds number problems turns out to be impracticable. Hence, in the current work a streamline upwind Petrov-Galerkin (SUPG) type stabilization technique is employed. The general idea of SUPG is to consistently add a residual-based balanced amount of numerical dissipation to theFE formulation in the streamline direction. To demonstrate how SUPG stabilization is defined, let us use model problem (3.1) by adding first a convection term and rewriting it in the residual form:

2 R(Θ) = Θt + u · ∇Θ − α∇ Θ − s. (3.35)

This residual now contains the convective term u · ∇Θ with the ﬂow velocity u. The modiﬁed weighting function w˜, which adds a form of upwinding to standard Galerkin-type formulations, can be written as:

w˜ = w + P(w)τ. (3.36)

In this expression, P(w) and τ are known as the perturbation term and intrinsic time, respectively. While many expressions exist for τ, SUPG deﬁnes the perturbation term as:

P(w) = u · ∇w, (3.37) which naturally results in an added weighting in the streamline upwind direction. The intrinsic time τ in Equation (3.36) acts as a scaling of the amount of added diffusion or upwinding. The ideal τ adds only the minimum required amount of numerical diffusion to the discrete system in order to obtain a monotonous solution. Many variations of τ have been proposed in the literature. A selection of deﬁnitions for τ can be found in [30]. The modiﬁed weighting function in Equation (3.36) can be applied to the weak form of a standard Galerkin-type formulation by simply adding the following term:

(nel)n X Z P(w)τR(Θ) dΩ. (3.38) e=1 Ωe

Notice that this leads to a consistent form of stabilization since the perturbation term P(w) is applied to each term in the strong form residual R(Θ). Further R(Θ) → 0 when Θh approaches the exact solution. By modifying the weighting functions, an intrinsic characteristic of Galerkin’s method, the concept of using the same basis for both weighting and interpolation function, no longer holds. Instead, methods with modified weighting functions are classified as Petrov-Galerkin methods; hence, the naming of this stabilization technique (SUPG). As will be demonstrated in Sections 5.2 and 5.3, SUPG can be applied to both semi-discrete, andST FEM. Depending on whether steady or unsteady problems are assumed, minor modifications to P(w), R(Θ), and τ might be required. For the interested reader, a full history on the development of stabilization techniques for FEM can be found in [30]. More residual based and other stabilization techniques such as Galerkin / least-squares (GLS) and variational multiscale (VMS) are also presented therein.

29 3.5 Consistent Boundary Flux Method

In the context of FSI, coupling quantities such as tractions and heat fluxes are often not directly available. Additional steps are needed to derive these quantities from the flow solution itself. A natural choice to obtain such quantities is by post-processing the discrete solution using the already available interpolation functions. Most such schemes use a Gauss point evaluation of the derivatives of the basis functions along the boundary. This approach is not consistent and can lead to inaccurate flux computations [68]. In this work, we therefore apply a consistent boundary flux (CBF) method, as first proposed in [68] for conventional Galerkin methods, and later on extended to Petrov-Galerkin formulations in [69, 70]. The CBF method is consistent with the set of discretized equations in the sense that if the fluxes obtained via the CBF method are used as boundary conditions instead of the imposed Dirichlet boundary condition, the original discrete solution is obtained [68]. The simplified weak form of model problem (3.1) is used to formulate the CBF method. The result gives us the normal heat flux along a Dirichlet boundary. As a starting point, we use the discretized weak form (3.8). For simplicity, in this section, we drop the superscript h indicating the finite-dimensional spaces used. The full integral form of Equation (3.8) is stated as: Find Θ ∈ S, such that ∀w ∈ V the following holds: Z Z Z ∇w · (α∇Θ) dΩ = w s dΩ + w qn dΓ, (3.39) Ω Ω Γqn with trial and weighting spaces as defined in Section 3.1 and repeated for convenience here:

1 S = { Θ | Θ ∈ H (Ω), Θ = g on ΓD}, (3.4a) 1 V = { w | w ∈ H (Ω), w = 0 on ΓD}. (3.4b)

Solving (3.39) will give us the unknown temperature Θ. In the next step, by inserting the now known solution Θ, into the weak form and assuming w 6= 0 on ΓD, a new variational form can be constructed.

As a result of setting w 6= 0 on ΓD the integral over the Dirichlet boundary remains non-zero. Hence, with Θ being known, the normal heat ﬂux qnD on ΓD becomes the unknown value to be solved for: h Find qnD ∈ SCBF , such that ∀w ∈ VCBF the following holds: Z Z Z Z

∇w · (α∇Θ) dΩ − w s dΩ − w qn dΓ = + w qnD dΓ. (3.41) Ω Ω Γqn ΓD

As we are solving for qnD and modiﬁed the requirements on w, the function spaces used for (3.41) are deﬁned as:

1 SCBF = { qnD | qnD ∈ H (ΓD)} (3.42a) 1 VCBF = { w | w ∈ H (Ω)}. (3.42b)

Discretization of Equation (3.41) is done in a similar fashion as the model problem itself, which provides the normal boundary ﬂux of our model problem. The added computational cost of the CBF method is minimal. In fact, it is merely a post-processing step that is performed only once after the ﬁnal solution for Θ is found.

30 Numerical Example of the Consistent Boundary Flux Method

To demonstrate the favorable characteristics of the CBF method, it is applied here to the well-known Couette flow problem. As depicted in Figure 3.4, the Couette flow problem involves a flow between two infinite parallel walls, separated by a distance D in the y-direction. The bottom wall is fixed, while the top wall moves at velocity U. The temperatures on the top and bottom wall are fixed at temperatures Θtop and Θbot, respectively.

utop = U, Θ¯ top

D u = uout uout y

x ubot = 0, Θ¯ bot

Figure 3.4: Computational domain of the two-dimensional Couette ﬂow problem.

The analytic solution of the temperature distribution in non-dimensional form is given by:

P rEc Θ¯ = y∗ + y∗ (1 − y∗) , (3.43) 2

∗ where Θ¯ = (Θ − Θbot)(Θtop − Θbot) and y = y/D are the non-dimensional temperature and length, 2 respectively. P r = ν/α and Ec = U /cp(Θtop − Θbot) are the Prandtl and Eckert number with α, ν and cp representing the thermal diffusivity, kinematic viscosity, and speciﬁc heat. To obtain a numerical solution for the Couette problem, the incompressible Navier-Stokes equations and the heat equation presented in Section 2.3 are solved using aSTFE formulation complemented with SUPG-type stabilization. For this, the computational domain is discretized using 20 by 20 bi-linear elements and the problem parameters given in Table 3.2 are used.

Table 3.2: Parameters used for the 2D Couette ﬂow problem.

Parameter Magnitude Dimensions P r 100.0 [−] Ec 1.0 [−] Re 7.75 × 10−6 [−] Θ¯ top 1.0 [−] Θ¯ bot 0.0 [−]

The analytic and numerical temperature distributions perpendicular to the ﬂow direction are depicted in Figure 3.5. The numerical solution shows to be close to the exact analytical solution.

31 1

0.9

0.8

0.7

0.6 ] −

[ 0.5 y D 0.4

0.3

0.2 Θ¯ analytic 0.1 Θ¯ numerical 0 0 2 4 6 8 10 12 14 16 Θ−Θbot [−] Θtop−Θbot

Figure 3.5: Dimensionless temperature proﬁle for P r = 100 and Ec = 1. The solid line illustrates the analytic solution, the "+" represent the numerical solution.

The analytic non-dimensional heat ﬂux can be computed by simply taking the derivative of (3.43) with respect to y:

∂Θ¯ P rEc = 1 + (1 − 2y∗) . (3.44) ∂y 2 The heat flux based on the numerical solution is computed using the CBF method and the conventional approach using the finite-element basis itself. The latter approach can be specifically problematic when linear finite elements are employed due to the piece-wise constant spatial derivatives over individual elements. The analytically and numerically computed heat fluxes are presented in Table 3.3.

Table 3.3: Computed normal heat ﬂux at the top and bottom wall of the 2D Couette ﬂow problem.

∂Θ¯ ∂Θ¯ Method ∂y |bot ∂y |top Analytic 51.000 −49.000 CBF 51.052 −48.954 Conventional 49.051 −46.549

The results presented in 3.3 demonstrate the strength of the CBF method. Additionally, the available weak form of the model problem makes the CBF method a justiﬁable choice for boundary ﬂux evaluation. Especially considering FEM solvers and their use in the context of FSI problems where accuracy can affect convergence behavior and robustness.

32 Chapter 4

Spline-Based Methods

The term spline has been around for many years. Long before the arrival of digital computers, slender wooden strips known as splines were used as a drafting tool by naval architects. To draw smooth continuous lines, draftsmen forced the splines into the desired shape using lead weights known as ducks. The first instance of the term spline in a mathematical context dates back to 1946 [71]. However it was the work by automotive engineers Pierre Bézier and Paul de Faget de Casteljau, in which they utilized Bernstein polynomials to define curves and surfaces [72–74]. This research opened the door for the development of modern-day CAD tools, which eventually replaced the traditional drafting tools such as the wooden splines and ducks. Following this first step, many computational geometry technologies were developed. For engineering design tools, the most widely used are NURBS[75]. This is due to their ability to exactly represent all conic sections, as well as their convenient free-form surface modeling capabilities. As briefly mentioned in Chapter1, several methods have been developed to use NURBS directly within numerical analysis. The motivation for this development is driven by the lacking ability of classical isoparametric element formulations to represent complex and smooth geometries in an accurate manner. Additionally, direct analysis on CAD geometries would omit the meshing stage required to obtain a discrete computational domain. One such spline-based numerical method is IGA[19]. With IGA, the CAD geometry represented by a NURBS is used directly for analysis. This is done by using the NURBS-basis of the geometry to also define the numerical solution in the context of finite element analysis. This approach is shown to be very powerful compared to the classical C0-continuous representations for many different applications, such as solid mechanics [76], structural vibrations [77], free-surface flows [26] , and fluid-structure interaction [28, 29, 78, 79]. Although IGA is a very powerful tool in numerical analysis, generating IGA-suitable CAD geometries is still an ongoing challenge. E.g., properly defining geometries with complex features such as holes and trimmed edges can be extremely difficult. The NEFEM, first introduced in [24], fills the gap between standard isoparametric finite element formulations and fully spline-based isogeometric analysis. The NEFEM applies standard Lagrangian shape functions to all elements in the computational domain. However, the elements with a face or edge on the spline boundary are enhanced by incorporating a NURBS parametrization of the boundary itself. As shown in Figure 4.1, this leads to a geometrically exact domain with respect to the CAD surface model. The remaining majority of elements in the domain are

33 NURBS-based Standard NURBS-enhanced CAD geometry FE mesh FE mesh Figure 4.1: Steps of the domain discretization procedure of a NURBS-based CAD geometry for NEFEM simulations. treated in the classical sense as standard Lagrangian elements. By using this approach, the computational efﬁciency of C0 Lagrangian ﬁnite elements is maintained, while at the same time, the exact geometry is made available. As shown in Chapter7, this replaces the polynomial approximation of the boundary via the isoparametric concept. Furthermore, NEFEM only requires a NURBS representation of the domain boundary. Consequently, three-dimensional problems only require surface geometries and the need for volume splines is therefore omitted. In the remainder of this chapter, the concept of IGA, NEFEM, and the newly proposed three- dimensional non-Cartesian NEFEM formulation are presented. However, before continuing on the topic of NEFEM, a good understanding of NURBS is necessary. Therefore a brief review on this topic is given next.

4.1 Non-Uniform Rational B-Splines (A Very Brief Introduction)

Geometries can be described by means of NURBS[80], or T-splines [81]. Due to their favorable properties, such descriptions are commonly used by CAD tools to represent geometric objects. A geometric NURBS curve in the physical space C(θ) is a function of parametric coordinate θ and

nsd p describes a geometric curve in R . Such a parametrization is constructed using a NURBS basis Ri and n a set of control points Pi(x) ∈ R sd :

ncp X p C(θ) = Ri (θ)Pi(x). (4.1) i=1

Here, ncp is the number of control points, and superscript p denotes the degree of the NURBS basis. p th p Ri (θ) are rational functions which are computed from p -degree B-spline basis functions Bi (θ), using a set of ncp control weights wi:

p p Bi (θ)wi Ri (θ) = Pn p . (4.2) j=1 Bj (θ)wj

34 3 Bi

3 3 B1 B7 3 3 3 B4 3 3 B2 B3 B5 B6

θ 1 2 3 4

3 Figure 4.2: Cubic B-Spline basis for Ξ = [0, 0, 0, 0, 1, 2, 3, 4, 4, 4, 4]. Note, in this example only B4 has the maximum basis function support of p + 1 knot spans. Recreated from [80].

p The B-spline basis functions Bi (θ) itself are generated using the Cox-de Boor recursion formula:  0 1 if θi ≤ θ < θi+1 Bi (θ) = (4.3a) 0 otherwise

p θ − θi p−1 θi+p+1 − θ p−1 Bi (θ) = Bi (θ) + Bi+1 (θ). (4.3b) θi+p − θi θi+p+1 − θi+1

The values θi are referred to as knots and are collected in an ordered knot-vector Ξ = [θ1, θ2, ··· θnk ], where θi ≤ θi+1, and nk is the number of knots given by nk = ncp + p + 1. The knot spans [θi, θi+1), ∞ i = 1, ..., nk − 1, deﬁne element domains where the basis functions are C continuous. At the knots themselves, the basis functions are Cp−k continuous. Here, k is the multiplicity of a knot. p Provided that the multiplicity of the knots in Ξ does not exceed p, the B-spline basis functions Bi (θ) are continuous. An example of such a case is presented in Figure 4.2 for a cubic NURBS curve. Elements, in the context of IGA, can refer to a NURBS patch or a single knot span. Since in the current work only single-patch splines are considered, the term element refers to knot spans. E.g., in Figure 4.2 the parameter space is subdivided into four equally spaced elements between 0 ≤ θ ≤ 4. The B-Spline basis functions come with a number of important characteristics for numerical analysis. First of all, the basis functions constitute a partition of unity and are non-negative for all θ. Each basis function is Cp−k continuous across the element boundaries. Hence, continuity can be controlled by the order and knot-multiplicity of the spline. Furthermore, the support of the basis functions is at most p + 1 knot spans. Consequently, higher-order splines have increased support of the basis functions. A NURBS curve corresponding to the basis from Figure 4.2 is depicted in Figure 4.3. Analogous to Equations (4.1) and (4.2), a NURBS surface S(θ) can be obtained by taking the tensor p,q product of two NURBS curves [27]. The resulting description for S(θ) is deﬁned using the basis Ri,j ( θ ) and a net of control points Pij(x):

ncp mcp X X p,q S( θ ) = Ri,j ( θ )Pij(x). (4.4) i=1 j=1

Note that, in this case, two parametric coordinates are used along the principal directions of the NURBS surface, i.e., θ = (θ1, θ2)T (see Figure 4.4). Since the NURBS surface is constructed by means of a

35 P4

P1 P2 P7 P6

Figure 4.3: NURBS curve corresponding to the knot vector used in Figure 4.2. The dashed and solid lines denote the elements on the curve and correspond to that indicated in Figure 4.2. Recreated from [80].

y x θ2

θ1

Control point

3 1 2 T Figure 4.4: NURBS surface in R , and its corresponding parametric coordinate system θ = (θ , θ ) .

p,q tensor product, the order of the basis Ri,j ( θ ) can be chosen independently in each parametric coordinate direction given by p and q. Similarly the number of control points ncp, and mcp can be chosen independently. A similar approach as presented here can be derived for three or more dimensions. An extensive discussion on NURBS and the underlying mathematical foundation is given in [27, 80].

4.2 Isogeometric Analysis

As already stated, IGA was first introduced in [19]. The general idea of IGA is to directly use the spline basis of a CAD geometry for numerical analysis in the context of FEM. Although spline types such as T-Splines [20, 82, 83] or locally refined B-Splines [84] can be used, in the current work, we restrict ourselves to the use of NURBS only. The isoparametric approach discussed in Section 3.2.1 uses the basis of the unknown solution fields to approximate the known geometry. In isogeometric analysis, this concept is turned around in that the NURBS basis of the exact geometry is used to approximate the numerical solution. This means that not only the geometry (e.g., given by the expressions in Equation (4.1) or (4.4) for p,q curves or surfaces), but also the solution d(θ) of the finite element problem is interpolated using Ri,j (or

36 η

1 ξ

Ωref −1 Υ −1 1

2 θj+1

Ωθ 2 2 θ Ωe θ j y

1 1 x θi θi+1 θ1 Figure 4.5: Mapping from the physical space to the parametric space, and from the parametric space to the reference element.

p equivalently Ri for 1D problems):

ncp mcp ˜ X X p,q ˆ d( θ ) = Ri,j ( θ )dij. (4.5) i=1 j=1

Here, the control points Pij are replaced with control variables dîj, which represent the discrete solution. Furthermore, instead of a curve or a surface, d˜( θ ) in Equation (4.5) now represents the spline-based numerical solution. Note, however, due to the non-interpolatory nature of the basis, the control variables dîj do not have a direct physical meaning as is the case with nodal solutions of standard finite element formulations. To obtain a numerical solution d in the physical space Ωe (see Figure 4.5), the exact geometry x( θ ) can be used:

d = d˜ ◦ x−1. (4.6)

Note, when the unknown solution d represents the displacement, e.g., for the elastic problem in Section 2.4, the solution in Equation (4.5) is already deﬁned in the physical space. In this case, the deformation needs to be combined with the undeformed state of the geometry in order to obtain the new deformed state of the structure. The isogeometric concept can be applied directly to a Galerkin or Petrov-Galerkin formulation. Within the assembly procedure, this leads to element integral terms. In this work, integration of these terms is done via Gaussian quadrature. For this, as shown in Figure 4.5, the integrals are mapped to a reference element via the parametric domain of the NURBS using x and Υ. This leads to a change of variables similar to Equation (3.16).

Note that the NURBS bases are polynomials only if wi = 1 ∀i. However, using the same Gauss quadrature rule for a pth order NURBS as done for polynomials of the same order has shown to be very

37 effective [27]. Depending on the order of a given NURBS, quantities such as derivatives can be computed accurately and in a straightforward fashion. This also holds for derivatives of the IGA solution d( θ ), as it is constructed using the same NURBS. Typically, the NURBS basis of the geometry is not sufficient for accurate numerical analysis. That is why a larger function space is typically required. This can be achieved through h-, k-, and p-refinement techniques [19, 85]. Although this inevitably results in larger linear systems, a higher accuracy per degree of freedom is obtained compared to standard finite element formulations [77, 86, 87]. A full and detailed discussion on IGA is given in [19, 27, 88].

4.3 NURBS-Enhanced Finite Elements

While a wide variety of mesh generation tools are available for classical Lagrangian finite element meshes, the contrary is true for tools to generate IGA-suitable NURBS geometries. In fact, for many complex engineering geometries, the generation of a suitable NURBS is not always practical. Despite the recent effort to tackle these issues [20, 21], a robust and user-friendly approach for real-world engineering geometries is yet to be found. To circumvent the need for complex volume splines and fill the gap between classic isoparametric finite element methods and IGA, the NEFEM was introduced in [24]. Instead of the full geometry, the NEFEM only requires a CAD surface geometry represented by a NURBS. Such geometries can be provided by most standard industry CAD tools. Hence, with the NEFEM the need for volume NURBS is omitted. For the NEFEM, a standard standard finite element mesh is generated from a CAD model represented by a NURBS surfaces. Subsequently, the CAD model is used to NURBS-enhance the elements with an edge or face on the boundary (see Figure 4.1). This yields two groups of elements. The first group consists of interior elements, which are treated as standard finite elements in the classic sense. The remaining elements are the NURBS-enhanced elements. The enhanced elements have either an edge or face on the NURBS, as depicted in Figure 4.6. The NURBS geometry itself is made available during the element integration using a suitable mapping. By using such a mapping, the positions of integration points are no longer determined by an approximate geometry but by the exact NURBS geometry instead. This leads to a shift of the quadrature points as shown in Figure 4.7. Furthermore, the NURBS geometry can be exploited for the evaluation of boundary integrals. In this case, integration points and surface normals are defined along the NURBS geometry rather than the approximated geometry, and the NURBS basis can be used for interpolation. Doing so increases the accuracy of the evaluation of these integrals. This can be particularly beneficial for mapping fluid forces along a coupling interface when considering FSI problems, as is demonstrated for two-dimensional problems in [28].

4.3.1 Cartesian NEFEM

In the original formulation of the NEFEM[24], the test and interpolation functions are Lagrange polynomi- n als deﬁned in the physical space using coordinates x in R sd . This so-called Cartesian approach ensures

38 x2 x2

x1(θ1) x3(θ3) x3(θ3)

x4(θ4) x1(θ1) θ2

θ1

Figure 4.6: Possible common face or edge conﬁgurations of boundary elements. Note that the nodal coordinates of the element nodes touching the NURBS can be expressed using the parametric coordinates and the NURBS itself.

Figure 4.7: Integration point positioning within an NEFEM element along a curved NURBS surface. Here, nodes x1, x3 and x4 are the nodes on the NURBS surface.

39 the reproducibility of polynomials in both the reference and the physical space. This is independent of the order of the polynomials itself [89]. An oddity that occurs with the Cartesian NEFEM approach, as discussed in [28], is that of non-zero interior shape function contributions along the NURBS boundary. This may cause the partition of unity property along the element interface on the NURBS no longer to be fulﬁlled, as shown in Figure 4.8a. Such behavior introduces unwanted errors when evaluating integrals along the boundary, which is of particular importance when evaluating boundary quantities, e.g., for FSI problems.

φ(x, y) φ(x, y) 1 1 φ(x2)=0 φ(x2)=0

x2 y x2 y

φ(x) 6= 0 φ(x)=0 0 0

x1 x1 x x φ(x1)=0 φ(x1)=0 (a) Cartesian NEFEM. (b) Non-Cartesian NEFEM.

Figure 4.8: Shape function φ(x) corresponding to the interior node of a 2D Cartesian and non-Cartesian NEFEM element. As can be seen here, the non-zero shape-function contributions along the domain boundary are avoided when using the proposed non-Cartesian NEFEM formulation.

4.3.2 Non-Cartesian NEFEM

The non-zero boundary integral contributions of the internal shape functions discussed in the previous section can be avoided by using a non-Cartesian approach instead. Within such an approach, e.g., the p-FEM formulation [89], or the non-Cartesian NEFEM formulation introduced for 2D problems in [28], shape functions are no longer defined in the global space. Instead, as is done for the integration points, the shape functions are defined on the reference element. As a result, the partition of unity property is fulfilled, and shape functions corresponding to the interior element nodes are zero along the NURBS boundary (see Figure 4.8b). The non-Cartesian NEFEM uses a mapping between the reference and the physical coordinates, which includes the NURBS geometry. Due to this mapping, which will be presented in the next section, the non-Cartesian NEFEM might lead to distorted shape functions in the physical space when using higher-order polynomials [89]. However, in the current work this is avoided by limiting ourselves to the use of first-order Lagrange polynomials only.

40 σ

Λ ϑ Ψ

λ H ζ x4 Ωe Ωref 4 x3 3 η

x2 2 1 x1 ξ −1 Φface = Λ ◦ Ψ

Figure 4.9: THT mapping from a reference tetrahedron to a tetrahedral element with a spline face in the physical space. Here, the element face ξ + η + ζ = 1 corresponds to the NURBS surface of the physical element spanned by vertices x1, x3 and x4. Vertex x2 corresponds to the interior element node.

Tetrahedron-Hexahedron-Tetrahedron Geometric Mapping

To include the NURBS geometry into the 3D non-Cartesian NEFEM formulation, a geometric mapping needs to be deﬁned. Using the concept of degeneration, a standard tri-linear hexahedral element H is mapped to a tetrahedral element by coalescing certain nodes. The inverse of this mapping Φ−1 is used

ﬁrst to map from a reference tetrahedron Ωref to a hexahedron H. In the next step, a similar mapping Λ is employed to map from the hexahedron H to a tetrahedral element Ωe in the physical space (see Figure 4.9). Combining both mappings yields Φ = Λ ◦ Ψ−1, which yields a mapping from the reference element

Ωref to a corresponding element Ωe in the physical space. This mapping is analogous to the mapping presented for two-dimensional problems in [28] and is termed tetrahedron-hexahedron-tetrahedron (THT) mapping.

As stated before, THT mapping is constructed with two mappings. First, the mapping Ψ: H 7→ Ωref from the reference hexahedron H to a reference tetrahedron Ωref is deﬁned. The deﬁnition for Ψ is

41 obtained by coalescing nodes and is given as:

Ψ: H 7→ Ωref , 1 ξ = [1 + λ][1 − ϑ][1 − σ], 8 1 η = [1 − λ][1 − ϑ][1 − σ], (4.7) 8 1 ζ = [1 + ϑ][1 − σ]. 4

Here ξ, η and ζ are the coordinates on Ωref , whereas λ, ϑ and σ are the coordinates on H. Similar to Equation (4.7), a mapping from H to the elements in global space Ωe can be deﬁned:

Λ: H 7→ Ωe, 1 1 Λ(λ, ϑ, σ) = (1 − σ) S (θ(λ, ϑ)) + (1 + σ) x . (4.8) 2 2 2 Note that this expression already includes the spline geometry deﬁned by NURBS surface S(θ). The parametric coordinates θ of the spline are, in this case, aligned with the coordinates λ and ϑ of H. To obtain θ(λ, ϑ) for the particular element in the global space, a linear interpolation of the parametric coordinate is used: (1 − λ)(1 − ϑ) (1 + λ)(1 − ϑ) (1 + ϑ) θ(λ, ϑ) = θ + θ + θ , (4.9) 4 1 4 2 2 4 where θi are the parametric coordinates of the element nodes touching the spline. Combining Ψ and Λ results in mapping Φ = Λ ◦ Ψ−1 for elements that have a common face with the NURBS boundary surface: Åθ ξ + θ η + θ ζ ã Φ (ξ, η, ζ) = (1 − ξ − η − ζ) x + (ξ + η + ζ) S 1 3 4 . (4.10) face 2 ξ + η + ζ

Here, Φface is a function of the local coordinates ξ, η and ζ on the reference element Ωref . As already stated, Equation (4.10) is specific for elements with a face on the NURBS geometry (e.g., the left-hand side element in Figure 4.6). Equation (4.10) is not defined for ξ + η + ζ = 0 which corresponds to the interior node x2. However, while the quadrature points used for integration are located on the interior of the element, this is of no concern. For elements with only an edge on the NURBS boundary (see right-hand side element in Figure 4.6), a slightly modified mapping is needed, as shown in Figure 4.10. In this particular case, only one edge of the NEFEM element is touching the NURBS geometry. Following a similar derivation as for Φface, the THT mapping for edge elements is as follows: Åθ ξ + θ η ã Φ (ξ, η, ζ) = (1 − ξ − η − ζ) x + (ξ + η) S 1 3 + ζ x . (4.11) edge 2 ξ + η 4

Note that this mapping only contains two points in the parametric space (θ1 and θ3) since only two element nodes touch the NURBS surface. Similar to the mapping in Equation (4.10), the edge mapping in

Equation (4.11) is not deﬁned for ξ + η = 0 which corresponds to the element edge x2 − x4. Also in this case it holds that the quadrature points used for integration are located on the interior of the element and thus the singularity along x2 − x4 is of no concern. Using Equations (4.10) and (4.11), the three-dimensional non-Cartesian NEFEM approach can be incorporated within a standardFE framework. The THT mappings are then used to reposition the

42 σ

Λ ϑ Ψ

H ζ x λ 4 Ωe Ωref 4 x3 3 η

x2 2 1 x1 ξ −1 Φedge = Λ ◦ Ψ

Figure 4.10: THT mapping from a reference tetrahedron to a global tetrahedral element with a spline edge. Here, the element edge ξ + η = 1 corresponds to the NURBS surface of the global element spanned by vertices x1, and x3. Vertices x2 and x4 correspond to the interior element nodes.

integration points. Additionally, the mappings are used when deﬁning the element Jacobians JΦ. These are needed for coordinate transformations and evaluating derivatives within the weak form of the model problem.

Mappings (4.10) and (4.11) can be used directly for computing the element Jacobians in space. By applying the non-Cartesian NEFEM approach to space-time ﬁnite elements, a linear combination of the mappings at multiple time levels is constructed. These additional steps are discussed in Section 4.3.3 and are analogous to those presented in [26] for the Cartesian NEFEM.

Due to the convenient expression of domain boundaries by NURBS, geometric properties such as tangents, normals, and curvature along such boundaries can be evaluated in an exact manner. Note, however, since strictly linear basis functions are used, properties such as higher-order spatial derivatives along the boundary are not directly available. For this, additional post-processing of the numerical solution is required.

Apart from the curved boundary faces and edges, the non-Cartesian NEFEM elements that follow from using (4.10) and (4.11) are also allowed to have curved interior faces (e.g. surface (x1, x3, x4) of domain Ωref in Figure 4.10). Consequently, it is naturally assured that no gaps or overlaps occur between neighboring elements.

43 THT Mapping and The Finite Element Formulation

To incorporate the NURBS geometry into an existingFE formulation, the ﬁnite elements selected for

NURBS-enhancement are mapped from the physical space to the reference element using Φface or Φedge. This results in a simple change of variables similar to (3.16) for standard isoparametric ﬁnite element formulations: Z Z f(x) dx dy dz = f(Φ(ξ, η, ζ)) |detJΦe | dξ dη dζ, (4.12) Ω Ωref

where, depending on the orientation of element e, JΦe represents the Jacobian of either Φface or Φedge. The same approach is used in post-processing procedures such as the CBF method discussed in Section 3.5. Note, Equation (4.12) is applied only to elements with a face or edge on the NURBS boundary. The remaining elements use the conventional isoparametric transformation in Equation (3.16).

4.3.3 On NEFEM and Space-Time Formulations

So far, the presented work on the three-dimensional non-Cartesian NEFEM only considers spatial finite elements. For space-time finite elements the THT mappings given by Equations (4.10) and (4.11) need to be extended with an additional coordinate for the time-dimension. This approach is similar to that presented for the two-dimensional Cartesian-NEFEM proposed in [26, 28, 29]. In the context of space-time slabs, as discussed in Section 3.3.2, a linear interpolation between the mappings at the lower and upper time level, tn and tn+1, is constructed for space-time finite elements. Using an additional reference coordinate for the time dimension τ ∈ [−1, 1] yields the following for Equation (4.10):

Ñ θ1ξ+θ3η+θ4ζ é st 1−τ (1 − ξ − η − ζ)x2 + (ξ + η + ζ) Sl ξ+η+ζ Φface(ξ, η, ζ, τ) = 2 (4.13) tl

Ñ θ5ξ+θ7η+θ8ζ é 1+τ (1 + ξ − η − ζ)x6 + (ξ + η + ζ) Su ξ+η+ζ + 2 . tu

Here, subscripts l and u refer to the lower and upper time levels n and n + 1, respectively. Furthermore, as is the case for standard space-time ﬁnite elements, the total number of element nodes is doubled. This follows from the extrusion of the spatial elements into the time-dimension. A graphical explanation of a two-dimensional triangular NEFEM element using a NURBS curve is shown in Figure 4.11. Note from this ﬁgure, that by applying the space-time NEFEM formulation to deforming domains, a separate

NURBS description of the upper and lower level is needed (i.e., Cl(θl) and Cu(θu) in Figure 4.11, or analogously Sl(θl) and Su(θu) in Equation (4.13)). When a non-deforming mesh is considered, a single NURBS deﬁnition can be used for both time levels. For the edge-only NEFEM element, the situation is similar, resulting in the following mapping:

Ñ θ1ξ+θ3η é st 1−τ (1 − ξ − η − ζ)x2 + (ξ + η) Sl ξ+η + ζx4 Φedge(ξ, η, ζ, τ) = 2 (4.14) tl

Ñ θ5ξ+θ7η é 1+τ (1 − ξ − η − ζ)x6 + (ξ + η) Su ξ+η + ζx8 + 2 . tu

44 t

tn+1

xu (θ4)

xu (θ3) Cu (θu)

xl (θ2)

xl (θ1) Cl (θl) x Figure 4.11: Space-time slab for a two-dimensional non-Cartesian NEFEM element. The NURBS boundary is now deﬁned by a linear interpolation of the NURBS geometry between the upper and lower time level n and n + 1, respectively.

As for the semi-discrete case, the THT mappings in Sections 4.3.3 and 4.3.3 can directly be used for the positioning of integration points and the evaluation of element Jacobians. Furthermore, the shape functions are again deﬁned on the reference element.

4.3.4 On Numerical Integration and Non-Cartesian NEFEM

The inverse and determinant of the Jacobian JΦe , which contains either Φface or Φedge, are non-polynomial functions. Consequently, exact integration using a standard Gauss quadrature rule is not possible. This issue is discussed in more detail for the Cartesian NEFEM and p-FEM in [89]. Similar observations are made for standard Gauss quadrature rules applied to IGA in [27].

Furthermore, it is shown that non-polynomial mappings combined with non-Cartesian formulations can result in a loss of consistency for higher-order elements (p > 1)[89] as it does not fulﬁll the higher-order patch test [90].

However, this is of no concern for the formulation presented here since only linear P1 ﬁnite elements are considered. Choosing a suitable integration rule, however, remains of great importance to obtain an integral evaluation of the highest possible accuracy. Therefore, the performance of various quadrature rules, such as that proposed in [91], is studied in the context of the non-Cartesian NEFEM next.

45 Flow Over a 3D Cylindrical Segment

To demonstrate the importance of integration rules in the context of spline-based methods, consider the volumetric domain Ω depicted in Figure 4.12. This domain consists of a cuboid with characteristic length L, from which a cylindrical segment of 2π/3 is subtracted.

L L L L

y x

z 2π/3

Figure 4.12: Cuboid with a cylindrical 2π/3 segment cut-out.

The cylindrical boundary can be represented exactly with a NURBS surface, making this problem very suitable for NEFEM. In fact, the cylindrical cut-out geometry cannot be represented exactly using linear ﬁnite elements and would require a large number of elements. This test case is used to study the numerical volume computation of the presented domain. For this, the volume is computed using the non-Cartesian NEFEM and SFEM for a series of mesh reﬁnements as presented in Table 4.1.

Table 4.1: Grids used for the volume computation comparison. ne and ne,face represent the total number of elements, and the number of NEFEM face elements along the cylindrical segment of the domain.

Mesh # ne ne,face 0 47 2 1 376 8 2 3,008 32 3 24,064 128 4 192,512 512 5 1,540,096 2048

The wall of the cylindrical segment is represented by a NURBS that is spanned by a 3 by 3 control net, as shown in Figure 4.12. This NURBS surface is of second degree, or p = q = 2, along both parametric coordinate directions (see Equation (4.4)). The computed domain volumes are compared against the exact volume for which the analytic

46 expression based on the characteristic length L is given by: √ Ç π 3å V = 3 − + L3. (4.15) Ω 9 12

The results of the reﬁnement study are presented in Figure 4.13. Here, the relative error εrel, between the numerical and exact volume (V˜Ω and VΩ) is presented.

100 SFEM −1 10 NEFEMtensor

] 10−2 NEFEMsym − [

| −3 Ω

| 10 V Ω − V | Ω −4 ˜ V 10 |

= 10−5 rel ε 10−6 10−7

100 101 102 103 104

ne,face [−]

Figure 4.13: Relative error εrel between the numerical and exact volume V˜Ω and VΩ.

The non-CartesianST NEFEM solution presented in 4.13, are obtained with two different integration rules. The given NEFEMtensor results use a tensor product of a standard 1D Gauss integration rule projected onto the reference tetrahedron. This integration rule leads to clustering of the integration points within the element, as shown in Figure 4.7. The NEFEMsym results are obtained using the symmetric quadrature rule introduced in [91]. This quadrature rule is speciﬁcally designed to be symmetric, which means that the integration points are distributed symmetrically within the reference element. The presented SFEM solutions are also obtained using the symmetric integration rule. From Figure 4.13 it is evident that, when comparing the NEFEM and SFEM, the NURBS-based method requires signiﬁcantly fewer elements along the spline for a certain level of accuracy. Furthermore, the presented results show that choosing a suitable integration rule can further improve the accuracy of the method. Especially, when using the symmetric integration rule from [91].

4.3.5 Visualization of Spline-Based Solutions

Non-Cartesian NEFEM solutions consist of nodal values corresponding to the element nodes of a standard ﬁnite element mesh. In principle, these results can be graphically visualized using workﬂows designed for conventional FEM-based simulations. However, compared to the approximated geometry of theFE mesh, this would lead to a mismatch between the discrete and exact geometry. The THT mapping introduced in Section 4.3.2 can be used to visualize NURBS-based solutions using the geometrically exact CAD model instead. For this, the solution and physical coordinate pairs within a

47 p˜

0.25 0.5 0.750.0 1.0

Figure 4.14: Normalized pressure ﬁeld p˜ within the NEFEM mesh corresponding to the example given in Figure 4.14. This mesh consists of linear ﬁnite elements combined with NEFEM elements along the curved NURBS surface.

NURBS-enhanced element can be interpolated using Section 4.3.3 or (4.14):

x(ξ, η, ζ) = Φ(ξ, η, ζ), (4.16) n Xen u(ξ, η, ζ) = Ni(ξ, η, ζ)ui. (4.17) i=1 Note that the THT mapping directly provides the physical coordinates as a function of the reference coordinates ξ, η, and ζ. The corresponding numerical solution is obtained by linear interpolation using the shape functions Ni. From Equation (4.17) it can be seen that the non-Cartesian NEFEM does not apply the same interpolation to define the geometry and the solution. The above approach can be used for each NEFEM element to generate a triangulation of the enhanced element and its corresponding solution using a user-specified resolution. However, the singularities observed in Equations (4.10) and (4.11) need to be addressed by setting θ = 0 when ξ + η + ζ = 0 and ξ + η = 0 for the face and edge element mapping, respectively. A graphical example is given in Figure 4.14. This figure shows a pressure gradient on the computational domain corresponding to the numerical example presented in Section 4.3.4. Although this example is based on a coarse mesh, it can be observed that the number of NEFEM elements is small compared to the standard (interior) finite elements. This is typically the case for analyses using the non-Cartesian NEFEM formulation. Hence, the potential impact of the non-Cartesian NEFEM on computational efficiency of FEM solvers can be assumed to be limited.

48 Chapter 5

Numerical Methods

This chapter covers the discretization techniques needed to numerically solve FSI problems such as the examples given in Chapter1. Both fluid and structural sub-problems involved with FSI are discretized using anFE approach. The fluid and structural problem are solved using the non-Cartesian NEFEM and the IGA methods described in Chapter4, respectively. As stated before, these methods allow us to exploit NURBS geometries and their favorable geometric properties. Before discussing the numerical formulations of the specific FSI sub-problems in Sections 5.2 to 5.5, a general overview of the solver framework and the necessary spatial and temporal coupling strategies are presented in Section 5.1. The complete solver framework given in this chapter is subsequently applied to a series of generic benchmark problems in Chapters6 and7.

5.1 FSI Solver Framework and Coupling Strategies

As stated in Chapter1, FSI problems are classically studied using analytical or semi-analytical methods. Although this gives engineers a tool to describe the complex physics of FSI, these methods are often complex and cumbersome in their use. Furthermore, they are typically valid only under strong assumptions, such as inviscid fluid flow and harmonic motions (e.g., [5–7]). With the continuous development in the field of computational engineering, however, numerical analysis of FSI problems is now common practice. In line with this development, engineers nowadays perform detailed FSI simulations with higher physical fidelity and increased accuracy. In this work, the numerical solver framework presented in Section 1.1 is applied. It is based on a black-box concept, which allows for the coupling of arbitrary fluid and structural solvers using a specifically designed coupling module. Due to their modular nature, partitioned solution strategies require additional methods to properly handle the spatial and temporal coupling. These methods and why they are needed is presented next in Sections 5.1.1 and 5.1.2

5.1.1 Spatial Coupling

The physical phenomena of fluid flow and structural dynamics typically have different requirements on the discretization in space and time. For flexibility and computational efficiency, it is often beneficial to permit non-matching grids at the coupling interface, which means that the nodes of the respective fluid and

49 structural meshes do not have to be coalescing at the coupling interface. This, however, requires additional measures to correctly transfer the interface data between the sub-problem domains in a consistent and energy-conserving manner. In literature, different methods to transfer data between non-matching meshes can be found, such as interpolation methods [92], projection methods [93–95], and a variety of spline-based methods [18, 96]. While other methods exist, the above-mentioned methods have in common that they, in principle, only need discrete interface data. Hence, they are very suitable for the partitioned black-box approach used in this work (see Figure 1.1). The coupling module does not have access to or cares about the internals of the individual black-box solvers. When using projection methods, generally speaking, it is assumed that energy should be conserved during the data transfer between the ﬂuid and structure. This assumption results in global conservation of virtual work over the interface [97]. To avoid inconsistent non-physical behavior, two different transformation matrices are used. While this approach solves the consistency issue, it does not guarantee energy conservation. In principle, however, when the introduced spatial coupling error is smaller than the spatial and temporal discretization error, it does not affect the stability and accuracy of the simulation [18].

Note on Consistency and Conservation of Energy

Independent of the chosen coupling method, the discrete coupling conditions derived from Equations (2.1) to (2.3) are written as:

df = Hsf ds, Tsns = HfsTf nf , (5.1) where Hsf and Hfs represent the transformation matrices between the ﬂuid and the structure. These transformation matrices are related through:

î óT Hfs = Mff Hsf (Mss)−1 , (5.2) with discretization matrices Mff and Mss depending on the method used. Assuming anFE-type discretization, these are deﬁned as: Z Z ff Tf nf df ss Tsns ds Mij = Ni Nj ds, Mij = Ni Nj ds (5.3) Γf Γs

Tn d where Nj and Ni are the basis functions used for the tractions and displacements, respectively. To globally satisfy conservation of energy, the amount of work applied to the interface Γ must be equal on Γf and Γs: Z Z df · Tf nf ds = ds · Tsnsds. (5.4) Γf Γs Replacing the continuous displacements and tractions by their discrete counterparts given by Equa- tion (5.1), using the discretization matrices in Equation (5.3), and simplifying the result yields the following relation to ensure global energy conservation:

î óT Tsns = Mff Hsf (Mss)−1 Tf nf . (5.5)

Notice that the above corresponds with the discrete dynamic coupling condition in Equation (5.1) combined with the expression of transformation matrix (5.2).

50 Regarding consistency of the considered coupling method, constant displacements and loads must be interpolated exactly over the interface, which is similar to the patch test in Lagrange multiplier methods (see, e.g., [44]). This means for the conservative approach that the row-sum of the transformation matrices fs sf H and H in Equation (5.2) is equal to one. Or, with constant vectors βA and βB the following must hold:

BA AB βA = H βB, βB = H βA. (5.6)

Where A, B refer to the ﬂuid or structural mesh and vectors βA and βB have the dimension corresponding to the number of nodes on the interface of mesh A and B respectively. For the interested reader, a full and detailed discussion on energy conservation and consistency for spatial coupling procedures is given in [18, 97].

Finite Interpolation Elements

A common choice for spatial coupling of non-matching meshes is that of ﬁnite interpolation elements (FIE)[98]. Although the concept of FIE is very simple in nature, it is very effective. The FIE procedure to transfer ﬂuid data to the structure is presented next using a two-dimensional example depicted in Figure 5.1.

f Fi fs,f s Γ Fa Γfs Γfs,s ξ f s e,Fi f Ωe Fi s s Na (ξ) Nb (ξ)

s Fb s s Fa ξ f Fb e,Fi

f s Figure 5.1: FIE data transfer procedure. Here, ﬂuid load Fi is projected onto structural element Ωe. s s Subsequently, the shape functions Na,b on Ωe are evaluated at parametric coordinate ξe,F f of the projection f s point. The shape function values are then multiplied with the ﬂuid load F and the resulting loads Fa,b are added to the corresponding structural nodes. In this case, the subscripts a, b correspond to the interface s nodes of the element under consideration (Ωe).

Reconstruct Nodal Fluid Forces First, the nodal ﬂuid loads F f at the interface need to be constructed f f using tractions (T n )i corresponding to the current state of the ﬂow solution:

Z nen f f X î f f f ó Fi = Ni Nj (T n )j ds, (5.7) Γf i=j = Mff Tf nf , (5.8) where the nodal fluid tractions are interpolated using the FIE basis on the fluid interface N f (ξ). The fluid f f tractions (T n )i can be obtain by the CBF approach introduced in Section 3.5. In fact, the known terms

51 of the weak form that is acquired by applying the CBF method to the ﬂuid problem, directly provides the f ﬂuid loads Fi (c.f. Equation (3.41) for the example problem used in Section 3.5).

Find Projection Point on Target Mesh To transfer the discrete loads F f from the fluid onto the structure, the data points are projected to the fluid-structure interface of the structural mesh Γs. To do so, s the structural element Ωe on which to project the loads needs to be found, and subsequently the parametric coordinate ξ f of the projection point within that element. e,Fi An orthogonal projection algorithm (e.g., [99,100]) is used to provide the parametric coordinate of the projection point. In the case of spline-based methods such as IGA or NEFEM, the initial procedure to find the projection element can be neglected for single-patch spline geometries. In such a case, the spline basis of the interface can be viewed as being the element onto which to project the fluid data.

s s Distribute Loads on Target Mesh With the projection point on Γe known, theFE basis of Γe is used f s s to compute the nodal contribution of the ﬂuid load Fi to the structural nodal ﬂuid loads Fa and Fb :

s s f s s f (Fa )i = Na (ξe,Ff )Fi , (Fb )i = Nb (ξe,Ff )Fi . (5.9)

s To obtain the complete nodal load Fa at structural interface node a, the contributions of all nodal ﬂuid loads within the support of the corresponding basis function need to be summed:

nn Γf s X s f Fa = Na (ξFf,i )Fi . (5.10) i=1 s Here, for simplicity, nnΓf represent all the ﬂuid nodes within the support of the basis function Na that corresponds to structural node a. Note, in case of IGA the NURBS basis can be used:

nn Γf s X s f Fa = Ra(θFf,i )Fi , (5.11) i=1

where now a refers to a control point of the spline and the parametric coordinate θFf,i of projection the point is used. Note that from Equations (5.10) and (5.11) the transformation matrix Hfs follows. Moreover, following from the partition of unity property of both N s and Rs, the total ﬂuid load applied at the interface is conserved.

Transferring of Structural Displacements The above procedure for transferring nodal loads from the ﬂuid to the structural mesh can be applied in the reverse direction to transfer the structural deformations from the structural mesh to the ﬂuid mesh. The resulting expression for the displacement yields:

nnΓs f X f s da = Na (ξds,i )di , (5.12) i=1 where in this case a represents a ﬂuid mesh node and similar to Equation (5.10), nnΓs denotes all structural f nodes within the support of the basis function Na . In case of IGA, the NURBS basis is applied instead. Here, the NURBS can simply be evaluated for all non-zero basis-functions at ﬂuid node a:

nnΓs f X s s da = Ra(θds,a )di , (5.13) i=1

52 where θds,a represents the parametric coordinate of the fluid node a. Both Equations (5.12) and (5.13) correspond to transformation matrix Hsf . Note that, in principle, the FIE method does not require access to any information of the individual fluid or structural solvers. The nodal coupling data (i.e., nodal loads, displacements, and physical coordinates) combined with a user specified basis are sufficient to provide the desired mapping. However, whenFE-based methods are coupled, use can be made of theFE bases provided by the coupled solvers themselves. The same holds when combining spline-based methods, such as those discussed in this work. In fact, the FIE approach can be used for such methods by using NURBS basis from Equation (4.2) directly in Equations (5.9), (5.10) and (5.12). When applying FIE, conservation of the loads follows directly from the partition of unity of theFE basis functions, or similarly, for the NURBS basis used by IGA, or non-Cartesian NEFEM. Note that a Cartesian NEFEM formulation does not satisfy the partition of unity. Hence in such case conservation of the loads cannot be guaranteed. When inserting fluid loads (5.8) into Equation (5.10) and rewriting it in matrix form using the transformation and discretization matrices, we can write the expression in terms of the tractions on the fluid and the structure side of the coupling interface Γ:

î óT Tsns = Mff Hsf (Mss)−1 Tf nf . (5.14)

Comparing the above with Equation (5.5) in the previous discussion on conservation of energy and the work presented in [18, 97], it follows that the FIE approach is energy-conserving in the global sense.

Direct NURBS-Based Data Transfers

When using spline-based methods to solve both sub-problems, the spline geometry can be utilized effectively in the spatial coupling. In fact, when combining NEFEM and IGA, both methods can have access to the same identical NURBS geometry. Hence, this allows us to compute the ﬂuid forces directly using the NURBS-basis of the interface. Although in this case, the discretizations of Γf and Γs are non-matching, compatibility of the spline-geometry is guaranteed. Using theFE and NURBS bases, this leads to a set of relations for the displacements and tractions:

f nn f f X f f f T n = Ni (T n )i, (5.15a) i=1 s nn s s X s s s T n = Ri (T n )i, (5.15b) i=1 f nn f X f f d = Ri di , (5.15c) i=1 s nn s X s s d = Ri di . (5.15d) i=1

f s Here, nn and nn indicate the number of nodes or control points on the ﬂuid and the structure side of the f f interface. Ni and Ri represent the LagrangeFE and NURBS bases. Notice that apart from T n , all quantities are interpolated using the NURBS basis.

53 Since the displacement d of both the ﬂuid and structure are interpolated using the spline basis, the transformation matrix for the structural displacements is reduced to the identity matrix:

Hsf = I (5.16)

Given this fact, consistency directly follows. Furthermore, by inserting Mff , Mss, and Equation (5.16) into Equation (5.1), we obtain:

Hfs = (Mss)−1Mff = [Mff Hsf (Mss)−1]T . (5.17)

Using Hsf = I proves global conservation of energy according to Equations (5.4) and (5.5). Consistency in coupling NEFEM-IGA using FIE follows from the given partition of unity of both LagrangeFE and NURBS bases, N f,s and Rf,s respectively. A full proof of the consistency and conservation of energy using the direct integration using the NURBS is presented in [29].

5.1.2 Temporal Coupling of Space-Time and FD Methods

As previously stated, in this work, semi-discrete and space-timeFE formulations are combined. The special treatment of the coupling conditions in this regard is therefore elaborated next. Recall from Section 3.3 that the semi-discrete time-integration schemes discussed in Sections 3.3.1 and 3.3.3 are deﬁned using single time instances per time level (tn and tn+1). On the other hand, the ± DSD/SST formulation presented in Section 3.3.2 splits the time instances in an upper and lower level (tn ± and tn+1). Hence, to couple semi-discrete and space-time ﬁnite element methods, the coupling conditions presented in Section 2.1 need to be treated adequately. t

tn+1

− tn+1 fs Γn+1 tn−1 Qn + tn − y tn fs Γn Qn−1

+ tn−1

fs x Γn−1 Figure 5.2: Time-level handling of spline-based space-time and semi-discrete meshes.

As depicted in Figure 5.2, a single space-time slab Qn contains the unknown nodal solutions of the + − time levels tn and tn+1. Hence, a boundary or coupling condition is needed for both these time levels.

54 For this, we deﬁne the following interface boundaries at the upper and lower time level of Qn:

fs,f f fs,s f + Γn = Pn ∩ Γn = (Γn) , (5.18a) fs,f f fs,s f − Γn+1 = Pn ∩ Γn+1 = (Γn+1) , (5.18b)

f where Pn is the FSI-coupled portion of the boundary Pn of Qn. Using these definitions for the upper and lower time-level boundaries, the coupling conditions enforced at the fluid-structure interface can be defined in a general form as:

f s fs αn = αn on Γn , (5.19a) f s fs αn+1 = αn+1 on Γn+1, (5.19b) where α represents an arbitrary coupling condition (d, u, a, Tn, T , or ∂T/∂n). While the fluid mesh is deformed according to the structural deformations, the flow problem uses the structural velocity of the deforming boundary. Since the mesh velocity is not directly available, it needs to be computed from the available information. For this, a simple finite difference of the deformed coupling interface at time tn and tn+1 is used: ds − ds uf = n n−1 on Γfs, (5.20a) n ∆t n ds − ds uf = n+1 n on Γfs . (5.20b) n+1 ∆t n+1 Subsequently, depending on the coupling used, these velocities can be applied as Dirichlet boundary conditions to the flow problem. In this work a linear elastostatic equation is solved to obtain the deformed mesh at time n + 1. These deformations are applied to the space-time slab Qn as defined in Section 3.3.2 and can directly used in the context of DSD/SST (see [50]). With the spatial and temporal coupling procedures defined, both the fluid and structure problem can be coupled within the solver framework presented in Section 5.1. For this, different coupling types exist to enforce the coupling conditions discussed. These different methods are presented in the next section.

5.1.3 Iterative Solution Procedure and Field Partitioning

As discussed in Section 5.1, a black-box type partitioned approach is employed in this work. This means the single-field solvers for the fluid and structure are coupled iteratively. More specifically, to consider the potentially strong interdependence between the single field problems, a strong coupling procedure is utilized. This procedure ensures that the coupling conditions are fulfilled after each time step by employing an energy-conserving fixed-point iteration scheme. A schematic of the partitioned procedure is given in Figure 5.3, where two coupling approaches are provided in which slightly different data sets are transferred between the individual solvers. Both approaches are discussed next.

Dirichlet-Neumann Coupling

There exist various coupling schemes to ensure that the interface conditions are met at each time step. The most common approach for FSI problems is the Dirichlet-Neumann (DN) scheme. A schematic of theDN coupling scheme within a single time step is depicted in Figure 5.3a and explained next.

55 Solution at tn Solution at tn k = 1 k = 1

Solve ﬂuid on Ωf : k = k + 1 k f f f Ä f ä Ω k = k + 1 u , p = f Ω Solve ﬂuid on k: k k k f f Ä f s f ä uk, pk = f α , Tk, Ωk

f Tractions tk f Tractions tk Update mesh f s Update mesh Ωk+1 = f(dk+1) f s Ωk+1 = f(dk+1) Solve structure: s s Ä f ä Solve structure: dk+1, uk+1 = f tk s s s Ä f ä dk+1, uk+1,Tk+1 = f tk

no no Converged? Converged? Interface deformation Interface deformation, and velocity ds , us yes k+1 k+1 yes velocity, and tractions s s s dk+1, uk+1, Tk+1

Solution at tn+1 Solution at tn+1

(a) Dirichlet-Neumann coupling. (b) Robin-Neumann coupling.

Figure 5.3: Schematic of the strong temporal coupling procedures. Here, k represents the current coupling iteration.

f f f f f For each fixed-point iteration k, the fluid tractions tk(uk, pk) = T n are used to impose a Neumann s boundary condition onto the structural problem. Vice versa, the deformations dk+1 and the corresponding s velocities uk+1 obtained from the structural problem are applied to the fluid mesh and fluid problem as Dirichlet boundary conditions, respectively. The Dirichlet and Neumann boundary conditions naturally enforce the kinematic and dynamic coupling conditions presented in Section 2.1 once the convergence of the coupling loop is reached. Note that the example in Figure 5.3a does not consider any thermal coupling. When thermal transport between the sub-problems needs to be considered, theDN coupling is extended by imposing a flow- f induced heat flux (qn) to the structural problem. Subsequently, the resulting structural temperatures are imposed as Dirichlet-boundary conditions to the fluid problem. As already discussed, aDN coupling scheme is suitable for most common FSI problems but leads to numerical instabilities when considering fully Dirichlet-bounded fluid domains. In fact, without taking extra measures, such problems are not solvable at all using aDN coupling [28, 101].

Robin-Neumann Coupling

To successfully compute fully-enclosed Dirichlet-bounded ﬂuid domain problems, a Robin-Neumann (RN) type coupling can be used as depicted in Figure 5.3b. The Robin boundary condition can be interpreted as a linear combination of the kinematic and dynamic coupling conditions and is proposed as an alternative

56 toDN schemes in [29, 102 –104]. In this particular case, both the tractions and the local velocity are combined for the ﬂuid and structural problem as follows: ∂ds αf uf + Tf n = αf − Tsn, on Γfs,f , (5.21a) ∂t ∂ds αs + Tsn = αsuf − Tf n, on Γfs,s, (5.21b) ∂t where for scalar coefﬁcients αf 6= αs and αf , αs ≥ 0 must hold. It can be observed that theDN coupling is a special case of Equations (5.21a) and (5.21b), where αs = 0 and αf → ∞. The RN coupling is obtained when αs = 0 and αf > 0: ∂ds αf uf + Tf n = αf − Tsn, on Γfs,f , (5.22a) ∂t Tsn = −Tf n, on Γfs,s. (5.22b)

Equation (5.22a) results in the following boundary condition for the fluid problem: Å∂ds ã Tf n = αf − uf − Tsn. (5.23) ∂t Inserting this expression into the boundary integral of the weak form of Equation (2.25) or (2.27) results in a weakly enforced condition on the fluid velocity at the coupling interface. The benefit of using the Robin condition is that the violation of mass conservation caused by errors in the structural solution is counterbalanced by allowing an artificial flux over the FSI interface. During the fixed-point coupling iterations within a single time step (k in Figure 5.3), this flux is minimized until a converged and mass- conserving solution is obtained. A detailed discussion on the RN coupling approach, as presented here, is given in [29].

Some Remarks on Added-Mass and Stability

While the partitioned solution approach is a popular approach, its major downside is added-mass instability [105, 106]. Depending on the problem solver procedure used, these instabilities can become so severe that additional measures are needed to successfully obtain a numerical solution. The added mass effect is caused by defective structural deformations in the iterative procedure. Consequently they do not match with correct loads at the new time-level. This results in an artificial load acting on the structure, which, in many cases, is amplified with each iteration. A simple way to control these instabilities is by applying relaxation to the exchanged data between the solvers such as Aitken’s method [107, 108] or more advanced quasi-Newton methods [109]. Additionally, Robin boundary conditions, due to the introduced boundary flux, can reduce added-mass instabilities [102, 103]. This also holds for theRN-type coupling used in this work. The artificial boundary flux of theRN-coupling, although minimized during the iteration procedure, has similar effects as applying the previously mentioned relaxation methods. For the numerical examples presented in this work, the added-mass effects are within the limits of numerical stability. Hence, no extra relaxation techniques or other measures are applied.

5.2 Stabilized FEM for Compressible Flows

In this section, the ﬁnite element formulation used for solving the compressible Navier-Stokes equations is presented. Following [33, 34], the deformable spatial domain/stabilized space-time (DSD/SST) for-

57 mulation is derived for the quasi-linear form in Equation (2.14). The starting point in this derivation is the pressure-primitive form deﬁned in Section 2.2.2. For convenience, this pressure-primitive form is repeated here:

adv\p p sp Res(Y) = A0Y,t + Ai + Ai + Ai Y,i − (KijY,j),i − S = 0 on Ω ∀ t ∈ (0,T ). (2.25)

To construct the weak form, Equation (2.25) is multiplied by a set of test functions Wh and integrated h over the space time slab Qn (see Figure 3.3). The resulting Galerkin formulation, after integrating by parts h − h h h h the higher order derivative terms, states: Given (Y )n , ﬁnd Y ∈ (SY )n, such that ∀ W ∈ (VW )n: Z h h h adv\p sp h hi W · A0Y,t + Ai + Ai Y,i − S dQ+ Qn Z h î h p hó W,i · KijY,j − Ai Y dQ + (5.24) Qn Z Z h + Ä h + h −ä h (W )n · A0 (Y )n − (Y )n dΩ = W · hdP. Ωn (Pn)h

Here, the Neumann boundary condition h, in the right-hand side integral, is given by:   0    p n + τ n   1 1i i    h =  p n + τ n  , (5.25)  2 2i i     p n3 + τ3ini    −qini where ni is the outward normal unit vector, not to be confused with the time level n. The jump term, given by the third left-hand side integral in Equation (5.24), is added to weakly impose continuity over the time-slab interface Ωn. Here, subscripts ± refer to the upper and lower time-slab solutions at time n.

The weak form in Equation (5.24) is solved sequentially for all space-time slabs Q1,Q2,...,QN−1 with h − initial condition (Y )0 = Y0 and N indicating the ﬁnal discrete solution time. The discreteFEST function spaces for the trial and weighting functions U and W are given by:

h ¶ h h î 1h óndof h h © (SY)n = Y |Y ∈ H (Qn) , Y · ed = gd on (Pn)g,d , d = 1...ndof , (5.26) h ¶ h h î 1h óndof h © (VW)n = W |W ∈ H (Qn) , W · ed = 0 on (Pn)h,d , d = 1...ndof . (5.27)

Stabilization and Shock Capturing

As stated in Section 3.4, when considering convection-dominated flow problems, the use of a FEM is not straightforward and stabilization is necessary. To obtain a global monotone solution of Equation (5.24), SUPG stabilization is employed. Additionally a discontinuity-capturing formulation is used to avoid oscillations near discontinuities such as shock waves in supersonic flow problems. The original SUPG formulation is developed for incompressible flow problems [110]. It is then further extended for compressible flows using conservation variables in [111, 112]. In [52, 113, 114] this formulation is modified to solve the symmetric form of the Navier-Stokes equations using entropy variables, and subsequently in [35,36] for arbitrary variable sets. Since different problems require different types of boundary conditions, the work on arbitrary variable sets is especially useful. A combination of an improved version of SUPG for compressible flows [115] and the DSD/SST method [50, 51, 56, 57] is

58 proposed in [116, 117] for a range of aerospace applications. The speciﬁc formulation used in the current work is based upon the formulations in [34] and subsequently [33], where it is applied to a unstructured space-time formulation. This last formulation is based on the pressure-primitive variable set as discussed in Section 2.2.4. The residual-based stabilization operator, following the procedure in Section 3.4, is added to Equa- tion (5.24). The operator is deﬁned as follows:

(nel)n Z X Ä ˆ T h ä h (Am) W,m · τ Res(Y ), (5.28) e e=1 Qn ˆ where m = 1, . . . , nnsd + 1 for the space and time dimensions. Matrices Ai are the combined advection matrices for each spatial direction using the conservation variables:

ˆ adv\p p sp −1 Am = Ai + Ai + Ai A0 , i = 1, . . . , nsd. (5.29)

For the remaining time-dimension it is simply the identity I:

ˆ ˆ Ansd+1 = At = I. (5.30)

For the stabilization matrix τ , various deﬁnitions are proposed in the literature [34,37,118]. Here, the formulation deﬁned in [33] is used:

− 1 Ä ˆ ˆ ˆ ˆ ä 2 τ = GmpAmAp + CinvGijGklKikKlj , i, j, k, l = 1, . . . , nsd, m, p = 1, . . . , nsd + 1. (5.31)

ˆ −1 This expression contains the diffusion matrices Kij = KikA0 for the conservation variables, as well as the pairwise products of the advection matrices from Equations (5.29) and (5.30). The metric tensor

Gij is used to incorporate the directional element length information into the stabilization parameter [34, 36, 37, 119]. Using the inverse Jacobian J−1 = ∂ξk , the metric tensor is deﬁned as: ∂xk

d X ∂ξk ∂ξk G = , i, j = 1, . . . , d, (5.32) ij ∂x ∂x k=1 i j where x(ξ) denotes the mapping from the parametric element to the element in physical space. For Cinv in Equation (5.31), the deﬁnition

2 Cinv = (nsd + 1) (nsd + 2), (5.33) is motivated by an inverse estimate inequality for a P1 simplex reference element of the spatial discretization as defined in [120]. Note that the formulation for τmom presented above includes information about the shape, size, and orientation of the respective element, as well as the local flow direction. When constructing τ, the square root inverse of a 4 × 4 or 5 × 5 matrix needs to be computed (depending on the spatial dimensions of the problem considered). This can be done using various methods, such as the iterative Denmam-Beavers method [34]. The Denman–Beavers method is an iterative method based on the Newton–Raphson approach [121, 122]. In the current work, the square root inverse is computed through a Schur decomposition method [123]. When stabilization methods are applied to compressible flow problems, additional spurious modes can occur near shocks and other sharp features in the flow solution. To account for such oscillatory

59 behavior, shock- or discontinuity-capturing methods can be applied, as first proposed in the context of entropy variables in [52,124]. These methods add a form of artificial viscosity based on the local mesh and flow solution. As these methods typically are residual-based, consistency of the complete formulation is preserved. In [53] the original compressible-flow SUPG formulation presented in [112] is supplemented with the shock-capturing method similar to that in [52, 124]. The 2D computations using the Euler equations presented in [115] show similar results to those obtained with the original formulation. Further development of shock- and discontinuity formulations includes a generalized form for arbitrary variable sets in [36,125], and the Y Zβ shock-capturing formulation based on a scaled residual and control of shock smoothness [126–133]. A very nice and brief overview of the development of stabilizedFE methods for compressible flows and the various related shock- and discontinuity-capturing methods can be found in [134]. In the current work, the discontinuity-capturing formulation as proposed in [34], which is inspired by the work in [135], is used. Similar to Equation (5.28), the discontinuity-capturing makes use of the residual to maintain consistency of the formulation. The discontinuity-capturing term for the conservation variables is defined as:

(nel)n Z X h ˆ ˜ W,i · KDC U,i dQ. (5.34) e e=1 Qn

Changing from U˜ to Y using the mapping A0 yields:

(nel)n Z X h ˆ W,i · KDC A0Y,i dQ. (5.35) e e=1 Qn

ˆ In this expression KDC is the diffusivity matrix for discontinuity-capturing, which for nsd = 3 is given by:

ˆ ˆ ˆ ˆ ˆ Kˆ DC = diag(kC , kM , kM , kM , kE). (5.36)

The diagonal entries in (5.36) are deﬁned as:

h|RES | kˆ = C 1 , (5.37a) C C |∇ρ| ˆ h|RES2:d+1| kM = CM , (5.37b) |∇(ρud)| h|RES | kˆ = C d+2 , (5.37c) E E |∇(ρe)|

where CC , CM , and CE are positive constants of O(1) which correspond to the continuity, momentum, and energy equations of the model problem. Note that the formulation is consistent due to the residual used in Equation (5.37a) to (5.37c).

60 5.3 Stabilized FEM for Incompressible Flows

Next, the DSD/SST formulation from Section 3.3.2 is applied to the incompressible Navier-Stokes equations. For the reader’s convenience, these equations are repeated here from Section 2.3:

∇ · u = 0 on Ω ∀t ∈ (0,T ), (2.27a) ∂u − ν∇2u + (u · ∇)u + ∇p − ρb = 0 on Ω ∀t ∈ (0,T ), (2.27b) ∂t

u = uD on ΓD ∀t ∈ (0,T ), (2.27c) T −p n + 2 ν n · ( ∇u + (∇u) ) u = t on Γt ∀t ∈ (0,T ), (2.27d)

u(x, 0) = u0 on Ω0. (2.27e)

The characteristics of Equation (2.27) make it useful to apply separate weighting functions for the velocity and pressure. In this particular case, the function spaces from which the weighting and interpolation functions are chosen are deﬁned as follows:

h h h 1h nsd h (Su)n = { u | u ∈ [H (Qn)] , u = uD on (Pn)D}, (5.39a)

h h h 1h nsd h (Vw)n = { w | w ∈ [H (Qn)] , w = 0 on (Pn)D}, (5.39b) h h h 1h (Sp )n = { p | p ∈ H (Qn)}, (5.39c) h h h 1h (Vq )n = { q | q ∈ H (Qn)}, (5.39d)

where wh and qh are the weighting functions for the velocity and pressure, respectively. Choosing separate function spaces in Equation (5.39) allows us to use different bases for the velocity and pressure. However, equal-order elements are typically preferred to maintain a constraint ratio of the pressure-velocity pairs close to nsd. By multiplying the mass and momentum equation in Equation (2.27) with the weighting functions, integrating over the space-time slab Qn, and integrating by parts the higher-order derivative terms yield h − h h h h h h the following Galerkin form: Given (u )n , find u ∈ (Su)n and p ∈ (Sp )n, such that ∀w ∈ (Vw)n h h and ∀q ∈ (Vq )n the following holds: Z h Ä h h h hä w · ρ ut + u · ∇u − f dQ Qn Z + ∇wh : ∇Th(ph, uh) dQ Qn Z Z h h h + Ä h + h −ä + q ∇ · u dQ + (w )n · ρ (u )n − (u )n dΩ Qn Ωn (nel)n Z (5.40) X 1 î Ä h h hä h h h ó + τmom ρ wt + u · ∇w − ∇ · T (q , w ) e ρ e=1 Qn î Ä h h h hä h h h ó · ρ ut + u · ∇u − f − ∇ · T (p , u ) dQ (nel)n Z Z X h h h + τcont∇ · w ρ∇ · u dQ = w · h dP. e e=1 Qn (Pn)t The first three integrals on the left-hand side, combined with the right-hand side integral, represent the Galerkin weak form of the Navier-Stokes equations. The fourth integral is the jump term, to enforce continuity between consecutive time-slabs in a weak sense. The fifth and sixth integrals are added to

61 stabilize the formulation using the consistent Galerkin/Least Squares method. A detailed discussion on this specific formulation and the corresponding stabilization parameters τmom and τcont is given in [119]. An extensive discussion on alternative stabilization techniques can be found in [30]. Note that in the case of linear finite elements, the higher-order spatial derivatives present in the stabilization term in Equation (5.40) can be recovered using a least-squares technique such as proposed in [136]. The temperature equation of the flow problem given by Equation (2.28) is very similar to the model problem used in Chapter3. Hence, only the resulting weak formulation is presented here. Using the functions spaces in Equation (3.23), the weak form of Equation (2.28) states: Find Θ ∈ (S)n, such that

∀ w ∈ (V)n the following holds: Z Z Z + Ä + −ä w (Θt − u · ∇Θ − Φ) dQ + ∇w · (α∇Θ) dQ+ wn Θn − Θn dΩ Qn Qn Ωn

(nel)n (5.41) X Z Z Z + P(w)τR(Θ)dQ = w s dQ + w qn dP. e e=1 Qn Qn (Pn)qn Depending on the specific discrete formulation, Equations (5.40) and (5.41) are solved using a strongly or weakly partitioned solution approach. When no coupling is assumed, e.g., through the Boussinesq approximation or a viscous dissipation term Φ, it suffices to solve the temperature equation only once after the flow field itself is solved.

5.4 FEM for Elastodynamic Problems

The elastodynamic model problem introduced in Section 2.4 is discretized using a GalerkinFE formulation. Such formulations are a popular choice in solving structural mechanics problems. When applied to such self-adjoint elliptic PDEs, the Galerkin FEM leads to symmetric stiffness matrices. There exists a quadratic functional in such a case, whose minimum satisﬁes the partial differential equation of the model problem at hand. This holds for linear elastic problems, where the minimum of the functional represents the minimum potential energy in the system. Or similarly, in steady heat conduction problems, the minimum of the quadratic functional represents the minimal total thermal energy in the system [30]. Structural model problem (2.31) is discretized using a semi-discrete GalerkinFE formulation combined with the generalized-α time-integration scheme as presented in Section 3.3.3. Following the h h h h procedure in Chapter3, the variational form of Equation (2.31) states: Find d ∈ Sd such that ∀w ∈ Vw the following holds: Z Ç 2 h å Z Z h ∂ d h T h h w · ρ 2 − ρb dΩ + ∇0w : SF dΩ = w · h dΓ. (5.42) Ωo ∂t Ω Γh Rewriting the above in a matrix-vector form yields the following semi-discrete form:

Md¨ + N(d) = R, (5.43) where for simplicity superscript h is omitted. A time discretization needs to be applied to obtain a fully discrete problem. Following Section 3.3.3, the generalized-α scheme yields the discrete in space and time system of equations: Å1 − α 1 − α 1 − α − 2β ã M m (d − d ) − m d˙ − m d¨ + N = R , (3.34) β∆t2 n+1 n β∆t n 2β n n+1−αf n+1−αf

62 with dn+1 being the unknown to solve. The parameters used in Equation (3.34) are those presented in Section 3.3.3 as suggested in [58].

5.5 FEM for Heat Transport in Solids

Similar to the argumentation in Section 5.4, the thermal problem for the structure is discretized using a Galerkin formulation in space, whereas the time integration is done using a second-order Crank-Nicholson FD scheme [30]. The weak form of the thermal problem is obtained by multiplying the strong form in Equation (2.35) by a set of weighting functions, integrating over the domain and applying integration by parts to the higher-order spatial derivative terms. The resulting ﬁnite-element formulation states: For Θh ∈ Sh, such that ∀wh ∈ Vh the following holds:

Z Ç h å Z Z h ∂Θ 1 h h h h w − s G dΩ + ∇w (α∇Θ ) dΩ = w qndΓ. (5.44) Ω ∂t ρ cp Ω Γqn

The ﬁnite-dimensional sub-spaces Sh and Vh are identical to those described in Section 3.2. Note that the formulation in Equation (5.44) is continuous in space. As mentioned, here a Crank-Nicholson ﬁnite-difference discretization is applied to integrate in time. For this the θ-scheme from Section 3.3.1 is applied using θ = 0.5 (see Table 3.1). The resulting fully discretized formulation is given by: Å ∆Θã w, − θa (w, ∆Θ) = −a(w, Θn) ∆t (3.22)

+ (w, θ sn+1 + (1 − θ ) sn) + (w, θ (qn)n+1 + (1 − θ )(qn)n)Γqn ,

h s with s = G /(ρ cp). Furthermore, ∆Θ = Θn+1 − Θn, and ∆t = tn+1 − tn are the ﬁnite difference of temperature and the time step size, respectively. Note here that we in fact solve for the temperature increment from tn to tn+1. The unknown solution Θ at tn+1 itself is reconstructed by Θn+1 = Θn + ∆Θ. For simplicity, the superscript h is dropped in Equation (3.22).

Chapter 6

Application of Non-Cartesian Space-Time NEFEM to Compressible and Incompressible Flows

Many engineering applications involve smooth curved geometries. Such geometries are designed using spline-based CAD tools and simply can not be represented exactly through linear finite elements in a practical sense. As already discussed, the non-Cartesian NEFEM allows for the exact geometric representation via a NURBS-based mapping. The effect of applying a Cartesian space-time NEFEM method to two-dimensional incompressible flow problems is studied in [26]. Later, a non-Cartesian space-time NEFEM formulation is applied to similar two-dimensional problems in [29]. In both works, smooth curved geometries are involved. In the current chapter, the non-Cartesian space-time formulation presented in Chapter4 is applied to two- and three-dimensional flow problems in the compressible and incompressible regime. The presented problems consider rigid, curved and smooth geometries; all of which cannot be represented exactly by means of linear Lagrange finite elements.

6.1 Application to 2D Compressible Flow Problems

The material in this section is based upon the work presented in [137].

6.1.1 Flow Around a 2D Cylinder

This first test case considers a supersonic flow around a 2D cylinder. The problem is computed using the non-CartesianST NEFEM and SFEM formulations. The goal is to study how the non-CartesianST NEFEM formulation compares to the SFEM formulation when looking at boundary quantities such as the drag coefficient. The given problem involves a flow with a Mach and Reynolds number of M = 1.7 and Re = 2.0×105 respectively. Since the Mach number is in the supersonic regime, shock waves are expected. The problem is based upon the work presented in [138, 139], for which the characteristic flow parameters are presented in Table 6.1. The computational domain is depicted in Figure 6.1. The dimensions are characterized by

65 the cylinder radius R. On all the side walls, the perpendicular velocity component is set to zero, leading to a slip boundary conditions. On the cylinder, all ﬂow velocity components are set to zero to obtain a no-slip boundary condition.

uy = 0 uin

2R 25R

NURBS Geometry Control Polygon Control Points

y u = 0 25R x uy = 0

25R 50R Figure 6.1: Computational domain and boundary conditions for a supersonic ﬂow around a 2D cylinder.

Table 6.1: Parameters used for the supersonic ﬂow around a 2D cylinder.

Parameter Variable Magnitude Dimension Mach number M 1.7 [−] Reynolds number Re 2.0 × 105 [−] Cylinder radius R 1.0 [m] −1 2 2 Inflow pressure pin 2.4715 × 10 [kg/s /m ] Inflow velocity uin 1.0 [m/s] −4 Inflow temperature Tin 8.6117 × 10 [K]

In case the non-CartesianST NEFEM formulation is used, the cylinder is represented by a second- order curve (p = 2 in Equation (4.1)), with the control points distributed as depicted in Figure 6.1. The drag and pressure coefficient used in this example are defined as F p − p C = D ,C = in , (6.1) D 1 2 p 1 2 2 ρ||u|| S 2 ρ||u|| where FD represents the drag force and S = 2R the frontal area of the cylinder. The flow solution for both methods, presented by means of the pressure coefficient Cp is given in Figure 6.2. From this figure, it can be seen that the non-CartesianST NEFEM and the SFEM result in qualitatively similar flow solutions. For validation, both formulations are also compared against experimental data from [139]. As shown in Figure 6.3a, quantitatively, the non-CartesianST NEFEM and SFEM solutions agree with the reference data. Hence, both implementations can be considered valid for this test case. However, the detailed plot in Figure 6.3b shows small differences between the non-CartesianST NEFEM and the SFEM results in the pressure coefficients along the cylinder wall. Given that apart from the NURBS-enhanced elements, both methods use the same formulation presented in Section 5.2, these discrepancies are likely attributed to the difference in the geometric error. This follows from the improved geometric representation of the cylinder, as included by means of the NURBS in the non-CartesianST NEFEM formulation.

66 8

6 0.5 0.3 0.1 0.7 4 •0.1 0.9 •0.1 2 1.1 •0.3 1.3 NEFEM

y 0 y 1.3 SFEM •2 1.1 •0.3 •0.1 0.9 •4 0.7 •0.1 0.5 •6 0.3 0.1

•8 0 5 10 x

Figure 6.2: Contour lines of the pressure coefﬁcient for non-CartesianST NEFEM and SFEM for the supersonic ﬂow around the cylinder problem.

To demonstrate the reduction of this geometric error, and with that the performance of the non-

Cartesian NEFEM compared to the SFEM, a mesh refinement study is done. The drag coefficient CD is evaluated for a series of mesh refinements using both the non-Cartesian NEFEM and the SFEM. The mesh refinements used in the study are presented in Table 6.2 and the corresponding numerical results are depicted in Figure 6.4.

Table 6.2: Meshes used for the ﬂow around a 2D cylinder problem. Parameters ne and ne,face represent the total number of elements and the number of non-Cartesian NEFEM elements along the cylinder wall.

Mesh ne ne,face 0 6,720 64 1 26,880 128 2 107,520 256 3 430,080 512 4 1,720,320 1,028

The relative error εrel illustrated in Figure 6.4 is based upon the numerical solution computed with the finest mesh, i.e., mesh 4 in Table 6.2. The presented relative error in the drag coefficient shows a similar convergence rate for both methods. However, the non-CartesianST NEFEM has a reduced error for all meshes. This effect is present more significantly when FSI phenomena are taken into account, as shown in Chapter7.

67 2.0 SFEM SFEM NEFEM −0.20 NEFEM 1.5 Murthy et al. [139] Murthy et al. [139] −0.30 ]

] 1.0 − − [ [ p p −0.40 C C 0.5

−0.50 0.0

−0.60 −0.5 0 30 60 90 120 150 180 90 100 110 120 130 140 θ in degrees θ in degrees (a) Pressure coefﬁcient over half the cylinder (b) Detailed view near θ = 125◦. wall.

Figure 6.3: Pressure coefﬁcient Cp as a function of the angular coordinate θ along the cylinder wall for non-CartesianST NEFEM, SFEM and the reference solution from [139].

SFEM NEFEM ] − [ −2 | 10 4 | D 4 C D − i C | D C | = rel ε

10−3 102 103

ne,face [−]

Figure 6.4: Mesh convergence for supersonic ﬂow around a cylinder. Errors ε are relative to the results of mesh 4 in Table 6.2. Here, CDi refers to the solution of mesh i in Table 6.2.

6.1.2 Flow Around a 2D NACA0012 Wing Section

The next example demonstrates the use of spline-based methods for transonic inviscid flow problems. Compared to the previous case in Section 6.1.1, it involves a more complex, higher-order, and partially smooth geometry of a NACA0012 wing section (see, e.g., [140]). The test case corresponds to one of the problems described in the extensive numerical study presented in [141], and involves a transonic inviscid flow with a Mach number of M = 0.8. Such a flow, of which the characteristic parameters are presented in Table 6.3, involves a shock-induced pressure jump at the wing section wall. At the boundary representing the wing section wall, a no-slip boundary is imposed by setting all of the velocity vector components to zero, or, u = 0.

68 Table 6.3: Parameters used for the transonic ﬂow around a NACA0012 wing section.

Parameter Variable Magnitude Dimension Mach number M 0.8 [−] Section chord c 1.0 [m] −1 2 2 Inflow pressure pin 1.1161 × 10 [kg/s /m ] Inflow velocity uin 1.0 [m/s] −3 Inflow temperature Tin 3.8893 × 10 [K]

The computational domain, depicted in Figure 6.5, is characterized by the chord length c of the wing section. Notice that the outer wall of the computational domain is defined using two semicircles. By doing so, only an inflow and outflow boundary condition needs to be specified. This greatly simplifies the complicated process of defining boundary conditions for problems involving transonic flows.

uin

25c

u = 0

NURBS Geometry Control Polygon Control Points 25c

10c 15c Figure 6.5: Computational domain and boundary conditions for the transonic ﬂow around a NACA0012 wing section.

The computational domain is discretized using 68, 224 P1 linear triangular ﬁnite elements. In total, 688 elements are located along the wing section wall, and are treated as NURBS-enhanced for the non- CartesianST NEFEM. The NACA0012 wing section geometry is represented by a closed fourth-order

69 NURBS geometry using nine control points. The transonic flow, the presence of shock-waves, and a smooth yet well-defined geometry of the NACA0012 wing section make this example interesting for analysis in conjunction with non-Cartesian ST NEFEM. Furthermore, the available reference data in [141] allow us to investigate the validity of the method for such problems. A detailed numerical study on NACA0012 wing sections in Euler flows is provided in [141], in which the numerical results of multiple finite volume (FV) solvers are presented. In the current work, the example problem is solved with the formulation provided in Section 5.2 using the non-CartesianST NEFEM and SFEM. The resulting numerical solutions are compared against the reference data in [141]. The non-CartesianST NEFEM and SFEM results are presented in Figures 6.6 and 6.7 through the pressure coefficient Cp defined by Equation (6.1). The Cp distribution and the large local gradient in the vicinity of the wing section in Figure 6.6 shows the shock wave that follows from the transonic flow. From Figure 6.6 it can be concluded that the non-CartesianST NEFEM and SFEM solutions are qualitatively in close agreement.

NEFEM •0.2 •0.1 •0.3 0.5 0.0 •0.4

0.1 •0.6 •0.1 0.0 0.2 0.1 •0.8 0.3 0.2 0.0 y 0.3 0.2 •0.8 0.2 0.1 •0.1 0.0 0.1 •0.6

•0.5 0.0 •0.4 •0.1 •0.3 •0.2 SFEM

0.0 0.5 1.0 x

Figure 6.6: Contour lines of the pressure coefﬁcient Cp for non-CartesianST NEFEM and SFEM solution of the transonic ﬂow around a NACA0012 wing section.

To provide a quantitative comparison, the distribution of Cp along the wing section wall is presented in Figure 6.7. This ﬁgure also showsFV-based numerical reference data from [141]. For both non-Cartesian ST NEFEM and SFEM solutions, Figure 6.7 depicts a close agreement to the reference data. The detailed view in Figure 6.7b shows a slight difference in the location of the jump in Cp caused by the transition from supersonic to subsonic ﬂow conditions. As for the cylinder test case, these differences are attributed

70 −1.5

−1.0 −0.30

−0.5 ] ] −0.20 − − [ [ 0.0 p p C C 0.5 −0.10 SFEM SFEM 1.0 NEFEM NEFEM Vassberg et al. [141] Vassberg et al. [141] 1.5 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.48 0.50 0.52 0.54 0.56 0.58 0.60 c [m] c [m]

(a) Cp along the chord of the wing section. (b) Detail of Cp behind the shock wave.

Figure 6.7: Pressure coefﬁcient Cp along the wing section for non-CartesianST NEFEM, SFEM and the reference solution from [141]. to the application of the NURBS-enhanced elements and improved geometry representation in case of the non-CartesianST NEFEM.

6.2 Application to 3D Incompressible Flow Problems

In this section, the non-Cartesian NEFEM is applied to a three-dimensional problem involving incompressible flow. The test case considered involves the flow around a 3D cylinder, as first proposed in [142] and further studied, e.g., in [143, 144]. Using a NURBS surface, it is possible to describe the geometry of a cylinder exactly. Similar to the previous examples, this problem is interesting as it is not possible to describe the geometry exactly using linear finite elements. The computational domain, as documented in [142] is shown in Figure 6.8 and is parametrized by the cylinder diameter D. A similar case is studied in 2D using non-Cartesian NEFEM[28], where it is also extended in the context of interface-coupled FSI simulations. The cylinder wall is represented by a NURBS that is spanned by a 9 by 3 control net, as shown in Figure 6.8. Note that this NURBS surface has two edges that coincide to obtain the cylindrical shape. For this, one row of control points along the Z-axis is therefore doubled. The cylinder is represented by second-order NURBS surface (i.e., p = q = 2) along both parametric coordinate directions (see also Equation (4.4)).

6.2.1 Steady Flow

The first studied case concerns a steady inflow condition, which leads to a stationary flow field. Using the finite element formulation described in Section 5.3, the incompressible Navier-Stokes equations are solved for a Reynolds number Re = 20 and thermal effects are neglected. The Reynolds number itself is based on the cylinder diameter and the mean inflow velocity U¯:

ρDU¯ Re = . (6.2) µ

71 u = 0 4.1D

uin u = 0 2.1D

2D D y x z 5D 20D Figure 6.8: Computational domain for the ﬂow around a 3D cylinder problem.

T The inﬂow velocity proﬁle is given by uin = (U, 0, 0) with:

4 U = 16 Um y z(H − y)(H − z)/H . (6.3)

Here, H is set to 4.1D, and Um is the maximum inflow velocity. The flow velocity is set to zero on the cylinder wall and all the side walls, resulting in a no-slip boundary condition. The problem parameters used for the presented benchmark problem are shown in Table 6.4. These parameters yield a steady flow that excludes any type of vortex shedding.

Table 6.4: Parameters used for the steady ﬂow around a 3D cylinder benchmark case.

Parameter Variable Magnitude Dimension Reynolds number Re 20 [−] Cylinder diameter D 0.1 [m] Fluid density ρ 1.0 [kg/m3] Dynamic viscosity µ 1.0 × 10−3 [kg/m/s] Maximum inﬂow velocity Um 0.45 [m/s]

To quantify the performance of the non-CartesianST NEFEM formulation in relation to the SFEM, a mesh reﬁnement study is performed. For this, a set of four meshes, as shown in Table 6.5, are computed.

Table 6.5: Meshes used for the steady ﬂow around a 3D cylinder problem. Parameters ne and ne,face represent the total number of elements, and the number of non-Cartesian NEFEM face elements along the cylinder wall of the domain.

Mesh ne ne,face 0 7,437 496 1 59,496 1,984 2 475,968 7,936 3 3,807,744 31,744

72 The obtained numerical results are compared by looking at the drag coefﬁcient given by: F C = D , (6.4) D 1 2 2 ρ ||u|| S where FD is the resulting ﬂuid force acting on the cylinder in x-direction, and S the frontal surface area of the cylinder given by 4.1D2 (see Figure 6.8).

SFEM NEFEM ] − [ | | ref D ref C −2 D

− 10 i C | D C | = rel ε

10−3 103 104

ne,face [−]

Figure 6.9: Mesh convergence for the steady ﬂow around a 3D cylinder. The relative error εrel of the drag coefﬁcient CD relative to the reference solution in [143]. Here, CDi refers to the solution of mesh i in Table 6.2.

The reference value that serves as a basis for the comparison between non-Cartesian NEFEM and

SFEM is the drag coefﬁcient, CDref = 6.18533, as provided in [143]. In both cases, non-Cartesian NEFEM and SFEM, the drag coefﬁcient computed on mesh 3 in Table 6.5 is in good agreement with the reference value (non-CartesianST NEFEM: CD = 6.17527, SFEM: CD = 6.17279).

For the complete set of computed meshes, the error εrel of the drag coefficient CD relative to the reference value from [143] is presented in Figure 6.9. It can be seen that for the complete range of meshes, the non-Cartesian NEFEM solution shows a reduced relative error. Similar behavior was previously observed for 2D compressible flows in Section 6.1, and 2D incompressible flows in [28].

6.2.2 Unsteady Flow

Next, an unsteady variant of the cylinder benchmark case is presented, following the work in [145]. The setup is analogous to the steady case in Section 6.2.1. Here, however, the inﬂow velocity is varied over time according to the following expression:

4 U = 16 Um sin(πt/8) y z (H − y)(H − z)/H , (6.5) where again H = 4.1D. Furthermore, the maximum inﬂow velocity is Um = 2.25 [m/s]. The resulting mean velocity U¯(t) = sin(πt/8) [m/s], which is now time dependent, yields a maximum Reynolds number Re = 100.

73 The flow problem is computed for a total time of T = 8 seconds, resulting in the flow changing in a sinusoidal fashion over a half period. The parameters used for the time-dependent problem are given in Table 6.6 and the corresponding time history of the drag coefficient is presented in Figure 6.10 for both the non-CartesianST NEFEM and SFEM. For these simulations, mesh 3 from Table 6.5 is used.

Table 6.6: Parameters used for the unsteady ﬂow around a 3D cylinder benchmark case.

Parameter Variable Magnitude Dimension Reynolds number Re 100 [−] Cylinder diameter D 0.1 [m] Fluid density ρ 1.0 [kg/m3] Dynamic viscosity µ 1.0 × 10−3 [kg/m/s] Maximum inﬂow velocity Um 2.25 [m/s] Time increment ∆t 0.01 [s]

4.0 3.40 SFEM SFEM NEFEM NEFEM John [145] John [145] 3.0 3.35 ] ] − − [ 2.0 [ 3.30 D D C C

1.0 3.25

0.0 3.20 0.0 2.0 4.0 6.0 8.0 3.8 3.9 4.0 4.1 4.2 Time t [s] Time t [s]

(a) Total simulation time T = 8.0 seconds. (b) Detailed view for simulation time t = 3.8 to t = 4.2 seconds.

Figure 6.10: Time history of the cylinder’s drag coefﬁcient CD of the non-CartesianST NEFEM, SFEM, and reference solution in [145].

The comparison of the drag coefficients between the non-Cartesian NEFEM, SFEM and reference data in Figure 6.10 are in good agreement. Of the three solutions, the non-Cartesian NEFEM and SFEM are very similar. Both, however, differ slightly from the reference solution. This difference can be explained by the specific spatial and temporal discretizations used. The space-time method used in the current work provides third-order accuracy, whereas the Crank-Nicolson scheme used in the reference solution yields only second-order accuracy (see also Table 3.1)). A quantitative comparison between the methods is obtained by utilizing the maximum drag coefficient. The maximum occurs at time t = 4.0 seconds. For the the non-Cartesian NEFEM and SFEM simulations, the respective maximum drag coefficient CD,max = 3.272 and CD,max = 3.275, are in good agreement with the reference solution CD,max = 3.2968 as provided by John [145].

74 Chapter 7

Application of Spline-Based Methods to FSI Problems

Following the fluid problems presented in Chapter6, a series of coupled FSI examples will be presented next. These examples are used to demonstrate the effect of geometrically exact spline-based methods on the numerical solutions. In addition to the individual methods used for the fluid and structure problem (i.e., SFEM, IGA, and non-CartesianST NEFEM), specific coupling procedures are employed (DN or RN). The test cases in this chapter are categorized as pure elastic and thermoelastic FSI problems which are presented in Sections 7.1 and 7.2, respectively.

7.1 Elastic Problems

Before considering fully thermoelastic FSI problems, the proposed partitioned solver framework is demonstrated for its application to pure elastic problems. First, an incompressible fluid inside a cavity with a flexible bottom is considered. In this case, the commonly knownDN coupling is applied. Next, a fully enclosed Dirichlet bounded problem where a thin-walled structure is inflated through a prescribed boundary condition is analyzed. StandardDN-type coupling procedures cannot solve such problems due to their enclosed nature. Hence, here aRN coupling is used instead. For all examples in this section, thermal effects are not considered.

7.1.1 3D Cavity With Flexible Bottom

In this example, we study a cubic fluid domain with rigid walls and a flexible bottom as depicted in Figure 7.1. Due to imposed gravitational body forces, the weight of the fluid will cause the flexible bottom to deflect. The deflecting bottom eventually reaches a steady-state deflection for t → ∞. The simple geometry of the fluid domain and the large curvature of the deformed flexible bottom make this example suitable to demonstrate the NURBS-based FSI approach presented in Chapter5. The characteristic parameters of the presented FSI problem are given in Table 7.1. The structure of the flexible bottom is discretized in a geometrically exact manner using a second-order NURBS shell of 11 by 11 elements in the x-y plane. The structural problem is solved using IGA as presented in Section 4.2. By representing the flexible bottom using a shell, the need for a volumetric spline is omitted. The use of shells is allowed under certain circumstances, depending on the structural

75 L

u = 0 Ωf

z y a x Ωs Figure 7.1: Fluid and structural domain of the 3D cavity with ﬂexible bottom problem.

Table 7.1: Parameters used for the 3D cavity with ﬂexible bottom problem.

Parameter Variable Magnitude Dimension General Time increment ∆t 5.0 × 10−3 [s] Characteristic length L 1.0 [m] Fluid Problem Dynamic viscosity µf 1.0 [kg/m/s] Density ρf 750 [kg/m3] Gravitational acceleration g (0.0, 0.0, −9.81)T [kg/m/s2] Structural Problem Density ρs 500 [kg/m3] Young’s modulus Es 5.0 × 105 [kg/s2/m2] Poisson ratio νs 0.3 [−] Bottom thickness a 0.02 [m] dimensions, the expected deformations, and the shell model used. Here, Reissner-Mindlin shell elements are used. Such elements have been previously applied in the context of IGA in, e.g, [86]. For further detail on the speciﬁcs of the Reissner-Mindlin shell element, the reader is referred to the derivations and the references therein. The ﬂuid problem is solved using both the non-CartesianST NEFEM and the SFEM formulation.

The computational domain is discretized using P1P1 linear tetrahedral elements. In the case of the non-CartesianST NEFEM, the elements with a face or edge on the bottom wall of the domain are

76 NURBS-enhanced using the second-order NURBS surface, as used for the structural problem. Hence, both fluid and structure utilize the NURBS-based geometry. The fluid and structure problems are spatially coupled at the fluid-structure interface using the FIE approach from Section 5.1.1. Furthermore, the strongly coupled partitioned approach with aDN-type coupling is used, as shown in Figure 5.3a. The strong coupling procedure ensures fulfillment of the coupling conditions (see Section 2.1) within each time-step.

In Figure 7.2 the transient maximum deflection Dm of the flexible cavity bottom is presented for the non-CartesianST NEFEM and the SFEM. The data in this figure is obtained with the finest fluid mesh given in Table 7.2. The non-CartesianST NEFEM and the SFEM solutions show a similar maximum deflection

Dm over time. Note, however, the differences between the methods presented in Figure 7.2b. Additionally, the absolute error between the steady-state maximum deﬂection of the non-CartesianST NEFEM and the −4 −1 −1 SFEM t =→ ∞ is 4.1465 × 10 m (with Dm = −1.4336 × 10 m and Dm = −1.4294 × 10 m respectively).

0.00

−0.135 −0.05

−0.140 ] ] −0.10 m m [ [ m m −0.145 D D −0.15

−0.150 −0.20 SFEM SFEM NEFEM −0.155 NEFEM −0.25 0.0 0.2 0.4 0.6 0.8 1.0 0.380 0.390 0.400 0.410 0.420 Time t [s] Time t [s]

(a) Simulation time t = 0 to t = 1.0 seconds. (b) Simulation time t = 0.375 to t = 0.425 seconds.

Figure 7.2: Maximum bottom deﬂection Dm over time.

When obtaining the fluid loads along the fluid-structure interface, the accuracy of the integration is crucial. A geometric error is introduced using linear Lagrange finite elements to obtain the fluid loads along curved boundaries. This follows directly from the polygonal approximation of the curved geometry by linear finite elements. This effect can also be seen when taking a closer look at the maximum deflection over time in Figure 7.2. Here, a difference in the magnitude of the maximum deflection, resulting from the geometric error, can be observed. By refining the spatial discretization, it is expected that the maximum deflection of the SFEM solution approaches that obtained with the non-CartesianST NEFEM formulation. To demonstrate that, a mesh refinement study is performed. In this study, the fluid domain is discretized for a range of spatial refinement levels, while the time-step size and structural discretization remain unchanged for all simulations. The mesh refinement study is performed using a total of four meshes presented in Table 7.2. The maximum deflections Dm obtained with the meshes from Table 7.2 are depicted in Figure 7.3 by means of their error relative to the non-CartesianST NEFEM solution of the finest spatial discretization (mesh 3 in Table 7.2).

77 Table 7.2: Fluid mesh refinements used for the 3D cavity with flexible bottom problem. Parameters ne and ne,face represent the total number of elements and the number of non-CartesianST NEFEM face elements along the flexible bottom of the domain.

Mesh ne ne,face 0 1,069 98 1 8,552 392 2 68,416 1,568 3 547,328 6,272

SFEM NEFEM

] 10−2 − [ | 3 | m 3 D m − −3 i D 10 | m D | =

rel −4

ε 10

102 103

ne,face [−]

Figure 7.3: The error ε of the maximum deformation Dm relative to the maximum deﬂection Dm3 of mesh 3 in Table 7.2.

Compared to the SFEM solution, a reduced error of the NURBS-enhanced solution for all discretization levels can be observed in Figure 7.3. However, when the mesh is refined, differences between both methods become smaller. This is in line with the fact that the geometric error of the linear finite element approximation vanishes when ne,face → ∞, and it justifies the use of the non-CartesianST NEFEM, especially for highly curved geometries. Hence, the NURBS-based approach allows fewer elements for the same error, which results in reduced computational cost.

7.1.2 Inﬂation of a Fully Enclosed 3D Cylindrical Domain

The three-dimensional example shown in Figure 7.4a involves an incompressible fluid entering a cylindrical domain enclosed by a thin-walled structure. The entering fluid causes the enclosed domain to inflate and consequently increase the radius of the thin-walled structure. This problem is specifically interesting, as it is a fully-enclosed Dirichlet-bounded problem. Such problems are not solvable using conventional DN-type coupling procedures [101]. Instead, to successfully solve such problems, aRN coupling can be employed, as suggested in [29, 102–104]. Furthermore, a discretization of the curved domain boundaries using linear finite elements leads to geometric errors, as previously discussed in Chapter6. Hence, this example is very suitable to study the effects of using geometrically exact spline-based methods.

78 y ] 0.308 m x [ 3D z exact 0.306 r a 0.304 6D at t = 0

0.302 uin D

Fluid domain radius 0.3 Ωf 0 0.2 0.4 0.6 0.8 1

Ωs Time t [s] (a) Problem setup. (b) Analytic time-dependent solution of the cylinder radius rexact.

Figure 7.4: Three-dimensional inﬂatable circular domain enclosed by a thin-walled structure.

The cylindrical fluid domain is enclosed by a thin-walled shell structure and two rigid walls at the end parts of the cylinder. At these ends, a no-slip boundary condition is enforced. At the remaining inner boundary, an inflow condition is applied, as depicted in Figure 7.4a. The characteristic properties of the problem are given in Table 7.3. Note, in this table, the inflow velocity is defined as the normal velocity using the inward unit normal n.

Table 7.3: Parameters used for the 3D inﬂatable cylindrical domain problem.

Parameter Variable Magnitude Dimension General Characteristic length D 0.1 [m] Time increment ∆t 1.0 × 10−3 [s] Fluid Problem

Inﬂow velocity uin(t) 0.1 · t · n [m/s] Dynamic viscosity µf 1.0 [kg/m/s] Density ρf 1000 [kg/m3] Structural Problem Density ρs 10000 [kg/m3] Young’s modulus Es 1.4 × 106 [kg/s2/m2] Poisson ratio νs 0.3 [−] Cylinder wall thickness a 0.02 [m]

The given inﬂow boundary condition results in an increase of the domain radius r(t). As the domain is fully enclosed by the structure and no-slip boundary conditions and since the inﬂow condition is known, r(t) can be computed exactly as a function of time using the following expression:

Z t 1 2 rexact(t) = Ain u(t)dt + R0. (7.1) π 0

Here, Ain is the area of the inﬂow boundary, and R0 represents the initial radius at t = 0. The exact transient solution for the presented example is shown in Figure 7.4b.

79 As mentioned previously, discretizing the ﬂuid domain using linear Lagrangian ﬁnite elements introduces a geometric error with respect to the exact domain. This error depends on the number of linear elements n along the circumferential direction of the outer domain boundary. The resulting piecewise linear approximation of the domain yields a slightly different radius given by

s Z t 1 2 rpolygon(t) = n 2π Ain u(t)dt + R0. (7.2) 2 sin n 0 Note that the geometric error, and hence, the error in r(t), will vanish when n → ∞. However, this would require undesirably large computational meshes. The use of the proposed NURBS-based methods can be powerful in this case. To numerically solve the presented problem, the cylindrical structure is represented in a geometrically exact manner using a second-order NURBS shell of 136 by 10 Reissner-Mindlin shell elements [86] in the circumferential and axial direction, respectively. The type of shell elements is identical to those used in the example in Section 7.1.1. The discretized fluid domain consists of 11, 540 structured P1P1 tetrahedral elements. This yields a polygonal-type geometry along the circumferential direction with n = 28.A rendering of the corresponding mesh is given in Figure 7.5. For compatibility, the inflow boundary is discretized using standard finite elements for both the non-CartesianST NEFEM and SFEM case. By doing so, the difference between the latter methods is purely at the fluid-structure interface.

X Z

Figure 7.5: Computational mesh with n = 28 linear tetrahedral elements in the circumferential direction.

Note that the volume of the discretized fluid domain is slightly smaller than the exact domain when using linear finite elements. The resulting discrepancy in structural deformations with respect to the exact solution is shown in Figure 7.6. In this figure, a comparison between the non-CartesianST

NEFEM and SFEM simulations is presented via the absolute error εabs over time. The non-CartesianST NEFEM solution shows an error concerning the exact solution, which can be largely attributed to the time integration.

0 0 0.2 0.4 0.6 0.8 1 Time t [s]

Figure 7.6: The error absolute εabs of the three-dimensional cylindrical domain radius with respect to the exact analytic radius.

This is clearly observed when reducing the time-step size ∆t. By doing so, the time integration error vanishes, as is shown in Figure 7.7. Additionally, for ∆t → 0, the non-CartesianST NEFEM solution at t = 1.0 approaches the exact solution given by Equation (7.1). The SFEM solution on the other hand approaches its best possible approximation, the polygonal solution of Equation (7.2).

10−3

] −4

m 10 [ abs ε

10−1 10−2 10−3 10−4 Time-step size ∆t [s]

Figure 7.7: The absolute error εabs of the three-dimensional cylindrical domain radius at time t = 1.0 seconds.

The presented results show that the employedRN coupling allows us to simulate fully-enclosed Dirichlet-bounded FSI problems. Furthermore, the study shows the importance of using an accurate geometric representation within a numerical formulation. This is true especially for fully-enclosed

81 Dirichlet bounded problems, as shown in this example. The accuracy of standard linear finite element methods heavily depends on the element type and its ability to represent the exact computational domain. For the curved domain used here, a geometric error remains even for highly refined SFEM meshes. Employment of non-CartesianST NEFEM elements can significantly reduce the geometric error, as the accuracy of the NURBS-enhanced finite element formulation is mainly affected by the time-discretization error.

7.2 Thermal Problems

Following the pure elastic examples, the thermal and thermal-mechanical FSI problems are considered next. The first example in Section 7.2.1 involves a natural convective cavity flow induced by a heated structure. With this example, the effect of spline-based methods on the coupling of non-matching meshes is investigated. The second example in Section 7.2.2 considers a showcase problem which demonstrates the introduced solver framework for thermoelastically coupled FSI problems. The problem consists of a hot fluid flow that induces thermal expansion of a fixed elastic structure.

7.2.1 Natural Convection through Heated Wall

Inspired by a well-studied example (see, e.g., [30, 38]), the ﬁrst thermal coupled FSI case considers a two-dimensional buoyancy-driven ﬂow on a unit square. As becomes clear from the presented results, this case is particularly interesting in the context of projection techniques for non-matching interfaces.

As depicted in Figure 7.8, while the right wall of the ﬂuid domain has a set temperature ΘR, the left wall of the domain is thermally coupled with a rectangular structure. The structural domain itself has an imposed temperature ΘL on the outer left wall. At the top and bottom walls of the ﬂuid and structure domains, an adiabatic boundary condition is imposed (∂Θ/∂n = 0). Hence, as can be observed in

Figure 7.8, by setting ΘL > ΘR, the ﬂuid has a cooling effect on the structure or vise versa the structure has a heating effect on the ﬂuid.

∂Θ ∂n = 0

Γ ΘL fs ΘR 4L

y ∂Θ x ∂n = 0

L 4L Figure 7.8: Problem setup for natural convection through 2D heated wall problem.

The characteristic parameters used in the presented example are given in Table 7.4. The expansion ratio βf , as used by the Boussinesq approximation for the buoyancy force (see Equation (2.29)), results in

82 a circulating ﬂow within the cavity, which cools the structural domain. This particular example involves purely thermal FSI, meaning that elastic deformations are not considered in this case.

Table 7.4: Parameters used for the natural convection through 2D heated wall problem.

Parameter Variable Magnitude Dimension General Characteristic length L 0.25 [m] Fluid Problem

Right wall temperature ΘR 0.0 [K] Dynamic viscosity µf 1.0 [kg/m/s] Density ρf 1.0 [kg/m3] f 2 2 Speciﬁc heat cp 1.0 [kg m /s /K] Thermal conductivity κf 1.0 [kg m2/s/K] Thermal expansion coefﬁcient βf 1 × 103 [m/K] Gravitational acceleration g (0.0, −1.0, 0.0)T [kg/m/s2] Structural Problem

Left wall temperature ΘL 1.0 [K] Density ρs 7.0 [kg/m3] s 2 2 Speciﬁc heat cp 4.0 [kg m /s /K] Thermal conductivity κs 1.0 [kg m2/s/K]

The thermal FSI problem is solved numerically using the non-CartesianST NEFEM and IGA formulations on the ﬂuid and structural domain respectively (NEFEM-IGA). To investigate the performance of this spline-based solver framework, the numerical results are compared against the results obtained with a purely SFEM based solver framework (SFEM-SFEM). The numerical setup, used in this study is discussed next. For both SFEM-SFEM and NEFEM-IGA simulations aDN-type coupling is employed within a strongly coupled partitioned solver framework (c.f. Figure 5.3a). The non-matching meshes of the ﬂuid and structure are coupled using the FIE method. In this particular problem, the CBF method described in

Section 3.5 is used for the recovery of the normal heat flux qn at the boundary. The fluid domain is spatially discretized using an 80 by 80 linear triangular finite element mesh. On the structural side, the domain is spatially discretized using either a structured 5 by 20 linear triangular finite element mesh or a second-order, 3 by 18 elements NURBS surface. When the non-CartesianST NEFEM is used, the edge of this NURBS surface is used for the enhancement of the fluid mesh elements. Note that using a 3 by 18 elements spline results in a total of 21 control points along the coupling boundary, which is identical to the number of coupling nodes on the structural SFEM mesh. The computed temperature distribution in the fluid and structure is depicted in Figure 7.9a. In this figure, the cooling effect of the fluid is observed. Furthermore, as can be seen in Figure 7.9b, the previously discussed natural convection flow can be observed within the fluid domain. This is a direct result of the buoyancy forces induced by the temperature gradient within the domain. The local fluid temperature at the coupling interface is given in Figure 7.10a for NEFEM-IGA and SFEM-SFEM. Both solver frameworks are in good qualitative agreement. The discrepancies between the two approaches become clear when plotting the difference using the absolute error εabs in Figure 7.10b.

83 0.0 Θ[K] 1.0

(a) Temperature distribution. (b) Velocity vector ﬁeld.

Figure 7.9: Natural convection through a 2D heated wall solution obtained with the NEFEM-IGA setup.

Although small, oscillatory behavior can be observed. Furthermore, these oscillations occur in the direct vicinity of the structural nodes indicated by the vertical lines in Figure 7.10b.

·10−4

0.82 | ΘNEFMEM−IGA − ΘSFEM−SFEM | ] K

fs,s ] 2 L Θ[ Nodes on Γ 0.8 (Θ) [ abs 0.78 ε 1 Error Wall Temperature 0.76 SFEM-SFEM NEFEM-IGA 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 y [m] y [m]

(a) Temperature Θ. (b) Absolute error of the wall temperature.

Figure 7.10: Thermal quantities at the ﬂuid side of the ﬂuid-structure interface Γfs,s.

This effect is more pronounced for the normal heat flux of the fluid at the coupling interface Γfs. The fs normal heat flux qn at Γ is plotted in Figure 7.11a. From this figure, it is apparent that the oscillatory behavior only appears in the SFEM-SFEM solution. Looking closer at these results in Figure 7.11b, it is observed that the oscillations occur near the structural nodes. This behavior is a direct result of using FIE for thermal coupled problems. Recall from Section 5.1.1 that with FIE the transferred data is projected onto the nodes for which the basis function is locally non-zero at the projection point (c.f. Figure 5.1). In case of SFEM-SFEM, the compact support of theFE basis functions leads to a local and uneven distribution of the data transferred from a coarse to a fine mesh. Hence, in the given example, the structural temperature solution is only distributed to the element nodes of the element within which the projection point is located. While the transferred structural temperatures are applied as a Dirichlet boundary condition to the fluid

84 1 1 Nodes on Γfs,s ] ] K/m K/m [ [ 0.99 n 0.9 n q q 0.98 0.8 0.97 Normal heat ﬂux 0.7 SFEM-SFEM Normal heat ﬂux 0.96 SFEM-SFEM NEFEM-IGA NEFEM-IGA 0 0.2 0.4 0.6 0.8 1 0.1 0.15 0.2 0.25 0.3 y [m] y [m]

(a) Complete interface. (b) Sub-region y = 0.1 to y = 0.3 of the interface.

fs Figure 7.11: Normal heat flux qn on the fluid side of the fluid-structure interface Γ . The vertical lines in these plots correspond to the location of the structural nodes or control points of the SFEM-SFEM mesh or NEFEM-IGA spline, respectively. problem, the oscillations remain in the computed fluid temperature. In turn, this introduces the oscillations in qn, as observed in Figure 7.11b. Depending on the mesh resolution and specific problem characteristics, the oscillations can cause difficulties in obtaining converged numerical solutions. Contrary to using SFEM and FIE, the NEFEM-IGA approach allows us to use the NURBS basis for data projection (c.f. Section 5.1.1). The wide support of the NURBS basis leads to an improved and smooth distribution of the projected data. This leads, as can be observed in Figure 7.11, to smoother projected data and corresponding numerical solutions in case of NEFEM-IGA.

Extension to 3D Problems

The previous example is extended to a 3D natural convection problem to investigate further the effect of spline-based methods on the spatial coupling of non-matching meshes. Again, the NEFEM-IGA and SFEM-SFEM formulations are used and their numerical solution are compared. The current example uses similar characteristic parameters to the 2D case presented in Table 7.4. The fluid mesh is discretized using a uniform tetrahedral mesh with 31 elements in the three principal directions. In the case of NEFEM-IGA, the structure is represented by a second order volume NURBS containing 8 by 8 elements along the coupling interface and 3 elements in the direction perpendicular to the coupling interface. For the SFEM-SFEM computations, the mesh is discretized using 21 by 21 linear tetrahedral elements along the coupling interface and 5 element perpendicular to the coupling interface. The numerical solution for the thermal-coupled natural convection problem in 3D is given in Fig- ure 7.12. In 7.12a, the temperature distribution within the structure and the fluid is presented in which the similarities between Figure 7.12a and Figure 7.9a are visible. The control net in Figure 7.12a also shows a volume NURBS as used by the IGA formulation. Note that the non-CartesianST NEFEM formulation only requires a surface NURBS. By fixing the parametric coordinate of the structural volume NURBS corresponding to the direction perpendicular to the coupling interface, the necessary Γfs,f surface geometry spline is obtained.

85 Θ[K] 0.0 1.0 0.65 qn [K/m] 0.95

(a) Temperature distribution in fluid and structure ob- (b) Normal heat flux at the fluid-structure interface us- tained with the NEFEM-IGA setup. ing NEFEM-IGA (left) and SFEM-SFEM formulations (right).

Figure 7.12: 3D heated cavity solution.

As for the 2D problem, oscillatory behavior of the coupling data in the SFEM-SFEM solution can be observed in Figure 7.12b. This figure shows the normal heat flux at the coupling interface Γfs. Whereas a smooth heat flux distribution is obtained for the spline-based NEFEM-IGA solution, the SFEM-SFEM formulation results in an oscillatory heat flux. This is due to the same reasons as for the 2D case. Hence, for three-dimensional thermal coupled problems, spline-based methods can be useful to the data transfer between non-matching meshes.

7.2.2 Thermal Expansion of a Heated 3D Supported Plate

In this example, both thermal and elastic FSI phenomena are combined in a coupled aerothermoelastic problem. A schematic of the 3D example is presented in Figure 7.13 and a corresponding cross-section is shown in Figure 7.14. As can be seen in these figures, the thermoelastic problem considers a flow over a plate which is fixed at its lower edges. The temperature of the fluid entering the domain at the inflow boundary is higher than the initial temperature of the plate and the fluid. Hence, due to a flow-induced temperature increase, the plate expands and elastically deforms in the positive y-direction. As depicted in Figure 7.13, a rectangular flow channel with uniform inflow velocity and temperature

(uin and Θin) is considered. At the bottom wall of the fluid domain, the fluid is thermoelastically coupled with the plate. Apart from the inflow, outflow, and fluid-structure boundaries, no-slip adiabatic boundary conditions are imposed everywhere. The plate is fixed along its bottom edges in x- and z-direction, adiabatic boundary conditions are applied on the boundaries that are not coupled to the fluid, and a stress-free temperature Θ0 is imposed as an initial condition. The dimensions of the coupled problem are defined by the characteristic length L. The corresponding characteristic parameters of the problem are given in Table 7.5. Note that the fluid and structure use the same initial (stress-free) temperature Θ0. In this example, the NEFEM-IGA as well as the SFEM-IGA formulations are compared. Hence,

86 L uin, Θin

1 4L

x L 20 Γfs z 3 4L

3 4L Figure 7.13: Problem setup for the thermal expansion of a heated 3D supported plate problem.

uin, Θin

Γfs

Figure 7.14: Cross-section of the thermal expansion of a heated 3D supported plate problem. contrary to the previous thermal-coupled examples in this chapter, the standard finite element formulation for the fluid problem is used in conjunction with the IGA formulation for the structure. The flow domain is discretized using 64504 unstructured linear tetrahedral finite elements, of which 2072 elements have a face on the fluid-structure interface. In case of NEFEM-IGA, the elements with a face on the fluid-structure interface are NURBS-enhanced. The structure is represented by a second order volume NURBS containing 16 by 16 elements on the coupling interface in the x- and z-direction, and 4 elements in the y-direction perpendicular to the coupling interface (see the coordinate system in Figure 7.13). The outline of the flow domain combined with the control net used for the NURBS volume representing the plate is shown in Figure 7.15. Over time, the temperature in the fluid and structure increase toward the inflow temperature. With this temperature increase, the plate starts to deform in the positive y-direction. The complete system reaches a steady state deformation when t → ∞ and a uniform temperature distribution is achieved. In Figure 7.16 the initial and final plate configuration at z = 0.5L are shown red and gray respectively. Furthermore,

87 Table 7.5: Parameters used for the thermally expanding supported plate.

Parameter Variable Magnitude Dimension General Characteristic length L 1.0 [m] Time increment ∆t 1.0 [s] Fluid Problem

Inflow velocity uin 0.01 [m/s] Initial/stress-free temperature Θ0 280.0 [K] Inflow temperature Θin 300.0 [K] Dynamic viscosity µf 1.0 × 10−3 [kg/m/s] Density ρf 998.0 [kg/m3] f 2 2 Specific heat cp 4.18 [kg m /s /K] Thermal conductivity κf 61.5 [kg m2/s/K] Structural Problem Density ρs 7.8 × 103 [kg/m3] s 2 2 Specific heat cp 200.0 [kg m /s /K] Thermal conductivity κs 40.0 [kg m2/s/K] Young’s modulus Es 2.1 × 108 [kg/s2/m2] Poisson ratio νs 0.3 [−] Thermal expansion coefficient βs 1.22 × 10−3 [m/K] in Figure 7.17, the expansion process at the centerline of the domain is presented for a series of time instances. Here both NEFEM-IGA and SFEM-IGA solutions are given. The NEFEM-IGA and SFEM-IGA solutions in Figure 7.17 show only small differences in the temperature distribution. Furthermore, the observed deformation of the plate is indistinguishable between the two methods. Hence, from Figure 7.17, it can be concluded that both methods are in good qualitative agreement with each other.

An additional comparison of the maximum plate deflection Dm over time in Figure 7.18 confirms the previous observations. In this figure it can be seen that the maximum plate deformation of the NEFEM- IGA and SFEM-IGA solutions are both approaching a steady state. A closer look at a short time interval in Figure 7.18b shows only minor differences. The final computed maximum steady state deflections at t = 2000 seconds are 4.6833 × 10−2 m and 4.6836 × 10−2 m for NEFEM-IGA and SFEM-IGA, respectively. Despite these small differences, it is expected that the t → ∞ solutions of NEFEM-IGA and SFEM- IGA approach roughly the same steady state. This since both NEFEM-IGA and SFEM-IGA make use of the same structural problem setup, and the temperature distribution eventually becomes uniform. Furthermore, due to chosen problem parameters, fluid forces acting on the structure only have a limited influence. The difference between the maximum deflections of the NEFEM-IGA and SFEM-IGA also shows the expected behavior for t → ∞ in Figure 7.19.

88 Θ[K] 280 300284 288 292 296

Figure 7.15: The outline of the ﬂow domain, the NURBS volume, and its corresponding control net representing the plate. The colors indicate the temperature distribution within the plate and at the centerline section of the ﬂuid domain at t = 100 seconds.

Figure 7.16: Initial and ﬁnal plate deformation of the NEFEM-IGA solution.

89 Θ[K] 280 300285 290 295 Θ[K] 280 300285 290 295

(a) t = 50 s (b) t = 50 s

Θ[K] 280 300285 290 295 Θ[K] 280 300285 290 295

(e) t = 200 s (f) t = 200 s

Θ[K] 280 300285 290 295 Θ[K] 280 300285 290 295

(g) t = 500 s (h) t = 500 s

Figure 7.17: Temperature distribution at z = 0.5L for several time instances of the NEFEM (left) and SFEM (right) solutions.

90 ·10−2 ·10−2 5

4 4.48 ] ] 3 m m [ [ 4.46 m m D D 2 4.44 1 SFEM SFEM NEFEM 4.42 NEFEM 0 0 500 1,000 1,500 2,000 1,000 1,020 1,040 1,060 1,080 1,100 Time t [s] Time t [s]

(a) t = 0 s to t = 2000 s. (b) t = 1000 s to t = 1100 s.

Figure 7.18: Maximum plate deﬂection Dm over time.

·10−3 ] -

[ 4.0 | | SFEM 3.0 m D − NEFEM

m 2.0 D | NEFEM m D | 1.0 = rel ε 0.0 0 500 1,000 1,500 2,000 Time t [s]

Figure 7.19: Relative error of the maximum plate deﬂection εrel of the NEFEM-IGA and SFEM-IGA solutions.

Chapter 8

Summary and Outlook

8.1 Summary

This thesis investigates the application of spline-based methods to three-dimensional, aerothermoelastic problems involving both compressible and incompressible flows. More specifically, the combination of NEFEM and IGA for the fluid and structure problems within a partitioned solver framework is explored. CAD tools are commonly used in the design stage of technical systems. Typically for numerical analysis, a CAD geometry is solely used for defining the discrete computational domain. Especially for curved domains, this step could result in a geometric error between the exact spline-based and discretized geometry. With the arrival of IGA, an attempt is made to close the gap between CAD and numerical analysis. IGA uses the spline-basis of the CAD geometry directly for numerical analysis. Hence, the possible geometric error resulting from the domain discretization step is omitted. As most CAD tools represent three-dimensional objects using their external surfaces and curves, IGA on volumetric splines is not always directly possible. Additionally, when geometries contain a large number of detailed components, generating IGA-suitable volumetric splines becomes extremely complex, if at all possible. While for some structural problems, this issue can be avoided by using shell theory, for three-dimensional flow problems, such workarounds do not exist. The NURBS-based NEFEM approach provides a good alternative to this issue. The NEFEM incor- porates the surface geometry of a CAD model directly into a standardFE formulation. This is achieved by enhancing the elements that have a face on the NURBS boundary. The enhancement takes place by applying a mapping which includes the NURBS. This approach allows for standard mesh generators to be used as the enhancement takes place afterward during theFE assembly procedure.

In this work, a coupled partitioned FSI solver framework is introduced, which combines both NEFEM and IGA formulations for the fluid and the structure, respectively. On the structural domain, IGA is employed within a semi-discreteFE formulation to solve thermal, geometrically non-linear elastodynamic, and weakly-coupled thermoelastic problems. Time integration is achieved by applying a θ-scheme for the thermal problem, and a generalized-α scheme for the elastodynamic problem. On the fluid domain, an NEFEM-approach is used within a DSD/SSTFE formulation to solve both compressible and incompressible flow problems. An FSI andST suitable NURBS-enhancement is

93 achieved by the introduction of a new space-time suitable geometric mapping between a three-dimensional element in physical space to the corresponding reference element. The proposed mapping results in a non-Cartesian NEFEM formulation that is specifically suitable for FSI. Both black-box fluid and structural solvers are strongly coupled in a partitioned solver framework by employingDN- orRN-type coupling. This proposed solver framework leads to a single NURBS-based, geometrically compatible fluid-structure interface which is used by the fluid and structural solvers. This joint use of the interface description allows improved projection procedures for data exchange between two non-matching discretizations at the interface. In this work, a consistent and energy-conserving FIE-type data mapping is used in which the NURBS basis is directly used for interpolation of nodal data.

Before exploring the use of spline-based methods in the context of aerothermoelasticity, the non- CartesianST NEFEM is first tested for pure fluid problems. Through a series of numerical examples involving curved computational domain boundaries, the performance of the newly proposed non-Cartesian ST NEFEM formulation is demonstrated. The non-CartesianST NEFEM is compared against the SFEM which differs solely by the exclusion of NURBS-enhanced elements. When available, the numerical solution of the non-CartesianST NEFEM and the SFEM show good agreement with the analytical and reference data in the available literature. For the demonstrated examples, involving incompressible and compressible flows, the non-Cartesian ST NEFEM shows a reduced numerical error compared to the SFEM. This holds for steady and unsteady problems, as well as compressible flows in the supersonic and transonic regimes involving shock waves. Small discrepancies in flow features are observed between the non-CartesianST NEFEM and the SFEM. The shock-wave location in the transonic flow over the NACA0012 wing section, for example, is slightly shifted down stream when the exact geometry is considered. Similar effects are observed in the wall pressure distribution for the cylinder flow examples. Since shock-waves and flow separation can significantly affect the aerodynamic performance of engineering systems, it is important to simulate such flow features accurately. The use of the proposed non-CartesianST NEFEM can contribute to further improve numerical accuracy when analyzing such flow phenomena.

After validation of the applied mapping within non-CartesianST NEFEM, the spline-based FSI solver framework is applied to a series of examples involving elastic, thermal, and fully coupled thermoelastic FSI problems. The purely elastic examples demonstrate the use of the spline-based framework forDN andRN coupling in combination with shell structures. For bothDN andRN problems, a difference in deformations is observed between the non-Cartesian ST NEFEM and the SFEM. A number of simulations and mesh reﬁnement study, show that the enhanced geometric representation of the spline-based approach results in improved accuracy. In fact, in the case of the fully-enclosed Dirichlet bounded problem, the SFEM solution approaches the best possible solution using linear elements, i.e., the analytic solution based upon a piecewise linear polygonal cylinder. Apart from this clear beneﬁt, the proposed spline-based formulation combined withRN shows that it is capable of solving fully-enclosed Dirichlet bounded problems without the common stability issues arising with such problems. The application of the spline-based solver framework to thermal and thermoelastic FSI problems

94 shows an important advantage in terms of the spatial coupling between non-matching meshes. While the spline-based methods are compatible in the geometric sense, the individual discretizations are non- matching. The wide support of the spline bases used in the examples result in an improved distribution of the transferred data. This leads to smoother boundary conditions at the coupling interface of the individual single-field problem. Node-to-node oscillations in the transferred data observed for a conventional SFEM- SFEM solver framework vanish when the spline-based approach is applied. Similar observations are made for SFEM and IGA when used for the fluid and structural problem, respectively. This is so since in such cases the spline-basis is utilized for the distribution of structural loads. Hence, the benefit of the wider support of the spline-basis remains.

8.2 Outlook

Since the work provided in this thesis acts as a proof of concept, additional research is needed for the proposed spline-based approach to be applicable to real-world problems. Moreover, additional extensions in several directions can help to further the improvement of the presented methods. The following points are to be considered in deﬁning future development of spline-based methods for FSI:

• The success of using spline-based approaches for numerical analysis depends heavily on how well these approaches can handle splines with increasing complexity. While the spline geometries used in the presented examples are rather simple, it is imperative that the spline-based analysis tools support the use of complex, trimmed, and multi-patch splines. Additionally, to further close the gap between CAD and numerical analysis, proper workﬂows and spline-data handling tools are needed. A combined effort could be of interest in future research.

• The accuracy of non-CartesianST NEFEM formulations rely on the applied numerical integration rules. The presented example in Section 4.3.4 shows that the choice of the integration rule can reduce geometric errors by several orders of magnitude. Further work towards the best possible integration rule for numerical accuracy and computational efﬁciency is therefore desired.

• Accurate representation of the problem domains can strongly influence fluid phenomena such as turbulence, cavitation, acoustics, and shock waves. Numerical simulation of such complex phenomena can benefit from accurate geometric representation, as provided by non-CartesianST NEFEM. Applying the presented methods to such problems could provide valuable insights and further expand our understanding of the numerical performance of spline-based methods.

• The incorporation of spline-geometries within non-CartesianST NEFEM opens the door for their use in the context of sliding interfaces. When employing NURBS-enhanced elements along curved internal sliding interfaces, the geometric compatibility issue can be resolved. Such concepts can be subsequently applied to problems involving rotating sub-domains or large relative translations.

• With recently developed unstructured space-time formulations, it is possible to apply local time- step reﬁnement. Extensions of the non-Cartesian ST NEFEM through new spline-based mapping strategies could be further investigated to allow for exact geometry representation in conjunction with local time-step reﬁnement.

95 • Fully-enclosed Dirichlet bounded FSI problems have shown to beneﬁt signiﬁcantly from the use of spline-based solver frameworks. The use of such frameworks to real-world applications involving fully-enclosed Dirichlet bounded domains can therefore be of interest.

• Another interesting ﬁeld of research in which spline-based methods could be exploited for numerical accuracy is lubricated contact. E.g., it could be applied to the analysis of temper rolling processes in the production of high grade steel, in which FSI plays an important role.

• The application of spline-based methods for thermoelastic coupled FSI problems can be further investigated. Especially, the combination of thermal expansion and exact geometry representation can be of interest for applications where thermal management is investigated. Some exemplary problems which involve compressible ﬂows in the trans- and supersonic regime are rocket propulsion systems and internal combustion engine design. Predicting the behavior of shock waves and thermal efﬁciency within such systems can greatly improve the lifespan and operational cost. Research towards using the proposed methods for such applications can further investigate numerical performance of various spatial coupling-procedures applied to thermal-coupled problems.

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