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Advanced Lattice BoltzmannModels for the Simulation of

Additive Manufacturing Processes

Hochentwickelte Lattice BoltzmannModelle fur die Simulation von additiven

Fertigungsprozessen

Der Technischen Fakultat

der Friedrich-Alexander-UniversitatErlangen-Nurnberg

zurErlangung des Doktorgrades Dr.-Ing.

vorgelegt vonRegina Degenhardt

aus Landshut.

Als Dissertation genehmigtvon der Technischen Fakultatder Friedrich-Alexander-Universitat Erlangen-NurnbergTag der mundlichen Prufung: 06.11.2017

Vorsitzender des Promotionsorgans: Prof. Dr.-Ing. Reinhard Lerch

Gutachter: Prof. Dr. Ulrich RudeProf. Dr. Carolin Korner

Verlag Dr. Hut, Munchen 2017ISBN 978–3–8439–3418–3

Abstract

This thesis presents the three–dimensional modeling, discretization, implemen-tation, and simulation of additive manufacturing processes on the example ofelectron beam melting (EBM). The fluid dynamics of the liquified melting poolare modeled by the incompressible Navier–Stokes equations and the incorpora-tion of energy by the heat equation. The applied numerical scheme is a thermalmulti–distribution lattice Boltzmann method (LBM) allowing an efficient par-allel implementation. The liquid phase of the melting pool and the gas phaseof the atmosphere are separated by the free surface lattice Boltzmann method(FSLBM) that does not compute the dynamics of the gas phase explicitly butsets a boundary condition at the interface. Furthermore, the electron beamgun and the metal powder particles are explicitly modeled. A realistic particlesize distribution is achieved by using an inverse Gaussian distribution. Forthe absorption of energy two different algorithms are derived depending on theacceleration voltage of the electron beam.

Most of the EBM specific algorithms are embedded in the highly parallel lat-tice Boltzmann framework WALBERLA. The metal powder particles are simu-lated by the also highly parallel physics engine pe. Within the coupling particlesare represented as rigid bodies in the pe and treated as boundaries in the LBMscheme of WALBERLA. Both frameworks work on state–of–the–art supercomput-ers. The EBM application and its implementation are validated by benchmarkswhere analytical solutions are common knowledge. Moreover, the simulationresults are compared to experimental data with respect to quality of the prod-uct in order to avoid porosity, and ensure dimensional accuracy. Since thenumerical and experimental data are highly concordant the implemented EBMmodel is suitable to develop new processing strategies in order to improve thequality of the products. The simulations support machine users and developersin order to find an optimal parameter set for specific parts.

Lastly, the accuracy order of the applied free surface boundary condition isexamined via the Chapman–Enskog expansion since it has a huge influenceon the simulation results. It is established that the original FSLBM bound-ary condition is just first order accurate for general cases and since the LBMis second order accurate the overall accuracy is reduced by applying FSLBM.In order to overcome this deficiency an improved second order FSLBM bound-ary condition is derived successfully. The importance and correctness of thisnew FSLBM boundary condition is finally underlined by a thorough validationagainst analytical calculations and experiments.

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Zusammenfassung

Diese Arbeit befasst sich mit der dreidimensionalen Modellierung, Diskreti-sierung, Implementierung und Simulation von additiver Fertigung anhand desBeispiels Elektronenstrahlschmelzen (EBM). Die Fluiddynamik des flussigenSchmelztiegels wird durch die inkompressiblen Navier–Stokes Gleichungen unddie Energieeinbringung durch die Warmeleitungsgleichung modelliert. Die nu-merische Diskretisierung erfolgt durch die thermische Lattice Boltzmann Me-thode (LBM) mit mehreren Verteilungsfunktionen, um eine effiziente Parallel-isierung zu ermoglichen. Die Trennung der flussigen Phase des Schmelztiegelsvon der Gasphase der Atmosphare wird durch die freie OberflachenmethodeFSLBM erreicht. Diese berechnet die Gasdynamik nicht direkt, sondern setzteine Randbedingung an der Grenzflache. Zudem werden die Elektronenstrahlka-none und die Partikel des Metallpulvers explizit modelliert. Hierbei wird eine re-alistische Großenverteilung der Partikel mit Hilfe einer inversen Gaußverteilungerreicht. Zwei verschiedene Absorptionsalgorithmen werden abhangig von derBeschleunigungsspannung des Elektronenstrahls hergeleitet.

Ein Großteil der EBM Algorithmen ist in das WALBERLA Softwarepaket einge-bettet, einem hochparallelen Lattice Boltzmann Loser. Die Partikel des Me-tallpulvers werden mit der ebenfalls hochparallelen Physics Engine pe erzeugt.Die Kopplung beider Softwarepakete erfolgt uber die Partikel als Starrkorperin der pe und uber Randbedingungen in der Lattice Boltzmann Methode vonWALBERLA. Das EBM Modell und seine parallele Implementierung werdendurch Vergleichsgroßen validiert, deren analytische Losungen allgemein bekanntsind. Zudem werden die Simulationsergebnisse mit experimentellen Daten hin-sichtlich der Qualitatseigenschaften des Bauteils verglichen. Hier werden vorallem Porositat und Maßhaltigkeit untersucht. Da die numerischen und experi-mentellen Daten sehr gut ubereinstimmen, eignet sich das EBM Modell hervor-ragend, neue Prozessstrategien zu entwickeln, um die Qualitat der Bauteile zuverbessern. Somit unterstutzen die Simulationen Nutzer als auch Entwicklerder EBM Maschinen um optimale Parameter fur spezielle Bauteile zu finden.

Zuletzt wird die Fehlerordnung der verwendeten Randbedingung der freienOberflachen durch eine Chapman–Enskog Entwicklung untersucht. Fur allge-meine Falle ist die ursprungliche FSLBM Randbedingung nur erste Ordnunggenau, was die allgemeine Fehlerordnung, welche durch den Lattice Boltz-mann Methode vorgegeben ist, reduziert. Um diesen Nachteil zu beseitigen,wird eine verbesserte FSLBM Randbedingung entwickelt, welche zweite Ord-nung genau ist. Die Bedeutung und Korrektheit dieser neuen FSLBM Randbe-dingung wird schließlich mit einer ausfuhrlichen Validierung gegen analytischeLoungen verdeutlicht.

v

Acknowledgements

The whole is greater than thesum of its parts.

(Aristoteles (384 - 322))

It takes a long time until all the blank pages of this thesis were filled withideas, explanations, and meaningful simulation results. Writing a thesis is alsonot only a product of sitting lonesome in an office and program models andalgorithms, in fact it is encouraged by a lot of people which I want to honor onthis page.

First of all, I would like to thank Prof. Dr. Ulrich Rude for being my super-visor. He trusted in me, founded my research, and always took time for megiving me new inspiring ideas. Furthermore, I would like express my thanks toProf. Dr. Carolin Korner for sharing her expertise related to additive manufac-turing processes and the corresponding physical questions with me. It was apleasure to work with her for the FastEBM project.

Moreover, this work would not have been possible without an essential amountof compute time on compute clusters. I am very grateful for the compute timeon LiMa and SuperMUC granted by the Regionales Rechenzentrum Erlangen(RRZE) and the Leibniz Rechenzentrum Munich (LRZ), respectively. Addition-ally, my work was supported by European Union Seventh Framework Program- Research for SME’s with full title ”High Productivity Electron Beam MeltingAdditive Manufacturing Development for the Part Production Systems Market”and grant agreement number 286695.

Next, I am much obliged to Dr. Matthias Markl not only for being a pleasantoffice mate for over two years but also for instructing me into the FastEBMproject. I like also to express my gratitude to Simon Bogner to give me alwaysa sympathetic hearing for mathematical discussions and especially belongingthe free surface problems. It would be also unpardonable not to thank theadmin team, namely Frank Deserno, and the office team, namely Iris Weiß andAlexandra Lukas-Rother at this point. They grant practical relief to me in everytechnical emergency situation one can imagine in the handling with computersor bureaucracy. Special thanks goes also to my caring friends Dr. KristinaPickl, Dr. Daniela Anderl, and Dr. Daniela Fußeder whose help and support Icannot sum up in a few sentences; suffice it to say, thank you for proofreadingall my papers as well as the whole thesis and always motivating me patientlywhen I struggled. Furthermore, I thank Christian Godenschwager and FlorianSchornbaum, and all other colleagues of the Chair of System Simulation. It was

vii

nice to have you all not only as helpful colleagues but also as good friends whomade work enjoyable.

Finally, I would like to express my sincere thanks to my family, especiallyto my lovely parents, Jan and Cosima. Thank you for believing in me all thetime and grant me the scope for development and time for doing my researchalthough it was not easy for you. I dedicate this disseration to you.

Regina Degenhardt

viii

Contents

Contents

1 Motivation 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I Fundamentals using LBM for Additive Manufacturing Processes 5

2 Additive Manufacturing Processes 72.1 What is Additive Manufacturing? . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Process Chain of AM Technology . . . . . . . . . . . . . . . . . 92.1.2 Advantages and Strategies of AM Technology . . . . . . . . . 102.1.3 Industries using AM Technology . . . . . . . . . . . . . . . . . 11

2.2 Classification of Additive Manufacturing Processes . . . . . . . . . . 132.2.1 Photopolymerization Processes . . . . . . . . . . . . . . . . . . 132.2.2 Extrusion-based Methods . . . . . . . . . . . . . . . . . . . . . 142.2.3 Material Jetting Processes . . . . . . . . . . . . . . . . . . . . 152.2.4 Binder Jetting Processes . . . . . . . . . . . . . . . . . . . . . 162.2.5 Sheet Lamination Process . . . . . . . . . . . . . . . . . . . . . 162.2.6 Directed Energy Deposition Processes . . . . . . . . . . . . . 172.2.7 Powder Bed Fusion Processes . . . . . . . . . . . . . . . . . . 18

2.3 EBM - Electron Beam Melting . . . . . . . . . . . . . . . . . . . . . . 192.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Mathematical Model for Thermal Fluid Flow 233.1 Eulerian and Lagrangian Description of Fluid Flow . . . . . . . . . . 233.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Conservation of Mass and Continuity Equation . . . . . . . . 283.2.2 Conservation of Linear Momentum and Navier–Stokes Equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Conservation of Energy and Energy Equation . . . . . . . . . 313.2.4 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Dimensionless Balance Laws and Dimensionless Numbers . . . . . 333.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Numerical Discretization 374.1 Numerical Discretization by LBM . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Spatial Discretization by Stencils in the Mesoscopic Range . 394.1.2 LBM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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Contents

4.1.3 Chapman-Enskog Ansatz . . . . . . . . . . . . . . . . . . . . . 454.1.4 Thermal LBM by Multispeed Approach . . . . . . . . . . . . . 49

4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Summary of Numerical Foundations . . . . . . . . . . . . . . . . . . 57

5 Free Surfaces 595.1 Mathematical Description of the Interface . . . . . . . . . . . . . . . 605.2 Review of LB Multiphase Methods . . . . . . . . . . . . . . . . . . . . 635.3 Thermal Free Surface Lattice Boltzmann Method . . . . . . . . . . . 66

5.3.1 Cell Types and Conversion Rules . . . . . . . . . . . . . . . . 665.3.2 Free Surface Boundary Condition . . . . . . . . . . . . . . . . 675.3.3 Thermal Free Surface Extension . . . . . . . . . . . . . . . . . 70

5.4 Summary of LBM Multi–Phase Methods . . . . . . . . . . . . . . . . 71

II Coupled Multiphysics Simulations 73

6 Specific EBM Modeling Aspects 756.1 Functionality and Properties of the Electron Beam . . . . . . . . . . 766.2 Modeling of Metal Powder Bed . . . . . . . . . . . . . . . . . . . . . . 78

6.2.1 Properties of Powder Particles . . . . . . . . . . . . . . . . . . 786.2.2 Mathematical Model of Powder Particles in the EBM Process 81

6.3 Model of Energy Absorption . . . . . . . . . . . . . . . . . . . . . . . 846.3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . 846.3.2 Numerical Approximation of Absorption Model . . . . . . . . 866.3.3 Different EBM Absorption Models . . . . . . . . . . . . . . . . 87

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7 The Software Frameworks WALBERLA & pe 917.1 The WALBERLA Framework . . . . . . . . . . . . . . . . . . . . . . . . 92

7.1.1 Software Quality in WALBERLA . . . . . . . . . . . . . . . . . . 927.1.2 Applications in WALBERLA . . . . . . . . . . . . . . . . . . . . 937.1.3 Domain Decomposition and Communication Concepts in

WALBERLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.2 pe -The Physics Engine for Rigid Body Simulations . . . . . . . . . . 96

7.2.1 DEM Solver in the pe . . . . . . . . . . . . . . . . . . . . . . . . 977.2.2 Parallelization Concepts of the pe . . . . . . . . . . . . . . . . 997.2.3 pe Solver for EBM Process . . . . . . . . . . . . . . . . . . . . . 100

7.3 The General Coupling of WALBERLA – pe . . . . . . . . . . . . . . . . 1007.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8 EBM in WALBERLA and its Performance 1058.1 Integration of the EBM Model in WALBERLA and pe . . . . . . . . . . 105

8.1.1 Powder Generation for EBM by pe . . . . . . . . . . . . . . . . 1068.1.2 Parallel Absorption Algorithms for EBM in WALBERLA . . . . 109

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Contents

8.1.3 Absorption of Energy by Particles in the Powder Bed . . . . . 1118.2 Scaling Tests of the EBM Model . . . . . . . . . . . . . . . . . . . . . 113

8.2.1 Metrics and Conventions . . . . . . . . . . . . . . . . . . . . . 1148.2.2 Test Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.2.3 Scaling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.2.3.1 Weak Scaling . . . . . . . . . . . . . . . . . . . . . . . 1178.2.3.2 Strong Scaling . . . . . . . . . . . . . . . . . . . . . . 120

8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

9 Simulation for Application: EBM 1259.1 Validation of the EBM Model . . . . . . . . . . . . . . . . . . . . . . . 126

9.1.1 Validation of the Energy Equation . . . . . . . . . . . . . . . . 1279.1.2 Validation of Solid – Liquid Phase Transition by the Stefan

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.1.3 EBM Validation by Comparison with Experimental Data . . . 131

9.1.3.1 Definitions and Validation Set–Up . . . . . . . . . . . 1319.1.3.2 Experimental Process Window . . . . . . . . . . . . . 1329.1.3.3 Numerical Process Window . . . . . . . . . . . . . . . 132

9.2 Improvements of the EBM Process by the Use of Simulations . . . . 1379.2.1 Extension of Process Window . . . . . . . . . . . . . . . . . . . 1379.2.2 Increase of Beam Diameter . . . . . . . . . . . . . . . . . . . . 1399.2.3 Decrease of Line Offset . . . . . . . . . . . . . . . . . . . . . . 140

9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

III Extension: Advanced Models and Methods 147

10 Accuracy Analysis of FSLBM Boundary Condition 14910.1Analyzing Boundary Conditions via Chapman–Enskog Expansion . 150

10.1.1TRT Collision Model . . . . . . . . . . . . . . . . . . . . . . . . 15010.1.2Chapman–Enskog Expansion of TRT Collision Model . . . . . 153

10.2Examination of the Accuracy Order of the FSLBM B.C. . . . . . . . 15610.2.1Boundary Closure Relations . . . . . . . . . . . . . . . . . . . 15610.2.2Chapman–Enskog Expansion of FSLBM B.C. . . . . . . . . . 157

10.3Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

11 Second Order FSLBM Boundary Condition 16111.1Shear Rate and Pressure B.C. of Second Order Accuracy . . . . . . 16211.2Second Order Accurate FSLBM B.C. . . . . . . . . . . . . . . . . . . 16511.3Numerical Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11.3.1Plate–Driven Planar Flow . . . . . . . . . . . . . . . . . . . . . 16611.3.2Linear Couette Flows . . . . . . . . . . . . . . . . . . . . . . . 16911.3.3Steady Parabolic Film Flow . . . . . . . . . . . . . . . . . . . . 16911.3.4Breaking Dam . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

11.4Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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Contents

12 Conclusion and Future Work 17512.1Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17512.2Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

13 Journal and Conference Publications 179

14 Talks and Trainings 181

15 Projects and Supervised Student Theses 183

List of Figures 185

List of Tables 189

Bibliography 191

xii

Chapter 1: Motivation

1 Motivation

One of the greatest pains tohuman nature is the pain ofa new idea.

(Walter Bagehot (1826-1877))

1.1 Motivation

Industries always search for strategies to improve and optimize their productionprocesses. These strategies should reduce production time or costs in order tolower the overall costs for the consumer and gain a competitive edge. Moreover,new developments may achieve cost and time savings. One way to find opti-mization techniques is based on real experiments. However, real experimentsare not always possible because e.g., they are too expensive or the process itselfis too fast or the products are too small. In order to overcome these short-comings numerical simulations are established since the 1960’s and becameincreasingly important with growing compute power.

PhysicalConditions

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Figure 1.1: Simulation pipeline.

The first step of numerical simulations is the derivation of a suitable mathe-matical model that covers the corresponding process thoroughly. Since modelsare only capable of abstracting and reducing the reality the decision whichaspects and parameters are modeled is significant. The mathematical modelprovides the governing equations of the process. A suitable numerical scheme

1


discretize the mathematical model. The choice of this scheme deeply dependson the process and the model. The algorithms can be implemented in alreadyexisting software frameworks or new ones have to be developed. Afterwards, thesimulations of the process can be started. According to the size and time of thesimulation, smaller compute machines or even state–of–the–art supercomput-ers are required. The simulation results have to be validated in order to checktheir correctness. In the best case benchmarks exist where the solution is givenanalytically or experimental data is available. The results can be compared andif they are concordant the simulation can be regarded as reliable. Simulationscan be used to gain a profound knowledge of the process, and even optimize theprocess by, e.g., finding an optimal parameter set for the process. It is furtherpossible to simulate future machines or new process sequences which have nottested in reality. This enables time and cost savings in the development pro-cess. After these steps have been completed improved and optimized productscan be achieved.

In this thesis the process of electron beam melting (EBM) is examined andthe corresponding steps of the simulation pipeline in Figure 1.1 are thoroughlyexecuted. Electron beam melting belongs to the class of additive manufacturing

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(AM) methods. Various products of daily life, like catalysts in cars or turbinesin airplanes or also medical implants for teeth or artificial hips, are producedby AM methods. These procedures allow the building of intricate compositionsof various materials as well as laborious building techniques in a more facileway than classical approaches. Their products can relieve our daily life. How-ever, since the AM approach is a quite new technology there is plenty potentialfor improvement. In order to stay competitive on the market an extensive un-

2


derstanding of the process is essential. This can also be reached by numericalsimulations.

The outline of this thesis is as follows: in Chapter 2 the AM technology isexplained. Industries which already use AM technology are presented as wellas their products. In particular, the EBM method is described, modeled, im-plemented, and simulated in the following. In the subsequent Chapter 3 themathematical model for the EBM process is introduced based on the conser-vation laws for mass, momentum, and energy. These partial differential equa-tions describe the behavior of the melting pool dynamics and the incorporationof the energy by the electron beam gun. The discretization schemes put lifeinto the partial differential equations. For the simulation of the EBM processthe lattice Boltzmann method (LBM) is chosen to discretize the mathematicalmodel. LBM is a mesoscopic approach and its derivation on the basis of theBoltzmann equation is given in Chapter 4. Since the incorporation of energyby the electron beam gun is essential in the EBM process a thermal LBM andcorresponding boundary conditions are further presented. In Chapter 5 thenumerical treatment of the free surface between the melting pool and the gasof the build chamber is explained. Several free surface methods, which areusually combined with the LBM, are compared and a free surface thermal LBMapproach is prescribed which neglects the dynamics of the gas phase by settinga special boundary condition at the interface. This closes Part I of the thesis(cf. Figure 1.2). Part II starts with Chapter 6 paying attention to EBM specificmodeling aspects. Here, it is focused on the modeling of the electron beamgun, a realistic powder size distribution, and the absorption of energy. Sub-sequent Chapter 7 introduces the two software frameworks WALBERLA and pewhich are used to embed the EBM algorithms. The structure and functionalityare explained regarding the high performance aspects. Both frameworks offerthe implementation infrastructure for simulations close to reality. The par-allel implementation of the EBM specific algorithms in these frameworks andthe examination of parallel scalability by scaling experiments follows in Chap-ter 8. Various meaningful simulation results of the EBM process are shown inChapter 9, which closes Part II. Validation cases show the correctness of themodel and its implementation. Extensive simulations deliver optimal parame-ter sets for current EBM machines and further provide strategies for improvedEBM processes. In the last Part III of this thesis, the accuracy order of theapplied free surface approach is examined (see Chapter 10) since a theoreticalfoundation of the accuracy is missing. This is achieved via a Chapman–Enskogexpansion. In Chapter 11 a higher order free surface boundary condition isintroduced and validated by various numerical test cases.

1.2 Overview

After Part III the main conclusions of the this thesis are summarized and anoutlook for further research topics is given in Chapter 12.

3


Some parts of the chapters in this work are mainly based on the followingpublications:

• Section 6.2 and Section 9.1 are based on,

∗ R. Ammer, M. Markl, U. Ljungblad, C. Korner, U. Rude,”Simulating fast electron beam melting with at parallel thermal freesurface lattice Boltzmann method.”, Computers & Mathematics withApplications, 67(2), 318 - 330 (2014)

• Section 6.3 and Section 8.2 show the results of,

∗ M. Markl, R. Ammer, U. Ljungblad, U. Rude, C. Korner,”Electron beam Absorption for Electron Beam Melting Processes Sim-ulated by a Three-Dimensional Thermal Free Surface Lattice Boltz-mann Method in a Distributed and Parallel Environment.”, ProcediaComputer Science, 18, 2127 - 2136 (2013).

• Section 9.1 makes use of,

∗ R. Ammer, M. Markl, C. Korner, U. Rude,”Validation experiments for LBM simulations of electron beam melt-ing.” International Journal of Modern Physics C, 25(11), 1441009–1- 1441009–9 (2014).

• Section 9.2 is based on,

∗ M. Markl, R. Ammer, U. Rude, C. Korner,”Numerical Investigations on hatching process strategies for powder-bed-based additive manufacturing using an electron beam.” The In-ternational Journal of Advanced Manufacturing Technology, 78(1-4),239 - 247 (2015).

• and Section 10.2, Section 11.1, and Section 11.3 work with,

∗ S. Bogner, R. Ammer, U. Rude,”Boundary conditions for Free Interfaces with the Lattice BoltzmannMethod.”International Journal of Computational Physics, 297, 1 – 12 (2015).

4

Part I

Fundamentals using LBM forAdditive Manufacturing

Processes

5

Chapter 2: Additive Manufacturing Processes

2 Additive Manufacturing Processes

In the new era, everyonecan potentially be their ownmanufacturer as well as theirown internet site and powercompany. The process iscalled 3-D printing.

(Jeremy Rifkin (1945))

Futurologists and sociologists like Jeremy Rifkin interpret the development ofthe technology of additive manufacturing (AM) as part of the third industrialrevolution. The first revolution denotes the change of an agricultural to an in-dustrial society in the 18th century and the second revolution starts with theuse of electricity and mass production at the beginning of the 20th century.Flagships of the first two revolutions were the steam engine and the conveyorbelt. Following the ideas in Rifkin [2011] the third industrial revolution happensright now and additive manufacturing allows the production of durable goodsby everyone. The AM technology started in 1987 under the name of rapid proto-typing (RP). Gibson et al. [2015] defines RP as the fast production of a physicalprototype directly from digital data before its final release or commercialization.This prototype is a basic model for the final product in order to test ideas andinnovations during the development process. By the improvement of the qualityof RP technology the process could not only generate prototypes but also sim-ple final products and many parts could directly be manufactured and thus, theconcept of rapid manufacturing (RM) arises. Further improvement of the qual-ity of RM and the successful application of RM in many industries demands aformal standardization of the new technology. The Technical Committee withinASTM (American Society for Testing and Materials) International pointed outthat all processes are united by using an additive approach and hence intro-duced the term AM (cf. ASTM). All processes combine materials layer-by-layerand therefore the term additive is used. The timeline of the development fromRP up to AM is depicted in Figure 2.1. However, it has to be mentioned that allthese terms are also used simultaneously.

Beyond these three terms there exist also the following concepts: automatedfabrication is defined by Marshall Burns in Burns [1993] in the early 1990’s.Burns focuses on the concept of automation to manufacture products. Further-more, the idea of freeform fabrication or solid freeform fabrication is to fabricatecomplex, geometric shapes, with the advantage that the complex geometries are

7


RapidPrototyping

RapidManufacturing

AdditiveManufacturing

mid of 1980’s mid of 1990’s today

Figure 2.1: Timeline of development of additive manufacturing.

gained for free, i.e., a simple tube or a complex anatomical structure will takethe same production time. This is a huge difference and advantage comparedto most conventional manufacturing processes.

In the beginning of this chapter AM processes are defined, the eight steps ofthe process chain are described, and the extraordinary potentials for variousindustry branches of the new technology compared to traditional manufactur-ing methods are highlighted. In the second part of the chapter, the different AMprocesses are classified and compared to each other with a special focus on theelectron beam melting process.

2.1 What is Additive Manufacturing?

Gebhardt [2012] defines AM as ”layer-based automated fabrication process formaking scaled 3-dimensional physical objects directly from 3D-CAD data with-out using part-depending tools”. Similar definitions can be found in Faster-mann [2012] and Gibson et al. [2015]. The book of Gibson et al. [2015] isrecommended as an overview of AM processes. The basic principle of AM isthe production of complex 3D parts directly from digital CAD data without pro-cess planning before. This is in contrast to other traditional manufacturingmethods, where a careful and detailed planning of the geometry of the productis necessary. Furthermore, traditional manufacturing methods, like casting,require the specification of the order in which things of the part have to be fab-ricated, and which tools and processes are necessary. The key feature of all AMprocesses is adding material in layers without any tools. Each layer is definedas a thin cross-section of the part which is derived by the original CAD data.Thus, the final product is an approximation of the original data and, similarto the discretization error, the thinner the layer the better the final approxi-mation. All commercialized AM processes use this layer-based approach. Theprocesses differ in the used materials, in the way the layers are created andmaybe bonded to each other. These details determine the accuracy, materialproperties, mechanical properties, built time, and how much post-processing isrequired. The following explanations of the AM technology is mainly based onGibson et al. [2015]. Other sources are stated explicitly.

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2.1.1 Process Chain of AM Technology

Following the classification of Gibson et al. [2015] eight general steps of theAM process can be distinguished which are depicted in Figure 2.2. Every AMprocess starts with at a computer-aided-design (CAD) model that is achievedas an output of a software. The CAD model describes the external geometry ofthe product as accurate as possible. In the next step, the external surface ofthe original CAD model is converted into a STL file format, which is supportedby all AM machines. In step three, the STL file has to be loaded into the AMmachine and some additional manipulations might be necessary to ensure thecorrect size, position, and orientation of the part. Subsequently, the correctparameters for the AM machine are adjusted, i.e., material constraints, energysource, layer thickness and timing are set. In step five the actual part is built inthe AM machine and afterwards, the part has to be removed from the machine(step six). In the optional post-processing step, additional cleaning or otherprocedures, like manual manipulation, maybe be required. In the last step, theproduct is finalized up to an additional treatment of the surface texture. Byapplying this process chain the final physical part grows layer–by–layer frombottom to top.

1. CAD model

2. STL conversion

3. Transfer to machine

4. Machine setup

5. Part building

6. Part removal

7. Post-processing

8. Application Finalproduct

Start

Figure 2.2: Process chain of AM processes.

After explaining the mode of operation for AM processes, the extraordinarybenefits and advantages of the AM process compared to traditional manufac-turing methods are explained in Section 2.1.2. These strengths of AM processes

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underline the huge potential of AM processes in the future and justify AM aspart of the third industrial revolution.

2.1.2 Advantages and Strategies of AM Technology

AM offers the opportunity for everyone to manufacture products of arbitraryshape and geometry. These products can consist of almost every material andcan be manufactured in various numbers. Hence, the basis of customized massproduction is established.

One of the advantages of AM technology is its rapid character. AM reduces theprocess steps independent of the complexity of the product and AM machinesbuild the product in only one step. Furthermore, AM reduce the amount ofresources required. Traditional manufacturing methods require multiple anditerative stages for such complex parts and even a slight change in the designmay increase the time to completion. A second advantage is the possibilityto generate products of very complex geometry which cannot be fabricated bymost of the conventional manufacturing processes. This enables integratedfunctionality which is defined as geometry based on kinetic functions and thiscan be produced in one build. In traditional processes, the original complexgeometry has to be simplified for fabrication, i.e., the building of the part isdivided into smaller parts and steps, building of these separately and assemblethese sub-parts at the end. Using complex tools is another solution to produceparts of complex geometries with conventional methods, but this is expensiveand highly specialized craftsmen are needed. A third advantage is the diversitywhich is only possible by using AM technologies. It enables the use of variousmaterials during the manufacturing process and leads to completely differentpart properties of the final product. Most AM products consist of only onematerial, but AM machines can also handle several materials at the same time.Different colors and material properties in one product are also possible.

These advantages of AM processes compared to traditional manufacturingprocesses result in the following strategies for the new AM industry. AM enablescustomization, i.e., buyers demand individual, unique parts which cannot bemanufactured in mass production in an economical manner. Mass productionrequires tools which build identical parts of a product and even the productionof these tools is costly and takes a long time. On the contrary, AM produc-tion is also called tool–less fabrication, since all individual requests of the finalcustomer can be directly included into the building process without additionaleffort. Gebhardt [2012] denotes this paradigm shift in production technologyas ”AM changes production from mass production of identical parts to massproduction of individual parts” (cf. p. 117). This customized mass productionenables personal fabrication which is an important aspect of AM technology.Another important strategy of AM technology is the already mentioned individ-ualization of the production. The design of the product can be immediatelychanged according to the taste or needs of the final customer. This individ-ualization allows a lot of different variations of one product without higher

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costs. This property of AM supports especially the competitiveness of small andmedium sized companies. Gebhardt [2012] indicates a future strategy whichis called distributed customer production or coproducing. Coproducing meansthat all original customers of AM machines have the opportunity to act as pro-ducers for other customers. The producers linked via the internet can establisha worldwide production, finally resulting in cloud manufacturing.

These strategies provide several benefits for numerous industries and com-panies. The main industries that use AM and the advantage gained by this newtechnology are shown in the next subsection.

2.1.3 Industries using AM Technology

Initially, AM products were used to create visualization models for products inorder to better understand the design and functionality. However, with the im-provement of quality AM can shorten product development times and costs, andthe ability of using high-power laser technology allows to directly manufactureparts made up of various metals.

Various branches of industry apply AM technology in order to strengthentheir business (cf. Figure 2.3). One of the first users of AM technology was theautomotive industry and their suppliers. Car manufacturers use CAD systemsfor the design and redesign and thus, the CAD data set of the required partsalready exists. The high demands of diversification and individualization of thecustomer have forced to produce the AM parts directly in order to avoid toolingto reduce costs and time. Interior and exterior components of a car are typicalexamples of parts that are built with AM technology. Examples for interior partscan be components of the multimedia or air condition control installations oralso a complete fuel tank which is an example for a quite large part producedwith AM. Exterior produced parts can be the front spoiler as well as metal partsfor power train or engine components of a car.

The aerospace industry is interested in AM technology due to its characterof tool-less fabrication. Tool-less fabrication is well-suited for small series ofairplane manufacturer. The products generated by AM also refer to the inte-rior equipment of the plane similar those in cars, e.g., lamps and conditioningsystems. Furthermore, metal and ceramic AM processes allow the direct fabri-cation of technical parts for the cell and engine as well as for elements of thecombustion chamber.

Foundry and casting industries use AM technology in order to manufactureprototypes. Furthermore, lost cores and cavities can be produced quickly andeasily compared to traditional manufacturing techniques. An improved com-plexity of the geometry can be achieved. In contrast this is not possible manu-ally without special tools with an increase in time and costs.

Medical engineering discovers AM also as an opportunity to meet the individ-ual needs of their customers. The human body is very individual and medicalaids have to be designed to fit to the human body in the best possible way.Medical implants, such as artificial hip joints, epitheses and orthoses require

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Industries usingAdditive Manufacturing

Technologies

AutomotiveIndustry

& Suppliers

AerospaceIndustry

CustomerGoods

ToyIndustry

Art

Foundry &Casting

Technology

MedicalEngineering

Architecture

Figure 2.3: Branches of industry interested in AM technologies.

a very high degree of accuracy of fit. Computer tomography, 3D ultrasonics ormagnetic resonance imaging provide the necessary 3D data for the AM produc-tion. The implants can substitute entire bones and a complex surface structureallows a successful acceptance of the implant by the body.

Besides these highly technical goods, customer goods also benefit by AMtechnology. Following Gebhardt [2012] customer goods can be categorized intoelectronic consumer goods and life style products. Examples for electronic con-sumer goods produced by AM technology are components of mobile phones. Lifestyle products change rapidly and should be adapted to the variable demandsof the customers. AM technology provides the possibility to design new prod-ucts like vases, lamps, or sunglasses with very individual features, in any size,with any material. In this way, the designers could overcome the geometricalrestrictions of traditional manufacturing methods.

Another industry which is interested in using AM technology is the toy indus-try. They benefit from the ability of customized models for toy cars, trains andplanes with a high degree of accurateness for details which require a sensitivescaling.

Besides these producing industries artists of several disciplines use the free-

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dom of unlimited geometries. Examples for these artworks are human sculp-tures based on 3D body scanning data of persons or even clothes which provideintegrated functionality like measurements of the body movement.

Moreover, even architects use AM technology to produce models of new build-ings. These so produced models can have various textures, colors, and shapes.Even 3D decorations and ornaments are possible. With AM the produced mod-els can be built faster and new models with small modifications are producedwith less effort.

The benefits and strategies of building products by AM have now been ex-plained. However, there are several different AM processes using diverse mate-rials with varying properties. Hence, Section 2.2 presents the main classes ofAM processes.

2.2 Classification of Additive Manufacturing Processes

There are several classifications of AM technology. One categorization is basedon the main technology of the AM process, i.e., whether the process uses lasers,printer technology, or extrusion technology as in Kruth et al. [1998], Burns[1993]. On the other hand one can also categorize the different methods basedon the type of raw material input like in Chua C. K. [1998]. In order to overcomesome shortcomings of these classifications Pham and Gault [1998] suggests atwo-dimensional classification approach. The first dimension is used to de-scribe how the layers are generated and the second dimension is related to theraw material which is used for the process. This classification distinguishesbetween four raw materials, like liquid polymers, discrete particles, molten ma-terial, and laminated sheets.

A newer classification method is given in Gibson et al. [2015] as depicted inFigure 2.4. This categorization differentiates seven methods, namely photopoly-merization, powder bed fusion methods, material extrusion, material jetting,binder jetting, sheet lamination, and directed energy deposition.

The following subsections elucidate how these different processes work andwhat disadvantages and advantages exist.

2.2.1 Photopolymerization Processes

Photopolymerization processes correspond to AM technology and use liquidphotopolymers as raw input material. The liquid photopolymer is containedin a vat and specific regions of the cross section are illuminated by an energysource, i.e., the photopolymers react by irradiation and the part gets solid. Thisprocess is called photopolymerization. These photopolymers were developed inthe late 1960’s for the coating and printing industry as well as for dentistry toseal the top surfaces of teeth. In the mid 1980’s Charles Hull has conductedexperiments with photopolymers by using a scanning laser as an illuminantand discovered the stereolithography (SL). SL was the first photopolymerization

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Photopolymerization

Powder Bed Fusion

Material Extrusion

Material Jetting

Binder Jetting

Sheet Lamination

Directed Energy DepositionAM processes

Figure 2.4: Classification of AM processes.

technique and one of the first AM technologies. In particular, SL was used forthe product development industry. SL produces solid polymer patterns layer-by-layer and 3D parts can be generated. The advantages of SL or photopoly-merizaton processes compared to other AM processes are the high accuracy andthe good surface finish of the final product and therefore, the SL can be usedfor various applications. Furthermore, the process itself is very flexible andfast. On the other side, there are not many photopolymers which can be ap-plied commercially. Only arcylates and epoxies are profitable and their limitedmaterial properties cannot be used for all applications.

2.2.2 Extrusion-based Methods

Extrusion-based systems are the most popular on the AM market. The processitself needs a reservoir filled with liquified material. This material is pressedthrough a special shaped nozzle by a tractor-wheel arrangement. Then theformed material has to solidify and it has to be ensured that the material mustbe bonded to that material which has been already extruded. There are twodifferent approaches for the liquidation. The most frequently used approach isbased on temperature which controls the material state. The second approachis based on the chemical change that causes solidification, e.g., a curing agent,residual solvent, reaction with air, or simply drying of wet material. After liq-uidation, pressure ensures that the liquid material is transported to the nozzlewhere the actual extrusion happens. After leaving the nozzle, the shaped mate-rial is plotted in a controlled manner due to a predefined path. At this point, itis important to ensure the bonding of the actually extruded material with the al-ready finished material. The inclusion of support structures has to be providedfor offering complex geometrical features. The most common extrusion-basedAM technology is fused deposition modeling (FDM) which was patented by ScottCrump in 1992. FDM uses heating for liquidation of polymers which are stored

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in a reservoir. This bulk is pushed by a tractor-wheel arrangement which leadto the required extrusion pressure.

The wide range of possibly used materials, the effective mechanical propertiesof the final product and the fact, that parts produced by FDM are among thestrongest of polymer-based AM technologies, are the benefits of FDM and thus,many industrial users are interested in FDM. In contrast, the slow build speed,the bad accuracy (sharp external edges are impossible by the process itself) arelimitations of FDM. contour crafting (CC) tries to overcome this shortcoming ofslow build time. The exterior surface is the most critical part in considerationof accuracy for extrusion-based methods. Thus, CC uses a scraping tool tosmooth the surface after production in order to interpolate between the layers.This feature allows to use very thick layers compared to the FDM.

An important application for extrusion–based methods is bioextrusion. Bioex-trusion denotes the process of creating biocompatible components. These bio-compatible parts are used to generate frameworks which are called scaffolds.The scaffolds are used for tissue engineering. Modified FDM machines are ableto create these scaffolds.

2.2.3 Material Jetting Processes

The 2D printing technology is very successful since the 1960s and the firstpatents for using printing in order to generate 3D parts were registered in the1980s for ballistic particle manufacturing. The material jetting (MJ) processis one of the printing technologies in AM processes. The material is printedby a print head to the previous layer or the bottom of the starting plate. Theraw materials which are used for the material jetting are waxy polymers, ce-ramics, and also metals. However, almost all commercial MJ processes usepolymers due to their wide range of mechanical properties and applications.The technical challenge of MJ processes is the liquidation of the raw material.The original solid material has to be liquified with the help of heat, using asolvent or a chemical reaction. In the next step, the droplet formation itselfis challenging since the continuous liquid material has to be converted into afinite number of small discrete droplets. The deposition of these droplets hasto be controlled. The difficulty of the deposition is that the printing head ismovable and its movements have to be taken into account in the calculationof the droplet trajectory. After the deposition of the droplet, the phase changeof the liquid droplets into a solid product has to be ensured and nonuniformsolidification has to be avoided. Finally, the correct deposition of the droplet ontop of the previous layer influences the stability of the final product.

Advantages of MJ are the low costs for the machines compared to AM pro-cesses which involve lasers or electron beams, the rapid build and the scalabil-ity. In this context, scalability means that several printing heads can increasethe amount of products or the build time. Another benefit of MJ is the possibil-ity to use multiple materials and different colors. Up to the mid of the 1990s itwas only possible to print 3D parts with one color. On the other side, MJ tech-

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nologies are limited in the choice of materials and the part accuracy is worsecompared to photopolymerization and extrusion-based systems.

2.2.4 Binder Jetting Processes

Binder jetting (BJ) processes were founded in the early 1990s by printing abinder onto a powder bed in order to form the cross section of a part from theCAD model. In the first instance, this process was called 3D printing (3DP).In contrast to MJ only a small portion of the raw material is delivered throughthe printing head whereas the larger part of the raw material is kept in thepowder bed. After the printing process small binder droplets form spherical ag-glomerates of binder liquid and powder particles. After finishing one layer, thepowder bed is lowered and a new layer of powder is spread onto it. Materialsused for the powder range from starch, poly-methyl methacrylates, sand pow-der, metals like stainless steel and Inconel 625 and ceramics. The binder canbe water-based or inorganic.

BJ and MJ have a lot of the advantages in common. However, BJ has afaster build speed since only a small portion of the material has to be dispensedthrough the printing heads. On the other hand, the distribution of the powderis an extra step in the process and takes time. Another advantage gain ofBJ is the possibility of combining powder particles and additives in binders.This enables combinations which are not possible in MJ. Moreover productswith higher solid loadings are possible compared to MJ, resulting in a betterquality of ceramic and metal parts. In contrast to that, poorer accuracies andsurface finishes require infiltration steps to fabricate dense parts which is adisadvantage compared to MJ.

2.2.5 Sheet Lamination Process

Laminated object manufacturing (LOM) was the first commercialized AM tech-nique in 1991. LOM works by laminating paper material sheets layer-by-layerand a cutting tool forms the cross section of each sheet due to the CAD model.The process requires bonding techniques between the single sheets, i.e., forthe lamination. Lamination can be enabled by gluing or adhesive bonding,thermal bonding, clamping, or, ultrasonic welding. Furthermore, the order ofcutting and bonding involves classification: there is the bond-then-form tech-nique where first the sheets are bonded and then cut in the corresponding crosssection or the form-then-bond approach where the order is vice versa. The firstLOM machines use the bond–then–form approach, consisting of three steps.First, the laminate sheet is placed, followed by the bonding to the substrate andfinally, the sheet is cut according to the predefined slice contour. The unusedmaterial remains after the cutting in the machine as support material whichimpedes the removal of the product. The advantages of the bond–then–formprocess are the small shrinkage of the product during the process, the smallresidual stresses and distortion problems. Furthermore, the fast production of

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large parts is possible using various materials like metal, ceramics and com-posites. The low material, machine and process costs are also advantageous.On the other side, the controlling of the accuracy in z-direction is difficult, themechanical and thermal properties are inhomogeneous according to the bond-ing material between the sheets, and preservation of small details is difficultdue to the manual removal. The form–then–bond technique works by placingthe shaped laminate on top of previously deposited layers and their bondingto them. This approach lowers the danger of cutting into previous layers andis faster since the removing step is eliminated. Moreover, the construction ofparts with internal features and channels is more feasible in contrast to theform–then–bond approach. Raw materials for both approaches are plastics,metals, ceramics, and papers.

In general, sheet lamination processes produce products with high accuracyand the production costs are low. Unfortunately, the removing of the final partcan be difficult and a strategy for recycling the surplus material is needed.

2.2.6 Directed Energy Deposition Processes

Directed energy deposition (DED) creates 3D parts by melting materials as theyare deposited. In contrast to powder bed fusion processes (cf. Section 2.2.7),DED are not used to melt a material that is contained in a powder bed but israther used to melt materials which are deposited instantaneously. DED usesa focused heat source like an electron beam or laser in order to the melt thematerial and builds with this melted material a 3D part similar to the extrusion-based method (cf. Section 2.2.2). Thus, each line of the DED head creates atrack of solidified material and layers are build by adjacent lines of solidifiedmaterial. The used raw material is either in the form of powder or wire. Adisadvantage of powder is that not all powder is captured in the melt pool andexcess powder has to be utilized. Thus, the layer thickness converges to asteady state. Beyond this, powder is a very multifunctional feedstock in theDED process. When using wire as raw material hundred percent of capture ef-ficiency is achieved. Wire feeding is very effective for simple geometries withoutmany transitions. However, the control of geometry in relation to the processparameters is very difficult and it is impossible to achieve both, high accuracyand low porosity using a wire. As already mentioned, the focused heat sourcecan be a laser as well as an electron beam. When using a laser the deposi-tion head consists of an integrated collection of laser optics, powder nozzles,inert gas tubing, and sensors. The laser generates a small melt pool on thesubstrate. Known laser based DED methods are laser engineering net shaping(LENS), direct metal deposition (DMD) and laser freeform fabrication (LFF). Onthe other side, the use of electron beams is also advantageous, especially for theaerospace industry. DED processes with electron beams are capable of rapiddeposition under high current flows and electron beams are much more effi-cient in converting electrical energy into a beam than most lasers. The advan-tage of using electron beams for aerospace industry is its property of working

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effectively in a vacuum and not in the presence of inert gas.Summarizing, DED produce fully dense parts with microstructural features.

Benefits of DED parts are the good accuracy when using small beam sizes anddeposition rates, their capability for control of microstructures by having thepossibility of changing material composition and solidification rates by simplychanging the process parameters. Furthermore, DED methods can be usedfor repairing, overhaul and modernization of metallic structures or to improvethe present structure by adding features. Disadvantages of DED parts are thepoor resolution and surface finish and the slower build time. Compared topowder fusion processes (cf. Section 2.2.7), DED is not capable of producingsuch complex structures with small features and is less accurate.

2.2.7 Powder Bed Fusion Processes

All powder bed fusion (PBF) processes require three characteristics. They haveone or more thermal sources for inducing the fusion between powder particles,they have a feasible solution to control the powder fusion to a predefined crosssection, and they have a mechanism for adding and smoothing powder particlesin one layer. The first PBF processes were based on the laser sintering machinesto produce plastic prototypes. Besides polymers, ceramics and metals can alsobe raw materials for PBF processes and can produce prototypes or end-useproducts as well. All PBF processes use a powder delivery system to generate athin powder layer within an enclosed building chamber. This chamber can befilled with nitrogen gas in order to minimize oxidation of the powder particlesand degradation of powder. In order to guarantee the leveling of the powderlayer there exist the following approaches. A counter-rotating action of a rollerfixed at the powder delivery system can level new powder layers. This actionleaves previous layers undisturbed. Another solution is powder spreading bydoctor blade which results in bigger shear forces than by the counter-rotatingroller. A third approach is using the hopper feeding system that supplies powderfrom above and deposits powder in front of a roller or doctor blade.

Ceramics, metals and polymers are some of the raw materials used for thepowder particles. More generally speaking, all materials which can be meltedand solidified can be used. After the deposition and leveling of the powder,the powder bed is maintained at an elevated temperature below the meltingpoint. This preheating minimizes power requirements and avoids warping ofthe product. The heat source which induces the fusion into the powder layercan be a focused laser or a focused electron beam. Lasers and electron beamswork in a different way but this will be explained in the further course. Theheat source is focused directly onto the powder bed and moves according to thecross section given by the CAD model. The surrounding powder remains looseand acts as a support material. Four different mechanisms, based on variantsof sintering and melting exist for the powder fusion. Solid-state sintering im-plies fusion without melting the powder particles. The sintering itself is drivenby the minimization of the total free energy of powder particles. In order to ac-

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complish low porosity, long sintering times or high sintering temperatures arenecessary. A second fusion process is the chemically–induced sintering. Thisinvolves thermally activated chemical reactions between two types of powdersor between powder and atmospheric gases and is mostly utilized for ceram-ics. This process can lead to higher porosity in the product. Another fusionmechanism is the liquid-phase sintering or partial melting. The powder con-sists of powder particles and a portion of constituents. The constituents canbe melted whereas a part of the particles remains solid. The constituents actas an adhesive which binds the solid particles together. The advantage of thismechanism is that particles with a very high melting temperature can be boundtogether without melting or sintering those directly. The fourth fusion processis called full melting which is mostly used for metal alloys. This fusion meltsthe entire powder layer completely by heat energy up to a depth exceeding thelayer thickness. The full melting creates high-density structures and ensures alow porosity. After building one layer by the specific fusion process a cool-downperiod is necessary. Afterward, the building platform is lowered and a new pow-der layer is delivered and leveled due to the leveling mechanism. These stepsare repeated until the entire part is built. The most famous PBF processes areselective laser sintering (SLS), selective laser melting (SLM), direct metal lasersintering (DMLS), laser cusing and electron beam melting (EBM) depending onthe powder fusion process and the energy source.

An advantages of the PBF is a huge variety of possible raw materials, rangingfrom polymers, and ceramics up to metals. The accuracy and surface finishof PBF depends on the particle size. The larger the average particle size theeasier the powder delivery process. However, the resulting surface quality isworse and limits the minimum feature size and minimum layer size. For finerparticles, a better surface quality can be achieved and it is possible to generatefeature details of a very small size. However, PBF is a competitive AM processfor low–to–medium volume series of products with a high degree of geometricalcomplexity and extraordinary material properties of the end-products. AlthoughPBF is a very versatile AM process class it can be improved to ensure less costsand less production time. PBF requires an increased production time comparedto other AM processes due to the powder delivery, the preheating and the cool-down period. Especially for polymer–based PBF it can be improved from point–wise to line–wise and layer–wise compilation, respectively. This would increasethe build speed dramatically and reduce the production costs.

After the classification of the AM processes Section 2.3 will describe the EBMprocess in detail since the focus of this thesis lies on the numerical simulationof the EBM exemplary for all AM processes.

2.3 EBM - Electron Beam Melting

All PBF processes differ in their powder delivery method, the heating process,energy input type, and the atmospheric conditions. The way EBM handles these

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PBF characteristics is explained in the following. The EBM process, which wasdeveloped at the Chalmers University of Technology in Sweden and patented2001 by Arcam, executes almost the same steps as a general PBF method(cf. Section 2.2.7). Figure 2.5 shows the vacuum chamber where the processtakes place. The build platform is arranged in the build tank where the currentpowder layer is treated. On the build platform powder particles are spread bythe powder hoppers which also take the leveling of the new layer. The powerparticles are melted by the electron beam which is located above the build plat-form. This process is repeated for the following layers until the entire part isproduced. As the name already implies, EBM uses a focused high-energy elec-

Electron beam gun

PowderHopper

PowderHopper

Powder

Start Plate

VacuumChamber

Build tank

Buildplatform

IR – Camera

Figure 2.5: Schematic of EBM process (taken from Ammer et al. [2014a]).

tron beam in order to induce fusion between the metal powder particles. This isin contrast to SLM or SLS where a laser is the heat source. Other characteristicdifferences between SLM and EBM are given in Table 2.1. The laser heats thepowder particles by sending photons which are absorbed by the particles. Incontrast, the electron beam heats the powder by the transfer of kinetic energyfrom the incoming electrons into the powder particles. The particles absorb theelectrons and gain a negative charge. This increased negative charge can beproblematic: if the repulsive force of neighboring negatively charged particlesexceeds gravitational and frictional forces powder particles can explode fromthe powder bed and a powder cloud can occur. Furthermore, an increasingnegative charge tends to repel the incoming electrons from the beam and thefocused electron beam gets more diffuse. These drawbacks are not given atlasers where photons are sent to the powder. Generally, the electron beam is

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CHARACTERISTIC EBM SLMThermal source electron beam laserAtmosphere vacuum inert gasScanning deflection coils galvanometersEnergy absorption conductivity–limited absorptivity–limitedPowder pre-heating use electron beam use infrared heatersScan speeds very fast, magnetically-

drivenlimited by galvanometerinertia

Energy costs moderate highSurface finish moderate to poor excellent to moderateFeature resolution moderate excellentMaterials metals (conductors) polymers, metals and ce-

ramics

Table 2.1: Differences between EBM and SLM (Gibson et al. [2015]).

more diffuse than the laser beam. This can result in a larger melt pool sizeand a larger heat–affected zone. Thus, the minimum feature size, the averagepowder particle size, the layer thickness, resolution, and the surface finish ofEBM produced parts are larger than by parts produced by SLM. Another dif-ference is that EBM needs a conductive powder bed because only conductivematerials, e.g., metals, can absorb electrons, whereas lasers can deal with ar-bitrary materials that is capable to absorb energy at laser wavelength, i.e., itcan work with metals, ceramics and polymers as well. One decisive advantageof EBM is its higher efficiency compared to SLM. In EBM systems most of theelectrical energy is converted into the electron beam and higher beam ener-gies, above 1kW are available at moderate costs. In contrast, when using lasersonly 10-20 % of the total electrical energy input is converted into beam energyand the remaining energy is lost in the form of heat, thus the energy costs forlasers are higher. Furthermore, in the EBM process the powder bed remains ata higher temperature than for laser sintering or melting methods because thehigher energy input of the electron beam results in a heat up of surroundingpowder. This elevated temperature leads to a different microstructure for EBMproducts. Moreover, the scan lines in SLS are distinguishable but this doesnot hold for EBM. Another difference between the two PBF processes refers tothe scan speed. EBM has the advantage that it can be moved instantaneouslyand new improvements can increase the build rate by using different scanningstrategies.

2.4 Summary

In this chapter the new technology of AM processes is introduced which is in-terpreted as a part of the third industrial revolution by famous economists.

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The huge potential of AM processes lies in the wide range of numerous in-dustrial and research applications and the possibility of individualization andcustomization of final end–user products without increased costs, compared totraditional manufacturing methods.

Following Gibson et al. [2015] the existing AM methods can be classified intoPhotopolymerization, Powder Bed Fusion, Material Extrusion, Material Jetting,Binder Jetting, Sheet Lamination, and Directed Energy Deposition processes.The differences concerning raw material, bonding of layers, forming of the crosssection, and end-user products have been described thoroughly.

In the remainder of this thesis the requirements for simulating PBF processes,with a special focus on EBM, is derived. The requirements include the underly-ing mathematical model, given in the subsequent chapter, followed by the usednumerical discretization methods. EBM specific modeling aspects and the in-tegration of the implementation in a fluid flow framework are also described.With this framework simulations of the EBM process are executed which pre-dict improvements of the process in order to increase the performance of EBMwithin the PBF processes.

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Chapter 3: Mathematical Model for Thermal Fluid Flow

3 Mathematical Model for ThermalFluid Flow

It is the aim of science toestablish general rules whichdetermine the reciprocalconnection of objects andevents in time and space.For these rules, or laws ofnature, absolutely generalvalidity is required - notproven.1

(Albert Einstein (1879 – 1955))

In Chapter 2 additive manufacturing is explained and its physical conditionsare identified. Following the simulation pipeline of Figure 1.1 an appropriatemathematical model describing the AM processes, especially the EBM process,by partial differential equations will be explained in this chapter. Since EBMessentially deals with the flow of molten metal powder particles, fundamentalsof hydrodynamics like the Eulerian and Lagrangian description of fluid are re-viewed in this chapter. Both points of view are justified. Whereas Lagrangiancoordinates simplify the derivation of conservation of mass, momentum, andenergy, Eulerian coordinates are equally important and practicable for simu-lations where the user is rather interested in velocity or pressure at a specificposition and time. The link and transformation of both approaches is explainedvia the substantial derivation. Referring to the conservation laws the continuityequation, the Navier–Stokes equations and the energy equation are derived andpresented in differential and integral form. The dimensionless form of theseconservation laws results in dimensionless numbers which enable comparabil-ity of simulations being independent of any quantities. The derivation of themathematical model in this chapter is based on the book of Aris [2012]. Fur-thermore, the books of Durst [2007] and Spurk and Aksel [1996] act as classicalintroductory literature for fluid mechanics.

3.1 Eulerian and Lagrangian Description of Fluid Flow

Fluid kinematics concentrates on the study of fluid motion and describes thismotion without considering the forces which cause motion. The fluid is the

23


superordinate concept for liquids and gases because both substances show noresistance to change shape when the flow velocity tends to zero. Furthermore,liquids and gases have the same dynamical behavior as long as their volumedoes not change during the flow. All fluids consist of molecules and thus, thebehavior of both, fluids and solids can be explained by their molecular structureand the interaction of molecules influences the flow. Therefore, equations forthe fluid flow may be obtained by integrating over all molecules the fluid is madeup. However, this concept does not work for mainly these practical reasons.Since due to the large amount of molecules it is not feasible to calculate the mo-tion of all molecules even with the help of a large supercomputer. In addition,following the uncertainty principle of Heisenberg two independent variables likeposition and momentum cannot be known at the same time. Therefore, aver-aging over molecules is indispensable in order to get working formulations forcomputing fluid flow. According to the continuum hypothesis, the fluid is con-sidered as a continuum and its macroscopic quantities like density, velocity, orenergy are considered as continuous functions depending on space and time. Amaterial volume is a part of the fluid which is under observation and consistsalso of infinitesimal many material points. The motion of the continuous fluidflow is described by partial differential equations. The continuum theory cannotexpress real materials but idealizes their behavior sufficiently accurate. In thisideal material model the fluid velocity at any point averages the velocity of themolecules in the surrounding volume of this point. The continuum hypothesisof the fluid is manageable computationally and serves as a basis for classicalfluid motion. In kinetic theory a particle exists at position ξ at time t and thecontinuous function ξ is defined by,

ξ = ξ(x, t). (3.1)

The coordinates ξ of the particles are called material coordinates or also La-grangian coordinates. Using the Lagrangian frame the motion of one specificmaterial particle in the domain is considered over time. A different point ofview is the Eulerian coordinate system where a specific position x of the fluiddomain is observed when time passes. The properties of the particles which arelocated at this specific position are considered,

x = x(ξ, t). (3.2)

Both points of view are depicted in 3.1. Each of these points of view has differentadvantages. Using the Lagrangian frame the derivation of fundamental equa-tions of fluid flow is more intuitive but in the field of applications researchersare more interested in knowing fluid properties like velocity, momentum, ortemperature at a particular position of the domain. Transformation from La-grangian into Eulerian or vice versa is consequently essential. A necessary andsufficient condition for an inverse transformation is a non-vanishing Jacobian

24


x (x, t)

fluid element

Eulerian frame

fluid particle

ξ(ξ, t)

Lagrangian frame

Figure 3.1: Eulerian and Lagrangian description of fluid flow.

determinant,

det J = det(∂(x1, x2, x3)

∂(ξ1, ξ2, ξ3)

)6= 0. (3.3)

We define F(x, t) as the transformation function from Lagrangian into Euleriandescription and vice versa F(ξ, t),

F(x, t) = F [ξ(x, t), t] , F(ξ, t) = F [x(ξ, t), t] . (3.4)

With these two different descriptions we have to differ also the following twoderivatives with respect to time t,

∂

∂t=

(∂

∂t

)x

”keeps Eulerian position x constant”, and (3.5)

d

dt=

(∂

∂t

)ξ

”keeps Lagrangian particle ξ constant”. (3.6)

The time derivative in Eq. (3.6), depending on the material coordinates, is calledmaterial derivative (and also known as substantial or total derivative). Thevelocity of a particle is its change of position in material coordinates over time,i.e.,

vi =d

dtxi =

∂

∂txi(ξ1, ξ2, ξ3, t). (3.7)

With Eq. (3.7) both derivatives are connected by,

dFdt

=∂

∂tF(ξ, t) =

∂

∂tF [x(ξ, t)] =

∂F∂xi

(∂xi∂t

)ξ

+

(∂F∂t

)x

= vi∂F∂xi

+∂F∂t. (3.8)

The following abbreviation DDt of Eq. (3.8) is more popular,

D

DtF =

∂

∂tF + (v · ∇)F , (3.9)

25


and is called total derivative or substantial derivative and plays an importantrole in the balance laws in the following section.

3.2 Conservation Laws

In this section the conservation laws for mass, momentum, and energy are de-rived in integral and differential form. The deduction follows the more generalprinciple of Reynold’s transport theorem. Reynold’s transport theorem can beinterpreted as a general conservation law applicable to arbitrary scalars or ten-sors. First, the change of the initial volume dV0 to an arbitrary volume dV isconsidered from Lagrangian to Eulerian coordinates,

dV =δ(x1, x2, x3)

δ (ξ1, ξ2, ξ3)dξ1dξ2dξ3 = J dV0, (3.10)

where ξ is the material vector at time t = 0 and dξ1dξ2dξ3 denotes the volumedV0 of an elementary parallelepiped. This elementary parallelepiped moves withthe fluid and is located at the point x = x(ξ, t). The new volume dV is given by,dV = JdV0, and J is the ratio of elementary material volume to its initial volumeand is already defined in Eq. (3.3). This change in volume by moving with theflow is depicted in Figure 3.2 and is called dilatation.

x1

x2

x3

ξ dV0

dVx = x(ξ, t)

Figure 3.2: Dilatation of initial material volume to elementary material volume.

Due to Schwarz’s law,

d

dt

(∂xi∂ξj

)=

∂

∂ξj

dxidt

=∂vi∂ξj

, (3.11)

and thus, the material derivative of J is given by,

dJ

dt=

(∂v1

∂x1+∂v2

∂x2+∂v3

∂x3

)J = (∇ · v) J, (3.12)

26


which is known as the Euler expansion formula. This is used for the derivationof Reynolds’ transport theorem which let us consider the rate of change of anarbitrary volume integral instead of only an infinitesimal small one. Thus, letF(x, t) be any function and V (t) a volume moving with the fluid, the averageover all fluid particles in this volume F (t) as a function of time t is given by,

F (t) =

∫∫∫V (t)

F(x, t)dV. (3.13)

When computing the material derivative dF (t)/dt the differentiation and in-tegration cannot be swapped. If the integration were with respect to a volumein the material coordinate space it would be possible to interchange differenti-ation and integration since d/dt is differentiation with respect to time keepingξ constant. The transformation x = x(ξ, t) with dV = JdV0 allows this with V0

being a fixed volume at time t = 0,

dF

dt=

d

dt

∫∫∫V (t)

F(x, t)dV =d

dt

∫∫∫V0

F [x(ξ, t), t] JdV0

=

∫∫∫V0

(dFdtJ + F dJ

dt

)dV0

=

∫∫∫V0

(dFdt

+ F (∇ · v)

)JdV0

=

∫∫∫V (t)

(dFdt

+ F (∇ · v)

)dV. (3.14)

Using material derivatives and the product rule, Eq. (3.14) can be rewritten to,

d

dt

∫∫∫V (t)

F(x, t)dV =

∫∫∫V (t)

(∂F∂t

+∇ · (Fv)

). (3.15)

The following reformulation of Eq. (3.14) using the Green’s theorem is also help-ful for many applications,

d

dt

∫∫∫V (t)

F(x, t)dV =

∫∫∫V (t)

∂F∂tdV +

∫∫S(t)

Fv · ndS, (3.16)

where S(t) is the surrounding surface of the moving volume V (t). According toEq. (3.16) the total rate of change of F consists of the rate of change at a pointand the net flow of F over the surface S.

Eq. (3.16) provides the main principle in order to derive the conservation of

27


mass, momentum, and energy which follow in the next subsections where Fcan be interpreted as a dummy variable which can be a scalar or a tensor ofhigher dimensionality.

3.2.1 Conservation of Mass and Continuity Equation

The conservation of mass, here of a fluid, assumes that there are no sourcesor sinks in the corresponding medium and that the mass does not change overtime. ρ(x, t) denotes the density of the fluid at position x and time t per unitmaterial volume V . Then, the mass m of any finite volume V is computed byintegrating the density over the specific control volume,

m =

∫∫∫V

ρ(x, t)dV. (3.17)

By using Eq. (3.14) and setting F = ρ as a dummy variable, the mass balanceis set to zero by,

dm

dt=

∫∫∫V

(dρ

dt+ ρ(∇ · v)

)dV

!= 0. (3.18)

Furthermore, assuming that Eq. (3.18) does not only hold for one specific vol-ume V but also for any arbitrary volume, thus, the integrand itself must beequal to zero and the conservation of mass in differential form is given by,

dρ

dt+ ρ(∇ · v) =

∂ρ

∂t+∇ · (ρv) = 0. (3.19)

Eq. (3.19) is also called equation of continuity and holds for compressible andincompressible fluids as well.

In the case of incompressible fluids, i.e., with constant density ρ, the equationof continuity simplifies to,

∇ · v = 0, (3.20)

which is also called divergence free constraint in computational fluid dynamics.Due to the Euler expansion formula (Eq. (3.12)), the flow is volume preservingfor incompressible fluids.

3.2.2 Conservation of Linear Momentum and Navier–StokesEquations

The conservation of linear momentum is based on Newton’s second law of mo-tion,

F = m · a. (3.21)

Hence, the balance of momentum means that the rate of change of momentumof a body is equal to the forces applied on this body. Forces acting on the body

28


are classified into two categories, external or body forces, and surface or contactforces. Body forces act on the whole material and have their source in forcefields. Gravity and electromagnetic forces are examples for body forces becausethey act on every molecule in the fluid particle. Surface forces or also calledinternal forces are exerted from the surrounding fluid or from other bodies onthe surface of the fluid particle. Stress is an example for a surface force. Theapplied force acting on the fluid particle is the sum of these two kinds. Thus,the change of linear momentum of the volume is equal to the sum of these twoforces,

D

Dt

∫∫∫V

ρv =

∫∫∫V

ρF dV

︸︷︷︸total external force

+

∫∫S

t(n)dS︸︷︷︸total internal force

, (3.22)

where F denotes a body force like gravity and t(n) is called stress vector. Stressitself is defined as the force per unit area on the surface S of the volume V andhave two directions; one direction of the force and one normal to the area. Dueto the principle of Cauchy from 1823 (see Gurtin and Martins [1976]) stress canbe represented by a tensor T of second order. Let n be the unit normal vector,then,

t(n) = n · T . (3.23)

With Eq. (3.23), the balance of momentum in each space dimension on i can bewritten to,

D

Dt

∫∫∫V

ρvi =

∫∫∫V

ρDviDt

dV =

∫∫∫V

ρFidV +

∫∫S

TjinjdS. (3.24)

Using Green’s theorem which relates the integral over a bounding surface to avolume integral Eq. (3.24) is rewritten to,∫∫∫

V

ρDviDt

=

∫∫∫V

(ρFi +∇ · Tji) dV. (3.25)

Because V is an arbitrary volume, Eq. (3.25) can be formulated in Cauchy’sequation of motion,

ρa = ρDv

Dt= ρF +∇ · T , (3.26)

where a denotes the acceleration of the fluid. Eq. (3.26) is true for Newtonianand non-Newtonian fluids as well. At this point it is necessary to examine thestress tensor T in detail. First of all, the velocity gradient tensor ∂vi

∂xjis consid-

ered. Its components can be split into a symmetric eij and an antisymmetricpart Ωij to,

∂vi∂xj

=1

2

(∂vi∂xj

+∂vj∂xi

)︸︷︷︸

eij

+1

2

(∂vi∂xj− ∂vj∂xi

)︸︷︷︸

Ωij

. (3.27)

29


eij is called strain rate or deformation tensor and Ωij represents a rigid bodyrotation. A vanishing eij means that the motion is non-deforming. Followingthe derivation of the stress tensor given in Aris [2012], the components of thestress tensor T are defined by,

Tij = −pδij + Pij , (3.28)

where Pij is called viscous stress tensor and p denotes the pressure. If there is avanishing viscous stress tensor, Eq. (3.28) reduces to the hydrostatic equation.In the case of incompressible fluids, the components of the viscous stress tensorPij are defined by,

Pij = 2µeij , (3.29)

where µ defines the proportionality constant relating shear stresses to velocitygradient ∂vi

∂xjand is called dynamic viscosity of the fluid. Inserting Eq. (3.29) into

Eq. (3.28) defines a relation between the stress tensor and the strain rate tensorand thus, Eq. (3.28) is called constitutive equation. The diagonal entries of thestress tensor Tii, i = 1, ..., 3, are called direct stresses and the non-diagonalentries are Tij, i 6= j, i, j = 1, . . . , 3, are called shearing stresses. Nonpolar fluidsconsidered in this thesis have a symmetric and diagonalizable stress tensor,i.e., Tij = Tji (the proof is given in Spurk and Aksel [1996]). If the definition ofthe stress tensor and the constitutive equation applied for a Newtonian fluid isinserted in Eq. (3.26) the well-known Navier-Stokes equations are derived. Inthe case of Newtonian fluids, stress tensor and strain rate tensor are relatedlinearly. The spatial derivative of the strain rate tensor is given by,

∂eij∂xj

=1

2

∂

∂xj

(∂vi∂xj

+∂vj∂xi

)=

1

2

∂2vi∂xj∂xj

+1

2

∂

∂xj

∂vj∂xi

=1

2∇2vi +

1

2

∂

∂xi(∇ · v). (3.30)

Using Eq. (3.30) the derivative of the stress tensor for incompressible fluidsis then,

∂

∂xjTij = − ∂p

∂xi+ µ∇2vi. (3.31)

Insert Eq. (3.31) into Eq. (3.26) and divide through ρ, the conservation of linearmomentum for incompressible fluids is given by,

a =Dv

Dt=∂v

∂t+ v · ∇v = F − 1

ρ∇p+ ν∇2v, (3.32)

where ν is called the kinematic viscosity of the fluid and defined by,

ν =µ

ρ. (3.33)

30


For incompressible fluids, the kinematic viscosity is a constant. Eq. (3.32) isalso called incompressible Navier-Stokes equations.

3.2.3 Conservation of Energy and Energy Equation

The physical foundation of the conservation of energy is the first law of thermo-dynamics. This law states that the total energy of a closed system is constant,i.e., energy can be transported from one material point to another but it can-not be created or destroyed. The rate of change of energy is computed by theamount of heat transferred to the volume minus the amount of work done bythe volume.

In order to obtain the total energy of a fluid particle the kinetic energy has tobe added to the internal energy E. Note, that only kinetic and internal energiesare considered here. This means, that the total energy is defined by,∫∫∫

V

ρ(E +

vivi2

). (3.34)

The rate of change of total energy is then obtained by,

D

Dt

∫∫∫V

ρ(E +

vivi2

)=

∫∫V

ρF · vdV +

∫∫S

t(n) · vdS −∫∫S

q · ndS, (3.35)

where the first two integrals on the right hand side express the power of appliedforces, i.e., surface and body forces. Furthermore, q denotes the heat fluxvector and −q ·n the heat flux into the volume with n being the outward normalof the surface. The minus sign of the heat flux means that the inflowing energyis considered to be positive. Take into consideration that the heat transferis restricted to conduction and other ways of heat transfers like radiation areneglected.

The thermal energy equation is obtained by subtracting the mechanical en-ergy equation from total energy equation. The mechanical energy equation isobtained by taking the dot product of the momentum equation Eq. (3.32) and vto, ∫∫∫

V

ρ1

2

D

Dt(vivi) dV =

D

Dt

∫∫∫V

1

2ρvividV

=

∫∫∫V

ρfivi −∫∫∫V

Tij∂vi∂xj

dV +

∫∫S

viTijnjdS, (3.36)

where the first integral on the right hand side denotes the rate at which the bodyforces do work, the second integral the rate at which internal stresses perform,and the last one the rate at which surface stresses have an effect. The overallrate at which kinetic energy changes regarding internal stresses consists of a

31


reversible interchange with strain energy and a dissipation by viscous forces.The thermal energy equation, which only involves the internal energy E, isobtained by subtracting the mechanical energy equation Eq. (3.36) from totalenergy equation Eq. (3.35) and applying Green’s theorem,∫∫∫

V

(ρDE

Dt+∇ · q − Tij

∂vi∂xj

)dV = 0, (3.37)

Assuming that Eq. (3.37) does not only hold for a specific volume V but also forany arbitrary volume, the integrand itself has to be zero,

ρDE

Dt= −∇ · q + Tij

∂vi∂xi

. (3.38)

The heat flux is described by Fourier’s law for heat conduction,

q = −κ∇T, (3.39)

where κ is the thermal conductivity and ∇T is the temperature gradient. For aStokesian fluid it can be assumed that,

Tij∂vi∂xj

= −p (∇ · v) + Φ, (3.40)

where Φ denotes the viscous dissipation and is defined by,

Φ = −4µΥ. (3.41)

Υ is the second invariant of the deformation tensor and is given by,

Υ = e22e33 − e23e32 + e33e11 − e31e13 + e11e22 − e12e21. (3.42)

Substituting Fourier’s law (Eq. (3.39)) and Eq. (3.40) into Eq. (3.38) in the caseof incompressible fluids, i.e., ∇ · v = 0, leads to the thermal energy equation indifferential form,

ρDE

Dt= −∇ · (κ∇T ) + Φ. (3.43)

Alternative versions of the thermal energy equation (Eq. (3.43)) can be derivedby using the definition of specific entropy or enthalpy. Furthermore, the energyequation for incompressible flows can be expressed directly depending on thetemperature T by,

ρcvDT

Dt= κ∇2T + Φ, (3.44)

where cv denotes the specific heat capacity of the fluid at constant volume. Forincompressible fluids the assumption cv ≈ cp is valid with cp being the specificheat capacity of the fluid at constant pressure.

32


3.2.4 Equation of State

So far, the system contains six unknowns, namely density ρ, pressure p, threevelocity components vi, i = 1, 2, 3, and the energy E (or temperature T ). Thesystem is composed of five equations, the continuity equation, the three Navier-Stokes equation, and the energy equation. The gap of one equation in order toclose the system is filled by the equation of state.

This equation of state relates density, temperature, and pressure and cou-ples the three conservation laws. For compressible flows, the commonly usedequation of state is the ideal gas law, i.e.,

p = ρRT, (3.45)

where R denotes the specific gas constant and T the absolute temperature.Eq. (3.45) was first stated by Clapeyron in 1834 (cf. Wisniak [2000]). This equa-tion is an idealization of real gases and is most accurate for monatomic gases athigh temperatures and low pressures. The coupling of the three conservationlaws implies that all six equations have to be solved simultaneously. Togetherwith this equation of state there exists a closed system of equations, i.e., thesame amount of equations and unknowns. This closed system is only a neces-sary condition that the system can be solved but not sufficient.

In the case of incompressible flows the concept of equation of state does notreally exist and the energy equation is decoupled from the equation of continuityand the Navier–Stokes equations. This means that the continuity equation andNavier–Stokes equations are solved in order to find the velocity and pressuredistribution. Afterward the energy equation is solved to find the temperaturedistribution. Since the density is constant for incompressible fluids, there areonly five unknowns and the equation of state is not needed to close the sys-tem. In the following the dimensionless conservation laws are introduced usingdimensionless quantities like for example the Reynolds number.

3.3 Dimensionless Balance Laws and DimensionlessNumbers

Before passing onto the numerical discretization methods dimensionless num-bers are introduced which are used to describe the properties of fluid flows andmake them comparable.

By writing the governing equations in a dimensionless form characteristicvariables, like characteristic length L, characteristic velocity V0, and character-istic temperatures T1 and T0, are required. The characteristic variables dependon the problem, e.g., when the flow in a channel is considered the characteris-tic length is the length of the channel. With these characteristic variables thefollowing dimensionless variables and parameters marked with ? are defined

33


by,

xi = x?iL,

t = t?L

V0,

vi = V0v?i ,

∂

∂xi=

1

L

∂

∂x?i,

∂

∂t=V0

L

∂

∂t?,

p = p?ρV 20 ,

T = T ?(T1 − T0) + T0, (3.46)

where T0 and T1 denote reference temperatures. The continuity equation (Eq. (3.20))is written in a dimensionless form to,

V0

L

∂v?i∂x?i

= 0⇒ ∂v?i∂x?i

= 0. (3.47)

For the sake of simplicity, the incompressible momentum equation (Eq. (3.32))is nondimensionalized by neglecting the force term F . After substituting thedimensionless variables the Navier-Stokes equation is given by,

V 20

L

∂v?i∂t?

+V 2

0

Lv?j∂v?j∂x?j

=V 2

0

L

∂p?

∂x?i+ ν

V 20

L

(∂

∂x?i

(∂

∂x?iv?i

)). (3.48)

Dividing by V 20 /L leads to,

∂v?i∂t?

+ v?j∂v?i∂x?j

=∂p?

∂x?i+

ν

V0L︸︷︷︸1Re

∇2v?i , (3.49)

where the coefficient ν/V0L defines the reciprocal value of the Reynolds numberRe. The Reynolds number is also interpreted as the ratio of inertial forces toviscous forces, given by,

Re =ρV0L

µ=V0L

ν=

inertial forcesviscous forces

. (3.50)

The advantage of the dimensionless Re is that two different flows with the sameReynolds number can be compared without considering any quantities. TheRe can also be seen as a measure how convective and diffusive terms of theNavier–Stokes equations are balanced. Another dimensionless quantity is in-troduced by the dimensionless formulation of the energy equation (Eq. (3.44)).The following simplified version of the energy equation neglects the dissipation

34


function,

ρcv

(∂T

∂t+ vi

∂T

∂xi

)= κ∇2T. (3.51)

Using the dimensionless variables given in Eq. (3.46) and parameters and thefact that the reference temperatures T0 and T1 are constants, Eq. (3.51) isrewritten as,

V0 (T1 − T0)

L

∂T ?

∂t?+ v?i

V0 (T1 − T0)

L

∂T ?

∂x?i=κ (T1 − T0)

ρcvL2

(∂2T ?

∂2x?i

). (3.52)

After dividing by V0(T1 − T0)/L it follows,

∂T ?

∂t?+ v?i

∂T ?

∂x?i=

κ

ρcvV0L︸︷︷︸1Re

1Pr

= 1Pe

∂2T ?

∂2x?i, (3.53)

where the Prandtl number Pr and Peclet number Pe are introduced besides theReynolds number. These dimensionless quantities are defined in the following.

Definition 3.3.1 (Prandtl number Pr):The Prandtl number characterizes how fast heat can diffuse compared to thevelocity,

Pr =cpµ

κ=

viscous diffusion ratethermal diffusion rate

, (3.54)

where cp denotes the specific heat of the fluid, µ the dynamic viscosity and κ thethermal conductivity.

Definition 3.3.2 (Peclet number Pe):The Peclet Pe number measures the ratio of advection and diffusion rate and isdefined by,

Pe =LV0ρcpκ

, (3.55)

or, as the product of Reynolds and Prandtl number.

Another important dimensionless number for this work will be the Mach num-ber Ma.

Definition 3.3.3 (Mach number Ma):The Mach number Ma is defined as the ratio of the velocity of the fluid and thespeed of sound cs,

Ma =v

cs. (3.56)

The speed of sound cs is also defined in the following.

Definition 3.3.4 (Speed of sound cs):The speed of sound cs of a fluid defines the rate of propagation of small distur-

35


bance pressure pulses (”sound waves”) through a fluid.

A ”small” Mach number of a fluid flow classifies the flow as incompressibleflow because compressibility effects are then also ”small” and can be neglected.

There exist numerous additional dimensionless quantities in order to describeand compare fluid flows and a different choice of characteristic variables wouldlead to other dimensional quantities. However, they are not considered in thefurther course of this thesis.

3.4 Summary

In this chapter, the mathematical model has been derived which is required tosimulate the EB process. The basis of the model are the continuity equation(conservation of mass, Eq. (3.20)), the Navier–Stokes equations (conservation oflinear momentum, Eq. (3.32)) and the energy equation (conservation of energy,Eq. (3.43)) for incompressible fluids, i.e., the velocity field is solenoidal and thevolume of the fluid is preserved. In addition a Newtonian fluid is considered, i.e.,there is a linear relationship between viscous stresses and strain rates. Theseconservation laws build a closed systems for a incompressible fluid with thesame amount of equations and unknowns which is a necessary preconditionthat a solution may exist. It should be noted that this mathematical modelonly idealizes the reality, but in a sufficiently accurate manner. These partialdifferential equations lay the foundations for simulating the EBM process.

The next task, following the simulation pipeline in Figure 1.1, will be thenumerical discretization of the mathematical model, i.e., the discretization ofthe partial differential equations describing the balance laws. Different dis-cretization methods will be compared and the lattice Boltzmann method will bederived, explained, and applied for the EBM process in detail in Chapter 4.

36

Chapter 4: Numerical Discretization

4 Numerical DiscretizationWhen eating an elephanttake one bite at a time.

(Unknown Origin)

In Chapter 3 the conservation laws for mass, momentum, and energy describethe fluid motion and its related processes in an abstract mathematical way bypartial differential equations. Following the simulation pipeline in Figure 1.1the next step is the choice of an appropriate numerical discretization methodin order to animate these partial differential equations by simulations. At thispoint it has to be noted, that the choice of the discretization method dependsexplicitly on the process which has to be simulated, in this case the EBM pro-cess.

There already exist several numerical methods for which it was shown thatthey are suitable for the discretization of the conservation laws of Eq. (3.20),Eq. (3.32), and Eq. (3.44). Based on commercial software frameworks the mostfrequent and established methods are Finite Difference Methods, Finite ElementMethods, and Finite Volume Methods.

The Finite Difference Methods (FDM) is one of the oldest discretization tech-niques where derivatives of the partial differential equations are replaced byfinite differences. Their approximation at fixed nodes solves the equation. Thework of Courant et al. [1928] defines approximate solutions by a five–point ap-proximation of the Laplace equation and shows the convergence of the approx-imated solution as the mesh width goes to zero. Furthermore, it defines theCFL-condition and proves its necessity for convergence. After the introductionof implicit FDMs in Crank and Nicolson [1947] several numerical analysts con-tinued to develop FDMs and standard references for FDM are Collatz [1955],Forsythe and Wasow [1960], and Richtmyer and Morton [1967]. The mathe-matical foundation of FDMs was developed during the 1950s by exploring theconcept of stability. The advantage of FDMs is the simple understanding of themethod and its fast implementation. However, dealing with complex domainswith unstructured grids and discontinuities is difficult.

Finite Element Methods (FEM) overcome these drawbacks. FEMs use theweak variational formulation of the boundary value problem and projects it withthe aid of a collocation or Galerkin methods into finite dimensional subspaces,the so-called Sobolev spaces for the numerical solution. Thus, the original largecontinuous problem is split up into a lot of small finite problems. The idea of avariational formulation goes already back to lord Rayleigh (Rayleigh [1894]) but

37


the work of Courant [1943] can be seen as the real starting point of FEM. Theterm Finite Element was firstly used in the work of Clough [1960]. Contem-poraneous structural engineers connected analysis with variational methods incontinuum mechanics into a discretization method (cf. Argyris [1955]) and ap-plied FEM to other classes of continuum mechanics. The work of Zienkiewicz[1971] is a standard reference in the context of engineers using FEM. Referencesfor the mathematical analysis of FEM are Strang and Fix [1973] and Babuskaand Aziz [1972]. The book of Atkinson and Han [2005] is also recommended atthis point since it offers a detailed and comprehensible survey of FEM.

The Finite Volume Method (FVM) can be seen as special case of FEM using agrid of cells and piecewise constant test functions (see Grossmann et al. [2007]).The FVM belongs to the class of integral schemes just as FEM. The idea of FVMsis the integration of the underlying PDEs over volume and the use of piecewiseconstant variables in this volume. These volume integrals are converted intosurface integrals using the Green’s theorem. These surface integrals define thefluxes between the volumes and the evaluation of these fluxes is an essentialpart of FVM. These application of fluxes guarantees also that FVM is conser-vative because the flux entering a given volume is identical to that leaving theadjacent volume (more information is found in LeVeque [2002]). Based on theconcept of fluxes the FVM is advantageous in solving problems where large gra-dients or even discontinuities occur. There also exist combinations of FEM andFVM like for example the Discontinuous Galerkin Method (for detailed infor-mation see Hesthaven and Warburton [2007]) which is a finite element methodusing discontinuous basis functions for more robustness for discontinuous pro-cesses.

Summarizing, these classical discretization techniques, no matter whetherthey are based on differential or integral basis, lead to a global system of lin-ear equations which has to be solved iteratively. But this global handling is ahandicap when parallelizing the algorithms. Although there exist parallelizationtechniques for these methods FEM grapples with it. Furthermore, even thoughZah and Lutzmann [2010] use FEM for the discretization of the EBM processFEM is not used in this thesis since due to the continuum approximation thereis no information on the surface geometry available. An exact recovering of thesurface is inevitable since the dynamics of the melting pool are influenced bysurface effects. Thus, macroscopic methods like FEM are not suitable for thediscretization of the EBM process.

In contrast the Lattice Boltzmann Method (LBM) was developed in the endof 1980s as an alternative mesoscopic approach for the numerical solution ofconservation laws which is focused on in this thesis. LBM can be parallal-ized easier than other traditional discretization methods whose computation isbased on the conservation of macroscopic quantities. The reason for this willbe explained later in this chaper. It has also less computational storage andwork requirements than molecular dynamic methods which consider micro-scopic physics.

In this Chapter the Lattice Boltzmann Method is explained and derived via

38


the Chapman–Enskog expansion. The multi–speed approach for incorporatingthe temperature is portrayed and compared with other methods like the multi–distribution approach. The Chapter ends with a elementary listing of differentboundary conditions in particular for LBM.

4.1 Numerical Discretization by LBM

In 1973 the first lattice gas cellular automata (LGCA) was introduced by Hardyet al. [1973] named HPP. It is the simplest LGCA and does not lead to the con-tinuity equation and Navier-Stokes equations in the macroscopic limit becauseof its anisotropic behavior due to its insufficient degree of rotational symmetryof the lattice. In order to overcome this defect Frisch, Hasslacher and Pomeaupresent the FHP LGCA in Frisch et al. [1986]. The FHP model has a hexagonalsymmetry and tends to the continuity equation and Navier-Stokes equations inthe macroscopic limit. Although, the LGCA suffers from various other draw-backs and therefore, the lattice Boltzmann methods were introduced as an in-dependent class of numerical methods for solving conservation laws given inChapter 3 in 1988 by the work of McNamara and Zanetti [1988]. Thus, LBMcan be interpreted as an improvement of LGCA.

In the following the original LBM, which is normally used for the solution ofisothermal flows is derived and explained. In addition, the extensions of theLBM are discussed for solving the energy equation. The theoretical founda-tion underlying the capability of the LBM to solve the conservation laws in themacroscopic limit is also explained in by means of Chapman–Enskog expan-sion.

4.1.1 Spatial Discretization by Stencils in the Mesoscopic Range

The LBM and the LGCA belong to the class of mesoscopic methods which are be-tween macroscopic and microscopic methods (cf. Figure 4.1). Numerical meth-ods like molecular dynamics (MD) or Direct Simulation Monte Carlo (DSMC) acton the microscopic length scale, i.e., compute the movement and position ofthe each single particle directly. Thus, they are based on intermolecular forcesand use a Lagrangian coordinate system. MD tracks all simulated molecules atthe same time and is deterministic. The corresponding macroscopic quantities,like velocity, are computed by averaging and integrating, respectively. Due tothe computational effort microscopic numerical methods are only suitable formicrofluidic simulations. Compared to that, macroscopic methods like FEM orFDM solve the partial differential equations directly on a given structured orunstructured, Eulerian mesh as it is explained in the beginning of this chapter.

Mesoscopic methods can be interpreted as the golden mean between both ap-proaches and combine their advantages. Mesoscopic methods like LBM operateon the same physical scale as microfluidics but do not track every moleculebut group of molecules. This reduces the computational amount dramatically

39


compared to the microscopic ansatz. On the other side, mesoscopic methodsresolve complex and multiphase flows more easily than macroscopic methods.

increasing scale

Microscopicmethods

(MD, DSMC)

Mesoscopicmethods

(LGCA, LBM)

Macroscopicmethods

(FEM, FDM)

ρ(x, t),v(x, t),T (x, t)

molecules

Figure 4.1: Classification of numerical discretization methods into scale cate-gories.

It makes no difference whether macro-, meso-, or microscopic methods arechosen each numerical discretization method separates the temporal and spa-tial domain in order to reduce the infinite set of points of a continuous domainto a finite number of points. The grid size of the spatial domain is denotedby ∆x and the time step size is denoted by ∆t analogously. Looking directlyat the discretization of the continuous 3D space, a lattice cell is defined as acubic volume of unit length ∆x centered around a lattice node. In mesoscopicmethods so-called stencils are used for the spatial discretization. Each of thelattice cells is related to a stencil. The structure of the used stencils is re-sponsible for if the future discretization method being Galilean invariant andisotropic or not. These requirements raise the question how the stencil hasto be assembled in order to guarantee isotropy and Galilean invariance for thecorresponding numerical discretization. Thus, a sufficient condition for gettingreasonable results by discretizing the conservation laws of Chapter 3, espe-cially of conservation of linear momentum (Eq. (3.32)), is the isotropy of thelattice tensors of second and fourth rank. This prerequisite will be resolved indetail in Section 4.1.3. First, isotropic tensors have to be defined.

Definition 4.1.1 (Isotropic tensor):A tensor Tα1α2...αn of rank n is called isotropic if it is invariant with respect toarbitrary orthogonal transformations O like rotations and reflections,

Tα1α2...αn = Tβ1β2...βnOα1β1...αnβn . (4.1)

40


How isotropic tensors up to rank four look like is described by the followingtheorem.

Theorem 4.1.1 (Properties of isotropic tensors):

1. There are no isotropic tensors of rank 1, i.e., vectors.

2. An isotropic tensor of rank 2 is proportional to δαβ.

3. An isotropic tensor of rank 3 is proportional to εαβγ, where εαβγ denotes theLevy-Civita symbol with ε123 = ε231 = ε312 = 1 and ε132 = ε321 = ε213 = −1 and0 otherwise.

4. There are three linear independent tensors of rank 4, namely, δαβδγδ, δαγδβδand δαδδβγ and can be combined in the general form of,

Tαβγδ = aδαβδγβ + bδαγδβδ + cδαδδβγ (4.2)

Theorem 4.1.1 and its proof are found in Jeffreys [1965]. Compared to LGCAthe introduction of lattice tensors establishes the LBM.

Definition 4.1.2 (Lattice tensor):A lattice tensor is a tensor Lα1α2...αn of rank n given by,

Lαaα2...αn =∑i

eiα1eiα2 . . . eiαn , (4.3)

where eiα is the Cartesian component of the lattice velocity ei.

01

2

3

4

56

7 8D2Q9

1

2

3

4

5

6

78

910

11

12

13

14

1516

17 18

D3Q19

Figure 4.2: Selected stencils in 2D and 3D for LBM.

In this thesis the general notation given by DnQm for the stencils is usedwhere n denotes the dimension and m the number of lattice velocities. This

41


style of notation and corresponding stencils are introduced by the work of Qianet al. [1992]. The most common stencils are the D2Q9 for two dimensions andD3Q19 for three dimensions. Both are depicted in Figure 4.2.

In order to guarantee isotropy sufficiently it has to be examined if the latticetensors Lαβ and Lαβγδ are isotropic. Exemplary, this is shown for the D3Q19model since it is used for the simulations in this thesis. The D3Q19 stencil hasthe following lattice velocities,

e0 = (0, 0, 0),

e1,2, e3,4e5,6 = (±1, 0, 0), (0,±1, 0), (0, 0,±1), (4.4)

e7,8,9,10, e11,12,13,14, e15,16,17,18 = (±1,±1, 0), (±1, 0,±1), (0,±1,±1).

The second order lattice tensor of the D3Q19 stencil is computed by,

LD3Q19αβ =

∑i=1,2

eiαeiβ =

=

0 0 00 0 00 0 0

+ 2 ·

1 0 00 0 00 0 0

+ 2 ·

0 0 00 1 00 0 0

+ 2 ·

0 0 00 0 00 0 1

+ 2 ·

1 1 01 1 00 0 0

+ 2 ·

1 −1 0−1 1 00 0 0

+ 2 ·

1 0 10 0 01 0 1

+ 2 ·

1 0 −10 0 0−1 0 1

+ 2 ·

0 0 00 1 10 1 1

+ 2 ·

0 0 10 1 −10 −1 1

=

= 10δαβ. (4.5)

Due to Theorem 4.1.1 LD3Q19αβ is isotropic because it is proportional to δαβ which

is isotropic by definition.

The calculation of the fourth order lattice tensor LD3Q19αβγδ is a little bit more

arduous but not really difficult and is skipped at this point. The result is,

LD3Q19αβγδ = 4 (δαβδγδ + δαγδβδ + δαδδβγ)− 2δαβγδ (4.6)

which is unfortunately not isotropic because it has not the form of an isotropicfourth order tensor given in Theorem 4.1.1. LD3Q19

αβγδ is usually anisotropic be-cause the symmetry group of the underlying lattice stencil is not large enough.So, how can the lattice tensors be modified to get LD3Q19

αβγδ isotropic? The answerare lattice weights ωi to guarantee that the symmetry group is large enough.Following the definition of Wolf-Gladrow [2005], the generalized lattice tensorsare introduced.

Definition 4.1.3 (Generalized lattice tensors):The generalized lattice tensors are obtained by extending the lattice tensors by

42


weights ωi,

Gα1α2...αn =∑i

ωieiα1eiα2 . . . eiαn (4.7)

where eiα is the Cartesian component of the lattice velocity ei.

The weights ωi for the D3Q19 stencil in order to receive generalized latticetensors are introduced by Qian et al. [1992] and are given by,

ωi =

13 i = 0118 i = 1, . . . , 6136 i = 7, . . . , 18.

(4.8)

It can be shown that the D3Q19 stencil has an isotropic generalized latticetensor of rank two and four (see Qian et al. [1992]). This stencil is used forall simulations in this thesis. Note, that the choice of weights is not unique.In Qian et al. [1992] the weights are summed up to one but there also existdifferent lattice weights for the D3Q19 stencil as given in Wolf-Gladrow [2005].

After deriving the foundations for the spatial discretization by lattice stencilsthe LBM algorithm is deduced from the Boltzmann equation. Finally, it is shownthat the LBM algorithm just consist of two equations and this serves as a basisfor its success over the last three decades.

4.1.2 LBM Algorithm

The first lattice Boltzmann models were introduced as alternative numericalmethods for hydrodynamic problems by McNamara and Zanetti [1988] and arebased on the discretization of the Boltzmann equation He and Luo [1997a]. Fol-lowing He and Luo [1997b] and Chen and Doolen [1998] LBM solves the Boltz-mann equation in the hydrodynamic limit in the physical momentum space.The Boltzmann transport equation describes the statistical behaviour of a ther-modynamic system and is given by,

D

Dtf (x,v, t) =

∂f

∂t+ v · ∇f + F · ∂f

∂v= Ω (4.9)

where f (x,v, t) is called particle distribution function (pdf) and denotes theprobability to find a particle at time t at position x with velocity v. F is anexternal force, e.g., gravity. The term on the right hand side denotes a collisionterm accounting the forces acting between the particles during collisions.

The most basic collision operator is the BGK-approximation Bhatnagar et al.[1954] proposed by Bhatagnar, Gross and Krook which replaces the complexright hand side of Eq. (4.9) by a linear expression. The pdf f is relaxed towardsthe equilibrium with relaxation time τ ,

∂f

∂t+ v · ∇f + F · ∂f

∂v= −1

τ[f − feq] , (4.10)

43


where feq denotes the Maxwell equilibrium distribution He and Luo [1997a].Chen et al. [1992] and Qian et al. [1992] show that the approximated BGKcollision operator proposes enough freedom for the equilibrium distributionto satisfy the necessary requirements as, Galilean invariance and a velocity-independent pressure to simulate fluid flow.

For the singe relaxation (SRT) collision model, the relaxation parameter τ isonly related to the viscosity ν of the fluid by,

ν = c2s∆t(τ − 0.5), (4.11)

where cs denotes the speed of sound (cf. Definition 3.3.4). For the lattice speedof sound, often the following value is used,

cs =1√3. (4.12)

However, this value is used for the EBM simulation but it is rather a rule ofthumb. There exist several other collision operators to enhance stability andaccuracy of the LBM. Most famous are the TRT (two-relaxation times) intro-duced by Ginzburg [2005a], Ginzburg et al. [2008] and MRT (multi-relaxationtimes) d’Humieres et al. [2002], Premnath and Abraham [2007] which workwith more than one relaxation parameter in order to get a better adjustment.The TRT model is explained in Chapter 10 and Chapter 11 since it is used forthe analysis and improvement of the free surface treatment. An additional butdifferent approach for the collision operator is the entropic lattice Boltzmannmethod (ELBM) introduced in Ansumali et al. [2006] which is based on the ideaof adherence of the H–theorem. Wagner [1998] shows that this ensures sta-bility. Using the SRT collision operator the discretized version of continuousEq. (4.10) is written as,

∂fi(x, t)

∂t+ ei · ∇fi(x, t) + Fi

∂fi∂ei

= −1

τ

(fi(x, t)− feq

i (x, t))

(4.13)

where fi(x, t) = f(x, ei, t). Discretized by a Taylor series expansion (see He andLuo [1997b]) the discrete Maxwell distribution f

eqi is given by,

feqi (x, t) = ωiρ

[1 +

(ei · v)

c2s

+(ei · v)2

2c4s

− v2

2c2s

]. (4.14)

There exist several approaches for the integration of the external force (cf.Eq. (4.13)). In this thesis for the simulation the one introduced by Luo [2000]is used and given by,

Fi = ωiρ

((ei − v)

c2s

+(ei · v)ei

c4s

)· g, (4.15)

where g denotes gravity. The macroscopic quantities density and momentum

44


are computed as moments of zeroth and first order and they are invariant ofcollision,

ρ =∑i

fi =∑i

feqi and ρv =

∑i

eifi =∑i

eifeqi . (4.16)

In the LBM the pressure is determined by the equation of state which is theideal gas law (cf. Eq. (3.45)), i.e., it is linearly related to the density,

p = c2sρ. (4.17)

Here it should be noted that the LBM itself is quasi–compressible (see Latt et al.[2008]) and thus, an equation of state exists.

Algorithmically, LBM can be divided into a stream and a collide step. Thestream step propagates the local pdfs of each cell along the corresponding latticelink into the neighboring cells and the resting particle with zero velocity remainsin the cell center. It is given by,

f′i (x+ ei∆t, t+ ∆t) = fi (x, t) . (4.18)

In the collide-step, the relaxation towards the local equilibrium is performed foreach cell for the pdfs f

′i by,

fi (x, t+ ∆t) = f′i (x, t+ ∆t) +

1

τ

[f

eqi (ρ,v)− f ′i (x, t+ ∆t)

]− Fi. (4.19)

Eq. (4.18) and Eq. (4.19) are called LBGK-scheme. It has to be noted that thecollide step is local and just the stream steps needs communication with thenearest neighbors. This straightforwardness of these two equations justifies thetriumph of LBM in modern computational fluid dynamics and its simplicity inparallelization.

A first review of the lattice Boltzmann method is given by Benzi et al. [1992],followed by a work of Chen and Doolen [1998] which discusses also furthertopics such as multiphase flows and fluid-particle interaction models. Detaileddiscussions of the derivation of LGA and LBM is also given in the book of Wolf-Gladrow [2005] which can be recommended for a first understanding. Anothergeneral survey about LBM can be found in the book of Succi [2001].

4.1.3 Chapman-Enskog Ansatz

The LBM algorithm has been derived and is summarized in Eq. (4.18) andEq. (4.19). It was shown that it is based on the discretization of the Boltz-mann equation (Eq. (4.9)). Here, the question arises whether LBGK solves theconservation laws of Chapter 3? At this point just the isothermal LBGK willbe considered, i.e., LBGK should solve the equation of continuity (Eq. (3.20))and the conservation of linear momentum (Eq. (3.32)) for incompressible fluids.The question can be answered with the aid of a Chapman–Enskog expansion.This multi–scale expansion, named after its developers, can be used to de-

45


rive the conservation laws from the Boltzmann equation (cf. Chapman [1916,1918] and Enskog [1917, 1922]). How the Boltzmann equation and the con-servation laws are linked by the expansion is depicted in Figure 4.3. On theright side the derivation of the LBGK via the discretization of the Boltzmannequation is visualized. Here, adequate Ma and Kn shall guarantee that thecompressibility effects are negligible. In order to simulate the incompressibleNavier-Stokes equations it has to be guaranteed that the velocity is ”small”to keep the compressibility errors ”small”. The latter depends on the velocitywhich will be shown in the following. Physicists demand that the Ma numberhas to be Ma < 0.1, a rule of thumb but this is not a real border; the higher thevelocity the larger are the compressibility errors and the simulated results can-not be used for predictions of incompressible flows since the numerical resultsare inaccurate and error-prone. From the LBGK scheme the incompressibleNavier-Stokes equation and the equation of continuity are derived using theChapman-Enskog expansion. Contrary to this, the conservation laws can alsobe deduced from the Boltzmann equation via the Chapman–Enskog expansionas shown on the left side of Figure 4.3. In the following, the Chapman–Enskogexpansion (CEE) is shortly explained for the BGK, following He and Luo [1997c].Here, the LBGK is used as a starting point to deduce the conservation of massand linear momentum. The following expansions are used,

fi (x+ ei∆t, t+ ∆t) =∞∑n=0

εn

n!Dnt fi (x, t) , (4.20)

fi =∞∑n=0

εnf(n)i , (4.21)

∂t =∞∑n=0

εn∂tn , (4.22)

where Dt = ∂∂t + ei

∂∂xi

denotes the material derivative (cf. Eq. (3.8)). The LBGKalgorithm is given by (see Eq. (4.18) and Eq. (4.19)),

fi (x+ ei∆t, t+ ∆t)− fi(x, t) = −1

τ

[fi(x, t)− feq

i (x, t)]

(4.23)

Eq. (4.20), Eq. (4.21), and Eq. (4.22) are inserted into Eq. (4.23), resulting in,

∞∑n=0

εn

n!Dnt fi(x, t)−

∞∑n=0

εnf(n)i =

1

τ

[ ∞∑n=0

εnf(n)i − feq

i (x, t)

]∞∑n=0

εn

n!Dnt

( ∞∑n=0

εnf(n)i

)= −1

τ

∞∑n=0

εnf(n)i +

∞∑n=0

εnf(n)i +

1

τf

eqi (x, t)

∞∑n=0

εn

n!Dnt

( ∞∑n=0

εnf(n)i

)=

(1− 1

τ

) ∞∑n=0

εnf(n)i +

1

τf

eqi (x, t) (4.24)

46


Boltzmann equation:∂f∂t +v ·∇f+F · ∂f∂v =

(∂f∂t

)coll

Chapman-Enskog

expansion

BGK-approximation

Continuity equation:∇ · v = 0

Navier-Stokes eq.s:DvDt = F − 1

ρ∇p+ ν∇2vEnergy equation:ρcv

DTDt = κ∇2T + Φ.

∂f∂t + ξ ∂f∂x = − 1

τ (f − feq)

discretization ofmomentum space

Chapman-Enskog

expansion

velocity discrete BGK eq.∂fi∂t + ei

∂fi∂x = − 1

τ

(fi − feq

i

)

discretization ofspace and time

fi (xi + ei∆t) = fi (x, t) − 1τ

(fi (x, t)− feq

i (ρ,v))

suitable Ma, Kn!

Figure 4.3: Connection between the LBM and the conservation laws by theChapman–Enskog expansion.

47


In the next step every sum is separated into the zeroth summand and the rest,

ε0

0!D0t

( ∞∑n=0

εnf(n)i

)+∞∑n=1

εn

n!Dnt

( ∞∑n=0

εnf(n)i

)=

=

(1− 1

τ

)(ε0f

(0)i +

∞∑n=1

εnf(n)i

)+

1

τf

eqi . (4.25)

ε0

(ε0f

(0)i +

∞∑n=1

εnf(n)i

)+ ε0f

(0)i

∞∑n=0

εn

n!Dnt +

∞∑n=1

εnf(n)i

∞∑n=1

εn

n!Dnt =

= ε0f(0)i −

1

τε0f

(0)i +

∞∑n=1

εnf(n)i − 1

τ

∞∑n=1

εnfni +1

τf

eqi (4.26)

ε0,2f(0)i + ε0

∞∑n=1

εnf(n)i + ε0f

(0)i

∞∑n=1

εn

n!Dnt +

∞∑n=1

εnf(n)i

∞∑n=1

εn

n!Dnt =

= ε0f(0)i −

1

τε0f

(0)i +

∞∑n=1

εnf(n)i − 1

τ

∞∑n=1

εnf(n)i +

1

τf

eqi . (4.27)

Eq. (4.27) is divided by ε0 and yields to,

ε0f(0)i +

∞∑n=1

εnf(n)i + f

(0)i

∞∑n=1

εn

n!Dnt +

1

ε0

∞∑n=1

εnf(n)i

∞∑n=1

εn

n!Dnt =

= f(0)i −

1

τf

(0)i +

1

ε0

∞∑n=1

εnf(n)i − 1

τε0

∞∑n=1

εnf(n)i +

1

τε0f

eqi . (4.28)

In the last step of the CEE, Eq. (4.28) in the asymptotic limit of ε→ 0 becomes,

f(0)i = f

eqi for O(ε0). (4.29)

Analogously, the following two equations of higher orders are derived using theCEE,

Dt0f(0)i = −1

τf

(1)i for O(ε1), (4.30)

∂t1f(0)i +

2τ − 1

2τDt0f

(1)i = −1

τf

(2)i for O(ε2). (4.31)

The moments of Eq. (4.30) lead to the Euler equations. Further, the momentsof Eq. (4.31) result in the incompressible Navier-Stokes equations, i.e., the con-servation of mass and linear momentum for incompressible fluids where thecontinuity equation is accurate to the order of O(Ma2) and the conservation oflinear momentum, accurate to the order of O(Ma3) (for detailed information see

48


the appendix of He and Luo [1997c]).This derivation of the conservation laws from the lattice Boltzmann equation

using CEE shows that the smaller the Ma the more accurate is the numeri-cal solution of the conservation laws. Here it has to be noted that the originalLBM always suffers from compressibility errors when used it for simulatingincompressible conservation laws. In the best case, incompressible fluids aresimulated by assuming the density to be constant. But this cannot be achievedwith the LBM since it always simulates a compressible fluid with a varying spa-tial density. Thus, when simulating incompressible fluids with the LBM a lowMa limit should be kept. Because of this LBM is also called quasi-compressible.

Hanel [2006] and Chapman and Cowling [1970] claim that the expansionparameter ε in the multi–scale analysis correspond with the Knudsen numberKn which is defined in following.

Definition 4.1.4 (Knudsen number):The Knudsen number Kn is defined by the ratio of molecular mean free pathlength λ and physical length scale L, i.e.,

Kn =λ

L. (4.32)

The validity of the continuum hypothesis requires Kn 1. This assertionexplains the ”small” Kn in Figure 4.3. It has to be noted that the assumptionof the equality of ε and Kn is not necessary for the validity of the Chapman–Enskog expansion. The expansion only requires 0 < ε 1.

So far, it has been shown that LBGK covers the continuity (Eq. (3.20)) aswell as the Navier–Stokes equations (Eq. (3.32)) without energy conservationand thermal effects. In order to solve thermal fluid flow the energy equation(Eq. (3.44)) has to be integrated into the LBM algorithm. This is done in thefollowing.

4.1.4 Thermal LBM by Multispeed Approach

The previous isothermal LB model is insufficient to simulate the EBM processwhere the temperature and energy distribution play a decisive role. Thus, thissection explains how the conservation of energy (Eq. (3.43)) can be integratedinto the LBGK model.

A first review of thermal LBM is given in Lallemand and Luo [2003c]. Gen-erally, thermal LBMs are classified into three groups: the multi–speed LBM(cf. Alexander [1993], Chen et al. [1994]) , the double-distribution or multi–distribution approach (Massaioli et al. [1993], Shan [1997], He et al. [1998])and the hybrid approach where the scalar temperature equation is solved byfinite differences (Lallemand and Luo [2003a]) or finite volume methods. Themulti–speed LBM can be seen as an extension of the isothermal LBM wherethe equilibrium distribution function depends also on the temperature and hashigher order nonlinear velocity terms. To obtain the macroscopic temperature

49


field additional speeds are required. This method suffers from numerical insta-bilities which have to be stabilized as it is shown in McNamara and Alder [1995].The possible temperature variations are also restricted in a narrow range. Addi-tionally, multi–speed LBMs are limited in the simulation of one Prandtl number(see Definition 3.3.1). This limits the range of applications which is rather un-desirable. Therefore, multi–speed algorithms have been generalized in 2D forsimulating arbitrary Prandtl numbers Pr as shown in the work of McNamaraand Alder [1995] and Soe et al. [1998]. The second group of multi–distributionLBMs or also called passive scalar approach overcomes these drawbacks byusing a separate distribution function for the temperature or energy field. Themulti–distribution approach has enhanced numerical stability compared to themulti–speed method. Viscous heat dissipation and compression work are typ-ically not integrated into multi–distribution models (cf. Bartoloni et al. [1993],Shan [1997]). This deficiency is resolved in the work of He et al. [1998] by theuse of a separate evolution function of the internal energy density distributionfunction. Furthermore, fluid flows with arbitrary Prandtl numbers can be simu-lated. It can be interpreted as a shortcoming that the two distribution functionsfor velocity and that one for the internal energy are coupled whereas the conser-vation laws for mass and momentum on the one side and the energy equationon the other side are decoupled. Furthermore, critical computer scientist mayalso adduce that the multi–distribution approach would waste computationalstorage for having a second pdf set and the corresponding fields but they shouldbe convinced by the unpretentious and facile handling described in the follow-ing. In this thesis, the multi–distribution approach for thermal LBM is used tosimulate the EBM process. In addition to the pdf set for hydrodynamic variablesf(x,v, t) a second pdf set h(x,v, t) for the numerical solution of the temperaturefield is used. The collision operator for the thermal LBM is also a SRT modelgiven by,

Ωh = − 1

τh(h− heq) , (4.33)

where heq denotes the Maxwell equilibrium distribution (see He and Luo [1997b])and τh is the relaxation parameter belonging to the thermal pdf set and is re-lated to the thermal diffusivity k by,

k = c2s∆t(τh − 0.5). (4.34)

For the spatial discretization the D3Q19 stencil is also used (Figure 4.2) with thevelocities of Eq. (4.5) and weights of Eq. (4.8) (Qian et al. [1992]). Following thework of He and Luo [1997b] the Maxwell equilibrium distribution is discretizedby a Taylor series expansion,

heqi (x, t) = ωiE

(1 +

(ei · v)

c2s

), (4.35)

where v denotes the velocity of the liquid and cs the speed of sound (see Defini-

50


tion 3.3.4). It is sufficient to use a linearized equilibrium distribution functionh

eqi for the temperature field. The macroscopic conserved quantity is the energy

density E and is computed as the zeroth moment of the second distributionfunction,

E =∑i

hi. (4.36)

Both pdf sets are one-way coupled by the velocity of the fluid v which is usedfor the computation of heq

i . The 3D coupled-thermal LBGK can be written in astream-collide algorithm,

stream

f′i (x+ ei∆t, t+ ∆t) = fi(x, t),

h′i(x+ ei∆t, t+ ∆t) = hi(x, t),

(4.37)

collide

fi(x, t+ ∆t) = f

′i (x, t+ ∆t) + 1

τf(f

eqi (ρ,v)− fi(x, t+ ∆t)) + Fi,

hi(x, t+ ∆t) = h′i(x, t+ ∆t) + 1

τh(h

eqi (E,v)− hi(x, t+ ∆t)) + Φi,

(4.38)

where Fi denotes an external force term, e.g., gravity, given in Eq. (4.15) and Φi

is the energy source of the system given by,

Φi(x, t) = ωiEb(x, t), (4.39)

where Eb denotes the energy source by the electron beam.This multi–distribution thermal LBM (TLBM) is used for the discretization

of the three conservation laws for mass, linear momentum, and energy whichbuild the mathematical model of the EBM process. In order to conclude thenumerical discretization boundary conditions due to the LBM are shown anddiscussed next since only boundary conditions make the existence and unique-ness of a solution possible.

4.2 Boundary Conditions

For the numerical simulation the right choice of boundary conditions (BC) is asimportant as the algorithm itself. BC are essential for stability and accuracy ofany numerical solution because BC with a lower accuracy than the algorithmcould reduce the order of the overall simulation. The setting of BC for kineticmethods like LBM is completely different compared to traditional macroscopicmethods like FEM or FD. In these traditional methods BC can be explicitlyset on the nodes at the boundary. However, in the LBM the pdfs leaving thecomputational domain have to be replaced by pdfs entering it. But these pdfsare unknown because there is no computation inside the wall. The proper-ties of these pdfs define the BC and can either be given a priori or computedfrom those leaving the domain. Unfortunately, a perfect knowledge of thesepdfs is unknown and approximations are required. The inaccuracy of these ap-

51


proximations creates a region near the boundary which Cercignani [1988] callsKnudsen layer or accommodation layer. This term is well-known in the con-text of the rarefied gas theory but the phenomenon also exists in the LBM (seeGinzburg and d’Humieres [2003]). Thus, for LBM the setting of the BC is morecomplex than for other numerical methods like FEM because LBM requires thesetting of the pdf values corresponding to determine macroscopic quantitiesρ,v, E at the boundary. Before the most frequently used BC are describedfluid and boundary nodes of LBM have to be defined in the following.

Definition 4.2.1 (Boundary nodes):If Ω denotes the computational domain and δΩ denotes the boundary of the do-main then,

• x is a fluid node if ∀ei|x+ ei∆t ∈ Ω ∪ ∂Ω and

• x is a boundary node if ∃ei|x+ ei∆t /∈ Ω ∪ ∂Ω.

On the boundary nodes, pdfs assigned to velocities ei pointing out of the com-putational domain in the streaming step and the ones assigend to the opposingvectors are underdetermined because there are no nodes which the pdfs comefrom (cf. Figure 4.4). The red, yellow, and green dashed arrows in Figure 4.4

boundary nodes

streaming

wall

fluid nodes

Figure 4.4: Situation of a solid–liquid interface for D2Q9 stencil: boundarynodes () have unknown pdf’s after the streaming step (right: un-knwon pdf’s are dashed).

visualize the unknown pdfs after the streaming step, because there are no pdfvalues available in wall cells. For the sake of simplicity the visualization is donein two dimensions like most of the visualizations of this chapter. The role ofBC is to find a substitution for missing pdfs in two ways: only missing pdf arereplaced or all pdfs are replaced on boundary node. For boundary nodes thesame dynamics should be valid and implemented as for fluid nodes. BC shouldalso conserve mass, momentum, and energy. In order to solve this problem

52


there exist several elementary and more elaborate boundary conditions for thesolid–liquid interface for LBM which are described in the following.

Periodic boundary conditions are one of the most commonly used BC fornumerical methods. Edges are treated as if they are attached to the oppositesite. Usually, this kind of BC are applied when the influence of the boundaryis small or computational domains of infinite order should be simulated. Theimplementation is realized by copying the values of boundary nodes from oneside to ghost nodes on the other side and then letting the fluid propagate. Thus,it is quite straightforward to use them for the LBM (see Succi [2001]). Themathematical formulation of the periodic boundary condition is given by,

vin = vout on ∂Ω. (4.40)

Figure 4.5 visualizes this idea. For the implementation of BC in the LBM thecomputational domain is surrounded by ghost cells which build a ghost layer.The pdf values are copied at the opposite ghost layer and then streamed intothe computational domain.

· · ·

fluid nodes

· · ·copy pdf

Figure 4.5: Periodic boundary conditions in x-direction for LBM (D2Q9).

No–slip bounce–back is the most popular Dirichlet BC for velocity mimick-ing a no–slip velocity. It is appropriate for solid–liquid interfaces for both wallsand objects. The idea of bounce–back is that a wall reflects the coming parti-cles back where they originally came from and no flux crosses the wall. Thisapproach has its origins in LGA (Wolfram [1986], Lavallee et al. [1991]) and isimplemented by simply reversing the velocities at the boundary. This ensuresthat the velocity normal to the wall is zero whereas the tangential part of thevelocity can correspond to a velocity of the wall. The comfortable implementa-tion of no-slip velocities for wall-boundary conditions promotes the handling ofLBM for complex boundaries.Cornubert et al. [1991], Ziegler [1993], and Ginzbourg and Adler [1994] showedthat the standard no–slip bounce–back boundary condition for LBM is only of

53


first order accuracy whilst the LBM itself for the nodes in the bulk phase is ofsecond order for general flows.

Figure 4.6: No–slip bounce–back boundary condition for D2Q9.

The bounce–back method can be implemented in two ways. Full–way bounce–back inverts the particle velocity during the collision step. At the collision withan obstacle or the wall the fluid particles simply reverse their direction of mo-tion. In the next stream step the fluid propagation leaves the boundary nodeand gets back to the fluid node only having a reversed direction of motion. Thefull–way bounce–back process takes two steps. The second possibility is calledhalf–way bounce–back. Here, the inversion of particle velocities takes placeduring the streaming step. The boundary surface is assumed to lie halfwaybetween a fluid node and a boundary node. The pdfs leaving the fluid nodeand encountering a boundary node are reflected and returns in one time stepto their original location pointing now in the opposite direction. The half–waybounce–back only takes one step.

Advantages of this simple bounce–back BC is that it is mass conserving, localand stable even for high Reynolds numbers. Furthermore, it is flexible in thehandling of walls, edges, and corners and easy to implement.

On the other side the order of accuracy may be decreased from second orderto first order. Using the discrete BGK collision operator the momentum is notexactly conserved. An additional shortcoming of the bounce–back BC is thatthe location of the boundary or wall is not exactly defined and it cannot directlymodel a general curvilinear surface because it results in a staircase shapedapproximation of the surface (cf. Figure 4.7). Interpolation based BC resolvethese problems where second order and third order accuracy respectively forvelocity for general flows in geometries of arbitrary shape are guaranteed.

Free–slip bounce–back for velocity is an example for a Cauchy BC, i.e., amixture of Dirichlet and Neumann BC. It models a fixed boundary where thevelocity normal to the boundary is zero (as for the no–slip BC) and for thetangential part of the velocity the gradient is set to zero. This is different to theno–slip BC where all velocity components are reflected.

The free–slip BC is also implemented by bouncing back the correspondingpdfs. Transverse directions towards a boundary are set differently and effect the

54


Figure 4.7: Staircase approximation of curvilinear surface by bounce–backmethod (dots visualize the boundary nodes).

adjacent fluid node in tangential direction. Free–slip is visualized in Figure 4.8.

Figure 4.8: Free–slip boundary condition for D2Q9.

Interpolation based Boundary Conditions is a possibility to represent sec-ond order accurate boundary conditions for problems where the wall is notlocated in the middle of a fluid and a boundary node. There exist linear andquadratic interpolation schemes proposed by the work of Bouzidi et al. [2001]and multi-reflection schemes introduced by Ginzburg and d’Humieres [2003].Interpolation schemes ensure stability compared to extrapolation boundaryschemes Chun and Ladd [2007].

The structure of interpolation schemes are visualized in Figure 4.9 in onedimension. The wall or boundary is located in point C. The ratio of distancefrom the boundary node to the wall and the wall to next regular node which isalready located in the wall is given by the value d.

If a pdf leaves the boundary node A and bounces back from the wall it willnot reach a fluid node except d = 0, 1

2 ,or 1. Thus, after the collide step, thepdf at A with ei = −1 (be aware of the 1D example!) is unknown. In orderto overcome this problem two cases are differed due to the value of d; first, ifd < 1

2 (cf Figure 4.9, top) a new node with pdf values at location D is computed.These pdf values stream to A after bouncing back on the wall at C. On theother hand, if d ≥ 1

2 (see Figure 4.9, bottom) the pdf values leaving A andarriving in D together with the new, post-stream situation at fluid nodes E

55


wall at C

A BEF

D Cd = |AC|

|AB| <12

wall at C

A BEF

D Cd = |AC|

|AB| ≥ 12

Figure 4.9: Overview of interpolation based boundary conditions in 1D.

(and F) is used to calculate unknown pdf values in A. Both cases use for linearinterpolation the values of nodes A and E. For quadratic interpolation threenodes are used, namely, A, E, and F. Chun and Ladd [2007] shows that bothlinear and quadratic interpolation schemes of Bouzidi et al. [2001] are of secondorder of accuracy at the boundaries. The interpolation BC of Bouzidi can beinterpreted as a correction of the bounce-back BC.

The so-called multi–reflection boundary condition of Ginzburg and d’Humieres[2003] is third order accurate for general flows. It is a generalization of the pre-vious bounce-back and interpolation schemes. The idea of this BC is writingthe BC as a closure relation between the unknown pdfs entering the fluid fromthe wall and some others which are known from the fluid dynamics. Thesepdfs are replaced by their second order approximations in the closure relation.The Taylor expansion of this result at the boundary node yield a second orderapproximation of the ideal solution at the boundary. It is also shown that theerror in the second order is equal to zero and thus, the BC is formally of thirdorder accuracy. A schematic overview in one dimension of the closure relation isvisualized in Figure 4.10. The closure relation Ginzburg and d’Humieres [2003]

κ−2 κ−1 κ−1 κ0 κ1?

xx− ei∆tx− 2ei∆t x+ ei∆t

x+ δiei

Figure 4.10: Overview of multi-reflection boundary condition in 1D.

56


is given by,

fi,t+1 = κ1fi (xb + ei, t+ 1) + κ0fi (xb, t+ 1) + κ−1fi (xb − ei, t+ 1)

+ κ−1fi (xb − ei, t+ 1) + κ−2fi (xb − 2ei, t+ 1)− ωt?i jiw + t?Fp.c.i

, (4.41)

where the overlined terms refer to quantities with ei = −ei and κ0, κ1, κ−1, κ−1,and κ−2 are the coefficients used for interpolation or multi–reflection BC. Theexpression ωit

?i jqw sets the Dirichlet BC. It should be noted that the closure

relation Eq. (4.41) involves the same pdfs as the interpolation based methodsintroduced by Bouzidi et al. [2001]. Since multi–reflection BCs are a generaliza-tion of bounce–back BC they can be expressed by the following multi–reflectioncoefficients,

Fp.c.i

= 0,

κ1 = 1,

ωi = 2, (4.42)

κ0 = κ−1 = κ−1 = κ−2 = 0.

The listed BC differ in accuracy and computational effort. It has to be decidedwhich BC is the most suitable one depending on the problem, i.e., if a curvedboundary occurs or accuracy is important.

4.3 Summary of Numerical Foundations

In this chapter the numerical discretization of the conservation laws in Chap-ter 3 is derived by the lattice Boltzmann method. The LBM is classified as amesoscopic method which is in between macroscopic methods (e.g., FEM) andmicroscopic methods (e.g., MD) using pdf. The stencils responsible for the spa-tial discretization have to fulfill certain properties in order to ensure isotropyof the subsequent LB model. LBM is derived based on the discretization of theBoltzmann equation where the collision operator is approximated by the BGKoperator. The algorithm just consists of a stream and collide step. The macro-scopic quantities like density and velocity are calculated by the zeroth and firstorder moments of the pdf values.

In order to persuade an interested reader that the Boltzmann equation basedLBM solves the conservation laws, the idea of the Chapman–Enskog expansionis showen and explained. The Chapman–Enskog multi–scale analysis is used toshow that the lattice Boltzmann equation solves the conservation laws assum-ing that the Ma and Kn number are small enough that compressibility errorsare negligible. The extension of the traditional, isothermal LBM by a secondpdf set establishes the multi–distribution LBM which can be used to solve theenergy equation (Eq. (3.43)) in addition. The multi–distribution approach hassome advantages compared to the multi–speed approach which are a betterstability and the option to simulate arbitrary Prandtl numbers.

57


In order to close the discretization of the mathematical model, elementaryboundary conditions for the LBM are shown and explained. For mesoscopic orkinetic approaches it is more complex to set BC than for macroscopic meth-ods since there is no communication of pdf values in the wall or boundaryand the inverse process of setting pdf values when knowing the macroscopicquantities is underdetermined. The Most used boundary conditions for LBMare no–slip and free–slip bounce–back BC and interpolation based methods likemulti–reflection BC.

One may think that the implementation of the EBM process can be startedat this point because the numerical discretization of the mathematical modelis finished so far. However, the numerical treatment of the free surface is stillmissing. This belongs also to the numerical discretization but it is explainedseparately in Chapter 5 because it goes beyond the considerations of this chap-ter.

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Chapter 5: Free Surfaces

5 Free Surfaces

Aut inveniam viam autfaciam.(I shall either find a way ormake one.)

(Hannibal Barka(247 – 183 BC))

Multi–component and multi–phase problems play an important role in variousindustrial applications where different fluids, like for example water and oil orwater and air interact and/or cross over. Their scope ranges from the modelingof combustion processes used for the development of diesel engines (Falcucciet al. [2010]), oil recovery (Gunstensen and Rothman [1993]), the developmentof protein foams in the food industriy (Anderl et al. [2014a]), and additive man-ufacturing processes like EBM. The simulation of such physical phenomena isnecessary but it is also a very computationally challenging task for high densityratio cases.

The EBM process consists of a phase transition indicated by the melting ofsolid metal powder particles and the interface between the liquid metal and thegas phase in the vacuum chamber (Figure 2.5). This interface is interpreted asa special boundary which has to be treated numerically. The mathematical andnumerical description is given in this Chapter.

Generally, one distinguish between multi–component and multi–phase prob-lems. Multi–component problems consist of different, immiscible fluids withvarying densities and viscosities. On the other side, multi–phase problems areproblems where the fluid undergoes phase transitions, e.g., a liquid turns intogas with increasing temperature or vice versa with decreasing temperature.

In order to solve the interface boundary numerically there are two categoriesfor multi–phase/multi–component models. Front capturing methods track themovements of the fluids/phases and capture the interface in a second step. Thephases/components are modeled as a single continuum assuming discontinu-ous properties at the interfaces. Examples for front capturing methods are theMarker– and–Cell (MAC) method, the Volume–of–Fluid (VOF) method, and thelevel set method. On the other hand front tracking methods track the location ofthe interface directly and this enables a more accurate calculation of curvatureof the the interface. Examples for tracking methods are the Boundary ElementMethod (Brebbia [1978]) and the Boundary–fitted Grid (BFG) method based onthe work of Barfield [1970] (for more information see Thompson et al. [1985]).

59


Thus, the numerical treatment of the multi–component/multi–phase problemsrequires the decision whether sharp or diffuse interfaces, and whether trackingor capturing methods are chosen.

The explanations in this chapter are limited to the treatment of two differentcomponents which are immiscible. The equilibrium state of this problem aretwo layers of fluid which are separated by a thin region called interface. Themathematical description of the interface layer is given and a brief overview ofthe numerous numerical interface handling methods based on LBM is provided.The interface methods differ in how they separate phases, how they track theinterface, if they form sharp or diffuse interface, and if they are mass conserva-tive. The thermal free surface lattice Boltzmann method is explained in detailsince it is used for simulation of the EBM process in Part II. The chapter isconcluded by a short overview and comparison of the different LB multi–phasemethods declaring their differences and agreements.

5.1 Mathematical Description of the Interface

Shikhmurzaev [2007] defines interface or also called surface of discontinuity as asurface of zero thickness, separating different fluids. The in reality very thin in-terface layer is abstracted to a interface of zero thickness. The necessary phys-ical phenomena due to the fluid dynamics have to be contributed by boundaryconditions. The numerical interface should be sharp because the molecularlength scales together with those associated with intermolecular forces get in-finitesimal by the continuum hypothesis. Boundary conditions given at theinterface should ensure the conservation laws belonging to the liquid phasesand the specific physics of the process itself.

Let f(x, t) be the equation of a moving smooth interface that separates fluid1 (f > 0) and fluid 2 (f < 0) (cf. Figure 5.1 ) and f = 0 at the interface. The unit

f(x, t) = 0

n

ρ1,v1, p1

ρ2,v2, p2

vs

f(x, t) < 0

f(x, t) > 0

Figure 5.1: Two-phase problem with fluid 1 and fluid 2, separated by interfacelayer (for example, liquid and air).

60


normal n points from fluid 2 to fluid 1 and is defined by,

n =∇f|∇f | . (5.1)

The moving interface is described by the following transport equation,

∂f

∂t+ vs · ∇f = 0, (5.2)

where vs is the velocity of the interface itself. The boundary conditions at theinterface have to fulfill the requirements of the conservation laws. The conser-vation of mass assumes the continuity of the mass flux at the interface, i.e.,

ρ1(v1 − vs) · n = ρ2(v2 − vs

)· n at f(x, t) = 0, (5.3)

and in the case of an impermeable interface,

v1 · n = v2 · n. (5.4)

The conservation of linear momentum has also to be ensured for the fluid phase.Thus, the momentum flux Πij is defined by,

Πij = ρvivj − Tij , (5.5)

where T is the stress tensor given by Eq. (3.28) in Chapter 3. The momentumflux across the interface is defined by,

Πi = ρi(vi − vs

) (vi − vs

)− T i for i = 1, 2. (5.6)

Sources or sinks of the momentum flux can be external forces acting on theinterface as well as the surface tension. Forces can compensate the zero thick-ness of the interface and incorporate dynamic properties. However, it should benoted that gravity is negligible because gravity depends on mass and the massof the very thin interface can be neglected. On the other side, surface tensioncan be considered as the asymmetric action on the interface layer caused by in-termolecular forces of the bulk phase and compensates the negligible thicknessof the interface layer. Surface tension can be interpreted as a force exerted bya part of the interface on a line restricting it along the inward normal. Thus,surface tension is a two–dimensional analogue of pressure but it differs in thatmanner that pressure expands the volume of a fluid and surface tension con-tracts the interface. The surface tension is characterized by the surface tensioncoefficient σ which defines the magnitude of force acting on line per unit lengthand depends on temperature. σ is a function defined along the interface andhas to be included in a two–dimensional surface stress tensor T s. In orderto imbed T s into three–dimensional space the δij − nn tensor is used whichpicks out the tangential components of a vector. For the surface stress tensor

61


T s = σ (δij − nn), the following relations can be set,

n · T s = 0, (5.7)

t1 · T s · t2 = 0 (5.8)

t1 · T s · t1 = σ. (5.9)

The stress tensor itself describes a force which is directed along the tangentialdirection t1 with magnitude σ per unit length of the line and therefore,

n · ∇σ = 0. (5.10)

The momentum flux across the interface is given by,

n ·(Π1 −Π2

)= ∇ · T s + F s (5.11)

where F s denotes the density of external forces per unit area acting on theinterface. Assuming no external force on an impermeable interface Eq. (5.11)can be simplified by,

p2 − p1 + n ·(P 1 − P 2

)· n = σ∇ · n, (5.12)

which is known as the capillary equation and P i, i = 1, 2, are the viscousstresses (see Eq. (3.29) in Chapter 3). σ∇ · n is the capillary or Laplacian pres-sure and ∇ · n defines the mean curvature of the interface. If the gradient ofsurface tension is zero, there is only the continuity of tangential stress acrossthe interface,

n · P 1 · (δij − nn) = n · P 2 · (δij − nn). (5.13)

In order to complete the set of boundary conditions, it is assumed that thetangential components of the bulk velocity on both sites of the interface arecontinuous, i.e.,

(δij − nn)v1 = (δij − nn)v2, (5.14)

using the tangential projection of the velocities. Eq. (5.14) is based on the as-sumption that the interface layer has a similar behaviour as the liquid phaseof a viscous fluid. This assumption is already mentioned in Batchelor [1993]because if the discontinuity in the velocity is assumed across the interface thiswould lead to a very large stress at the surface and would eliminate the relativevelocity of the two masses. If the thickness of the interface goes to zero, the ve-locity variation across it due to finite forces will also tend to zero and Eq. (5.14)is reached. In the case of an impermeable interface with constant surface ten-sion and no external surface force, the equations summarized in Eq. (5.15) arenecessary boundary conditions for the description of a interface layer a givenmathematically,

∂f

∂t+ vs · ∇f = 0,

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v1 · n = v2n,

p2 − p1 + n ·(p1 − p2

)· n = σ∇ · n, (5.15)

n · P 1 · (δij − nn) = n · P 2 · (δij − nn) ,

(δij − nn) · v1 = (δij − nn) · v2.

In the following a brief literature review of numerical LBM methods for solvingmultiphase/multicomponent problems is given.

5.2 Review of LB Multiphase Methods

Generally, numerical multi–phase methods consists of two steps, the segrega-tion step where the different phases are separated, and the identification of theinterface. The methods differ in the realization of the interface whether sharpor diffuse, or if the resulting method is mass conservative or not. One of thefirst systematic studies of multi–phase methods is given by Hirt and Nichols[1981], based on the work in Nichols and Hirt [1971]. Furthermore, there ex-ist overview papers for the numerical treatment of free surfaces or multiphaseflows, like Scardovelli and Zaleski [1999] and a more recent one of Shyy et al.[2012] which are recommended for general explanations. An overview of multi–phase LB models is given in Nourgaliev et al. [2003].

In the following the most popular multi–phase/multi-component models forLBM like the Gunstensen approach, the Shan & Chen model, the free energyapproach, the method of He et al., and hybrid approaches are described brieflyand compared with each other.

The Gunstensen approach or Color–gradient method (see Gunstensen et al.[1991]) was the first multi–phase approach for LBM which introduces a two–phase LBM description for modeling free interfaces of immiscible binary fluidsin two dimensions. It is based on the two–phase model of Rothman and Keller[1988] for LGA and was originally applied for the simulation of porous media.In both approaches, the two phases are differed by coloring the particles of onephase red and the particles of the other phase blue. The different phases in-teract with each other via local color gradients. The approach has a two–stepcollision model which facilitates interfacial dynamics like surface tension vianearest–neighbor particles interaction. In the first step a perturbation is addedto the particles close to the interface to guarantee the correct surface tensiondynamics. In a second step, the re–color step, mass is recolored in order tomaintain the mass of both fluids. The different particles are forced to movetoward the fluid with the same color. This handles the phase separation. Con-servation of mass and momentum are enforced at each node during collisionbut the Gunstensen approach is unstable for large density ratios and suffersfrom spurious currents (see Kehrwald [2002]). Grunau et al. [1993] extended

63


this model to allow variations of density and viscosity but the ratio is still re-stricted.

The Shan & Chen model introduced by Shan and Chen [1993] uses an inter–particle potential and belongs to the pseudo–potential methods. This approachenables both, systems with multiple components and with multiple phase tran-sitions. These components allow non–ideal gas equations of state and differentmass densities. The model was applied for simulating blood flow in Dupin et al.[2003], flow in microchannels in Yu et al. [2007], and for thermal flows (Zhangand Chen [2003]). The model has no explicit forcing term but rather an implicitone via the definition of equilibrium velocity. LBM itself is a quasi–compressiblescheme, i.e., local pressure is coupled to density via the equation of state (seeEq. (4.17)). If no intermolecular interaction occurs, fluid behaves like an idealgas and Eq. (3.45) is valid. The introduction of interaction potentials implies anon–ideal component in the equation of state and the definition of interactionpotentials determine what form the non–ideal term has in the equation of state.The definition of interaction potentials uses Green functions which act only on anearest–neighbor level and model phase segregation and surface tension. Thislocality permits and simplifies a parallel implementation of the model. The col-lisions do not conserve the net momentum of each site, but total momentum ofthe system.

Drawbacks of the Shan & Chen model are the non–thermodynamic behaviorlike the dependence of surface tension on the same parameter as equation ofstate, spurious currents, and a limited gas–liquid ratio. The use of differentequations of state result in a decrease of spurious currents and obtain in anincreased gas–liquid ratio.

The free energy approach of Swift et al. [1995] uses a free energy functionalfor the phase separation of two phases in the LBM. The idea of the free energyfunctional is based on the work of Landau and Lifshitz in Landau and Lifshitz[1987]. It introduces also a non–ideal pressure tensor coupled into the collisionoperator via an extension of the equilibrium function. This ensures the phaseseparation with correct bulk and interface dynamics at low temperature but aconsequence of this is a diffuse interface of several lattice cells. Inamuro et al.[2000] shows an improvement of the work of the original approach by modifyingthe equilibrium distribution such that the pressure tensor is consistent with thetensor derived from the free energy function of non–uniform fluids.

The free energy approach conserves the total energy but is unstable for largegas–liquid density ratio problems.

The approach of He et al. [1999] models interfacial dynamics like phase seg-regation and surface tension with the aid of molecular interactions and uses in-dex function for identifying the interface between the phases. It transforms theclassical LBM for a single component/phase from a mass and momentum to a

64


pressure and momentum formulation. Thus, one pdf set is used for calculatingvelocity and pressure, one for calculating the index function which is analoguesto the level set function. This reformulation reduces the potential of instabilitiesdue to high gradients in fluid density near the interface. The resulting interfaceis diffuse with a thickness of about three to four lattice cells without an artificialreconstruction step. Like Shan & Chen model, the approach of He et al. cannotsimulate large gas–liquid density ratio problems.

Hybrid approaches combine LBM with standard numerical front tracking orinterface capturing methods. LBM solves the fluid flow problem and a fronttracking/capturing scheme computes the motion of the interface. Both ap-proaches are coupled. One of the most well–known interface capturing meth-ods is the Level Set Method. It is introduced by Osher and Sethian [1988] anda first overview is given in Osher and Fedkiw [2001]. Mulder et al. [1992] ana-lyzes the coupling of the level set formulation to the system of conservation lawsfor compressible gas laws. In the level set method a smooth, at least Lipschitzcontinuous function ϕ(x, t) is defined to present the interface as a set of pointswhere ϕ(x, t) = 0. The interface is captured by locating the set Γ(t) for which ϕis zero. This formulation is advantageous because topological changes such asbreaking and merging are well defined. The evolution of the interface itself isdescribed by,

∂ϕ

∂t+ vs · ∇ϕ = 0, (5.16)

which is equal to the transport equation describing the interface mathemati-cally (see Eq. (5.2)). For the evolution itself only the normal component of theinterface velocity, i.e., vn = v · ∇ϕ|∇ϕ| is needed, and the evolution Eq. (5.16) be-comes,

∂ϕ

∂t+ vn|∇ϕ| = 0. (5.17)

Eq. (5.17) can be seen as a first order Hamilton – Jacobi equation,

∂ϕ

∂t+ |∇ϕ|γ(n) = 0, (5.18)

where γ is a function depending on the normal of the surface n = ∇ϕ|∇ϕ . The

advantages of the level set method are that changes in the topology are han-dled easy. and it produces also a sharp interface. Furthermore, there existalready efficient implementation techniques, like Adalsteinsson and Sethian[1999] based on a Fast Marching Method. On the other side, level set methodscan suffer from mass loss due to numerical errors.

After this review of multi–phase LBMs the free surface approach of Korneret al. [2005] called Free Surface LBM (FSLBM) is explained in detail since it isused for the numerical results in Chapter 9 and the extension of the models inChapter 10 and Chapter 11.

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5.3 Thermal Free Surface Lattice Boltzmann Method

The FSLBM of Korner et al. [2005] reduces the complex two–component problemconsisting of liquid and gas phase within the EBM process to a one–componentproblem. FSLBM neglects the dynamics of the gas phase and sets an appropri-ate boundary condition at the interface. Thus, the approach leads to a straight-forward treatment of the free surface with high computational efficiency but itcan only be used for the simulation of liquid–gas systems where the influence ofthe gas phase on the liquid phase is negligible. For the the application of liquid–liquid systems it has to be at least modified. This section is mainly based onthe work of Korner et al. [2005], Attar [2011], and Ammer et al. [2014a].

5.3.1 Cell Types and Conversion Rules

In this method each lattice cell of the computational domain has one of thefour specific cell types, namely, fluid (F), interface (I), gas (G) or solid/wall (W).These different cell types are depicted in Figure 5.2. Fluid cells are completelyfilled with fluid and have no gas cells as direct neighbors. In the fluid cellsthe LBM is computed as described in Section 4.1.2. Contrary to this, gas cellsare completely filled with gas and have no fluid cells next to them. In the gascells no LBM or other method is computed. Interface cells are filled with fluidpartially and build a sharp boundary between fluid and gas cells of one gridspacing. In these cells, the boundary condition to balance the force has to beexerted. Finally, in solid/walls cells a predescribed boundary condition (seeSection 4.2) is executed, for example the no–slip boundary condition where thepdf values are bounced back.

Fluid cells

Interface cells

Gas cells

Wall cells

– Free Surface

Figure 5.2: Example of a deformed drop on a bottom wall boundary. Gas phaseis separated from the fluid phase by an interface layer ( taken fromAmmer et al. [2014a]).

Since FSLBM is a volume–of–fluid approach, each interface cell has an addi-tional variable called mass fraction or fill level ϕ denoting the portion of areafilled with fluid. The value of ϕ ranges from 0 to 1. Fluid cells which are com-pletely filled with fluid have a fill level of ϕ(F ) = 1, gas cells which are empty

66


have a fill level of ϕ(G) = 0, and for interface it is 0 ≤ ϕ(I) ≤ 1. The fluid masscontent M(x, t) depends on the volume fraction and the density of the cell, i.e.,

M(x, t) =

0 x ∈ G,ρ(x, t)ϕ(x, t) x ∈ I,ρ x ∈ F.

(5.19)

The conservation of mass is ensured by the following conversion rules for cellstypes depending on M and ϕ, respectively. The direct conversion of a liquid cellinto a gas cell or vice versa is not allowed, this is only possible via interface cells.If M ≥ 1, then the interface cell converts into a liquid cell and the surroundingliquid cells convert into interface cells to assure a closed interface layer. Onthe other side, if M ≤ 0, the interface cell is transformed into a gas cell. Itshould be noted that density and mass are completely decoupled for interfacecells. The density is defined by the pressure boundary condition and the fluiddynamics, whereas M is calculated by the mass exchange of interface cells withthe surrounding fluid cells. Mass exchange between gas and interface cells isalso not allowed to guarantee mass conservation and an impermeable interface.The mass exchange ∆Mi is calculated by the particle distribution function,

∆Mi(x, t) =

0 x+ ei ∈ G,fouti

(x+ ei, t)− fouti (x, t) x+ ei ∈ F,

ϕ(x,t)+ϕ(x+ei,t)2 (fout

i(x+ ei, t)− fout

i (x, t)) x+ ei ∈ I,(5.20)

where fouti denotes the pdfs after collision and before streaming and i the inverse

direction, i.e., ei = −ei. Eq. (5.20) ensures that the mass which a cell receivesfrom a neighboring cell is automatically lost there. The temporal evolution ofthe mass content is given by,

M(x, t+ ∆t) = M(x, t) +19∑i=1

∆Mi(x, t). (5.21)

When new interface cells are generated, their initial distribution functions areextrapolated from the cells in the normal direction towards the fluid. Leftoverfluid fraction is evenly distributed to new interface cells to ensure exact massconservation. This mass redistribution only influences M but does not affectthe density or the pdf values.

5.3.2 Free Surface Boundary Condition

Due to the fact that in the FSLBM approach the gas phase is not directly simu-lated, after the streaming step only pdf values coming from fluid, interface, andsolid cells are known. However, pdfs coming from gas cells are unknown andhave to be reconstructed in such a way that the pressure boundary conditions

67


are fulfilled. This challenge is visualized in Figure 5.3. This means that theforce performed by the gas phase has to be balanced by the force performed bythe fluid. The disregard of the gas phase makes a special boundary condition at

gas

liquid

Figure 5.3: Missing pdfs (dashed) after streaming for a D2Q9 stencil.

the interface necessary which assures that the velocity of both bulk phases isequal and that the force induced by the gas phase is balanced by the force of theliquid, i.e., that the conditions of Eq. (5.15) are fulfilled (see Figure 5.4). In the

pG

gas

liquid

Figure 5.4: Idea of interface boundary condition (unknown pdf values comingfrom the gas phase visualized in red, dashed arrows).

approach by Korner et al. [2005] the pdf’s have to be adjusted to comply theseconditions. After the streaming step, the pdfs from liquid and interface cellsare defined and the pdfs coming from gas cells are unknown (cf. Figure 5.3).These unknown pdfs have to be determined by a special reconstruction ap-proach based on the momentum exchange method of Ladd [1994a,b]. The re-construction is based on the knowledge of the gas pressure and thereby, theforce exerted from the gas phase to the liquid phase. The total force F consistsof particles – coming from the liquid phase with velocity n · ei < 0 – crossing theinterface – and also of particles coming from the gas phase, i.e., with velocityn · ei ≥ 0. F exerted by the fluid on a surface element A(x) = n · A(x) hasits origin in the the momentum transported by the particles streaming throughthis element during one time step,

Fα =− nβA(x)

∑i,n·ei<0

fouti (x, t)(ei,α − vα)(ei,β − vβ)

︸︷︷︸known pdfs coming from fluid going into gas

(5.22)

68


+∑

i,n·ei≥0

fouti (x− ei, t)(ei,α − vα)(ei,β − vβ)

︸︷︷︸unknown pdfs coming from the gas phase into liquid

. (5.23)

Exploiting the fact, that the gas pressure and the velocity of the fluid are knownthe unknown pdfs are reconstruced by,

fouti (x− t · ei, t) = f

eqi (ρG,vs) + f

eqi

(ρG,vS)− fouti (x, t), ∀i|n · ei ≥ 0, (5.24)

where the gas density is given by ρG(t) = 3pG(t) and vS denotes the velocity ofthe interface. Putting Eq. (5.24) into Eq. (5.23) leads to,

Fα/A =− nβ∑

i,n·ei<0

fouti (x, t)(ei,α − vα)(ei,β − vβ)

− nβ∑

i,n·ei≥0

[f

eqi (ρG,v) + f

eqi

(ρG,v)− fouti

](ei,α − vα)(ei,β − vβ)

=− nβ∑

i,n·ei≥0

[f

eqi (ρG,v) + f

eqi

(ρG,v)]

(ei,α − vα)(ei,β − vβ)

=− nβ∑i

feqi (ρG,v)(ei,α − vα)(ei,β − vβ)

=− nβδαβpG

=− nαpG. (5.25)

The resulting fluid pressure has the same value as the gas pressure with anopposite sign, −n. Eq. (5.25) shows that the balance of forces at the interfaceis guaranteed and the boundary condition is fulfilled.

The effect of surface tension is integrated as a local modification of the gaspressure pG by the Laplace pressure,

pG(t) =1

3ρG(t)− 2κσ, (5.26)

where κ is the mean curvature and σ the surface tension. Details for the compu-tation of the curvature based on a triangulation of the interface via a marchingcube algorithm is found in Korner et al. [2005].

The authors of Korner et al. [2005] note that in FSLBM not only the missingpdfs coming from the gas phase are reconstructed but all pdfs with ei · n ≥ 0.This seems a little bit peculiar since information from pdfs coming from neigh-boring interface cells is not contributed but the maintenance of symmetry issubstantial for FSLBM. Moreover, the order of accuracy of this FSLBM bound-ary condition is not resolved in Korner et al. [2005] but it is examined in Chap-ter 10. A comparison of the level set method and FSLBM is shown in Rude andThurey [2004]. The extension of the free surface approach for thermal LBM isexplained in the following.

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5.3.3 Thermal Free Surface Extension

In the thermal FSLBM more cell types are necessary. The class of cell types hasto be extended by solid and solid interface cells in order to consider the phasetransition between liquid and solid cells. The different cell types are depictedin Figure 5.5.

wall cells

solid cells

solid interface cells

gas cells

fluid interface cells

liquid cells

liquid–solid transition

free surface

Figure 5.5: Cell types in thermal FSLBM explained by the example of a meltedspot.

A careful consideration of the solid–liquid phase transition (see Chatterjee andChakraborty [2006]) and the related cell conversions (Figure 5.6, red-dashedarrows) is necessary for the EBM process. The phase transition is controlled bythe energy density E (see Eq. (4.36)) and can be interpreted as an analogue ofthe fill level ϕ for the fluid interface handling. If E exceeds a material specificvalue the former solid cell converts to a liquid cell. The conversion goes theother way round if E falls below the other specific threshold. The change in E isgoverned by the computation of the thermal LBGK of Eq. (4.37) and Eq. (4.38)in the solid cells where the fluid velocity v is set to zero. In order to guaranteea stable behavior of thermal cell conversions the boundary between two cellstates is not a fixed value. It is implemented as a continuous hysteresis region,i.e., the phase transition between solid and liquid is done between a meltingrange between the solidus and liquidus temperature depending on the materialof the simulated powder.

Solid Interface Liquid Interface Gas

Solid Liquid

E ϕ

Figure 5.6: All possible cell conversions due to liquid–gas interface and solid–liquid phase transition.

70


The conversion from liquid or solid cells to liquid- or solid-interface cells doesnot require any additional initialization steps. For conversions from gas intoliquid/solid-interface cells, however, a reinitialization of the pdfs fi and hi isnecessary, since the data is undefined for the gas phase in the free surfaceapproach. At this point it has to be noted that there is no heat exchange fromthe particles to the atmosphere of the gas phase by the free surface just asthere is no exchange from liquid to gas phase. Thermal boundary conditionsare given by a reflection condition at the interface. Unknowns pdfs hi enteringfrom the gas phase are approximated by the pdf values hi outgoing from theliquid phase. More details are found in Attar [2011] and Korner et al. [2011].

The Thermal FSLBM is a straightforward multi–phase approach which ismass conserving and leads to a sharp interface. The disregard of the explicitcomputation of the dynamics of the gas phase allows large density ratios of thetwo phases which is essential when using it for the simulation of the EBM pro-cess since the melted powder and the vacuum have a large density ratio. Thus,thermal FSLBM is the appropriate choice of mutli–phase method for simulatingEBM process and all simulations in Chapter 9 make use of it.

5.4 Summary of LBM Multi–Phase Methods

In this chapter a mathematical description of the interface between two phasesis explained and the boundary conditions are derived for the special case of animpermeable interface. These boundary conditions assume the same velocitiesof the two phases at the interface and a sharp interface of zero thickness.

The two categories of solving the interface problems numerically are inter-face capturing and interface tracking methods. Most of the multi–phase/multi–component LBM methods of the last two decades belong to the class of cap-turing methods. A short overview of the most popular methods is given. Theydiffer in their handling of phase separation, whether they are mass conserva-tive or not, whether they create a sharp or diffuse interface layer, and whetherthey can handle large density ratios or not. An overview of these multi–phasemethods is given in Table 5.1.

METHOD PHASE SEGREGATION SHARP/DIFFUSE

INTERFACE

CONS. OF

MASS

Gunstensen blue/red pdf diffuse XShan & Chen interaction potentials diffuse —Free energy free energy functional diffuse XHe et al. molecular interactions diffuse —Level Set root of level set function sharp —FSLBM fill level sharp X

Table 5.1: Comparison of multi–phase LB models.

71


A detailed description of Thermal FSLBM based on the work Korner et al.[2005] and Korner et al. [2011] is given since it is used for the simulation ofEBM processes in Chapter 9. It is a Volume–of–Fluid based approach and ne-glects the dynamics of the gas phase. On the one side this disregard limitsthe application of the FSLBM approach only for liquid–gas problems but onthe other hand it is also the origin of its efficient realization and enables thehandling of high density ratios.

Part I of this thesis ends with the numerical treatment of the free surfacesby the thermal FSLBM. So far, the mathematical model describing the EBMprocess via partial differential equations and its numerical discretization by athermal LB method are derived. Following the simulation pipeline in Figure 1.1the implementation of the algorithm and the performing of simulations is ap-proaching in Part II.

72

Part II

Coupled MultiphysicsSimulations

73

Chapter 6: Specific EBM Modeling Aspects

6 Specific EBM Modeling Aspects

Essentially, all models arewrong, but some are useful.

(George Box (1919 – 2013))

The general mathematical model of the fluid dynamics of the melting pool ofthe EBM proces are derived in Chapter 3 and the corresponding numericaldiscretization is described in Chapter 4 and Chapter 5. However, for a realisticsimulation of the overall EBM process specific aspects of the application haveto be included. Therefore, the mathematical model has to be extended.

EBMmodeling

energyabsorp-

tion

electronbeamgun

powderparticles

Figure 6.1: EBM requirements.

There are three different areas which constitute the resulting model (see Fig-ure 6.1). Firstly, properties regarding the core of the EBM machine, the electronbeam gun, have to be described mathematically. The characterization of theelectron beam and the energy distribution model is discussed. Furthermore,the incorporation of energy and its absorption in the metal powder bed is for-mulated in mathematical equations. Different absorption algorithms which are

75


derived in an empirical way are discussed. The numerical solution of the result-ing absorption algorithms follows. The third integral part of the EBM processis the metal powder. The composition and distribution of the powder particlesexpressed mathematically in order to accomplish the model.

The results of this Chapter are mainly based on the collaborated work of Dr.Matthias Markl and me which is published in Ammer et al. [2014a] and Marklet al. [2013].

6.1 Functionality and Properties of the Electron Beam

The EBM process and its components are briefly described in Section 2.3 andvisualized in Figure 2.5. In the following the machine itself and its functionalityis explained in detail. The electron emitter and the induction coils are placed inthe vacuum chamber. The induction coils are used to modify the electron beam.The electron emitter itself is composed of a hot cathode providing free electrons,i.e., the cathode is the electron source, and a anode accelerating the electrons.The electrons come through a nozzle in the anode to the coils which focus anddeflect the electron beam by electromagnetic fields. It is a thermionic emissionprocess where the thermal induced energy overcomes the electron binding en-ergy. There exist different cathodes made of different materials, like for exampleplasma or ceramic. After the emission of the electrons they are attracted by thepositive potential of the anode which is generated by the acceleration voltage.

The resulting electron trajectories are assumed to be parallel. The electronbeam can be changed by electromagnetic fields which are arisen by the coils.The coils are used to focus the electron beam in order to one single spot. Bythe deflection of the electron beam different positions on the powder bed can behit. Since electromagnetic fields can be changed instantaneously the electronbeam is able to move with high velocities or even jump at various positions.

The electron beam in the EBM process is determined by its shape, energy dis-tribution, and size. These properties are discussed in the following. Since theelectron beam gun is radial symmetric the resulting shape of a non–deflected,focused electron beam is assumed to be circular. The shape gets more andmore elliptical by deflection. In order to have equivalent and reliable beamproperties on the whole building domain it is tried to minimize or prevent thiseffect by the use of astigmator coils. However, for the numerical model shapedistortions are neglected and a perfect circular shape of the electron beam isassumed. The spot size of the electron beam influences the resulting meltingpool size and shape. The spot size is measured by using the Full Width HalfMax (FWHM) value. This value denotes the width of the electron beam wherethe power intensity is just half of its maximum. The energy distribution of thespot is also radial symmetric for the case of a non–deflected beam. However,the main intensity is in the center since a lot of electrons overlay here. Further-more, it is assumed that the total beam power is transferred completely to theproduct of its acceleration voltage U and the beam current I. Following Ammer

76


−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

distance from beam center in [mm]

0

2

4

6

8

10

12

14

16

beam

pow

erin

tens

ityin

[kV

mm

2]

FWHM

σb σb

Figure 6.2: Cross section of assumed electron beam power intensity based ontwo–dimensional Gaussian profile of a 1kV electron beam (red) witha standard deviation of σb = 100µm (green). The resulting FWHMbeam diameter is dbeam = 235.5µm (blue).

et al. [2014a] electron beam intensity Ib is approximated by a two–dimensionalGaussian distribution,

Ib (xb, t) =UI

2πσ2b

exp

(− 1

2σ2b

1∑i=0

(xi − xb,i(t))2

), (6.1)

where the xb denotes the center position of the electron beam and σb standsfor the standard deviation. If a Gaussian function is assumed the standarddeviation σb depending on a predetermined FWHMbeam diameter is computedby,

σb =db

2√

2 ln 2. (6.2)

Figure 6.2 shows the interaction of the Gaussian intensity profile of a 1kV beamwith a standard deviation σb = 100 µm and the corresponding FWHM db. InChapter 9 the influence of different FWHM values are discussed and simulated.

In order to complete the model definition of the electron beam gun, beammovements representing the center position of the electron beam are defined.Two basic geometric shapes, namely a line and a sphere, are sufficient to ap-proximate even complex geometric shapes by combination. A line movement isdefined by start and end point. For a circle movement the circle center point,start point, angle and circular direction is necessary. In combination with the

77


beam velocity the total time duration for the movement and the complex layeris computable.

6.2 Modeling of Metal Powder Bed

Since the EBM process belongs to the powder bed fusion processes (see Chap-ter 2) the modeling of the metal powder bed is an integral part. Therefore, theproperties of the real powder particles used for the EBM process have to beexamined. They are determined by the production process which influencesshape, size distribution, surface topography, flow rate, and impurity.

This subsection is mainly based on the work of Markl [2015] and the resultsof Ammer et al. [2014a].

6.2.1 Properties of Powder Particles

An overview of the different production processes like atomization, chemicaland mechanical methods is given in Lawley [1978]. The material which is mostused for the EBM process is the Ti–6Al–4V alloy. This is also used for the EBMsimulations in Chapter 9. Since titanium has a high reactivity the atomizationmethod EIGA (electrode induction–melting gas atomization) is an appropriatepowder production method to ensure flowability of the particles. Here, a metalrod is heated by an induction coil and when the liquidus temperature is reachedthe melted material drops down and is atomized in a gas nozzle by a high pres-sure argon gas stream. The generated particles are sieved to ensure an intervalof particle sizes with upper and lower limits. Detailed information of the EIGAprocedure is given in Pleier et al. [2004] and Hohmann and Ludwig [1992].The EIGA method produces powder particles having sizes of a lower and uppersize limit. The EIGA production process also influences the resulting shape ofthe powder particles. Most particles generated by atomization processes areof spherical shape which can be examined by electron microscope images. Al-though, there also occur cylindrical or rod–like particles, for the powder modeland the following simulation spherical particles are assumed.

The next serious powder characteristic is the size distribution. The distri-bution of spherical powder particles can be examined by the light scatteringmethod (for more details see Schatt et al. [2007]). Here, the deflected and scat-tered light is converted into electric signals by photodetectors in order to exam-ine the size distribution of spherical particles. Due to the spherical shape of theparticles the size distribution is mainly characterized by their diameters. Thisprocedure results in a cumulative relative frequency distribution Q0(dp) usingthe notation introduced in Markl [2015]. It describes,

Q0(dp) =#of all particles with < dp

total # of all particles. (6.3)

where n diameter intervals are assumed ]dp,i, dp,i+1[ for i ∈ (1, . . . , n− 1) and the

78


difference of the cumulative frequency distribution ∆Q0 is defined by,

∆Q0(dp,i, dp,i+1) = Q0(dp,i+1)−Q0(dp,i), (6.4)

and the corresponding relative frequency density distribution q0 is given by,

q0(dp,i, dp,i+1) =∆Q0(dp,i), dp,i+1

dp,i+1 − dp,i. (6.5)

With these notations the mean diameter of the size distribution dp,50 indicateswhere the relative frequency is one half. dp,min and dp,max stand for the minimumand maximum diameter, respectively. Since the deviation of the powder sizedistribution is unknown an estimator for it is required. For a random sampleof bulk powder an estimator for the standard deviation can be computed andthis is called sample standard deviation s. Various strategies exist to computethe sample standard deviation. The most used one is called corrected samplestandard deviation based on Bessel’s correction and is given by,

s(X) =

√√√√ 1

N − 1

N∑i=1

(Xi −X

)2, (6.6)

where the sample mean value is computed by,

X =1

N

N∑i=1

Xi, (6.7)

where N stands for the size of the sample, i.e., because of statistical fluctua-tions the measurements of the bulk powder have to be repeated N times. Thestandard error of the mean s(X) is then defined by,

s(X) =1√Ns(X). (6.8)

In this special case the random variable X describes the diameter of the pow-der particles in [m]. The so–called variation intervals are used to examine thepowder distribution. The 95 % variation interval ±1.96σ(X) represents the oc-curring of the random variable X in this interval with a probability of 95 %.The standard error shows the certainty of the mean value by the confidenceinterval where the 95 % confidence interval of ±1.96s(X) is used. With Eq. (6.6),Eq. (6.7), and Eq. (6.8) the arithmetic mean ∆Q0, the sample standard devia-tion s(∆Q0) and the sample standard error s(∆Q0) for each single measurementinterval of the relative frequency distribution as well as corresponding valuesq0, s(q0) and s(q0) of the relative frequency density distribution can be com-puted. Data of bulk powder are provided by Thorsten Scharowsky. Figure 6.3shows the mean frequency density distribution produced of Ti–6Al–4V powder

79


20 40 60 80 100 120 140 160

particle diameter dp in [µm]

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Rela

tive f

requency

densi

ty q 0

in [1/µm

]

dp,min dp, 50 dp,max

∆Q0

q0 ± 1. 96S0(q0)

Figure 6.3: Ti–6Al–4V powder particle size distribution generated by EIGA andmeasured by light scattering method. The histogram represents themean relative frequency ∆Q0.The mean relative frequency q0 is given in its 95 % confidence inter-val by ±1.96s(q0) (data provided by Thorsten Scharowsky).

which is produced by the EIGA atomization approach. The diameter rangesfrom 26.3 µm and 158.5 µm and the mean diameter is 57.8 µm (see the ver-tical line denoted by dp,50). The histogram represents ∆Q0 while the errorbarand the ? denote the 95 % confidence interval of q0. If the powder is producedby atomization techniques the resulting particles have a smooth surface andhave a circular shape. Thus the frictional forces between the particles and theflowability of the powder depend on the overall surface topology The flowingbehavior of the powder can be categorized into 4 classes, namely non–flowing,cohesive, free–flowing, and excellent flowing (see De Jong et al. [1999], Lumayet al. [2012]). In order to determine to which category the powder belongs thebulk density is examined. The bulk density is defined by the ratio of the massto its total volume where also the space between the particles is included. TheHausner ratio introduced in Hausner [1967] giving a connection between thebulk density and the flowability can be used to determine the flowability grade.It depends on the particle distribution size. Assuming a mean diameter of 57.8µm the flowability is classified as free–flowing. Thus, the flowability of the metalpowder is influenced by the shape, size distribution and the surface topology ofthe particles. The last property which describes powder particles is impurity.By using EIGA produced Ti–6Al–4V particles the high pressure argon gas whichis used by the atomization can remain in the particles as pores. The duration ofthe liquified melting pool is often too short to dissolve the pores. Furthermore,

80


an oxidized surface of the powder particles is often observed since the Ti–6Al–4V alloy is highly reactive and reacts also with air. The properties of the EIGAproduced Ti–6Al–4V powder particles are modeled meticulously in the following.

6.2.2 Mathematical Model of Powder Particles in the EBM Process

For the mathematical model of the metal powder particles it is assumed thatthe particles are perfect spheres and agglomerates, satellites, and gas pores areneglected. Thus, the surface topology is not modeled explicitly and impurity ofpowder is neglected. How the particle size distribution is modeled is explainedin the following.

Figure 6.3 shows that the sizes of particle diameter are not normally dis-tributed but skewed. For such data the lognormal, Weibull, Gamma, or inverseGaussian distribution can be applied. In this work the inverse Gaussian dis-tribution (IG) is used to approximate the particle size distribution. IG has beenalready used for the approximation of aerosol size distributions (see Alexandrovand Lacis [2000]). The three parameter IG distribution is given by,

f(dp, dp,0, µ, λ) =

√

λ2π(dp−dp,0)3

exp(−λ(dp−dp,0−µ)2

2µ2(dp−dp,0)

), if (dp − dp,0) > 0

0 else,(6.9)

where µ > 0 denotes the mean value, λ > 0 the scale vector, dp,0 < ∞ thediameter beam offset, and dp the particle diameter. Usually, the IG distributionis just a two–parameter function and it can be derived by Eq. (6.9) by settingdp,0 = 0. For the model of the metal powder particles the three parameter IGdistribution function is used in order to get more accurate approximations forthe range of small diameters (see Chhikara [1988]). The skewness of the IGdistribution function is measured by,

φ = 3

√λ

µ. (6.10)

If φ → 0 and λ → ∞ the IG distribution functions tends to the normal distribu-tion function and is symmetric around µ. Following the work of Markl [2015] themean value µ and the scale parameter λ describing the IG distribution functionare estimated by a maximum likelihood method (see Tweedie [1957]). Havinga sample dp,k, k ∈ 1, . . . , N of N independent measurements of powder particledistributions the mean and scale parameter are estimated by,

µ =1

N

N∑k=1

dp,k, (6.11)

1

λ=

1

N

N∑k=1

(1

dp,k− 1

µ

). (6.12)

81


The maximum likelihood estimator applied to the data of the measurements ofbulk powder results in,

dp,0 = dp,min, (6.13)

µ =1

2

n−1∑i=1

(dp,i+1 + dp,i − 2dp,0) ∆Q0 (dp,i, dp,i+1) , (6.14)

1

λ= 2

n−1∑i=1

∆Q0 (dp,i, dp,i+1)

dp,i+1 + dp,i − 2dp,0− 1

µ. (6.15)

Eq. (6.13), Eq. (6.14), and Eq. (6.15) form the initial condition for a least–squareanalysis in order to fit the IG distribution for a determinate bulk powder. The

20 40 60 80 100 120 140 160

particle diameter dp in [µm]

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Rela

tive f

requency

densi

ty q 0

in [1/µm

]

dp,min dp, 50 dp,max

f(dp)

q0 ± 1. 96S0(q0)

Figure 6.4: The IG distribution function f(dp, dp,0 = 17µm, µ = 44µm, λ = 273µm)approximates the relative frequency density distribution q0 with95 % confidence interval 1.96s(q0).

cumulative inverse Gaussian distribution is given by,

Ψ(dp) =Φ

(√λ

dp − dp,0

(dp − dp,0

µ− 1

))+

exp

(2λ

µ

)Φ

(−√

λ

dp − dp,0

(dp − dp,0

µ+ 1

)), (6.16)

where Φ denotes the cumulative standard normal distribution.The IG distribution function received by the estimators for the mean and

scale is depicted in Figure 6.4. The measurements ranges from dp,min = 26.3 µm

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and dp,max = 158.5 µm and a mean diameter dp,50 = 57.8 µm. The measureddata result in an estimator for the mean value µ = 44 µm, the scale parameterλ = 273 µm and dp,0 = 17 µm. The measurements of the bulk powders are pro-vided by Thorsten Scharowsky and listed in Markl [2015]. In order to generatethe powder layer for the EBM simulation spherical particles with a diameter dueto the IG distribution are initialized. Thus, uniformly distributed random num-bers between zero and one are generated and the corresponding diameter iscomputed by the quantile of the cumulative IG distribution given in Eq. (6.16).The quantile defines the diameter where the cumulative distribution function

20 40 60 80 100 120 140 160

particle diameterdp in [µm]

0.0

0.2

0.4

0.6

0.8

1.0

Ψ(dp)

Figure 6.5: Relationship of particle diameter db and quantile value.

exceeds the quantile value, for example the mean diameter of dp,50 correspondswith a quantile of 1

2 . The relationship of particle diameter and the quantile valueis depicted in Figure 6.5. The computation is repeated if the a diameter value iscomputed by this approach which is outside ]dp,min, dp,max[. Markl [2015] showedthat the numerical modeled particle diameters and the ones measured by realexperiments are highly concordant and the deviation is less than 1 %.

For the simulation of a complete powder layer the number of particles nprequired to fill the layer is computed by the following formula,

np =RDV1

V2=

RD · V1 ·(

4

3πn−1∑i=1

1

2

((1

2dp,i

)3

+

(1

2dp,i+1

)3)

∆Ψ(dp,i, dp,i+1)

)−1

, (6.17)

where RD denotes the relative density, V1 the volume of the powder layer, V p

83


the mean volume of the powder particle size distribution, and n the numberof measurement intervals. The relative density of a powder layer is defined asthe ratio of bulk density to the mass density. Following Attar [2011] a relativedensity of around 55 % can be assumed in the EBM process. Markl [2015]showed that the relative frequency difference of sampled and approximated sizedistribution for np = 369 particles is less than 5 %.

Following the results given in Ammer et al. [2014a] and Markl [2015] theapproximation of the particle size distribution by an IG distribution is suffi-ciently accurate to model the real metal powder particles processed by the EIGAmethod. For the simulation only the data of the corresponding bulk powder isrequired, i.e., information about the minimum, maximum and mean diameter,in order to determine the IG distribution function. With this model of the par-ticle size distribution the Ti–6Al–4V powder particles can be simulated as rigidbodies using the pe framework (see Chapter 7 and Chapter 8).

In the following the incorporation of the energy of the electron beam is derivedby the use of absorption algorithms.

6.3 Model of Energy Absorption

The mathematical model for the energy absorption of the electron beam ismainly based on the work of Klassen et al. [2014a] where especially the ab-sorption on inclined surfaces is examined. The model of Klassen et al. [2014a]was validated in a 2D software framework. The discretization and approxima-tion of this model is based on the work of Markl et al. [2013] for the 3D EBMprocess.

This Section describes the mathematical absorption model for the energy ofthe electron beam. This model is rather semi–empirical than analytical anddescribes a function depending on the acceleration voltage, the target materialand the angle of the surface. The discretization and approximation of this modelis used for the final thermodynamic coupling with the LBM via the source term.

6.3.1 Mathematical Model

The mathematical model assumes a electron beam which sends parallel elec-trons of the same energy to the surface of the powder bed. The incoming elec-trons react with the nuclei and electrons of the powder particles by elastic andinelastic collisions at the surface. In the case of elastic collisions of the elec-trons with the nuclei, the electrons can be deflected by a large angle and theiroriginal direction is changed (see Turner and Hamm [1995]). Following Tabataet al. [1999] electrons loose some part of their kinetic energy by radiating pho-tons. Through the scattering of the electrons the energy is distributed in alarger region. Although the main part of the kinetic energy is spent by the in-elastic collisions with the atomic electrons of the material by absorption. Sincethe width of the electron beam σb is much larger than the interaction volume of

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the electron beam with the powder for typical acceleration voltages (see Klassenet al. [2014a]) the absorption is only examined in the one spatial dimensionand depends on the z-direction. The situation is visualized in Figure 6.6. The

R

σb

electronssent by the electron beam gun

zx

y

Figure 6.6: Energy dissipation volume of an electron beam with width σb andpenetration depth R.

electrons coming form the electron beam gun can be absorbed by the materialsurface, backscattered due to large angle deflections, or transmitted into otherlayers of the powder bed. The fractions for absorbed ηA, backscattered ηB, andtransmitted ηT electrons can be summed up to one and are declared by,

ηA + ηB + ηT = 1. (6.18)

The transmission coefficient ηT itself decreases exponentially in the z–direction(cf. Fitting [1974]). In this context the maximum penetration depths R is definedas the depth where no transmission occurs and the energy from transmittedelectrons is negligible respectively, i.e., ηB ≈ 0. The value of the backscatter-ing coefficient ηB increases up to a saturation value ηB,0 at a region where noelectrons can backscatter any longer. The backscattering fraction depends onthe incidence angle θb of the electron beam to the material surface. The largerthe incidence angle is the less energy can be absorbed. Following Cosslett andThomas [1965] the fraction ηA of absorbed energy can be derived provided thatηB,0 and ηT are known,

ηA(z, θB) = (1− ηB,0(θb)eB,0(θb)) (1− ηT (z)eT (z)) , (6.19)

where eB,0 and eT energy fractions of backscattered and transmitted electrons.The complete empirical model of absorption is derived in Klassen et al. [2014a].

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6.3.2 Numerical Approximation of Absorption Model

The absorption model of Section 6.3.1 and the power intensity of the electronbeam gun (described in Section 6.1) have to be coupled in order to compute theenergy source term of the thermal LBM Eq. (4.38). Therefore, the electron beamintensity in Eq. (6.1) has to be discretized in space and time. Since the energyhits on the surface of the powder particles, the spatial discretization occurs inthe horizontal lattice grid space. This discretization is based on the followingintegral formula,

Eb(x, t) =

UI

2πσ2b

∫ x0+ 12

∆x

x0− 12

∆x

∫ x1+ 12

∆x

x1− 12

∆x

∫ t+∆t

texp

(− 1

2σ2b

1∑i=0

(x?i − xb,i(t?))2

)dt?dx?1dx

?0, (6.20)

where U denotes the acceleration voltage, I the beam current, ∆x and ∆t theLBM corresponding spatial and temporal step, xb the position of the beam, andσb the width of the beam. Assuming δx σ the space integrals can be approx-imated by sampling the original distribution function at the cell center coordi-nates. In current EBM machines the smallest beam diameter is in the range ofpowder particles. Since in the LBM approach each powder particle is resolvedby several lattice cells this assumption is fulfilled. Thus, the integration formulaleads to the following approximation,

Eb(x, t) =UI∆t∆x2

2πσ2b

exp

(− 1

2σ2b

1∑i=0

(xi − xb,i (t+ 0.5∆t))2

). (6.21)

The error of the time approximation influences only the position of the energyabsorption. Since the electron reaction in lateral direction is neglected in thisabsorption model possible deviations from the integral solution are insignifi-cant. However, the spatial discretization error may modify the overall absorbedenergy and therefore, this error should be minimized. This absorption modelis adapted to the corresponding lattice grid. For each lattice cell at position xthe depth z(x) is measured from the uppermost surface in vertical direction bysumming up all fill levels ϕ,

z(x) = z(x+ ∆x) +1

∆x2ϕ(x), ∆x =

00

∆x

. (6.22)

Because of the lattice approach the area affected by the electron beam has to becut on the left and right hand side. However, this effect is insignificant becauseof the Gaussian profile of the power intensity and the assumption ∆x σb .The difference between the absorption fraction of two consecutive cells at x and

86


x+ ∆x based on Eq. (6.19), gives the remaining fraction of absorbed energy,

∆ηA(x, t) =ηA(z(x, t) cos(θb(x, t)), θb(x, t))−ηA(z(x+ ∆x, t) cos(θb(x, t)), θb(x, t)). (6.23)

Following the work of Klassen et al. [2014a] the depth in vertical direction hasto be adapted by the beam incidence angle θb. Eq. (6.23) and Eq. (6.21) arecombined in,

E1,i(x, t) = ωi∆ηA(x, t)Eb(x, t)

ϕ, (6.24)

where E1,i denotes the source term in Eq. (4.38).

6.3.3 Different EBM Absorption Models

This subsection two different absorption models are compared and examinedand is mainly based of the common work published in Markl et al. [2013]. Thefirst is an exponential absorption model for an electron beam gun with an ac-celeration voltage of 60 kV and the second is a constant absorption model fora 120 kV electron beam. The lower voltage is typically used in EBM machinesand the higher voltage is a current research topic. In Figure 6.7 the unfilled

0 10 20 30 40 50 600

2

4

6

8

10

penetration depth z in [µm]

abso

rpti

onco

effici

ent

[%] U = 60 kV

U = 60 kV, approx.U = 120 kVU = 120 kV, approx.

(a) lattice cell length c = 1µm

0 10 20 30 40 50 600

10

20

30

40


abso

rpti

onco

effici

ent

[%] U = 60 kV

U = 60 kV, approx.U = 120 kVU = 120 kV, approx.

(b) lattice cell length c = 5µm

Figure 6.7: Relation between penetration depth and absorption coefficient for60 kV and 120 kV (cf. Kanaya and Okayama [1972]) and suitableapproximations (graphs taken from Markl et al. [2013]).

markers show the relation of absorption coefficients and penetration depths(cf. Kanaya and Okayama [1972]) for the two acceleration voltages. The filledmarkers visualize the approximated value functions where the 60 kV is approxi-mated by a monotonically decreasing exponential function. This approximationtakes into account that more energy is consumed by cells that are closer to theelectron beam gun. The second absorption type for a 120 kV electron beam is

87


approximated by a constant function. Thus, each cell in z–direction absorbsthe same amount of energy until the complete energy is used up. The constantabsorption type has a higher penetration depth. The artificial absorption pa-rameter λ is deduced by the maximum penetration depth that is determined bythe minimization of the approximation error. Figure 6.7a shows that the choiceof approximations for a cell size of 1 µm are inaccurate but for a cell size of 5 µm(cf. Figure 6.7b) the approximations fit better. Since in the EBM simulationsof Chapter 9 use just cell sizes of 5 µm the approximations for the absorptioncoefficients are sufficiently accurate. It is important to mention that they donot differ in the amount of energy which is absorbed, but rather in the depth.In both cases there is more energy deported into deeper cells. Furthermore,it has to be noted that there should be a relation between absorption coeffi-cient and the fill level ϕ of the lattice cell. Completely filled cells should absorbmore energy than less filled cells. In the following the exponential and constantabsorption type are examined as well as the fill level relation.

Exponential electron beam absorption is used to approximate the absorp-tion of the electron beam Eb(x, y) for acceleration voltages at 60 kV by the ex-ponential law. In order to model the absorption on the lattice grid the spatialdimensions are discretized. For the x–y–plane of the powder bed surface this isdone in Eb of Eq. (4.38) and the discretization of the z direction leads to,

Ea(xi, yj , zk) = Eb(xi, yj) ·(

1− e−λc)· e−λzk , (6.25)

where Ea denotes the beam energy absorbed in one lattice cell and λ the ma-terial absorption parameter. Eq. (6.25) assumes that the energy of the electronbeam is completely absorbed at the cell stack (xi, yj) where the cells are com-pletely filled with liquid cells. However, in the simulation of the EBM processthis assumption cannot always be fulfilled since the powder layers consist ofpartly filled interface cells and also gas cells where ϕ = 0. Thus, it is necessarythat the computation of the absorption does not depend on any global relationbut can be computed cell–by–cell. Therefore, Eq. (6.25) is given in a recur-sively by using the auxiliary function ψ (xi, yj , zl) representing the exponentialfunction,

Ea (xi, yj , zk) = Eb (xi, yj) · ψ (xi, yj , zk) , (6.26)

where

ψ (xi, yj , zk) =(

1− e−λc)(

1−k−1∑l=0

ψ (xi, yj , zl)

). (6.27)

By induction of Eq. (6.25), Eq. (6.26) and Eq. (6.27) can be proved. In a next stepby using the volume–of–fluid method the fraction of absorbed energy picked by

88


each cell is determined and ψ is replaced by,

χ (xi, yj , zk) =(

1− e−λc)(

1−k−1∑l=0

χ (xi, yj , zl)ϕ (xi, yj , zl)

). (6.28)

Constant electron beam absorption approximates beam energy absorptionfor acceleration voltages of 120 kV. The constant function is discretized spa-tially,

Ea (xi, yj , zk) =

λEb (xi, yj) · c k < n

λEb (xi, yj) · (R− zk) k = n

0 k > n

, λ = 1/R, n = bR/cc, (6.29)

with the material absorption parameter λ and maximum penetration depth R.For constant absorption approximation a cell–by–cell computation is requiredusing the auxiliary function χ based on the volume–of–fluid approach,

Ea (xi, yj , zk) = Eb (xi, yj)χ (xi, yj , zk) , (6.30)

χ (xi, yj , zk) = min

(λc, 1−

k−1∑l=0

χ(xi, yj , zl)ϕ (xi, yj , zl)

). (6.31)

0 10 20 30 40 50 600

10

20

30

40


abso

rpti

onco

effici

ent

[%] ϕ = 1.0

ϕ = 0.9ϕ = 0.8

(a) acceleration voltage 60 kV

0 10 20 30 40 50 600

2

4

6

8

10

12


abso

rpti

onco

effici

ent

[%] ϕ = 1.0

ϕ = 0.9ϕ = 0.8

(b) acceleration voltage 120 kV

Figure 6.8: Absorption behavior for varying fill levels (taken from Markl et al.[2013]).

Fill level effects have to be examined since they influence the absorption be-havior. Figure 6.8 visualizes the absorption behavior for both absorption typesfor the first twelve lattice cells having a cell size of 5 µm. For both absorption

89


types the graphs for a fill level ϕ = 1 match with the approximations in Fig-ure 6.7. If ϕ < 1 the absorption behavior changes as described in the following.For the exponential absorption type the first cells absorb less energy but whenthe third cell is reached the fill level effect is negligible since these cells absorbmore energy than the former cells. For the constant absorption an analoguebehavior is observed. The maximum penetration depth is indirect proportionalto the fill level ϕ. The parallel version of these absorption types is discussed inSection 8.1.2.

6.4 Summary

In this chapter the requirements and properties of the electron beam gun arediscussed. The electron beam is defined by the acceleration voltage, the beamcurrent, and the beam width. Furthermore, the second essential part of theEBM process – the metal powder particles – are modeled. The particle sizedistribution is examined on the basis of real bulk powder data measured bythe light scattering method. For the EBM model the powder size distributionis modeled by an inverse Gaussian distribution function. Via the cumulativedistribution function and the quantile function the random parameters for thenumerical powder particles can be computed. In a third part the absorptionof energy by the electron beam is modeled. The absorption couples the beampower intensity into the LBM by the energy source term. It has to be notedthat the absorption is only considered in one spatial direction. Two differentabsorption types – namely exponential and constant – are derived in order tomodel machines with an acceleration voltage of 60 kV and 120 kV. Since thefill level influences the absorption behavior the absorption models have to becomputed cell–by–cell. Serial absorptions are derived. Their parallel version isdeduced in Chapter 8.

Chapter 6 closes the part of modeling and discretizing the EBM process. Inthe next steps the software frameworks are introduced (cf. Chapter 7) that areused to implement the EBM specific algorithms, and the corresponding parallelimplementation is explained in Chapter 8.

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Chapter 7: The Software Frameworks WALBERLA & pe

7 The Software Frameworks WALBERLA& pe

Everyone is trying toaccomplish something big,not realizing that life is madeup of little things.

(Frank Clark (1860 – 1936))

The transfer of numerical algorithms into software makes the partial differen-tial equations alive and enables a better understanding of an application bysimulations in the next steps. The huge growth and immense development ofcompute power during the last decades favored the research field of computa-tional fluid dynamics (CFD) and its corresponding software. There exist severalCFD software frameworks for solving Navier–Stokes equations based on finiteelement methods, e.g., NASTRAN (cf. Nastran [2016]) which is one of the old-est, available finite element methods software, and ANSYS (ANalysis SYStem,detailed information at ANSYS [2016]) which is a multi-purpose program, whereproblems in structural mechanics, fluid dynamics, acoustics and electromag-netism can be solved. Both commercial frameworks have been evolved overyears and have been successfully used in various applications. There is alsoa finite volume based software called FLUENT available by ANSYS (see FLU-ENT [2016]) and Star-CCM+ of the company cd-adapco (cf. Star-CCM+ [2015]).Most of these software frameworks utilize also the possibility of parallel com-puting to exploit the strength of modern supercomputers. This parallelizationof software permits simulations of larger computational domains or a finer gridresolution. In this way, simulations of turbulent flows in a high Reynolds num-ber regime are possible, or the corresponding numerical results can be achievedin a shorter time.

The straightforward parallelization of the LBM, due to its explicit form andlocality during the stream step (cf. Section 4.1.2) gives the LBM a competi-tive edge compared to classical numerical discretization methods like FEM andFVM. These properties pioneer the formation of several commercial and opensource LBM based software frameworks. Palabos (see Palabos [2016]) is anopen–source framework for CFD simulations with a LBM kernel written in C++.It covers related physics, basic fluid models, boundary conditions, particles,pre- and post-processing and check-pointing. OpenLB (see OpenLB [2015]) is

91


another open source LBM framework written in C++. This software is modularand enables high performance computing. On the other side, PowerFLOW is anexample for a commercial software package for solving fluid dynamics includ-ing turbulence modeling (cf. PowerFLOW [2015]). XFlow is another commercialLBM solver (detailed information can be found at XFlow [2015]).

The simulations of the EBM process is all done by the WALBERLA frame-work. Thus, in this chapter the structure and composition of WALBERLA areexplained. The name WALBERLA stands for Widely Applicable Lattice Boltzmannsolver from Erlangen and is developed at the Chair for System Simulation at theFriedrich–Alexander university in Erlangen (cf. waLBerla [2016]). Further, thepe – The physics engine is described, another software framework developed atthe Chair for System Simulation handling the simulation of rigid bodies. Bothframeworks can be coupled and this coupling provides an successful fluid–structure interaction which is employed for the simulation of the EBM process.

7.1 The WALBERLA Framework

The WALBERLA framework is a CFD solver (cf. Gotz et al. [2010b], Feichtingeret al. [2011], Kostler and Rude [2013], Godenschwager et al. [2013]), based onthe lattice Boltzmann method. Four phd students started this software projectin 2006. Almost one decade later WALBERLA is not only able to handle LBM sim-ulations but is also suitable for all kinds of numeric algorithms working withuniform domain decompositions, e.g., it has embedded solvers for linear equa-tions systems and also phase field methods as shown in Hotzer et al. [2015].

7.1.1 Software Quality in WALBERLA

As other successful software frameworks WALBERLA has to maintain compre-hensible quality aspects. The main principles of software quality are usabil-ity, reliability, portability, maintainability, and efficiency (following the work ofLakos [1996]). Feichtinger [2012] shows that WALBERLA fulfills these guide-lines. Usability is provided by guidelines and tutorials for new users to getin contact with the software structure of WALBERLA. Furthermore, input filesfor all the parameters of an application facilitates the handling with WALBERLA

for end-users who are interested in performing parameter studies or solving aspecific problem and do not want to extend or change the software. In thisway, end-users can handle applications of WALBERLA as a black box. On theother side, users who want to extend WALBERLA with more functionality canmake use of the underlying concepts. This is especially facilitated by the strictindependence of the individual modules.

Further WALBERLA ensures reliability by providing an automated build andtest system. After the extension of new functions in WALBERLA and their committo the git repository (see Git [2016]), specific tests are executed in order toensure the correctness of the overall framework. The search of errors is also

92


possible by a DEBUG mode which can be activated by a compilation switch.Moreover, portability of WALBERLA is approved from the level of laptops withdifferent operating systems – Windows as well as Linux is supported in the samemanner – and compilers as well as up to the level of supercomputers. It hasbeen ported to various current state–of–the art clusters, like JUGENE in Julich(Germany), Tsubame2 in Tokyo (Japan), or SuperMuc in Munich (Germany).Additionally, WALBERLA fulfills the aspect of maintainability since it minimizesphysical dependencies. Abstraction makes redesign or augmentation of newfunctions possible in an ease way. In Figure 7.1 the base structure of WALBERLA

is visualized. It provides a huge class of different modules that are optimizedand tested to guarantee fast and correct behavior. Examples of these modulesare the lbm package which is responsible for the stream–collide step of the LBM-algorithm, the communication package which organizes the MPI and OpenMPparallelization and also the vtk package that handles the visualization outputof the simulations for ParaView.

ApplicationsModules and core in WALBERLA

blockforest boundary communication

domain decomposition field gather

geometry gui core lbm pde

pe coupling post processing simd

python coupling stencil timeloop vtk

EBM

Protein Foams

Free Surfaces

Self Propelled Particles

Electrokinetic Flows

Continuous Casting

. . .

Figure 7.1: Structure of WALBERLA.

7.1.2 Applications in WALBERLA

On the right side of Figure 7.1 a few current applications are shown which useWALBERLA as a software framework. Besides the EBM application, which isthe main focus of this thesis, there exist an application modeling protein foamsfor the food industry. In this project the stability of protein foams is exam-ined via simulations. Detailed information of these results is found in Anderlet al. [2014a], Anderl et al. [2014b], and in Anderl [2016]. This application uses

93


the free surface application as a basis as well as the EBM application. Selfpropelled particles is another application to model and simulate low Reynoldsnumber swimmers. Pickl et al. [2012] describes the used model and Pickl et al.[2013] explains the parallelization of this model in WALBERLA. Another applica-tion is the simulation of electrokinetic flows with WALBERLA, which is describedin Bartuschat and Rude [2015] and in Bartuschat [2016]. In this application,charged particles in fluids under the influence of electric fields are simulated.These simulations can be used for electrostatic filters to remove particles fromnon-conductive liquids like oil and electrostatic precipitators for filtering pollu-tants from gases.

Besides these applications and the applications listed in Figure 7.1 there ex-ist numerous applications which are not explained in this thesis. However, therange of applications should persuade the reader of the wide usability and ca-pability of WALBERLA as a simulation framework for general problems in fluiddynamics. All this justifies why the author of this thesis has chosen WALBERLA

for simulating the EBM process. In the following the functionality and paral-lelization concepts of WALBERLA are explained.

7.1.3 Domain Decomposition and Communication Concepts inWALBERLA

WALBERLA evaluates the parallelized LBM in four steps. First of all, it evalu-ates the collision operator whereby the user can freely choose between threedifferent collision models, namely SRT, TRT or MRT. Information about the dif-ferent LBM collision models is given in Chapter 4 and Chapter 10. The collideoperation is cell-local and therefore, parallel treatment is not required. In asecond step all cell values which are on the process boundary are exchangedwith neighboring processes by the use of so-called ghost layers. Afterward, theperformance of the boundary treatment follows, i.e., the values in the boundarycells are disposed for the stream step, before in the fourth step the stream stepis executed, which is a nearest–neighbor operation.

Figure 7.2 shows the concept of domain decomposition in WALBERLA, i.e.,how the simulation domain is decomposed into blocks (cf. Godenschwager et al.[2013], Schornbaum and Rude [2016]). These blocks are data structures, i.e.,each block can store arbitrary C++ data. One block is always distributed toexactly one process and the blocks are the smallest quantity of workload. Sub-sequently, load balancing is responsible for the distribution of all blocks toall available processes. Although grid refinement is possible in walberla (de-tailed information about grid refinement can be found in Schornbaum and Rude[2016]), it will not be explained here because it is not used for the EBM simu-lations of this thesis. The simulation domain is split into uniform blocks andeach block stores a Cartesian grid related to its part of the simulation space(cf. Figure 7.2).

The MPI communication concepts is based on ghost layers of the grid on eachblock in order to synchronize neighboring blocks in parallel simulations. The

94


undistributedsimulation domain

uniformdomain decomposition

block

Figure 7.2: Uniform domain decomposition.

communication itself consists of two parts. First, all data is packed and sent tothe buffer and in a second step, this data is unpacked from the buffer in orderto exchange data between processes. Figure 7.3 visualizes this concept. The

copy

copy

ghost layer

Figure 7.3: Ghost layer concept for parallelization in WALBERLA.

grid assignment of each block is extended by at least one additional ghost layerin order to synchronize data. Figure 7.3 shows that during communication PDFvalues which are stored in the outermost inner cells (cf. red PDFs on the lefthand side) are copied to the corresponding ghost layer cells of the neighboringblocks (red and blue arrows in Figure 7.3). This synchronization process has tobe executed for all neighbors of a block (all blocks which have a common face,corner or edge). Only one message is exchanged between two processes andpacking and unpacking is organized in parallel.

The original modular design of the communication layer enables the imple-mentation of various communication schemes depending on the requirementsof the application. This sophisticated communication scheme and the modu-

95


larity of the WALBERLA framework facilitates high performance computing andfully distributed data structures make the scalability to hundreds of thousandsof processes and beyond possible as it is shown in Godenschwager et al. [2013].

7.2 pe -The Physics Engine for Rigid Body Simulations

The dynamics of fluid, which corresponds in the EBM process to the melt pool,can be handled by WALBERLA. However, the numerical treatment of the metalpowder particles is missing. We assume that the metal powder particles actas non–deformable rigid bodies and therefore algorithms are required, whichare responsible for the collision and contact detection. Serendipitously, the pe–physics engine is another in–house software framework of the Chair of SystemSimulation. It is mainly developed and implemented by Iglberger and for aprofound interest in the functionality and details of the implementation thework of Iglberger [2010] is recommended.

The pe-physics engine is developed for all kinds of rigid body simulations rang-ing from real–time requirements to large scale simulations with billions of in-teracting non–deformable particles. Other state–of-the–art frameworks fulfillreal–time requirements but are not designed for massively parallel simulationssince they do not involve necessary data structure and concepts for distributedmemory simulations. However, the pe provides on the one side a specializedinfrastructure for parallelism Iglberger and Rude [2009] as well as suitabilityfor all kinds of rigid body methods by considering high physical accuracy andvirtual reality environments. Furthermore, the pe in combination with LB flowscan simulate the behavior of rigid bodies in a flow Iglberger et al. [2008] as, e.g.,for simulating fluidization or sedimentation processes.

There exist several numerical methods for simulating granular media. Ingeneral, two categories can be classified: methods based on point masses andmethods with fully resolved particles. Smooth Particle Hydrodynamics (SPH),Molecular Dynamics (MD) or Direct Simulation Monte Carlo (DSMC) are exam-ples for the former category, the Discrete Element Method (DEM) or Rigid BodyDynamics (RBD) (Anitescu [2006]) are examples for the latter one. pe uses onlymethods with fully resolved particles because of a higher degree of reality andthe ability to simulate particles of arbitrary shape. RBD considers the discreteparticles as perfectly rigid and allows no deformations. The forces acting on theparticles are calculated by a set of motion constraints in order to avoid penetra-tions. The numerical solution of these constraints can be distinguished into twocategories. Firstly, methods based on the resolution of a linear complementaryproblem (LCP) and non–linear complementary problems which has to be set upglobally for all particles (find detailed information in Anitescu and Potra [1997]).LCP based methods are very accurate but suffer from huge computational effortbecause of the global structure of the LCP which is difficult to parallelize. Sec-ondly, there exists also a parallel solver for RBD, the Fast Frictional Dynamics(FFD) algorithm developed by Kaufman et al. [2005]. The FFD solver treats col-

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lisions of particles locally, concentrating only on particles which get in contact.This locality allows efficient parallelization, results in linear complexity and isappropriate for large scale simulations. The parallelization of FFD in the pe isdescribed in Iglberger and Rude [2009]. Since the DEM solver is used for thesimulation of the EBM process it is explained briefly in the following section.

7.2.1 DEM Solver in the pe

In contrast to RBD, the DEM assumes rigid bodies with deformable contacts.The DEM was introduced as Distinct Element Method by Cundall [1971] forproblems in rock mechanics and described as Discrete Element Method by Cun-dall and Strack [1979]. This approach does not impose any constraints on thedisplacements or rotations but allows overlaps between the rigid bodies. Theseoverlaps can be seen as virtual deformations of the colliding particles by theuse of a spring-dashpot system in between. Figure 7.4 visualizes this mainconcept of the DEM solver. Fictitious elastic materials are introduced in orderto develop repulsive forces to resolve the required collisions of particles. Heene[2010] describes the parallel implementation of the DEM solver in the pe.

r1p1

r2

Ft

−Ft

Fn −Fnn

overlap

ω1

ω2

a)

γnγt

kn

b)

Figure 7.4: a) Contact detection of two spheres with center positions p1 and p2,radii r1 and r2, angular velocities ω1, ω2.b) Spring–dashpot system between two bodies where kn is the springstiffness, γn and γt denote the damping coefficients in normal andtangential directions. Penetrations are dealt in normal directionsand friction is determined in tangential direction.

The DEM solver itself consists of four steps. Firstly, the initialization of theparticles, then the detection of contacts between particles, the calculation offorces which act on the rigid bodies and finally the time integration. The lastthree form the main loop of the DEM solver. In the contact detection each pairof particles in contact is identified. When the location of the collision is known

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the forces acting on these colliding particles can be calculated. An intuitiveapproach for the detection is to check each pair of particles. However, thisis infeasible for a huge amount of particles and is also not necessary sincethe probability that particles far away from each other are in contact is goingto zero. Hence, the pe provides a more efficient contact detection and differsbetween fine and coarse collision detections. During the coarse detection phaseit is decided which particles are close enough to have a contact (this can berealized by bounding volumes together with hierarchical hash grids). Duringthe fine collision detection these possible contacts are further evaluated forcreating the actual contacts. A contact between two particles i and j, which areallowed to have a small overlap, is identified by,

ξij = ri + rj − |pi − pj | > 0. (7.1)

ξij is not only an overlap but rather a compression or deformation of the twocolliding particles. It is important that these overlaps or deformations have tobe small and reversible when particles dissolve in order to ensure the originalshape of the particles. The normal vector pointing from the center of sphere ito the center of the colliding sphere j is computed by,

nij =pj − pi|pj − pi|

, (7.2)

and can be used for computing the velocity of the compression to,

∂ξij∂t

= (vi − vj) · nij . (7.3)

The position of the contact itself is computed by,

zij = pi +

(ri −

ξij2

)nij , (7.4)

which lies in the middle of the contact area. If the contact region and overlapare identified, the forces F ij and torques τij acting on the colliding particles arecalculated, where the forces consists of normal part F n

ij and tangential part F tij,

F ij =

F nij + F t

ij if ξij > 0,

0 else,(7.5)

and,

τ ij =

(zij − cα)xF t

ij if ξij > 0,

0 else.(7.6)

Tangential and normal part of the acting force is decoupled and there existseveral methods for their computation corresponding to the properties of thesimulated material. For an overview of normal force models and their valida-

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tion with experimental results refer to the work of Kruggel-Emden et al. [2007].Most common models are the linear viscoelastic and the nonlinear viscoelasticmodels which can be interpreted as spring–dashpot system. Therefore, the twofree parameters of the spring–dashpot, namely kn as the stiffness of a linearspring and γn as the velocity damper can be set (cf. Figure 7.4,b) individually.The tangential part of the contact force is interpreted as roughness of the par-ticles affected by collision and can be seen as friction. The tangential contactforce is limited by the Coulomb’s friction law to,

|F t| ≤ µ|F n|, (7.7)

where µ denotes the material dependent friction parameter. The motion of theparticles is governed by Newton’s second law of motion (see Eq. (3.21)). Thisleads to a coupled system of second order ordinary differential equations whichhave to be solved numerically. Another overview of different time integrationschemes is given in Kruggel-Emden et al. [2008]. Contact detection, force cal-culation and time integration form the main loop of the DEM solver.

7.2.2 Parallelization Concepts of the pe

In the following the main concepts of the parallelization of the pe and the DEMsolver are described. The most decisive task during parallelization is to avoidglobal data because the software should scale for the simulation of an arbi-trary large number of particles on an arbitrary number of processor cores. Inthe most application of WALBERLA the simulation domain is divided into sub-domains and each process controls only a part of the overall domain. Com-munication takes only place between neighboring processes for sending andreceiving information of entering or leaving particles over the process border.The communication within the pe used in this thesis is organized by MPI fordistributed memory systems. This nearest–neighbor communication guaran-tees a minimum number of MPI communications. The fact that the pe resolvesthe particles in a fully resolved volume representation makes the parallelizationmore difficult compared to algorithms which consider only point masses. It canhappen that a particle resides on two different subdomains at the same timeand it has to be clarified which processor is responsible for this particle. The pesolves this problem by differing between local and remote bodies. If the centerof mass lies on the specific subdomain, the corresponding particle is definedas a local body. The neighboring processes, on which the particle also resides,holds a copy of this particle and this is a so–called remote body. The parallelversion of the main loop of the DEM starts also with the contact detection, tak-ing into account all remote and local bodies of the corresponding subdomain.After the detection of all contacts, all forces and torques acting on the particlesare stored in the local and remote bodies. At this point, the first MPI com-munication has to take place, the necessary synchronization of the forces andtorques between all neighboring processes has to be made. Therefore, the MPI

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send ensures that for remote bodies the corresponding forces and torques aresent to that processes and a MPI receive is used that these information is re-ceived by the processor who holds the local copy. The updates of positions andvelocities have to be done next which requires the second MPI communication.MPI send and receive ensure that the velocities and positions of each localcopy are sent to the corresponding remote body such that the local copies canbe updated. Altogether, the parallel DEM solver needs two MPI communicationsteps.

7.2.3 pe Solver for EBM Process

The EBM simulations of this thesis use the parallel DEM solver due to severalreasons. Firstly, solver based on LCP can predict the contact forces betweenthe rigid bodies with a high accuracy but result in high computational costsbecause of its global parallelization structure, which is a very difficult task.However, for the sake of completeness it has to be mentioned, that a paral-lel version of LCP is implemented in the pe by now. Preclik and Rude [2015]presents details of its implementation and its scaling results on state-of-the-artsupercomputers.

The pe also provides a parallel version of the FFD solver (PFFD). The advan-tage of the PFFD is its ability of fast real time collision algorithms with a strictlylocal collision treatment. Therefore, PFFD can scale linearly with the numberof contacts. Because there is no free lunch PFFD suffers from a reduced degreeof accuracy. Furthermore the PFFD requires four MPI communication steps,i.e., 50 % more communication steps compared to the parallel DEM solver. Al-though, communication is one of the most time consuming parts, the questionwhich one of the two solvers is faster cannot be easily answered. The DEMrequires only two communication steps but the size of the time steps does notonly need to be small for stability reasons but also for detecting the collisionsas very small overlaps while the FFD allows higher sizes for timesteps for pro-ducing reasonable results. Heene [2010] showed that the parallel DEM requiresonly 57 % of the computation time compared to the PFFD. These reasons shouldjustify the use of the parallel DEM solver for the generation of metal powderparticles in EBM simulations.

Before explaining how EBM specific models are integrated in WALBERLA andpe, the explicit coupling algorithm of the two frameworks is explained in thesubsequent section.

7.3 The General Coupling of WALBERLA – pe

For the simulation of the EBM process WALBERLA and pe are necessary. There-fore, the data handling the fluid dynamics of the melting pool are stored withinthe LB solver of WALBERLA and has to be communicated to the pe solver whichis responsible for the contacts and collisions between the metal powder parti-

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cles. This essential communication occurs in the explicit four–way coupling ofboth frameworks. This was was firstly introduced by Gotz et al. [2010a] andtested for a large number of particles in Gotz et al. [2010b]. The coupling wasalso already applied for bacterial swimmers in Pickl et al. [2012]. Figure 7.5visualizes this four–way coupling algorithm.

rigid bodies act as obstacles

fluid results in external forces

Figure 7.5: General four–way coupling of WALBERLA and pe.

In this coupling algorithm, rigid bodies of the pe are represented as movingboundaries in the fluid flow and the flow itself corresponds to hydrodynamicforces acting on the rigid bodies. The coupling consists of four algorithmicsteps that are explained in the following. First of all, all rigid particles of thepe which uses a Lagrangian description are mapped onto the Eulerian LB grid.The objects or particles are represented as flag fields in WALBERLA. These flagfields result in a staircase approximation of the particle. On the other hand,each lattice node with a cell center inside the particle is marked as an obstaclecell. The interface between the obstacle and the fluid is treated by a movingboundary condition,

fi (xb, t) = fi (x, t) + 6ωiρwei · vw, (7.8)

where fi denotes the post–collision value, index w denotes values from the wall.The wall value ρw is obtained by extrapolation of the adjacent fluid nodes orby just setting ρw = ρ(xF ) since an incompressible fluid is assumed. Eq. (7.8)is given in Yu et al. [2003] and can be interpreted as a variation of the well–known no–slip boundary condition for moving boundaries (cf. Section 4.2).This boundary condition is only first order accurate for velocity, but an im-proved higher order boundary condition for moving walls is given in Bouzidiet al. [2001]. By movement of the particles, the flags identifier change in two

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possible ways. A former fluid cell can convert into a object cell. The reversecase is more difficult since the missing pdf values have to be reconstructed ifa boundary cell converts into a fluid cell. There exist several methods for set-ting the pdf values and the most general way is to set the equilibrium at thesecells. Therefore, the velocity of the moving particles and the average densityof the surrounding fluid cells is taken to compute the equilibrium like it is de-scribed in Iglberger et al. [2008]. Caiazzo [2008], Lallemand and Luo [2003b]and Lorenz et al. [2009] describe more accurate and sophisticated methods forsetting the pdf values. In order to ensure an efficient mapping and to avoidadditional communication effort, the rigid bodies have to reside on the sameprocessors as in the LBM block structure.

The second step of the explicit coupling algorithm executes the LBM stepsof Eq. (4.37) and Eq. (4.38) and guarantees that the fluid flow acts throughhydrodynamic forces on the rigid objects. Pdfs of fluid cells stream to theirneighboring cells. In the case, they enter a cell occupied by a moving particle,they are reversed and cause a momentum exchange between the fluid and theparticle. The momentum exchange method is described in Ladd [1994a,b] andYu et al. [2003]. The total force F which results from this momentum exchange,is then computed to,

F =∑xb

18∑i=0

ei[2fi(xf , t) + 6ωiρw(xf , t)ei · vw(xf + ei, t)

] ∆x

∆t︸︷︷︸=1

. (7.9)

A comparison of different momentum exchange methods is given in Lorenz et al.[2009]. In the last step of the coupling algorithm, the position and velocitiesof particles in the pe have to be updated due to the force acting on them inWALBERLA. Afterward these new positions have to be mapped again onto theEulerian LB grid in the next step.

In order to ensure the correctness of this described coupling, the work ofBinder et al. [2006] is recommended which uses the coupling for simulating thedrag force on agglomerated particles, and further, the work of Iglberger et al.[2008] which validates the coupling algorithm itself.

7.4 Summary

In this chapter the two software frameworks WALBERLA and pe of the Chair forSystem Simulation are introduced and explained. WALBERLA is a LBM basedfluid flow solver and fulfills the quality aspects usability, reliability, maintain-ability, and efficiency. The sophisticated communication concept and the do-main decomposition approach form the basis for the scaling of WALBERLA whichruns on laptops as well as on modern state–of–art supercomputers. A lot ofdifferent applications show its broad bandwidth. WALBERLA is used for thesimulations of the fluid dynamics in the melting pool of the EBM process.

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In addition the pe handles rigid body dynamics, i.e., contacts and collisionsof rigid bodies. The pe and its various solving methods are explained, especiallythe DEM solver. The DEM solver is used for the simulation of the metal powderparticles in the EBM process. Since both frameworks are needed for the sim-ulation a coupling of WALBERLA and pe is required. Thus the general two–waycoupling of both is described.

Chapter 8 goes into the details of the integration and implementation of theEBM model in WALBERLA and describes a simplified coupling algorithm. It willalso show the performance of the EBM model in WALBERLA by scaling experi-ments on the state-of–the–art supercomputers LIMA and SUPERMUC.

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Chapter 8: EBM in WALBERLA and its Performance

8 EBM in WALBERLA and itsPerformance

If you have build castles inthe air, your work need notbe lost, that is where theyshould be.Now put foundations underthem

(Henry David Thoreau(1817 – 1862) )

The fluid solver WALBERLA and the physics engine pe, handling the dynamics offully resolved rigid bodies, are selected for the simulation of the additive manu-facturing process EBM. Both are developed at the Chair for System Simulationand the parallelization concepts of both are described in Chapter 7. In orderto conclude the implementation part of the simulation pipeline in Figure 1.1in this chapter the details of the integration of the implementation of the EBMmodel in the WALBERLA framework and a simplified coupling with the pe is suf-ficient for the simulation of the EBM process. Additionally, the focus is placedon the generation of realistic powder particles with the pe.

Before executing simulations scalability is defined in general. Hereby strongand weak scaling experiments of the EBM application in the WALBERLA frame-work are run on the state–of–the–art supercomputers SUPERMUC and LIMA.Weak scaling results show which computational domain size can be solved andinforms how fast the EBM problem can be simulated. A brief summary of theEBM implementation in WALBERLA concludes this chapter. Chapter 7 and thischapter prepare the ground for the subsequent simulations in Chapter 9.

8.1 Integration of the EBM Model in WALBERLA and pe

Due to the software aspects of WALBERLA given in Section 7.1.1 especially theaspect of maintainability enables the straightforward integration of the EBMmodel. Figure 7.1 visualizes the structure of WALBERLA and shows the embed-ding of the application EBM which accesses to particular modules for examplelike lbm, stencil, communication, or vtk. The standard LBM kernel lbm ofWALBERLA is used for the melting pool simulation in the EBM application. Itexecutes the classical LBM regarding the flow dynamics (see Section 4.1.2). In

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order to enable the computation of the energy by the input of the electron beamthe existing LBM in WALBERLA is extended by the multi–distribution approachgiven in Section 4.1.4. This means that for every lattice cell another field of anadditional pdf set has to be stored and communicated. The stencil moduleprovides the foundations for the use of the D3Q19 stencil. For the simula-tion of the free surface between the liquid melting pool and the atmosphereof the vacuum chamber the free surface application is used which is alsoextended in order to fulfill the requirements of the thermal FSLBM describedin Section 5.3.3. After the simulation the vtk package facilitates the visualpostprocessing of the computational data.

In the following subsection the EBM specific extension of the WALBERLA

framework are derived with a special focus on the powder generation and theabsorption algorithms, which are responsible for transmitting the energy of theelectron beam into the powder bed.

8.1.1 Powder Generation for EBM by pe

The method for describing the realistic distribution of metal powder particles isdescribed in Chapter 6. How this method is embedded in the pe and how thesevalues are coupled into the WALBERLA framework is described in the following.

For the EBM process a first powder layer is generated by the pe (cf. Am-mer et al. [2014a]). The pe represents the particles as rigid bodies on a La-grangian grid and handles the detection of collisions between the rigid bodiescorresponding to the chosen solver. In the case of the EBM simulation this isthe DEM solver (see Section 7.2.1). The rigid bodies of the pe are treated asboundaries within the LBM scheme of WALBERLA. In the next step the incorpo-ration of energy by the electron beam liquifies the metal powder particles andthen WALBERLA computes the hydrodynamic quantities like velocity, density,and energy density. Furthermore it is responsible for the fluid node update andthe corresponding forces. After a solidification process the melting pool hard-ens and the process starts again. The surface of the last layer (consisting ofunmelted and melted solid particles) is approximated and then the next powderlayer is generated by the pe. This is a simplification of the general four–waycoupling described in Section 7.3 and is visualized in Figure 8.1. The powderbed generation algorithm couples the LBM solver in WALBERLA and the DEMsolver in the pe and converts the free surface properties into powder particle in-formation. The corresponding algorithms are described in detail in Markl [2015]which is the main reference for the following subsection. The coupling of bothmethods and frameworks consists of three steps. Firstly, the free surface of theprevious layer is approximated by fixed spheres generated by the DEM solver.This algorithm requires the maximum sizes of the computational domain andthe desired height of the powder layer confining the approximation region andreturns the maximum height of all approximated spheres. The application ofthis algorithm results in a closed grid of approximated spheres.

In the second step of the coupling new powder particles have to be initial-

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rigid bodiesact as obstacles

... is respsonsible for:• collision detection• collision response

creates new powder layer

... is responsible for:• update of fluid nodes• calculation of hydrodynamic forces• calculation of free surface

after solidification process

Figure 8.1: Simplified WALBERLA – pe coupling for the EBM application.

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ized according to the distribution explained in Chapter 6 and have to find theirfinal position above the approximated spheres of the first step. The final posi-tion is achieved by a free fall simulation of the particles with an initial velocityup to the approximated surface. During the initialization phase the horizontaloffsets and the number of particles are defined. In order to guarantee a com-pletely filled layer the estimated amount of necessary particles is doubled. Theparticles are arranged in a regular grid above the maximum height of the previ-ous layer and their diameters are initialized via a cumulative inverse Gaussiandistribution function described in Chapter 6 and Ammer et al. [2014a]. Theparticles are initialized with a negative, i.e., downward oriented velocity andrandom velocities in the other directions in order to improve the particle mix-ture during free fall. In the free fall simulation the downward velocity and theeffect of gravity bring the particles towards the approximated free surface ofthe previous powder layer. During the free fall simulation the detection of allcontacts of the approximated spheres, particles, and the boundaries is done.The resulting forces due to collisions are computed and integrated. The freefall simulation algorithm stops if the velocity of all particles reaches a thresholdor stops automatically ofter the execution of 50 000 time steps. In a last step

PARAMETER VALUES FOR TI–6AL–4V UNITS

Constant coefficient of restitution ε 1.0 · 10−4 –collision duration tc 0.5 · 10−3 sstiffness in normal direction k(n) 0.849 · 10−3 N

mdamping in normal direction γ(n) 8.257 · 10−6 kg

sdamping in tangential direction γt 3.783 · 10−6 kg

stime step tc

50 50 –

Table 8.1: Particle parameters for the DEM solver in the pe for the Ti–6Al–4Vpowder.

the information of the powder particles of the finished layer is converted ontothe Eulerian LBM grid in WALBERLA. Therefore the fill level, the cell type, andthe macroscopic quantities have to be modified depending on the center posi-tion of the particles. For this mapping all particles which belong to the currentpowder layer have to be identified. A necessary condition for belonging is if theparticle center position lies inside the layer and the particle top height does notexceed the layer for more than 10 µm or 20 % of its diameter. This conditionexpresses the behavior of the rake (cf. Figure 2.5) whose tines are bended byparticles exceeding the layer. All particles which are identified by this conditionare converted into the LBM grid of WALBERLA. This conversion is done by themodification of the fill level ϕ, i.e., for each cell which overlaps with the particlethe intersection volume is computed. Afterward, in order to have LBM man-ageable cells corresponding cell types and macroscopic quantities like density,momentum, and energy density are set to initial values of solid cells. This ini-

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tialization is justified and essential since the raking process and the preheatingof the EBM process are not simulated explicitly. Moreover the temperature dis-tribution of the preceding powder layers is overwritten by the preheating initialtemperature in all cells when the new powder layer is treated. Finally the celltypes are replaced for cells with ϕ > 0 by a solid interface cell depending on thepresence of gas cells in the local neighborhood.

The force model in the DEM solver (cf. Section 7.2.1) requires the siffnessin normal direction k(n), the damping parameter in normal γ(n) and tangentialγ(t) direction, and the dynamic friction coefficient µd. The parameters whichare necessary for the computation of the particles in the DEM solver for theTi–6Al–4V material are given in Table 8.1. Detailed information how they arecomputed is found in Markl [2015].

Figure 8.2: Powder particles generated by pe.

Figure 8.2 shows the particles generated by the pe framework via the inverseGaussian distribution during the free fall. How these particles absorb the en-ergy offered by the electron beam gun in order to be melted and build partslayer–by–layer is explained in the following.

8.1.2 Parallel Absorption Algorithms for EBM in WALBERLA

The following subsection on absorption algorithms is mainly based on Marklet al. [2013]. The absorption algorithms are responsible for the incorporationof energy from the electron beam into the metal powder bed. The mathematicalmodel of the absorption algorithm is given in Section 6.3. In this subsection theparallelization and implementation of this model is described.

In order to parallelize the absorption algorithms the global computational do-main is split in uniform blocks and those are distributed to different processesas visualized in Figure 8.3. This splitting follows the domain decomposition

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and communication concepts of the WALBERLA framework described in Sec-tion 7.1.3. Thus, in a streaming step of the LBM WALBERLA communicates pdfvalues to the directly neighboring cells (red arrows in Figure 8.3). However, forthe absorption of energy of the electron beam guns makes the iteration over thewhole domain in z–direction in one time step necessary. Generally, the solutionof Eq. (6.26), Eq. (6.28), Eq. (6.30), and Eq. (6.31) allows this. However, this pro-cedure results in a completely sequential computation of the absorption valuesfrom top to bottom. A sequential execution of the algorithm is unwanted be-cause it leads to long waiting times for other processes which is ineffective andallows only a smaller computational domain or makes longer simulation timesnecessary. An improved parallelization requires a reformulation of the algo-rithm. Following Markl et al. [2013] the main idea of the parallelization concept

Figure 8.3: Communication schemes for the absorption algorithms. Local com-munication for the streaming of pdf values (red). Top–to–bottomcommunication for beam absorption values (green) (taken fromMarkl et al. [2013]).

for the absorption algorithm is the splitting of the computation in a pre– andpost–compute step which is evaluated parallel on each block and a commu-nication step in between to exchange the required data. This communicationconcept (cf. Figure 8.3, green arrows) is a global top–to–bottom approach whereeach block sends its information to each lower block. Using this approach eachblock post–computes the absorption coefficient. The communication concept ofWALBERLA is extended by this top–to–bottom approach in the EBM application.

For the following explanations one specific position (xi, yj) is assumed tobe fixed and the top–to–bottom approach makes another index m for the z–direction necessary which denotes the corresponding block m and results fi-

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nally in z(k,m). The auxiliary function χ (cf. Eq. (6.28)) is split into χpre and χpostfor the computation of the pre– and the post–absorption values, respectively. InChapter 6 an exponential and constant absorption algorithm are derived in or-der to model different electron beam powers of 60 kV and 120 kV, respectively.The modifications in order to improve the parallelization lead to the followingequations for an exponential absorption algorithm with Eb denotes the beamenergy and Ea the absorbed energy,

Ea(z(k,m)) = Ebχpost(z(k,m)), (8.1)

χpost(z(k,m)) = χpre(z(k,m))

m−1∏n=0

(1−

∑l

χpre(z(l,n))ϕ(z(l,n))

), (8.2)

χpre(z(k,m)) =(

1− e−λc)(

1−k−1∑l=0

χpre(z(l,m)ϕ(z(l,m))

). (8.3)

The parallel computation starts at Eq. (8.3) that is computed on each block. Af-ter the communication of the second term of Eq. (8.2), Eq. (8.2) and Eq. (8.1) areevaluated in the post–computation step. Analogously the algorithm describingthe constant absorption are derived for the improved parallelization,

Ea(z(k,m)) = Ebχpost(z(k,m)) (8.4)

χpost = min

(χpre(z(k,m)),max

(0, 1−

m−1∑n=0

∑l

χpre(z(l,n))ϕ(z(l,n))

−k−1∑l=0

χpost(z(l,m))ϕ(z(l,m))

))(8.5)

χpre(z(k,m)) = min

(λc, 1−

k−1∑l=0

χpre(z(l,m))ϕ(z(l,m))

)(8.6)

Again Eq. (8.6) is computed on each block. Afterward the sum over the pre–computation values of the maximum function of Eq. (8.5) is communicated andEq. (8.5) and Eq. (8.4) are evaluated.

The parallelization concepts based on a top–to–bottom approach guaranteea fast parallel execution of exponential as well as constant absorption algo-rithm. This is essential for the fast execution of the overall EBM simulationwith WALBERLA and pe and avoids waiting times for the other processes.

8.1.3 Absorption of Energy by Particles in the Powder Bed

The generation of the powder particles by pe and the absorption of the energyof the electron beam incorporated by WALBERLA are coupled. Figure 8.4 showsa slice through a generated powder bed, whereby the colors represent the ab-sorption coefficient from a maximum value (red) to zero (blue) in fluid cells. This

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(a) χpre (b) χpost

Figure 8.4: Absorption coefficients in powder slice for 60 kV acceleration volt-age: the global domain is split into three blocks. The absorptionalgorithm computes the pre values (a), then values are commu-nicated, and correct post–values can be computed (b) (taken fromMarkl et al. [2013]).

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expresses the coupling of both frameworks. Gas cells and the lattice grid arenot shown. For this example the exponential algorithm is used with a maximumabsorption coefficient of 0.38. The left subfigure represents the absorption co-efficients in the pre–computation step of a domain split into three blocks. Theright subfigure shows the absorption coefficients after the communication andpost–computation step. The values at the top of the lower blocks are correctand the values show a monotonous decrease from top to bottom. In both fig-ures it is obvious that the gas cells do not influence the absorption behavior.Furthermore the absorption coefficients of interface cells are not affected bytheir fill levels since Ea is the absorbed energy per lattice cell volume. Thereforewe model the energy source as a volumetric force of the first cells

Φi (x, t) = ωiϕ (x, t)Ea (x, t) , (8.7)

where x is the lattice cell center with the corresponding amount of absorbedenergy Ea and Φi the source term in Eq. (4.39) which takes the fill level intoaccount.

In this section the embedding of the EBM model in the WALBERLA is explainedand the generation of the powder particles by the pe where the DEM solverresolves the contacts and collisions. Parallel absorption algorithms for expo-nential and constant absorption behavior are described using a top–to–bottomapproach in order to guarantee an efficient parallelization. The coupling of bothframeworks is tested by generating particles and incorporating energy and vi-sualized. In the following the implementation of the EBM process is examineddue to performance via scaling examinations.

8.2 Scaling Tests of the EBM Model

Before performance tests of the EBM implementation are executed parallel com-puting, scaling, and corresponding terms and concepts have to be defined andexplained. The book of Hager and Wellein [2010] is recommended for givinga comprehensive overview of the history of parallelization techniques and thecurrent state–of–the–art methods. Parallel computing is defined as solving anumerical problem by using a number of compute elements (cores) in a coop-erative way. The question arises why a numerical problem should be solved inparallel and not in serial? The answers are on the one side that one single coreis too slow to perform the solution for a complex numerical algorithm like it isrequired in applications for fluid dynamics. On the other side the computationof the solution may have high memory requirements and these cannot be ful-filled by the available main memory of a single core. These reasons make thedevelopment of parallel algorithms and implementations inevitable.

For algorithmic parallelization the numerical problem is split into several sub-tasks which communicate with each other in order to guarantee data exchange.The data exchange has to be enabled by the programmer depending on the cor-

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responding memory nature of the particular supercomputer. There exist twomain kinds of memory managements. Shared–memory computers have a com-mon, shared physical address space where the cores work on. Contrary tothis in distributed–memory computers each processor is connected to an ex-clusive local memory and no other processor has direct access to the addressspace. As a consequence, a network interface is required in order to ensurethe communication between different processors where messages can be sent.MPI (Message Passing Interface) is the most used (cf. Gropp et al. [1999] fordetailed information). The simultaneous use of many cores was only possibleby the steady increase of CPUs (central processing units) in modern large–scalesupercomputers.

So far, these are the ideas behind parallelization. However, the user is alwaysinterested in the scalability which indicates the efficiency of his or her paral-lel implementation. Scalability metrics measure how successful the tasks areparallelized and how the necessary communication scheme can influence theperformance. The user is interested in answering the question how much fastera given problem can be solved using more processes instead of one or also howmuch more work, i.e., a how much larger simulation domain can be solved.In the following the ideas behind strong and weak scaling are explained andcorresponding results for the EBM application in the WALBERLA framework fortwo state-of-the-art supercomputers are shown.

8.2.1 Metrics and Conventions

Assuming that all execution units, i.e., all N used cores, execute their workloadin exactly the same time and the overall time to solve the problem in serialis T , the parallel solution should take ideally, T/N and allows a speedup ofN . Unfortunately, this is the ideal case and cannot hold in reality. Not allprocesses might take the same time for execution because not all subtaskshave the same complexity and this can lead to load imbalances. Furthermore,some of the compute resources have to be shared, because they have to be usedby all processes and therefore all processes have to wait for these resources.Additionally, the parallel working processes have to communicate among eachother to transfer data and this communication time produces some overheadwhich is absent in the serial implementation. All these causes may limit thespeedup in the parallel case.

Following the definitions in Hager and Wellein [2010] the overall problem sizecan be denoted by,

T = s+ p = 1, (8.8)

where s stands for the serial part of the implementation and p for the perfectparallel part. Reasons for the existence of a nonzero serial part have alreadybe mentioned. Subtasks of the algorithm depend on further results and limitthe parallelization, bottlenecks in the resources which have to shared by allprocesses, and the inevitable communication time for data exchange. The serial

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solution time T ss of a overall problem is defined by,

T ss = s+ p = 1 = T. (8.9)

If instead of one process N processes are used the overall solution time is com-puted by,

T ps = s+p

N. (8.10)

This measurement uses a fixed problem size which is solved by N processes.Consequently, an increasing number of processors results in a decreasing localproblem size for each processor. Strong scaling experiments should lead to lin-ear scaling and the runtime is indirect proportional to the number of processors(t ∼ 1/N ) while the workload is constant (W const.). Strong scaling experimentsshow how fast a problem can be solved using more compute power. On theother side if the time to solution is not the primary target objective and the useris more interested in solving larger problem sizes weak scaling is the prevailingscaling experiment. In this case the problem size is scaled with the same powerof N and the total amount of work is given by,

T sw = s+ pNα (8.11)

where α is a constant, free parameter and s denotes again the serial fraction ofthe solution. The parallel runtime is,

T pw = s+ pNα−1. (8.12)

In the weak scaling experiment the local problem size for each process is fixed,and with an increasing number of processes the global, overall problem size isincreased. A linear scaling is expected, the runtime should stay constant in theideal case while the workload is direct proportional to the number of processes(W ∼ N ).

After these general definitions and explanations the performance of the EBMimplementation in WALBERLA and pe is examined using strong and weak scalingexperiments. The best unit of measurement of the LBM are MLUPS or MFLUPSwhich stands for million lattice cell updates per seconds/ million fluid latticecell (see Resch et al. [2006], Zeiser et al. [2008]).

8.2.2 Test Machines

The weak and strong scaling experiments of the EBM model are executed on theSuperMUC (SuperMUC [2015]) and the LiMa (LiMa [2015]) compute cluster. TheSuperMUC machine is located at the Leibniz-Rechenzentrum in Garching nearMunich and the first version was installed 2011. Next phases of SuperMUCwere installed 2012, 2013 and 2015. The scaling experiments of the EBM modelwere executed on thin nodes in 2013. SuperMUC itself is a combination of so–called thin nodes, fat nodes, and many core nodes. This hybrid composition

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should satisfy the various demands of different applications due to memory. Atits installation SuperMUC was on the 6th place on the Top 500 list (see TOP500 [2013]).

LiMa is located at the Regionales Rechenzentrum Erlangen (RRZE) in Erlan-gen. It was installed at the end of 2010 and was in November 2010 on rank130 in the Top 500 list of supercomputers (cf. TOP 500 [2013]). In contrast toSuperMUC LiMa has a uniform structure of equal nodes. A detailed overview ofthe characteristics of Lima and SuperMuc is presented in Table 8.2.

SUPERMUC(THIN NODES)

LIMA

Year of installation 2012 2010Total # of cores 147,456 500Total # of compute nodes 9216 6000# of physical cores per node 16 12# of cores per node using SMT/Hypertreading

32 24

Main memory per node [GByte] 32 24Memory bandwidth per node[GByte/s]

102.4 40

CPU generation/processor type Sandy Bridge-EP, Xeon E5-2680 8C

Xeon 5680Westmere

Total peak performance (com-plete system)[PFlops/s]

3.2 0.064

Interconnect system InfinibandFDR10

Infiniband

Table 8.2: Characteristics of Lima and SuperMUC compute cluster.

8.2.3 Scaling Results

In the following scaling experiments of the EBM model in WALBERLA with theuse of the pe framework for the powder generation are shown. Strong scalingexperiments as well as weak scaling are examined since due to the demands ofthe EBM project it was necessary to know how fast the solution can be producedwith a higher amount of processors as well as the fastest possible solution forone problem size.

The scaling set–up can be seen as a minimal working example supplied withall necessary EBM aspects, like the powder generation, the incorporation ofthe energy by the electron beam, the corresponding melting of the particlesand the computation of the melting pool dynamics by the LBM. Furthermore,the free surface boundary condition is calculated at the liquid–gas interface.

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The set–up cannot be used for building a realistic part but it is sufficient forthe examination of the performance of the implementation since it contains allmethods and aspects.

8.2.3.1 Weak Scaling

In the case of a weak scaling experiment the user is interested in how fast aspecific problem size can be simulated. Therefore, the workload per processoris fixed and the global problem size is increased with increasing number ofcores. The parameter values for the weak scaling experiments executed on LiMaand SuperMUC are given in Table 8.3. The experiments start with a minimum

PARAMETERS VALUES

# of timesteps 250dx 5.0 ·10−6 m# of cells in x-direction 64·(xP)# of cells in y-direction 64·(yP)# of cells in z-direction 48·(zP)# of processors in x-direction xP# of processors in x-direction yP# of processors in x-direction zP

Table 8.3: Parameters of the EBM setup for weak scaling experiments on LiMaand SuperMUC.

domain size and this scales with a increasing number of processors in x- andy-direction, i.e., xP and yP in Table 8.3 denote the number of processors usedin x- and y-direction, respectively. z–direction remains unmodified because48 lattice cells are just necessary to fulfill the requirements for the powdergeneration. Less cells do not enable the generation of a realistic powder layer.With this increase of the domain size the workload for each processor remainsconstant.

The following weak scaling experiments are executed on LiMa and Super-MUC compute cluster. All scaling results are visualized by graphs where thex–axis indicates the number of processes and the y–axis the execution or run-time of the experiment in seconds. For a detailed performance analysis thescaling behavior within one LiMa node, starting with one processor is showenin Figure 8.5a up to 24 processors. The execution time increases when usingtwo processes instead of one since communication time is required. Thus theruntime increases further slightly when using more processes within one node.Since the EBM model scales quite well within one compute node at the LiMacluster the performance can be examined when using more compute nodes.These results are shown in Figure 8.5b. For weak scaling results a constantruntime is expected when doubling the number of nodes. Figure 8.5b conveys

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20 21 22 23 24

number of cores

20

40

60

80

100

runt

ime

in[s

]

(a) Intra-node weak scaling on LiMa.

20 21 22 23 24

number of nodes

20

40

60

80

100

runt

ime

in[s

]

(b) Inter-node weak scaling on LiMa.

Figure 8.5: Weak scaling results of EBM model on Lima compute cluster.

this as the runtime increases a little bit as a consequence of the communica-tion time but not that much. Figure 8.5 shows that the EBM implementationin WALBERLA and pe allows the user an efficient simulation of an increasing,growing computational domain by using more cores.

20 21 22 23 24 25

number of cores

0

500

1000

1500

2000

2500

time

[s]

Walltime completeLBMthermal LBM

(a) Intra–node weak scaling SuperMUCusing 1-32 tasks.

5 10 15 20 25 30number of cores

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

MLU

PS

(b) Intra–node weak scaling SuperMUCusing 1-32 task, MLUPS.

Figure 8.6: Weak scaling results of the EBM model on SuperMUC cluster.

In a next step the weak scaling performance of the EBM implementation is ex-amined on SuperMUC. Analogously the weak scaling behavior inside one nodeusing a different number of tasks and later the performance of several nodes isexamined. Figure 8.6a shows the walltime in seconds and the number of coreswithin one node. For further investigations the execution time for the compu-tation of the hydrodynamic LBM and the thermal LBM are depicted explicitly.

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It can be observed that the thermal and hydrodynamic LBM require almost thesame amount of time as it was expected since both use a D3Q19 stencil. Thescaling results present 1, 2, 4, 8, 16, and 32 cores of one node. When two coresare used the runtime increases slightly because of the communication overheadwhich is necessary when more than one process is used. The other results of4, 8, and 16 cores build a constant line as it is expected for weak scaling. Theresult for 32 cores is noticeable. As explained in Table 8.2 32 cores can only beemployed by using hyperthreading. Hyperthreading or also called simultane-ous multithreading (SMT) implies that the architectural state of a core is presentmultiple times whereas the execution resources like arithmetic units, caches,queues and memory interfaces are not multiplied. Hyperthreading or SMT canenhance the instructions which are executed per cycle, e.g., if different coresuse different execution resources. But in the case of the EBM application whichis based on the solution of the memory–bound LBM, SMT does not bring anyprofit but rather an impairment of the scaling result. This experiment shoulddemonstrate that the use of hyperthreading is not reasonable for simulating theEBM process.

Figure 8.6b shows intra–node weak scaling of one node at the SuperMUC onthe basis of MLUPS. As expected the MLUPS increase linearly the more coresare used and reach a maximum at around 1.7 MLUPS when using 16 cores. Adecrease of MLUPS occurs again when using hyperthreading (cf. Figure 8.6b,32cores) which has the same reasons like explained above.

After examining the scaling behavior within one compute node the questionarises how scaling results are effected when using more than one computenode. Figure 8.7a and Figure 8.7b present the use of several nodes examiningruntime and MLUPS, respectively. It should be noted that for both results inFigure 8.7a and Figure 8.7b no hypertreading was used, i.e., only the physicalcores of a node are used for the simulation. Figure 8.7a depicts the runtinefor the thermal LBM by the line with the downwards triangles and uses bluecrosses for the hydrodynamic LBM. The larger runtime of the hydrodynamicLBM, compared to the runtime of thermal LBM, can be also found by the highereffort of computing the quadratic equilibrium distribution function of the hy-drodynamic case (cf. Eq. (4.14)) compared to the linear equilibrium function inthe thermal LBM (see Eq. (4.35)). The purple squares representing the overallwalltime indicate a satisfying weak scaling because the line is almost constant.A slight increase of walltime is acceptable because of the communication over-head when using a higher amount of nodes.

Figure 8.7b presents the MLUPS. A constant growth of MLUPS is observedwith a increasing number of nodes. Figure 8.6a up to Figure 8.7b demonstratethat the parallel implementation of the EBM model in WALBERLA coupled withthe pe is efficient and enables 3D simulations of large simulation domains withthe aid of supercomputers. This was one of the two fundamental questions ofa user who is interested which size of simulation domains can be solved. Thesecond question, how fast the original problem can be solved to get as soonas possible the required simulation results is answered in the next part by the

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20 21 22 23 24 25 26 27

number of nodes

100

150

200

250

300

350

400

450

500

550tim

e[s

] Walltime completeLBMthermal LBM

(a) Inter–node weak scaling Super-MUC: 1-128 nodes.

20 40 60 80 100 120number of nodes

0

50

100

150

200

250

MLU

PS

(b) Inter–node weak scaling Super-MUC: 1-128 nodes, MLUPS.

Figure 8.7: Weak scaling results of the EBM model on SuperMUC cluster.

strong scaling results.

8.2.3.2 Strong Scaling

The simulation domain in strong scaling experiments is fixed for all runs whilethe number of processes is increased. Consequently the workload per processdecreases and therefore the overall runtime decreases. First the scaling re-sults on the LiMa cluster are shown (see Figure 8.8). Figure 8.8a visualizes

1 2 4 6 8 12 16 24

number of cores

0

200

400

600

800

1000

runt

ime

in[s

]

(a) Intra–node strong scaling LiMa.

1 2 4 6 8 12 24

number of nodes

20

40

60

80

100

runt

ime

in[s

]

size to smallfor powder generation

(b) Inter–node strong scaling LiMa.

Figure 8.8: Strong scaling results of EBM model on LiMa cluster.

the intra–node performance of one LiMa node. It is observed that the executiontime is reduced by a factor of two when using two processes instead of one.

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SETUP # OF NODES # OF CORES SIMULATION DOMAIN (LATTICE CELLS)1. 1 16 512 x 384 x 48

2 32 256 x 384 x 484 64 256 x 192 x 488 128 128 x 192 x 48

16 256 128 x 96 x 482. 4 64 1024 x 768 x 48

8 128 512 x 768 x 4816 256 512 x 384 x 4832 512 256 x 384 x 4864 1024 256 x 192 x 48

3. 16 256 2048 x 1536 x 4832 512 1024 x 1536 x 4864 1024 1024 x 768 x 48

128 2048 512 x 768 x 48256 4096 512 x 384 x 48

Table 8.4: Details of three setups for strong scaling on SuperMUC.

This is exactly the behavior which was expected for strong scaling experiments.This constant reduction of runtime by a factor of two is further observed up totwelve cores. However when using 24 cores of one compute node the runtimeis not further reduced. This is caused by the effect of hyperthreading whichwas already observed at the weak scaling results. Since EBM implementationis memory–bound the use of virtual cores does not lead to a reduction of theruntime. Therefore the execution time of twelve cores is the same as the exe-cution time for 24 cores. Because the strong scaling of the EBM model withinone compute node works well, the strong scaling behavior of several computenodes at the LiMa cluster is examined in Figure 8.8b. For these experiments nohyperthreading is used. Once again it is observed that the execution time is re-duced by a constant factor of two when the number of nodes is doubled. It hasto be noted that the strong scaling experiments go only up to 8 node because ofthe powder bed generation. The pe needs a minimum domain size of around 40x 40 x 40 lattice cells in order to generate a realistic powder layer. The simula-tion domain of each node is too small to for the corresponding algorithm whenusing 16 or more nodes.

In a second step strong scaling experiments of the EBM implementation areexecuted on the SuperMUC. Since the effect that the powder bed generationalgorithm requires a minimal computational domain size, three different setupsof different domain sizes are used for the scaling experiments. These setups areshown in detail in Table 8.4. These are used to examine the behavior from 1node up to 256 nodes (16 cores - 4096 cores) which corresponds to half anisland of SuperMUC.

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50 100 150 200 250number of nodes

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600

800

1000

1200

1400

1600tim

e[s

]1-16 nodes (512x384x48 cells)4-64 nodes (1024x768x48 cells)16-256 nodes (2048x1536x48 cells)

(a) Inter–node strong scaling on Super-MUC, 1-256 nodes.

0 500 1000 1500 2000 2500 3000 3500 4000number of cores

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200

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300

350

400

MLU

PS

16-256 cores (512x384x48 cells)64-1024 cores (512x384x48 cells)256-4096 cores (2048x1536x48 cells)

(b) Inter-node strong scaling on Super-MUC, 16-4096 cores.

Figure 8.9: Strong scaling results of the EBM model on SuperMUC for threedifferent setups.

Figure 8.9a shows the scaling results for the three setups. In the ideal case,the execution time is reduced by a constant factor of two by doubling the num-ber of nodes apart from the necessary communication overhead. The first setup(cf. purple squares in Figure 8.9a) shows that the runtime is almost reduced bya constant factor of two, e.g., the execution time required for one node is about1350 s and the execution time for two nodes is about 700 s, and this bisection ofruntime is conserved up to 16 nodes. In almost the same manner, the bisectionof runtime can be observed for the second (cf. blue diamonds in Figure 8.9a)and third (cf. green triangles in Figure 8.9a) setup. In Figure 8.9b the devel-opment of MLUPS of the EBM implementation is examined. The three setupsproduce linear curves with a constant slope. Figure 8.9a and Figure 8.9b indi-cate the almost perfect strong scaling behavior of the EBM model in WALBERLA

together with the pe–coupling. This answers the second important question ofthe user to be sure that the simulation can be solved faster when using morecompute nodes (up to a certain minimum domain size where the powder canstill generated).

8.3 Summary

In this chapter it is explained how the EBM model and the corresponding al-gorithms are embedded in the WALBERLA framework. The EBM model is anapplication of the framework which uses several of its modules, e.g., the lbm orstencil module. The powder generation is carried out by the pe. The modelof Chapter 6 for a realistic powder size distribution is applied and embedded.Parallel algorithms which are responsible for the powder layer generation areexplained, like the free fall approach of regular ordered particles with negative

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downward oriented velocities. The contact detection between the particles iscomputed by the DEM solver of the pe. The required parameter for the DEMsolver based on the properties of Ti–6Al–4V powder are given. Both frameworksare connected by a simplified one–way coupling where the powder particles actas obstacles in WALBERLA. An efficient parallelization of the two types of ab-sorption algorithms (cf. Chapter 6) is introduced which handle the transfer ofthe energy brought into the powder bed based on a top–to–bottom approach.

In the second part of the chapter the performance and parallel scalability ofthe embedded EBM implementation on two state–of–the–art compute clusters,namely LiMa and SuperMUC, is examined. Good weak scaling results allowto simulate a larger computational domain by using more processors. On theother side convincing strong scaling results enable to get simulation results ofa fixed domain size faster when using more compute nodes. Due to the scalingresults it can be summarized that the EBM implementation results in a goodscaling behavior. However, a minimum computational domain is necessary forthe generation of the metal powder bed by the pe. Further, the EBM applicationis memory–bound, i.e., that hyperthreading is futile.

With Chapter 7, and Chapter 8 all essential parts and information for thesimulation of the EBM process have been established. At this point the simu-lation step of Figure 1.1 is reached and thus the simulation results are given inthe following Chapter 9.

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Chapter 9: Simulation for Application: EBM

9 Simulation for Application: EBM

Computers are useless. Theycan only give you answers.

(Picasso(1881 – 1973) )

The mathematical model of the EBM model consisting of partial differentialequations together with the necessary boundary conditions is discretized byLBM. These algorithms are implemented in WALBERLA and pe and this all pavethe way for the simulations of the EBM process. Meaningful simulation re-sults are a milestone of this thesis and emphasize its significance at the sametime. Simulations are more and more an integral part of the development andimprovement of machines and products and belong to digital prototyping. How-ever in order to justify the use of simulations as a helpful and convincing toolthe validation of the simulation results is indispensable. On the one side vali-dation means that simple test cases or benchmarks are run where the solutionsof other methods already exist or even analytical solutions are known. Thesebenchmarks just serve as a coarse checking of single aspects of the model, forexample the heat transfer or in particular the solidification of the melting poolin the EBM process. Such validation experiments are not sufficient to convinceany producer or manufacturer of EBM machines to believe in the correctness ofthe overall simulation. Hence, on the other side validation means the compar-ison of real experimental data with simulation results in a qualitative and/orquantitative way. Sometimes this is not possible since data cannot always beextracted and adapted from real experiments in order to be suited for numeri-cal simulations. In the case of the EBM process there exists data provided bymaterial scientists. At this point I have to thank for the good cooperation withthe Chair of Material Science and Engineering for Metals. Especially I wouldlike to express my gratitude to Dr. Matthias Markl who worked with me foraround two years and Prof. Carolin Korner for sharing her profound expertiseof the EBM process. Furthermore, I would also like to thank Vera Juchter andThorsten Scharowsky who provided the real experiments by the electron beammelting machine and the corresponding data. Without their work and expertisethe following validation and simulation results would not exist.

This chapter validates the EBM model and implementation by traditionalbenchmarks for example the Stefan problem and by the comparison of sim-ulation data with experimental data. The simulated parts are compared to

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manufactured parts with regard to porosity and surface quality in order tocheck if simulation and real build process generate components with similarproperties when using the same beam parameters and initial conditions. Sincethe results of simulation and real experiments are highly concordant the EBMimplementation can be taken to predict results of non–existing machines bysimulations. Thus various parameter studies are conducted in order to helppointing the way for the development of a new electron beam melting machinewith improved properties in order to enable a faster production rate or a betterquality of the products. The results and the structure of this chapter is mainlybased on the publications of Ammer et al. [2014a], Ammer et al. [2014b], andMarkl et al. [2015].

At this point it should be noted that the derived model and implementationof this thesis is only used for the simulation of the EBM process and the Ti-6Al-4V material. However, the model can be adapted in order to use it forthe simulation of other additive manufacturing processes with other materials.Therefore the absorption algorithms have to be exchanged depending on theapplied energy source. Further, the material parameters have to be modified inthe parameter file of WALBERLA. Another material may require different numer-ical handling of the pe. The adjustment to other powder bed fusion processes (cf.Section 2.2.7), such as selective laser sintering is straightforward and requiresthe modification of the energy source model and the penetration depth.

9.1 Validation of the EBM Model

In order to convince customers and machine developers to trust in simulationsthe correctness of the model, the implementation, and the simulation has to beshown and proven. Therefore, the concepts of model verification and validationexist. Both terms are often mixed and an inaccurate use of them is confusing.This thesis uses the definition of validation given in Schlesinger et al. [1979]which states that the implementation of a mathematical model within its do-main of applicability has a sufficient range of accuracy which is consistent withthe intended application of the model. On the other side model verificationshould ensure that the programming of the model is correct (see Schlesingeret al. [1979]). In other words, verification answers the question if the simu-lation results in the correct application and validation indicates whether theimplementation simulates the application correctly. Unfortunately, no specifictests and checks exist which can be applied to arbitrary simulation models todecide if the simulation produces the correct results. It is a rather challengingtask for every new simulation model to determine suitable validation and veri-fication tests, which provide a general insight into the credibility of the results.Verification tests go beyond the scope of this thesis and in this chapter the focusis placed on the validation of the EBM simulation. A first overview of the generaltheory of validation for simulation is given in Kleindorfer and Geneshan [1993]and Naylor and Finger [1967] even put the problem of validation of simulation

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at the same level as validating new scientific theories. At this point it also has tobe noted that the guarantee that the simulation is absolutely valid is too costlyand time consuming since this would require to test all possible occurring vari-ables and parameters over the complete computational domain. Therefore, in afirst step of validation the significant output variables of the simulation have tobe identified and their required accuracy has to be determined. Kleijnen [1995]identifies four basic approaches in order to decide if a simulation is valid ornot. In a first approach the model developer and simulation engineer decideby him– or herself if the results are valid. This very subjective approach is notrecommendable. In the second approach the users of the simulation resultswhich are already involved in the model development and simulation decide ifthe simulation is valid. A third possibility is that a third independent party,i.e., neither the simulation engineer nor the user, determines if the simula-tion produces valid results. This third party should of course have a profoundknowledge and understanding of the application process. That approach is ap-propriate especially for large–scale simulation projects. The last approach usesa scoring model to decide whether the simulation results are valid, i.e., pointsare awarded for predescribed steps and also for the overall results. However,this procedure is not often used since it is laborious and there is no logicalstrategy how many points are sufficient for the validity of the simulation.

Furthermore Kleijnen [1995] gives a detailed overview of validation techniques.The most used one is the animation of simulation data, i.e., the operational be-havior is displayed graphically as the simulation moves when time passes. Thecomparison to other models, analytic solutions, and experimental data is alsovery often applied. The simulation results are compared to simulation resultsof already validated models and implementations, or compared to known an-alytical solutions of testcases. Another validation technique is based on thecomparison of events between the simulation model and the real system. In ad-dition parameter variability or sensitivity analysis can be done by changing theinput values and internal parameters of a model to determine the effect uponthe model’ s behavior or output.

For the validation of the EBM process the validation tests are determined bythe developers and users of the EBM machine as well as by material scientists.Thus the second and third validation approach are combined. The used valida-tion techniques are the application of the standard benchmarks like the Stefanproblem and the comparison of simulation data with experimental data.

9.1.1 Validation of the Energy Equation

This section is mainly based on the work of Ammer et al. [2014a]. A first valida-tion is to check if the numerical transfer and its implementation of the energyequation (cf. Section 3.2.3) are correct. This is examined with the aid of a smallsimulation set–up visualized in Figure 9.1. A box filled with fluid has an initialfluid temperature of 2000 K. A free surface is involved since the rest of the boxis filled with gas. The wall boundary in the (yz)-plane is heat up to a temper-

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yx

z

Figure 9.1: Sketch of validation scenario of energy equation (taken from Ammeret al. [2014a]).

ature of 3000 K. Dirichlet boundary conditions, explained in Section 4.2, areused. The open thermal boundary condition is used for setting temperaturesat the two walls and for the other walls the closed thermal boundary conditionis set. The no–slip boundary condition (cf. Section 4.2) is set for the hydrody-namic flow field. The corresponding analytical solution of this specific test caseof the energy equation is given by,

g(xj ,∆T, L, k, tj) =2∆T

π

∞∑n=1

(−1)n

nsin(nπ

xiL

)exp

(−kn2π2 tj

L2

)+

∆T

Lxj + T0, (9.1)

where ∆T = 1000 K, L = 625 · 10−6m, and thermal diffusivity k = 9.93 · 10−6 m2

s .Figure 9.2 shows the analytical and numerical solutions of Eq. (9.1) for threedifferent times. Since Figure 9.2 shows only a qualitative agreement of simu-lated and analytical data, Table 9.1 shows the norm of the L1–error of analyticaland numerical solutions in order to examine the numerical test case also quan-titatively. The L1–erros are in the range of 10−9 and indicate that modeling

TIME in [ms] ‖err‖10.88 ms 3.1487 · 10−9

2.63 ms 3.0684 · 10−9

8.77 ms 3.0791 · 10−9

Table 9.1: L1–error of numerical and analytical solution of the energy equation.

and implementation of energy equation is in a sufficient range and that theresolution is appropriate. This numerical example shows that the model andimplementation regarding the solution of the energy equation works well. In thenext step the correctness of the liquid–solid phase transition is validated by theStefan problem.

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0 100 200 300 400 500 600 700

Position in 10−6m

2000

2200

2400

2600

2800

3000Te

mpe

ratu

reT

in[K

]T(t=0.88ms)-numericalT(t=2.63ms)-numericalT(t=8.77ms)-numericalT(t=0.88ms)-analyticalT(t=2.63ms)-analyticalT(t=8.77ms)-analytical

Figure 9.2: Comparison of numerical and analytical solutions of heat equation.

9.1.2 Validation of Solid – Liquid Phase Transition by the StefanProblem

The original Stefan problem (SP) introduced in Stefan [1891] describes the for-mation of ice in the Arctic Ocean and is interpreted as a special solid–liquidboundary problem with a moving interface between the two phases when timepasses. The SP is a traditional benchmark problem for melting as well as forsolidification processes (see Rubinshtein [1971]). Huber et al. [2008] alreadyapplied SP for LBM in order to validate the phase change. In SP the conserva-tion of energy regulates the corresponding phase change process. Parametersinvolved in the SP are the phase change material – here it is the metal powder –with constant density ρ, the latent heat L, liquidus temperature Tl, specific heatcapacity cp, and thermal conductivity k. In the context of SP the Stefan numberSt also arises. Like other dimensionless quantities defined in Section 3.3 theStefan number can be used to compare different systems. It characterizes thespecific phase transition problem and is defined by the ratio of sensible heat tolatent heat, i.e.,

St =cp · (T − Tl)

L. (9.2)

The validation set–up of SP consists of a cubic box that is filled with meltedmetal. One wall boundary is cooled down to a fixed temperature below the so-lidification temperature of the metal. Thus, a linear temperature profile shouldbe adjusted. The solidification of liquid metals begins at the interface between

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the cooled wall and the liquid metal. The resulting analytical depth of the so-lidified metal is given by the function h(t) depending on time. Stefan showsin Stefan [1891] that h(t) is defined to,

h(t) =

√2λat

∆Hρ, (9.3)

with parameters given in Table 9.2. Figure 9.3 shows the analytical solution

PARAMETER VALUE UNIT

latent enthalpy ∆H 2.9 · 105 ms

s2

T − Ts a 6.5 · 101 K

heat conductivity λ 3.0 kg·ms3K

specific heat capacity cp 7.0 · 102 m2

s2K

density ρ 4.0 · 103 kgm3

Stefan number St 1.5689 · 101 –

Table 9.2: Data for the Stefan problem.

0 2 4 6 8 10 12 14 16 18

time in [ms]

0

50

100

150

200

250

dept

hin

[10−

6m

]

numericalanalytical

Figure 9.3: Comparison of numerical and analytical solution of the Stefan prob-lem.

of Eq. (9.3) and the numerical solution of EBM simulation, i.e., using parame-ter values given in Table 9.2. The minimum relative error of both solutions for

130


h(t) is 0.0062. This sufficiently small error of the Stefan problem indicates thatthe phase–transition treatment of the EBM implementation is physically cor-rect. After these general validation examples considering energy equation andphase transition specific EBM validation tests are executed in the following bycomparing experimental data with simulation data.

9.1.3 EBM Validation by Comparison with Experimental Data

The following validation tests and results are mainly based on the work of Am-mer et al. [2014b]. Before beginning with the tests a meaningful validationset–up has to be determined. The set–up should not only cover single aspects,like the solid–liquid phase transition and its qualitative accordance, but ratherputs the accent on the overall EBM process in a qualitative way. Furthermorethe range of parameters and variables are determined in order to ensure a cor-rect and accurate simulation result. In the following a suitable set–up with thecorresponding variable ranges are described. Real parts are generated via thisset–up layer–by–layer by the EBM process and in a subsequent step the partsare generated by simulations and compared with experimental data.

9.1.3.1 Definitions and Validation Set–Up

For the EBM specific validation tests the following set–up shown in Figure 9.4 isused. A cuboid of size (15x15x10) mm3 is generated by hatching the Ti-Al6-V4

15 mm15mm

10

mm

Figure 9.4: EBM specific validation set–up (taken from Ammer et al. [2014b]).

powder particles layer–by–layer. The hatching process happens by moving theelectron beam gun line wise (cf. the red arrows in Figure 9.4). The hatchingprocesses differ by line energy and scan velocity of the electron beam. The lineenergy is defined by,

EL =U · Ivscan

=Pbeam

vscan, (9.4)

where U denotes the acceleration voltage in [V], I the current in [A] and vscan thescan velocity in [m

s ] of the electron beam. The parameter set (EL,vscan) definesthe properties of the used electron beam gun.

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The developer of the EBM machine as well as the end consumers which usethe EBM machine for producing advanced products (see Section 2.1.3) are in-terested in a high quality of the products. The quality of EBM products is clas-sified into three categories, namely porous, good, and products with swellingsurfaces. Good samples excel in smooth surface and a relative density higherthan 99.5 %. On the one side a high relative density requires a certain energyincorporation. One the other side if the temperature during the process is toohigh swelling may arise and the dimensional accuracy of the product cannotbe guaranteed. On the other side, if the line energy is too low and the relative

porous good swelling

Increasing line energy EL

Figure 9.5: Categories of samples.

density is smaller than 99.5 %, then the sample is named porous and is uselessfor customers. Figure 9.5 visualizes the relation between line energy EL andsample properties. In the following an experimental process window generatedby material scientists is discussed. A process window relates the relation be-tween scan velocity and line energy of the electron beam gun graphically. Forthe validation experimental and numerical process windows are compared.

9.1.3.2 Experimental Process Window

Figure 9.6 shows the process window generated by real experiments done bythe material scientist Vera Juchter. The process window shows the quality ofsamples with different line energies in [kJ

m ] and scan velocities in [ms ] of the

electron beam gun. Each symbol stands for one part according to the set–updescribed in Figure 9.4 with different parameters. The red circles symbolizeparts being porous, the blue squares stand for an optimal, good parameterset and the yellow rhombi for a sample where swelling occurs because of toohigh temperatures at the surface. Summarizing, the higher the scan velocity isthe lower the embedded line energy has to be to ensure a part with sufficientproperties to satisfy the requirements of customers.

This experimental process window is now used for the validation tests of theEBM model and simulation which is explained in the following.

9.1.3.3 Numerical Process Window

In this section the 3D numerical EBM model (described in Chapter 4 and Chap-ter 6) and its implementation in WALBERLA (cf. Chapter 8) is validated againstthe experimental data of a process window shown in Figure 9.6. The validationis performed with the set–up given in Section 9.1.3.1. However, the completeset–up leads to high computational costs for a fully resolved 3D simulation.

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0 1 2 3 4 5 6 7

Scan velocity [m/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Line

ener

gy[k

J/m

]porousgoodswelling

Figure 9.6: Process window for experimental generated EBM parts (taken fromAmmer et al. [2014b]).

simulated powder bedhatching lines

beam offset

Figure 9.7: Sketch of simplified simulation scenario (cf. Ammer et al. [2014b]).

Hence to reduce these costs the numerical transfer of the validation set–up isdone by hatching only one powder layer instead of multiple layers. In orderto demonstrate the computational effort it is noted that even for one simplifiedscenario with an exemplary parameter set of (3.2 m

s , 0.2Jm) already eight compute

nodes of LiMa (for details see Section 8.2.2) are occupied for four wall clockhours. The LiMa cluster is used for all subsequent 3D EBM simulations. Nev-ertheless, this simplification is in good agreement with the situation expectedfor the full set–up as it is shown in the following. In addition the simulated

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powder particle layer is minimized, i.e., just a rectangular powder layer is fo-cused and has just seven hatching lines (cf. Figure 9.7). Thus, a beam offsetper layer is defined where the electron beam is on but outside the simulatedpowder particle bed.

The boundary conditions are given in the following. A thermal Dirichletboundary condition (see Ginzburg [2005b]) is defined at the bottom of the com-putational domain in order to ensure a temperature of 1000 K and a no–sliphalf–way bounce–back (see Section 4.2) condition is used for the hydrodynamicpart. Additionally, periodic boundary conditions in x– and y–direction are ap-plied to reduce the influence of the boundary treatment on the melting pool.The numerical errors evoked by the periodic boundary conditions are delimitedby clipping the borders of the measurements and images.

Figure 9.8 visualizes the numerical simulation results of the validation set–up with different line energies and scan velocities. In order to determine if the

0 1 2 3 4 5 6 7

Scan velocity [m/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Line

ener

gy[k

J/m

]

swellingporousgood

Figure 9.8: Process window of numerical data. ∆x = 5 · 10−6 m, ∆t = 1.75 · 10−7 s,simulated powder domain (1.44 x 0.64 x 0.24)·10−3 m3, beam offset =13.56 · 10−3 m, and a line offset of 100 µm (taken from Ammer et al.[2014b]).

numerical sample is porous the relative density of the sample is measured. Avalue less than 99.5% results in a porous classification for the sample. On theother side, a definition for numerical swelling is less straightforward. So far,the EBM model and implementation does not contain evaporation and thus no

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evaporation pressure. Hence, numerical swelling effects do not occur explicitly.It is necessary to determine regions of numerical swelling effects by a differ-ent manner. The solution for this problem is based on temperature computednumerically. The numerical temperature is higher compared to the real EBMprocess since the cooling effects due to the process of evaporation are not in-cluded in the model. Additionally numerical artefacts result in outliers in thenumerical temperature field and distort the results. In order to overcome and tobalance these numerical influences an averaging process is applied for the tem-perature values. With this strategy, the numerical swelling effects are detectedif the averaged temperature values are still higher than 7500 K.

Since the numerical porosity and swelling effects can now be determined thenumerical process can now be validated against the experimental process win-dow. In comparison with the experimental data shown in Figure 9.6 it is ob-served that all porous samples are consistent in the numerical process window(cf. Figure 9.8). It has also to be noted that the numerical and experimentalresults for scan velocities of 3.2 m

s , 4.0 ms and 6.4 m

s are almost identical. Small de-viations between numerical and experimental process windows are only foundfor velocities of 0.8 m

s , 1.6 ms , 2.4 m

s , 4.8 ms , and 5.6 m

s for higher line energies. Fora scan velocity of 0.8 m

s in experimental data swelling already occurs for 0.4

and 0.5 kJm . However, in the numerical process window no swelling effects are

observed. The same effect is observed for scan velocities 1.6 and 2.4 ms where

swelling occurs for smaller line energies in the experiment than in the numer-ical tests. For higher scan velocity values of 4.8 and 5.6 m

s the reverse effectis observed, i.e., more swelling occurs numerically than experimentally. Fora scan velocity of 4.8 m

s swelling arises numerically already for a line energy of0.25 kJ

m while the experimental setup is still good. The same effect is observablefor 5.6 m

s . Line energies higher than 0.25 kJm lead to swelling in numerical sim-

ulations but in experimental data only energies equal 0.3 kJm result in swelling

effects. Both types of deviation can be explained by the integration of the focusof the electron beam gun into the numerical model. In the real EBM machinethe focus is more precise for smaller scan velocities imposed by a smaller elec-tron beam power Pbeam, e.g, vscan = 1.6 m

s or vscan = 2.4 ms result in a precise,

small focus and thus, energy is brought onto a smaller area. For these veloc-ities swelling occurs for smaller line energies experimentally than numericallysince the numerical EBM gun focus is constant for different beam power andscan velocities, respectively. For higher scan velocities (larger Pbeam) like 4.8 m

sthe focus of the gun spreads and, subsequently, the same amount of energy isbrought into a larger area of powder particles and the maximum temperature issmaller. Consequently, in the experimental process window less swelling occursfor higher scan velocities (larger Pbeam) than in the numerical process window.

Figure 9.9 shows the numerical simulation results of the hatching lines forthe exemplary parameter set

(6.4m

s , 0.15kJm

)at six different time steps. For the

visualization of the numerical data POV–Ray (for detailed information see POV-Ray [2016]) is used, an open source ray tracing software. The free surface of

135


(a) (b)

(c) (d)

(e) (f)

Figure 9.9: Numerical hatching results for the parameter set(

6.4ms , 0.15kJ

m

).

136


the metal powder particles is visualized by an isosurface on the fill level. InFigure 9.9a the electron beam is located at the beginning of the first scan lineand the affected powder particles are not yet completely melted. Figure 9.9bshows the situation after scanning the first line. The particles affected by thisscan line are melted. In addition, on the left side unaffected particles are partlycovered by the melting pool and on the right side a groove of the melting pool isobserved. Figure 9.9c, Figure 9.9d, and Figure 9.9e demonstrate the proceedingbeam. In Figure 9.9f the complete powder bed has been scanned by the electronbeam and is already solidified. Unmelted particles or layer bonding defectsbetween the bottom plane and the original powder layer are observed.

These very EBM specific validation tests by the comparison of the experimen-tal and numerical process window for the hatching of a cuboid shows the qual-itative and quantitative correctness of the EBM model and its implementationin WALBERLA. In addition, the validation of the liquid–solid phase transition bythe Stefan problem and the incorporation of the energy equation give a quanti-tative validation. All these tests should persuade critical users and developersof the EBM machine by the meaningfulness of the corresponding simulationresults although a reduced model is incorporated which does not account forevaporation and temperature dependent surface tension. Therefore, the EBMsimulation has a sufficient accuracy to produce simulation results of futuremachines and can assist developers by the improvement and development ofadvanced EBM machines as shown in the following.

9.2 Improvements of the EBM Process by the Use ofSimulations

This section is mainly based on the cooperate work of Dr. Matthias Markl andme and is published in Markl et al. [2015]. In the following it is shown howthe EBM process can be further examined via simulations and how the processitself may be improved. These investigations are achieved by several modifica-tions of the original EBM process due to used beam diameter and line offset. Atthe beginning the process window (cf. Figure 9.8) is extended, i.e., higher scanvelocities are simulated in order to see which is the fastest possible parameterset. At this point it has to be noted that the following process windows are allgenerated by numerical simulations and not by real experiments.

9.2.1 Extension of Process Window

Faster production rates require higher scan velocities. Thus the examination ofthe extension of the process window is essential in order to check the qualityof products generated with faster scan velocities. Existing EBM machines havepower restrictions since reliable beam characteristics are only available up toa beam power of 1.2 kW (see Juchter et al. [2014]). Assuming a line energy of0.1 kJ

m only a scan velocity of 12 ms can be achieved. This limitation does not

137


enable a speed up of the build times. However, developers and customers areinterested in faster production times since they reduce the overall costs andincrease the success of the EBM process in other industry branches. In thefollowing the process window in Figure 9.8 which shows scan velocities up to6.4m

s is extended numerically up to 30 ms . The scan velocities are increased

while studying porosity and swelling limits in order to find out the best possibleparameter configuration of scan velocity and line energy.

0 5 10 15 20 25 30 35

scan velocity m/s

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Line

Ene

rgy

kJ/m

swellingporous

good1.2 kW

2.4 kW4.8 kW

Figure 9.10: Extended numerical process window of the EBM process with100 µm line offset.

Figure 9.10 shows the numerical extended process window with scan veloci-ties up to around 30 m

s . It contains also the previous simulation results whichwere produced for the validation in Section 9.1.3.3 (see left hand side of thevertical line). The blue downward oriented triangles stand for porous samples,the green circles for samples with sufficient properties, and the red upwarddirected triangles for samples where swelling effects may occur. In additionthree strictly decreasing functions denote three different beam powers wherethe beam diameter is reliable. 1.2 kW stands for the existing electron beam gunand 2.4 kW and 4.8 kW for future electron beam guns.

In Figure 9.10 the last numerical simulation classified as ”good” has a scanvelocity of 29 m

s and the range of possible good samples closes at around 30 ms .

An almost constant lower porosity border and an upper swelling border can beidentified. Higher scan velocities than 30 m

s result in samples which are porous

138


and have swelling effects at the same time. The reason for this behavior is thatthe energy, which is integrated into the metal powder bed, is not high enough tomelt the complete powder bed but at the same time the absorbed energy resultsin too high surface temperatures. The two borders which separate good andporous products on the one side and good and parts with swelling effects on theother side have to be examined. The upper border between good samples andsuch with swelling effects depends on the maximum calculated temperature onthe surface. The increase of the scan velocity with constant line energy, i.e.,considering a constant line in the process window, results in the incorporationof the same amount of energy during a shorter duration. Thus, the remainingtemperature of the melting pool is higher when the electron beam proceeds toin the next scan line and the temperature of the melting pool is increased fromscan line to scan line. This border between good products and products whichhave a reduced surface quality can be predicted quite well.

In the next step the lower border separating good products and porous prod-ucts is considered. This border forms an almost constant line since the nec-essary amount of energy to melt the powder layer depends only on its corre-sponding volume. During the hatching only a small fraction of the integratedenergy is lost by diffusion and for higher scan velocities this amount of lostenergy decreases. Therefore, the lower border decreases over the range of scanvelocities. At this point it should be noted that there are statistical fluctuations,i.e., porous and good results can be very close together. Since it is important toproduce deterministic parts some kind of safety distance from the lower bordershould exist.

Summarizing, a maximum manageable parameter set for EBM productionhas a scan velocity of 20 m

s and a line energy of 0.125 kJm . If the lines denoting

the beam power are considered it is observed that so far the available settingscannot exploit higher beam powers. Thus, improved hatching strategies will benecessary for using faster scan velocities and the improvements of the overallpotential of the EBM process. Several new hatching strategies are examined inthe following with respect to beam diameter and line offset.

9.2.2 Increase of Beam Diameter

A first strategy in order to improve the quality of the EBM parts is to use higherscan velocities and at the same time increase the beam diameter. The energydistribution of the electron beam is modeled by a two–dimensional Gaussiandistribution which leads to a lower maximum temperature at the surface. Thisreduced maximum temperature allows for an increase of the line energy and anacceleration of the scan velocity by using higher beam powers.

Two beam shapes with increased beam diameters are examined with an in-creased heat affected area by 50% and 100%. Here the term of FWHM comes intoplay. It is already defined in Section 6.1 and visualized in Figure 6.2. In thecase of the EBM process this function is the density of the normal distributionand thus, the FWHM is given by Eq. (6.2). The resulting standard deviations

139


are 0.122 mm and 0.141 mm leading to a FWHM of 0.287 mm and 0.332 mm, re-spectively. Figure 9.11 shows simulation results of the two beam diameters.The increase of heat affected area by 50 %, shown in Figure 9.11a, enables apossible line energy of 0.125 kJ

m with a scan velocity of 40 ms for producing good

products. Thus, a potential electron beam power of 5 kW can be exploited (cf.Figure 9.11a). A further increase of the heat affected zone by 100 % enables amaximum parameter set of (0.125 kJ

m , 50 ms ) to generate products with satisfying

properties. This parameter set results in a maximum beam power of 6.25 kW (seeFigure 9.11b). The strategy of increasing the beam diameter by 100 % results ina build time reduction of 60 % since scan velocities of 50 m

s are affordableThe hatching strategy of the increase of the beam diameter can speed up the

possible scan velocity of the electron beam and makes a better exploitation ofhigher beam powers possible. However, the increase may lead to further chal-lenges due to the borders of the powder layers. Since the larger beam diameterresults in a larger melting pool width the dimensional accuracy may be wors-ened. In order to overcome this problem the contour can be melted separatelybefore or after the hatching. The required time for the hatching process of onecomplete powder layer of the validation set–up in Section 9.1.3.1 is 56.25 ms byusing a scan velocity of 40 m

s . Compared to the basic hatching strategy whichrequires 112.5 m

s with a scan velocity of 20 ms , the production time is almost

halved. Unfortunately, this speed up is consumed by the additional slower con-tour melting, which requires 120 ms when using a speed of 0.5 m

s . One possibilityto solve this problem is the multi–beam approach. Here many different loca-tions are melted simultaneously. Summarizing, the strategy of an increasedbeam diameter may lead to faster production rates if the hatching geometry issimple, the hatching area itself is very large compared to the contour, or whenno further contour scanning is required at all. As explained in Chapter 2 asubsequent check of dimensional accuracy can be executed.

9.2.3 Decrease of Line Offset

A second strategy to achieve a speed up of the build time is the decrease ofthe line offset between the scan lines. Therefore the total scan length has tobe increased because this enables higher scan velocities and a reduction of thetotal build time. For this strategy the line offset as well as the line energy haveto be halved in order to maintain the total energy constant that is brought intothe metal powder bed by the electron beam. The hatching set–up of Figure 9.4is changed by hatching now 13 scan lines of the computational domain. Theconfiguration sets are comparable with the basic hatching strategy by doublingthe scan velocity.

Figure 9.12 shows the numerical process window for a decreased line offset of50 µm. This process window increases to 70 m

s and has also an upper and lowerborder like the process window of Figure 9.10. Using this strategy the beampower, which can be exploited increases to 5 kW. Furthermore, the reductions

140


20 25 30 35 40 45 50

scan velocity [m/s]

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Line

Ene

rgy

[kJ/

m]

Numerical Process Windowswellingporousgood2.4 kW4.8 kW

(a) Increase of the heat affected area by 50%.

20 25 30 35 40 45 50

scan velocity [m/s]

0.10

0.12

0.14

0.16

0.18

0.20

0.22

Line

Ene

rgy

[kJ/

m]

Numerical Process Windowswellingporousgood2.4 kW4.8 kW

(b) Increase of the heat affected area by 100%.

Figure 9.11: Partial process window with increased beam parameters.

141


of build time are about 43 % by using a beam parameter set of(

0.075kJm , 70m

s

).

The reason why the range of good parts in the process window is shifted to the

10 20 30 40 50 60 70

scan velocity m/s

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Line

Ene

rgy

kJ/m

swellingporous

good1.2 kW

2.4 kW4.8 kW

Figure 9.12: Numerical process window with a scan line offset of 50 µm

right is analogue to that explained for the increased beam diameter strategy.Since a faster scan velocity is used the energy is brought on a larger area ofthe powder bed on the same time compared to the basic strategy using 100 µm.This results in a high averaged temperature in the heat affected area over thecomplete scan length but the maximum peak values of the temperature aresmaller. In addition the form of the melting pool is changed from a drop toa shape where the powder along the scan line is liquified. Consequently, themelting pool dynamics are reduced and less swelling effects occur.

In fact, the parameter set of this process window with increased beam offsetis obtained by doubling the scan velocity and halving the line energy of processwindow of Figure 9.10. A parameter set (0.1 kJ

m , 20 ms ) of the increased line offset

leads to a part with sufficient properties whereas the corresponding parameterset (0.2 kJ

m , 10 ms ) with a line offset of 100 µm yield to swelling effects on the part.

This example shows how the decrease of the line offset enables an increase ofthe scan velocities to reduce the overall build time.

Figure 9.13 shows the maximum temperatures for both line offsets, i.e., forboth hatching strategies. Here, the lines with the same colors in both figurescan be compared since they symbolize configurations with constant beam powerbut different offsets. The gray, dashed, constant line indicates the border be-

142


tween good products and products where numerical swelling effects may arise.Following this condition a maximum line energy of around 0.1 kJ

m is usable forthe basic hatching strategy with a line offset of 100 µm (see Figure 9.13a) inorder to prevent swelling. However, if these parameter configurations are com-pared with the results in the process window shown in Figure 9.10 almost allgenerated parts are porous. This corresponds to the line energy of 0.05 kJ

m in Fig-

5 10 15 20 25 30

‖ub‖ in m/s

2000

3000

4000

5000

6000

7000

8000

9000

10000

max

(T)

inK

0.0500 kJ/m

0.0750 kJ/m

0.1000 kJ/m

0.1250 kJ/m

0.1500 kJ/m

0.1750 kJ/m

0.2000 kJ/m

(a) line offset of 100 µm.

20 30 40 50 60 70

‖ub‖ in m/s

2000

3000

4000

5000

6000

7000

8000

9000

10000

max

(T)

inK

0.0250 kJ/m

0.0375 kJ/m

0.0500 kJ/m

0.0625 kJ/m

0.0750 kJ/m

0.0875 kJ/m

0.1000 kJ/m

(b) line offset of 50 µm.

Figure 9.13: Maximum temperatures for different line offsets. Lines with thesame colors can be compared by having the same beam power.The gray shaded area shows the experimentally validated data upto 6.4 m

s (taken from Markl et al. [2015]).

ure 9.13b. Here the maximum temperature value does not reach the temper-ature threshold. It is further possible to increase the line energy up to 0.075 kJ

mand still produce good products. If these parameter sets are put into the pro-cess window with decreased line offset (see Figure 9.12) almost all simulationsgenerate non–porous parts.

Figure 9.14 visualizes the temporal evolution of the melting pool duration.Here, the volume of the melting pool depending on the simulation time for threedifferent scan velocities and a line energy of 0.05 kJ

m is given. This specific lineenergy is examined since almost all simulations lead to porous parts. It is ob-served that the volume of the melting pool increases with higher scan velocitieswhile the duration of the melting pool is almost constant. For the a scan veloc-ity of 30 m

s the melting pool is almost solidified when the electron beam comes

143


0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

t in ms

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035m

elt

poo

lvo

lum

ein

mm

3

30 m/s

50 m/s

70 m/s

Figure 9.14: Time–dependent melting pool volume for different scan velocitieswith 0.05 kJ

m : at a scan velocity of 50 ms and 70 m

s the previous scanline is still liquid and the melt pool grows continuously (taken fromMarkl et al. [2015]).

back. Finally, this leads to a porous part and there is no advantage of thishatching. However, if the scan velocity is increased up to 50 m

s the melting poolstays over the whole simulation time, i.e., it does not solidify between differentscan lines. The melting pool volume has a maximum value after ten scan linesthat is 1.6 times larger than the previous example. The maximum of the volumeis obtained at this point since the following scan lines are partially outside thecomputational domain and melt less particles. However, this behavior of themelting pool growth indicates that the natural maximum volume with this scanvelocity is almost obtained. In addition the depth of the melting pool is notsufficient to get a non–porous part by re–melting the top part of the previouslayer. The last example of a scan velocity of 70 m

s conveys the same behavior,although, the melting pool volume is 2.6 times larger and leads to a non–porous,dense part. After the first three scan lines there is an almost constant growthof the melting pool per scan line. This indicates that the maximum value of themelting pool volume is solely obtained due to the border of the computationaldomain.

Concluding, the strategy of decreased line offset allows a significant enlarge-ment of the process window. In addition this strategy enables more reliableand reproducible parts with reduced build time of at least 43%. Hence, thehigh energy potential of electron beams of future generations can be exploited.The dimensional accuracy should be the same as in the basic hatching strategysince the beam diameter is the same.

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9.3 Summary

In this chapter the EBM model and its implementation in the WALBERLA andpe framework is validated. Firstly, the correctness of single parts of the modelis checked. The numerical transfer of the energy equation is validated usinga specific analytical solution and the liquid–solid phase transition is checkedby the Stefan problem. These tests examine quantitatively the correctness ofsingle aspects of the model. Even though these basic validation results are ofsufficient accuracy no general statement of the correctness of the overall EBMmodel can be made. It is necessary to check if all single components interactcorrectly together. Therefore, EBM specific validation tests are essential to con-vince also skeptical users of the gain of simulation results. Process windowsform the basis of these all–encompassing validation tests. In process windowsthe final quality of EBM products is classified into porous, good, and swellingeffects depending on the used line energy and scan velocity of the electron beamgun. A process window generated by experiments that are executed at the EBMmachine by material scientists is used to validate the numerical simulationresults. Here numerical and experimental parts generated by the same line en-ergy and scan velocity are compared with each other referring to porosity andsurface quality. Both types of process windows – numerically as well as exper-imentally generated – are highly concordant with only a few exceptions, whichcan be explained by the assumption of a constant beam focus in the numericalmodel.

Since these validation experiments indicate the correctness of the EBM modelintroduced in this thesis it can be used for further examinations and improve-ments of the process. In order to speed–up the overall process it is examinedhow fast the scan speed can be to still generate good EBM products. The re-sulting numerical process window shows that there is a limit of scan velocitiesat which no good products can be generated. At this point the question ariseshow this limit can be shifted and higher scan velocities can be applied in orderto produce faster. A first strategy is the increase of the beam diameter in orderto distribute the amount of energy to a larger area. This results in a shifted pro-cess window where higher scan velocities can be used. However, the increaseof the beam diameter is subject to some difficulties like a worse dimensionalaccuracy. A second strategy is the decrease of the line offset, i.e., a reductionof the distance between two scan lines is applied. In order to compare pro-cess windows the beam offset for these tests is halved, the scan velocities aredoubled, and the line energy is also halved in order to keep the beam energyconstant which is brought into the powder bed. The results of this strategyare promising. Even though more scan lines are necessary higher scan veloci-ties are possible without having swelling effects at the surface and therefore, aspeed–up of the production rate can be achieved.

Concluding, in this chapter the two steps of the simulation pipeline in Fig-ure 1.1 ”validation” and ”simulation” are executed. Additionally, several en-hancements of the hatching strategies, which can lead to a better exploitation

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of the electron beam power, are derived numerically. These improvements of theEBM machine and application belong to the last part of the simulation pipelineand therefore, the end of the pipeline is reached.

Part II ends also with Chapter 9. This part of the thesis gives a detailed de-scription of the implementation of the EBM specific modeling aspects into theWALBERLA framework and explains the numerical generation of the metal pow-der particles by the pe framework. The connection of both frameworks by theabsorption of energy in the powder bed is realized by the one–way coupling.Weak and strong scaling experiments executed on two state–of–the–art super-computers LiMa and SuperMUC demonstrate the parallel efficiency of the im-plementation of the EBM model. This efficiency facilitates the time–consumingsimulations of the EBM process and with the use of simulations the improvedhatching strategies can be identified.

At this point it has to be noted that the meaningfulness and predictions ofthe simulations depend directly on the accuracy of the corresponding model.Thus, a better accuracy of the model would lead to better numerical results.A starting point for the possible improvement of the EBM model is the usedthermal FSLBM. Since the order of accuracy is still unknown (cf. Section 5.3.3)it has to be examined if it has the same as the LBM. If the FSLBM boundarycondition has a lower accuracy order it will diminish the overall accuracy. Thus,a first step is the examination of the order of the FSLBM boundary conditionwhich is done in Part III of the thesis. In addition an improvement of the FSLBMboundary condition is presented in order to ensure a free surface method withthe same order of accuracy like LBM.

146

Part III

Extension: Advanced Modelsand Methods

147

Chapter 10: Accuracy Analysis of FSLBM Boundary Condition

10 Accuracy Analysis of FSLBMBoundary Condition

Das Leben bildet eineOberflache, die so tut, als obsie so sein musste, wie sie ist,aber unter ihrer Haut treibenund drangen die Dinge1.

(Robert Musil (1880 – 1942))

There are three main possible ways of proceeding in order to get improved re-sults (see Figure 10.1). A first strategy is the extension of the mathematicalmodel, i.e., the incorporation of additional model aspects like for example theevaporation pressure which is missing in the EBM model. Another way is thefurther optimization of the implementation. This could make the use of highergrid resolutions feasible and may produce more accurate solutions. A more ef-ficient implementation could also enable faster simulations and therefore moresimulations. A third option is the enhancement of the numerical methods. Dis-

ImprovedSimulation

Results

Extension ofMathematical Model

Optimization ofImplementation

→ Finer Grid Resolution

Improvement ofNumerical Methods

Figure 10.1: Possible strategies in order to improve accuracy of simulation re-sults.

cretization techniques of higher order accuracy or which result in more efficientalgorithms can support the generation of more accurate simulation results. In

149


Part III of the thesis the effect of improved numerical methods, more preciselya higher order free surface boundary condition is shown. FSLBM introducedby Korner et al. [2005] and explained in detail in Section 5.3 is a free surfacemethod which produces a sharp interface by tracking the motion of the bound-ary. It does not simulate the gas dynamics explicitly and the original two–phaseproblem is simplified to a one–phase problem. This reduction is responsiblefor its efficient implementation. The analysis of the FSLBM boundary condi-tion (B.C.) is essential since a theoretical foundation of the accuracy is missing.Furthermore, its successful use in various different applications (see Anderlet al. [2014b], Bogner and Rude [2013], and Janßen [2010]) demands the clar-ification of the accuracy order. Thus, the order of accuracy of the FSLBM B.C.used in the EBM simulations is examined in Chapter 10. In a second step animproved version of the FSLBM B.C. is introduced in Chapter 11.

The procedure and results of the following chapter are mainly based on thework of Bogner et al. [2015]. This publication is a common and cooperatework of Simon Bogner and me. We applied a Chapman–Enskog expansion onthe FSLBM boundary condition. With this procedure the spatial order of theaccuracy is determined and compared to the spatial order of accuracy of theLBM which is of second order.

10.1 Analyzing Boundary Conditions viaChapman–Enskog Expansion

Based on the work of Ginzburg et al. [2008] the free surface boundary conditionof FSLBM is analyzed via the Chapman–Enskog expansion which is introducedin Section 4.1.3. The Chapman–Enskog expansion is applied to the FSLBMformulation using the two relaxation times (TRT) collision operator. The TRTcollision operator can be interpreted as a generalization of the known BGK oralso known SRT collision model (cf. Section 4.1.2) by taking the same value forboth relaxation times. Using the TRT collision operator instead of SRT enablesa simpler analysis. Nevertheless, the analysis can be generalized to other col-lision operators without having any effect of the convergence order. The ideaof taking two different relaxation times for the collision is explained in the fol-lowing. Afterward, FSLBM is developed using TRT collision operator and theChapman–Enskog expansion is applied to this formulation in order to deter-mine the accuracy.

10.1.1 TRT Collision Model

An essential part of the LBM algorithm is the numerical treatment of the colli-sion integral of the Boltzmann equation given in Eq. (4.9). There exist severalLBM collision models. The most common collision modes are the BGK (see Qianet al. [1992]) or also called SRT model, the TRT model (Ginzburg et al. [2008]),and the MRT model (see d’Humieres [1992] and d’Humieres [2002]). The three

150


models are visualized in Figure 10.2. The BGK model is explained in detail inSection 4.1.2 and used for all EBM simulations of the previous Chapter 9. It isthe most basic one and uses just one relaxation parameter. This relaxation pa-

more computational effortmore accuracy/stability

BGK - 1 relaxation timeτ ∼ ν

TRT - 2 relaxation timesλ1 ∼ ν and λ2 free parameter

MRT - 19 relaxation timesusing collision matrix

Figure 10.2: The MRT collision model as a generalization of TRT and SRT/BGKcollision model.

rameter is related to the kinematic viscosity ν. For a lot of applications – as forexample the EBM process – this simple collision model is of sufficient accuracyand stability. However, BGK suffers from errors depending on this relaxationparameter. Thus, if the simulation set–up and also the kinematic viscosity arechanged different spatial errors can result. The multi–relaxation–times (MRT)collision model may overcome these problems since it uses different relaxationparameters for the kinetic modes and enables a fine–tuning of the free relax-ation times. Thus, numerical stability and accuracy may be improved. TheMRT uses a collision matrix to reflect the underlying physics of collision as arelaxation process. The MRT collision model can be interpreted as a generaliza-tion of the BGK collision model by using ΩBGK = 1

τ 1 where 1 ∈ R19×19 denotesthe unit matrix. Although the MRT collision model can repair stability problemsthe use of a collision matrix is costly. The TRT collision model, using just tworelaxation times, can be regarded as a bridge between BGK and MRT. It sharesthe simplicity of BGK but has also the advantages of one free collision parame-ter to adjust the model with regard to accuracy and stability. The main conceptof the TRT model is based on the theorem that every function or tensor can bewritten as the sum of symmetric/even (+) and antisymmetric/odd (-) functionsand tensors respectively. In the case of LBM the pdf fi can be presented as,

fi (x, t) = f+i (x, t)︸︷︷︸

symmetric part

+ f−i (x, t)︸︷︷︸antisymmetric part

, (10.1)

151


and the symmetric f+i (x, t) and antisymmetric part f−i (x, t) are defined to,

f+i (x, t) =

1

2(fi (x, t) + fi (x, t)) and f−i (x, t) =

1

2(fi (x, t)− fi (x, t)) , (10.2)

where the index i denotes again the diametrically opposite direction of latticelink i. The LBM algorithm consisting of a stream and collide step is writtenusing the TRT collision model of Ginzburg [2005a] and Ginzburg et al. [2008]to,

fi (x+ ei∆t, t) = fi (x, t) (10.3)

fi (x, t) = fi (x, t) + λ+

(f+i (x, t)− feq,+

i (x, t))

︸︷︷︸n+i

+

+ λ−

(f−i (x, t)− feq,−

i (x, t))

︸︷︷︸n−i

+Fi, (10.4)

where fi is the post–collision value and λ+, λ− ∈ (−2, 0) denote the two relaxationtimes. The symmetric relaxation parameter λ+ is related to the lattice viscosityν by,

ν = −1

3

(1

λ+ + 12

), (10.5)

and λ− can be chosen freely in order to guarantee numerical stability. Therelaxation times are used to define Λ±,

Λ± = −(1

2+

1

λ±). (10.6)

For a further characterization of the parametrization of the model the followingproduct is defined,

Λ = Λ+Λ−. (10.7)

Ginzburg et al. [2008] calls Λ magic parameter and for example Λ = 14 stands

for the best stability. n±i denotes the even and odd part the non–equilibriumfunction of the pdf. Fi stands for an external force term. For the sake of simplic-ity, the term of the symmetric/antisymmetric equilibrium density distributionfunction are abbreviated in the following by,

feq,±i = e±i . (10.8)

Generally, the TRT collision model can also be interpreted as a special caseof the MRT collision model. Here, only two different relaxation values are givenin the diagonal matrix and the collision operator is split into symmetric andantisymmetric part. Furthermore, the TRT collision model is a generalization of

152


the SRT/BGK model which can be obtained by assuming,

λ− = λ+ =1

τ, (10.9)

i.e., both relaxation times have the same value. Summarizing, BGK is a specialcase of TRT, and further the TRT model is a special case of the MRT collisionmodel. Each generalization has higher demands of computational costs butresults also in a more stable behavior (see Figure 10.2). In the following, theChapman–Enskog expansion of the TRT collision model is derived.

10.1.2 Chapman–Enskog Expansion of TRT Collision Model

The significance of the Chapman–Enskog expansion as a theoretical founda-tion for using LBM as a solver for the incompressible Navier–Stokes equationshas already been shown in Chapter 4. The idea and ansatz of this asymptoticanalysis is already explained in Section 4.1.3 for the example of the BGK colli-sion model. In the following the asymptotic analysis is applied to the LBM–TRTmodel consisting of Eq. (10.3) and Eq. (10.4). Following the work of Ginzburget al. [2008], using the asymptotic expansions of Eq. (4.20), Eq. (4.21), andEq. (4.22), and by the splitting of pdfs fi and the collision operator into sym-metric and antisymmetric parts it follows,

fi =

∞∑n=0

εn(fn,+i + fn,−i

), (10.10)

ΩTRT = λ+(f+i − e+

i

)− λ−

(f−i − e−i

). (10.11)

These expressions are inserted into the LBM–TRT algorithm of Eq. (10.3) andEq. (10.4) and result in,

∞∑n=0

εn

n!Dnt f

ni (x, t)−

∞∑n=0

εnfni =

=λ+

( ∞∑n=0

εnfn,+i − e+i

)+ λ−

( ∞∑n=0

εnfn,−i − e−i

). (10.12)

Following the asymptotic analysis the zeroth ε-term is split up,

ε0

0!D0t f

0i (x, t) +

∞∑n=1

εn

n!Dnt f

ni (x, t)− ε0f0

i −∞∑n=1

εnfni =

λ+

(ε0f0,+

i +

∞∑n=1

εnfn,+i − e+i

)+ λ−

(ε0f0,−

i +

∞∑n=1

εnfn,−i − e−i

). (10.13)

153


After dividing Eq. (10.13) by ε and for remaining terms ε→ 0 lead to,

0 = λ+(f+,0i − e+

i

)+ λ−

(f−,0i − e−i

). (10.14)

Eq. (10.14) presents the asymptotic analysis of order of O(ε0) of the LBM–TRTmodel. In order to express the second order of the expansion the process hasto be repeated. The assumption starts now at n = 1,

∞∑n=1

εnfni +∞∑n=1

εnDnt

n!

( ∞∑n=1

εnfni

)−∞∑n=1

εnfni =

= λ+f+,0i + λ+

∞∑n=1

εnfn,+i − λ+e+i +

+ λ−f0,−i + λ−

∞∑n=1

εnfni − λ−ε−i , (10.15)

which can be transformed to,

εDt

(f0i +

∞∑n=1

εnfni

)+∞∑n=2

εnDnt

n!

(f0i +

∞∑n=1

εnfni

)=

=∞∑n=2

(λ+εnfn,+i + λ−εnfn,−

)+ λ+εf1,+

i + λ−εf1,−i . (10.16)

Eq. (10.16) is divided by ε and after ε→ 0 it results in,

Dt0

(e+i + e−i

)= λ+f1,+

i + λ−f1,−i , (10.17)

and this represents the asymptotic analysis of O(ε1) of the LBM–TRT model. Theasymptotic analysis has to be done up to second order since the LBM itself hasalso a second order accuracy. Thus, the process has to be continued using theinformation of the zeroth and first order expansion Eq. (10.14) and Eq. (10.17),

εDt

(f0i +

∞∑n=1

εnfni

)+

∞∑n=2

εnDnt

n!

(f0i +

∞∑n=1

εnfni

)=

=∞∑n=2

(λ+εnfn,+i + λ−εnfn,−i

)+ λ+εf1,+

i + λ−εf1,−i . (10.18)

Using Eq. (10.17) it can be written as,

εDtf0i︸︷︷︸

(?)

+εDt

∞∑n=1

εnfni +

∞∑n=2

εnDnt

n!

(f0i +

∞∑n=1

εnfni

)=

154


∞∑n=2

(λ+εnfn,+i − λ−εnfn,−i

)+ εDt0

(f0i

)︸︷︷︸(?)

. (10.19)

The two terms marked by (?) can be combined but therefore the asymptoticanalysis of the temporal expansion,

∂t = ∂t0 + ε∂t1 + ε2∂t2 + . . . , (10.20)

and the definition of the total differential of Eq. (3.9) has to be kept in mind.Using these relationships it can be written,

ε2∂t1f0i + ε2Dtf

1i +εDt

∞∑n=2

εnfni +

(ε2D2

t

2+

∞∑n=3

εnD2t

n!

)(f0i +

∞∑n=1

εnfni

)=

= λ+ε2f2,+i + λ−ε2f2,−

i +∞∑n=3

(λ+εnfn,+i + λ−εnfn,−i

). (10.21)

After dividing by ε2 and for ε→ 0 the last equation is simplified to,

∂t1f0i +Dtf

1i +

D2t2

2f0i︸︷︷︸

12Dt(Dtf0i )

= λ+ε2f2,+i + λ−ε2f2,−

i . (10.22)

Using Eq. (10.17) the marked expression is written as,

1

2Dt(Dtf

0i ) =

1

2Dt

(λ+f1,+

i + λ−f1,−i

)+O(ε2). (10.23)

By inserting Eq. (10.23), Eq. (10.22) is simplified to,

∂t1f0i +Dtf

1i +

1

2Dt0

(λ+f1,+

i + λ−f1,−i

)= λ+f2,+

i + λ−f2,−i . (10.24)

Since the higher order derivatives of Dt can be ignored the last equation iswritten to,

∂t1f0i +Dt0f

1i +

1

2Dt0

(λ+f1,+

i + λ−f1,−i

)= λ+f2,+

i + λ−f2,−i (10.25)

The asymptotic expansion of the LBM–TRT model of second order O(ε2) is sum-marized to,

∂t1f0i +Dt0

(f1i +

1

2λ+f1,+

i +1

2λ−f1,−

i

)= λ+f2,+

i + λ−f2,−i . (10.26)

Concluding, the first three parts of the Chapman–Enskog expansion of theLBM–TRT algorithm are given by Eq. (10.14), Eq. (10.17), and Eq. (10.26). Thisasymptotic analysis is expanded independent of the form of the equilibrium dis-

155


tribution function e±i . Here, the polynomial version for the equilibrium functionis used and thus, the symmetric and anti–symmetric part of it is given by,

e+i =

ωic2s

Πi, (10.27)

e−i =ωic2s

ρ0ei,αuα, (10.28)

where ωi denote the lattice weights of Eq. (4.8) and cs the lattice speed of soundEq. (4.12). Πi is defined by,

Πi = p︸︷︷︸pressure

+Ni = p+1

2ρ0vαvβ

(ei,αei,βc2s

− δαβ)

︸︷︷︸non–linear contribution

. (10.29)

This asymptotic expansion of the LBM–TRT model and the even and odd rep-resentation of the polynomial equilibrium distribution functions form the basisfor the following examination of the accuracy of the FSLBM B.C..

10.2 Examination of the Accuracy Order of the FSLBMB.C.

The following results are mainly based on the publication of Bogner et al. [2015].For the examination the link–wise representation of the boundary conditions isintroduced based on the work of Ginzburg and d’Humieres [2003] and Ginzburget al. [2008]. The FSLBM B.C. is formulated in the link–wise closure relationand this form is used for the Chapman–Enskog expansion in order to determinethe general order of accuracy.

10.2.1 Boundary Closure Relations

The Boundary nodes xb are defined in Definition 4.2.1. Following the explana-tions in Section 4.2 the value fi (xb, t+ 1) cannot be computed by the streamand collide steps of the LBM algorithm. A closure relation is needed to solvethis problem. For the following derivation the link–wise multi–reflection closurerelation of Ginzburg and d’Humieres [2003] and Ginzburg et al. [2008] is used,already given in Eq. (4.41),

fi (xb, t+ 1) = a0fi (xb, t) + a0fi (xb, t) + a1fi (xb − ci, t) +

+ fp.c.i (xb, t) + f bi (xw, t) , (10.30)

when assuming for the coefficients κ1 = a0, κ−1 = a0, κ0 = a1, κ−1 = κ−2 = 0. Thepost–collision term fp.c.i depends on the local non–equilibrium term ni (xb, t) andf bi depends on the macroscopic boundary values at the wall point xw.

156


First, the free surface boundary condition introduced by Korner et al. [2005]is expressed in the link–wise closure relation Eq. (10.30) as,

fi (xb, t+ 1) = −fi (xb, t) + 2 · e+i (ρb,vb) , (10.31)

where vb denotes the velocity at the interface. The boundary value of the densityρb is determined by the the ideal gas law given in Eq. (4.17) with the pressureboundary value pp. In the next steps, the Chapman–Enskog expansion is ap-plied to the FSLBM boundary condition of Eq. (10.31) in order to determine theorder of accuracy.

10.2.2 Chapman–Enskog Expansion of FSLBM B.C.

Based on the work of Ginzburg et al. [2008] the Chapman–Enskog approachis applied for the FSLBM B.C. for the incompressible case. It is important tonote that based on the diffusive time scaling (see Junk and Yang [2005]) thetime derivatives of the first order in expansion parameter ε are dropped and thespatial and time step are scaled by ∆x2 = ∆t = O

(ε2). For brevity, the following

terms are introduced,

∂i = ci,α∂α, and ji = ci,αjα = ci,αρvα. (10.32)

Assuming a constant external force the even and odd part of the non–equilibriumfunction up to third order is defined as,

n±i =1

λ±

[∂i(e

∓i − Λ∓∂ie

±i ) + ∂te

±i

]+O(ε3), (10.33)

where Λ is given in Eq. (10.7) and Λ± of Eq. (10.6). The symmetric (Eq. (10.27))e+i and anti–symmetric (Eq. (10.28)) e−i part of the polynomial equilibrium func-

tions are inserted into Eq. (10.33) and a constant external force is assumed, nican be expressed depending on the macroscopic variables Πi and ji,

n+i =

1

λ+

ωic2s

[∂i (ji − Λ−∂iΠi) + ∂tΠi] +O(ε3), (10.34)

n−i =1

λ−

ωic2s

[∂i(Πi − Λ+∂iji) + ∂tji] +O(ε3). (10.35)

The approximated solution of the non–equilibrium function (Eq. (10.33)) isused for the examination of the boundary conditions. After the Taylor–expansionof all occurring fi around (xb, t) Eq. (10.33) is inserted into the correspondingclosure relation and is expressed depending on macroscopic variables Π, j. Ithas to be noted that space derivatives are of O(ε) and the time derivatives areof O(ε2) and only terms up to O(ε2) need to be included in the expansion.

The closure relation of the FSLBM B.C. Eq. (10.31) is expanded around (xb, t)on the left hand side. Substituting fi = ei + ni in order to split equilibrium andnon–equilibrium parts of the solution, and further separating into even and odd

157


parts, one obtains up to the order ε2,[e+i − Λ+λ+n

+i +

λ−2n−i + ∂t(e

+i − e−i )

](xb, t) = e+

i (xw). (10.36)

Eq. (10.34) and Eq. (10.35) are inserted into the last Eq. (10.36) and it is ob-tained,[

e+i − Λ+λ+

(1

λ+

ωic2s

(∂iji − Λ−∂iΠi) + ∂tΠi

)+

+λ−2

(1

λ−

ωic2s

(∂i (Πi − Λ+∂iji) + ∂tji)

)+ ∂te

+i − ∂te−i

](xb, t) = e+

i (xw, t) . (10.37)

It has to be mentioned that all terms associated with (xb, t) are placed on theleft hand side and on the right hand side only the intersection wall point xwremains. Eq. (10.37) can be summarized to,[(

1 +1

2∂i + Λ∂2

i + (1− Λ+)∂t

)e+i −

−(

Λ+∂i +Λ+

2∂2i +

1

2∂t

)e−i

](xb, t) = e+

i (xw, t) . (10.38)

In a last step the polynomial equilibria functions Eq. (10.27) and Eq. (10.28)are inserted, the non–linear terms can be neglected and all time derivatives aredropped, [

(1 +1

2∂i + Λ∂2

i )P − Λ+(1 +1

2∂i)∂iji

](xb, t) = pb(xw, t). (10.39)

Following Bogner et al. [2015] the left hand side of Eq. (10.39) can be interpretedas a combination of the Tayler–series expansion approximation of the pressurep and shear rate ∂iji at position xb + 1

2ci. For the case of a distance to the wallof d = 1

2 (for more information see Figure 4.9) this results in a second orderaccurate agreement (analogues, third order accurate for Λ = 1

8 ) of the pressureboundary value pb combined with a second order condition of vanishing shearstress in xw. It can be shown analytically (or by numerical tests) that the spe-cific case of a steady parabolic force–driven tangential free surface over a latticealigned plane with no–slip boundary condition is solved without a numericalerror by FSLBM B.C. of Korner et al. [2005] and assuming the film thickness isequal an integer value as d = 0.5 (see more details in Chapter 11). However, thespatial accuracy for both pressure and shear rate reduces to first order if d 6= 1

2 .In Chapter 11 it is shown how this deficiency is improved in order to guaranteesecond order accuracy for general distances.

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10.3 Summary

In Part III of the thesis general model extensions and improvements for thesimulation of the EBM process are examined. The three fields consisting ofnumerical methods, mathematical modeling and implementation offer the op-portunity to improve the accuracy of the simulation results. Since the order ofaccuracy of the FSLBM B.C. is unknown so far an examination of this is es-sential. Boundary conditions have a great influence on the simulation resultsand if they have a minor accuracy than the numerical solution method theycan diminish the overall accuracy. Thus, the FSLBM B.C. is examined in thischapter. It is analyzed via the Chapman–Enskog expansion. For the sake ofsimplicity, for the analysis the collision operator is changed from BGK to TRTsince the splitting of the pdf values into symmetric and antisymmetric part en-ables a simpler treatment. However, the convergence results of the analysis arenot affected by this change, i.e., the result of the accuracy is the same usingBGK, TRT, or MRT as numerical collision operator. In a first step, the boundarycondition of the FSLBM approach is written in a closure relation and this clo-sure relation is expanded by Chapman–Enskog. This results in a general firstorder accuracy for the pressure and the shear rate and a second order accuracyif the distance to the wall is equal to 1

2 . Since this dependence of the distance tothe boundary is restrictive and impracticable a general second order accurateboundary condition for the FSLBM approach is derived in Chapter 11.

159

Chapter 11: Second Order FSLBM Boundary Condition

11 Second Order FSLBM BoundaryCondition

When one door closes,another opens;but we often look so longand so regretfully upon theclosed door that we do notsee the one that hasopened for us.

(Alexander Graham Bell(1847 – 1922))

The free surface boundary approach has a huge impact on the accuracy of thesimulation results of the EBM process. In Chapter 10 the FSLBM B.C. is de-termined of first order accurate for general distances from the lattice boundarynode to the wall. Since the overall accuracy of the LBM is second order the useof the FSLBM B.C. may diminish the overall accuracy. Thus, it is inevitable toincrease the accuracy order of the FSLBM B.C. in order to guarantee an overallsecond order accuracy of the simulation results independent of the distancefrom the boundary nodes to the wall. Of course there exist also other pitfalls ofdiminishing overall accuracy. It has to be ensured that all numerical methodsare of second order, i.e., all other boundary conditions or possible data inter-polations have to be of second order. However, this chapter focuses just onthe derivation of a second order accurate free surface boundary condition. Inorder to construct new, higher order boundary conditions the following strat-egy is applied; the approximated solution of the non–equilibrium function upto third order is inserted into the general closure relation. In a second step theunknown coefficients are matched in order to ensure the desired accuracy forthe macroscopic variables. The strategy can be applied for TRT as well as BGKcollision operators and in this chapter the TRT model is chosen for the sake ofsimplicity.

The reminder of this chapter is the following. Firstly, an increased boundarycondition for shear rate and pressure is derived. In a next step a second orderaccurate FSLBM B.C. is deduced based on the same strategy. This boundarycondition is tested extensively by various numerical experiments, for exampleby a Couette flow and a breaking dam test case. The results of these numericalexperiments support the correctness of the new FSLBM B.C., and show also its

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functionality. The results of the original and new FSLBM B.C. are compared inorder to demonstrate the differences of both conditions. The derivation of thenew B.C. and the simulation results are mainly based on the collaborated workof Simon Bogner and me, published in Bogner et al. [2015]

11.1 Shear Rate and Pressure B.C. of Second OrderAccuracy

Before a higher order accurate B.C. for the shear rate and pressure is deducedgeneral conditions for free surfaces (explained in detail in Chapter 5) are re-vised. Eq. (5.12) can be written as,(

p− pb + σ∇ · n︸︷︷︸κ

)nα = Pαβnβ, (11.1)

where Pαβ denotes the viscous stress tensor defined in Eq. (3.29) and n thenormal vector. In the following the indices α and β are used in order to avoidconfusion arising by the pdf values fi. The right hand side of Eq. (11.1) can bereformulated by using Eq. (3.33) and Eq. (3.30),

(p− pp + σκ)nα = 2µ

(∂vα∂xβ

+∂vβ∂xα

)= 2ν

1

2

(∂jα∂xβ

+∂jβ∂xα

)︸︷︷︸

Sαβ

, (11.2)

and leads to a formulation depending on pressure values p and pb and theshear rate tensor Sαβ. jα stands for the momentum as defined in Eq. (10.32).Eq. (11.2) is projected on n,

p− pp + σκ = 2ν∂njn, (11.3)

and also projected on tangential vector,

0 = ∂τ jn + ∂njτ , (11.4)

which leads to conditions for normal and tangential viscous stresses. It hasto be noted that the abbreviations for the equilibrium distribution function e±i ,non–equilibrium part n±i as well as the relaxation parameters belonging to theTRT collision model which are defined in Chapter 10 are still valid in this Chap-ter.

The construction of a higher order B.C. for pressure and shear rate is basedon the link–wise multi–reflection closure relation of Eq. (10.30). Herein, thefollowing local correction term of post–collision value fp.c.i is assumed,

fp.c.i (xb, t) = C · n+i (xb, t), (11.5)

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with C denoting a constant parameter. Furthermore, the boundary value termin Eq. (10.30) is of the form,

f bi = α+e+i (ρb,vb) +D · ci,αci,βSbαβ, (11.6)

which enables the prescription of the pressure boundary value pb = c2sρb and

shear rate boundary value Sbαβ in xw. D stands also for a constant parameter.The following Taylor–expansion,

fi(xb − ci, t) = fi(xb, t+ 1) ≈ fi(xb, t) + ∂tei(xb, t), (11.7)

is used to replace fi (xb − ci, t) in the closure relation. Afterward, Eq. (10.30) issorted by placing all terms except the boundary value f bi (xw, t) on the left handside. Using the Chapman–Enskog approximation from Eq. (10.33) and rear-ranging terms with respect to the ∂nt (for n = 0, 1, 2), e±i , and n±i , it is obtained,(

α+e+i + β+n

+i + α−e

−i + β−n

−i + αt+∂te

+i + αt−∂te

−i

)(xb, t) = f bi (xw), (11.8)

with

α+ = 1− a0 − a0 − a1, (11.9a)

β+ = 1− (1 + λ+)(a0 + a0)− a1 − C, (11.9b)

α− = a0 − a0 − a1 − 1, (11.9c)

β− = (1 + λ−)a0 − (1 + λ−)a0 − a1 − 1, (11.9d)

αt+ = 1− a1, (11.9e)

αt− = −(1 + a1). (11.9f)

The main idea of the construction is to match the coefficients α±, β±, and αt±with the spatial Taylor–series expansion around xb up to the second order forpressure and shear rate, respectively. Since spatial derivatives of pressure andmomentum are contained in the non–equilibrium parts, Eq. (10.34)–Eq. (10.35),the system of equations leads to,

α− = 0, (11.10a)

β− = α+dλ−, (11.10b)

β+ = −α+Λ+λ+, (11.10c)

keeping α+ as free parameter. Here, β− is chosen to fit the coefficient of thefirst order derivative of the pressure in n−i . β+ is chosen to fit the coefficient of∂iji from n+

i with the second order derivative ∂2i ji from n−i . The closure relation

coefficients follow from the Eq. (11.9a), Eq. (11.10a), and Eq. (11.10b) as

a0 = 1− α+(1

2+ d), (11.11a)

a0 = 1− 1

2α+, (11.11b)

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a1 = dα+ − 1. (11.11c)

The coefficient C in the correction term fp.ci derived from Eq. (11.10c) is givenby,

C = α+λ+(1

2+ d)− 2λ+. (11.11d)

Eq. (11.8) is expressed depending on gradients of the equilibria with Eq. (10.33),

α+

(1 + d∂i + Λ∂2

i

)e+i − α+Λ+ (1 + d∂i) ∂ie

−i +

+ (1− a1 − α+Λ+) ∂te+i − (1 + a1 − α+d) ∂te

−i = f bi , (11.12)

and after substituting a1 and f bi it follows,

α+

(1 + d∂i + Λ∂2

i

)e+i − α+Λ+ (1 + d∂i) ∂ie

−i + (2− α+(d+ Λ+)) ∂te

+i =

= α+e+i (ρb,vb)−α+Λ+

ωic2s︸︷︷︸

=D

ci,αci,βSbαβ. (11.13)

On the right hand side of Eq. (11.13) the unknown coefficient of the boundaryterm Eq. (11.6) is determined as D = −α+Λ+

ωic2s

in order to fit with the lefthand side. Since the spatial approximation matches up to second order forboth pressure and shear rate this boundary condition is classified of secondorder accurate in space for both pressure and momentum. This means that theoverall accuracy is not diminished by the free surface boundary condition whenusing LBM.

At this point the spatial and temporal error terms of e+i receive special at-

tention since it contains the non–linear terms (see Eq. (10.27) and Eq. (10.29))which may cause numerical instabilities. The spatial error of second order isdetermined by α+(d

2

2 − Λ)∂ie+i . It is bounded and independent of the viscosity if

Λ is a fixed, constant value. This is a usual requirement for parametrization ofthe TRT collision model (cf. Ginzburg et al. [2008]). The error in time expressedby α+(1− 2

α++d+Λ+)∂te

+i is influenced by the lattice viscosity by Λ+/3 = ν. Usu-

ally there are two different cases. In the first case there is a high Mach number(see Definition 3.3.3) and low viscosity, i.e., it is the case of a high Reynoldsnumber (see Eq. (3.50)) regime which lead to Λ+ 1. The second case consistsof high Mach number and high viscosity which implies a low Reynolds number.However, in this Stokes–like regime non–linear terms does not play a role anddo not have to be included in the equilibrium function Ladd [1994a]. Thus themomentum–dependent error in time may be eliminated.

It should be noted that the coefficients a0, a0 and a1 are identical to the lin-ear interpolation based pressure boundary condition “PLI” of Ginzburg et al.[2008] (cf. Section 4.2). This is a direct consequence of the construction basedon matching the coefficients of the pressure gradients in the closure relation.However, the coefficients C and D are different from the PLI–rule. They areessential to obtain the ∂iji term on the left hand side of Eq. (11.13) and to de-

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fine the boundary value for the shear rate in the right hand side, respectively.With this preliminary work a general second order accurate FSLBM B.C. is con-structed in the following independent on the distance from lattice boundarynode to wall.

11.2 Second Order Accurate FSLBM B.C.

The boundary condition for pressure and shear rate derived in Section 11.1 isused to increase originally first order accurate FSLBM B.C. of Eq. (10.31) upto second order. Thus a second order accurate B.C. is obtained by definingEq. (11.11a) up to Eq. (11.11d) and setting Sbαβ = 0 at the boundary. However,for full consistency with the physical model due to Eq. (11.3) and Eq. (11.4) itis necessary to control the tangential and normal shear stresses individually.Let t1, t2,n be a local orthonormal basis with n normal to the free surfaceboundary. Using the indices α′, β′ for the corresponding coordinate system,related to the standard coordinates by rotation lαβ, the shear rate tensor can beexpressed in the local basis by,

Sα′β′ = lαα′ lββ′Sαβ. (11.14)

The respective entries of the shear tensor Sα′β′ can now be set individuallyaccording to Eq. (11.3) and Eq. (11.4) leaving the remaining components un-changed. In practice, Sαβ has to be obtained by extrapolation from the liquidphase. In the following the original first order accurate FSLBM B.C. is abbre-viated by FSK and the new, second order accurate B.C. is abbreviated by FSL.The FSK B.C. can reach second order accuracy for a plane aligned interface atdistance dx/2 from the boundary nodes, and is equivalent to the original FSLBMclosure relation Eq. (10.31) when setting D = 0. The FSL B.C. is the free surfacecondition based on the construction of Section 11.1. It has second order spatialaccuracy, and is fully consistent with Eq. (11.3) and Eq. (11.4). The coefficientsfor both free surface conditions FSK and FSL are depicted in Table 11.1.

a0 a0 a1 α+ C D

FSK −1 0 0 2 0 −2Λ+ωic2s

FSL 12 − δ 1

2 d− 1 1 λ+(12 + d)− 2λ+ −Λ+

ωic2s

Table 11.1: Coefficients of for both closure relations Eq. (10.30) with correc-tion term fp.c.i from Eq. (11.5) and boundary value term f bi fromEq. (11.6).(Table taken from Bogner et al. [2015].)

It is noted, that by setting D = 0 simplified boundary conditions are re-ceived which are consistent only with Eq. (11.4) but neglect the normal viscousstresses in Eq. (11.3). The importance of these terms has been discussed forinstance in McKibben and Aidun [1995], Hirt and Shannon [1968] and depends

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on the respective problem. However, for D = 0 all shear stress componentsvanish at the boundary. Numerical simulations of free surface flows often usethis simplified free surface condition. In this case the Sbαβ in Eq. (11.6) dropsout, and the condition can be implemented without the construction above andwithout extrapolation of Sαβ. In the following Section 11.3 numerical examplesshow the correctness and functionality of the general second order accurateFSL boundary condition.

11.3 Numerical Test Cases

All numerical examples and test cases of this Section use the TRT collision oper-ator described in Section 10.1.1 and the D3Q19 stencil (see Section 4.1.1). Thenumerical test cases for different channel flows are visualized in Figure 11.1.Assuming the flow variables like velocity v constant along the y–axis the rota-tion of the channel by angle α about the y–axis is considered. All test cases arerealized by the WALBERLA framework described in detail in Chapter 7.

x

z

α

Figure 11.1: Sketch of channel flow, rotated by angle α with respect to the lat-tice (taken from Bogner et al. [2015]).

11.3.1 Plate–Driven Planar Flow

In this validation case the transient behavior of a planar flow with initial condi-tion v(x, 0) = 0 is considered. The domain is periodic in the x– and y–directionhaving a free surface boundary at z = 0 and a solid wall at z = h that movesin x–direction with constant tangential velocity of vwall = 0.001 lattice units (cf.Figure 11.1 with α = 0). For this test case Yin et al. [2006] proposed the analyticFourier series solution,

vid(z, t)

vwall= 1−

∞∑k=0

4(−1)k

(2k + 1)πe−(2k+1)2π2µt/(4ρh2) × cos

((2k + 1)πz

2h

). (11.15)

This analytic solution is used to validate a free–slip boundary condition (fordetails see Section 4.2) which has same solution as the free surface conditionfor this case. For t 1 the flow profile gets quickly uniform because the free

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surface does not induce any friction. The introduction of the dimensionlesstime scale T = µt/(ρh2) enables the evaluation of the flow at time T = 1/64, 1/8,3/8 and 3/4. The parameters ρ = 1 and µ = 1

6 (both values are given in latticeunits) are used in the simulations for channel heights h = 8, 16, 32, and 64.The corresponding simulation results are shown qualitatively in Figure 11.2.It is observed that for FSK and FSL B.C. the resulting flow profiles are verysimilar because the channel height is an integer value (in lattice cells). Since

Figure 11.2: The velocity profile at non-dimensional times T = 1/64, 1/8, 3/8.FSK B.C. and FSL B.C. show a good agreement with analyticalsolution of Eq. (11.15) (taken from Bogner et al. [2015]).

Figure 11.2 conveys just qualitatively the behavior of both B.C. and says noth-ing about the quantitative accuracy order. Therefore, a quantitative evaluationof the error is necessary. The error ε is defined as,

ε(h, T ) =1

vwall

√1

h

∑zi

(vx(zi, T )− vid(zi, T ))2, (11.16)

where zi ranges over all lattice node positions along the z-axis. Figure 11.3shows that both boundary conditions yield correct transient behavior and theexpected second order rate of convergence is demonstrated. The results havebeen obtained with a TRT magic parametrization value of Λ = 1/4 (cf. Eq. (10.7)).This numerical example shows the special case where the FSK B.C. also leadsto second order accurate results. In the following numerical situations are ex-amined showing that for general cases FSK is only first order accurate whereasFSL is of second order convergence.

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(a) FSK

(b) FSL

Figure 11.3: Grid convergence of the FSK-rule (see a)) and the new FSL-rule (seeb)), based on linear interpolation.Because the channel width is an integral number both approacheshave a second order rate of convergence (both figures taken fromBogner et al. [2015]).

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11.3.2 Linear Couette Flows

Due to analysis linear flow profiles should be exactly recovered if the secondorder boundary condition of Section 11.1 is used (FSL-rule with prescribedboundary value Sbαβ). In the following steady flows are evaluated. In a cubicdomain there are no–slip boundary conditions (bounce-back, see Section 4.2) atz = 0 and the position of the first lattice node is at plane z = 0.5 (cf. Figure 11.1with α = 0). The shear rate condition of Section 11.1 is imposed at z = h. A firstvalidation set–up imposes a tangential shear rate with Sbxz(z = h) = 0.001. Theresulting steady Couette profile is recovered without a numerical error, indepen-dent of the choice of equilibrium function and film thickness h, in accordancewith the analytical properties of the B.C..

The next test case considers a rotated linear film flow where bottom andtop boundary planes are placed with a slope of ∆z/∆x = 1/4 (i.e., α ≈ 14

in Fig. 11.1). In order to realize the skewed no–slip boundary the CLI B.C.introduced in Ginzburg et al. [2008] is used. It is a second order link–wiseB.C. similar to the one proposed by Bouzidi et al. [2001] which is based onlinear interpolation (see Section 4.2). This B.C. enables the exact solution ofsteady Couette flows with arbitrary rotated channels when using linear equi-libria. Applying again a tangential shear rate ∂nvt = 0.001 the exact profileis recovered. The use of non–linear equilibrium functions lead to a spuriousKnudsen layer at the boundary nodes of the skewed channel where the shearrate is set by Eq. (10.39). Regarding the analysis this error should be of secondorder. A grid convergence study with a fixed lattice viscosity ν and Reynoldsnumber Re = 0.064 is conducted, with the following varying channel widths hi =h0, 2h0, 4h0, 8h0 and imposed shear rates ∂nvt = 0.001, 0.00025, 6.25e−5, 1.5625e−5.For the grid convergence study the relative errors are computed either by theL2–norm,

L2(Φ) =

√∑x(Φ(x)− Φid(x))2∑

xΦid(x)2, (11.17)

or the Tchebysheff norm,

L∞(Φ) =maxx |Φ(x)− Φid(x)|

maxx |Φid(x)| , (11.18)

where Φ(x) and Φid(x) are the respective numerical and the ideal values at thenode position x. Figure 11.4 shows second order grid convergence as it wasexpected.

11.3.3 Steady Parabolic Film Flow

In a third numerical experiment a force–driven slow flow of finite thicknessover a planar no–slip surface is used for validation since it enables an analyticsolution. The simulation set–up consists of a cubic domain where a no–slipbounce–back B.C. is imposed at the bottom z = 0 plane of the domain. This

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Figure 11.4: Second order convergence rate for linear shear flow when imposingconstant shear rate on the top boundary in a skewed channel withslope ∆z/∆x = 1/4.

means that the first lattice nodes are located at a distance 0.5 from the bottomplane. At the plane z = h a free boundary is set by the FSL B.C. of Section 11.1with Sbαβ = 0. Furthermore, periodic boundary conditions in the x and y direc-tion are used. The magic parametrization Λ = 3/16 proposed by Ginzbourg andAdler [1994] for parabolic straight channel flows is set in order to eliminate theerror of the bounce–back condition. It can be verified that the shear boundarycondition leads to the correct steady state profile without numerical error, in-dependent of the film thickness h and the choice of the equilibrium function.Applying additional gravity directed towards the bottom plane yields an addi-tional linear hydrostatic pressure gradient not influencing the solution providedthat the “incompressible equilibria” He and Luo [1997c] are used. It is impor-tant to note that the FSLBM B.C. of Korner et al. [2005] given by Eq. (10.31)is exact in this test case only if h is divisible by the grid spacing. If it is notdivisible accuracy is only of first order O(δx). Figure 11.5 shows that the mea-sured error convergence with the FSLBM B.C. of Korner et al. [2005] given byEq. (10.31) is reduced to first order for h = 8.33.

The test case is repeated with the flow direction rotated about a slope of∆z/∆x = 1/7 (α ≈ 8.1 in Figure 11.1) with respect to the lattice. The CLI B.C. isused for the skewed no–slip wall to ensure a second order rate of convergenceand fix the parametrization with Λ = 1/4. Similar to Couette flow a certainerror is inevitable. Using the interpolated FSL B.C. at the free boundary anerror convergence rate of order O(δ2

x) is expected, independent of the flow direc-tion, opposed to a first order error for the original free surface condition fromEq. (10.31) (FSK with D = 0, Table 11.1). For the convergence study the grid

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Figure 11.5: First order convergence rate for a planar film flow of height h = 8.33with the FSK B.C..FSL B.C. solves the test case exactly (taken from Bogner et al.[2015]).

Figure 11.6: Comparison of the convergence behavior in a rotated planar filmflow. The rate of convergence with the proposed FSL B.C. is secondorder. The behavior of the FSK B.C. with D = 0 is below secondorder (taken from Bogner et al. [2015]).

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spacings are δx = 1, 0.5, 0.25, 0.125, 0.0625 and Reynolds number is kept constantby adjusting the accelerating force according to g = g0 × δ−3

x at a constant re-laxation time τ = 2. Figure 11.6 visualizes grid convergence of the two differentB.C.. The new FSL B.C. shows a second order behavior whereas for the originalFSK B.C. the rate of convergence is below second order in this test case.

11.3.4 Breaking Dam

In the last numerical experiment the effect of the viscous stress term in the freesurface condition of Eq. (11.3) and Eq. (11.4) in the instationary case of a col-lapsing rectangular column of liquid under gravity is examined. This scenariois well–known as breaking dam. At the surge front the condition ∂njn > 0 isvalid. Although the simplified boundary rule with D = 0 determines ∂njn = 0at the free surface a lower acceleration of the surge front is accepted for thisscenario. The initial size of the dam column is 80 x 40 lattice cells, the latticeviscosity is ν = 1

3 , and the maximal flow velocity is 0.05. These flow parametersdetermine a Reynolds number of Re = 12.

Figure 11.7: Simulation of a breaking dam after time step 2000, 4000, 6000,8000. The faster surge front (black) is obtained from using the fullboundary condition, while the slower front (gray blue) is obtainedfrom the simplified boundary scheme with D = 0 in Eq. (11.6)(taken from Bogner et al. [2015]).

Figure 11.7 shows that the collapse of the column at four different time stepsfor two different conditions at the free surface. The black line shows the resultswhen using the full B.C. whereas the blue lines visualize the collapsing frontby imposing the simplified B.C. with D = 0. The velocity of the collapse issignificantly slower if the terms are neglected in the boundary rule. This showsthe importance of the D term.

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For the simulation of the breaking dam scenario first order FSK B.C. rule andonly first order (next–neighbor) extrapolation of the macroscopic flow quantitiesρ, v and Sαβ are used for the computation of the boundary values. The interfacetracking implementation is directly based on the original works Korner et al.[2005], Thurey et al. [2006].

11.4 Summary

After the accuracy analysis of the FSLBM B.C. in Chapter 10 an improved spa-tial second order accurate free surface B.C. called FSL is derived in this chap-ter. Therefore, the construction of second order accurate B.C. for shear rateand pressure are derived based on the Chapman–Enskog analysis of the lat-tice Boltzmann equation. This approach is further extended to deduce a spatialsecond order accurate free surface boundary condition to replace the FSLBMapproach of Korner et al. [2005]. The main strategy of the construction of theseB.C. is the matching the coefficients of a expanded Chapman–Enskog expan-sion up the desired order. Various numerical experiments confirm the resultsof the analytical derivation, i.e., that the proposed new boundary scheme FSLis of second order accuracy whereas the FSK model is only first order accurate.For these validations analytic solutions as well as grid convergence studies areused. However, it is also shown that full second order accuracy is only reachedby defining the free interface position geometrically with the same accuracyorder in order to get the d values of the link intersection with the boundary.However, this is not possible with the interface tracking approach used in theoriginal FSLBM of Korner et al. [2005]. In order to guarantee second order ac-curate determination of the interface determination higher order interface re-construction methods or alternative methods such as let set methods of Osherand Fedkiw [2001], Setian and Smereka [2003] have to implemented in orderto represent the free surface boundary. Since the FSK B.C. is purely local theother interface tracking methods may increase the algorithmic complexity andtherefore also the computational costs but ensure the overall accuracy as dis-cussed in Nichols and Hirt [1971]. Thus, the FSL B.C. can serve as the basisfor currently developing a second order accurate implementation of free surfaceflows with the LBM when it is combined with second order accurate interfacetracking.

At this point further notes have to be made. The full consistency with theequations describing the free surface requires that the scheme needs an ap-proximation of the shear stress at the boundary in order to set the correctboundary values on the LBM data. For dam break problem with Re = 12 thesignificance of the viscous stresses at the boundary becomes evident. For flu-ids with lower viscosities this term may become less important. For under–resolved free surface simulations it is often more accurate to employ the simpli-fied boundary scheme because physical viscosity and simulated viscosity do notmatch. Janssen et al. [2010] showed a good agreement of high Re - breaking

173


dam simulations with experimental data using the original FSLBM neglectingthe viscous terms. This effect in a well–known issue for free surface simulationsand has already been described in Hirt and Shannon [1968].

174

Chapter 12: Conclusion and Future Work

12 Conclusion and Future WorkFor the future - there isalways one!

(Miller Mendoza Jimenez(1985))

12.1 Conclusion

This thesis showed successfully how additive manufacturing processes likeelectron beam melting given in Chapter 2 can be modeled using Navier–Stokesand heat equation (see Chapter 3). The discretization was done by a multi–distribution lattice Boltzmann method (cf. Chapter 4) and the liquid–gas in-terface was treated with a LBM specific free surface approach as explained inChapter 5. The parallelized algorithms corresponding to the fluid dynamicswere embedded in the LBM based WALBERLA software framework which pro-vides already the necessary infrastructure to run highly parallel simulationson state–of–the–art supercomputers (see Chapter 7 and Chapter 8). The metalpowder particles of the powder bed were modeled as rigid bodies in the pe frame-work (Chapter 6) which is a highly parallelizable physics engine. Parallel ab-sorption algorithms were deduced using both frameworks coupled.

A detailed and extensive validation of the EBM model was described in Chap-ter 9 and implementation was carried out. Benchmarks examining the phasechange from solid to liquid and the incorporation of the heat equation were ex-amined as well as the comparison of numerical with experimental data. Sincethese validation studies showed that the EBM model is sufficiently close to re-ality and the simulation results are highly concordant with experimental data,simulations can be used to derive new process strategies. This thesis focusedthe hatching process since it occurs often in reality. Different hatching strate-gies were examined with regard to the quality properties of the part and toachieve a speed–up of the build time. The quality of a build part is classifiedinto porous, good, and parts with swelling effects at the surface. A reduced sim-ulation set–up was applied to scrutinize the quality depending on the processparameter like scan velocity and beam power using different process strategies.The results were visualized in so-called process windows. The first strategy de-creases the line offset, more precisely the line offset was halved and the scanvelocity was doubled in order to achieve a better range for process parameters todefine good products. A second strategy was based on the increase of the beamwidth. The strategies were promising to improve the quality and to shorten

175


the build time of the process. The process windows also indicate that there isa constant border between the parameter range resulting in good and porousbuild parts. This behavior is based on the fact that each powder layer has a cer-tain minimum amount of powder particles which have to be melted. All thesesimulation results support EBM machine developers adjusting the machinesand developing also new machines as well as the manufacturer of the powderparticles.

In the third and last part of this thesis the accuracy order of the appliedFSLBM approach was examined (cf. Chapter 10) which treated the liquid–gasinterface by setting a boundary condition at the surface and did not computethe gas dynamics explicitly. So far, a theoretical foundation of FSLBM to deter-mine the order was missing. The analysis was based on a Chapman–Enskogexpansion and resulted in a spatial first order accuracy for general cases. Asecond order accurate boundary condition for the free surface was derived inChapter 11 since the accuracy of the LBM is of second order and a lower orderfree surface approach would reduce the overall accuracy. Several benchmarksand numerical test cases of stationary and instationary problems show the spa-tial order accuracy of the new boundary condition for general cases.

12.2 Future Work

The last part of this thesis dealt already with model extensions focusing on thefree surface approach. However, there are further model extensions possibleregarding the EBM model itself which could further improve the quality of sim-ulations. The modular architecture of WALBERLA enables a smooth integrationof further modeling aspects.

A first extension of the EBM model would be a temperature dependent sur-face tension (see Kayser [1976]). The EBM simulations in this thesis work on asurface tension which is independent of the temperature. Furthermore, an in-corporation of the evaporation phenomena and the corresponding evaporationpressure is missing (cf. Klassen et al. [2014b]). The evaporation itself results ina cooling effect during the process. Since it is missing, the computed tempera-ture values are too high and thus, swelling effects on the part surface have tobe estimated numerically by a threshold. However, the evaporation effect leadsto a recoil pressure and high pressure gradients at the free surface. Thesehigh gradients may result in instabilities and in high surface forces. The cor-responding high velocities violate the low Mach number assumption which isessential for the LBM solving the incompressible Navier–Stokes equations. Thenumerical solution of this problem in three dimensions is a challenging taskand requires a smaller grid resolution in order to prevent instabilities and highMach numbers. This requires a higher computational effort and increases thesimulation time. In order to limit the computation time grid refinement maybe applied. Here, the region of the interface is resolved with smaller grid sizesand the region of the melting pool uses larger grid sizes. Summarizing, the

176


integration of the evaporation is extensive but indispensable.This thesis gives an overview of the different classes of additive manufacturing

methods but the simulation focuses the EBM process. The EBM model can beused as a starting point to simulate other additive manufacturing processes,especially other powder bed fusion processes. For example, when simulatingthe laser sintering process the modeling of the powder particles can be re–usedand just the absorption algorithms have to be adapted for lasers. Thus, theadapted models and implementations can be used to gain a profound knowledgeof the third industrial revolution by simulations.

Another topic of future research is a more accurate reconstruction of theinterface position. A second order accurate boundary condition for the freesurface was derived in this thesis but it was also mentioned that for the ensurean overall second order accuracy the position of the interface has to be deter-mined more precisely, for example by the level set method (see Adalsteinssonand Sethian [1999]). A more accurate interface reconstruction would furtherimprove the free surface approach and strengthen its usability for other appli-cations.

177

Chapter 13: Journal and Conference Publications

13 Journal and ConferencePublications

1. M. Markl, R. Ammer, U. Ljungblad, U. Rude, C. Korner.Electron Beam Absorption Algorithms for Electron Beam Melting ProcessesSimulated by a Three-Dimensional Thermal Free Surface Lattice BoltzmannMethod in a Distributed and Parallel EnvironmentProcedia Computer Science, 18, 2127 – 2136 (2013).

2. R. Ammer, M. Markl, U. Ljungblad, C. Korner and U. Rude.Simulating fast electron beam melting with a parallel thermal free surfacelattice Boltzmann method.Computers & Mathematics with Applications, 67(2), 318 – 330 (2014)

3. R. Ammer, R. Rude, M. Markl, V. Juchter and C. Korner.Validation experiments for LBM simulations of electron beam melting.International Journal of Modern Physics C, 25(11), 1441009 -1 – 1441009-9 (2014)

4. M. Markl, R. Ammer, U. Rude, C. Korner.Numerical Investigations on hatching process strategies for powder-bed-based additive manufacturing using an electron beam.The International Journal of Advanced Manufacturing Technology, 78(1-4),239 – 247 (2015)

5. S. Bogner, R. Ammer, U. Rude.Boundary Conditions for Free Interfaces with the Lattice Boltzmann Method.Journal of Computational Physics, 297, 1 –12 (2015)

179

Chapter 14: Talks and Trainings

14 Talks and Trainings

1. 25.07.2012, The Ninth International Conference for Mesoscopic Methodsin Engineering and Science (ICMMES), Taipei (Taiwan),Simulating Fast Electron Beam Melting with a Parallel Thermal Free SurfaceLattice Boltzmann Method.

2. 17.07.2013, The 22nd International Conference on the Discrete Simula-tion of Fluid Dynamics (DSFD), Yerevan (Aremenia),Validation Experiments for LBM Simulations of Electron Beam Melting.

3. 19.06.2014, The 4th European Seminar on Computing (ESCO), Pilsen(Czech Republic),Simulating Additive Manufacturing Processes With a 3D Free Surface LatticeBoltzmann Method.

4. 31.07.2014, The 23rd International Conference on Discrete Simulation ofFluid Dynamics (DSFD), Paris (France),Modeling of Thermodynamic Phenomena with Lattice Boltzmann Method forAdditive Manufacturing Processes.

5. 10.03.2015, High Performance and Parallel Computing for Material De-fects and Multiphase Flows, Singapore (Singapore),Complex Flows with the Lattice Boltzmann Method.

6. 14.07.2015, The 24th International Conference on Discrete Simulation ofFluid Dynamics (DSFD), Edinburgh (United Kingdom),Boundary Conditions for Free Interfaces with the Lattice Boltzmann Method.

181

Chapter 15: Projects and Supervised Student Theses

15 Projects and Supervised StudentTheses

1. Markus Siko, Diplom Thesis: Numerical Modeling of Cell Conversions re-garding Conservation Laws, 2013.

2. Ghulam Mustafa Majal, Master Thesis: Entropic stabilization of the latticeBoltzmann method, 2014.

183

List of Figures

List of Figures

1.1 Simulation pipeline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Structure of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Timeline of development of additive manufacturing. . . . . . . . . . 82.2 Process chain of AM processes. . . . . . . . . . . . . . . . . . . . . . 92.3 Branches of industry interested in AM technologies. . . . . . . . . . 122.4 Classification of AM processes. . . . . . . . . . . . . . . . . . . . . . 142.5 Schematic of EBM process (taken from Ammer et al. [2014a]). . . . 20

3.1 Eulerian and Lagrangian description of fluid flow. . . . . . . . . . . 253.2 Dilatation of initial material volume to elementary material volume. 26

4.1 Classification of numerical discretization methods into scale cate-gories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Selected stencils in 2D and 3D for LBM. . . . . . . . . . . . . . . . . 414.3 Connection between the LBM and the conservation laws by the

Chapman–Enskog expansion. . . . . . . . . . . . . . . . . . . . . . . 474.4 Situation of a solid–liquid interface for D2Q9 stencil: boundary

nodes () have unknown pdf’s after the streaming step (right: un-knwon pdf’s are dashed). . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Periodic boundary conditions in x-direction for LBM (D2Q9). . . . . 534.6 No–slip bounce–back boundary condition for D2Q9. . . . . . . . . . 544.7 Staircase approximation of curvilinear surface by bounce–back

method (dots visualize the boundary nodes). . . . . . . . . . . . . . 554.8 Free–slip boundary condition for D2Q9. . . . . . . . . . . . . . . . . 554.9 Overview of interpolation based boundary conditions in 1D. . . . . 564.10Overview of multi-reflection boundary condition in 1D. . . . . . . . 56

5.1 Two-phase problem with fluid 1 and fluid 2, separated by interfacelayer (for example, liquid and air). . . . . . . . . . . . . . . . . . . . . 60

5.2 Example of a deformed drop on a bottom wall boundary. Gasphase is separated from the fluid phase by an interface layer (taken from Ammer et al. [2014a]). . . . . . . . . . . . . . . . . . . . 66

5.3 Missing pdfs (dashed) after streaming for a D2Q9 stencil. . . . . . . 685.4 Idea of interface boundary condition (unknown pdf values coming

from the gas phase visualized in red, dashed arrows). . . . . . . . . 685.5 Cell types in thermal FSLBM explained by the example of a melted

spot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

185

List of Figures

5.6 All possible cell conversions due to liquid–gas interface and solid–liquid phase transition. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.1 EBM requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Cross section of assumed electron beam power intensity based ontwo–dimensional Gaussian profile of a 1kV electron beam (red)with a standard deviation of σb = 100µm (green). The resultingFWHM beam diameter is dbeam = 235.5µm (blue). . . . . . . . . . . . 77

6.3 Ti–6Al–4V powder particle size distribution generated by EIGA andmeasured by light scattering method. The histogram representsthe mean relative frequency ∆Q0. The mean relative frequency q0

is given in its 95 % confidence interval by ±1.96s(q0) (data providedby Thorsten Scharowsky). . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4 The IG distribution function f(dp, dp,0 = 17µm, µ = 44µm, λ = 273µm)approximates the relative frequency density distribution q0 with95 % confidence interval 1.96s(q0). . . . . . . . . . . . . . . . . . . . 82

6.5 Relationship of particle diameter db and quantile value. . . . . . . . 83

6.6 Energy dissipation volume of an electron beam with width σb andpenetration depth R. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.7 Relation between penetration depth and absorption coefficient for60 kV and 120 kV (cf. Kanaya and Okayama [1972]) and suitableapproximations (graphs taken from Markl et al. [2013]). . . . . . . . 87

6.8 Absorption behavior for varying fill levels (taken from Markl et al.[2013]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.1 Structure of WALBERLA. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2 Uniform domain decomposition. . . . . . . . . . . . . . . . . . . . . . 95

7.3 Ghost layer concept for parallelization in WALBERLA. . . . . . . . . . 95

7.4 a) Contact detection of two spheres with center positions p1 andp2, radii r1 and r2, angular velocities ω1, ω2.b) Spring–dashpot system between two bodies where kn is thespring stiffness, γn and γt denote the damping coefficients in nor-mal and tangential directions. Penetrations are dealt in normaldirections and friction is determined in tangential direction. . . . . 97

7.5 General four–way coupling of WALBERLA and pe. . . . . . . . . . . . 101

8.1 Simplified WALBERLA – pe coupling for the EBM application. . . . . 107

8.2 Powder particles generated by pe. . . . . . . . . . . . . . . . . . . . . 109

8.3 Communication schemes for the absorption algorithms. Local com-munication for the streaming of pdf values (red). Top–to–bottomcommunication for beam absorption values (green) (taken fromMarkl et al. [2013]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

186

List of Figures

8.4 Absorption coefficients in powder slice for 60 kV acceleration volt-age: the global domain is split into three blocks. The absorptionalgorithm computes the pre values (a), then values are communi-cated, and correct post–values can be computed (b) (taken fromMarkl et al. [2013]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.5 Weak scaling results of EBM model on Lima compute cluster. . . . 1188.6 Weak scaling results of the EBM model on SuperMUC cluster. . . . 1188.7 Weak scaling results of the EBM model on SuperMUC cluster. . . . 1208.8 Strong scaling results of EBM model on LiMa cluster. . . . . . . . . 1208.9 Strong scaling results of the EBM model on SuperMUC for three

different setups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

9.1 Sketch of validation scenario of energy equation (taken from Am-mer et al. [2014a]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9.2 Comparison of numerical and analytical solutions of heat equation. 1299.3 Comparison of numerical and analytical solution of the Stefan

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.4 EBM specific validation set–up (taken from Ammer et al. [2014b]). . 1319.5 Categories of samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.6 Process window for experimental generated EBM parts (taken from

Ammer et al. [2014b]). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.7 Sketch of simplified simulation scenario (cf. Ammer et al. [2014b]). 1339.8 Process window of numerical data. ∆x = 5·10−6 m, ∆t = 1.75·10−7 s,

simulated powder domain (1.44 x 0.64 x 0.24)·10−3 m3, beam offset =13.56 · 10−3 m, and a line offset of 100 µm (taken from Ammer et al.[2014b]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.9 Numerical hatching results for the parameter set(

6.4ms , 0.15kJ

m

). . 136

9.10Extended numerical process window of the EBM process with 100 µmline offset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.11Partial process window with increased beam parameters. . . . . . 1419.12Numerical process window with a scan line offset of 50 µm . . . . . 1429.13Maximum temperatures for different line offsets. Lines with the

same colors can be compared by having the same beam power.The gray shaded area shows the experimentally validated data upto 6.4 m

s (taken from Markl et al. [2015]). . . . . . . . . . . . . . . . . 1439.14Time–dependent melting pool volume for different scan velocities

with 0.05 kJm : at a scan velocity of 50 m

s and 70 ms the previous scan

line is still liquid and the melt pool grows continuously (taken fromMarkl et al. [2015]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

10.1Possible strategies in order to improve accuracy of simulation re-sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10.2The MRT collision model as a generalization of TRT and SRT/BGKcollision model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

187

List of Figures

11.1Sketch of channel flow, rotated by angle α with respect to the lat-tice (taken from Bogner et al. [2015]). . . . . . . . . . . . . . . . . . . 166

11.2The velocity profile at non-dimensional times T = 1/64, 1/8, 3/8.FSK B.C. and FSL B.C. show a good agreement with analyticalsolution of Eq. (11.15) (taken from Bogner et al. [2015]). . . . . . . . 167

11.3Grid convergence of the FSK-rule (see a)) and the new FSL-rule(see b)), based on linear interpolation. Because the channel widthis an integral number both approaches have a second order rateof convergence (both figures taken from Bogner et al. [2015]). . . . 168

11.4Second order convergence rate for linear shear flow when imposingconstant shear rate on the top boundary in a skewed channel withslope ∆z/∆x = 1/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

11.5First order convergence rate for a planar film flow of height h = 8.33with the FSK B.C.. FSL B.C. solves the test case exactly (takenfrom Bogner et al. [2015]). . . . . . . . . . . . . . . . . . . . . . . . . 171

11.6Comparison of the convergence behavior in a rotated planar filmflow. The rate of convergence with the proposed FSL B.C. is secondorder. The behavior of the FSK B.C. with D = 0 is below secondorder (taken from Bogner et al. [2015]). . . . . . . . . . . . . . . . . . 171

11.7Simulation of a breaking dam after time step 2000, 4000, 6000,8000. The faster surge front (black) is obtained from using the fullboundary condition, while the slower front (gray blue) is obtainedfrom the simplified boundary scheme with D = 0 in Eq. (11.6)(taken from Bogner et al. [2015]). . . . . . . . . . . . . . . . . . . . . 172

188

List of Tables

List of Tables

2.1 Differences between EBM and SLM (Gibson et al. [2015]). . . . . . . 21

5.1 Comparison of multi–phase LB models. . . . . . . . . . . . . . . . . 71

8.1 Particle parameters for the DEM solver in the pe for the Ti–6Al–4Vpowder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.2 Characteristics of Lima and SuperMUC compute cluster. . . . . . . 1168.3 Parameters of the EBM setup for weak scaling experiments on

LiMa and SuperMUC. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.4 Details of three setups for strong scaling on SuperMUC. . . . . . . . 121

9.1 L1–error of numerical and analytical solution of the energy equation.1289.2 Data for the Stefan problem. . . . . . . . . . . . . . . . . . . . . . . . 130

11.1Coefficients of for both closure relations Eq. (10.30) with correc-tion term fp.c.i from Eq. (11.5) and boundary value term f bi fromEq. (11.6). (Table taken from Bogner et al. [2015].) . . . . . . . . . 165

189

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Index

Index

WALBERLA, 923D Printing, 16

accommodation layer, 52Additive Manufacturing, 7, 8additive manufacturing methods, 2automated fabrication, 7

Ballistic Particle Manufacturing, 15BGK collision operator, 43Binder Jetting, 13, 16Bioextrusion, 15body force, 29Boltzmann equation, 43bond-then-form, 16bounce–back, 53Boundary Element Method, 59boundary node, 52Boundary–fitted Grid Method, 59breaking dam, 161, 172

capillary equation, 62capillary pressure, 62Cauchy’s equation of motion, 29Chapman-Enskog expansion, 45communication, 95Computational Fluid Dynamics, 91computer aided design (CAD) model,

9conservative, 38constitutive equation, 30continuity equation, 28continuum hypothesis, 24Contour Crafting, 15coproducing, 11Couette flow, 161, 169, 170customized mass production, 10

deformation tensor, 30dilatation, 26Direct Metal Deposition, 18Direct Metal Laser Sintering, 19Direct Simulation Monte Carlo, 39,

96direct stress, 30Directed Energy Deposition, 13, 17Discontinuous Galerkin Method, 38discrete element method (DEM), 96distributed–memory computer, 114domain decomposition, 94dynamic viscosity, 30

EIGA, 78Electron Beam Melting, 19electron emitter, 76Entropic lattice Boltzmann method,

44equation of state, 33Euler equations, 48Euler expansion formula, 27Eulerian coordinates, 24

Fast Frictional Dynamics, 96Fast Marching Method, 65fill level, 66Finite Difference Method, 37Finite Volume Method, 38first law of thermodynamics, 31form-then-bond, 17Fourier’s law, 32free–slip, 54Freeform Fabrication, 8front capturing methods, 59front tracking method, 59

207

Index

full–way bounce–back, 54Fused Deposition Modeling, 15FWHM, 76

Galerkin method, 37Galilean invariance, 40ghost cells, 53ghost layer, 53, 94Green’s theorem, 27, 38

H–theorem, 44half–way bounce–back, 54Hamilon–Jacobi equation, 65heat conduction, 32hydrostatic equation, 30hyperthreading, 119, 121

ideal gas law, 33induction coil, 76integrated functionality, 10interface, 60inverse Gaussian distribution, 81isotropy, 40

JUGENE, 93

kinematic viscosity, 30kinetic energy, 31Knudsen layer, 52, 169

L2–norm, 169Lagrangian coordinates, 24Laminated Object Manufacturing, 16Laser Cusing, 19Laser Engineering Net Shaping, 18Laser Freeform Fabrication, 18Lattice Boltzmann Method, 38, 39lattice Boltzmann method, 3lattice cell, 40Lattice Gas Cellular Automata, 39lattice tensors, 41lattice weights, 42Level Set Method, 65light scattering method, 78line energy, 131linear complementary problem, 96

Mach number, 35, 164magic parameter, 152Marker–and–Cell Method, 59mass exchange, 67mass fraction, 66material coordinates, 24material derivative, 25Material Extrusion, 13Material Jetting, 13, 15mechanical energy equation, 31medical implants, 2Message Passing Interface, 114MFLUPS, 115MLUPS, 115Molecular Dynamics, 96molecular dynamics, 39MPI communication, 94MRT, 44, 150multi–beam, 140multi–component problems, 59multi–phase problems, 59multi–speed LBM, 49multi-distribution LBM, 49

Navier-Stokes equations, 30, 31Newton’s second law, 28Newtonian fluid, 29, 30no–slip, 53non-Newtonian fluid, 29

OpenLB, 91

Palabos, 91parallel computing, 113particle distribution function (pdf), 43Peclet number, 35penetration depth, 85photopolymer, 13Photopolymerization, 13photopolymerization, 14Powder Bed Fusion, 13Powder Bed Fusion Process, 18PowerFLOW, 92Prandtl number, 35, 50process window, 132pseudo–potential method, 64

208

Index

quasi–compressible, 45, 49

Rapid Prototyping, 7Reynolds number, 34Reynolds’ transport theorem, 26, 27Rigid Body Dynamics, 96

sample standard deviation, 79scalability, 114second invariant, 32Selective Laser Melting, 19Selective Laser Sintering, 19sensitivity analysis, 127Shan-Chen model, 64shared–memory computer, 114shearing stress, 30Sheet Lamination, 13simultaneous multithreading, 119Sintering, 19Smooth Particle Hydrodynamics, 96Sobolev space, 37software quality, 92Solid Freeform Fabrication, 8speed of sound, 35SRT, 43, 150Stefan number, 129Stefan problem, 128stencil, 40stereolithography, 14Stokes flow, 164strain rate tensor, 30strong scaling, 115substantial derivative, 25, 26SuperMUC, 115surface force, 29surface tension, 61, 69surface tension coefficient, 61

Tchebysheff norm, 169thermal conductivity, 32Third Industrial Revolution, 7tissue engineering, 15total derivative, 25, 26total energy, 31TRT, 44, 150two–collision times (TRT), 150

uncertainty principle, 24

validation, 125viscous stress tensor, 30Volume–of–Fluid Method, 59Volume-of-Fluid Method, 66

waLBerla, 92weak scaling, 115

XFlow, 92

209

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