EBM Seminarski Rad

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Simulation of Selective Electron Beam Melting Processes Simulation der selektiven Elektronenstrahlschmelzprozesse Der Technischen Fakult¨ at der Universit¨ at Erlangen-N¨ urnberg zur Erlangung des Grades DOKTOR-INGENIEUR vorgelegt von Elham Attar Erlangen 2011

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EBM Melting

Transcript of EBM Seminarski Rad

  • Simulation of Selective Electron

    Beam Melting Processes

    Simulation der selektiven

    Elektronenstrahlschmelzprozesse

    Der Technischen Fakultat der

    Universitat Erlangen-Nurnberg

    zur Erlangung des Grades

    D O K T O R - I N G E N I E U R

    vorgelegt von

    Elham Attar

    Erlangen 2011

  • Als Dissertation genehmigt von der Technischen Fakultat

    der Universitat Erlangen-Nurnberg.

    Tag der Einreichung: 24.01.2011

    Tag der Promotion: 01.06.2011

    Dekan: Prof. Dr.-Ing. Reinhard German

    Berichterstatter: PD Dr.-Ing. habil. Carolin Korner

    Prof. Dr. Ulrich Rude

  • Contents

    Contents iii

    Abstract v

    Kurzfassung vii

    List of Symbols and Abbreviations xi

    Introduction 1

    1 Beam and Powder Based Additive Manufacturing 5

    1.1 Selective Laser Sintering/Melting . . . . . . . . . . . . . . . . . . . 6

    1.2 Selective Electron Beam Melting . . . . . . . . . . . . . . . . . . . . 9

    1.3 Physical Aspects of the SEBM Process . . . . . . . . . . . . . . . . 12

    1.4 Materials and Applications . . . . . . . . . . . . . . . . . . . . . . . 15

    1.4.1 Titanium Alloys . . . . . . . . . . . . . . . . . . . . . . . . . 17

    1.4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2 Physical Model 23

    2.1 Random Powder Bed Generation . . . . . . . . . . . . . . . . . . . 24

    2.2 Beam Definition and Absorption in 2D . . . . . . . . . . . . . . . . 26

    2.3 Energy Transfer and Conservation Equations . . . . . . . . . . . . . 27

    2.4 Capillarity and Wetting . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

  • iv Contents

    3 Numerical Implementation 31

    3.1 Lattice Gas Automata . . . . . . . . . . . . . . . . . . . . . . . . . 33

    3.2 The Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . 34

    3.3 Thermal Lattice Boltzmann Method . . . . . . . . . . . . . . . . . 36

    3.4 Multi-distribution Function Method . . . . . . . . . . . . . . . . . . 37

    3.5 Free Boundary Treatment . . . . . . . . . . . . . . . . . . . . . . . 39

    3.5.1 Missing Distribution Functions . . . . . . . . . . . . . . . . 42

    3.5.2 Curvature Calculation . . . . . . . . . . . . . . . . . . . . . 48

    3.6 Wetting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    4 Verification Experiments 57

    4.1 Thermal Hydrodynamic Problems with Free Surface . . . . . . . . 57

    4.1.1 Rising Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . 58

    4.1.2 Collision between a Droplet and Solid Object . . . . . . . . 59

    4.1.3 Rising Bubble in a Solidifying Liquid . . . . . . . . . . . . . 61

    4.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    4.2 Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    4.2.1 Droplets in Equilibrium . . . . . . . . . . . . . . . . . . . . 64

    4.2.2 Spreading of a Droplet . . . . . . . . . . . . . . . . . . . . . 65

    4.2.3 Capillary Rise/Depression . . . . . . . . . . . . . . . . . . . 66

    4.2.4 Comparison with Experiments . . . . . . . . . . . . . . . . . 69

    4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    5 Electron Beam Melting Simulation 73

    5.1 Experimental Procedure and Simulation Parameters . . . . . . . . . 73

    5.2 Single Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    5.2.1 Wetting Conditions . . . . . . . . . . . . . . . . . . . . . . . 76

    5.2.2 Relative Powder Density . . . . . . . . . . . . . . . . . . . . 77

    5.2.3 Energy Input . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    5.2.4 Stochastic Powder Layer . . . . . . . . . . . . . . . . . . . . 79

    5.2.5 Processing Map for Single Layer Fabrication . . . . . . . . . 80

  • Contents v

    5.3 Multilayer Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    5.3.1 Layer Thickness . . . . . . . . . . . . . . . . . . . . . . . . 87

    5.3.2 Powder Particle Size Distribution . . . . . . . . . . . . . . . 87

    5.3.3 Beam Shape and Spot Size . . . . . . . . . . . . . . . . . . 88

    5.3.4 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . 90

    5.3.5 Processing Map for Multi-layer Fabrication . . . . . . . . . . 92

    5.3.6 Refill Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    5.3.7 Compact Parts . . . . . . . . . . . . . . . . . . . . . . . . . 94

    5.3.8 Comparison with Experiments . . . . . . . . . . . . . . . . . 96

    5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

    6 Summary and Conclusion 105

    Bibliography 109

    A Particle Packing Algorithm 123

    B Dimensionless Numbers For SEBM process 127

    C Publications

    related with this work 129

  • Abstract

    Metal powders are selectively molten layer by layer during beam based additive manu-

    facturing processes. The density of the resulting material, the spatial resolution as well

    as the surface roughness of the completed components are complex functions of material

    and processing parameters.

    The purpose of this dissertation is to achieve a better understanding of the beam based

    additive manufacturing process with the help of numerical simulations. For the first time,

    numerical simulations of selective beam melting processes are presented where individual

    powder particles are considered. The proposed model is based on a lattice Boltzmann

    method. New methods to treat thermal hydrodynamic problems with free surface and

    wetting are presented and evaluated with theoretical and experimental benchmarks.

    A two-dimensional lattice Boltzmann model (LBM) is developed to investigate melting

    and re-solidifying of a randomly packed powder bed under the irradiation of a Gaus-

    sian beam. This approach makes many physical phenomena accessible which can not be

    described in a standard continuum picture, e.g. the influence of the relative powder den-

    sity, the stochastic effect of a randomly packed powder bed, the powder size distribution,

    capillary effects and the wetting conditions.

    The potential of the proposed model to simulate the selective electron beam melting pro-

    cess (SEBM) is demonstrated by means of some examples for single tracks and multilayer

    parts. The effect of the beam power, scan speed and layer thickness, which are considered

    as dominant parameters for the process, are investigated numerically. The simulation

    results are compared with experimental findings during selective electron beam melting.

    The comparison shows that the proposed model, although 2D, is able to predict the main

    characteristics of the experimental observations.

  • Kurzfassung

    In strahlbasierten additiven Herstellungsverfahren werden Metallpulver schichtweise se-

    lektiv geschmolzen. Die Dichte des auf diese Weise erhaltenen Materials, die raumliche

    Auflosung, wie auch die Oberflachenrauigkeit der fertigen Komponenten resultieren aus

    einem komplexen Zusammenspiel von Material- und Prozessparametern.

    Das Ziel dieser Arbeit ist es, ein besseres Verstandnis des strahlbasierten additiven Her-

    stellungsprozesses mit Hilfe numerischer Simulation zu erreichen. Erstmals wird eine

    numerische Simulation selektiver Strahlschmelzprozesse aufgezeigt, bei der einzelne Pul-

    verpartikel berucksichtigt werden. Das vorgeschlagene Modell basiert auf einer Lattice-

    Boltzmann-Methode. Neue Methoden zur Behandlung thermisch-hydrodynamischer Pro-

    bleme mit freier Oberflache und Benetzung werden aufgezeigt und hinsichtlich theoreti-

    scher und experimenteller Mastabe bewertet.

    Es wird ein zweidimensionales Lattice-Boltzmann-Modell (LBM) entwickelt, um das

    Schmelzen und Wiedererstarren von zufallig gepackten Pulverbetten unter der Einwir-

    kung eines Gauschen Strahls zu untersuchen. Dieser Ansatz macht zahlreiche physikali-

    sche Phanomene zuganglich, welche nicht in einem Standard-Kontinuum-Abbild beschrie-

    ben werden konnen, wie z. B. den Einfluss der relativen Pulverdichte, den stochastischen

    Effekt eines zufallig gepackten Pulverbetts, die Pulvergroenverteilung, Kapillareffekte

    und die Benetzungsbedingungen.

    Das Potenzial des vorgeschlagenen Modells zur Simulation des selektiven Elektronen-

    strahlschmelzprozesses wird mit Hilfe einiger Beispiele fur Einzelbahnen und mehrschich-

    tige Bauteile aufgezeigt. Der Einfluss der Strahlleistung, Scangeschwindigkeit und Schicht-

    dicke, welche als bestimmende Parameter des Prozesses angesehen werden, wird nume-

    risch untersucht. Die Ergebnisse der Simulation werden mit experimentellen Erkennt-

  • x Kurzfassung

    nissen aus dem selektiven Elektronenstrahlschmelzen (SEBM, Selective Electron Beam

    Melting) verglichen. Der Vergleich zeigt, dass das vorgeschlagene Modell trotz Zwei-

    dimensionalitat in der Lage ist, die wesentlichen Charakteristika der experimentellen

    Beobachtungen vorherzusagen.

  • Acknowledgments

    I am deeply grateful to a number of people for their assistance and support to make this

    study possible.

    First and foremost, I wish to convey my deep appreciation to my supervisor, PD Dr.- Ing.

    Carolin Korner, for her encouragement, support and guidance during my research work.

    Her expertise and invaluable comments helped greatly in the completion of this disser-

    tation.

    I would like to thank Prof. Dr.-Ing. Singer for giving me an opportunity to work as

    research associate in institute for Material Science and Technology of Metals (Lehrstuhl

    Werkstoffkunde und Technologie der Metalle) at the University of Erlangen. I am also

    grateful to Prof. Dr. Rude for accepting the referee of my work.

    Special thanks to Dipl.-Ing. Peter Heinl, for his close collaboration in the experimental

    part of this work as well as the valuable discussion during different phases of my work.

    I am also grateful to the people who helped me writing this thesis by proofreading parts

    of it: Dr.-Ing. Hanadi Ghanem, M. Sc. Atefeh Yousefi Amin, Dipl.-Ing Peter Heinl

    and Dipl.-Inf. Matthias Markl. Especially Matthias helped me a lot in formatting and

    finalizing this dissertation.

    I would like to thank my colleagues in the light weight material group and specially

    my roommates Dr.-Ing Andre Trepper, Dipl.-Inf. Matthias Markl, Dipl.-Ing. Alexander

    Klassen, and Jorg Komma.

    I have to thank many other people from our department, the secretaries (Mrs. Anneli

    Dupree and Mrs. Ingrid Hilpert) as well as, Mrs. Kerstin Zinn and Mrs. Beate Rohl for

    preparation of metallography samples.

  • xii Acknowledgments

    Many friends have helped me during the last four years. I wish to thank all my friends

    in Erlangen and specially Mrs Irmgard Schurmann for her kind support.

    I can not thank my family enough for all their encouragements and support. I wish

    to thank my parents for instilling in me the principles which carried me through this

    journey.

    Finally, I am deeply grateful to my husband, Pouria, for all his exceptional love, support,

    and understanding. I wish to thank him for being compassionate friend during these

    years.

  • List of Symbols

    and Abbreviations

    Symbols

    Greek letters

    gradient operator angle which is depicted in figure 3.12

    1 the wall angle relative to the building plate

    angle between the normal vector and wall surface in triple point

    distance between cell center and interface

    H latent enthalpy of a computational cell

    I beam absorbed energy within a numerical cell

    m mass scale

    M mass exchange between an interface cell and its neighbor

    t time scale

    T temperature scale

    x length scale

    Kronecker Symbol

    x distance which is depicted in figure 3.12

    fraction of material (solid or liquid) within the numerical cell

    volume fractions of the wall cells within the template sphere

    d dynamic wetting angle

    eq equilibrium wetting angle

    mean curvature

    heat conductivity

    abs absorption coefficient

  • xiv List of Symbols and Abbreviations

    viscosity

    kinematic viscosity

    liquid fraction in a cell

    correction factor

    velocity

    momentum-flux-tensor

    density

    standard deviation

    surface tension

    relaxation time

    f dimensionless relaxation times for the velocity field

    h dimensionless relaxation times for the temperature field

    energy source

    i boolean variables

    i energy deposited in each cell under beam radiation

    i weights

    i collision operator

    Roman letters

    a base of a droplet

    A fraction of gas in the template circle

    b number of particle velocity directions

    Bj bubble

    cs speed of sound

    cp specific heat at constant pressure

    cp effective specific heat

    Ca capillary number

    dA surface element

    D characteristic length

    D beam width

    ei set of lattice vector

    E thermal energy density

    EL line energy

    Eo Eotvos number

    f particle distribution functions

    fi density distribution function

  • List of Symbols and Abbreviations xv

    feq equilibrium distribution function

    Fi body force

    Fwet wetting force

    FG The gas force per unit area

    F the force field per unit mass acting on the particle

    F force acting at the triple-point

    Fcap capillary force

    g gravity

    h height of a droplet

    hi energy distribution function

    H hight during the capillary rise

    H0 equilibrium capillary rising height

    i index for different velocity directions

    i/2 index for set of distribution functions pointing to the gas

    i/2 index for set of distribution functions pointing to the liquid

    I beam power density

    k thermal diffusivity

    K an additional force

    l the depth of immersed capillary tube

    L latent heat

    L1 predefined beam scanning cross section

    m number of dimensions

    M mass of the interface cell

    Ma Mach number

    Mr center of the template circle

    MR center of circle which approximates interface curvature

    nj the gas content

    n normal vector belonging to surface element

    pj bubble pressure

    p pressure

    P total beam power

    P pressure tensor

    qF heat flux

    Q unknown heat current

    r radius of the template circle

    r0 radius of the capillary tube

    R ideal gas constant

    R radius of circle which approximates interface curvature

  • xvi List of Symbols and Abbreviations

    R radius of a droplet

    R the radius of the liquid zone

    Re Reynolds number

    S collision operator

    St Stefan number

    t a discrete time

    t time

    T temperature

    Tj bubble temperature

    Tl liquidus temperature

    Ts solidus temperature

    T1 wall thickness

    u the macroscopic velocity

    U rising velocity of the bubble

    v speed of the beam

    V velocity

    Vj the bubble volume

    x a vector in the lattice space

    x location

    Subscript

    G gas

    i an index for different velocity directions

    L liquid

    n component in normal direction

    S solid

    t component in tangential direction

  • List of Symbols and Abbreviations xvii

    Superscript

    * dimensionless quantities

    eq equilibrium

    F fluid

    G gas

    in incoming

    out outgoing

    Abbreviations

    2D Two Dimensional

    3D Three Dimensional

    3DP 3D Printing

    BGK Bhatnagar, Gross and Krook Approximation

    CFD Computational Fluid Dynamic

    CT Computed Tomography

    D2Q9 Two-dimensional LB Model with Nine Velocities

    D3Q15 Three-dimensional LB Model with Fifteen velocities

    D3Q19 Three-dimensional LB Model with Nineteen velocities

    D3Q27 Three-dimensional LB Model with Twenty Seven velocities

    CAD Computer Aided Design

    DLF Direct Laser Fabrication

    DMD Direct Metal Deposition

    FHP Frisch, Hasslacher and Pomeau Model

    GA Gas Atomization

    IJP Ink Jet Printing

    LB Lattice Boltzmann

    LBM Lattice Boltzmann Model

    LENS Laser Engineered Net Shaping

    LGA Lattice Gas Automata

    MD Molecular Dynamics

    MRT Multi Relaxation Time Model

    N-S Navier-Stokes

    PLIC Piecewise Linear Interface Construction

    PREP Plasma Rotating Electrode Process

    RM Rapid Manufacturing

  • xviii List of Symbols and Abbreviations

    RP Rapid Prototyping

    SBM Selective Beam Melting

    SEBM Selective Electron Beam Melting

    SFF Solid Free Fabrication

    SLA Stereolithography

    SLM Selective Laser Melting

    SLS Selective Laser Sintering

    VOF Volume of Fluid

  • Introduction

    Nowadays, numerical modeling provides a powerful means of analyzing various

    physical phenomena occurring in a complex process. It allows researchers to ob-

    serve and quantify what is not usually visible or measurable during the real pro-

    cesses and it is also inexpensive in comparison with laboratory experiments.

    The rapid improvement of high performance computers help us to use numerical

    modeling for solving the problems which seemed to be unsolvable few years ago.

    The results of numerical modeling include some uncertainties arising from the

    mathematical model, or the numerical techniques. Therefore, it is necessary to

    validate models with theoretical and/or experimental benchmarks.

    The main manufacturing technologies such as casting, forging, and machining ex-

    hibit long development times. These technologies are typically tool based. Todays

    developments in the field of production technologies are mainly focused on shorter

    cycles of innovation. In recent years, additive manufacturing technologies have

    been implemented in many aspects of industry, especially in the area of new prod-

    uct development due to the opportunity of manufacturing without a specific tool

    and greatly reduced fabrication time and cost. Additive manufacturing technolo-

    gies enable the industry to produce complex parts on the basis of 3D CAD data

    in one process step [1, 2].

    Beam and powder based layered manufacturing methods are relatively novel addi-

    tive manufacturing technologies that can build parts from powdered material via

    layer-by-layer melting induced by a directed electron or laser beam [1]. Examples

    of commercialized selective beam melting processes are Selective Laser Melting

  • 2 Introduction

    (SLM) and Selective Electron Beam Melting (SEBM). During SLM or SEBM pro-

    cess, the surface of a powder bed is selectively scanned by a beam. Thin molten

    tracks develop and combine to form a 2D layer of the final part. After completion

    of one layer, the whole powder bed is lowered and a fresh powder layer is spread on

    the building zone. The selective melting process is repeated until the component

    is completed.

    Generally for different materials, different powder consolidation mechanisms are

    essential [3,4]. For metal powders, melting and re-solidification are the underlying

    mechanisms to consolidate the powder particles for building a functional part.

    Typical process defects associated with SLM/SEBM process are porosity, residual

    powder and non-connected layers. State of the art to find the optimal processing

    parameters for a new material is still based on the expensive trial and error process

    [2,4]. This makes the range of applicable materials strongly limited. Therefore it is

    important to have a fundamental and broad understanding of how process variables

    relate to final part quality.

    The SLM/SEBM process is rather complex and involves many different physical

    phenomena [5] such as absorption of the beam in the powder bed and the melt

    pool or the re-solidified melt, melting and re-solidification of a liquid pool, wetting

    of the powder particles with the liquid, diffusive and radiative heat conduction in

    the powders, diffusive and convective heat conduction in the melt pool, capillary

    effects, gravity, etc. The melt pool caused by the beam is highly dynamic, and

    it is driven by the high surface tension in combination with the low viscosity of

    liquid metals. This leads to the development of tracks with irregular, corrugated

    appearance which might result in typical process defects.

    Reviewing the literature, several authors apply numerical simulation methods in

    order to develop a better understanding of the underlying consolidation process.

    Williams and Deckard [6] developed a 2D finite difference model to study process

    parameters in selective laser sintering of polymers. There are also finite element

    models presented by Bugeda et al. [7] and Shiomi et al. [8] to simulate selective

    laser sintering process. Zhang and Faghri [9] developed a model for melting of two

  • Introduction 3

    component metal powders with significantly different melting points. Tolochko

    et al. [10] used simulations and experiments in order to investigate the effects

    of process parameters on sintering mechanism of titanium powders. Kolossov et

    al. [11] developed a three dimensional finite element model which considers the

    nonlinear behavior of thermal conductivity and specific heat due to temperature

    and phase transformation. The results of this model were experimentally tested by

    direct temperature measurements. In order to have a more realistic model, Konrad

    et al. [12] and Xiao and Zhang [13,14] divided the powder bed into different regions

    from the bottom to the heating surface and for each region the effective thermal

    conductivity is defined separately. Recently, Zah et al. [15] developed the finite

    element method for the simulation of the electron beam melting process. In all

    mentioned approaches, the underlying model is based on a homogenized picture,

    i.e. the powders are considered as a homogeneous material with effective properties,

    e.g. an effective thermal conductivity which depends on the relative density.

    Though Zhou et al. [16] consider a bimodal randomly packed particle bed for

    the simulation of the radiative heat transfer in a selective laser sintering process,

    melting and the development of the melt pool geometry are not described.

    An essential challenge for the homogeneous approaches is to model about 50%

    powder shrinkage during the solid-liquid phase transformation. It is well known

    that shrinkage has an enormous influence on the melt pool geometry and the

    local thermal properties. All available shrinkage models are solely a function of

    the powder packing density. None of these models consider the shrinkage of a

    real random powder bed. The resulting melt pool geometries are thus always

    well defined without the stochastic behavior which is experimentally observed [10].

    That is, the existing models in literature are still far away from the experimental

    findings. One reason for that discrepancy is certainly that these models dont

    consider individual powder particles.

    The general purpose of this thesis is to gain a much better understanding of the

    beam based additive manufacturing process with the help of numerical simulation.

    In contrast to existing models in the literature, we have developed a numerical

  • 4 Introduction

    tool where the effect of individual powder particles is considered. A sequential

    addition packing algorithm is employed to generate 2D randomly packed powder

    layers composing of spherical particles.

    Our method is based on a lattice Boltzmann model (LBM) [1719] which is an

    alternative for ordinary computational fluid dynamic methods. The LBM approach

    is especially beneficial in problems with complex interfaces such as flows in porous

    media [20, 21] or the development of foams [22, 23]. The beam is absorbed by the

    powder layer, heats the powder and eventually melts it. Due to capillary, gravity,

    and wetting forces, a complex and strongly changing melt pool geometry develops

    until solidification freezes the current state. It gives an insight into the details

    of fluid flow, heat transfer, and solidification. The comparison with experimental

    results from SEBM, demonstrates the predictive power of the proposed numerical

    model. This method can also be utilized to model other powder based rapid

    prototyping processes like SLM.

    This dissertation is organized into six chapters. The focus of chapter 1 is to give

    an introduction into the selective beam melting process as an example of additive

    manufacturing processes. The second chapter describes the underlying physical

    models of the electron beam melting process. Chapter 3, the main part of the thesis,

    is concerned with the numerical implementation of the physical models based on the

    lattice Boltzmann method. The theory of the LBM is introduced and the method

    is extended by free surface boundary conditions for the fluid flow and solidification.

    The wetting algorithm is also described in detail. The numerical implementation

    of wetting and free surface boundary conditions is verified by numerous tests in

    chapter 4. Electron beam melting simulations are presented in chapter 5. The

    simulation results and the influence of material and process parameters on the final

    structure produced by the selective electron beam melting process are extensively

    discussed and compared to experiments. Conclusions and an outlook of future

    work are presented in chapter 6.

  • Chapter 1

    Beam and Powder Based

    Additive Manufacturing

    Rapid Prototyping (RP) and Rapid Manufacturing (RM) refer to the automatic

    construction of three-dimensional parts using additive manufacturing technology.

    Rapid prototyping, which is also known as Solid Free Fabrication (SFF), was de-

    veloped due to an increased demand for shortened product development cycles.

    The aim of rapid prototyping and rapid manufacturing processes is to fabricate

    three-dimensional, fully functional parts directly from different materials (i.e. met-

    als, polymers or ceramics) without using additional processing steps before or after

    the rapid prototyping operation.

    During additive manufacturing processes, parts are made by adding material in

    layers which each layer is a thin cross section of the part defined from original

    CAD data. Figure 1.1 describes different steps of additive manufacturing process.

    The 3D CAD model is sliced into layers with constant thickness to generate layer

    information.

    A large number of additive manufacturing technologies are available, their main dif-

    ferences are found in the way the layers are built. Each of these different technolo-

    gies uses different materials and has different advantages. Some examples of addi-

    tive manufacturing technologies are Selective Laser Sintering/Melting SLS/SLM),

  • 6 Beam and Powder Based Additive Manufacturing

    3D CAD Model Slicing In layers assembly Complete part

    Figure 1.1: Different steps of the additive manufacturing process

    Stereolithography (SLA), 3D Printing (3DP), Ink Jet Printing (IJP), Direct Laser

    Fabrication (DLF), Direct Metal Deposition (DMD) and Selective Electron Beam

    Melting (SEBM) [1,2].

    Additive manufacturing technologies can be divided into two main categories: non-

    melting and melting processes [1,2]. Beam and powder based layered manufactur-

    ing methods are a family of melting processes that involves a layer-wise shaping

    and consolidation of material [1]. After a short review of beam based layered man-

    ufacturing processes such as SLS/SLM and SEBM, applications and limitations of

    the Electron Beam Melting process are described in more detail.

    1.1 Selective Laser Sintering/Melting

    The selective laser sintering process was the first commercialized powder bed fu-

    sion process and it became the most popular rapid prototyping process used for a

    wide range of materials (polymers, metals, ceramics and composites) in rapid man-

    ufacturing. All powder bed fusion processes include at least one thermal source

    to induce fusion between the powder particles. A three-dimensional object is built

    layer by layer out of a powder which is selectively heated by beam radiation. The

    molten material solidifies when the temperature decreases. The solidified melt

    pool forms the part while the unmolten powder remains at its place to support the

    structure. After the build process is completed the residual powder is removed.

    The schematic of the SLM process is shown in figure 1.2.

    Laser based additive manufacturing processes utilized a high power laser (CO2

    or Nd-YAG) as a heat source. In addition a computer aided design model is

  • 1.1 Selective Laser Sintering/Melting 7

    Powder delivery piston

    Powder delivery System

    Laser

    Scanner System

    Fabricated PartPowder Bed

    Fabrication Piston

    Roller

    Figure 1.2: Schematic of the build chamber of the SLM process

    used for building the components from powder material layer by layer. Typically,

    this process is performed in an inert atmosphere (e.g. argon) to reduce oxidation

    effects. In SLS, the laser scans the desired areas of the powder bed. After scanning

    one layer, a new layer of powder is spread on the previous layer. This process is

    repeated until completion of the final shape. The layer thickness usually ranges

    between 50m and 200m depending on the powder size and material. The Laser

    Engineered Net Shaping (LENS) process is an alternative process for SLS with the

    ability of feeding powder into the melt pool produced by the laser, therefore alloy

    composition may be adjusted where needed [1].

    There are some different powder consolidation mechanisms such as solid state sin-

    tering, liquid phase sintering, partial melting and full melting [3, 4]. Solid state

    sintering is a consolidation process below the melting temperature and occurs when

    diffusion of atoms forms a neck between solid particles. This mechanism is slow

    and requires a long time for completion and is rarely applied in layer manufacturing

    while the process is not economically viable [3,4,24]. During liquid phase sintering

    or partial melting, part of the powder material is molten and spreads between the

  • 8 Beam and Powder Based Additive Manufacturing

    solid particles. The presence of the liquid phase results in rapid sintering since

    mass transport can occur by liquid flow and particle rearrangement. This allows

    much higher scan speeds of the laser. An example of liquid phase sintering is a

    material system which consists of a two component powder (high and low-melting

    materials). The low melting phase melts while the high melting phase remains

    solid. As a result the solid powders are bonded together by the melt [4, 24].

    Full melting is the mechanism most commonly associated with powder bed fusion.

    This technology is known as selective beam melting. During full melting, the

    entire region of material subjected to heat energy is molten to a depth exceeding

    the layer thickness. The energy input is typically sufficient to remelt a portion of

    the previously solidified structure. Thus this type of full melting is very effective

    for creating well-bonded, high density structures from engineering materials. For

    metal powders (such as titanium, stainless steel or cobalt chromium) full melting

    is used to produce parts. The rapid melting and solidification of these metal alloys

    results in unique properties, which can sometimes be better than cast/wrought

    parts made from the same alloys [25].

    Full melting has the main advantage to produce almost full dense products in one

    step. Nevertheless, it also has general problems like internal stresses, part dis-

    tortion due to high temperature gradients and shrinkage. Typical process defects

    associated with SLM processes are porosity, residual powder, and non-connected

    layers, but a more substantial problem is balling phenomenon. Balling is the for-

    mation of small spheres approximately the diameter of the beam and may result in

    the formation of discontinuous scan tracks [24]. The risk of balling of the melt pool

    may also result in bad surface finishes [3,4]. The processed material can also suffer

    from the effects of vaporization. Using a shielding atmosphere or pre-heating the

    powders to higher temperatures may help to overcome these problems. Also pow-

    ders with bimodal distributions for optimum packing and using additives to reduce

    surface tension also suggested for decreasing process defects [24]. Using different

    laser strategies may lead to reduce thermal stresses, porosity, and shrinkage [26].

    Maintaining part accuracy is another factor which makes further complications

    when using high power lasers.

  • 1.2 Selective Electron Beam Melting 9

    Today the SLM process is in the focus of many researches [24, 27] in order to find

    the optimum process parameters for different materials. Despite the advantages of

    the SLM technologies, some restrictions exist regarding the use of different metal

    materials and the achievable building speed. Therefore, the economic use of this

    technology is limited in the aerospace industry and/or in the medical technology.

    For an industrial use, it is necessary to solve the described difficulties. The electron

    beam as an energy source shows advantages compared to the laser beam especially

    because of the high deflection speed realized by electromagnetic lenses and the

    high possible energy input. This leads to the idea of using an electron beam

    during additive manufacturing.

    1.2 Selective Electron Beam Melting

    The electron beam has been used for many years for welding, soldering, zone

    refinement, and reclamation of scrap. Recently Arcam AB has developed a new

    generative process called Selective Electron Beam Melting (SEBM) which is used

    for manufacturing special parts and complex-shaped objects that have comparable

    properties to those of cast components [2, 28].

    The SEBM process used for rapid component prototyping is operationally similar

    to the rastering of an electron beam in a scanning electron microscope and it can

    be considered as a variant of selective laser beam melting. Similar to the SLM

    process, metal powders are melted selectively in paths traced by the electron beam

    gun.

    The SEBM machine consists of a control panel and a processing chamber evacuated

    with a turbomolecular pump. The schematic of the build chamber of the SEBM

    machine is shown in figure 1.3, (a). The processing chamber consists of a building

    tank with an adjustable process platform, two powder dispensing hoppers, and a

    rake system for spreading the powders. The electron beam is generated by heating

    a tungsten filament. Electrons are accelerated to a velocity between 0.1 and 0.4

    times the speed of light using an accelerating voltage of 60 kV . The electrons are

    focused and deflected by electromagnetic lenses. They hit the powder particles in

  • 10 Beam and Powder Based Additive Manufacturing

    2. Melting of thecross section

    3. Lowering of theprocess platform

    1. Preheating of thepowder layer

    4 Application of anew powder layer

    .

    powderhopper

    powder

    start plate

    vacuumchamber

    elec

    tro

    n b

    eam

    gu

    n

    powderhopper

    rake

    buildingtank

    processplatform

    a)

    b)

    Figure 1.3: Schematic of the EBM machine a) build chamber b) layer by layercomponent generation [29].

  • 1.2 Selective Electron Beam Melting 11

    the building chamber and release their kinetic energy mostly as thermal energy.

    Depending on the energy of the electron beam, the powder particles are completely

    molten in the range of a certain layer thickness [29].

    The EBM-A2 system used for this study provides a maximum beam power of

    3500W , a spot size of about 0.1mm to 0.4mm, and a maximum build size

    of 200 200 350mm3. During the generation process a vacuum pressure of104 105mbar is employed. In principle, all conductive materials can be usedin the SEBM process but the popular materials are steel, titanium, and cobalt

    chromium alloys.

    Figure 1.3, (b) describes the layer by layer part generation. At first a layer of

    metal powder is spread homogeneously on the build platform providing a base

    for the part to be built. The SEBM process starts with the preheating of the

    powder layer using a relatively low beam current and a relatively high scan speed.

    The preheating step lightly sinters the metal powder to hold it in place during

    subsequent melting at higher beam powers. It also helps to reduce the thermal

    gradient between the melted layer and the rest of the part.

    Afterward the electron beam scans the powder surface according to the layer data,

    line by line on defined position and melts the loosely joined powder particles to a

    compact layer with the desired shape. Once the first layer has been melted, the

    build plate is lowered by one layer thickness, additional powder is delivered from

    the powder dispensing hopper and spread/raked over the previously solidified layer

    and the process will be repeated. After the building stage, the part is cooled down

    either under vacuum or helium flow. Cleaning of the parts from adherent partly

    molten powders is done by powder blasting with the same powders as used in

    the building process. The removed powders can be reused after sieving in a new

    process [29].

    Since the energy source in SEBM are electrons, there are a number of differences

    between SEBM and SLM. The electron beam consists of electrons moving near the

    speed of light but the laser beam consists of coherent photons. If an electron beam

    passes through a gas, the electrons interact with the gas atoms and are deflected.

  • 12 Beam and Powder Based Additive Manufacturing

    In contrast, the laser beam can pass trough a gas unaffected as long as the gas

    is transparent at the laser wavelength. Therefore, SEBM is practiced in a low

    partial pressure vacuum environment. Since the process takes place under high

    vacuum, material properties are excellent because of preventing degradation by

    the absorption of atmospheric gases [2].

    The deflection and focus of the photons is accomplished by mirrors whereas the

    electron beam is focused and reflected by electromagnetic lenses resulting in high

    scanning speeds and a high positioning accuracy. As a result, novel scanning

    patterns and melting strategies can be applied and high building speeds of 60 cm3/h

    can be realized [2, 15].

    The electron beam exhibits a high energy density and a high efficiency. When the

    voltage difference is applied to the heated filament most of the electrical energy is

    converted into electron energy. During SEBM, electrons heat a powder by trans-

    ferring their kinetic energy to the powder bed. On the other hand, laser beams

    heat the powder by photon absorption. Consequently, laser can be used with any

    kind of material that absorbs energy at the laser wavelength (e.g. metals, polymers

    and ceramics) [2, 29]. The temperature of the powder bed is much higher during

    SEBM compared to SLM, since the powder is preheated by the defocused electron

    beam.

    1.3 Physical Aspects of the SEBM Process

    The SEBM process is complex and involves many different physical phenomena

    (figure 1.4) [5]. The strong interplay between these physical mechanisms directly

    impacts the process and influences the properties of the processed material.

    As the beam is absorbed in the powder bed, the powder starts to melt and the

    volume reduces. Due to surface tension more powder may be dragged into the

    melt pool. The melt pool caused by the beam is highly dynamic and is driven

    by the high surface tension in combination with the low viscosity of liquid metals.

    Changes of the viscosity across the melt pool, due to changes of viscosity between

  • 1.3 Physical Aspects of the SEBM Process 13

    Heat Transfer

    Gravity

    Fluid Flow

    VaporizationWetting and Capillary forces

    Powder Packing

    Radiation

    Marangoni-convection

    Sintering

    Beam AbsorptionMelting

    Solidification

    Unmelted powder

    Melted powderPrevious layer

    Figure 1.4: Different physical phenomena during the selective electron beam melt-ing process.

    the liquidus and solidus temperature, may largely influence the shape of the tracks

    and the resulting surface smoothness. This leads to the development of stochastic

    melt tracks with irregular, corrugated appearance. On the other hand, the life

    time of the melt pool is rather short (only some milliseconds). After finishing

    one layer, a new powder layer is applied on the corrugated surface leading to a

    new powder layer with strongly varying thickness which might result in a typical

    process defect [4, 5].

    The wetting characteristics of the solid phase by the liquid phase are crucial for

    a successful processing. The melt pool should wet the previously consolidated

    material and the powder particles. The wettability of a solid by a liquid depends

    on surface tension which can be influenced by the material temperature, impurities,

    contamination, and atmosphere [5, 24].

    Even in the absence of contamination, there may be a problem of a liquid metal

    wetting its solid form, if the solid has almost the same temperature. In this homol-

    ogous wetting case, there is no driving force for wetting. It has been suggested that

    the processing parameters should be chosen to ensure that sufficient remelting of

    the previous layer takes place, and continuity of the solid-liquid interface under the

    moving beam is maintained [4, 5]. On the other hand, the required excess energy

  • 14 Beam and Powder Based Additive Manufacturing

    will have consequences known as Marangoni flow which will affect the quality of

    the upper surface of the melt pool. The flow instabilities in a melt pool can lead

    to break up of thin melt pools into spherical droplets, called balling and commonly

    denoted as Rayleigh instabilities [30]. Balling has occurred whenever the predicted

    track melt length-to-diameter ratio has exceeded a value in the range 2.8 - 3.3 [27].

    This range is similar to the value expected for Rayleigh instability. According to

    the Plateau-Rayleigh instability criteria the length at which a cylindrical column

    of liquid becomes unstable is pi times the diameter of the cylinder [31]. Balling

    results in a rough and bead-shaped surface obstructing a smooth layer deposition

    and decreasing the density of the produced part.

    In general, the binding mechanism in full melting is strongly driven by the fluid

    behavior of the melt which is related to surface tension (Raleigh instabilities),

    viscosity, wetting, thermocapillary effects (Marangoni convection), evaporation,

    and oxidation.

    Evaporation of elements is a well known phenomenon in vacuum metallurgy, weld-

    ing, and electron beam processing [5]. The evaporation from a molten metal is

    governed by four distinct regimes; mass transport of atoms from the interior of the

    melt to its surface, phase change to gaseous state at the surface, mass transport

    in the gas phase above the melt and, finally condensation. Although processing

    under high vacuum yields desirable results with respect to wetting, most of the

    metals vaporize due to low pressure. Since electron beam based systems require a

    vacuum environment, evaporation of alloying elements may be substantial.

    One of the well known phenomena during the SEBM process is powder spreading

    or pushing which means that the powders spread like in an explosion when

    hit by the electron beam [32, 33]. This phenomenon gets more serious for higher

    beam currents. Milberg et al. [32] considered three different physical effects for

    the powder spreading; water residues in the powder, momentum transfer into the

    powder, and electrostatic charge. They examined the listed effects and showed that

    water residues and momentum transfer are not relevant to the spreading and the

    reason for spreading is potentially the electrostatic charge of the powder particles.

  • 1.4 Materials and Applications 15

    On the other hand, Qi et al. [33] claim that the impact force produced by the

    electron beam is the main reason for spreading. Other thermodynamic effects such

    as sudden evaporation were excluded, otherwise spreading should also occur in

    SLM. Preheating of the powders reduces this effect in an efficient manner [32].

    1.4 Materials and Applications

    The SEBM process was originally developed for the tool and die making indus-

    try. The usage of a highly efficient computer controlled electron beam in vacuum

    provides high precision and quality. The production process is fast in comparison

    with conventional manufacturing methods [34]. The time, cost, and challenges of

    machining or investment casting are eliminated, which makes parts readily avail-

    able for functional testing or installation. The process occurs in a high vacuum,

    which ensures the part is without imperfections caused by oxidation. SEBM makes

    the fabrication of homogeneous dense metal components possible such as complex

    tools and functional prototypes. The SEBM process can also produce hollow parts

    with an internal strengthening scaffold. Impossible with any other method, SEBM

    can deliver the required mechanical strength with much less mass. This reduces

    the cost of raw materials and the weight of the component [34].

    As mentioned before, different metallic materials can be used for SEBM such as

    titanium alloys (Ti-6AL-4V), numerous steels and cobalt alloys (CoCrMo) and the

    SEBM method can be used to produce fully dense or porous parts. At present

    time SEBM has an increasing number of production applications within both the

    aerospace and the medical implant industries. Production cases usually revolve

    around parts with complex geometries or when the conventional methods are diffi-

    cult or expensive. In aerospace industry the light-weight designs also lead to parts

    with very complex geometries [35,36]. In medical field such challenges are present

    in two obvious areas, patient-specific implants based on Computed Tomography

    (CT) data, and implants with advanced cellular structures [37].

    Porous titanium draws attention as structural material for biomedical applications

    (bone substitutes and dental implants). Titanium alloys, specifically Ti-6Al-4V,

  • 16 Beam and Powder Based Additive Manufacturing

    are widely used as an implant material due to their relatively low modulus, good

    biocompatibility, and corrosion resistance compared to other conventional alloys

    [38].

    Various processes can be used to fabricate cellular structures such as powder met-

    allurgy, rapid prototyping, deformation forming, metal wire approaches and in-

    vestment casting [37, 39]. Direct manufacturing of metal foam structures is now

    possible through the use of laser and electron beam-based layered manufacturing

    systems [29, 40]. Selective Electron Beam Melting (SEBM), shows high capability

    for the fabrication of porous titanium with defined cellular structure [29]. Some

    open celled titanium structured produced by SEBM are shown in figure 1.5. Recent

    studies have reported mechanical properties of open cell structures [4043].

    25 mm

    Figure 1.5: Open-celled titanium structures produced by SEBM [29].

    The mismatch in stiffness of the human bone and the titanium implant leads to

    the so called stress shielding effect responsible for bone resorption and eventual

    implant loosening [29]. Cellular structures are expected to prevent stress-shielding

    due to the possibility in adapting the mechanical properties of the implant to the

    biomechanical properties of the bone. In addition, this means the opportunity

  • 1.4 Materials and Applications 17

    of manufacturing of highly porous parts which enable bone tissue to grow within

    the replacement part leading to a better fixation [37]. Titanium alloys, specif-

    ically Ti-6Al-4V, are widely used as an implant material due to their relatively

    low modulus, good biocompatibility, and corrosion resistance compared to other

    conventional alloys [38].

    Figure 1.6 shows an example of a cellular titanium interbody fusion cage. Titanium

    cages are used as spacers in spine surgery to keep the space of intervertebral discs,

    which have been removed because of degenerative disease or other pathological

    conditions [29].

    5 mm

    Figure 1.6: Interbody fusion cage consisting of two different cellular structures [29].

    1.4.1 Titanium Alloys

    Titanium is a material with excellent mechanical properties, low density, high

    chemical resistance, and good biocompatibility. Due to the high melting temper-

    ature and the extreme reactivity of liquid titanium with atmospheric gases, the

    production with standard methods is expensive and difficult. Traditionally, tita-

    nium alloy parts are fabricated by forging or casting. Parts which are too expensive

    or complex for forging are produced by casting (i.e. investment casting), although

    the strength and ductility are sacrificed. Titanium cast products are formed by

    melting the alloy in a vacuum furnace or in an inert atmosphere since titanium is

    highly reactive at high temperatures. The molten metal is poured into a mold of

    the desired shape. Casting of titanium can be difficult due to low fluidity and high

    reactivity. Thereby it causes casting defects like porosity, shrinkage, and surface

  • 18 Beam and Powder Based Additive Manufacturing

    defects. Careful selection of the mold materials is required to restrict the reac-

    tion between the mold materials and the molten titanium. Powder metallurgy can

    also be considered as an alternative fabrication method at much lower tempera-

    tures [39].

    Solid free fabrication is a new method of producing titanium alloys. High dimen-

    sional tolerance may be produced in parts because the components are built layer

    by layer with alloy powders. Recently laser and electron beam melting methods

    have been focused on the fabrication of Ti-6Al-4V near net shape parts. The final

    product contains a rough surface due to the part being in contact with the loose

    powder. It is possible that gas porosity is created by trapped gas within the pow-

    der particles from the shielding gas during atomization. The processing parameters

    must be adjusted in order to obtain adequate densification of the component.

    Using vacuum during SEBM makes this process ideal for highly reactive metals like

    titanium. On the other hand, evaporation of elements is a common problem for

    the SEBM system since fabrication takes place under high vacuum. Evaporation of

    alloying elements presents challenges for controlling the chemical composition. For

    Ti-6Al-4V the depletion of aluminum is substantial due to its high vapor pressure

    [5].

    Figure 1.7: SEM secondary electron image of gas atomized Ti-6Al-4V powderparticles.

    Pre-alloyed powders used for SEBM can be prepared either by a gas atomization

    (GA) process, or a plasma rotating electrode process (PREP). In this study we

  • 1.4 Materials and Applications 19

    used metal powders produced from gas atomization as shown in figure 1.7. It can

    be seen that the as-received powder particles are rather spherical.

    The SEBM method can be used to produce fully dense or porous titanium parts.

    The Ti-6Al-4V alloy is widely used as an implantable material. The combination

    of excellent properties of titanium alloys with a cellular structures opens new po-

    tential applications in the aerospace, medical, chemical, and process engineering

    industries.

    1.4.2 Limitations

    Problems facing acceptance of components produced by the Arcam EBM process

    include porosity issues (figure 1.8) and the surface quality (figure 1.9).

    0.5 mm

    Figure 1.8: Gas porosity within compact part produced with SEBM.

    Layer-additive processes create ridged surfaces corresponding to the deposited pow-

    der layers (stair stepping effect). Nonhorizontal part surfaces tend to be rough

    because the part is built of slices having a specific thickness. Therefore there is

    often a noticeable stair-stepping effect between consecutive slices of the part (fig-

    ure 1.10, (b)). The stair-stepping effect decreases with reduced slice thickness.

    Furthermore,the surface morphology of as-built parts is non-uniform with a high

    roughness because the part is in contact with the loose powder (figure 1.10, (c)).

    Different powder size distribution can results in better powder packing and increase

  • 20 Beam and Powder Based Additive Manufacturing

    1 mm

    Figure 1.9: X-ray computed tomography image of a wall produced with SEBM.

    the surface quality. Powder size distribution has a significant effect on the surface

    quality.

    a b c

    Figure 1.10: stair stepping effect a) CAD data b)Slicing and stair stepping effectc) produced part on the powders

    The processed material can suffer also from the effects of balling, vaporization, and

    reduced wetting between layers.

    Since electron beam melting is a layerwise additive process, it is important to bond

    different layers perfectly. Remelting the previous layer provides a clean solid-liquid

    interface at the atomic level. Chosen processing parameters should ensure the

    sufficient melt of the previous layer. Insufficient energy can lead to defects like

    non-connected layer and residual powder (figure 1.11).

  • 1.5 Summary 21

    m50

    Figure 1.11: Non-connected layer and residual powder in SEBM part.

    1.5 Summary

    Beam based layered manufacturing processes are described in this chapter. The

    quality of fabricated parts can be affected by process parameters and materials

    properties like beam power, beam speed, building strategy, powder size distribution

    and layer thickness. This dissertation attempts to investigate the effect of these

    process and material parameters on the surface quality and process defects with

    the help of simulation.

  • Chapter 2

    Physical Model

    Figure 2.1 gives a rough overview of the basic physical phenomena governing SEBM

    processes. In order to make the SEBM process accessible to numerical simulation,

    the real physical process has to be simplified in such a way that the dominant

    mechanisms (in bold fonts in figure 2.1) are taken into account while the secondary

    ones are neglected for the present model. In the following, details of the underlying

    physical model are described.

    Heat conduction

    Melting/solidification

    Capillary forces

    Gravity

    Convection

    Vaporization

    Wetting/dewetting

    Powder layer

    Radiation

    Marangoni-convection

    Sintering

    Solidification shrinkage

    BEAM

    Absorption

    Figure 2.1: Physical phenomena during selective beam melting.

  • 24 Physical Model

    2.1 Random Powder Bed Generation

    The generation of the random powder bed follows the so-called rain model for

    random packing (for details see [44]). This is a model in which the falling particle,

    after its first contact, searches with the help of gravity for a more favorable situation

    by rotation until another contact is realized. The particle can rotate as often as

    necessary (always decreasing its potential energy), to finally reach the nearest local

    minimum. When no contacted particle is found, the particle is deposited on the

    basal line. The algorithm is schematically depicted in figure 2.2, (a).

    The falling particle algorithm represents what happens in a very strong gravity

    field in 2D, and produces very dense powder beds (relative density 75%, figure2.2), (b). Different packing densities are realized by removing some of the powder

    particles after dense powder bed production (figure 2.2, (c)). From each n powder

    particles one has to be removed. Changing the n results in different packing density.

    It is also possible to add powders with different size distribution such as a Gaussian

    distribution or a bimodal distribution. An example of a real powder bed cross

    section is depicted in figure 2.2, (d).

    a) b)

    c) d)

    Figure 2.2: Random powder bed a) Schematic of the rain model for random packingwith rotations. b) Powder bed produced by the rain model. c) Adjusting therelative density by removing some of the particles. d) Cross section of a realpowder bed (titanium).

  • 2.1 Random Powder Bed Generation 25

    The packing density for the powders with Gaussian size distribution in the range

    of 45-115 m is about 55%. In order to reproduce similar packing density in

    simulation one of four powder particles has to be removed (figure 2.2, (c)).

    During the real process the powders are added layer by layer. After scanning

    one layer, a new layer of powder is applied. Figure 2.3 shows the schematic view

    of powder deposition layer by layer. The shrinkage associated with densification

    leads to a change in bed geometry around the solidified region. By adding the new

    powder layer the shrinkage area is also filled with new powders.

    Layer thickness

    Figure 2.3: Schematic view of adding the powder layer by layer.

    The modeling is extended to multi-layer processes by defining the layer thickness.

    Instead of lowering the platform after solidification of each layer, the defined layer

    thickness is added to the height of the existing powder bed and the new layer of

    powders will be applied on top of the solidified cross section (figure 2.4). Same

    algorithm for powder generation is used, the only difference is to find the surface

    of the solidified material or powder bed and define a new base line for each time

    adding the powder. Details of the powder packing algorithm are given in Appendix

    A.

    During the powder generation all the particles which their center placed above the

    desired layer thickness are removed (figure 2.4, (c)). Same algorithm for removing

    the powder particle is used to adjust the powder packing density (figure 2.4, (d)).

  • 26 Physical Model

    a b c d

    Layerthickness

    Figure 2.4: Schematic view of adding the new powder layer. a) Small particles areconsidered on the surface. b) New particles find their final position with the rainmodel. c) Powders which their center placed above the defined layer thickness areremoved. d) Some particles are removed in order to adjust powder packing density.

    2.2 Beam Definition and Absorption in 2D

    The moving beam is described by a Gaussian distribution (figure 2.5, left):

    I(x, t) =P2pi

    exp

    ((x v t)

    2

    22

    )(2.1)

    where I is the beam power density, v is the speed of the beam, is the standard

    deviation, and P is the total beam power. In order to characterize manufacturing

    processes the line energy, EL, has shown to be an important parameter:

    EL =P

    v(2.2)

    The radiation penetrates into the powder bed by the open pore system. In the

    case of an electron beam, the electron energy is nearly completely absorbed at the

    position where it has first contact with the material. The absorption process for

    laser radiation is much more complicated due to multi-reflection processes causing

    radiation transport in much deeper powder layers [16]. Our present model does not

    take reflection processes into account but it is able to handle the transient nature

    of the absorbing surface due to melting. Figure 2.5 shows schematically how the

    model treats the penetration of the beam into the powder layer and melt pool.

    When the beam touches a powder particle or the melt pool, energy absorption

    follows the exponential Lambert-Beer absorption law [45],

    dI

    dz= absI (2.3)

  • 2.3 Energy Transfer and Conservation Equations 27

    BEAM

    Figure 2.5: Beam absorption. Left: Absorption of the beam into the powder layerand melt pool. Right: Absorption of the beam within a powder particle. Thenumerical grid is schematically shown.

    where abs denotes the absorption coefficient. The absorption depth, which is

    1/abs in titanium alloys is about 0.125m for the electron beam [46] and is about

    0.669 m for the CO2 laser beam [47].

    Figure 2.5 shows the absorption of the beam through a single powder particle. In

    addition, the numerical grid is schematically shown. Numerically, the energy I

    is absorbed within a numerical cell with width x by

    I

    x= absI, (2.4)

    where denotes the fraction of material (solid or liquid) within the numerical cell.

    2.3 Energy Transfer and Conservation Equations

    The beam energy is absorbed in the powder bed, the powder temperature increases,

    and the thermal energy spreads by heat diffusion. When the temperature exceeds

    the solidus temperature of the metal, the solid-fluid phase transformation starts

    thereby consuming latent heat L. When the local liquid phase fraction exceeds a

    given threshold value, the solid starts to behave as a liquid. The liquid material

    is governed by the Navier-Stokes equations. Heat transport in the liquid is either

    by diffusion or convection. Radiation and convection heat transfer from the liquid

    surface are neglected. Therefore, the excess heat of the liquid must be dissipated

  • 28 Physical Model

    by heat conduction into the powder bed in order to re-solidify the melt pool. The

    neglect of convection heat transfer for the surface is justified since the EBM process

    is carried out under vacuum. Radiation could have an essential effect and will be

    taken into account in a future work.

    The underlying continuum equations of heat convection-diffusion transport are

    founded on an enthalpy based methodology. The single-phase continuum con-

    servation equations to simulate thermo-fluid incompressible transport comprising

    melting and solidification are given by:

    u = 0, (2.5)u

    t+ (u ) u = 1

    p+ 2u + g, (2.6)

    E

    t+ (uE) = ( kE) + , (2.7)

    where is the gradient operator, t the time, u the local velocity of the melt, p thepressure, the density and the kinematic viscosity. The thermal diffusivity is

    designated by k = k(E) and gravity is denoted by g. The energy source describes

    the energy deposited in the material by the beam. Viscous heat dissipation and

    compression work are neglected in the present model. The thermal energy density

    E is given by

    E =

    T0

    cp dT + H, (2.8)

    where cp is the specific heat at constant pressure, T is the temperature and H is

    the latent enthalpy of a computational cell undergoing phase change. For a multi

    component metal alloy, H is a complex function of the temperature. In a simple

    approximation it can be expressed as follows:

    H(T ) =

    L T TlTTsTlTs L Ts < T < Tl0 T Ts,

    (2.9)

    Where Ts and Tl are representing the solidus and liquidus temperature respectively.

    L is the latent heat of phase change. Denoting as the liquid fraction in a cell,

    (T ) =H(T )

    L. (2.10)

  • 2.4 Capillarity and Wetting 29

    The latent enthalpy is taken up into an effective specific heat cp

    E =

    T0

    cp dT + H =

    T0

    cp dT (2.11)

    with

    cp =

    cp T Tlcp +

    LTlTs Ts < T < Tl.

    cp T Ts(2.12)

    The thermal diffusivity k is related to the heat conductivity by

    k(E) =(E)

    cp(E).(2.13)

    2.4 Capillarity and Wetting

    Capillarity and wetting are strongly correlated and both phenomena are governed

    by the surface and interface energies. They play a crucial role in SLM/SEBM

    processes. It depends on the experimental conditions whether the liquid wets the

    still solid powder (or re-solidified melt pool) underneath (figure 2.6).

    Figure 2.6: Capillarity and wetting. a) Non-wetting melt pool on top of the powder.b) Wetting melt pool on top of the powder. c) Dynamic wetting angle andequilibrium wetting angle 0 with respect to the tangent direction t.

    A well-known phenomenon during SLM/SEBM processes is the break up of thin

    melt pools into spherical droplets called balling [4]. Commonly, balling is explained

    by the Plateau-Rayleigh capillary instability of a cylinder at length to diameter

    ratio greater than pi [27, 30]. A strong non wetting condition further amplifies

    balling (figure 2.6, (a)), while good wetting of the melt with the underlying powder

    (or re-solidified melt pool) works against balling. Capillary force, Fcap, exists if

    the surface curvature does not vanish:

    Fcap = dA n, (2.14)

  • 30 Physical Model

    where is the curvature, is the surface tension, dA denotes a surface element,

    and n is the normal vector belonging to dA.

    In order to describe dynamic wetting we have to consider the wetting force that can

    be derived from Youngs equation [48]. A wetting force is present if the dynamic

    wetting angle, d, is not equal to the equilibrium wetting angle eq. The tangential

    component of the force Fwett equals (see figure 2.6, (c)):

    Fwett = (cos d cos eq) . (2.15)

    This force vanishes when the dynamic wetting angle is equal to the equilibrium

    wetting angle. The wetting angle between fluid and solid powder can be adjusted

    between 0 and pi. It is also possible to define the wetting angle between fluid and

    re-solidified fluid. In this thesis, we assumed complete wetting between fluid and

    re-solidified fluid.

    2.5 Summary

    The SEBM process is complex and involves different phenomena. In order to make

    the SEBM process accessible to numerical simulation, the real physical process

    has to be simplified. Several assumptions have been made to simplify the process.

    These assumptions lead to a set of differential equations which describe the system.

    Generally, these equations can not be solved analytically. Therefore, numerical

    methods have to be employed. The numerical implementation is discussed in the

    next chapter.

  • Chapter 3

    Numerical Implementation

    Computational fluid dynamic has been developed systematically to solve and an-

    alyze problems involving fluid flows. Two groups of approaches were widely used

    in fluid modeling during past decades. One group which is known as macroscopic

    methods, including classical fluid mechanics and thermodynamics. Classical fluid

    mechanics study a fluid system from the macroscopic point of view. It means that

    although a fluid system consists of discrete particles, the detailed behavior of each

    individual molecule or atom is not considered. Theses methods can be used to

    obtain macroscopic variables, such as velocity, pressure and temperature, which

    characterize the state of the fluid system. Based on the continuum description of

    macroscopic phenomena, Navier-Stokes equations can be derived through conser-

    vation laws. Fluid mechanics researchers attempted to use different methods to

    solve Navier-Stokes equations with specific boundary conditions and initial con-

    ditions. Various numerical methods are available, e.g. finite element methods,

    finite volume methods, and finite difference methods [49]. These methods are used

    to transform the continuum description into a discrete one in order to solve the

    equations numerically on a computer [49].

    The other way to simulate a fluid behavior on a computer is to model the individual

    molecules which make up the fluid, and it is known as Molecular Dynamics (MD)

    approach. This method is based on the microscopic particle description provided

    by the molecular dynamics equations and is often used in material science and

  • 32 Numerical Implementation

    biological researches [50]. In this method at each time step, position and velocity

    of molecules in the system are calculated according to their previous position and

    velocity based on Newtons second law.

    This microscopic description is straightforward to program on a computer but

    it is time consuming and computationally expensive. As a result, the number

    of molecules that can be simulated is still very limited and this method can be

    used for very small systems and very short times. Two possible ways have been

    proposed to reduce the computational demands for MD methods. First, instead

    of considering each individual molecule at the microscopic scale, fluid particles at

    mesoscopic scale made of a group of particles are considered in the simulation.

    Second, the freedom degrees of the system can be reduced by forcing the fluid

    particles to move in specified directions. The lattice gas method and the lattice

    Boltzmann method are based on these concepts and have been successfully applied

    to simulate fluid flow and transport phenomena [18].

    The collective behavior of the particles in a system is used to simulate the contin-

    uum behavior of the system with the Lattice Boltzmann Method (LBM). Particles

    exist on a set of discrete points that are spaced at regular intervals to form a lat-

    tice. Time is also divided into discrete time steps. During each time step particles

    jump to the next lattice site and then scatter according to simple kinetic rules that

    conserve mass, momentum and energy. The method is based on the Boltzmann

    transport equation which simply says that the rate of change equals to difference

    between the number of particles scattered into that state from the number of par-

    ticles scattered out of that state. Since boundary conditions are imposed locally,

    lattice methods simulate flows in both simple and complex geometries with almost

    the same speed and efficiency. Therefore, they are suitable for modeling flows in

    extremely complex geometries involving interfacial dynamics and complex bound-

    aries [51]. Recently, the lattice Boltzmann method has attracted much attention

    because of having a remarkable ability to simulate single and multiphase fluids.

  • 3.1 Lattice Gas Automata 33

    This chapter describes the algorithm of the LBM. First, an overview of the method

    development will be given and afterward the method itself and the boundary con-

    ditions will be described.

    3.1 Lattice Gas Automata

    Historically, the LBE method belongs to the class of lattice gas automata (LGA).

    Frisch, Hasslacher and Pomeau provided the first two-dimensional LGA model

    known as the FHP model in 1986 [52]. The FHP model uses a triangular lattice

    and it can properly simulate the two-dimensional Navier Stokes equations [53].

    In order to construct the kinetic LGA model, a regular lattice of cells in m dimen-

    sions must be first considered and then suitable evolution rules must be established.

    At each lattice node, a set of Boolean variables i is used to describe the local state

    (x, t) = {1, ..., b}, where the subscript i is an index for velocity and denotesdifferent velocity directions, x is a vector in the lattice space, t denotes a discrete

    time and b is the number of particle velocity directions. The evolution equation of

    LGA can be written as

    i(x + ei, t+ 1) = i(x, t) + i((x, t)), (3.1)

    where ei are the local particle velocities and i is the collision operator. The

    evolution of LGA consists of two steps that take place during each time step:

    Streaming; advection of a particle to the nearest neighboring node along its

    velocity direction

    Collision; particles collide with each other and scatter according to collision

    rules.

    It is very important to construct correct collision rules for LGA. The collision rules

    must guarantee the conservation of mass, momentum and energy.

    The LGA has several advantages over traditional CFD methods, like simple evo-

    lution rules, which are easy to implement as parallel computations [53], and easy

  • 34 Numerical Implementation

    introduction of the boundary conditions. However, it also has several undesirable

    features. The most serious one is the inherent statistical noise in the simulations

    due to the large fluctuation in the Boolean variables. To overcome the intrinsic

    drawbacks of LGA, lattice Boltzmann equation models were introduced where the

    particle occupation variables (Boolean variables) in the evolution equation are re-

    placed by particle distribution functions which eliminates the statistical noise of

    the LGA. Particle distribution functions are real variables between zero and one.

    3.2 The Lattice Boltzmann Method

    McNamara and Zanetti [54] for the first time used the lattice Boltzmann equation

    as a numerical scheme in which the same form of the collision operator as in

    the LGA was adopted. Later on, Higuera and Jimenez [55] could show that the

    nonlinear collision operator, which is time consuming, can be approximated by a

    linear operator. Although the statistical noise was eliminated in both models, other

    problems still remained. Chen et al. [56] and Qian et al. [57] proposed LBM models

    which provided the freedom required for the equilibrium distribution to satisfy

    isotropy, Galilean invariance, and to possess a velocity-independent pressure. In

    their models, the single relaxation time approximation known as the Bhatnagar,

    Gross and Krook (BGK) approximation was applied to greatly simplify the collision

    operator. The LBE model with the BGK approximation is called lattice BGK

    (LBGK) model [58]. The LBGK model is the most widely used model in the

    lattice Boltzmann simulations. This thesis is focused on this model.

    It is well known that the BGK approximation often results in numerical instability

    when the fluid has a relatively low viscosity. Recent studies suggest using multiple

    relaxation times instead of the BGK model [59]. It is useful for improving the

    stability of the scheme [59,60].

    The fundamental principle of the LBM [18, 19] is to solve the microscopic ki-

    netic equation for single-particle distribution functions f(x, , t) in the physical-

    momentum space

  • 3.2 The Lattice Boltzmann Method 35

    f

    t+ Of + F f

    = S, (3.2)

    where f is defined as the number of particles or molecules at the time t positioned

    between x and x + dx with velocities between and + d. F is the force field

    per unit mass acting on the particle, and S is the collision operator which is the

    sum of all intermolecular interactions. This collision takes particles in or out the

    streaming trajectory. The Boltzmann equation has its foundations in gas dynamics

    and is a well-accepted mathematical model of a fluid at the microscopic level. It

    provides detailed microscopic information which is critical for the modeling of the

    underlying physics behind complex fluid behavior. This is more fundamental than

    the N-S equations. However, due to the high dimensions of the distribution and the

    complexity in the collision operator, direct solution of the full Boltzmann equation

    is a difficult task for both analytical and numerical techniques.

    One of the difficulties in dealing with the Boltzmann equation is the complicated

    nature of the collision operator. Therefore an important simplification of the colli-

    sion term was proposed by Bhatnagar, Gross and Krook in 1954 [58], and is known

    as the BGK approximation. The Boltzmann-BGK equation then takes the form

    f

    t+ Of + F f

    = 1

    [f f eq], (3.3)

    where f eq is the equilibrium distribution function and is the relaxation time.

    Equation (3.3) is first discretized in the momentum space using a finite set of

    velocities {i|i = 1, ..., b} without violating the conservation laws [19,61].fit

    + i Ofi = 1

    [fi f eqi ] (3.4)

    In the above equation, fi(x, t) f(x, i, t) and f eqi (x, t) f eq(x, i, t) are thedistribution function and the equilibrium distribution function of the i-th discrete

    velocity i, respectively.

    For 2D flow, the 9-velocity LBE model on the 2D square lattice, denoted as D2Q9

    model, has been widely used. For simulating 3D flow, there are several cubic lattice

    models such as D3Q15, D3Q19, and D3Q27 models [23]. Figure 3.1 presents the

    most common lattices.

  • 36 Numerical Implementation

    Figure 3.1: Velocity sets for different LBMs: D2Q4, D2Q9 and D3Q19 [23].

    3.3 Thermal Lattice Boltzmann Method

    Incorporating the effects of temperature into lattice Boltzmann models has turned

    out to be surprisingly difficult. LBM approaches dealing with thermal fluids can be

    categorized into multi-speed models [6264] and the multi-distribution functions

    models [6568].

    Multi-speed models introduce additional discrete velocities and higher order ve-

    locity terms in the equilibrium functions. The philosophy behind this approach is

    to define the internal energy as a moment of the lattice Boltzmann distribution.

    Disadvantages of this approach are severe numerical instabilities combined with a

    very restricted range of temperature variation.

    These limitations are not present for the multi-distribution function models where

    the temperature is treated as a passive diffusing scalar [65, 69]. That is, two sets

    of distribution functions are defined; one for the density and the velocity field

    and the other for the temperature. The advantage of this approach is that it

    can easily handle arbitrary Prandtl numbers (the ratio of kinematic viscosity to

    thermal diffusivity). Nevertheless, it is only applicable for systems where the fluid

    density is not strongly dependent on temperature.

    Lattice Boltzmann models where solid-liquid phase transition problems are treated

    are relatively rare [7072]. Miller and Succi [70] utilized a phase-field based method-

    ology for the evolution of the phase fractions. The model is applied to simulate

  • 3.4 Multi-distribution Function Method 37

    binary alloy solidification and dendritic growth into an undercooled melt. Chat-

    terjee and Chakraborty [71] introduced a hybrid technique by coupling a modified

    thermal LB model with a fixed-grid enthalpy-porosity approach. The macroscopic

    density and velocity fields are simulated by using a single-particle distribution func-

    tion, while the macroscopic temperature field is obtained from a total enthalpy

    density distribution function.

    3.4 Multi-distribution Function Method

    In this section a lattice Boltzmann algorithm for simulating thermal transport in

    fluids with free surfaces and a solid-liquid phase transformation is presented. This

    is a typical problem appearing in materials science where materials are produced by

    solidification of melts in which strong topological changes occur and free boundaries

    have to be treated. The underlying LB method is based on the multi-distribution

    function model, i.e. the internal energy is captured by a second distribution function

    that models the energy as a conserved scalar quantity analogous to the density.

    The treatment of the phase transformation follows the approach of Chatterjee and

    Chakraborty [71].

    In order to solve the macroscopic single phase continuum conservation equations

    (section 2.3), we apply a multi-distribution function method [66, 71]. Using a

    second distribution to model the energy density implies that we are following the

    passive-scalar approach. This is based on the fact that the temperature satisfies

    the same evolution equation as a passive scalar, if viscous heat dissipation and

    compression work would be negligible [65].

    At each lattice site, two sets of distribution functions, fi and hi, are defined. The

    distribution fi models mass and momentum transport, whereas the distribution hi

    represents the movement of the internal energy. The macroscopic quantities are

    given by

    =i

    fi, u =i

    eifi, E =i

    hi, (3.5)

  • 38 Numerical Implementation

    where is the density, u is the macroscopic velocity, and E is the energy density,

    i.e. the energy per unit volume.

    The collision and displacement of the distributions are summarized by the equa-

    tions of motion:

    fi(x + ei, t+ t) fi(x, t) = tf

    (f eqi (x, t) fi(x, t)) + Fi, (3.6)

    hi(x + ei, t+ t) hi(x, t) = th

    (heqi (x, t) hi(x, t)) + i, (3.7)

    where fi(x, t) and hi(x, t) represent the density and energy distribution functions

    in i-direction, respectively. The energy source i is the energy deposited in each

    cell under beam radiation which was calculated from equation (2.4).

    For the consideration of body forces (e.g. the gravity g) we use the method de-

    scribed by Luo [73]:

    Fi = wi

    [(ei u)

    c2s+

    (ei u) eic4s

    ] g (3.8)

    f eqi (x, t) and heqi (x, t) are the equilibrium distributions functions:

    f eqi (x, t) = i

    [1 +

    (ei u)c2s

    +(ei u)2

    2 c4s u

    2

    2 c2s

    ](3.9)

    heqi (x, t) = iE

    [1 +

    (ei u)c2s

    +(ei u)2

    2 c4s u

    2

    2 c2s

    ](3.10)

    For the two-dimensional D2Q9 model, the velocity vectors ei and the weights i

    are given by:

    ei =

    (0, 0) i = 0

    (c, 0) (0,c) i = 1, . . . , 4(c,c) i = 5, . . . , 8

    (3.11)

    i =

    4/9 i = 0

    1/9 i = 1, . . . , 4

    1/36 i = 5, . . . , 8

    (3.12)

    The speed of sound is given by c2s = c2/3. For small Mach numbers Ma = |u| /cs

    1, i.e. under the incompressible flow limit, the mass, momentum and energy equa-

  • 3.5 Free Boundary Treatment 39

    tions (equations ( 2.5), (2.6) and (2.7)) can be derived through a Chapman-Enskop

    expansion [66,71,74]. The viscosity and the thermal diffusivity k are given by

    = c2s t (f 0.5) k = c2s t (h 0.5) (3.13)

    where f and h are the dimensionless relaxation times for the velocity and tem-

    perature fields respectively. Equations (3.6) and (3.7) are solved in a two-step

    procedure:

    Collision:

    f outi (xi, t) = fini (x, t) +

    t

    f

    (f eqi (x, t) f ini (x, t)

    )+ Fi (3.14)

    houti (xi, t) = hini (x, t) +

    t

    h

    (heqi (x, t) hini (x, t)

    )+ i (3.15)

    Streaming:

    f ini (x + ei, t+ t) = fouti (x, t) (3.16)

    hini (x + ei, t+ t) = houti (x, t) (3.17)

    where f outi and fini denote the outgoing (i.e. after collision) and incoming (i.e. before

    collision) distribution functions, respectively. At equilibrium, the energy current

    is proportional to the mass current.

    3.5 Free Boundary Treatment

    The free surface lattice Boltzmann model is developed for simulating the moving

    interface between immiscible gas and liquid fluids. In contrast to the multiphase

    LB descriptions, capturing of the interface is necessary for the free surface model.

    It leads to a relatively simple treatment of free surface boundary conditions with

    high computational efficiency but without sacrificing the underlying physics.

    The limitation of the free surface model is that it cannot be used to study liquid-

    liquid or liquid-vapor systems where two phases affect each other. So the free

    surface model is suitable only for those liquid-gas systems where the gas phase has

    negligible influence on the liquid phase.

  • 40 Numerical Implementation

    Wall

    Interface

    Fluid

    Gas

    Figure 3.2: Different cell types assumed in simulation. Additional interface cellsare defined near wall cells in order to allow very small wetting angles.

    In the computational domain, each cell belongs to one of the following cell types

    (figure 3.2).

    Fluid cell: Cells completely filled with fluid and no gas cell as a direct neigh-

    bor.

    Gas cell: Cells completely filled with gas and no fluid cell as a direct neighbor.

    These cells are not considered in the fluid simulation.

    Interface cell: Cells representing either the boundary between fluid cells and

    gas cells, or the boundary between gas cells and wall/solid cells (fluid cells

    and wall/solid cells), if this cell has at least one not empty (not full) direct

    neighbor interface cell.

    Wall/Solid cell: No slip boundary condition, i.e. the density distribution

    functions (fi) are bounced back at wall/solid cells.

    The description of the liquid-gas interface is very similar to the volume of fluid

    (VOF) method. An additional variable, the volume fraction of fluid defined as

    the portion of cell area filled with fluid, is assigned to each interface cell. All cells

    are able to change their types but it is important to notice that direct state changes

    from fluid to gas and vice versa are not possible.

    To guarantee stability of the interface, it is required to have only a single layer

    of interface cells surrounding the fluid cells. This condition has to be modified at

    the fluid wall/solid interface. In order to realize very small (eq 5)or very largewetting angels (eq 175), interface cells without a neighboring fluid cell have to

  • 3.5 Free Boundary Treatment 41

    be tolerated. These additional cells (marked with a cross in figure 3.2) have to be

    generated with an additional algorithm.

    The used cell types, their state variables, and possible state transformations are

    listed in table 3.1. For more details see reference [22].

    Table 3.1: Cell types, state variables, and possible state transformations. (statevariab