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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
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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
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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
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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
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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
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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.
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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-
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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),
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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
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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
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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
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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.
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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
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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.
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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
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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)
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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-
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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.
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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