-
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
-
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
-
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
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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)
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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.
-
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
-
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-
-
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