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Accepted Manuscript Title: Mesoscopic Simulation of Selective Beam Melting Processes Authors: Carolin K ¨ orner, Elham Attar, Peter Heinl PII: S0924-0136(10)00386-9 DOI: doi:10.1016/j.jmatprotec.2010.12.016 Reference: PROTEC 13049 To appear in: Journal of Materials Processing Technology Received date: 6-10-2010 Revised date: 10-12-2010 Accepted date: 17-12-2010 Please cite this article as: K¨ orner, C., Attar, E., Mesoscopic Simulation of Selective Beam Melting Processes, Journal of Materials Processing Technology (2010), doi:10.1016/j.jmatprotec.2010.12.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Page 1: Korner - Mesoscopic simulation o selective beam melting processes

Accepted Manuscript

Title: Mesoscopic Simulation of Selective Beam MeltingProcesses

Authors: Carolin Korner, Elham Attar, Peter Heinl

PII: S0924-0136(10)00386-9DOI: doi:10.1016/j.jmatprotec.2010.12.016Reference: PROTEC 13049

To appear in: Journal of Materials Processing Technology

Received date: 6-10-2010Revised date: 10-12-2010Accepted date: 17-12-2010

Please cite this article as: Korner, C., Attar, E., Mesoscopic Simulation of SelectiveBeam Melting Processes, Journal of Materials Processing Technology (2010),doi:10.1016/j.jmatprotec.2010.12.016

This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.

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Mesoscopic Simulation of Selective Beam Melting

Processes

Carolin Korner*, Elham Attar** and Peter Heinl

Institute for Material Science and Technology of Metals, Department of MaterialsScience, University of Erlangen - Nuremberg, Martensstrasse 5, 91058 Erlangen,

Germany

* Corresponding author: [email protected]** [email protected]

Tel: +49 9131 85-27528 Fax: +49 9131 85-27515

Abstract

A 2D lattice Boltzmann model is developed to investigate melting andre-solidification of a randomly packed powder bed under the irradiation of aGaussian beam during selective beam melting processes. Numerical simula-tion results are presented where individual powder particles are considered.This approach makes many physical phenomena accessible which can not bedescribed in a standard continuum picture, e. g. the influence of the rela-tive powder density, the stochastic effect of a randomly packed powder bed,capillary and wetting phenomena. The proposed model, although still 2D,is able to predict many experimental observations such as the well knownballing effect.

A process map is used to demonstrate the effect of the process param-eters, beam powder and scan speed. The simulation results are comparedwith experimental findings during Selective Electron Beam Melting (SEBM).The comparison shows good agreement between simulation results and ex-periments.

Keywords: Additive manufacturing, Selective beam melting, LatticeBoltzmann method, Fluid dynamics, Heat transfer.

1. Introduction

Beam and powder based layered manufacturing methods are relativelynovel technologies that can build parts from powdered material via layer-

Preprint submitted to Journal of Materials Processing Technology December 10, 2010

*Manuscript

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by-layer melting induced by a directed electron or laser beam (Levy et al.(2003)). Examples of commercialised selective beam melting processes areSelective Laser Beam Melting (SLM) and Selective Electron Beam Melting(SEBM). During the SLM or SEBM process the surface of a powder bed isselectively scanned with a beam. Thin molten tracks develop and combineto form a 2D layer of the final part. After completion of one layer, the wholepowder 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 mecha-nisms are essential: solid state sintering, liquid phase sintering, partial melt-ing, full melting, chemical binding, etc. (Kruth et al. (2005, 2007)). Formetal powders, melting and re-solidification are the underlying mechanismsto consolidate the powder particles for building a functional part. Typicalprocess defects associated with SLM/SEBM processes are porosity, residualpowder and not connected layers. State-of-the-art to find the optimal pro-cessing parameters for a new material is still based on an expensive trialand error process. This makes the range of applicable materials still to bestrongly limited.

The SLM/SEBM process is rather complex and involves many differentphysical phenomena (Das (2003)): absorption of the beam in the powderbed and the melt pool or the re-solidified melt, melting and re-solidificationof a melt pool, wetting of the powder particles with the liquid, diffusiveand radiative heat conduction in the powder, diffusive and convective heatconduction in the melt pool, capillary effects, gravity, etc. The melt poolcaused by the beam is highly dynamic and driven by the high surface tensionin combination with the low viscosity of liquid metals. This leads to thedevelopment of stochastic melt tracks with irregular, corrugated appearance.On the other hand, the life time of the melt pool is rather short - only somemilliseconds. After finishing one layer, a new powder layer is applied onthe corrugated surface leading to a new powder layer with strongly varyingthickness which might result in a typical process defect.

Several authors apply numerical simulation methods in order to develop abetter understanding of the underlying consolidation process. Williams andDeckard (1998) developed the 2D finite difference model to study process pa-rameters in selective laser sintering of polymers. There are also finite elementmodels presented by Bugeda et al. (1999) and Shiomi et al. (1999) to simulateselective laser sintering process. Zhang and Faghri (1999) developed a modelfor melting of two component metal powders with significantly different melt-

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ing points. Tolochko et al. (2000) used the simulation and experiments inorder to investigate the effects of process parameters on sintering mechanismof Ti powders. Kolossov et al. (2004) developed a three dimensional finiteelement model which considers the nonlinear behaviour of thermal conduc-tivity and specific heat due to temperature and phase transformation. Theresults of this model were experimentally tested by direct temperature mea-surements. Recently, Zah and Lutzmann (2010) developed the finite elementmethod for simulation of electron beam melting process. In all mentionedapproaches the underlying model is based on a homogenised picture, i. e. thepowders are considered as a homogeneous material with effective properties,e. g. an effective thermal conductivity which depends on the relative density.

In order to have a more realistic model, Konrad et al. (2005) and Xiaoand Zhang (2007a,b) divided the powder bed into different regions from thebottom to the heating surface and for each region the effective thermal con-ductivity is defined separately. Though Zhou et al. (2009) consider a bimodalrandomly packed particle bed for the simulation of the radiative heat transferin a selective laser sintering process, melting and the development of the meltpool geometry is not described. A essential challenge for the homogeneousapproaches is to model the powder shrinkage of about 50% during the solid-liquid phase transformation. It is well known that shrinkage has an enormousinfluence on the melt pool geometry and the local thermal properties. Allshrinkage models available are solely a function of the packing density of thepowder. None of the models considers the shrinkage of a real random powderbed. The resulting melt pool geometries are thus always well defined withoutthe stochastic behaviour which is experimentally observed (Tolochko et al.(2000)). That is, the existing models in literature are still far away from theexperimental findings. One reason for that discrepancy is certainly that themodels are not considering individual powder particles.

The purpose of this paper is to gain a much better understanding of theSLM/SEBM process with the help of numerical simulation. Our method isbased on a lattice Boltzmann model (LBM) (Chen and Doolen (1998); He andLuo (1997)) which is an alternative to ordinary computational fluid dynam-ics models. The LBM approach is especially beneficial for problems withcomplex interfaces such as flows in porous media (Bernsdorf et al. (2000);Pan et al. (2004)) or the development of foams (Korner et al. (2005)). Incontrast to existing models in the literature we are developing a numericaltool where the effect of individual powder particles is considered. A se-quential addition packing algorithm is employed to generate 2D randomly

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packed powder layers that are composed of spherical particles. The beamis absorbed by the powder layer, heats the powder and eventually melts it.Due to capillary forces, gravity and wetting a complex shaped and stronglychanging melt pool geometry develops until solidification freezes the currentstate. The comparison with experimental results from SEBM demonstratesthe predictive power of our numerical model.

2. Physical Model

Figure 1 gives a rough overview of the basic physical processes governingselective beam melting processes. In order to make the process accessible tonumerical simulation, the real physical process has to be simplified in such away that the dominant mechanisms (in bold fonts in figure 1) are taken intoaccount while the secondary ones are neglected for the present model. In thefollowing, details of the underlying physical model are described.

Figure 1: Physical phenomena during selective beam melting.

2.1. Random Powder Bed Generation

The generation of the random powder bed follows the so-called rain modelfor random packing, for details see Meakin and Juillien (1987). This is a

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model in which the falling particle, after its first contact, searches, withthe help of gravity, for a more favorable situation by rotation until anothercontact is realized. The particle can rotate as often as necessary (alwaysdecreasing its potential energy), to finally reach the nearest local minimum.When no contacted particle is found, the particle is deposited on the basalline. The algorithm is schematically depicted in figure 2, (a). The falling

a) b)

c) d)

Figure 2: Random powder bed a) Schematic of the rain model for random packing withrotations. b) Powder bed produced by the rain model. c) Adjusting the relative densityby removing some of the particles. d) Cross section of a real powder bed (titanium alloy).

particle algorithm is representation of what happens in a very strong gravityfield in 2D and produces very dense powder beds (relative density ≈ 75%),figure 2, (b). Different packing densities are realized by removing some of thepowder particles after dense powder bed production, figure 2, (c). It is alsopossible to add powders with different size distribution such as a Gaussiandistribution or a bimodal distribution. An example of a real powder bedcross section is depicted in figure 2, (d). The packing density for the powderswith Gaussian size distribution in the range of 45-115 µm is about 55%. Inorder to reproduce similar packing density in simulation, some of the powderparticles have to be removed from the powder bed(figure 2, (c)).

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2.2. Beam Definition in 2D

The moving beam is described by a Gaussian distribution:

I(x, t) =P√2 πσ

exp

(

−(x− v t)2

2 σ2

)

(1)

where I is the beam power density, v is the velocity of the beam, σ is thestandard deviation and P is the total beam power.

In order to characterize manufacturing processes the line energy, EL, hasshown to be an important parameter:

EL =P

v. (2)

2.3. Beam Absorption

The radiation penetrates into the powder bed by the open pore system.In the case of an electron beam, the radiation energy is nearly completelyabsorbed at the position where it has first contact with the material. Theabsorption process for laser radiation is much more complicated due to multi-reflection processes causing radiation transport in much deeper powder layers(Zhou et al. (2009)). Our present model does not take reflection processes intoaccount but is able to handle the transient nature of the absorbing surfacedue to melting. Figure 3 shows how the model treats the penetration of thebeam into the powder layer and the melt pool. When the beam touches apowder particle or the melt pool energy absorption follows the exponentialLambert-Beer absorption law,

dI

dz= −λabsI, (3)

where λabs denotes the absorption coefficient. Figure 3 shows the absorptionof the beam through a single powder particle. In addition, the numerical gridis schematically depicted. Numerically, the energy ∆I is absorbed within anumerical cell with width ∆x:

∆I

∆x= −λabsI ǫ (4)

where ǫ denotes the fraction of material (solid or liquid) within the numericalcell.

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Figure 3: Beam absorption. Left: Absorption of the beam into the powder layer andmelt pool that develops. Right: Absorption of the beam within a powder particle. Thenumerical grid is schematically shown.

2.4. Energy Transfer and Conservation Equations

The beam energy is absorbed in the powder bed, the powder temperatureincreases and the thermal energy spreads by heat diffusion. When the tem-perature exceeds the solidus temperature of the metal, the solid-fluid phasetransformation starts thereby consuming latent heat L. When the local liq-uid phase fraction exceeds a given threshold value, the solid starts to behaveas a liquid. The liquid material is governed by the Navier-Stokes equations.Heat transport in the liquid is either by diffusion or by convection. Radi-ation and convection of heat from the liquid surface are neglected so thatthe excess heat of the liquid must be dissipated by heat conduction into thepowder bed in order to re-solidify the melt pool. The neglect of convectionis justified since the SEBM process is under vacuum. Radiation could havean essential effect and will be taken into account in a further work.The underlying continuum equations of heat convection-diffusion transportare founded on an enthalpy based methodology. The single-phase contin-uum conservation equations to simulate thermo-fluid incompressible trans-port comprising melting and solidification are given by:

∇ · u = 0, (5)

∂u

∂t+ (u · ∇)u = −1

ρ∇p+ ν∇2u+ g, (6)

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∂t+∇ · (uE) = ∇ · ( k∇E) + Φ (7)

where ∇ is the gradient operator, t the time, u the local velocity of the melt,p the pressure, ρ the density and ν the kinematic viscosity. The thermaldiffusivity is designated by k = k(E) and gravity is denoted by g. Theenergy source Φ describes the energy deposited in the material by the beam.Viscous heat dissipation and compression work are neglected in the presentmodel.

The thermal energy density E is given by

E =

∫ T

0

ρ cp dT + ρ∆H, (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 temper-ature. In a simple approximation it can be expressed as follows:

∆H(T ) =

L, T > TfT−Ti

Tf−Ti· L, Ti < T < Tf

0, T < Ti,

(9)

with Ti and Tf representing the beginning and the end of the phase trans-formation, respectively. L is the latent heat of phase change. Denoting ξ asthe liquid fraction in a cell,

ξ(T ) =∆H(T )

L. (10)

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

E =

∫ T

0

ρ cp dT + ρ∆H =

∫ T

0

ρ cp dT (11)

with

cp =

cp, T > Tf

cp +L

Tf−Ti, Ti < T < Tf

cp, T < Ti,

(12)

The thermal diffusivity k is related to the heat conductivity λ by:

k(E) =λ(E)

ρ cp(E)(13)

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2.5. Capillarity and Wetting

Capillarity and wetting are strongly correlated and both phenomena aregoverned by the surface and interface energies. Both play a crucial role duringSLM/SEBM processes. It depends on the experimental conditions whetherthe liquid wets the still solid powder (or re-solidified melt pool) underneath,see figure 4.

Figure 4: Schematic of capillarity and wetting. a) Non-wetting melt pool on top of thepowder. b) Wetting melt pool on top of the powder. c) Dynamic wetting angle θ andequilibrium wetting angle θ0 with respect to the tangent direction t.

A well-known phenomenon during SLM/SEBM processes is the break upof thin melt pools into spherical droplets, called balling (Levy et al. (2003)).Commonly, balling is explained by the Plateau-Rayleigh capillary instabilityof a cylinder at length to diameter ratio greater than π (Gusarov et al.(2007)). A strong non-wetting condition further amplifies balling (figure 4,a), while good wetting of the melt with the underlying powder (or re-solidifiedmelt pool) works against balling. Capillary forces, Fcap, develop if the surfacecurvature κ does not vanish:

Fcap = κ · σ · dA · n, (14)

where κ is the curvature, σ is the surface tension, dA denotes a surfaceelement, and n is the normal vector belonging to dA.

In order to describe dynamic wetting we have to take into account thewetting force that can be derived from Young’s equation. A wetting forceis present if the dynamic wetting angle, θ, is not equal to the equilibriumwetting angle, θ0. The tangential component of the force Fwet

t equals (seefigure 4, c):

Fwett = σ (cos θ − cos θ0) . (15)

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3. Lattice Boltzmann Model

Lattice Boltzmann approaches model physical problems (commonly fluiddynamical problems) by simulating the temporal evolution of one-particledistribution functions f(x,v, t), where f represents the probability of findinga (fluid) particle at position x, at time t, moving with velocity v. Knowing f ,local values of the density and momentum are given by evaluating momentsof f . The lattice Boltzmann distribution function fi is a discretized versionof the continuous function f , where space is divided up into a regular latticeand the velocities are represented by a finite number of displacement vectors∆t· ei to neighboring lattice sites, where i = 0, . . . , b, ∆t is the time step, andb represents the total number of displacement directions. Resting particlesare represented by a zero displacement vector e0.

3.1. Multi-distribution function method

In order to solve the macroscopic equations (5), (6), (7), we apply amulti-distribution function method (Palmer and Rector (2000); Chatterjeeand Chakraborty (2006)). Using a second distribution to model the energydensity implies that we are following the passive-scalar approach which isbased on the fact that the temperature satisfies the same evolution equa-tion as a passive scalar if viscous heat dissipation and compression work arenegligible (Shan (1997)).

At each lattice site, two sets of distribution functions, fi and hi, aredefined. The distribution fi models mass and momentum transport, whereasthe distribution hi represents the movement of the internal energy. Themacroscopic quantities are given by

ρ =∑

i

fi, ρu =∑

i

eifi, E =∑

i

hi, (16)

where ρ is the density, u is the macroscopic velocity and E is the energydensity, i. e. the energy per unit volume.

The collision and displacement of the distributions are summarized bythe equations of motion

fi(x + ei, t+∆t)− fi(x, t) =∆t

τf(f eq

i (x, t)− fi(x, t)) + Fi (17)

hi(x + ei, t+∆t)− hi(x, t) =∆t

τh(heq

i (x, t)− hi(x, t)) + Φi, (18)

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where fi(x, t) and hi(x, t) represent the density and energy distribution func-tions in i-direction .The energy source Φi is the energy deposited in each cellunder beam radiation which calculated from Eq. (4)

For the consideration of body forces, e. g. the gravity g, we use the methoddescribed by Luo (2000):

Fi = wi ρ

[

(ei − u)

c2s+

(ei · u) eic2s

]

· g (19)

f eqi (x, t) and heq

i (x, t) are the equilibrium distributions functions:

f eqi (x, t) = ωi ρ

[

1 +(ei · u)

c2s+

(ei · u)22 c4s

− u2

2 c2s

]

(20)

heqi (x, t) = ωiE

[

1 +(ei · u)

c2s+

(ei · u)22 c4s

− u2

2 c2s

]

(21)

For the two-dimensional D2Q9 model, the velocity vectors ei and the weightsωi are given by:

ei =

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

(22)

ωi =

4/9, i = 01/9, i = 1, . . . , 41/36, i = 5, . . . , 8.

(23)

The speed of sound is given by c2s = c2/3. For small Mach numbersMa = |u| /cs ≪ 1, i. e. under the incompressible flow limit, the mass, mo-mentum and energy equation can be derived through a Chapman-Enskopexpansion(Palmer and Rector (2000); Chatterjee and Chakraborty (2006);Shi and Guo (2009)). The viscosity, ν, and the thermal diffusivity, k, aregiven by:

ν = c2s ∆t (τf − 0.5); k = c2s ∆t (τh − 0.5). (24)

where τf and τh are the dimensionless relaxation times for the velocity andtemperature field respectively Equations (17) and (18) are solved in a two-step procedure - collision and advection:

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Collision:

f outi (xi, t) = f in

i (x, t) +∆t

τf

(

f eqi (x, t)− f in

i (x, t))

+ Fi (25)

houti (xi, t) = hin

i (x, t) +∆t

τh

(

heqi (x, t)− hin

i (x, t))

+ Φi (26)

Advection:f ini (x+ ei, t+∆t) = f out

i (x, t) (27)

hini (x+ ei, t+∆t) = hout

i (x, t) (28)

where f outi and f in

i denote the outgoing (i. e. after collision) and incoming (i.e. before collision) distribution functions, respectively. At equilibrium, theenergy current is proportional to the mass current.

3.2. Free Boundary Treatment

The description of the liquid-gas interface is very similar to that of vol-ume of fluid methods. An additional variable, the volume fraction of fluid ǫ,defined as the portion of the area of the cell filled with fluid, is assigned toeach interface cell. The representation of liquid-gas interfaces is depicted infigure 5. Gas cells are separated from liquid cells by a layer of interface cells.These interface cells form a completely closed boundary in the sense thatno distribution function is directly advected from fluid to gas cells and viceversa. This is a crucial point to assure mass conservation since mass comingfrom the liquid or mass transferred to the liquid always passes through theinterface cells where the total mass is balanced. Hence, global conservationlaws are fulfilled if mass and momentum conservation is ensured for interfacecells. The used cell types and their state variables and possible state trans-formations are listed in Table 1. For details see Korner et al. (2005).

Fluid cells and interface cells solidify when the temperature falls below acertain threshold value which corresponds to a certain solid fraction value.For our examples we use a solid fraction value of 55% as threshold value.Solidification is realized by putting the velocity of the cell back to zero. Asolid cell in the neighbourhood of a liquid cell is treated as a wall cell, i. e.the distributions functions fi arriving in a solid cell from a liquid cell underlythe bounce back condition.

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Figure 5: Representation of a free liquid-gas interface by interface cells. The real interface(dashed line) is captured by assigning the interface cells their liquid fraction.

cell type fi/hi fluid fraction ǫ gas pressure pG change of statefluid F • - - → I, → Sgas G - - • → Iinterface I • • • → G , → F, → Ssolid S • • - → I , → Fwall W • - - -

Table 1: Cell types: State variables and possible state transformations.

The effect of the surface tension is treated as a local modification of thegas pressure pG acting at the interface, i. e. the gas pressure is replace by

pG − κ · σ, (29)

where κ and σ denote the mean curvature and the surface energy, respectively.Equation (29) is valid for the 2D situation. In 3D κ has to be replaced by2 κ.

Wetting is also included in the numerical model. The basic idea of ourapproach is to bring this force into the numerical model as a kind of addi-tional capillary force. Details of the algorithm and its validation are givenin reference (Attar and Korner (2009)). The wetting angle between fluidand solid powder can be adjusted between 0 and π. It is also possible todefine the wetting angle between fluid and re-solidified fluid. In this paper,we assumed complete wetting between fluid and re-solidified fluid.

Interface cells separate gas cells from fluid or solidified cells. After stream-ing, only distribution functions from fluid, solid and interface cells are given.

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Distribution functions arriving from gas cells are not defined and have tobe reconstructed in such a way that the pressure boundary conditions arefulfilled. The detailed reconstruction algorithm is published somewhere else(Attar and Korner (2010)).

4. Results and Discussion

In the following the potential of the LBM to simulate selective beam melt-ing processes is demonstrated by means of some illustrative examples. Thenumerical simulation allows to investigate the influence of material and pro-cessing parameters which can not easily be experimentally realized. Thus,two classes of numerical investigations are presented. The first class of nu-merical experiments has predictive character since parameters are consideredwhich can not be easily adjusted in the experiment but might have stronginfluence on the result. The second class are simulations where experimentalresults are available.We consider Selective Electron Beam Melting as an example for a selectivebeam melting process. Details of the SEBM are described in reference (Heinlet al. (2007)). The raw material for the process is gas atomized pre-alloyedTiAl6V4 powder in a size range of 45 − 115 µm. In the experiment thebeam is moving relatively to the powder bed to create a single melt line ona pre-sintered powder bed. The electron beam power and beam velocity areadjusted to obtain lines with different widths. Cross sections of the experi-mental lines are considered and compared with the 2D simulation results.

The simulation parameter have to be adjusted to the physical system(table 2). For numerical simulation the material parameters have to be ex-pressed in dimensionless form (LBM parameters). The dimensionless num-bers for beam based additive manufacturing process, as described in Elsenet al. (2008), are considered to set the simulation parameters. The physicalsystem is first converted into dimensionless numbers and simulation param-eters are calculated with the help of these dimensionless numbers.

For all simulations length scale, mass scale, temperature scale and timescale are considered as ∆x = 5.0 ×10−6m, ∆m = 5.0 ×10−13kg, ∆T = 6.5×10−4K and ∆t = 2.2 ×10−7s respectively. The material parameters followby multiplying the dimensionless quantities with the relevant scales such asρ∗ = ρ∆x3

∆m, ν∗ = ν ∆t

∆x2 , σ∗ = σ∆t2

∆m, g∗ = g∆t2

∆x, k∗ = k ∆t

∆x2 and etc. Thedimensionless quantities are marked with ∗.

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iptTable 2: Physical properties of TiAl6V4 in experiments (Rai et al. (2009); Iida and Guthrie

(1988)) and simulations

Physical Properties Experiment LBMDensity of liquid 4000 kg/m3 1.0Viscosity 0.005 Pas 0.011Surface tension 1.65 N/m 0.15Gravity 9.81 m/s2 10−7

Thermal diffusivity of solid 7.83×10−6 m2/s 0.068Thermal diffusivity of liquid 9.93×10−6 m2/s 0.087Solidus temperature 1878 K 1.22Liquidus temperature 1928 K 1.25Preheat temperature 1023 K 0.664Latent heat 0.37×106 J/kg 0.3

4.1. Wetting Conditions

The impact of the wetting conditions is only numerically investigatedsince the wetting behaviour can not be easily influenced in the experiment.A system with 250×200 cells (1.25×1.0 mm) is considered where the beam(diameter: 72 cells: 350 µm ) is kept motionless relative to the powder bed.The beam energy is considered to be 2.5/∆t and the duration of the beamirradiation is 1440 time steps which is equivalent to 0.830 J/mm. Figure 6shows that the wetting condition has an enormous influence on the formationof the melt pool.

If the liquid metal does not wet the powder, pronounced balling takesplace. Balling is experimentally observed when the molten metal powderlayer does not wet the underlying substrate due to a contamination layer ofoxide being present on the surface of the powder and the melt (Das (2003)).Balling is one of the problems obstructing selective beam melting processes.It is important to notice that the wetting conditions do not only have influ-ence on the shape of the melt pool but also on the thermal conditions andtherewith on the size of the melt pool since wetting leads to a larger heat tothe surrounding solid powder.

In the following, it is assumed that liquid metal wets the powder andthe solidified liquid (i. e. θ0 = 30°) since our experimental SEBM approachis under high vacuum conditions. We will also see that balling and the

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100 cells =0.5 mm

Figure 6: Influence of different wetting conditions on the appearance of the melt pool fora static beam. Left: wetting condition: θ0 = 10°. Right : non-wetting condition: θ0 =160°.

appearance of the melt pool are not only a phenomenon governed by wettingbut also influenced by the relative density, randomness of the powder bedand the size of the melt pool relative to the powder particle diameter.

4.2. Energy Input

In order to investigate the influence of the extension of the pool theenergy input is increased while all other parameters are kept constant. Todefine the beam power and irradiation time in simulation, the dimensionlesspower (P ∗ = P

vρLD2 ) is considered where P is the beam power, v is the scan

speed, L is the latent heat and D is the beam diameter (Elsen et al. (2008)).Figure 7 shows the experimental and numerical alteration of the geometrywith increasing energy input.

As long as the total melt pool size is not much larger than the meanpowder diameter, the melt tries to assume a rather round shape in orderto minimize the surface energy. The geometry changes for larger melt poolssince the relative contribution of capillary, wetting and gravity forces changes.We observe an analogous effect for the numerical simulation, figure 7 (right).

4.3. Relative Powder Density

The influence of the powder packing density on the resulting melt poolgeometry is studied in figure 8. By increasing the particle packing densityfrom 39% to 74% the melt pool geometry gets more and more defined andapproaches a half circle. For low relative powder densities, the resultingmelt pool geometry is strongly dependent on the local powder configuration.This results in geometries which are dominated by an interplay between

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ìm500

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b)

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ìm500100 cells =

Figure 7: Size of the melt pool. Experiment: Moving beam, cross section, line energy(a) 0.355 J/mm and (b) 0.830 J/mm. Simulation: Static beam, 250× 200 cells (1.25×1.0mm), beam diameter: 72 cells (350 µm), Beam Power (a)2.13/∆t (b) 2.5/∆t and duration:(a)1440 (b) 2880 time steps

local wetting and capillary forces and which are far away from the half circlegeometry. Thus, in order to get a well-defined melt pool geometry highpowder densities are favorable.

4.4. Stochastic Powder Layer

In the following, the influence of the random character of the powderlayer on the final shape of the re-solidified melt pool is investigated. Fig-ure 9 (top) shows different cross sections of a molten line in TiAl6V4. Thegeometric appearance of the re-solidified melt pools is strongly dependenton the local powder density and arrangement, wetting of the melt with thepowder particles, etc. The surface curvature may be concave or convex andthe orientation of the surface may extremely deviate from the normal direc-tion. Altogether, the resulting melt pool geometry is not well-defined. Theidentical effect is observed for the numerical results, figure 9 (bottom). Inall simulations the packing density is approximately the experimental one ofabout 55%. The findings of the numerical simulation are completely consis-

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51 % 55%

74%63% 67%

39 %

100 cells =0.5 mm

Figure 8: Melt pool geometry after re-solidification for different particle packing densities.By increasing the particle packing density from 39% to 74% the melt pool shape gets moreand more well-defined. (250×200 cells (1.25×1.0mm), beam diameter: 72 cells( 350 µm),beam irradiation time: 3000 time steps, Beam power: 3.0/∆t (energy: 1.04 J/mm))

tent with the experimental observation. The stochastic nature of the powderlayer is thus a major cause for the chaotic melt pool geometry.

4.5. Track Formation and Process Map

In order to investigate the influence of the processing parameters, themorphology of molten lines in powder beds is analyzed by producing 2D lineswith different beam powers and scan speeds. The alteration of the moltenand re-solidified line with increasing line energy (decreasing scan speed atconstant beam power) is depicted in figure 10. For a too low line energy,the powder doesn’t or only slightly gets molten. Increasing the line energyleads at first to the formation of small droplets which grow with increasingenergy. Eventually, the droplets stick together and form a closed molten andre-solidified surface.

Figure 11 shows the consolidated line at constant line energy but fordifferent velocities and powers. The main characteristics of the lines, thedisintegration into droplets, seems not to be dependent on the processingparameters if the line energy is constant. This is also true for different powderbeds. Nevertheless, the resulting structure is unique for each parameter

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Figure 9: Stochastic effect of randomly packed powder layers on the melt pool geometryfor identical packing densities of 55 %. Top: Experimental result for the titanium alloyTiAl6V4 (beam mean diameter: 350 µm, line energy: 0.83 J/mm. Bottom: Numericalresults (250 × 200 cells (1.25×1.0 mm), beam diameter: 72 cell (350 µm), Beam power:2.5/∆t, duration: 2880 time steps)

combination since the dynamics of the melt pool driven by the capillaryforces plays a significant role during solidification (figure 11).

The dynamic behaviour of the melt can be observed during the formationof the track in Figure 12. It is important to notice that the formation of thedroplets (balling) doesn’t seem to be the result of a breakup of a long meltpool (Rayleigh instability). The droplets develop already during meltingunder the strong influence of the local powder arrangement, wetting, gravity,and capillary forces.

Figure 13 shows the resulting structures in a processing map. The processmaps was created to show the combinations of electron beam power and scanspeed at which the different track types formed (Childs et al. (2005)). Thestraight lines represent lines of constant energy. Obviously, the morphologyalong lines of constant energy is invariable. A classification in four differentareas is evident; A: not molten, B: small semicircle droplets, C: large semicir-cle droplets, D: closed line. The transition between the different areas is notsharp but continuous and it is governed by the line energy. Putting all to-gether, the line energy seems to dominate the resulting morphology althoughthe melt pool dynamics is rather vigorous.

Although the 2D numerical results and the 3D experimental results arenot comparable, the experimental results with equivalent line energy are also

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Increasingline energy

v

1mm

0.02 Äx/ Ät

0.03 Äx/ Ät

0.05 Äx/ Ät

0.07 Äx/ Ät

0.10 Äx/ Ät

0.15 Äx/ Ät

0.20 Äx/ Ät

Scan speed

Figure 10: Line morphology as a function of the line energy. (1000× 150 cells (5.0×0.75mm), moving beam, beam diameter: 72 cells (350 µm), Beam power: 3.0/∆t (600 J/sec),different scan speeds.

depicted in figure 13. The equivalent beam power is calculated by using thedimensionless power (P ∗ = P

vρLD2 ) (Elsen et al. (2008)). In order to have theequivalent beam power in 2D, the area under radiation is considered. As anexample, for the experiment with line energy 0.415 J/mm (beam current 8.30A and scan speed 1200 mm/s) the equivalent beam power and scan speedin simulation are 2.5/∆t and 0.05 ∆x/∆t respectively. The dashed linesare showing the equivalent line energy for the experiments. The simulationresults, although only 2D, show very good agreement with the experiments.

5. Summary

This paper provides a numerical approach to simulate beam and pow-der bed layered manufacturing processes at a mesoscopic scale. A numericaltool on the basis of a lattice Boltzmann model is developed which allows topredict local powder melting and re-solidification processes on the powderlevel in 2D. Physical mechanisms like wetting or capillary forces as well as

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v

1mm

0.1 Äx/ Ät

0.083 Äx/ Ät

0.066 Äx/ Ät

0.05 Äx/ Ät

0.033 Äx/ Ät

0.016 Äx/ Ät

6/ Ät

5/ Ät

4/ Ät

3/ Ät

2/ Ät

1/ Ät

Figure 11: Line morphology at constant line energy for different scan speeds and beampowers. (1000 × 150 cells (5.0×0.75 mm), moving beam, beam diameter: 72 cells (350µm) )

materials parameters like the influence of the stochastic powder bed get acces-sible within the framework of the numerical model. Numerical experimentsdemonstrate that the packing density of the powder bed has the most signif-icant effect on the melt pool characteristics. A Processing map shows thatthe line energy dominates the processing results at constant powder packingdensity. Comparison with experimental results from selective electron beammelting demonstrates good agreement. Thus, the powder-level simulationhas been revealed as an key to understand the mechanisms and parametersinfluencing consolidation and will be further developed in the future.

Acknowledgement 1. The authors gratefully acknowledge partial fundingof the German Research Council (DFG), which, within the framework ofits ’Excellence Initiative’ supports the Cluster of Excellence ’Engineering ofAdvanced Materials’ at the University of Erlangen-Nuremberg.

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Figure 13: Processing map for simulation results (1000×150 cells (5.0×0.75 mm), movingbeam, beam diameter: 72 cells (350 µm)) and the equivalent experiment results.

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