modeling of thermodynamic phenomena with lattice boltzmann ... · modeling of thermodynamic...

Modeling of ThermodynamicPhenomena with Lattice BoltzmannMethod for Additive ManufacturingProcessesRegina Ammer†,Matthias Markl∗, Carolin Körner∗, Ulrich Rüde†July 31th, 2014†Chair for System Simulation (LSS),∗Chair for Metal Science and Technology (WTM)

Outline

1 Additive Manufacturing

2 Mathematical and Numerical Models

3 Validation Experiments

4 Improvements for EBM Process

5 Evaporation - Condensation Problem

July 31th, 2014 | Regina Ammer et al. | LSS | Simulation of Thermodynamic Phenomena by LBM 2

Additive Manufacturing

AdditiveManufacturing

Methods

ElectronBeam

Melting

SelectiveLaser

Melting

SelectiveLaser

SinteringEB free

formfabrication

StereoLithog-raphy

DirectMetal

DepositionFusedLayer

Modeling

LayerLaminate

Manu-facturing


Our target application: Electron Beam Melting

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

ele

ctr

on

beam

gu

n

powderhopper

rake

buildingtank

processplatform

a) b)


From Mathematical Model to Numerical Discretization

Incompressible Navier-Stokes Equations

∇ · u = 0∂u∂t

+ (u · ∇) u = −∇p + ν∆u + f

∂E∂t

+∇ · (uE) = ∇ · (k∇E) + φ

Numercial Discretization Methods:% Finite Volume Methods% Finite Element Methods" Lattice Boltzmann Method


Thermal 3D LBM

Multi-distribution approach for thermal LBM

fi (x + ei, t + ∆t) = fi(x, t) +∆tτf

(f eqi (x, t)− fi(x, t)

)+ Fi(x, t)

hi (x + ei, t + ∆t) = hi(x, t) +∆tτh

(heq

i (x, t)− hi(x, t))

+ Φi(x, t)

0 1

2

3

4

5

6

78

9 10

11

12

13

14

1516

17 18

f eqi (ρ,u) = ωiρ

1 +ei · uc2

s+

(ei · u)2

2c2s− u2

2c4s

heqi (E,u) = ωiE

1 +ei · uc2

s

• Macroscopic quantities: ρ = ∑ifi ρu = ∑

ieifi E = ∑

ihi


Free Surface Treatment1

Liquid Liquid Interface

Solid Solid Interface

Gas – Free Surface

Wall ·· Phase Transition

Volume of Fluid Approach• Fill level for interface cells is defined byϕ, 0 ≤ ϕ ≤ 1

• Simulate only liquid phase andneglect the gas phase

→ Reconstruct unknown fi,hi values fromthe gas phase in the interface layer

→ Convert interface cells due to thedynamic melt pool surface

1Körner et al., 2005


Implemenation

collision detectioncollision response

update of fluid nodescalculation of hydrodynamic forcescalculation of free surface

rigid bodies act as obstacles

create new powder layer

after solidification process


Summary of Numerical Model

" Parallelized and optimized 3D model"Wetting effects" Free surface treatment" Different absorption types" Realistic metal powder distribution

% Evaporation model /% Temperature dependent surface tension


Validation – Experimental Setting

• Line energy

EL =uBIbeam

vscan=

Pbeam

vscan

kJm

with• acceleration voltage uB in V• beam current Ibeam in A• scan velocity vscan is scan

velocity in ms

• Examination of a sampleregarding• porosity• swelling

• Hatching of a cuboidconsidering

15mm 15mm

10m

m

simulated powder bedhatching lines

beam offset

• Simulation domain:(1.44x0.64x0.24)·10−3 m3

• One powder layer with0.05 mm thickness

• Define porosity/swellingnumerically!


Categorization of Test Settings

Porosity Good Surface Swelling

Line Energy


Hatching one Layer (6.4ms , 200 kJ

m)


Comparison of experimental and numerical processwindow2

0 1 2 3 4 5 6 7

Scan velocity [m/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Line

ener

gy[k

J/m

]

porousgoodswelling

Figure: Experimental process window.

0 1 2 3 4 5 6 7

Scan velocity [m/s]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Line

ener

gy[k

J/m

]

swellingporousgood

Figure: Numerical process window.2Ammer et al. 2014


Comparison of experimental and numerical processwindow

F Experimental and simulation values are highly concordant!(especially for mid-scan-velocity and all porous values areequal)

F Small differences for low and high scan velocities→ numerical EB focus constant, experimentallythe focus spreads out

F $$$ Time is money - can it be "faster"???


Advanced Hatching Strategies - achieved bynumerical simulations3

F Numerical extension of the process window up to 30 ms

I Numerical simulations show a decrease of "window" height up to aclosing at 30 m

sI Sharp temperature/evaporation border→ Small statistical variance

for the maximum temperature on the melt pool surfaceI Rough porosity border→ High statistical variance for the powder

distribution in one layerF Decrease of line offset (100µm→ 50µm)

I Increase of beam power and scan velocity→ faster production rate!

I Lower maximum temperature for the same beam power→ less evaporation rate!

3Markl et al. 2014


Evaporation–Condensation Problem4

dense phase

fe(ρe, Ts)fifr

vapour phase

O(λ)

beam energyI Knudsen layer = boundary layer with

a thickness of a few molecular meanfree path

I Classical Hertz-Knudsen formula fornet mass flux

mHK = me − mi = ρe

√√√√√√RTs

2π− ρv

√√√√√√RTv

2πI / lack of nonlinear convective

effects, limited in the range ofevaporation, not including the backpressure problem...

4Hertz 1882 and Knudsen 1915


Flow Structure of the Vapour Plume

condensedmatter

Knudsen layer

vapour

compressedambient gas

ambient gas

contactdiscontinuity

shock wave

subsonic flow supersonic flow

rarefaction fan


Evaporation and condensation fluxes

I Net mass transport from evaporating surface

jnet = j+ − j− =

j+ − j−

j+

· j+ = φ · j+

with evaporation coefficient φI Evaporation flux

j+ = ps

√√√√√√ mA

2πkBTs

where mA is atomic mass and kB Boltzmann’s constant


Evaporation and condensation fluxes cont’d

I Condensation flux

j− = pKn

√√√√√√ mA

2πkBTKn· β · F−

where β and F− account for collisional effects in downstreamflow and require jump conditions across the Knudsen layer

I evaporation coefficient φ

φ =√2πγν ·MaKn(Ts) ·

ρKn

ρs

√√√√√√√TKn

Ts

where γν is ratio of specific heats and MaKn(Ts) is flow Machnumber of outer Knudsen layer


Back pressure

I Conservation of momentum→ expanding vapour generates aback pressure pback onto evaporating surface

I For φ = 0→ state of thermodynamic equilibrium betweenvapour and condensed phase, flux of evaporating particlesmatches these of condensing ones:

pback = 0.5ps + 0.5pKn = ps

I For higher evaporation fluxes, i.e., φ > 0:

pback =12

ps +12

(1− φ)

12

pKn +12

ps

=12

ps ·1 +

12· (1− φ) ·

1 +ρKn

ρs

TKn

Ts


Numerical transfer

F Vapour layer is neglected→ remove evaproated mass andenergy from the free surface cells!

F Update the state variables by:∆pback(xs, t) = pback(xs, t)− pa

∆mvap(xs, t) = jnet(xs, t)∆t(∆x)2

∆Evap(xs, t) = ∆mvap(xs, t) · [Lvap(Ts(xs, t)) + Lmelt+

cp,sTliquidus + cp,l(Ts(xs, t)− Tliquidus)]

F Post-evaporation quantities are:pG,post(xs, t) = pG,pre + ∆pback

mpost(xs, t) = mpre(xs, t)−∆mvap(xs, t)

hE,post(xs, t) =hE,pre(xs, t)mpre(xs, t)−∆Evap(xs, t)

mpost(xs, t)


Conclusion & Outlook

" 3D model for simulating EBM processes ," Validation experiments show highly accordance with

experimental results ," Improvement of hatching strategies by decreased line offset→ find "fastest" parameter set (EL, vscan) ,

% Including evaporation and condensation problem in theWALBERLA-framework!

% Simulate more powder layers to achieve information aboutbeam-powder-bed-interaction!

% Use static grid refinement for the melt pool!% . . .


More Powder Layers!


References & Acknowledgments

EU Grand Agreement Number 28 66 95 – FastEBM

High Productivity Electron Beam Melting Additive Manufacturing Developmentfor the Part Production Systems Market

l M. Markl, R. Ammer, U. Rüde, C. Körner:Improving Hatching Strategies for Powder Bed Based AddivitiveManufacturing with an Electron Beam by 3D Simulationssubmitted (2014)

l R. Ammer, M. Markl, V. Jüchter, C. Körner, U. Rüde:Validation experiments for LBM simulations of electron beam meltingInt. J. Mod. Phys. C 25, 1441009 (2014)

l R. Ammer, M. Markl, U. Ljungblad, C. Körner, U. Rüde:Simulating Fast Electron Beam Melting with a Parallel Thermal Free SurfaceLattice Boltzmann MethodComput. Math. Appl. 67, 318 (2014)

l M. Markl, R. Ammer, U. Ljungblad, U. Rüde, C. Körner:Electron Beam Absorption Algorithms for Electron Beam Melting ProcessesSimulated by a 3D Thermal Free Surface LBM in a Distributed and ParallelEnvironmentProcedia Comput. Sci. 18 2127 (2013)


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