Simulation of Electron Beam Melting With the Lattice Boltzmann Method

Simulation of Electron Beam MeltingWith the Lattice Boltzmann Method

Rishi DuaIndian Institute of Technology Delhi

Tutor: R. AmmerSupervisor: Prof. U. Rude

Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 1 / 32

Simulation of EBM using LBM

Outline

Introduction

Derivation of the LBM

Algorithm

EBM using Lattice Boltzmann Method

High Performance Computing for EBM

Applications

Advantages of EBM

Conclusions


IntroductionElectron beam melting (EBM)

Additive manufacturing method

Melt metal powder layer by layer with electron beam in high vacuum

Parts are fully dense, void-free, and extremely strong

Figure: Electron beam melting [1]


IntroductionLattice Boltzmann Method [2]

Need for simulation of EBM to improve manufacturing process

LBM: models hydrodynamics + heat transfer

Originated from the lattice gas automata (LGA), a simplifiedmolecular dynamics model in which space, time, and particlevelocities are all discrete

Based on the discrete Boltzmann equationMeso-scale approach

Bridges the gap between micro-scale and macro-scale approachesBehavior of a collection of particles considered as a unitDistribution function: Representative for the collection of particles


Derivation of the LBMBoltzmann equation

Describes dynamics in terms of probability functions f in phase space:

∂f

∂t+ ξ · ∇f + F · ∂f

∂ξ= S

where:

ξ: microscopic velocities

f : number of particles/molecules at the time t positioned between xand x + dx with velocities between ξ and ξ + dξ

F : force field per unit mass acting on the particle

S : Collision operator (sum of all intermolecular interactions)

Solving the equation can be difficult due to

high dimensions of the distribution

complexity in the collision operator


Derivation of the LBMBhatnagar, Gross, Krook equation

Linear approximation for collision term [3]

External forces ignored

∂f

∂t+ ξ · ∇f + F · ∂f

∂ξ= −1

τ(f − f eq)

where

f eq: equilibrium distribution function

τ : relaxation time


Derivation of the LBMDiscretization

The BGK equation is continuous in the phase variables x and ξ

Momentum space discretization using finite set of velocities [4] [5]

{ξi |i = 1, ..., b}

∂fi∂t

+ ξi · ∇fi = −1

τ(fi − fi

eq)

where

fi ≡ fi (x , ξi , t), fieq ≡ fi

eq(x , ξi , t): Distribution function andequilibrium distribution function of i-th discrete velocity ξi


Derivation of the LBMDiscretization

DmQn lattice notation [6]

m: dimensionn: number of velocity directions

Real quantities as space and time converted to lattice units prior tosimulation

Nondimensional quantities as the Reynolds number remain the same


Derivation of the LBMThermal Lattice Boltzmann

Multi-speed models:

Additional discrete and higher order velocity termsDisadvantage: Numerical Instability

Multi-distribution models [7]:

Temperature treated as a passive diffusing scalarTwo sets of distribution functions:

fi models mass and momentum transporthi represents the movement of the internal energy

Macroscopic quantities given by

ρ =N−1∑i=0

fi ρu =N−1∑i=0

ei fi E =N−1∑i=0

hi

where ρ: density u: macroscopic velocity E : energy density


Derivation of the LBMProblem formulation

fi (x + ei , t + ∆t)− fi (x , t) =∆t

τf(fi

eq(x , t)− fi (x , t)) + Fi

hi (x + ei , t + ∆t)− hi (x , t) =∆t

τh(hi

eq(x , t)− hi (x , t)) + φi

where

fi (x , t) and hi (x , t): density and energy distribution functions ini-direction

φi : Energy deposited in each cell under beam radiation

Fi = ωiρ[ ei−uc2s

+ ei ·ueic4s

] · g [8]


Derivation of the LBMEquilibrium solution [7][9][10]

For local equilibrium, S = 0

In three dimensions, in continuous phase space, the distributions are:

fieq (x , t) = ωiρ[1 +

ei · uc2s

+(ei · u)2

2c4s

− u2

2c2s

]

hieq (x , t) = ωiE [1 +

ei · uc2s

]

neglecting higher order terms


Derivation of the LBMEquilibrium Solution D3Q19 model

Velocity vectors ei

(0,0,0) i=0

(±1,0, 0) i=1, 2

(0, ±1,0) i=3, 4

(0, 0, ±1) i=5, 6

(±1,±1, 0) i=7, ,10

(0,±1,±1) i=11, ,14

(±1,0, ±1) i=15, ,18

Weights ωi

2/36 i = 1, ..., 6

1/36 i = 7, ..., 18

12/36 i = 19


Derivation of the LBMEquilibrium Solution

Under low speedv = cs

2∆t(τf − 0.5)k = cs

2∆t(τh − 0.5)

v is kinematic viscosity k is thermal diffusivity

Collision

fiout (xi , t) = fi

in (x , t) + ∆tτf

(fieq(x , t)− fi (x , t)) + Fi

hiout (xi , t) = hi

in (x , t) + ∆tτh

(hieq(x , t)− hi (x , t)) + φi

Streamingfiin(x + ei , t + ∆t) = fi

out (x , t)

hiin (x + ei , t + ∆t) = hi

out (x , t)

where

fiout : outgoing (i.e. after collision) distribution function

fiin: incoming (i.e. before collision) distribution function


AlgorithmOutline

The numerical methods of solution of the system of partial differentialequations then gives rise to a discrete map

The map can be interpreted as

Streaming Step: particles jump from one lattice to next according totheir velocityCollision Step: particles collide, exchange energy and get a new velocity


AlgorithmStreaming Step

In the streaming step, particles are simply shifted in the direction ofmotion to the adjacent nodes


AlgorithmCollission Step

The collision step models the interactions between particles. The factor τcontrols the tendency of the system to return to equilibrium


AlgorithmBoundary Conditions

LBM allows intuitive and clear specification of Boundary Conditions

A flag array can be used to distinguish bulk and boundary cells

Two common Boundary Conditions are:


AlgorithmBoundary Conditions

Moving walls cause change of momentum due to friction. Change inmomentum is modeled by a term in the no-slip condition:

fα (x , t) = fα (x , t)− 2tiρ(3/c2)cαuw

α: direction towards wall, α : direction from wall, uw : wall velocity

Treating boundaries that are inclined to the direction of velocities: acurved boundary may be approximated by step-wise segments


AlgorithmOther Conditions

When do not want to observe the effect of free surfaces, PeriodicBoundary Conditions used

Wrap-around condition. Particles that exit one wall re-enter from theopposite wallMass and momentum are conserved

Body forces can be included. A constant acceleration can be modeledby a statement like ux = −a. The particle distribution will be seen torespond to the force


AlgorithmSummary

Flowchart of the most fundamental parts in an implementation of theLBM

The convergence is usually tested for a macroscopic variable


EBM using Lattice Boltzmann MethodElectron Beam Model

The electron beam energy Eb on the surface area of one lattice cell ismodeled by a two-dimensional Gaussian distribution:

Eb (x) = γUIc2∆t

2πσ2exp(− 1

2σ2< x − xb, x − xb >)

x : the lattice cell center position in the xyplane

xb: current beam center position

σ: the standard deviation

U: acceleration voltage

I : beam current

γ ∈ [0; 1]: the remaining fraction of the electron beam energypenetrating the material due to electrons not lost due to reflection atthe surface


EBM using Lattice Boltzmann MethodElectron Beam Absorption Model

The inclusion of beam energy has to be modelled by absorption,because absorption length of the electron beam is magnitudes higherthan the thermal length

The electron beam penetrates through the material nearlyinstantaneous. Therefore, we model the energy source as a volumetricforce of the first cells:

φi (x , t) = ωiφi (x , t)Eα(x , t)

where

x : lattice cell center

Eα: corresponding amount of absorbed energy

φi : source term


EBM using Lattice Boltzmann MethodElectron Beam Absorption Model

Exponential Electron Beam Absorption vs. Constant Electron BeamAbsorption

Figure: Relation between penetration depth and absorption coefficient for 60 kVand 120 kV and suitable approximations [11]


EBM using Lattice Boltzmann MethodLiquid-Void interface

Figure: Different cell types assumed in simulation

Fluid cellCompletely filled with liuid, no gas cell as a direct neighbor

Gas cellCompletely filled with gas, no fluid cell as a direct neighborNot considered in the fluid simulation

Interface cellBoundary cell

Wall/Solid cellNo slip boundary condition, i.e. fi bounced back


High Performance Computing for EBMParallelization of LBM

Approach: the global domain is split in several blocks and those aredistributed to different CPUs

Motivation:The Lattice Boltzmann Method is resource intensive

lack of memory resourceslong computation time

LBM generally needs only nearest neighbor information

In a single streaming step of the LBM the pdfs to the direct neighborcells have to be communicated (red)But for the absorption of the electron beam the iteration over thewhole domain in z direction in one time step is necessaryThus the computation of the absorption is completely sequential fromtop to bottomThis behavior will cause waiting times for other CPUs, so theseequations are reformulated for parallel computation


High Performance Computing for EBMConcept

Split the computation in a preand post compute step and acommunication step

The pre and post compute stepscan be evaluated parallel oneach block

The communication stepexchanges the necessaryinformation between 2 blocks

Communication schemes:

local for pdf streaming (red)top-to-bottom for beamabsorption (green)

Figure: Communication schemes: [11]

local for pdf streaming (red),top-to-bottom for beam absorption(green)


High Performance Computing for EBMSummary

Zk extended to Z(k,m),k: index in the corresponding block m

Split up the auxiliary function χ into χpre and χpostExponential beam absorption:

Ea

(z(k,m)

)= Ebχpost

(z(k,m)

)χpost

(z(k,m)

)= χpre

(z(k,m)

) m−1∏n=0

(1−k−1∑l=0

χpre

(z(l,m)

)ϕ(z(l,m)

)χpre

(z(k,m)

)= (1− e−λc)(1−

k−1∑l=0

χpre

(z(l,m)

)ϕ(z(l,m)

)Constant beam absorption:

Ea

(z(k,m)

)= Ebχpost

(z(k,m)

)χpost

(z(k,m)

)= min(χpre

(z(k,m)

),max(0, 1−

m−1∑n=0

k−1∑l=0

χpre

(z(l,n)

)ϕ(z(l,n)

)−

k−1∑l=0

χpost

(z(l,m)

)ϕ(z(l,m)

)χpre

(z(k,m)

)= min(λc, 1−

k−1∑l=0

χpre

(z(l,m)

)ϕ(z(l,m)

))


Applications

Applications of EMB

Medical implants like hip joints or artificial spinal discsComponents for aerospace or automotive industry

Applications of LBM

Fluid FlowHeat Transfer


Advantages

Accuracy

Possibilities to construct very complex structures which are strong andflexible using EBMBoth LGA and LBM developed as theoritical methods. However LBMis now used commercially and competes with classical fuid dynamicsmethod based on Navier-Stokes equationLBM includes hydrodynamic physical effects, like the flow of the meltpool, capillarity and wetting, as well as thermal effects, like heatconduction and transport, electron beam absorption and solidliquidphase transitions

Speed

Accelerate the build process and the production accuracy as LMB helpsin parallelization of the algorithm


Conclusions

This talk derives the fundamental equations for lattice Boltzmanndiscretization and develops an algorithmic outline for the example ofelectron beam melting

Particular importance was placed on the development of parallelabsorption algorithms to take account of the high computationalcosts of threedimensional simulations

The model for the electron beam consists of a definition of theelectron beam by the acceleration voltage and the current and enablesus to define different movements of it

Two different absorption types, constant and exponential, are derivedand their relation due to penetration depth and dissipated energy isexplained


Outlook for future

Future research topics will be the validation of the melt poolbehaviour, i.e. the comparison of its lifespan and size withexperimental data

At present, high-Mach number flows in aerodynamics are still difficultfor LBM, and a consistent thermo-hydrodynamic scheme is absent

Videos:

EBM Demo

Simulation Demo


References

1. Website Arcam AB - Additive Manufacturing for Implants and Aerospace, EBM. url: http://www.arcam.com/

2. S. Chen and G.D. Doolen. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30: 329-364,1998

3. P. Bhatnagar, E. Gross, and M. Krook. A model for collision process in gases I: small amplitude processes in chargedand neutral one-component systems. Physical Review, 50: 511-525, 1954

4. X. He and L.-S. Luo. A priori derivation of the lattice Boltzmann equation. Physical Review E, 55: R6333-6336, 1997

5. X. He and L.-S. Luo. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmannequation. Physical Review E, 56: 6811-6817, 1997

6. C. Korner. Integral Foam Molding of Light Metals, Technology, Foam Physics and Foam Simulation. Springer, 2008

7. D. Chatterjee and S. Chakraborty. A hybrid lattice Boltzmann model for solid/liquid phase transitions in presence offluid flow. Physics Letters A, 351: 359-367,2006

8. L.-S. Luo. Theory of the lattice Boltzmann method: Lattice Boltzmann models for non-ideal gases. Physical Review E,62: 4982-4996, 2000

9. B. J. Palmer and D. R. Rector. Lattice Boltzmann algorithm for simulating thermal flow in compressible fluids. Journalof Computational Physics, 161: 1-20, 2000

10. B. Shi and Z. Guo. Lattice Boltzmann model for nonlinear convection-diffusion equations. Physical Review E, 79:016701, 2009

11. Matthias Markl, Regina Ammer, Ulric Ljungblad, Ulrich Rude, Carolin Korner, Electron Beam Absorption Algorithms forElectron Beam Melting Processes Simulated by a Three-Dimensional Thermal Free Surface Lattice Boltzmann Methodin a Distributed and Parallel Environment, Procedia Computer Science, Volume 18, 2013, Pages 2127-2136, ISSN1877-0509


Simulation of Electron Beam Melting With the Lattice Boltzmann Method

Education

simulation of ebm

introductionelectron

time t

collision term

macroscale approachesbehavior

collision operator sum

manufacturing processlbm

lattice gas automata