Simulation of Electron Beam Melting With the Lattice Boltzmann Method
description
Transcript of 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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 2 / 32
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]
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 3 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 4 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 5 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 6 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 7 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 8 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 9 / 32
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]
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 10 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 11 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 12 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 13 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 14 / 32
AlgorithmStreaming Step
In the streaming step, particles are simply shifted in the direction ofmotion to the adjacent nodes
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 15 / 32
AlgorithmCollission Step
The collision step models the interactions between particles. The factor τcontrols the tendency of the system to return to equilibrium
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 16 / 32
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:
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 17 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 18 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 19 / 32
AlgorithmSummary
Flowchart of the most fundamental parts in an implementation of theLBM
The convergence is usually tested for a macroscopic variable
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 20 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 21 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 22 / 32
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]
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 23 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 24 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 25 / 32
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)
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 26 / 32
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)
))
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 27 / 32
Applications
Applications of EMB
Medical implants like hip joints or artificial spinal discsComponents for aerospace or automotive industry
Applications of LBM
Fluid FlowHeat Transfer
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 28 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 29 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 30 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 31 / 32
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
Rishi Dua (IIT Delhi) Simulation of EMB With LBM 14 December 2013 32 / 32