Efficient methods for solving the Boltzmann Introduction equation - PowerPoint PPT Presentation

Efficient methods for solving the Boltzmann Introduction equation for nanoscale transport applications Introduction II Direct simulation Monte Carlo Nicolas G. Hadjiconstantinou Variance reduction: killing two birds with one Massachusetts Institute of Technology stone Department of Mechanical Engineering LVDSMC BGK model Multiscale 8 November 2011 Implications VRDSMC Application: Phonon Acknowledgements: L. Baker, T. Homolle, H. Al-Mohssen Transport G. Radtke, C. Landon, J-P. Peraud Conclusions Financial support: Singapore-MIT Alliance NSF/Sandia National Laboratories, MITEI

Breakdown of Navier-Stokes description (gases) Interest lies in scientific and practical challenges associated with breakdown of Navier-Stokes description at small scales Introduction Breakdown of Navier-Stokes � = breakdown of continuum Introduction II assumption. Conservation laws, e.g. Direct simulation ρD u Dt = − ∂P ∂ x + ∂τ Monte Carlo ∂ x + ρ f Variance reduction: can always be written killing two birds with one Navier-Stokes description fails because collision-dominated stone LVDSMC transport models, i.e. constitutive relations such as BGK model τ ij = µ ( ∂u i /∂x j + ∂u j /∂x i ) , i � = j fail Multiscale Implications VRDSMC This failure occurs when the characteristic flow lengthscale Application: approaches the fluid “internal scale” λ Phonon Transport In a gas λ is typically identified with the molecular mean free Conclusions path. λ air ≈ 0 . 05 µ m (atmospheric pressure) ⇒ Kinetic phenomena appear in air at micrometer scale.

Motivation Small scale devices (sensors/actuators [Karabacak, 2007], pumps with no moving parts using thermal transpiration [Muntz Introduction et al., 1997-2009; Sone et al., 2002],...) Introduction II Direct Processes involving nanoscale transport (Chemical vapor simulation deposition [e.g. Cale, 1991-2004], flight characteristics of Monte Carlo hard-drive read/write head [Alexander et al., 1994], Variance reduction: damping/thin films [Park et al., 2004; Breuer, 1999],...) killing two birds with one stone Vacuum science/technology: Small-scale fabrication LVDSMC (removal/control of particle contaminants [Gallis et al., BGK model Multiscale 2001&2002],...) Implications VRDSMC Application: Similar challenges associated with nanoscale heat transfer in the Phonon Transport solid state (in silicon at T = 300 K , λ phonon ≈ 0 . 1 µ m) Conclusions [Majumdar (1993), Chen “Nanoscale Energy Transport and Conversion” (2005)]

Outline 1 Introduction Introduction 2 Introduction II Introduction II Direct 3 Direct simulation Monte Carlo simulation Monte Carlo Variance 4 Variance reduction: killing two birds with one stone reduction: killing two LVDSMC birds with one BGK model stone LVDSMC Multiscale Implications BGK model Multiscale Implications VRDSMC VRDSMC Application: 5 Application: Phonon Transport Phonon Transport Conclusions 6 Conclusions

Introduction II: Knudsen regimes Deviation from Navier-Stokes is quantified by Kn = λ/H Introduction H is flow characteristic lengthscale Introduction II Direct simulation Monte Carlo Flow regimes (conventional wisdom): Variance reduction: Kn ≪ 0 . 1 , Navier-Stokes (Transport collision dominated) killing two birds with one stone Kn � 0 . 1 , Slip flow (Navier-Stokes valid in body of flow, LVDSMC BGK model slip at the boundaries) Multiscale Implications VRDSMC 0 . 1 � Kn � 10 , Transition regime Application: Phonon Transport Kn � 10 , Free molecular flow (Ballistic motion) Conclusions

Introduction II: Knudsen regimes ∆ T/T 0 , Ma, ✻ etc. high-altitude Introduction hypersonic Introduction II flow 10 Direct Navier Stokes simulation collisionless ✛ ✲ Monte Carlo (slip flow) Variance reduction: 1 killing two birds with one stone LVDSMC BGK model Multiscale MEMS oscillating Knudsen 0 . 1 Implications acceler. microbeam pump VRDSMC Application: Phonon Transport Conclusions Frangi, 2007 Gallis, 2004 Han, 2007 ✲ 0 . 1 1 10 Kn = λ/L

Introduction II: Kinetic description Boltzmann Equation: Evolution equation for f ( x , c , t ) � d � � � Introduction ∂f ∂t + c · ∂f ∂ x + F · ∂f f ( f ∗ f ∗ 1 − f f 1 ) | c r | σ d 2 Ω d 3 c 1 ∂ c = = Introduction II dt coll Direct simulation Monte Carlo f ( x , c , t ) d 3 c d 3 x = number of particles (at time t ) in Variance reduction: phase-space volume element d 3 c d 3 x located at ( x , c ) killing two birds with one stone F = external force per unit mass LVDSMC BGK model 1 , t ) f ∗ = f ( x , c ∗ , t ) f 1 = f ( x , c 1 , t ) f ∗ 1 = f ( x , c ∗ Multiscale Implications VRDSMC Stars denote post-collision velocities Application: Phonon Transport | c r | = | c − c 1 | Conclusions σ = σ ( | c r | , Ω) = collision cross-section

Introduction II: Kinetic description Connection to hydrodynamics: � m fd 3 c ρ ( x , t ) = mn ( x , t ) = Introduction Introduction II � Direct 1 m c fd 3 c u ( x , t ) = simulation ρ ( x , t ) Monte Carlo Variance � 1 m ( c − u ( x , t )) 2 fd 3 c reduction: T ( x , t ) = killing two 3 k b n ( x , t ) birds with one stone � LVDSMC m ( c i − u i ( x , t ))( c j − u j ( x , t )) fd 3 c τ ij ( x , t ) = BGK model Multiscale Implications VRDSMC (Absolute) Equilibrium ( ∂f ∂t + c · ∂f ∂ x = 0 , [ d f/dt ] coll = 0 ): Application: Phonon Transport � � − c 2 n 0 � f 0 = Conclusions exp , c 0 = 2 k b T 0 /m π 3 / 2 c 3 / 2 c 2 0 0

Introduction II: Kinetic description Local equilibrium ( ∂f ∂t + c · ∂f ∂ x � = 0 , [ d f/dt ] coll = 0 ): Introduction � � − ( c − u loc ( x , t )) 2 Introduction II n loc ( x , t ) f loc = exp c 2 Direct π 3 / 2 c 3 / 2 loc ( x , t ) loc ( x , t ) simulation Monte Carlo � c loc ( x , t ) = 2 k b T loc ( x , t ) /m Variance reduction: killing two birds with one stone The BGK (relaxation-time) approximation: LVDSMC BGK model � � Multiscale Implications ( f ∗ f ∗ 1 − ff 1 ) | v r | σd 2 Ω d 3 v 1 ≈ − ( f − f loc ) /τ VRDSMC Application: Phonon λ Transport √ where τ = 8 kT 0 /πm = “mean time between collisions” Conclusions

Introduction II: A useful identity for numerical method development Introduction Introduction II � � Direct ( f ∗ f ∗ 1 − f f 1 ) | c r | σ d 2 Ω d 3 c 1 = simulation Monte Carlo � � � � 1 Variance � f 1 f 2 | c r | σd 2 Ω d 3 c 1 d 3 c 2 δ ′ 1 + δ ′ 2 − δ 1 − δ 2 reduction: 2 killing two birds with one stone where LVDSMC BGK model Multiscale δ i = δ ( c − c i ) , δ ′ i = δ ( c − c ′ Implications i ) VRDSMC Application: Phonon Transport Conclusions

Direct Simulation Monte Carlo Smart molecular dynamics : no need to numerically integrate essentially straight line trajectories [Bird]. Introduction Introduction II System state defined by { x i , c i } , i = 1 , ...N Direct Solves Boltzmann equation by splitting motion: simulation Monte Carlo Collisionless advection for ∆ t ( x i → x i + c i ∆ t ) Variance reduction: ∂f ∂t + c · ∂f killing two ∂ x = 0 birds with one stone LVDSMC BGK model Perform collisions for the same period of time ∆ t : Multiscale Implications � � � VRDSMC ∂f ∂t = 1 ( δ ′ 1 + δ ′ 2 − δ 1 − δ 2 ) f 1 f 2 | c r | σd 2 Ω d 3 c 1 d 3 c 2 Application: 2 Phonon Transport Collisions performed in cells of linear size ∆ x . Collision Conclusions partners picked randomly within cell

DSMC discussion Introduction Significantly faster than MD (for dilute gases) Introduction II Direct In the limit ∆ t, ∆ x → 0 , N → ∞ , DSMC solves the simulation Monte Carlo Boltzmann equation [Wagner, 1992] Variance Error in transport coefficients ∝ ∆ x 2 in the limit ∆ t → 0 reduction: killing two [Alexander et al,. 1998] birds with one stone Error in transport coefficients ∝ ∆ t 2 in the limit ∆ x → 0 LVDSMC BGK model Multiscale [Hadjiconstantinou, 2000] Implications VRDSMC Application: DSMC (Boltzmann) � = Lattice Boltzmann (solves NS) Phonon Transport Conclusions

DSMC Advantages DSMC has overshadowed numerical discretization approaches for problems of practical interest. Solution by numerical Introduction discretization only advantageous when very high accuracy is Introduction II required for special (low-dimensional, simple) problems e.g. Direct [Sone, Aoki & Ohwada (1989-)] simulation Monte Carlo Variance DSMC Advantages: reduction: killing two SIMPLICITY birds with one stone No need to discretize 6-dimensional phase space LVDSMC BGK model Multiscale Unconditionally stable Implications VRDSMC Importance sampling Application: � � � Phonon ∂t = 1 ∂f ( δ ′ 1 + δ ′ 2 − δ 1 − δ 2 ) f 1 f 2 | c r | σd 2 Ω d 3 c 1 d 3 c 2 Transport 2 Conclusions Natural treatment of discontinuities ∂f ∂t + c · ∂f ∂ x = 0 → ”Move”

DSMC Limitations Introduction Statistical error [Hadjiconstantinou et al., 2003] Introduction II � Direct σ u x 1 1 σ T 1 k B /c V simulation | u x, 0 | = √ N C N ens Ma √ γ , ∆ T = √ N C N ens Monte Carlo ∆ T/T 0 Variance reduction: killing two Resolution of a Ma = 0 . 01 flow to 1% uncertainty birds with one requires ∼ 10 8 INDEPENDENT samples stone LVDSMC BGK model Statistical uncertainty affects all molecular simulation Multiscale Implications methods VRDSMC Application: Multiscale problems .... Phonon Transport Conclusions