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

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Efficient methods for solving the Boltzmann equation for nanoscale transport applications

Nicolas G. Hadjiconstantinou

Massachusetts Institute of Technology Department of Mechanical Engineering

8 November 2011

Acknowledgements: L. Baker, T. Homolle, H. Al-Mohssen

• G. Radtke, C. Landon, J-P. Peraud

Financial support: Singapore-MIT Alliance NSF/Sandia National Laboratories, MITEI

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Breakdown of Navier-Stokes description (gases)

Interest lies in scientific and practical challenges associated with breakdown of Navier-Stokes description at small scales Breakdown of Navier-Stokes = breakdown of continuum

• assumption. Conservation laws, e.g.

ρDu Dt = −∂P ∂x + ∂τ ∂x + ρf can always be written Navier-Stokes description fails because collision-dominated transport models, i.e. constitutive relations such as τij = µ (∂ui/∂xj + ∂uj/∂xi) , i = j fail This failure occurs when the characteristic flow lengthscale approaches the fluid “internal scale” λ In a gas λ is typically identified with the molecular mean free

• path. λair ≈ 0.05µm (atmospheric pressure) ⇒ Kinetic

phenomena appear in air at micrometer scale.

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Motivation

Small scale devices (sensors/actuators [Karabacak, 2007], pumps with no moving parts using thermal transpiration [Muntz et al., 1997-2009; Sone et al., 2002],...) Processes involving nanoscale transport (Chemical vapor deposition [e.g. Cale, 1991-2004], flight characteristics of hard-drive read/write head [Alexander et al., 1994], damping/thin films [Park et al., 2004; Breuer, 1999],...) Vacuum science/technology: Small-scale fabrication (removal/control of particle contaminants [Gallis et al., 2001&2002],...) Similar challenges associated with nanoscale heat transfer in the solid state (in silicon at T = 300K, λphonon ≈ 0.1µm) [Majumdar (1993), Chen “Nanoscale Energy Transport and Conversion” (2005)]

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Outline

1 Introduction 2 Introduction II 3 Direct simulation Monte Carlo 4 Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

5 Application: Phonon Transport 6 Conclusions

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: Knudsen regimes

Deviation from Navier-Stokes is quantified by Kn = λ/H H is flow characteristic lengthscale Flow regimes (conventional wisdom): Kn ≪ 0.1, Navier-Stokes (Transport collision dominated) Kn 0.1, Slip flow (Navier-Stokes valid in body of flow, slip at the boundaries) 0.1 Kn 10, Transition regime Kn 10, Free molecular flow (Ballistic motion)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: Knudsen regimes

1 0.1 10 Kn = λ/L 1 0.1 10 Ma, ∆T/T0, etc. ✛ Navier Stokes (slip flow) ✲ collisionless ✲ ✻

high-altitude hypersonic flow MEMS acceler.

Frangi, 2007

Knudsen pump

Han, 2007

• scillating

microbeam

Gallis, 2004

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: Kinetic description

Boltzmann Equation: Evolution equation for f(x, c, t) ∂f ∂t +c·∂f ∂x+F·∂f ∂c = d f dt

• coll

= (f∗f∗

1 −f f1)|cr| σ d2Ω d3c1

f(x, c, t)d3cd3x = number of particles (at time t) in phase-space volume element d3cd3x located at (x, c) F = external force per unit mass f1 = f(x, c1, t) f∗

1 = f(x, c∗ 1, t) f∗ = f(x, c∗, t)

Stars denote post-collision velocities |cr| = |c − c1| σ = σ(|cr|, Ω) = collision cross-section

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: Kinetic description

Connection to hydrodynamics:

ρ(x, t) = mn(x, t) =

• m fd3c

u(x, t) = 1 ρ(x, t)

• mc fd3c

T(x, t) = 1 3kbn(x, t)

• m (c − u(x, t))2 fd3c

τij(x, t) =

• m (ci − ui(x, t))(cj − uj(x, t)) fd3c

(Absolute) Equilibrium ( ∂f

∂t + c · ∂f ∂x = 0, [d

f/dt]coll = 0): f 0 = n0 π3/2c3/2 exp

• −c2

c2

• ,

c0 =

• 2kbT0/m

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: Kinetic description

Local equilibrium ( ∂f

∂t + c · ∂f ∂x = 0, [d

f/dt]coll = 0): f loc = nloc(x, t) π3/2c3/2

loc (x, t)

exp

• −(c − uloc(x, t))2

c2

loc(x, t)

• cloc(x, t) =
• 2kbTloc(x, t)/m

The BGK (relaxation-time) approximation:

(f ∗f ∗

1 − ff1)|vr|σd2Ωd3v1 ≈ −(f − f loc)/τ

where τ =

λ

√

8kT0/πm = “mean time between collisions”

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Introduction II: A useful identity for numerical method development

(f∗f∗

1 − f f1)|cr| σ d2Ω d3c1 =

1 2 δ′

1 + δ′ 2 − δ1 − δ2

• f1f2|cr|σd2Ωd3c1d3c2

where

δi = δ (c − ci) , δ′

i = δ (c − c′ i)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Direct Simulation Monte Carlo

Smart molecular dynamics: no need to numerically integrate essentially straight line trajectories [Bird]. System state defined by {xi, ci}, i = 1, ...N Solves Boltzmann equation by splitting motion:

Collisionless advection for ∆t (xi → xi + ci∆t) ∂f ∂t + c · ∂f ∂x = 0 Perform collisions for the same period of time ∆t: ∂f ∂t = 1 2 (δ′

1 + δ′ 2 − δ1 − δ2) f1f2|cr|σd2Ωd3c1d3c2

Collisions performed in cells of linear size ∆x. Collision partners picked randomly within cell

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

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Application: Phonon Transport Conclusions

DSMC discussion

Significantly faster than MD (for dilute gases) In the limit ∆t, ∆x → 0, N → ∞, DSMC solves the Boltzmann equation [Wagner, 1992] Error in transport coefficients ∝ ∆x2 in the limit ∆t → 0 [Alexander et al,. 1998] Error in transport coefficients ∝ ∆t2 in the limit ∆x → 0 [Hadjiconstantinou, 2000] DSMC (Boltzmann)= Lattice Boltzmann (solves NS)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

DSMC Advantages

DSMC has overshadowed numerical discretization approaches for problems of practical interest. Solution by numerical discretization only advantageous when very high accuracy is required for special (low-dimensional, simple) problems e.g. [Sone, Aoki & Ohwada (1989-)] DSMC Advantages: SIMPLICITY No need to discretize 6-dimensional phase space Unconditionally stable Importance sampling

∂f ∂t = 1 2

(δ′

1 + δ′ 2 − δ1 − δ2) f1f2|cr|σd2Ωd3c1d3c2

Natural treatment of discontinuities

∂f ∂t + c · ∂f ∂x = 0 → ”Move”

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

DSMC Limitations

Statistical error [Hadjiconstantinou et al., 2003] σux |ux,0| = 1 √NCNens 1 Ma√γ , σT ∆T = 1 √NCNens

• kB/cV

∆T/T0 Resolution of a Ma = 0.01 flow to 1% uncertainty requires ∼ 108 INDEPENDENT samples Statistical uncertainty affects all molecular simulation methods Multiscale problems ....

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

A problem currently out of reach of DSMC

Temperature response to a laser pulse (∆T ∼ 1K)

Challenges: Temperature differences too small to discern (from noise) Domain too large to explicitly simulate (Loading)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Variance reduction: killing two birds with one stone

0.5 0.4 0.3 0.2 0.1 0.07 0.05

• 0.05
• 0.07
• 0.1
• 0.2

0.5

T − T0 ∆T q ρ0c0∆T 2λ

✛ ✲

LVDSMC DSMC Reduces noise by orders of magnitude Seamlessly transitions to continuum limit with NO APPROXIMATION

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Variance reduction Removes the cost associated with molecular motion that averages to known behavior

Observation: for low speed flows, the distribution function is very close to equilibrium (Maxwell Boltzmann distribution) Write f = fMB + fd, where fMB is an arbitrary Maxwell Boltzmann distribution [Baker & Hadjiconstantinou, 2005]

d f dt

• coll

= 1 2 (δ′

1 + δ′ 2 − δ1 − δ2)

• f MB

1

+ f d

1

• ×
• f MB

2

+ f d

2

• |cr|σ d2Ω d3c1 d3c2

= 1 2 (δ′

1 + δ′ 2 − δ1 − δ2)

• 2f MB

1

+ f d

1

• f d

2 ×

|cr|σ d2Ω d3c1 d3c2

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Variance reduction

What have we done here? Removed

(δ′

1 + δ′ 2 − δ1 − δ2) f MB 1

f MB

2

|cr|σd2Ωd3c1d3c2 = 0

from the calculation Recall that fMB

1

fMB

2

≫ fMB

1

fd

2 ≫ fd 1 fd 2

In other words, we choose to not perform the vast majority of collisions that have no effect and thus Save a lot of effort Not pollute the answer by statistical uncertainty of evaluating 0.0

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Variance reduction

This is more generally known as Control Variate Monte Carlo integration: Integral

• f(x)dx can be evaluated with

significantly smaller uncertainty by writing

• f(x)dx =
• (f(x) − g(x))dx +
• g(x)dx

where g(x) ≈ f(x)

• g(x)dx can be evaluated deterministically

because f(x) − g(x) is small The answer is still EXACT (no approximation)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Variance reduction

Relative statistical uncertainty=Standard deviation/ Signal magnitude

10

−5

10

−4

10

−3

10

−2

10

−1

10

−4

10

−3

10

−2

10

−1

10 10

1

wall velocity relative statistical uncertainty in flow velocity

direct method, 6400 collision events/timestep direct method, 32000 collision events/timestep typical DSMC

Computational cost scales with square of relative statistical uncertainty

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Deviational Particle methods

Like DSMC BUT simulate the motion of “deviational particles” that can be positive or negative

f = fMB + fd. If fMB = fMB(x) ∂f/∂t + c · ∂f/∂x = ∂fd/∂t + c · ∂fd/∂x → ”move” Case fMB = fMB(x) can also be treated with small

• changes. From now on, focus on collision integral.

Collide:

d f dt

• coll

= (δ′

1 + δ′ 2 − δ1 − δ2) f MB 1

f d

2 gσd2Ω d3c1 d3c2 +

1 2 (δ′

1 + δ′ 2 − δ1 − δ2) f d 1 f d 2 gσd2Ω d3c1 d3c2

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Deviational Particle methods

Let us look at the linear term

d f dt

• coll

= (δ′

1 + δ′ 2 − δ1 − δ2) f MB 1

f d

2 gσd2Ω d3c1 d3c2

In contrast to DSMC (fMB = 0, fd > 0)

d f dt

• coll

= 1 2 (δ′

1 + δ′ 2 − δ1 − δ2) f1f2gσd2Ω d3c1 d3c2

i.e. collision process is simply an update δ1, δ2 → δ′

1, δ′ 2

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Deviational Particle methods

Result: For Kn > 1, where wall collisions are dominant, method is very efficient For Kn < 1, where collisions dominate, number of particles diverges (after about one collision time) Fundamental limitation? [Brownian Dynamics, Wagner & Ottinger, 1996;Chun & Koch, 2005]

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Low variance deviational simulation Monte Carlo (LVDSMC)

Automatic and EXACT cancellation can be achieved using the property [Homolle & Hadjiconstantinou, 2007]: δ′

1 + δ′ 2 − δ1 − δ2

• fMB

1

fd

2 gσd2Ω d3c1 d3c2 =

• [K1(c, c1) − K2(c, c1)]fd(c1)d3c1 − ν(c)fd(c)

[Hilbert 1912]

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Low variance deviational simulation Monte Carlo (LVDSMC)

where

K1(c, c1) =

• 2

π3 1 λ0c2 1 |ˆ c − ˆ c1| exp

• −|ˆ

c|2 + (ˆ c × ˆ c1)2 |ˆ c − ˆ c1|2

• =
• 2

π3 1 λ0c2 1 |ˆ c − ˆ c1| exp

• −[ˆ

c · (ˆ c − ˆ c1)]2 |ˆ c − ˆ c1|2

• K2(c, c1)

=

• 1

2π3 1 λ0c2 |ˆ c − ˆ c1| exp

• −|ˆ

c|2 ν(c) =

• 1

2π c0 λ0

• exp
• −|ˆ

c|2 +

• 2|ˆ

c| + 1 |ˆ c| |ˆ

c|

exp

• −ξ2

dξ

• and ˆ

c = c/c0

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Low variance deviational simulation Monte Carlo (LVDSMC)

Collision integral interpretation: ∂f ∂t

• coll

=

• [K2 − K1]fdd3c1
• particle generation

−ν(c)fd

• particle deletion

Based on this arrangement, the collision algorithm proceeds as follows: Delete deviational particles of velocity c with probability proportional to ν(c)∆t. Generate deviational particles according to the distribution

• ∆t
• [K1 − K2] fdd3c1
• (c)

Generalized, convergence results [Wagner (2008)]

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

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BGK collision model

d f dt

• coll

≈ −(f − floc)/τ = νfloc − νf = ν(floc − fMB) − νfd Although “crude” as an approximation, BGK is widely used in many physics fields, most notably, phonon, electron transport. More to come. Similarity between Hilbert’s form and BGK model suggests BGK may be stable. Indeed it is [Radtke & Hadjiconstantinou, 2009]. Its simplicity leads to VERY simple, efficient algorithms [Radtke & Hadjiconstantinou, 2009; Hadjiconstantinou, Radtke &Baker, 2010]

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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LVDSMC implementations

Fixed equilibrium distribution ∂f ∂t

• coll

= f loc − f 0 τ

• generation

−f d τ

deletion

Simple, stable, easy to implement [Hadjiconstantinou, Radtke &Baker, 2010]

Spatially-variable equilibrium distribution ∂f ∂t

• coll

= f loc − f MB τ − ∆f MB

• part. generation

+∆f MB

change in equilibrium

−f d τ

• part. deletion

Stable and efficient No particle generation in the linearized regime (Ma → 0, etc.) [Radtke & Hadjiconstantinou, 2009] Highly efficient for continuum limit (Kn → 0) Multiscale implications....

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Numerical efficiency:

BGK LVDSMC methods Simulation example:

BGK, spatially-variable equilibrium distribution Transient shear problem Details

Kn = 0.1 u

• ± L

2

• = ∓U

U ≪ c0

CPU time: 70 sec. (3.0 GHz)

−0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

u U x/L

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Numerical efficiency:

BGK LVDSMC methods Simulation example:

BGK, spatially-variable equilibrium distribution Transient shear problem Details

Kn = 0.1 u

• ± L

2

• = ∓U

U ≪ c0

CPU time: 70 sec. (3.0 GHz)

— DSMC (Ma = 0.02) — LVDSMC

−0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

u U x/L

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Numerical efficiency:

BGK LVDSMC methods

Statistical error in temperature LVDSMC vs. DSMC

10

• 5

10

• 4

10

• 3

10

• 2

10

• 1

10 10

• 3

10

• 2

10

• 1

10 10

1

10

2

10

3

10

4

∆T/T0 σT ∆T DSMC LVDSMC For a single cell in the center of the simulation containing ≈ 950 particles (all methods)

LVDSMC: fixed vs. spatially-variable equilibrium

10

• 1

10 10

1

10

• 5

10

• 4

10

• 3

σ2

T

(∆T)2 k =

2 √π Kn

Gray: fixed equilibrium Colors: spatially-variable equilibrium

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

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Multiscale Implications

As we approach Kn → 0, variable equilibrium methods become more efficient. Normally molecular methods become “stiff” in the continuum limit (more and more particles, longer timescales) Recall Chapman-Enskog expansion:

ˆ f ≈ ˆ f loc + Kn ˆ g + O(Kn2)

As Kn → 0 we need less and less particles to describe system Basis for multiscale methods that seamlessly transition from molecular to continuum [Pareschi et al. 2004].

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

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An alternative approach

[Al-Mohssen & Hadjiconstantinou (2010)]

Basic approach

Auxiliary equilibrium simulation, correlated to non-equilibrium simulation (DSMC) used for variance reduction Both simulations share initial conditions and random variables Use one set of particles to describe both simulations. How?

Importance weights Wi = feq(ci) f(ci)

[Ottinger et al. 1996]

“A particle at ci in the non-equilibrium simulation, is worth Wi particles in the equilibrium simulation”

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VRDSMC method

[Al-Mohssen & Hadjiconstantinou (2010)]

Non-equilibrium property (as before) R =

• R(c)f(c)d3c ≃ ¯

R = 1 N

Ncell

• i=1

R(ci) Equilibrium property (from the same particle data + weights) Req =

• R(c)W(c)f(c)d3c ≃ Req = 1

N

Ncell

• i=1

WiR(ci) Variance-reduced property

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VRDSMC method

σ{u} UW UW/c0

Couette Flow Kn = 1

This method is best suited to the BGK model and thus very useful for the applications discussed next ...

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Applications to solid state heat transfer

A crash course in phonon transport. More details on modeling aspects: [G. Chen (2005)] In a large class of materials (broadly speaking non-metals) lattice vibrations are responsible for significant part of the heat transfer from hot to cold (not conduction!) The discrete nature of allowed wavemodes and energy levels makes a particle description convenient: phonon=”quantum of lattice vibration modes” At the device level (λ ≈ 100nm) phonon behavior may be modeled by a Boltzmann equation

∂f ∂t + vg · ∂f ∂x = d f dt

• coll

where f = f(x, k, t), or assuming isotropic dispersion relation ω(k) = ω(k), f = f(x, ω, t)

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Applications to solid state heat transfer

Crash course in phonon-transport continued... Equilibrium distribution: Bose-Einstein

f 0(ω) = 1 exp( ω

kbT0 ) − 1

Scattering (impurity, intrinsic): Relaxation Approximation d f dt

• coll

= f loc(ω) − f(ω) τ(ω) E = V

p ωf(ω)D(ω, p)dω, D(ω, p) = density of states

(assumed continuous) Tloc determined from [Cercignani, 1988;Hao et al., 2009]:

• ω

ω

• p

D(ω, p)f loc(ω) − f(ω) τ(ω) dω = 0

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Applications to solid state heat transfer

Governing equation very similar to BGK model discussed above (with τ = τ(ω) [Holland (1963)]) DSMC-like simulations have been developed [Mazumder & Majumdar, 2001; Hao et al. 2009] with similar limitations Variance reduction needed. Use VRDSMC (weights) to illustrate method. For deviational method in phonon transport see theses by [C. Landon & J-P. Peraud] Present application of our methodology to “state of the art” in Monte Carlo simulations of phonon transport

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Applications to solid state heat transfer

VRDSMC for phonon transport: Primary simulation simulates equilibrium at T0 Non-equilibrium simulation inferred through weights

W(ω) = f(ω) f 0(ω) E = V

p[W(ω) − 1]ωf 0(ω)D(ω, p)dω + E0

Weight evolution can determined from

d f dt

• coll

= dW(ω)f 0(ω) dt

• coll

= f loc(ω) − f(ω) τ(ω)

leading to

dW(ω) dt

• coll

= 1 τ(ω) f loc(ω) f 0(ω) − W(ω)

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Applications to solid state heat transfer

Relative statistical uncertainty=Standard deviation/ Signal magnitude

10

2

10

1

10 10

3

10

2

10

1

10 10

1

σ ∆T ∆T T0

Standard VarianceReduced

!"#$%%#&# '()*(+,-#)-./,01+#*2#

• 34-,"-.#/4#"1#5()6-#

6)(.*-+"2##

Computational cost scales with square of relative statistical uncertainty

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

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Applications to solid state heat transfer

Effective conductivity (Kn ≈ 1) of porous pure silicon. Temperature field in response to an applied temperature gradient.

298.5 299 299.5 300 300.5 301 301.5

!"#$%&'(&#)*$+ ,*-.)*-+ %&'(&#)*$+ ,*-.)*-+ /+ 0+ 123+#4+

Non-variance-reduced simulation needs 60+ years to reach same level of uncertainty (same computer)

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Applications to solid state heat transfer

Tuning the effective conductivity of porous pure silicon.

d

Ballistic (Kn ≈ 4) shading exploited to reduce effective thermal conductivity (2D periodic structure)

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Applications to solid state heat transfer

Thermal conductivity spectroscopy

Temperature response to a laser pulse (∆T ∼ 1K) Experiment from G. Chen (MIT)

Speedup: O(109)

(Loading)

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Conclusions

Proposed and developed a new class of methodologies for drastically reducing the statistical uncertainty (cost) of Monte Carlo simulations of kinetic transport phenomena (most importantly: resolution of debilitating stability problems [2007]) Formulation sufficiently general to apply to various kinetic models For typical applications, the speedup is sufficiently large [O(1,000-10,000)], to enable otherwise impossible simulations (gaseous thermal response, kinetic flow through porous media, solid-state heat transfer) Formulation able to capture arbitrarily small deviations from equilibrium (where speedup → ∞) Removing the part associated with molecular motion that averages to known behavior via algebraically decomposing the distribution function is a very effective approach towards multiscale simulation

Introduction Introduction II Direct simulation Monte Carlo Variance reduction: killing two birds with one stone

LVDSMC BGK model Multiscale Implications VRDSMC

Application: Phonon Transport Conclusions

Thanks/Apologies

Thanks for your attention Sorry for talking so fast