efficient methods for solving the boltzmann …...e cient methods for solving the boltzmann equation...

Introduction

Introduction II

DirectsimulationMonte Carlo

Variancereduction:killing twobirds with onestone

LVDSMC

BGK model

MultiscaleImplications

VRDSMC

Application:PhononTransport

Conclusions

Efficient methods for solving the Boltzmannequation for nanoscale transport applications

Nicolas G. Hadjiconstantinou

Massachusetts Institute of TechnologyDepartment of Mechanical Engineering

8 November 2011

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

G. Radtke, C. Landon, J-P. PeraudFinancial support: Singapore-MIT Alliance

NSF/Sandia National Laboratories, MITEI

Introduction

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Conclusions

Breakdown of Navier-Stokes description (gases)

Interest lies in scientific and practical challenges associated withbreakdown of Navier-Stokes description at small scales

Breakdown of Navier-Stokes 6= breakdown of continuumassumption. Conservation laws, e.g.

ρDu

Dt= −∂P

∂x+∂τ

∂x+ ρf

can always be written

Navier-Stokes description fails because collision-dominatedtransport models, i.e. constitutive relations such asτij = µ (∂ui/∂xj + ∂uj/∂xi) , i 6= j failThis failure occurs when the characteristic flow lengthscaleapproaches the fluid “internal scale” λ

In a gas λ is typically identified with the molecular mean freepath. λair ≈ 0.05µm (atmospheric pressure) ⇒ Kineticphenomena appear in air at micrometer scale.

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Conclusions

Motivation

Small scale devices (sensors/actuators [Karabacak, 2007],pumps with no moving parts using thermal transpiration [Muntzet al., 1997-2009; Sone et al., 2002],...)

Processes involving nanoscale transport (Chemical vapordeposition [e.g. Cale, 1991-2004], flight characteristics ofhard-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 thesolid state (in silicon at T = 300K, λphonon ≈ 0.1µm)[Majumdar (1993), Chen “Nanoscale Energy Transport andConversion” (2005)]

Introduction

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VRDSMC


Conclusions

Outline

1 Introduction

2 Introduction II

3 Direct simulation Monte Carlo

4 Variance reduction: killing two birds with one stoneLVDSMCBGK modelMultiscale ImplicationsVRDSMC

5 Application: Phonon Transport

6 Conclusions

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions

Introduction II: Knudsen regimes

Deviation from Navier-Stokes is quantified by Kn = λ/HH 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

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VRDSMC


Conclusions

Introduction II: Knudsen regimes

10.1 10 Kn = λ/L

1

0.1

10

Ma,∆T/T0,

etc.

�Navier Stokes(slip flow) -

collisionless

-

6high-altitudehypersonicflow

MEMSacceler.

Frangi, 2007

Knudsenpump

Han, 2007

oscillatingmicrobeam

Gallis, 2004

Introduction

Introduction II



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VRDSMC


Conclusions

Introduction II: Kinetic description

Boltzmann Equation: Evolution equation for f(x, c, t)

∂f

∂t+c·∂f

∂x+F·∂f

∂c=

[df

dt

]coll

=

∫ ∫(f∗f∗1−f f1)|cr|σ d2Ω d3c1

f(x, c, t)d3cd3x = number of particles (at time t) inphase-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

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Conclusions


Connection to hydrodynamics:

ρ(x, t) = mn(x, t) =

∫mfd3c

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, [df/dt]coll = 0):

f0 =n0

π3/2c3/20

exp

(−c

2

c20

), c0 =

√2kbT0/m

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Conclusions


Local equilibrium (∂f∂t + c ·∂f∂x 6= 0, [df/dt]coll = 0):

f loc =nloc(x, t)

π3/2c3/2loc (x, t)

exp

(− (c− uloc(x, t))

2

c2loc(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”

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Conclusions

Introduction II: A useful identity for numericalmethod 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

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VRDSMC


Conclusions

Direct Simulation Monte Carlo

Smart molecular dynamics: no need to numericallyintegrate essentially straight line trajectories [Bird].

System state defined by {xi, ci}, i = 1, ...NSolves 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. Collisionpartners picked randomly within cell

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Conclusions

DSMC discussion

Significantly faster than MD (for dilute gases)

In the limit ∆t,∆x→ 0, N →∞, DSMC solves theBoltzmann 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) 6= Lattice Boltzmann (solves NS)

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Conclusions

DSMC Advantages

DSMC has overshadowed numerical discretization approachesfor problems of practical interest. Solution by numericaldiscretization only advantageous when very high accuracy isrequired 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 =

12

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2) f1f2|cr|σd2Ωd3c1d3c2

Natural treatment of discontinuities∂f∂t + c ·

∂f∂x = 0→ ”Move”

Introduction

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VRDSMC


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% uncertaintyrequires ∼ 108 INDEPENDENT samplesStatistical uncertainty affects all molecular simulationmethods

Multiscale problems ....

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Conclusions

A problem currently out of reach of DSMC

Temperature response to alaser pulse (∆T ∼ 1K)

Challenges:

Temperature differences toosmall to discern (from noise)

Domain too large to explicitly simulate

(Loading)

movie3D_long.aviMedia File (video/avi)

Introduction

Introduction II



LVDSMC

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VRDSMC


Conclusions

Variance reduction: killing two birds with one stone

0.50.40.30.20.10.070.050-0.05-0.07-0.1-0.2

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

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LVDSMC

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VRDSMC


Conclusions

Variance reduction

Removes the cost associated with molecularmotion that averages to known behavior

Observation: for low speed flows, the distribution functionis very close to equilibrium (Maxwell Boltzmanndistribution)

Write f = fMB + fd, where fMB is an arbitrary MaxwellBoltzmann distribution [Baker & Hadjiconstantinou, 2005][df

dt

]coll

=1

2

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2)

(fMB1 + f

d1

)×(

fMB2 + fd2

)|cr|σ d2Ω d3c1 d3c2

=1

2

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2)

(2fMB1 + f

d1

)fd2 ×

|cr|σ d2Ω d3c1 d3c2

Introduction

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VRDSMC


Conclusions

Variance reduction

What have we done here? Removed∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2) fMB1 fMB2 |cr|σd2Ωd3c1d3c2 = 0

from the calculationRecall that

fMB1 fMB2 � fMB1 fd2 � fd1 fd2

In other words, we choose to not perform the vast majority ofcollisions that have no effect and thus

Save a lot of effort

Not pollute the answer by statistical uncertainty ofevaluating 0.0

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Conclusions

Variance reduction

This is more generally known as Control Variate Monte Carlointegration: 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 smallThe answer is still EXACT (no approximation)

Introduction

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Conclusions

Variance reduction

Relative statistical uncertainty=Standard deviation/ Signalmagnitude

10−5

10−4

10−3

10−2

10−1

10−4

10−3

10−2

10−1

100

101

wall velocity

rela

tive

stat

istic

al u

ncer

tain

ty in

flow

vel

ocity direct method, 6400 collision events/timestep

direct method, 32000 collision events/timesteptypical DSMC

Computational cost scales with square of relative statisticaluncertainty

Introduction

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VRDSMC


Conclusions

Deviational Particle methods

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

f = fMB + fd. If fMB 6= fMB(x)

∂f/∂t+ c · ∂f/∂x = ∂fd/∂t+ c · ∂fd/∂x→ ”move”

Case fMB = fMB(x) can also be treated with smallchanges. From now on, focus on collision integral.

Collide:[df

dt

]coll

=

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2) fMB1 fd2 gσd2Ω d3c1 d3c2 +

1

2

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2) fd1 fd2 gσd2Ω d3c1 d3c2

Introduction

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LVDSMC

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VRDSMC


Conclusions


Let us look at the linear term[df

dt

]coll

=

∫ ∫ ∫(δ′1 + δ

′2 − δ1 − δ2) fMB1 fd2 gσd2Ω d3c1 d3c2

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

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

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VRDSMC


Conclusions


Result:

For Kn > 1, where wall collisions are dominant, method isvery efficient

For Kn < 1, where collisions dominate, number ofparticles diverges (after about one collision time)

Fundamental limitation? [Brownian Dynamics, Wagner &Ottinger, 1996;Chun & Koch, 2005]

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions

Low variance deviational simulation Monte Carlo(LVDSMC)

Automatic and EXACT cancellation can be achievedusing the property [Homolle & Hadjiconstantinou, 2007]:∫ ∫ ∫ (

δ′1 + δ′2 − δ1 − δ2

)fMB1 f

d2 gσd

2Ω d3c1 d3c2 =∫

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

[Hilbert 1912]

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


where

K1(c, c1) =

√2

π31

λ0c20

1

|ĉ− ĉ1|exp

[−|ĉ|2 + (ĉ× ĉ1)

2

|ĉ− ĉ1|2

]=

√2

π31

λ0c20

1

|ĉ− ĉ1|exp

[− [ĉ · (ĉ− ĉ1)]

2

|ĉ− ĉ1|2

]K2(c, c1) =

√1

2π31

λ0c20|ĉ− ĉ1| exp

(−|ĉ|2

)ν(c) =

√1

2π

c0λ0

[exp

(−|ĉ|2

)+

(2|ĉ|+ 1

|ĉ|

)∫ |ĉ|0

exp(−ξ2

)dξ

]

and ĉ = c/c0

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions


Collision integral interpretation:[∂f

∂t

]coll

=

∫[K2 −K1]fdd3c1︸︷︷︸particle generation

−ν(c)fd︸︷︷︸particle deletion

Based on this arrangement, the collision algorithm proceeds asfollows:

Delete deviational particles of velocity c with probabilityproportional to ν(c)∆t.

Generate deviational particles according to the distribution{∆t

∫[K1 −K2] fdd3c1

}(c)

Generalized, convergence results [Wagner (2008)]

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions

BGK collision model

[df

dt

]coll

≈ −(f − f loc)/τ = νf loc− νf = ν(f loc− fMB)− νfd

Although “crude” as an approximation, BGK is widelyused in many physics fields, most notably, phonon,electron transport. More to come.

Similarity between Hilbert’s form and BGK model suggestsBGK 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

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VRDSMC


Conclusions

LVDSMC implementations

Fixed equilibrium distribution[∂f

∂t

]coll

=f loc − f0

τ︸︷︷︸generation

−fd

τ︸︷︷︸deletion

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

Spatially-variable equilibrium distribution[∂f

∂t

]coll

=f loc − fMB

τ−∆fMB︸︷︷︸

part. generation

+∆fMB︸︷︷︸change in equilibrium

−fd

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

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LVDSMC

BGK model


VRDSMC


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 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

u

U

x/L

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions


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 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

u

U

x/L

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


Statistical error in temperature

LVDSMC vs. DSMC

10-5

10-4

10-3

10-2

10-1

10010

-3

10-2

10-1

100

101

102

103

104

∆T/T0

σT

∆T

DSMC

LVDSMC

For a single cell in the center of thesimulation containing ≈ 950 particles(all methods)

LVDSMC: fixed vs.

spatially-variable equilibrium

10-1

100

101

10-5

10-4

10-3

σ2T(∆T )2

k = 2√πKn

Gray: fixed equilibrium

Colors: spatially-variable

equilibrium

Introduction

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LVDSMC

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VRDSMC


Conclusions

Multiscale Implications

As we approach Kn→ 0, variable equilibrium methodsbecome more efficient.

Normally molecular methods become “stiff” in thecontinuum limit (more and more particles, longertimescales)

Recall Chapman-Enskog expansion:

f̂ ≈ f̂ loc +Kn ĝ +O(Kn2)

As Kn→ 0 we need less and less particles to describesystem

Basis for multiscale methods that seamlessly transitionfrom molecular to continuum [Pareschi et al. 2004].

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions

An alternative approach[Al-Mohssen & Hadjiconstantinou (2010)]

Basic approach

Auxiliary equilibrium simulation,correlated to non-equilibriumsimulation (DSMC) used forvariance reduction

Both simulations share initialconditions and random variables

Use one set of particles to describeboth 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”

Introduction

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LVDSMC

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VRDSMC


Conclusions

VRDSMC method[Al-Mohssen & Hadjiconstantinou (2010)]

Non-equilibrium property (as before)

〈R〉 =∫R(c)f(c)d3c ' R̄ = 1

N

Ncell∑i=1

R(ci)

Equilibrium property (from the same particle data + weights)

〈R〉eq =∫R(c)W (c)f(c)d3c ' Req =

1

N

Ncell∑i=1

WiR(ci)

Variance-reduced property

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions

VRDSMC method

σ{u}UW

UW/c0

Couette Flow

Kn = 1

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

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions

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 theheat transfer from hot to cold (not conduction!)

The discrete nature of allowed wavemodes and energylevels makes a particle description convenient:phonon=”quantum of lattice vibration modes”

At the device level (λ ≈ 100nm) phonon behavior may bemodeled by a Boltzmann equation

∂f

∂t+ vg ·

∂f

∂x=

[df

dt

]coll

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

Introduction

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LVDSMC

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VRDSMC


Conclusions


Crash course in phonon-transport continued...

Equilibrium distribution: Bose-Einstein

f0(ω) =1

exp( ~ωkbT0 )− 1

Scattering (impurity, intrinsic): Relaxation Approximation[df

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

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


Governing equation very similar to BGK model discussedabove (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. Fordeviational method in phonon transport see theses by [C.Landon & J-P. Peraud]

Present application of our methodology to “state of theart” in Monte Carlo simulations of phonon transport

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


VRDSMC for phonon transport:

Primary simulation simulates equilibrium at T0

Non-equilibrium simulation inferred through weights

W (ω) =f(ω)

f0(ω)

E = V∫ ∑

p[W (ω)− 1]~ωf0(ω)D(ω, p)dω + E0Weight evolution can determined from[

df

dt

]coll

=

(dW (ω)f0(ω)

dt

)coll

=f loc(ω)− f(ω)

τ(ω)

leading to[dW (ω)

dt

]coll

=1

τ(ω)

(f loc(ω)

f0(ω)−W (ω)

)

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions


Relative statistical uncertainty=Standard deviation/ Signalmagnitude

10 2 10 1 10010 3

10 2

10 1

100

101

σ∆T

∆TT0

StandardVariance Reduced

!"#$%%#'()*(+,-#)-./,01+#*2#-34-,"-.#/4#"1#5()6-#6)(.*-+"2##

Computational cost scales with square of relative statisticaluncertainty

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


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

298.5 299 299.5 300 300.5 301 301.5

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

%&'()*$+,*-.)*-+

/+

0+

123+#4+ Kn = .97

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

Introduction

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LVDSMC

BGK model


VRDSMC


Conclusions


Tuning the effective conductivity of porous pure silicon.

d

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

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions


Thermal conductivity spectroscopy

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

Speedup: O(109)

(Loading)

movie3D_long.aviMedia File (video/avi)

Introduction

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LVDSMC

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VRDSMC


Conclusions

Conclusions

Proposed and developed a new class of methodologies fordrastically reducing the statistical uncertainty (cost) of MonteCarlo simulations of kinetic transport phenomena (mostimportantly: resolution of debilitating stability problems [2007])

Formulation sufficiently general to apply to various kineticmodels

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 fromequilibrium (where speedup →∞)Removing the part associated with molecular motion thataverages to known behavior via algebraically decomposing thedistribution function is a very effective approach towardsmultiscale simulation

Introduction

Introduction II



LVDSMC

BGK model


VRDSMC


Conclusions

Thanks/Apologies

Thanks for your attention

Sorry for talking so fast

IntroductionIntroduction IIDirect simulation Monte CarloVariance reduction: killing two birds with one stoneLVDSMCBGK modelMultiscale ImplicationsVRDSMC

Application: Phonon TransportConclusions

efficient methods for solving the boltzmann …...e cient methods for solving the boltzmann equation...

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