1

Acceleration Methods for Numerical Solution of the

Boltzmann Equation

Husain Al-Mohssen

2

Outline

• Motivation & Introduction• Problem Statement• Proposed Approach• Important Implementation Details• Examples• Discussion• Future Work

3

Motivation

• Nano-Micro devices have been developed recently with very small dimensions:

– DLP (Length) – HD read/write head (Gap Length)

• At STP an air molecule travels an average distance between collisions

• As may be expected the Navier-Stokes (NS) description of the flow starts to break down as system length becomes comparable to

• Accurate engineering models are essential for the understanding and design of such systems

m10~

m 1.0

m05.0~

4

5

10Kn

Motivation (cnt)

• The Knudsen number is defined as the ratio of the mean free path to a characteristic dimension (Kn= L). Kn is a measure of the degree of departure from the NS description

• Kn Regimes:

• Recent applications are at low Ma number

NS Description Valid

NS Holds inside the domain but slip corrections are needed at the domain boundaries

Transition Flow

Free molecular Flow

0Kn0.001

0.001Kn0.1

0.1Kn10

6

Introduction

A Kinetic Description for a dilute gas • A distribution function ),,( tcxf

is used to describe the gas state, s. t.

xdcdtcxf

),,( is the expected number expected at position x

with velocity c

at t. • “Macroscopic” properties are defined as averages over f , for example:

33; fdccufdcn x

• Evolution of f is governed by the Boltzmann Equation

• Air at STP is satisfies the dilute gas criterion )1( 3 n

7

Introduction (cnt)

The Boltzmann Equation (BE) in normalized form:

• Follows from the dilute gas assumption• Valid for all Kn• 7D(1time+3Space+3Velocity) nonlinear Integro-

differential equation

cddVffffIntegralCollision

IntegralCollisionC

fa

x

fc

t

f

Dt

Df

3211 )``(

2

..2

8

Introduction (cnt)

Numerical Methods of Solving the BE:• Particle based: DSMC

– Collisionless advection step + collision steps are successively applied. – Can be shown to simulate BE exactly in the limit of large numbers

[Wagner 1992].– Chronic sampling problems at low speeds [Hadjiconstantinou et al,

2003].» Low Ma lmit particularly troublesome

• Approximations of the BE– Linearized (has many advantages espcially when Ma<<1; still requires

numcerical solution)– BGK CI Replaced with

• Numerical solutions of the BE – Recently Baker and Hadjiconstantinou (B&H) proposed a method to

solve the BE at low Ma in a relatively efficient manner.

tff eq /)(

9

Introduction (cnt)B&H method of calculating the collision integral:

• Solves the nonlinear BE exactly • f is written as DMB fff

MBf is Maxwell-Boltzmann equilibrium distribution and

Df is deviation from MB distribution

• Since MBf is not changed by BE, effort is spent on solving Df

• Even when Df is large the solution is still correct only less efficient.

• Solution has constant relative noise that is quite small in contrast to DSMC

B&H solution methods for f: – Explicit time integration scheme:

• uses time splitting to apply convection step and collision step separately

• Stability condition limits us to relatively small time steps – Implicit scheme for finding steady state solutions:

• Scales badly with lower Kn. – New proposed method for finding SS solutions

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We want to find the steady state solution for the first few moments of the BE (velocity, temp, etc.)

• Consider the x-direction flow velocities in the plot and let us denote iu the

velocity at node i in a certain time • Furthermore, let )(tu

be the vector

T

ni uuuutu },.......,.....,,{)( 21

• If we define tuuF /)(

then we are interested in finding ssu

such that for

our system 0)(

ssuF

Problem Statement

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• We will solve the (in general) nonlinear system of equations 0)(

uF using Newton’s Method.

• In 1-D, Newton’s Method finds successive approximations to F(u)=0 using F(u) and dF/du=F’(u)

• Analogously in multi-dimensions: )(][ 1

1 iiii uFJuu

Where the ][ iJ is the Jacobian

matrix of partial derivatives • Each iteration of the method will need to evaluate )( iuF

and the

corresponding ][ iJ to find 1iu

. Since the Jacobian matrix is large and very

expensive to compute, a method to approximate new ][ iJ efficiently has

to be found for this approach to be practical • Broyden [Broyden] developed an update method that is very powerful

Proposed Solution Methodology

F(u)

x

ui

ui+1

F(ui) and F’(ui)

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Proposed Solution Methodology (cnt)

The Broyden update formula is a method of updating ][ iJ to ][ 1iJ such

that: ][ 1iJ will be consistent with the new “measured” )( iuF

][ 1iJ will retain as much information as possible from ][ iJ .

Using the Broyden update formula each Newton iteration will only need an evaluation of )( iuF

to get a new guess of the solution 1iu

and a new ][ 1iJ

In 1D, Broyden’s method reduces to the Secant Method

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Find

Simplified Flow Chart of Method

1iu

Start

Find

Estimate

Integrate BE to find

Converged?

End

][J

0u

)( iuF

Use Broyden to find from and ][ 1iJ ][ iJ )( iuF

No

Yes

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Important Implementation Details(for Broyden Portions)

Finding an Initial Jacobian Matrix Use continuum solution approximation ][ cJ

Fairly robust even when ][ cJ is not close to ][ exactJ

Noise o Due to the statistical nature of the method the value of )( iuF

will

have a noisy component o We can easily show that )}({|| xFNxx ExBr

Exx

Is exact solution

Brx

Is solution after many Newton-Broyden steps

N is system characteristic time constant (in steps).

Less noise is needed for systems with larger time constants if we want to maintain solution accuracy.

15

1D Graphical AnalogF[u]

u

)}({|| xFNxx ExBr

noiseinput)}({ xF

solinyuncertaint || ExBr xx

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Important Implementation Details (BE Portions)

-1 1 2 3

0.2

0.4

0.6

0.8

1

-1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

-1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3

IntegrateBE

Shift f to target mean

Initialization of a proper ),( cxf

to use with a certain iu

:

• A finite number of moments at any x

is not enough to specify ),( cxf

at that position

• Need to find a special ),( cxf

consistent with BE dynamics and

prescribed iu

• Use BE to “mature” ),( cxf

for us by:

1. starting with MBf

2. Integrating BE for a short time to get ),(' cxf

3. Modify f to keep the new shape but give the target u

4. Repeat 2 and 3 until we converge

• Notes: • Takes 1-4 molecular to converge

• Has to be done at every evaluation of )( ixF

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Use Broyden to find from and

Find

Flow Chart of Method

1iu

][ 1iJ ][ iJ

Start

Find

Estimate

Integrate BE Step1: Equilibrate fStep2: Sample Calculation to find

Converged?

End

][ cJ

0u

)( iuF

)( iuF

No

Yes

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Examples

100 200 300 400 500

-0.04

-0.02

0.02

0.04

U

Node #

Exaggerated KnLayer

Exaggerated Kn Layer

• Method successfully calculates 1D flows over the Kn Spectrum (both pressure driven and shear driven).

• Next results are for shear driven flows with a 0.05 (normalized) wall velocity at different Kn and discretization.

• Plot on right shows error bars for different discretization for a kn=0.1 Shear flow. Accuracy of solution is well within expected bounds

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20 40 60 80 100 120

-0.0015

-0.001

-0.0005

0.0005

0.001

0.0015

kn=0.1

512 nodes, kn =0.1

Examples (cnt)

Knudsen Layer

Convergence History

Exact layer

Broyden Solution

2.5 7.5 10 12.5 15 17.5 20

-3.75

-3.5

-3.25

-3

-2.75

-2.5

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Discussion

• To first order, cost is about time to integrate O(10) iterations. Which translates to the time to integrate the system about mol40 (where mol is

the mean time between collisions)

• To find solution of accuracy tg , the allowed error in estimating )( ixF

(which we will denote Br ) should be

Ntg

Br

• To increase accuracy, more accurate sampling is required

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Future Work

Reduce Broyden Integration time: o Reducing sampling steps by better understanding which

parameters affect the noise level the most. o Refine relaxation method for ),(* cxf

able to use method on lower Kn systems

Extend our approach to do 2D/3D grids which would allow more complex problems with staggered timescales

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The End

Questions?

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DSMC Performance Scaling• Noise in DSMC is well understood [Hadjiconstantinou et al] and scales as

Samplesof#

1 in general

• It can be shown that : • Time to find solution by direct integration

2

1)

1(

tgtg

LogN

• Time to find solution by Broden method for similar accuracy

)1

(

3/2

tgtg

LogN

• At large enough N Broyden method can be significantly faster than direct

integration using the DSMC to solve the BE. This however is only the case for fairly large N (of order 104 105)

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B&H Performance Scaling

• Direct integration cost scales in a similar way to DSMC • Broyden methods performance scales in a more complex manner:

– B&H noise verses cost scaling is more complicated than: – Noise= ,....)..,,,( parametersICxtNfun Samples and is

generally nonlinear. – Noise vs. cost scaling becomes similar to DSMC

scaling but only in the limit of very low levels of noise (& fine meshing). Our numerical experiments indicate that it is in general much better behaved.

– Advantage is even stronger when looking at problems of

engineering accuracy – In our runs Kn~0.1 breakeven point vs. direct integration

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Plot of Convergence Rates of Different Methods

• Plot of error for Direct integration, Broyden and Baker Implicit code. Kn=0.025 # of nodes 128. (log[Error] vs. log[CI evaluations])

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Error of Broyden vs. noise of F

• Show how sig=sig/N_inf in multidimensions

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Broyden Step

• Broden formula

• Formula constraints

• Broyden Formula derivation

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Backup slides+notes

• [[check conv. History 4 high kn and 512]]• “proper” kndsen layer with 100^3 and

lower noise kn=0.1 and at least 128 nodes. Replace one already in presentation

• Change Conv. History plto to 512 and kn0.025 and 30^3 cells

• N_inf vs. Kn for our pb’s to show our rough break point….

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DSMC Performance Scaling (Backup)

Direct Integration Cost:

Broyden Cost:

Slope Sampling Scaling is key:

Analysis assumes sampling a small portion of run =>

N LogUtgtg2 NLog1

tg1tg2

Sample 2Ns3 Ns

N 4tg23

102 Ntg

23 Molt

Ntg23Log1tg

3010 100 1000

1. 108

1. 107

1. 106

0.00001

0.0001dt.01,g.1Red dt.1g.1Green dt.1g90Blue dt.1,.01g1Orange

B&H Noise for Different Paramters(Backup)

For little extra computational Effort you get a dramatic decrease in measurement error. compare for example pt. A, B and C.

A

B

C

Kn=?

If only interested in eng.Accuracy N_inf=10^-4/sig_sample

Cost A=Cost BCost C=10 Cost A

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Distribution Function initilization (Backup)

• Plot of norm f vs. step [[Possibly for multiple kn

1000 2000 3000 4000

0.2

0.4

0.6

0.8

1

[[what kn? What state of F?]]

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2.5 7.5 10 12.5 15 17.5 20

-6.8

-6.6

-6.4

-6.2

-6

Scaling Arguments (Backup)

• Why is it always O(10)? Well possibly because of this: • As per Kelly Newton’s is q-Quadratic and secent is Q-superlinear; Broyden is

somewhere in between. • The other plot is the MMA result using [a] x/nnn + noise• Kelly says eps=K eps^2 not exp[-2t]

In[13]:= LinearLogPlot.05Expt1, .05Expt2,t, 0, 10, GridLinesRange10,103, 104, 105

0 2 4 6 8 10

1. 1010

1. 108

1. 106

0.0001

0.01

1

Out[13]= Graphics

MMA Model Problem in Multi-D with Noise

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Can u answer these Questions

• Is it possible that O(10) will increase with less noise Requrement

• If u reduce Dt sample to decrease noise, don’t u increase N_inf??!!!

• [[Re-initializing a Run after it reaches its minimum noise level with less noise as a method of Confirming convergance or reducing noise (NB: since we are somehow finding the null space of the Jacobian aren’t we somehow garanteed to have a sick matrix when we stall?)]]

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Can u Explain B&H?

• What is importance sampling? & how is it applied to CI? Write the appt. version of CI.

• What is control variate M/C interation?

• How is the finite volume Spliting method implemented? What are the various Stability conditions?

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Integration Stability Codnition

• CI step

• Convection Step

• Implicit step?

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-7 -6 -5 -4

-5

-4

-3

-2

-1

512

128

64

32

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Log10[Input Noise]

Log10[error]