Acceleration Methods for Numerical Solution of the Boltzmann Equation

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Acceleration Methods for Numerical Solution of
the Boltzmann Equation

• Husain Al-Mohssen

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Outline

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

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

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Motivation (cnt)

• The Knudsen number is defined as the ratio of the
  mean free path to a characteristic dimension (Kn
  l/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
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Introduction
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Introduction (cnt)

• The Boltzmann Equation (BE) in normalized form
• Follows from the dilute gas assumption
• Valid for all Kn
• 7D(1time3Space3Velocity) nonlinear
  Integro-differential equation

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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
  Maltlt1 still requires numcerical solution)
• BGK CI Replaced with
• Numerical solutions of the BE
• Recently Baker and Hadjiconstantinou (BH)
  proposed a method to solve the BE at low Ma in a
  relatively efficient manner.

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Introduction (cnt)
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Problem Statement
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Proposed Solution Methodology
F(ui) and F(ui)
F(u)
x
ui1
ui
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Proposed Solution Methodology (cnt)
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Simplified Flow Chart of Method
Start
Find
Estimate
Integrate BE to find
Use Broyden to find from and

Find
Converged?
No
Yes
End
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Important Implementation Details(for Broyden
Portions)
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1D Graphical Analog
Fu
u
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Important Implementation Details (BE Portions)
Shift f to target mean
Integrate BE
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2
3
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Flow Chart of Method
Start
Find
Estimate
Integrate BE
Use Broyden to find from and

Find
Converged?
No
Yes
End
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Examples
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Examples (cnt)
Knudsen Layer
Broyden Solution
Exact layer
Convergence History
512 nodes, kn 0.1
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Discussion
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Future Work
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The End

• Questions?

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DSMC Performance Scaling
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BH Performance Scaling
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Plot of Convergence Rates of Different Methods

• Plot of error for Direct integration, Broyden and
  Baker Implicit code. Kn0.025 of nodes 128.
  (logError vs. logCI evaluations)

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Error of Broyden vs. noise of F

• Show how sigsig/N_inf in multidimensions

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Broyden Step

• Broden formula
• Formula constraints
• Broyden Formula derivation

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Backup slidesnotes

• check conv. History 4 high kn and 512
• proper kndsen layer with 1003 and lower noise
  kn0.1 and at least 128 nodes. Replace one
  already in presentation
• Change Conv. History plto to 512 and kn0.025 and
  303 cells
• N_inf vs. Kn for our pbs 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 gt
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BH 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
Kn? If only interested in eng. Accuracy
N_inf10-4/sig_sample Cost ACost B Cost C10
Cost A
B
C
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Distribution Function initilization (Backup)

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

what kn? What state of F?
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Scaling Arguments (Backup)

• Why is it always O(10)? Well possibly because of
  this
• As per Kelly Newtons 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 epsK eps2 not exp-2t

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, dont 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 arent we somehow garanteed to have
  a sick matrix when we stall?)

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Can u Explain BH?

• 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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