Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators C. Theodoropoulos and I.G. Kevrekidis in collaboration with K. Sankaranarayanan and S. Sundaresan Department of Chemical Engineering, Princeton University, Princeton, NJ 08544

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Coarse Bifurcation Studies of Alternative
Microscopic/Hybrid SimulatorsC. Theodoropoulos
and I.G. Kevrekidisin collaboration with K.
Sankaranarayanan and S. SundaresanDepartment of
Chemical Engineering,Princeton University,
Princeton, NJ 08544
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Outline

• Motivation
• Basics of the Lattice Boltzmann method
• Bubble dynamics
• The Recursive Projection Method (RPM)
• The basic ideas
• Use of RPM for coarse bifurcation/stability
  analysis of LB simulations of a rising bubble
• Mathematical Issues
• Hybrid Simulations
• Gap-tooth scheme
• Dynamic simulations of the FitzHugh-Nagumo model
• Conclusions

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Motivation

• Bubbly flows are frequently encountered in
  industrial practice
• Study the dynamics of a rising bubble via 2-D LB
  simulations
• Oscillations occur beyond some parameter (density
  difference) threshold
• Objectives
• Obtain stable and unstable steady state solutions
  with dynamic LB code
• Accelerate convergence of LB simulator to
  corresponding steady state
• Calculate coarse eigenvalues and eigenvectors
  for control applications
• RPM technique of choice to build around LB
  simulator
• Identifies the low-dimensional unstable subspace
  of a few slow coarse eigenmodes
• Speeds-up convergence and stabilizes even
  unstable steady-states.
• Efficient bifurcation analysis by computing only
  the few eigenvalues of the small subspace.
• Bifurcation analysis although coarse equations
  (and Jacobians) are not explicitly available
  (!!!)

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Initialization
Happens in nature
Boltzmann
NS
Happens in computations
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Lattice-Boltzmann Method

• Microscopic timestepping
• By multi-scale expansion can retrieve macroscopic
  PDEs
• Obtain states from the systems moments

Streaming (move particles)
Collision
t1
t1
t
t
moments
Distribution functions
r(x,y)
states
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LBM background

• LBM units are lattice units
• Correspondence with physical world through
• dimensionless groups
• LBM?NS
• Reynolds
  number
• Eötvös number
• Morton number

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Dynamic LB Simulations
g
Ta2.407
Ta 13.61
Bubble rise direction
Stable Unstable
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Dynamic LB Simulations
g
Ta2.407
Ta 13.61
Bubble rise direction
Stable Unstable
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Bubble column flow regimes
Chen et al., 1994
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LBM single bubble rise velocity
Mo 3.9 x 10-10
Mo 1.5 x 10-5
Mo 7.8 x 10-4
FT Correlation Fan Tsuchiya (1990)
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Wake shedding and aspect ratio
VE Vakhrushev Efremov (1970)
Sr fd/Urise Ta Re Mo0.23
1/2
Sr 0.4(1-1.8/Ta)2 , Fan Tsuchiya, (1990)
based on data of Kubota et al. (1967), Tsuge
and Hibino (1971), Lindt and de Groot (1974) and
Miyahara et al. (1988)
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Recursive Projection Method (RPM)

• Treats timstepping routine, as a black-box
• Timestepper evaluates un1 F(un)
• Recursively identifies subspace of slow
  eigenmodes, P
• Substitutes pure Picard iteration with
• Newton method in P
• Picard iteration in Q I-P
• Reconstructs solution u from the sum of the
  projectors P and Q onto subspace P and its
  orthogonal complement Q, respectively
• u PN(p,q) QF

Reconstruct solution u pq PN(p,q)QF
Initial state un
Newton iterations
Picard iterations
Timestepper
F(un)
Picard iteration
NO
Subspace P of few slow eigenmodes
Subspace Q I-P
Convergence?
YES
Final state uf
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Subspace Construction

• First isolate slow modes for Picard iteration
  scheme
• Subspace
• maximal invariant subspace of
• Basis
• Vp obtained using iterative techniques
• Orhtogonal complement Q
• Basis
• Not an invariant subspace of M
• Orthogonal projectors
• P projects onto P, Q projects
  onto Q,
• Use different numerical techniques in subspaces
• Low-dimensional subspace P Newton with direct
  solver
• High-dimensional subspace Q Picard iteration

un1
Q
Q
P
P
QF
PN(p,q)
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RPM for Coarse Bifurcations
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Stabilization with RPM
g
Ta13.61
Unstable Stabilized
Unstable Steady State
Bubble rise direction
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Stabilization with RPM
g
Ta13.61
Bubble rise direction
Unstable Stabilized
Unstable Steady State
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Bifurcation Diagram
m2
m4
m6
Total mass on centerline
Hopf point
Ta
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Eigenspectrum Around Hopf Point
Ta 8.2
Ta 10.84
Stable
Unstable
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Eigenvectors near Hopf point
Stable branch
Ta8.85
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Density Eigenvectors near Hopf point
Ta9.25
Unstable branch
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X-Momentum Eigenvectors
Ta9.25
Unstable branch
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Mathematical Issues

• Shifting to remove translational invariance
• Need to find appropriate travelling frame for
  stationary solution
• Idea Use templates to shift RawleyMarsden
  Physica D (2000)
• Alternatively use Fast Fourier Transforms (FFTs)
  to obtain a continuous shift
• Conservation of Mass Momentum (linear
  constraints)
• In LB implicit conservation is achieved via
  consistent initialization
• RPM initialization with perturbed density and
  momentum profiles
• Total mass and momentum changes
• RPM calculations can be naturally implemented in
  Fourier space

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The Gap-tooth Scheme
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FitzHugh-Nagumo Model

• Reaction-diffusion model in one dimension
• Employed to study issues of pattern formation
• in reacting systems
• e.g. Beloushov-Zhabotinski
• reaction
• u activator, v inhibitor
• Parameters
  
• no-flux boundary conditions
• e, time-scale ratio, continuation parameter
• Variation of e produces turning points
• and Hopf bifurcations

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FD Intregration
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FD-FD and FD-LB Integration
FD-FD FD-LB
t
t
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Phase Diagram
t
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Conclusions

• RPM was efficiently built around a 2-D Lattice
  Boltzmann simulator
• Coupled with RPM, the LB code was able to
  converge
• even onto unstable steady states
• Coarse eigenvalues and eigenvectors were
  calculated
• without right-hand sides of governing equations
  !!!
• The translational invariance of the LB Scheme was
  efficiently removed using templates in Fourier
  space for shifting.
• Conservation of mass and momentum (linear
  constraints) was achieved by implementing RPM
  calculations in Fourier space.
• A hybrid simulator, the gap-tooth scheme was
  constructed
• and used to calculate accurate coarse dynamic
  profiles
• of the FitzHugh-Nagumo reaction-diffusion model.

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Acknowledgements

• Financial support
• Sandia National Laboratories, Albuquerque, NM.
• United Technologies Research Center, Hartford,
  CT.
• Air Force Office for Scientific Research (Dr. M.
  Jacobs)