Transcript matut 6004

Multiscale Simulation Methods
Russel Caflisch
Math & Materials Science Depts, UCLA
IPAM, Sept 13, 2005
Outline
• Simulations
– Equations often unavailable or cumbersome
– New multiscale strategies needed
• Perron-Cluster Cluster Analysis
– Automatically identifies metastable states
– Example of clustering algorithm
• Equation-Free Multiscale Method of Kevrekidis
– Using simulation results to form approximate model on fine scale
– Extend to coarse scale
• Interpolated fluid/Monte Carlo method for rarefied gas
dynamics
– Combine particle and continuum descriptions of gas in single hybrid
method
• Conclusions
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Perron-Cluster Cluster Analysis
(PCCA)
• Objective: Identify modes for reduced order
description of a complex system
– E.g. metastable states for bio molecule
• Method: clustering methods
– Principal component analysis (PCA)
• Principal orthogonal decomposition (POD)
– Independent component analysis (ICA)
– PCCA
• Nonlinear method
• Similar to Laplace projection
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Perron Vectors
• Stochastic matrix T
– Nonegative entries
– Rows sum to 1
– Assume eigenvalue 1 has multiplicity k
• Invariant sets
– Invariant set of dimension di
– Invariant measure -> eigenvector Xi with eigenvalue 1
– Matrix X=(X1 ,…, Xk )
• Characteristic “functions”
– Let χi with χ=(0,…,0,1,…1,0,…0)t with di 1’s
– For χi ∙ χj =0 for different i,j
– Matrix of e-vectors χ=(χ1 ,…, χk )
• Coordinate transformation A
– χ=XA
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PCCA
• Stochastic matrix T
– Nonegative entries
– Rows sum to 1
– Assume k eigenvalues close to 1
• Nearly-invariant measures
– eigenvector Xi with eigenvalue near 1
– Matrix X=(X1 ,…, Xk )
• Find
– transformation matrix A,
– characteristic matrix χ
– χ ≈ XA
• Robust algorithm for finding A, χ
– Deuflhard, Dellnitz, Junge & Schutte (1999)
– Deuflhard & Weber (2004)
– Schutte to speak in Workshop IV
• Project dynamics onto subspaces given by A, to find reduced
order approximation
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Equation-Free Multiscale Method
• For many processes, equations are not readily
available
– dynamics specified by an algorithm, difficult to write as a
set of equations
– Legacy computer code
• Multiscale modeling and simulation must proceed
without use of equations
– Method by Kevrekidis and co-workers
– Kevrekidis to speak in Caltech workshop following
Workshop IV
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Equation-free multiscale method
• Perform small number of fine scale
simulations
– computationally expensive
• Evolution of coarse-grained variables
determined by projection
– Sensitive to choice of coarse-grained variables
– Polynomial expansion used
– Perhaps PCCA would be of use
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Application of Equation-Free
Multiscale Method
• Diffusion in a random medium
• Comparison to Monte Carlo solution
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Overview of RGD
• Rarefied gas dynamics (RGD)
– RGD required when collisional effects are significant
– Key step (i.e. computational bottleneck) in many material processing and
aerospace simulations
– Direct Simulation Monte Carlo (DSMC) is dominant computational method
• Boltzmann equation for density function f
f  f (v , x , t )
ft  v  f   1Q( f , f )
– ε = Knudsen number = mean free path / characteristic length scale
• Applications
– Aerospace
– Materials processing
– MEMS
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Equilibrium and Fluid Limit of
Boltzmann
• Maxwellian equilibrium
– Q(f,f)=0 implies f=M(v;ρ,u,T)
M ( v)   (2 T )3 2 exp(( v  u)2  2T )
• Equilibration
–
–
–
–
Consider f=f(v,t) spatially homogeneous
Entropy
H ( f )   f log( f )dv
Boltzmann’s H-theorem
dH / dt  0
H-theorem implies f →M as t →∞
• Fluid Limit (Hilbert, Grad, Nishida, REC)
– ε→0
– f(v,x,t)→ M(v;ρ,u,T), with ρ= ρ(x,t), etc.
– ρ,u,T satisfy Euler (or Navier-Stokes)
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Rarefied vs. Continuum Flow:
Knudsen number Kn
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Collisional Effects in the Atmosphere
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DSMC
• DSMC = Direct Simulation Monte Carlo
– Invented by Graeme Bird, early 1970’s
– Represents density
function as collection of particles
N
F (v)    (v  vk (t )) ( x  xk (t ))
k 1
– Directly simulates RGD by randomizing collisions
• Collision v,w →v’,w’ conserving momentum, energy
• Relative position of v and w particles is randomized
dxk / dt  vk
– Particle advection
– Convergence (Wagner 1992)
• Limitation of DSMC
– DSMC becomes computationally intractable near fluid
regime, since collision time-scale becomes small
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Hybrid method
• IFMC=Interpolated Fluid Monte Carlo
– Combines DSMC and fluid methods
– Representation of density function as combination of
Maxwellian and particles
N
F (v)   M (v)  m  (v  vk (t ))
k 1
M (v)   (2 T )3/ 2 exp((v  u)2 / 2T )
•
•
•
•
ρ, u, T solved from fluid eqtns, using Boltzmann scheme for CFD
N = O(1- α)
α = 0 ↔ DSMC
α = 1 ↔ CFD
– Remains robust near fluid limit
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IFMC for Spatially Homogeneous Problem
• Implicit time formulation
• Thermalization approximation
• Hybrid representation
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Implicit time formulation
• From Pareschi’s thesis
– related to Bird’s “no time counter” (NTC) method
• Collision operator
– Rewrite with constant negative term (“trial collision” rate)
Q( f  f )  Q  ( f  f )  Q  ( f ) f
 P( f  f )   f 
  Q ( f )
• Implicit time formulation of Boltzmann
  t 
)
– New time variable   (1  e
– Boltzmann equation
df / dt   1Q( f  f )   1P( f  f )   1 f 
becomes
F 1
 P( F  F )
 
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F  fet  
Thermalization Approximation
• Spatially Homogeneous Problem
f k 0  f (t  0)
– Wild expansion

F ( )    k f k
f1 
1

P( f 0  f 0 )
k 0
f k 1  (k  1)
– fk includes particles having k collisions
1
k
1

 P( f h  f k h )
h 0
• Thermalization approximation
– Replace particles having 2 or more collisions in time step
dt by Maxwellian M
– Resulting evolution over dt
f (t )  Af (0)  Bf1  CM
A  (1   )
B   (1   )
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C  2
Hybrid Representation and Evolution
• Hybrid representation
f  (1   ) g   M
– g= {particles}, M=Maxwellian, β= equilibration coefficient
• Evolution equations
– From thermalization approximation
f (t )  Af (0)  Bf1  CM
– Equations for β and g
 n1  A n  B( n )2  C
g
n 1
  A  B(1   ) 
n
1
n
n
n
n
n
n
Ag

B
(1


)
f
(
g

g
)

2

Bf
(
g
 M )

1
1
– g eqtn has Monte Carlo (DSMC) representation
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Relxation to Equilibrium
• Spatially homogeneous, Kac model
• Similarity solution (Krook & Wu, 1976)
Comparison of DSMC(+) and IFMC(◊)
At time t=1.5 (top) and t=3.0 (bottom).
Number of particles (top) and number of collisions
(bottom) for IFMC with dt=0.5(◊) and dt=1.0 (+).
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IFMC for Spatially Inhomogeneous
Problem
• Time splitting
• Collision step as above
– Because of disequilibrium from advection, start with β=0
• Convection step: 2 methods
– Move particles by their velocity
– Move continuum part: 2 methods
• Direct convection of Maxwellians
• Use of Euler or Navier-Stokes equations for convection
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Computational Results
• Shock
• Leading Edge problem
– Flow past half-infinite flat plate
• Flow past wedge
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Comparison of DSMC (blue) and IFMC (red) for a
shock with Mach=1.4 and Kn=0.019
Direct convection of Maxwellians
ρ
u
T
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Comparison of DSMC (contours with num values)
and IFMC (contours w/o num values)
for the leading edge problem.
ρ
u
T
v
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Conclusions and Prospects
• Hybrid method for RGD that performs
uniformly in the fluid and near-fluid regime
• Applications to aerospace, materials,
MEMS
• Current development
– Generalized numerics, physics, geometry
– Test problems
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