Modern Particle Methods for Complex Flows
G. Amati (2), F. Castiglione (1), F. Massaioli(2), S. Succi (1)
Acks to:
G. Bella (Roma), M. Bernaschi(IAC), H. Chen (EXA), S. Orszag (Yale),
E. Kaxiras (Harvard), S. Ubertini (Roma)
1)
Istituto Applicazioni del Calcolo
Mauro Picone , CNR, Roma, Italy
2) CASPUR, Roma, Italy
Why Particle Methods?
Atomistic physics
PDEs with large distorsions (Astrophysics)
Moving geometries (Combustion)
Moving interfaces (Multiphase flows)
Classical Particle Methods
•Particle-Particle (Molecular dynamics, Monte Carlo)
•Particle-Mesh (Neutral Plasmas, Semiconductors)
•P3M (Gravitational,Charged Plasmas)
•Fluids?
J.Eastwood, R. Hockney: Computer Simulations using particles
Particle Methods: pros and cons
Pros:
•Geoflexibility (boundary conditions)
•Physically Sound
•No matrix algebra
Cons:
•Noisy
•Small timesteps
New Particle Methods for Fluid Flows
Simple fluids, complex flows:
The Navier-Stokes equations are very hard to solve:
  

 t u  u  u  u  p
Complex fluids, complex flows:
Fluid equations are often NOT KNOWN!
New Particle Methods for Fluid Flows
Fluids (3D)
Idea: Solve fluid equations using fictitious
quasi-particle dynamics
Kinetic Theory (6D)
Universality: Molecular details do NOT
count
Driver: Statistical Physics (front-end) ,
Numerical Analysis (back-end)
Phase – space Fluid (6N D)
•Lattice Gas Cellular Automata (LGCA)
•Lattice Boltzmann (LBE)
•Dissipative Particle Dynamics (DPD)
Atoms / Molecules
Details dont count: quasi-particle trajectories
Coarse-Graining via 'Superparticles':
B blocking factor: (Macro to Meso to Micro scale)
1 computational particle = B molecules
B
X I   xi , I  1, N / B
i 1
B
VI   vi , I  1, N / B
i 1
Coarse-grained equations
dX I
 VI
dt
dVI
M
  FIJ
dt
J
Modeling goes into FIJ
Free stream
Details dont count: kinetic theory
Pre-averaged distributions: Boltzmann
approach (Probabilistic)
f ( x, v, t )    ( x  xi (t )) (v  vi (t )) 
i
F

 t f  v  f    v f  C ( f , f )
m
Modeling goes into f and C(f,f)
Collision
Lattice Gas Cellular Automata
Boolean representation:
n_i=0,1
particle absence/presence
3
2
1
4
5
6
001001
Lattice Gas Cellular Automata
t
t+1
2
3
4
1
5
t+1+ε
i
6
streaming
ni =
i = 0,6

0
absence
1
presence

collision
 
ni x  c i ,t  1  ni x, t  Ci n 
Ci n  :
collisions
(Frisch, Hasslacher, Pomeau, 1986)
Boundary condition
From LGCA to Navier-Stokes
Conservation laws:
C
i
0
(mass)
i
C c  0
C c / 2  0
i
i
(momentum)
i
2
i i
(energy)
i
No details of molecular interactions
(true collision)
(lattice collision)
From LGCA to Navier-Stokes
Isotropy (Rotational invariance)
Ta ,b,c ,d   ci ,a ci ,bci ,c ci ,d  .......
i
such that:
3
T
c ,d 1
a ,b ,c ,d
a, b, c, d  x, y , z
ucud  ua ub
Von Karman street
LGCA: blue-sky scenario
•Exact computing (Round-off freedom)
•Ideal for parallel computing (Local)
•Flexible boundary conditions
LGCA: grey-sky scenario
•Noise (Lots of automata)
•Low Reynolds (too few collisions)
•Exponential complexity 2^b (3D requires b=24)
•Lack of Galilean invariance
From LGCA to (Lattice) Boltzmann
• (Boolean) molecules to (discrete) distributions
ni
fi = < n i >
• (Lattice) Boltzmann equations (LBE)

  
fi x  ci , t  1  fi x, t  Ci  f 
From (Lattice) Boltzmann to Navier - Stokes

M
ρ( x, t ) 
M
1
u ( x,t) 
ρ
E
1
( v  u )2
T( x,t)   f( x,v,t)
dv
ρ
2
P
P

f ( x, v, t )d v
f( x,v,t)vd v
 f ( x, v, t )vvd v
(density)
(speed)
v
(temperature)
(pressure tensor)
u
From (Lattice) Boltzmann to Navier - Stokes
Weak Departure from local equilibrium
ff f
eq
neq
f
f neq
f neq
Kn  eq  1
f
u
v
From (Lattice) Boltzmann to Navier - Stokes
 
M  t ρ  div ρu  0
LBE
 
 

M  t ρu  div ρuu   p  div μu  λdiv u
 
E  t  ρT   div ρuT  P : u  K T

THE LBE STORY
• Non-linear LBE (Mc Namara-Zanetti, 1988), noise-free
• Quasi-linear LBE (Higuera-Jimenez, 1989), 3D sim’s
• Enhanced LBE (Higuera-Succi-Benzi, 1989), High Reynolds,
TOP-DOWN approach
• G-invariant LBE (Chen-Chen-Mattheus, 1991), Galilean
invariant
LATTICE BGK
Since Re depends only on  , single time relaxation only

  
1
eq
f i x  ci ,t  1  f i x,t    f i  f i 
τ
Viscosity
ν  cs2 τ 
c
2
s
1

3
1
2

(lattice sound speed)
Qian, d’Humières, Lallemand, 1992
LBE assets:
Noise-free, high Reynolds
Flexible Boundary Conditions
Efficient on serial, even more on parallel
Poisson-freedom
Additional physics (beyond fluids)
Quick grid set up (EXA-Powerflow)
LBE liabilities
Later …
Who needs LBE?
DON’T USE: Strong heat transfer, compressibility (combustion)
CAN USE: Turbulence in simple geos
SHOULD USE: Porous media
MUST USE: Multiphase, Colloidal, External Aerodynamics
Parallel Speed-up
Amati, Massaioli, Bernaschi, Scicomp 2002
LBE
t=0
t=5000
SP
Ansumali et al, ETHZ+IAC
t=20000
Turbulent channel
APE-100: 10 Gflops sustained
(Amati , Benzi, Piva, Succi, PRL 99)
Porous media: random fiber networks
A.Hoekstra,P Sloot, A.Koponen, J Timonen, PRL 2001
Cristal Growth
Miller, Succi, Mansutti, PRL 1999
LBE-Multiphase, Demixing flow: Amati, Gonnella, Lamura, Massaioli
LBE: Multiphase
B. Palmer, D. Rector, pnl.gov
http://gallery.pnl.gov/mscf/bubble_web1/bubble_web.mpg
Local grid refinement
Different time scales and no. of time steps for different
refinement levels, interpolation between levels
Succi, Filippova, Smith, Kaxiras 2001,
LBE: Airfoils
Succi,Filippova,Kaxiras, Cise 2001
You can do something like this…
Bella, Ubertini, 2001
LBE: Car design
H Chen, S Kandasamy, R Shock, S. Orszag, S. Succi, V. Yakhot, Science (2003)
Powerflow, EXA
LBE: Reactive microflows
LBE: Multiscale microflows
Unstructured LBE
Ubertini,Succi,Bella, 2003
Unstructured LBE
LBE: Unstructured (soon moving) grids
Lattice Boltzmann: Future Agenda
* Better (non-linear) stability
* Turbomodels (boundary conditions)
* Thermal consistency, Potential energy
* High-Knudsen (challenge true Boltzmann?)
* Moving grids
* Multiscale coupling
Dissipative particle dynamics
LGCA: too stiff
MD: Too expensive
LBE: Grid-Bound
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
DPD thermodynamics
Pressure:
Viscosity:
P   ij 
    
2
ij
DPD applications
•Colloidal suspensions
•Dilute polymers
•Phase separation
•Model membranes
DPD: High-density suspension under shear
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
Phase separation
Prof Coveney’s group
DPD: Amphiphiles
http://www.lce.hut.fi/research/polymer/dpd.shtml
DPD: pros and cons
+ Thermodynamically consistent
+ Flexible (Grid-free)
+ Soft forces allow large dt
- Adaptive versions (Voronoi) are complex
- Theory still in flux (?)
Conclusions and Future Prospects
Strengths:
•Much faster than MD
•Comparable with grid methods
•Highly flexible
•Amenability to parallel computing
Future:
•Multiscale hybrids
•Grid computing