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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, Italy2) CASPUR, Roma, Italy
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Why Particle Methods?
Atomistic physics PDEs with large distorsions (Astrophysics) Moving geometries (Combustion) Moving interfaces (Multiphase flows)
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•Particle-Particle (Molecular dynamics, Monte Carlo)
•Particle-Mesh (Neutral Plasmas, Semiconductors)
•P3M (Gravitational,Charged Plasmas)
•Fluids?
J.Eastwood, R. Hockney: Computer Simulations using particles
Classical Particle Methods
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Particle Methods: pros and cons
Pros:•Geoflexibility (boundary conditions)•Physically Sound•No matrix algebra
Cons:•Noisy•Small timesteps
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New Particle Methods for Fluid Flows
Simple fluids, complex flows:
The Navier-Stokes equations are very hard to solve:
puuuut
Complex fluids, complex flows:
Fluid equations are often NOT KNOWN!
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New Particle Methods for Fluid Flows
Phase – space Fluid (6N D)
Atoms / Molecules
Fluids (3D)
Kinetic Theory (6D)
Idea: Solve fluid equations using fictitious quasi-particle dynamics
Universality: Molecular details do NOT count
Driver: Statistical Physics (front-end) , Numerical Analysis (back-end)
•Lattice Gas Cellular Automata (LGCA)•Lattice Boltzmann (LBE)•Dissipative Particle Dynamics (DPD)
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Details dont count: quasi-particle trajectories
Coarse-Graining via 'Superparticles':
B blocking factor: (Macro to Meso to Micro scale)
1 computational particle = B molecules
BNIxXB
iiI /,1,
1
BNIvVB
iiI /,1,
1
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Coarse-grained equations
J
IJI F
dt
dVM
II V
dt
dX
Modeling goes into FIJ
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Details dont count: kinetic theory
Free stream
Collision
Pre-averaged distributions: Boltzmann approach (Probabilistic)
i
ii tvvtxxtvxf ))(())((),,(
),( ffCfm
Ffvf vt
Modeling goes into f and C(f,f)
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Lattice Gas Cellular Automata
Boolean representation:
n_i=0,1 particle absence/presence
001001
1
23
4
5 6
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Lattice Gas Cellular Automata
0 absence
ni = i = 0,6 1 presence
nCtxntcxn iiii ,1,
:nCi
t t+1 t+1+ε2
1
65
4
3
i
streaming collision
collisions (Frisch, Hasslacher, Pomeau, 1986)
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Boundary condition
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From LGCA to Navier-Stokes
Conservation laws:
(mass) (momentum)
(energy) No details of molecular interactions
(true collision) (lattice collision)
i
iC 0
i
ii cC 0
02/2 ii
icC
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From LGCA to Navier-Stokes
Isotropy (Rotational invariance)
i
dicibiaidcba ccccT .......,,,,,,,
badc
dcdcba uuuuT
3
1,,,,
zyxdcba ,,,,,
such that:
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Von Karman street
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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
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From LGCA to (Lattice) Boltzmann
• (Boolean) molecules to (discrete) distributions ni fi = < ni >
• (Lattice) Boltzmann equations (LBE)
fCtxftcxf iiii ,1,
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M (density)
M (speed)
E (temperature)
P (pressure tensor)
From (Lattice) Boltzmann to Navier - Stokes
vdtvxftx ),,(),ρ(
vdv,t)v,xf(ρ
,t)xu 1
(
vd)uv(
,t)v,xf(ρ
,t)xT(2
1 2
uv
vdvvtvxfP ),,(
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From (Lattice) Boltzmann to Navier - Stokes
Weak Departure from local equilibrium
neqeq fff
1eq
neq
f
fKn
f
u v
neqf
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From (Lattice) Boltzmann to Navier - Stokes
0 uρdivρt
uλdivuμdivpuuρdivuρt
TKuPTuρdivρTt :
LBE
M
M
E
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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
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LATTICE BGK
Since Re depends only on , single time relaxation only
Viscosity
(lattice sound speed)
Qian, d’Humières, Lallemand, 1992
eqiiiii ff
τ,txf,tcxf
11
212 τcν s
3
12 sc
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LBE assets:Noise-free, high ReynoldsFlexible Boundary ConditionsEfficient on serial, even more on parallelPoisson-freedomAdditional physics (beyond fluids)Quick grid set up (EXA-Powerflow)
LBE liabilitiesLater …
Who needs LBE?
DON’T USE: Strong heat transfer, compressibility (combustion) CAN USE: Turbulence in simple geosSHOULD USE: Porous mediaMUST USE: Multiphase, Colloidal, External Aerodynamics
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Parallel Speed-up
Amati, Massaioli, Bernaschi, Scicomp 2002
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LBE
t=0 t=5000
SP
t=20000
Ansumali et al, ETHZ+IAC
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Turbulent channel
APE-100: 10 Gflops sustained(Amati , Benzi, Piva, Succi, PRL 99)
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Porous media: random fiber networks
A.Hoekstra,P Sloot, A.Koponen, J Timonen, PRL 2001
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Cristal Growth
Miller, Succi, Mansutti, PRL 1999
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LBE-Multiphase, Demixing flow: Amati, Gonnella, Lamura, Massaioli
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LBE: MultiphaseB. Palmer, D. Rector, pnl.gov
http://gallery.pnl.gov/mscf/bubble_web1/bubble_web.mpg
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Local grid refinement
Different time scales and no. of time steps for different refinement levels, interpolation between levels
Succi, Filippova, Smith, Kaxiras 2001,
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LBE: Airfoils
Succi,Filippova,Kaxiras, Cise 2001
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You can do something like this…
Bella, Ubertini, 2001
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LBE: Car design
Powerflow, EXA
H Chen, S Kandasamy, R Shock, S. Orszag, S. Succi, V. Yakhot, Science (2003)
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LBE: Reactive microflows
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LBE: Multiscale microflows
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Unstructured LBE
Ubertini,Succi,Bella, 2003
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LBE: Unstructured (soon moving) grids
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Lattice Boltzmann: Future Agenda
* Better (non-linear) stability
* Turbomodels (boundary conditions)
* Thermal consistency, Potential energy
* High-Knudsen (challenge true Boltzmann?)
* Moving grids
* Multiscale coupling
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LGCA: too stiff
MD: Too expensive
LBE: Grid-Bound
Dissipative particle dynamics
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
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Pressure:
Viscosity:
ijP
2ij
DPD thermodynamics
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DPD applications
•Colloidal suspensions•Dilute polymers•Phase separation•Model membranes
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DPD: High-density suspension under shear
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
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Phase separation
Prof Coveney’s group
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DPD: Amphiphiles
http://www.lce.hut.fi/research/polymer/dpd.shtml
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DPD: pros and cons
+ Thermodynamically consistent
+ Flexible (Grid-free)
+ Soft forces allow large dt
- Adaptive versions (Voronoi) are complex
- Theory still in flux (?)
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Conclusions and Future Prospects
Strengths:
•Much faster than MD•Comparable with grid methods•Highly flexible•Amenability to parallel computing
Future:
•Multiscale hybrids•Grid computing