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

Why Particle Methods?

Atomistic physics PDEs with large distorsions (Astrophysics) Moving geometries (Combustion) Moving interfaces (Multiphase flows)

•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

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:

puuuut

Complex fluids, complex flows:

Fluid equations are often NOT KNOWN!

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)

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

Coarse-grained equations

J

IJI F

dt

dVM

II V

dt

dX

Modeling goes into FIJ

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)

Lattice Gas Cellular Automata

Boolean representation:

n_i=0,1 particle absence/presence

001001

1

23

4

5 6

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)

Boundary condition

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

From LGCA to Navier-Stokes

Isotropy (Rotational invariance)

i

dicibiaidcba ccccT .......,,,,,,,

badc

dcdcba uuuuT

3

1,,,,

zyxdcba ,,,,,

such that:

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 = < ni >

• (Lattice) Boltzmann equations (LBE)

fCtxftcxf iiii ,1,

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 ),,(

From (Lattice) Boltzmann to Navier - Stokes

Weak Departure from local equilibrium

neqeq fff

1eq

neq

f

fKn

f

u v

neqf

From (Lattice) Boltzmann to Navier - Stokes

0 uρdivρt

uλdivuμdivpuuρdivuρt

TKuPTuρdivρTt :

LBE

M

M

E

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

Viscosity

(lattice sound speed)

Qian, d’Humières, Lallemand, 1992

eqiiiii ff

τ,txf,tcxf

11

212 τcν s

3

12 sc

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

Parallel Speed-up

Amati, Massaioli, Bernaschi, Scicomp 2002

LBE

t=0 t=5000

SP

t=20000

Ansumali et al, ETHZ+IAC

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: MultiphaseB. 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

Powerflow, EXA

H Chen, S Kandasamy, R Shock, S. Orszag, S. Succi, V. Yakhot, Science (2003)

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

LGCA: too stiff

MD: Too expensive

LBE: Grid-Bound

Dissipative particle dynamics

http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html

Pressure:

Viscosity:

ijP

2ij

DPD thermodynamics

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