Modern Particle Methods for Complex Flows G. Amati (2), F. Castiglione (1), F. Massaioli(2), S....
-
Upload
oliver-parrish -
Category
Documents
-
view
219 -
download
4
Transcript of Modern Particle Methods for Complex Flows G. Amati (2), F. Castiglione (1), F. Massaioli(2), S....
![Page 1: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/1.jpg)
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
![Page 2: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/2.jpg)
Why Particle Methods?
Atomistic physics PDEs with large distorsions (Astrophysics) Moving geometries (Combustion) Moving interfaces (Multiphase flows)
![Page 3: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/3.jpg)
•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
![Page 4: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/4.jpg)
Particle Methods: pros and cons
Pros:•Geoflexibility (boundary conditions)•Physically Sound•No matrix algebra
Cons:•Noisy•Small timesteps
![Page 5: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/5.jpg)
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!
![Page 6: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/6.jpg)
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)
![Page 7: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/7.jpg)
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
![Page 8: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/8.jpg)
Coarse-grained equations
J
IJI F
dt
dVM
II V
dt
dX
Modeling goes into FIJ
![Page 9: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/9.jpg)
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)
![Page 10: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/10.jpg)
Lattice Gas Cellular Automata
Boolean representation:
n_i=0,1 particle absence/presence
001001
1
23
4
5 6
![Page 11: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/11.jpg)
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)
![Page 12: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/12.jpg)
Boundary condition
![Page 13: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/13.jpg)
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
![Page 14: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/14.jpg)
From LGCA to Navier-Stokes
Isotropy (Rotational invariance)
i
dicibiaidcba ccccT .......,,,,,,,
badc
dcdcba uuuuT
3
1,,,,
zyxdcba ,,,,,
such that:
![Page 15: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/15.jpg)
Von Karman street
![Page 16: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/16.jpg)
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
![Page 17: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/17.jpg)
From LGCA to (Lattice) Boltzmann
• (Boolean) molecules to (discrete) distributions ni fi = < ni >
• (Lattice) Boltzmann equations (LBE)
fCtxftcxf iiii ,1,
![Page 18: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/18.jpg)
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 ),,(
![Page 19: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/19.jpg)
From (Lattice) Boltzmann to Navier - Stokes
Weak Departure from local equilibrium
neqeq fff
1eq
neq
f
fKn
f
u v
neqf
![Page 20: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/20.jpg)
From (Lattice) Boltzmann to Navier - Stokes
0 uρdivρt
uλdivuμdivpuuρdivuρt
TKuPTuρdivρTt :
LBE
M
M
E
![Page 21: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/21.jpg)
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
![Page 22: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/22.jpg)
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
![Page 23: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/23.jpg)
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
![Page 24: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/24.jpg)
Parallel Speed-up
Amati, Massaioli, Bernaschi, Scicomp 2002
![Page 25: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/25.jpg)
LBE
t=0 t=5000
SP
t=20000
Ansumali et al, ETHZ+IAC
![Page 26: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/26.jpg)
Turbulent channel
APE-100: 10 Gflops sustained(Amati , Benzi, Piva, Succi, PRL 99)
![Page 27: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/27.jpg)
![Page 28: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/28.jpg)
Porous media: random fiber networks
A.Hoekstra,P Sloot, A.Koponen, J Timonen, PRL 2001
![Page 29: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/29.jpg)
Cristal Growth
Miller, Succi, Mansutti, PRL 1999
![Page 30: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/30.jpg)
LBE-Multiphase, Demixing flow: Amati, Gonnella, Lamura, Massaioli
![Page 31: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/31.jpg)
LBE: MultiphaseB. Palmer, D. Rector, pnl.gov
http://gallery.pnl.gov/mscf/bubble_web1/bubble_web.mpg
![Page 32: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/32.jpg)
Local grid refinement
Different time scales and no. of time steps for different refinement levels, interpolation between levels
Succi, Filippova, Smith, Kaxiras 2001,
![Page 33: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/33.jpg)
LBE: Airfoils
Succi,Filippova,Kaxiras, Cise 2001
![Page 34: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/34.jpg)
You can do something like this…
Bella, Ubertini, 2001
![Page 35: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/35.jpg)
LBE: Car design
Powerflow, EXA
H Chen, S Kandasamy, R Shock, S. Orszag, S. Succi, V. Yakhot, Science (2003)
![Page 36: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/36.jpg)
LBE: Reactive microflows
![Page 37: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/37.jpg)
LBE: Multiscale microflows
![Page 38: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/38.jpg)
Unstructured LBE
Ubertini,Succi,Bella, 2003
![Page 39: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/39.jpg)
Unstructured LBE
![Page 40: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/40.jpg)
LBE: Unstructured (soon moving) grids
![Page 41: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/41.jpg)
Lattice Boltzmann: Future Agenda
* Better (non-linear) stability
* Turbomodels (boundary conditions)
* Thermal consistency, Potential energy
* High-Knudsen (challenge true Boltzmann?)
* Moving grids
* Multiscale coupling
![Page 42: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/42.jpg)
LGCA: too stiff
MD: Too expensive
LBE: Grid-Bound
Dissipative particle dynamics
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
![Page 43: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/43.jpg)
Pressure:
Viscosity:
ijP
2ij
DPD thermodynamics
![Page 44: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/44.jpg)
DPD applications
•Colloidal suspensions•Dilute polymers•Phase separation•Model membranes
![Page 45: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/45.jpg)
DPD: High-density suspension under shear
http://www.bfrl.nist.gov/861/vcttl/talks/talkG/sdl001.html
![Page 46: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/46.jpg)
Phase separation
Prof Coveney’s group
![Page 47: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/47.jpg)
DPD: Amphiphiles
http://www.lce.hut.fi/research/polymer/dpd.shtml
![Page 48: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/48.jpg)
DPD: pros and cons
+ Thermodynamically consistent
+ Flexible (Grid-free)
+ Soft forces allow large dt
- Adaptive versions (Voronoi) are complex
- Theory still in flux (?)
![Page 49: 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),](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649e595503460f94b52a07/html5/thumbnails/49.jpg)
Conclusions and Future Prospects
Strengths:
•Much faster than MD•Comparable with grid methods•Highly flexible•Amenability to parallel computing
Future:
•Multiscale hybrids•Grid computing