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TRANSCRIPT
8 June 2007
Gridless DSMC
Spencer Olson
University of MichiganNaval Research LaboratoryNow at: Air Force Research Laboratory
30 June 2009
Collaborator: Andrew Christlieb, Michigan State University
J Comp Phys, Vol 227, pp 8035-8064, 2008.
Gas Dynamics Simulation Approaches
Fig. 1.1, G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 1994.
: Local Knudsen number
: Mean free path
: Local Characteristic length
Dense
Rarefied
Free-ParticleLimit
InviscidLimit
Discrete ParticleModel
Continuous Solution?
: N-Body phase-space density
For cm/s and ~10µ m X 10µ m X 1mm resolution, need ~>> 1 TB of memory!
(and a lifetime to compute the collision integral)
What about a numerical solution of the Boltzmann equation?
: Collision integral
Direct Simulation of gas dynamics using Monte Carlo
Grid
Ansatz: Collisional processes separable from particle motion.
Collide Transport
Grid Mismatch: Coarse Grid
Grid
Captures only low-density dynamics well Allows non-probable collisions
Density variations could be due to:
Shockwaves Nonfree particle systems Transient processes
Grid Mismatch: Fine Grid
Grid Density variations could be due to:
Shockwaves Nonfree particle systems Transient processes
Captures only high-density dynamics well Wastes memory resources Thermally isolates particles that should be allowed to collide
Particle Method of Choice: DSMC
Direct Simulation of gas dynamics using Monte Carlo (DSMC)
Algorithm:
CollideSort (gridless)Update/Sample macroscopic gas quantities
Move (integrate)
Apply Boundary Conditions (reflections, absorption,etc.)
Binary Sort/Search Algorithm
B
A C
D
F
E
J
I K
L
N
M O
H
G
Execution time:Sort:Search: (averaged over all trees)
Binary Sort/Search Algorithm
B
A C
D
F
E G
J
I K
L
N
M O
H
Execution time:Sort:Search: (averaged over all trees)
Quadtree Sorting: Divisions about Geometric Center
Divide by geometric centerDisallow children with too few particlesAvoid bad aspect ratioA
H
GF
CB
A
HGF
CB
I
D E
Quadtree Sorting: Divisions about Geometric Center
Divide by geometric centerDisallow children with too few particlesAvoid bad aspect ratioA
H
GF
CB
A
HGF
CB
I
D E
Quadtree Sorting: Divisions about Geometric Center
Divide by geometric centerDisallow children with too few particlesAvoid bad aspect ratioA
H
GF
CB
A
HGF
CB
I
D E
Quadtree Sorting: Divisions about Center of Mass
A
H
JF
CB
L
A
HGF
CB
I
D E
LKJ M
Divide by center of massDisallow children with too few particlesAvoid bad aspect ratioShrink to avoid large empty space
Quadtree Sorting
Divisions aboutGeometric Center
Divisions about Center of Mass
Minimal Bounded VolumeLarge Empty Space Avoided
DSMC Validity Condition
F. J. Alexander, A.~L. Garcia, B. J. Alder, “Cell size dependence of transport coefficients in stochastic particle algorithms,” Phys. Fluids, 10:1540-1542, 1998.
: Mean collision separation
: Scale size of nodes
DSMC Validity Condition
F. J. Alexander, A.~L. Garcia, B. J. Alder, “Cell size dependence of transport coefficients in stochastic particle algorithms,” Phys. Fluids, 10:1540-1542, 1998.
Validity Metric
Getting the Collision Rate Correct (Grid Based)
• Time counting methods:In each cycle, evaluate random collision pairs until desired collision rate is met.–Collision probability and spacing mismatched in time.–Not vectorizable (multi-processor code not easy)
• G. A. Birds “NTC” method (1989)–Calculate number of tested collisions ahead of time (no time mismatch problems).–Vectorizable.
Getting the Collision Rate Correct (Tree Based)
#Select =
Grid
“~” --> time averaged gridless data
Gridless
Move Particles
Solve Using favorite integrator.
Free particles:
RK2, leap-frog, etc.
Non-free particles:
• Adaptive RK5 (embedded RK4 for error detection)
• RK4 (with small time step)
• etc.
Apply Boundary Conditions
Each boundary segment treated as an individual object with its own interaction logic
1. Scale boundary boxes
2. Test boundary boxes for overlap with tree nodes
3. Allow each overlapped boundary to interact with tree node particles
Test Cases
Couette flowVelocity diffusionThermal diffusion
Very low velocity flow past thin plate
Hypersonic flowSquare cylinderBiconic cylinder (2D) (static shock-shock interaction)
Evaporative cooling in ultracold gas
34
Hypersonic Flow
Mach 10Gas: Argon Compared to results from G. A. Bird's DS2V program
Case 1: Square Cylinder
Case 2: Biconic Cylinder (2D)
35
Hypersonic Flow: Square Cylinder
Mach 10Gas: Argon Compared to results from G. A. Bird's DS2V program
Case 1: Square Cylinder
Case 2: Biconic Cylinder (2D) 0.5
0.5
8 m
3 mv
Computational domain
36
Hypersonic Flow: Square CylinderG
ridle
ss D
SM
C
DS
2V D
SM
C
Temperature
Mach Number
Number Density
DSMC Validity Condition: Square Cylinder
Validity condition met:
Uniform validity throughout.
Validity condition met:
Non-uniform validity metric.Cell size might possibly be made too small—causes thermal isolation.
Gridless DSMC DS2V DSMC0.08
0.06
0.04
0.02
0.00
38
Hypersonic Flow: Biconic Cylinder (2D)
Mach 10Gas: Argon Compared to results from G. A. Bird's DS2V program
Case 1: Square Cylinder
Case 2: Biconic Cylinder (2D)
25 cm
20 cm
v
J. N. Moss, G. A. Bird, and G. N. Markelov, “DSMC Simulations of Hypersonic Flows and Comparison With Experiments”, Rarefied Gas Dynamics: 24th Intntl. Sym. on Rare. Gas Dyn, 2005.
Computational domain
39
DSMC Validity Condition: Biconic Cylinder (2D)
Validity condition NOT met:
Non-uniform validity metric—not enough particles
Validity condition partially met:
Non-uniform validity metric.Cell size may be too small—causes thermal isolation.
Gridless DSMC
DS2V DSMC0.4
0.3
0.2
0.1
0.00
0.4
0.3
0.2
0.1
0.00
40
DSMC Validity Condition: Biconic Cylinder (2D)
Validity condition met:
N ~ X13 N0
Validity condition met:
Non-uniform validity metric.Cell size may still be too small—causes thermal isolation.
N ~ X13 N0
Gridless DSMC
DS2V DSMC0.4
0.3
0.2
0.1
0.00
41
Future Work
In the middle of a major code-rewriteFaster:
Sort timeTree traversal time
Handles multiple particle typesArbitrary collision equation sets (only binary input now)
Integrate with multipole-expansion treecode field solversSimulate plasma+neutral shock-shock interactions
Continue boundary code development