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.

Outline

BackgroundAlgorithm OverviewTest Cases

Low velocity flowHypersonic flow

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

Direct Simulation of gas dynamics using Monte Carlo

Grid

Truncate 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.)

Simple Gridless Case: Binary Sort/Search Algorithm

NOHFMJ CEGIDKBAL

Binary Sort/Search Algorithm

B

A C

D

F

E

J

I K

L

N

M O

H

G

NOHFMJ CEGIDKBA

Binary Sort/Search Algorithm

B

A C

D

F

E

J

I K

L

N

M O

H

G

CEGIDKBA

Binary Sort/Search Algorithm

B

A C

D

F

E

J

I K

L

N

M O

H

G

CEG

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

Quadtree Example

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 (Grid Based)

Grid

#Select =

G. A. Birds “NTC” method (1989)

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

Example Boundary (collisional)

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

32

Flow between two infinite parallel plates.

Case 1:

Case 2:

Couette Flow

33Gridded DSMC data provided by Dr. Quanhua Sun

Couette Flow: v2 – v

1 = 300m/s

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