simulating the swimming of microorganisms towards swarming · simulating the swimming of...

Simulating the Swimming ofMicroorganisms towards Swarming

K. Pickl a,b J. Pande b,c H. Köstler a A.-S. Smithb,c U. Rüdea,b

DSFD 2014, Paris, FranceaLehrstuhl für Informatik 10 (Systemsimulation), FAU Erlangen-NürnbergbCluster of Excellence: Engineering of Advanced Materials, FAU Erlangen-NürnbergcInstitut für Theoretische Physik I, FAU Erlangen-Nürnberg

Flow Regimes

∗all images taken from www.wikipedia.com

DSFD 2014 | kristina.pickl@fau.de | FAU Erlangen-Nürnberg | Simulating the Swimming of Microorganisms towards Swarming 2

Flow at Low Reynolds Number: Purcell’s Scallop Theorem∗

Stokes flow

• domination of viscous forces• small momentum• always laminar• time reversible• no coasting⇒ we need asymmetric, non-time

reversible motion to achieve anynet movement

∗E.M. Purcell. Life at low Reynolds number. American Journal of Physics 45: 3-11 (1977)

Overall Goal: Simulation of a Swarm

Characteristics of a Swarm

• large-scale collective hydrodynamics• complex long-time dynamics• pattern formation

⇒ we want to study these effects⇒ compare analytical calculations with simulations⇒ not only of a single swimmer but many of them

Ingredients for a Simulation of a Swarm

“Physics” Ingredients

• model of a swimmer• non-time reversible cycling strategy

“Software” Ingredients

• fluid and rigid body simulation tool• coupling both tools consistently• allow for large scale computations

% model of a swimmer• non-time reversible cycling strategy

Physical Model of a Swimmer

• we choose the simplest possible design:Golestanian’s* swimmer

• connections between the objects:• linear spring-damper systems†

• angular spring-damper systems‡

∗A. Najafi and R. Golestanian. Simple swimmer at low Reynolds number: Three linked spheres. Phys. Rev. E, 69(6):062901 (2004)†K. Pickl et al. All good things come in threes – three beads learn to swim with lattice Boltzmann and a rigid body solver. JoCS 3(5):374 – 387 (2012)‡K. Pickl et al. Parallel Simulations of Self-propelled Microorganisms. Advances in Parallel Computing, Vol. 25 (2014)DSFD 2014 | kristina.pickl@fau.de | FAU Erlangen-Nürnberg | Simulating the Swimming of Microorganisms towards Swarming 7

! model of a swimmer% non-time reversible cycling strategy

Non-time Reversible Cycling Strategy

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000Time step

Force on body 2

Force on body 1

Force on body 3

• total applied force vanishes over one cycle (displacement ofswimmer over one cycle is zero in absence of fluid)

• applied along specified main axis of swimmer on center of massof each body (in this case: x-direction)

• net driving force acting on system at each instant of time is zero

! model of a swimmer! non-time reversible cycling strategy

% fluid and rigid body simulation tool• coupling both tools consistently• allow for large scale computations

Software

Fluid Simulation – WALBERLA

(widely applicable Lattice Boltzmann solver from Erlangen)

• suited for various flow applications• different fluid models (SRT, TRT∗, MRT)• suitable for homo- and heterogeneous architectures• large-scale, MPI-based parallelization

Rigid Body Simulation – pe• based on Newton’s mechanics• fully resolved objects (sphere, box, . . . )• connections between objects can be soft or hard constraints• accurate handling of friction during collision†

• large-scale, MPI-based parallelization

∗I. Ginzburg et al. Two-relaxation-time lattice Boltzmann scheme: About parametrization, . . . . Comm. in Computational Physics, 3(2):427–478, (2008)†P. A. Cundall and O. D. L. Strack. A discrete numerical model for granular assemblies. Geotechnique, 29:47–65, (1979)DSFD 2014 | kristina.pickl@fau.de | FAU Erlangen-Nürnberg | Simulating the Swimming of Microorganisms towards Swarming 11

! fluid and rigid body simulation tool% coupling both tools consistently• allow for large scale computations

Coupling both Frameworks: Four-Way Coupling

1. Object Mapping2. LBM Communication3. Boundary Handling

(including Hydrodynamic Forces)

4. Stream Collide5. Lubrication Correction

6. Physics Engine

! fluid and rigid body simulation tool! coupling both tools consistently% allow for large scale computations

Allow for Large Scale Computations

Parallel Discrete Element Method (DEM)∗

• handling of pair-wise spring-like interactions, extending not only overneighboring but also over multiple process domains

• for long-range interactions: only associated processes communicate

∗K. Pickl et al. Parallel Simulations of Self-propelled Microorganisms. Advances in Parallel Computing, Vol. 25 (2014)∗M. Hofmann. Parallelisation of Swimmer Models for Swarms of Bacteria in the Physics Engine pe. Master’s thesis, LSS, FAU Erlangen-Nürnberg (2013)

Allow for Large Scale ComputationsWeak Scaling Results on JUQUEEN∗

largest simulated setup:

131,072 cores

16,777,216 swimmers!

not displayed: Setup, Swimmer Setup and Lubrication Correction∗K. Pickl et al. Parallel Simulations of Self-propelled Microorganisms. Advances in Parallel Computing, Vol. 25 (2014)

! fluid and rigid body simulation tool! coupling both tools consistently! allow for large scale computations

We have all necessary ingredients forthe simulation of a swarm!

! fluid and rigid body simulation tool! coupling both tools consistently! allow for large scale computations

We have all necessary ingredients forthe simulation of a swarm!

Initial Configuration of the System

• characteristics of the fluid simulationviscosity 73.6 · 10−6 m 2/sresolution dx 1.0 · 10−6mrelaxation time 1.5

• characteristics of the channeldimensions 400× 200× 200 lattice cellsswimming direction along x-axisall boundaries free slip

• characteristics of the external forcespulse length 4692 time stepsphase shift π/2

• geometric characteristics of the swimmer

radius of spheres 4 lattice cellsmass of spheres 400 on the latticerest length 16 lattice cells

Oscillations of the Arms

0 4000 8000 12000 16000 20000 24000 28000 32000Time step

s]Leading Arm

Trailing Arm

⇒ leading arm is the dominating arm in terms of collisions⇒ system is in a steady state after 5 cycles

Results of the Initial System

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

Amplitude [10-5N]

amplitude 1.0 · 10−5 N

swimming velocity 0.515 · 10−4

distance in 1 cycle 0.25RE swimmer 0.00618

amplitude 2.6 · 10−5 N

*all other quantities given on the lattice

⇒ explore bounds of low RE⇒ maximize swimming velocity

Tuning the Swimmer Speed by Changing its Geometry

radius of spheres 4 lattice cellsmass of spheres 400 on the latticerest length 16

• resulting configurations

radiusrest length

channel dimensions(4∗radius)

4 16 400× 200× 2006 24 420× 204× 2048 32 440× 208× 208

radius of spheres 4 lattice cellsmass of spheres 400 on the latticerest length 16

• resulting configurations

radiusrest length

channel dimensions(4∗radius)

4 16 400× 200× 2006 24 420× 204× 2048 32 440× 208× 208

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

Amplitude [10-5N]

Rest length 16, R4, tau 1.5

radius 4, rest length 16

max. amplitude 2.6 · 10−5 Nswimming velocity 3.176 · 10−4

So far...

• we have a distance gain from 0.25 to 2.33 lattice cells per cycle• we have a velocity gain from 0.515 · 10−4 to 4.734 · 10−4

• and have reached a RE of 0.11363 (compared to 0.00618)

Can we go any faster?• rest length 32 lattice cells and radius 8 lattice cells→ swimmer x-size of 80 lattice cells⇒ we do not want to enlarge this any further!

• rest length 32 lattice cells and radius 6 lattice cells→ still sufficiently resolved→ allows for more oscillations of the arms

So far...

Can we go any faster?

• rest length 32 lattice cells and radius 8 lattice cells→ swimmer x-size of 80 lattice cells⇒ we do not want to enlarge this any further!

So far...

Maximizing the Oscillations of the Arms

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

Amplitude [10-5N]

Rest length 32, R8, tau 1.5radius 8, rest length 32

Conclusions of the Geometry Study

• we have a final distance gain from 0.25 to 3.81 lattice cells per cycle• we have a final velocity gain from 5.15 · 10−4 to 7.731 · 10−4

• and have eventually reached a RE of 0.17627 (compared to 0.00618)

Changing the Viscosity of the Fluid

• characteristics of the fluid simulationviscosity 73.6 · 10−5 m 2/sresolution dx 10−6mrelaxation time 10

⇒ compare with our initial system⇒ with higher viscosity theory predicts

slower swimmer in this regime

Changing the Viscosity of the Fluid

• characteristics of the fluid simulationviscosity 73.6 · 10−5 m 2/sresolution dx 10−6mrelaxation time 10

⇒ compare with our initial system⇒ with higher viscosity theory predicts

slower swimmer in this regime

Oscillations of the two Arms

0 15000 30000 45000 60000 75000 90000 105000 120000Time step

s]Leading Arm

Trailing Arm

⇒ leading arm is still the dominating arm in terms of collisions⇒ system is not in a steady state after 5 cycles but after 10 cycles⇒ after switching off the external forces, it takes longer for the springs to relaxDSFD 2014 | kristina.pickl@fau.de | FAU Erlangen-Nürnberg | Simulating the Swimming of Microorganisms towards Swarming 27

Comparing Viscosities

at amplitude 1.0 · 10−5 N:

viscosity 73.6 · 10−6 m 2/s

⇒ quantitative agreement isobtained at low RE and smalloscillations, where actually isthe limit of the theory

1.0 2.0 3.0 4.0

Amplitude [10-5N]

Simulation: nu = 73.6.10-6 m 2/s

Theory: nu = 73.6.10-6 m 2/s

Theory: nu = 73.6.10-5 m 2/s

1.0 2.0 3.0 4.0

Amplitude [10-5N]

Theory: nu = 73.6.10-6 m 2/s

Theory: nu = 73.6.10-5 m 2/s

1.0 2.0 3.0 4.0

Amplitude [10-5N]

Theory: nu = 73.6.10-6 m 2/s

Theory: nu = 73.6.10-5 m 2/s

Conclusions and Future Work

Conclusions• successfully achieved a higher swimming

velocity by changing the swimmer geometry• obtained quantitative agreement of the

viscosity dependence within one regime

• demonstrate flexibility of our framework byseveral parameter studies

Future Work• static grid refinement→ reflect infinite domain as good as possible

• analyze collective behavior of swimmerssystematically

• improvement of parallel I/O and associateddata analysis

*images courtesy of F. Schornbaum, E. Fattahi: “A Study of the Vocal Fold”

Thank you for your attention!Extract from the References• K. Pickl et al. Parallel Simulations of Self-propelled Microorganisms. Advances in Parallel Computing,

Vol. 25 (2014)• K. Pickl et al. All good things come in threes – three beads learn to swim with lattice Boltzmann and a

rigid body solver. Journal of Computational Science, 3(5):374 – 387, 2012.• C. Godenschwager et al. A Framework for Hybrid Parallel Flow Simulations with a Trillion Cells in

Complex Geometries. Proceedings of SC13: International Conference for High PerformanceComputing, Networking, Storage and Analysis. p. 35-1 – 35-12.

• A. Najafi and R. Golestanian. Simple swimmer at low Reynolds number: Three linked spheres. Phys.Rev. E, 69(6):062901, 2004.

• C. M. Pooley et al. Hydrodynamic interaction between two swimmers at low Reynolds number. Phys.Rev. Lett., 99:228103, 2007.

• D. Saintillan and M. J. Shelley. Instabilities and Pattern Formation in Active Particle Suspensions:Kinetic Theory and Continuum Simulations. Phys. Rev. Lett., 100:178103, 2008.

Acknowledgments

simulating the swimming of microorganisms towards swarming · simulating the swimming of...

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