simulation of self-propelled micro-devices with a … · simulation of self-propelled micro-devices...

Simulation of Self-propelled Micro-devices

Kristina Pickl — Chair for System Simulation — /21

Simulation of Self-propelled Micro-devices with aLattice Boltzmann Solver and a Rigid Body

Physics Engine

Kristina Pickl1,2, Jan Gotz1, Klaus Iglberger3, Jayant Pande2,4,Ana-Suncana Smith2,4, Ulrich Rude1,2,3

1Chair for System Simulation, University Erlangen-Nurnberg

2Cluster of Excellence: Engineering of Advanced Materials, University Erlangen-Nurnberg

3Zentralinstitut fur Scientific Computing, University Erlangen-Nurnberg

4Institute for Theoretical Physics I, University Erlangen-Nurnberg

July, 7th 2011



Numerical Method



Flow Regimes

Reynolds Numbers

104

109

102

10-4

Re

∗all images taken from www.wikipedia.com

www.wikipedia.com



Flow at Low Reynolds Numbers

Purcell’s Scallop Theorem∗

t2

t1

t

xx1

x2

Stokes flow

domination of viscous forces

small momentum

always laminar

time reversible

no coasting

⇒ we need asymmetric, non-timereversible motion to achieve anynet movement

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



Software

framework for physically accurate and virtualreality multibody simulations

based on Newton’s mechanics

fully resolved objects (spheres, boxes, . . . )

connections between objects can be soft(e.g. springs) or hard constraints (e.g. hinges)

accurate handling of friction during collision(Hertzian contact mechanics)

widely applicable Lattice Boltzmann solverfrom Erlangen

framework for physically correct fluidsimulations

suited for various flow applications

different fluid models (SRT, MRT, . . . )

suitable for homo- and heterogeneousarchitectures

⇒ coupled framework with large-scale, MPI-based parallelization



Coupling both Frameworks for Swimmers

1: //Mapping step2: for each swimmer B do3: Map B to lattice grid4: end for5:

6: //Perform LBM7: for each lattice cell do8: Stream and collide9: end for

10:

11: //Determine hydrodynamic forces12: for each surface cell do13: Add forces Fhydro from fluid to rigid objects14: end for15:

16: //Apply driving forces17: for each body of the swimmers do18: Add driving forces Fdri

19: end for

20: // Time step in the pe rigid body simulation21: for all rigid bodies do22:

23: //Collision detection24: for each rigid body B do25: Detect all contacts k26: Add them to set of constraints c27: end for28:

29: //Constraint resolution30: for each constraint c do31: Determine acting constraint forces Fc

32: end for33:

34: //Time integration35: for each rigid body B do36: Apply forces Ftot = Fhydro + Fdri + Fc:37: update position and velocity38: end for39:

40: end for



Modeling of a Swimmer

we choose the simplest possible design: Golestanian’s∗ swimmer

studied extensively in three spheres geometry by analytical methods

physical setup:

connections between the objects(e.g. stiff rods, springs)cycling strategy responsible for characteristic movement

(e.g. imposing known velocities)

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



Connections

Damped Harmonic Oscillators

two bodies Bi and Bj connected by a spring Sn:

~FBiosci = −k ∆~xSn − γ∆~uSn

~FBj

osci = k ∆~xSn + γ∆~uSn

standard way to classify a damped harmonic oscillator:

damping ratio D =γ

2√

m k

⇒ choose parameters of spring such that underdamped regime, i.e. D � 1(oscillate with re-normalizing frequency decreasing amplitude to 0 until equilibrium is reached)

k force constant ~xBi ,~xBj current positions

γ damping parameter ~uBi ,~uBj current velocities

m mass of each body ∆~xSn =~l0 − (~xBi −~xBj ) current deformation of spring

l0 restlength of the springs ∆~uSn = ~uBi −~uBj current difference in velocity



Driving Forces

8 12 16 20 24 28 32Time step [10 3s]

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

Forc

e (z

-com

pone

nt)

[10

-6 N

]

Force on Body 2Force on Body 1Force on Body 3

“~FB2

dri

”z

= −a · sin (ωt) = −a · sin`

2πtT

´“~FB3

dri

”z

= a · sin“

2π(t+ϕ)T

”“~FB1

dri

”z

= −““~FB2

dri

”z

+“~FB3

dri

”z

”a amplitude T oscillation period ω driving frequency ϕ phase shift (const. at T/4)

non-time reversible

total applied force should vanish over one cycle↔ displacement of swimmer over one cycle should be zero in absence of fluid

applied only along main axis of swimmer on centre of mass of each body

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



Cycling Strategy

Theoretical Positions Position Plot from the Simulation

lmin minimal extended armlength l0 restlength of the springs ∆ covered distance of the swimmerlmax maximal extended armlength l1, l2 current armlengths of the springs after one swimming cycle



Appropriate Reynolds Number Regime

general definition:

Re =~u L

ν� 1

with respect to whole swimming device:

Reswim =~u swim L swim

ν� 1

with respect to constituent bodies:

ReBi =~uBi LBi

ν� 1

ν kinematic viscosity of the fluid

~u characteristic maximum velocity

~u swim velocity of the swimmer

(~u swim = ∆/T ) of the last cycle

~uBi maximum velocity of constituent bodies

L characteristic length-scale

L swim rest length in direction of motion

LBi diameter of the relevant body

in direction of motion



Results



Vacuum Validation

Three-sphere Swimmer

Lagrangian of a non-dissipative assembly

L = m~x2

1 +~x22 +~x2

32

− k (~x1−~x2−~l0)2+(~x3−~x1−~l0)2

2+ ~x1

~FB1dri + ~x2

~FB2dri + ~x3

~FB3dri

equation of motion for each sphereddt

∂L

∂~xi= ∂L

∂~xi+ ~F

Bidamp for i = 1, 2, 3

0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time step [10 4 s]

20

25

30

35

40

45

50

55

60

65

70

75

80

sphe

re 2

sp

here

1

sphe

re 3

pe simulationAnalyticalcalculation

t 2 t 3

z-po

sitio

n

t 1

196812 total time steps

t1 = 140580 time steps

= time step at which force on left body

is switched off

t2 = 147609 time steps

= 3/4 of total time steps

= time step at which force on right body

(and hence middle body) is switched off

t3 ≈ 179000 time steps

= time step when springs roughly stop

relaxing



Vacuum Validation

Three-sphere Swimmer

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0ω0/ω

0

0.5

1.0

1.5

2.0

2.5

3.0

A/a

[10

5 ]

Analytical calculationpe simulation

ω0 =p

k/m

A amplitude of oscillation



Simulations in Fluid

totalDesign swimming appropriate Reynolds numbers Re

distance ∆tot ReB2 ReB1 ReB3 Reswim

0.281 0.080 0.061 0.083 2.33e-03

0.638 0.119 0.096 0.136 5.51e-03

0.651 0.051 0.041 0.053 3.53e-03

0.678 0.074 0.041 0.054 3.64e-03

0.755 0.143 0.047 0.130 5.72e-03

0.856 0.045 0.063 0.078 4.54e-03

0.887 0.078 0.058 0.050 4.64e-03

0.907 0.062 0.041 0.085 4.86e-03

0.909 0.053 0.123 0.081 5.75e-03

0.937 0.074 0.060 0.121 6.14e-03

0.978 0.128 0.047 0.130 6.32e-03

1.187 0.066 0.059 0.078 6.51e-03

narrow channel (x,y,z) = (100, 100, 200)lattice cells,

dx on the lattice: 10−6,

total time steps 196812,

kinematic viscosity 73.6 · 10−6m2/s,

microswimmer:

k = 1.72965 kg/s2,

γ = 1.57237 · 10−7 kg/s,

a = 0.10 · 10−4 kgm/s2,

ω = 296057.86703 1/s,

number of pulses: 5,

with pulse length 28116 time steps,

spheres: m = 5.44 · 10−13 kg ,

r = 4 · 10−6m,large spheres: m = 5.44 · 10−13 kg ,

r = 8 · 10−6m,capsules: m = 5.44 · 10−13 kg ,

r = 4 · 10−6m,l = 8 · 10−6m,

→ damping ratio D=0.07



Conclusion and Outlook



Conclusions

successfully integrated self-propelled micro-devices intocoupled pe-waLBerla framework

validation of simulations against analytical models

advantage of flexibility of our framework: symmetric andasymmetric designs

study is important for systems inaccessible via analyticalmethods



Outlook & Preliminary Results

Validation in Fluid

16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50Armlength

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

1,5

Sw

imm

ing d

ista

nce

c

over

ed i

n 4

c

ycl

es

By our simulations

By our calculations



Outlook & Preliminary Results

Average Flowfield of Three Three-sphere Swimmers

expansion of the flow field in order to minimize wall effects

parallelization handling springs and swimmers

⇒ address problem of swarming



Last but not Least: A Swimming Contest



Thank you for your attention!

Extract from the References

A. Najafi and R. Golestanian. Simple swimmer at low reynolds number: Threelinked spheres. Phys. Rev. E, 69(6):062901 (2004).

C.M. Pooley and J.M. Yeomans. Lattice boltzmann simulation techniques forsimulating microscopic swimmers. Computer Physics Communications,179(1-3):159-164 (2008).

More Information on both Frameworks

Homepage of the pe software framework:http://www10.informatik.uni-erlangen.de/de/Research/Projects/pe/

Homepage of the waLBerla software framework:http://www10.informatik.uni-erlangen.de/Research/Projects/walberla/

Acknowledgments

http://www10.informatik.uni-erlangen.de/de/Research/Projects/pe/

http://www10.informatik.uni-erlangen.de/Research/Projects/walberla/

simulation of self-propelled micro-devices with a … · simulation of self-propelled micro-devices...

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