a control algorithm for the multiscale simulation of liquid …...a continuum and molecular dynamics...

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A Control Algorithm for the Multiscale Simulation of Liquid

Water

Evangelos M. Kotsalis

with: I. Hanasaki, J.H. Walther and P. Koumoutsakos

CSE LabComputational Science & Engineering Laboratoryhttp://www.cse-lab.ethz.ch

http://www.cse-lab.ethz.ch



Outline• Atomistic-continuum simulations of Lennard-Jones fluids.

• Multiscale flow of argon past a CNT.• Need for a control algorithm.

• Coarse grained simulation of a lipid bilayer in aqueous solution.

• Extension to polyatomic liquids such as water.• Multiscale Couette flow of liquid water.


Need for multiscaling

• MD prohibitively expensive.

• Fully continuum simulation not possible due to the lack of correct boundary conditions for the velocity.

System:• Water @ 500bar, 300K• CNT: (256,0)• Flow: 100 m/s (Re ~ 3)Simulation:• 810.000 atoms• 1.5 ns• 64 CPUs (3 years CPU time)

64nm

Flow past a large carbon nanotube

22nm




Atomistic description of the liquid (MD)

System of 6N first order ODEs Boundary conditions: non-periodic

v : atom velocitiesr : atom positions

U : Interaction potential

f : force

• Atoms are modeled as interacting mass points whose trajectories are numerically integrated in time

U1 : atomistic-atomistic interactionU2: atomistic-continuum interaction

U(r) =∑

i !=j

U1(ri, rj) + U2(ri)




Continuum description of the liquid (NS)

Continuum: Incompressible Navier- Stokes Equations

€

∂U∂t

+U ⋅ ∇U = −1ρ∇p+ νΔU

∇ ⋅U = 0

Conservation of Momentum:

Conservation of Mass:

U : velocity, p : pressure, ρ : density, ν : viscosity

+ Boundary Conditions




Schwarz iteration

Iterate until convergence steady state solution

CFD simulation

Impose velocity BC on continuum

MD simulation

Measure velocity BC for the continuum

Impose continuum velocities on MD

Obtain next velocity BC for the MD

The solution in the hybrid domain is found iteratively




Boundary force

Specular wall

Missing Interactions Specular wall

Boundary force




How can we replace the particles in the red domain?The total force should be F = P x A

Boundary force

z

rCx

z

dens.

r

g(r)

215 K1.0 gcm-3

Take fluid structure into account: g(r)

ρ(r) =∫ r

04πr′2ρg(r′)dr′




Equilibrium (No Flow)Uniform distribution (O’Connell 1995A)

Forc

e [k

J/(m

ol n

m)]

Dens

ity [

g/cm

3 ]distance to wall [nm]distance to wall [nm]

No force

Use fluid structure (Werder 2005C)Repulsive (Nie 2004B)A) S. T. O’Connell and P. A. Thompson. Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flow. Phys. Rev. E, 52(6):R5792-R5795, 1995.B) X. B. Nie, S. Y. Chen, W. N. E and M. O. Robbins. A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. J. Fluid Mech., 500:55-64, 2004.C) T. Werder, J. H. Walther, and P. Koumoutsakos. Hybrid atomistic-continuum method for the simulation of dense fluid flow. J. Comput. Phys., 205: 373-390, 2005.




Flow of argon past a CNTPeriodic

Diric

hlet

u =

0.1

nm/p

s

Fluid•Argon•Temperature 215 K•Density 1.0 gcm-3

30 nmT. Werder, J. H. Walther, and P. Koumoutsakos. Hybrid atomistic-continuum method for the simulation of dense fluid flow. J. Comput. Phys., 205: 373-390, 2005.

Diric

hlet

u =

0.1

nm/p

s




MD vs Hybrid scheme

Hybrid solution Reference MD solution

202632 38

3844

44

50

50

56

56

6262

62

68

68 68

7474

74

8080

8080

8686

86

86

9292

92

9292

98

9898 98

98

98

98

98

98

104

104

104

104

104

104104

104

110

110

110

110

110

110

116

x [nm]

y [n

m]

0.5 5 10 15 20 25 30

30

25

20

15

10

5

0.5 20

30

40

50

60

70

80

90

100

110

2026 32 383844

44

50

50

56

56

62

62

62

68

68 68

7474

74 74

8080

80 80

8686

86

86

9292

9292 9298

98 98

98

9898

98

104

104

104

104104

104

110

11011

0

110

110

110

116

116

x [nm]

y [n

m]

0.5 5 10 15 20 25 30

30

25

20

15

10

5

0.5 20

30

40

50

60

70

80

90

100

110

Relative Error ~ 1.3%

A. Dupuis, E.M. Kotsalis, and P. Koumoutsakos. Coupling lattice Boltzmann and molecular dynamics for dense fluids. Phys. Rev. E, 75: 046704, 2007

The hybrid scheme is ~ (L/R)**3 times faster for a computational domain of size L and a MD subdomain of size R.




The problem with density variations

•Density variations depend on liquid state•Amplitude proportional to structural correlations in the liquid

distance to wall [nm] distance to wall [nm]

g(r)

Red.

Den

sity

€

T = 215K,ρ =1.0gcm−3

€

T =131K,ρ =1.35gcm−3

€

T = 84K,ρ =1.5gcm−3




Control approach to coupling

€

P-Controller MD system

€

+

€

−error Force

• Controlling of the external boundary force

• measured density => target density

E.M. Kotsalis, J.H. Walther, and P. Koumoutsakos. Control of density fluctuations in atomistic-continuum simulations of dense liquids. Phys. Rev. E, 76 016709, 2007.

ρmρt

ρtρm




Results with Control Approach

€

T = 84K,ρ =1.5gcm−3

€

RESULT WITH CONTROL

RESULT WITHOUT CONTROL

Forc

e [k

J/(m

ol n

m)]

distance to wall distance to wall

Dens

ity [

g/cm

3 ]

Algorithm converged after 1.7 ns

Equilibrium




Results with Control Approach

€ x [nm]

Velo

city

[m/s

]

20

25

30

0.0 1.0 2.0 3.0 4.0 5.0

YLA

BE

L

XLABEL

Dens

ity [

g/cm

3 ]

x [nm]

0.8

0.9

1.0

1.1

1.2

0.0 1.0 2.0 3.0 4.0 5.0

YLABEL

XLABEL

Parallel flow


RESULT WITH CONTROL

T=131K,ρ =1.35gcm−3




Multiscale Membrane Simulation

€

Continuum

Coarse-Grained MD

Continuum

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

YLABEL

XLABELx [nm]

Dens

ity [

g/cm

3 ]

RESULT WITH CONTROL


System:

•Coarse grained DPPC lipid surrounded by coarse grained water.

Size:

•12 nm x 20 nm x 20 nm

Simulation:

•26250 CG water molecules•1500 CG lipid molecules•Timestep: 20 fs•Berendsen Thermostat

•: Marrink, SJ; Risselada, HJ; Yefimov, S, et al., The MARTINI force field: Coarse grained model for biomolecular simulations Journal of Physical Chemistry B, July 2007

4 water molecules are mapped to one site interacting with a Lennard-Jones potential.


Water in non-periodic simulations Goal: Extend the technique for monoatomic liquids to water.

Issues:

• Electrostatic forces•Reaction field method

• Orientation of the water molecules

• Elastic Collision

• External Boundary Force

Interface

Non-periodic in 3 directions

rπ-φ

rπ-φ

Interface

Non-periodic in 1 direction




Non-periodic in 1 directionFo

rce

[kJ/

(mol

nm

)]

x [nm] x [nm]

RESULT WITH CONTROL

RESULT WITHOUT CONTROLDe

nsity

[g/

cm3 ]

Equilibrium

The control force is updated every 3 ps. The algorithm has converged after 0.4 ns

-10

-8

-6

-4

-2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

YLA

BE

L

XLABEL

0.8

0.9

1.0

1.1

0.0 0.2 0.4 0.6 0.8 1.0

YLABEL

XLABEL




Multiscale Couette Flow

x [nm]

MD

CFD

CFD

v

-v

Velo

city

[nm

/ps] CFD MD CFD

-0.10

-0.05

0.00

0.05

0.10

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

YLA

BE

L

XLABEL





RESULT WITH CONTROL


Equilibrium

Dens

ity [

g/cm

3 ]

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.6 0.8 1.0 1.2 1.4

Forc

e [k

J/(m

ol n

m)]

r [nm]-2

0

2

4

6

8

10

12

14

0.6 0.8 1.0 1.2 1.4

r [nm]


R=1.5 nm




Summary-Ongoing Work • A novel non-periodic boundary model for MD of dense fluids.• Coupling of MD to a continuum description for Lennard-Jones

fluids and liquid water.

RNA transport through CNT’s

U. Zimmerli, and P. Koumoutsakos. Simulations of electrophoretic RNA transport through transmembrane carbon nanotubes. Biophysical Journal,94: 2546-2557, 2008.




THANK YOU




References

E.M. Kotsalis, J.H. Walther, and P. Koumoutsakos. Control of density fluctuations in atomistic-continuum simulations of dense liquids. Phys. Rev. E, 76: 016709, 2007.

A. Dupuis, E.M. Kotsalis, and P. Koumoutsakos. Coupling lattice Boltzmann and molecular dynamics for dense fluids. Phys. Rev. E, 75: 046704, 2007

T. Werder, J. H. Walther, and P. Koumoutsakos. Hybrid atomistic-continuum method for the simulation of dense fluid flow. J. Comput. Phys., 205: 373-390, 2005.

E.M. Kotsalis, J.H. Walther, and P. Koumoutsakos. Multiphase water flow inside carbon nanotubes. Int. J. of Multiphase Flow, 30: 995-1010, 2004.

U. Zimmerli, and P. Koumoutsakos. Simulations of electrophoretic RNA transport through transmembrane carbon nanotubes. Biophysical Journal,94: 2546-2557, 2008.



a control algorithm for the multiscale simulation of liquid …...a continuum and molecular dynamics...

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