Analytical and Numerical Study of Coupled Atomistic ...Analytical and Numerical Study of Coupled...
Transcript of Analytical and Numerical Study of Coupled Atomistic ...Analytical and Numerical Study of Coupled...
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Analytical and Numerical Study ofCoupled Atomistic-Continuum Methods
for Fluids
Weiqing RenCourant Institute, NYU
Joint work with Weinan E
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Multiscale modeling for two types of problems:
• Complex fluids - Constitutive modeling
• Microfluidics - Atomistic-based boundary condition modeling
(i) Heterogeneous multiscale method: Macro solver + missing data from MD(ii) Domain-decomposition framework
Koumoutsakos, JCP ’05
Koplik et al. PRL ’88; Thompson & Robbins, PRL ’89Qian et al., PRE ’03; Ren & E, PoF ’07
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Multiscale method in the domain decompositionframework
C-region : Continuum hydrodynamicsP-region : Molecular dynamics
The two descriptions are coupled through exchangingboundary conditions in the overlapping region after eachtime interval .
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Two fundamental issues:
• What information need to be exchangedbetween the two descriptions?
(i) Fields (e.g. velocity):(ii) Fluxes of conserved quantities
• How to accurately impose boundary conditionson molecular dynamics?
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Existing multiscale methods for dense fluids
• Velocity coupling:O’Connel and Thompson 1995Hadjiconstantinou and Patera 1997Li, Liao and Yip 1999Nie, Chen, E and Robbins 2004Werder, Walther and Koumoutsakos 2005
• Flux coupling:Flekkoy, Wagner and Feder 2000Delgado-Buscalioni and Coveney 2003
• Mixed scheme: Ren and E 2005
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Present work:
• Stability and convergence rate of the differentcoupling schemes; Propagation of statisticalerrors in the numerical solution.
• Error introduced when imposing boundaryconditions in MD.
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Problem setup: Lennard-Jones fluid in achannel
(i). Static (U=0); (ii). Impulsively started shear flow
U
-U
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Four coupling schemes:
• Velocity - Velocity
• Momentum flux - Velocity
• Velocity - Flux
• Flux - Flux
U
-U
The two models exchange BCs after every time interval .
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The rest of the talk:
• Algorithmic details of the multiscale methodfor the benchmark problems;
• Numerical results;
• Assessment of the error introduced in theimposition of boundary condition in MD.
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Solving the continuum model
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Correspondence of hydrodynamics and moleculardynamics
( Irving-Kirkwood 1950)
Using these formulae to calculate the continuum BCs from MD.
for Newtonian fluids
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Boundary conditions for MD: Reflection BC + Boundary force
Mean boundary force:
Red = ReflectionBlue = Reflection+Black = Exact
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Matching with continuum - Imposing velocity BC on MD
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Matching with continuum: Imposing shear stress on MD
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Microscopic profile of shear stress
Left: Shear stress profile at various shear rates;Right: Shear stress profile rescaled by the shear rate.
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Summary of the multiscale method
• Continuum solver: Finite difference in space + forwardEuler’s method in time
• Molecular dynamics: (1) Velocity Verlet, Langevin dynamics to control temperature;
(2) Refection BC + Boundary force; (3) Projection method to match the velocity;
(4) Distribute the shear stress based on an universal profile.
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Numerical result for the static problem:
(i). Errors are due tostatistical errors in themeasured boundarycondition (velocity,or shearstress) from MD.
(ii). Errors are bounded inVV, FV, VF schemes, whileaccumulate in the FFscheme.
Velocity - Velocity
Flux - Velocity
Velocity - Flux
Flux - Flux
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Numerical result for the static problem:
Velocity - Velocity
Flux - Velocity
Velocity - Flux
Flux - Flux
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Numerical result for the static problem:
Velocity - Velocity
Flux - Velocity
Velocity - Flux
Flux - Flux
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Analysis of the problem forVelocity - Velocity coupling scheme:
where
--- Amplification factor
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Analysis of the problem forThe numerical solution has the following form:
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Analysis of the problem for finite
V-V
F-V
V-F
F-F
(iii). Flux-Flux scheme is weakly unstable.
(i). The VV and FV schemes are stable;(ii). Velocity-Flux is stable when , and unstablewhen
Conclusions:
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A dynamic problem: Impulsively startedshear flow
U
-U
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Numerical results:tim
e
time
Velocity-Velocity Flux-Velocity
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Numerical results:tim
e
time
Velocity-Flux Flux-Flux
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Steady state calculation: Comparison ofconvergence rate
V-V
F-V
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Assessment of the error from the impositionof velocity BC in MD
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Assessment of the error from the impositionof velocity BC in MD
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Assessment of the error for
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Error of the stress: Analysis vs. Numerics
Red curve:
Blue curve: Numerics
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Summary:(1). Stability of different coupling schemes.Schemes based on flux coupling is weaklyunstable. Flux-velocity scheme performs the best.
(2). Error introduced when imposing velocityboundary condition in MD.
Ongoing work:Boundary conditions for non-equilibrium MD;
Incorporating fluctuations in the BC of MD;Coupling fluctuating hydrodynamic with moleculardynamics.
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Improved numerical scheme: Using ghostparticles
• Less disturbance to fluid structure• Mass reservoir for 2d velocity field
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References:
• Heterogeneous multiscale method for the modeling ofcomplex fluids and microfluidics, J. Comp. Phys. 204, 1(2005)
• Boundary conditions for the moving contact lineproblem, Physics of Fluids, 19, 022101 (2007)
• Analytical and Numerical study of coupled atomistic-continuum methods for fluids, preprint