Discussion of Hybrid Atomistic-Continuum Methods for …web.mit.edu/ngh/www/Hettithanthrige.pdf · 2004. 8. 21. · pling methods nearly 50 years of continuum computational ﬂuid

International Journal of Multiscale Computational Engineering, 2(2)X-X(2004)

Discussion of HybridAtomistic-Continuum Methodsfor Multiscale Hydrodynamics

Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou *

Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02139

ABSTRACT

We discuss hybrid atomistic-continuum methods for multiscale hydrodynamicapplications. Both dense-fluid and dilute-gas formulations are considered. Thechoice of coupling method and its relation to the fluid physics as well as the needfor timescale decoupling is highlighted. In particular, by relating the molecularintegration timestep to the CFL timestep, we show that compressibility is im-portant in determining the choice of a coupling method. Appropriate couplingtechniques for various flow regimes are discussed and proposed. We also dis-cuss recently developed incompressible and compressible hybrid methods for di-lute gases. The incompressible framework is based on the Schwarz alternatingmethod, which provides timescale decoupling; the compressible method is a mul-tispecies, fully adaptive mesh and algorithm refinement approach that introducesthe direct-simulation Monte Carlo at the finest level of mesh refinement.

*Corresponding author. Email address: [email protected]

0731-8898/02/$5.00 © 2004 by Begell House, Inc. 1

2 HETTITHANTHRIGE ET AL

1. INTRODUCTION

By limiting the molecular treatment to regionswhere it is needed, a hybrid method allows thesimulation of complex thermo-fluid phenom-ena, which require modeling at the microscalewithout the prohibitive cost of a fully molec-ular calculation. In what follows we providean overview of this rapidly expanding fieldand discuss recent developments. We also de-scribe archetypal hybrid methods for incom-pressible and compressible flows; the hybridmethod for incompressible gas flow is basedon the Schwarz alternating coupling method;and the hybrid method for compressible flowis based on the recently developed [34] flux-coupling, multispecies adaptive mesh and algo-rithm refinement scheme that extends adaptivemesh refinement by introducing the moleculardescription at the finest level of refinement.

Over the years a fair number of hybrid simu-lation frameworks have been proposed leadingto some confusion over the relative merits andapplicability of each approach. Original hybridmethods focused on dilute gases [10, 20, 28, 29],which are arguably easier to deal with withina hybrid framework than dense fluids, mainlybecause boundary condition imposition is sig-nificantly easier in gases. The first hybrid meth-ods for dense fluids appeared a few years later[11, 16, 17, 26]. These initial attempts have ledto a better understanding of the challenges as-sociated with hybrid methods.

To a large extent, the two major issues in de-veloping a hybrid method is the choice of acoupling method and the imposition of bound-ary conditions on the molecular simulation.Generally speaking, these two can be viewedas decoupled, in the sense that the couplingtechnique can be developed on the basis ofmatching two compatible and equivalent oversome region of space hydrodynamic descrip-tions and can thus be borrowed from the al-ready existing and extensive continuum-basednumerical methods literature. The choice ofcoupling technique is further discussed in Sec.2.1–2.3. Boundary condition imposition canagain be considered in a decoupled sense and

can be posed as a general problem of impos-ing “macroscopic” boundary conditions on amolecular simulation. In our opinion, this isa very challenging problem that has not been,in general, resolved to date completely satis-factorily. Boundary condition imposition onthe molecular subdomain is discussed in Sec.2.4. Boundary condition imposition on thecontinuum subdomain is generally well under-stood, as is the process of extracting macro-scopic fields from molecular simulations (typ-ically achieved through averaging).

In Sec. 3, we give a brief description ofthe direct simulation Monte Carlo (DSMC),the dilute-gas simulation method used in thiswork. In Sec. 4, we demonstrate a hybridscheme suitable for low-speed, incompressiblegaseous flows based on the Schwarz alternat-ing method. The current paper introducesChapman-Enskog boundary condition impo-sition in incompressible hybrid formulations.Subsequently, in Sec. 5 we discuss a recently de-veloped multispecies compressible formulation[34] for gases that introduces the molecular sim-ulation at the finest level of refinement within afully adaptive mesh refinement framework. Wefinish with some concluding remarks.

2. DEVELOPING A HYBRID METHOD

2.1. The Choice of Coupling Method

Coupling a continuum to a molecular descrip-tion is meaningful in a region where both canbe presumed valid. In choosing a couplingmethod, it is therefore convenient to draw uponthe wealth of experience and large cadre of cou-pling methods nearly 50 years of continuumcomputational fluid dynamics have brought us.Coupling methods for compressible and incom-pressible formulations generally differ, sincethe two correspond to two different physicaland mathematical hydrodynamic limits. Thecompressible formulation lends itself naturallyto time-explicit flux-based coupling while in-compressible formulations are typically cou-pled using either state (Dirichlet) properties orgradient (Neumann) variables.

International Journal for Multiscale Computational Engineering

DISCUSSION OF HYBRID ATOMISTIC-CONTINUUM METHODS 3

Given that the two formulations have dif-ferent limits of applicability and/or physicalregimes in which each is significantly more ef-ficient than the other, care must be exercisedwhen selecting the ingredients of the hybridmethod. In other words, the choice of a cou-pling method and continuum subdomain for-mulation needs to be based on the degree towhich compressibility effects are important inthe problem of interest and not on a presetnotion that a particular coupling method ismore appropriate than all others. The latterapproach was recently pursued in a varietyof studies, which enforce the use of a com-pressible, time-explicit, flux-matching couplingscheme to steady and essentially incompress-ible physical problems. This approach is notrecommended. On the contrary, for an effi-cient simulation method, similar to the case ofcontinuum solution methods, it is importantto allow the flow physics to dictate the appro-priate formulation, while the numerical imple-mentation is chosen to cater to the particular re-quirements of the latter. Below, we expand onsome of the considerations that influence thechoice of coupling method under the assump-tion that the hybrid method is applied to prob-lems of practical interest and, therefore, the con-tinuum subdomain is appropriately large. Ourdiscussion extends to timescale considerationsthat are more complex, but equally importantto limitations resulting from length scale con-siderations, such as the size of the molecular re-gion(s).

2.2. Timescale Decoupling

It is well known [31] that the timestep for ex-plicit integration of the compressible Navier-Stokes formulation τc, scales with the physicaltimestep of the problem τ∆x(= ∆x/U , where∆x is the numerical grid spacing and U is thecharacteristic velocity), according to

τc ≤ M

1 + Mτ∆x (1)

where M is the Mach number. As the Machnumber becomes small, we are faced with the

well-known stiffness problem whereby a) thenumerical efficiency degrades due to dispar-ity of the timescales in the system of equationsand b) the accuracy of the compressible solu-tion degrades due to mismatch of magnitudesbetween fluxes in the original equations andcorresponding terms in the numerically addedartificial viscosity [36]. For this reason, whenthe Mach number is small, the incompressibleformulation is used that allows integration atthe physical timestep τ∆x. In the hybrid case,matters are complicated by the introduction ofthe molecular integration timestep, τm, whichis at most of the order of τc (in some cases ingases when ∆x ≤ λ, where λ is the molecu-lar mean free path) and in most cases signifi-cantly smaller. One consequence of Eq. (1) isthat as the global domain of interest grows, thetotal integration time grows, and transient cal-culations in which the molecular subdomain isexplicitly integrated in time become more com-putationally expensive and eventually infeasi-ble. The severity of this problem increases withdecreasing Mach number and makes unsteadyincompressible problems very computationallyexpensive. New integrative frameworks, whichcoarse grain the time integration of the molecu-lar subdomain, are therefore required.

Fortunately, for low-speed steady problemsimplicit (iterative) coupling methods exist thatprovide solutions without the need for ex-plicit integration of the molecular domain tothe global problem steady state. The par-ticular method used here is known as theSchwarz method and is discussed further inSec. 4. This method decouples the globalevolution timescale from the molecular evo-lution timescale (and timestep) by achievingconvergence to the global problem steady-statethrough an iteration between steady state so-lutions of the continuum and molecular sub-domains. Because the molecular subdomain issmall, explicit integration to its steady state isfeasible. Although the steady assumption mayappear restrictive, it is interesting to note thatthe vast majority of both compressible and in-compressible test problems solved to date byhybrid methods have been steady.

Volume 2, Number 2, 2004


A variety of other iterative methods may besuitable as they provide for timescale decou-pling. The choice of the Schwarz couplingmethod, which uses state variables instead offluxes to achieve matching, was motivated bythe fact (as explained below) that state variablessuffer from smaller statistical noise and are thuseasier to prescribe on a continuum formulation.

The above observations do not preclude theuse of the compressible formulation in the con-tinuum subdomain for low-speed flows. Infact, preconditioning techniques, which allowthe use of the compressible formulation at verylow Mach numbers, have been developed [31].Such a formulation can, in principle, be usedto solve the continuum subproblem while thisis being coupled to the molecular subprob-lem via an implicit (e.g., Schwarz) iteration.What should be avoided is a compressible,time-explicit, flux-based coupling procedurefor solving essentially incompressible steady-state problems.

The issues discussed above have not beenvery apparent to date because in typical testproblems published so far the continuum andatomistic subdomains are of the same size (and,of course, small). In this case the large costof the molecular subdomain masks the cost ofthe continuum subdomain and also typical evo-lution timescales (or times to steady state) aresmall. It should not be forgotten, however, thathybrid methods make sense when the contin-uum subdomain is significantly larger than themolecular subdomain.

2.3. Statistical Noise Considerations

The use of a compressible formulation with fluxcoupling in the M → 0 limit leads to two ad-ditional disadvantages. The first, continuumsubdomain stiffness (see Eq. 1), may be reme-died by implicit timestepping methods [38] orpreconditioning approaches [31]. The second,more serious disadvantage, is linked to adversesignal to noise ratios (compared to non-flux-based schemes) in connection with the averag-ing required for imposition of boundary con-ditions from the molecular subdomain to the

continuum subdomain. More specifically, inthe case of an ideal gas (where compressibleformulations are typical) it has been shown in[19] that, for the same number of samples, flux(shear stress, heat flux) averaging exhibits rel-ative noise Ef , which scales with Esv, the rel-ative noise in the corresponding state variable(velocity, temperature) according to,

Ef ∼ Esv

Kn(2)

Here Kn = λ/L is the Knudsen number basedon the characteristic length scale of the trans-port gradients L, and λ is the mean-free path,which is expected to be much smaller than L,because, by assumption, a continuum subdo-main is present. It thus appears that flux cou-pling will be significantly disadvantaged in thiscase, because the number of samples requiredto make Ef ≈ Esv scales as 1/Kn2 times thenumber of samples required by state-variableaveraging.

On the other hand, Schwarz-type iterativemethods based on the incompressible physicsof the flow require a fair number of iterationsfor convergence [O(10)]. These iterations re-quire the re-evaluation of the molecular solu-tion. This is an additional computational costthat is not shared by time-explicit approaches.At this time, the best choice for incompressibleunsteady problems appears to be an explicit in-compressible approach, such as the one used byO’Connell and Thompson [26]. We should re-call, however, that unless time coarse-grainingtechniques are developed, large, low-speed,unsteady problems are currently too expensiveto be feasible by any approach (see discussionin Sec. 2.2).

2.4. Boundary Condition Imposition

Consider the molecular region Ω on the bound-ary of which, ∂Ω, we wish to impose a set ofhydrodynamic (macroscopic) boundary condi-tions. Typical implementations require the useof particle reservoirs R (see Fig. 1) in whichparticle dynamics may be altered in such a waythat the desired boundary conditions appear on



FIGURE 1. Continuum to atomistic boundarycondition imposition using reservoirs.

∂Ω; the hope is that the influence of the per-turbed dynamics in the reservoir regions de-cays sufficiently fast and does not propagateinto the region of interest, that is, the relaxationdistance both for the velocity distribution func-tion and the fluid structure is small comparedto the characteristic size of Ω.

In a dilute gas, the nonequilibrium distri-bution function in the continuum limit hasbeen characterized [8] and is known as theChapman-Enskog distribution. Use of this dis-tribution to impose boundary conditions onmolecular simulations of dilute gases results ina robust, accurate, and theoretically elegant ap-proach. Typical implementations [14] requireparticle generation and initialization within R.Particles that move into Ω within the simulationtimestep are added to the simulation whereasparticles remaining in R are discarded. Moredetails on implementation are provided in Sec.4.

Unfortunately, for dense fluids where notonly the particle velocities but also the fluidstructure is important and needs to be imposed,no theoretical results for their distributions ex-ist. A related issue is that of domain termina-tion; due to particle interactions, Ω, or in thepresence of a reservoir, R needs to be termi-nated in a way that does not have a big effect

on the fluid state inside of Ω.As a result, researchers have experimented

with possible methods to impose boundaryconditions. It is now known that similarly toa dilute gas, use of a Maxwell-Boltzmann dis-tribution for the velocities leads to slip [16]. Liet al. [22] used a Chapman-Enskog distribu-tion to impose boundary conditions to gener-ate a dense-fluid shear flow. In this approach,particles crossing ∂Ω acquire velocities that aredrawn from a Chapman-Enskog distributionparametrized by the local values of the requiredvelocity and stress boundary condition. Al-though this approach was only tested for aCouette flow, it appears to give reasonable re-sults (within molecular fluctuations). Becausein Couette flow no flow normal to ∂Ω exists, ∂Ωcan be used as symmetry boundary separatingtwo back-to-back shear flows; this sidesteps theissue of domain termination.

In a different approach, Flekkoy et al. [11]use external forces to impose boundary con-ditions. More specifically, in the reservoir re-gion they apply an external field of such mag-nitude that the total force on the fluid parti-cles in the reservoir region is the one requiredby momentum conservation. They then ter-minate their reservoir region by using an adhoc weighing factor for the distribution of thisforce on particles within R that prevents parti-cles from leaving the reservoir region from itsouter edge. In particular, they chose a weigh-ing factor that diverges as particles approachthis boundary such that particles do not es-cape the reservoir region while particles intro-duced there move towards Ω. Particles intro-duced into the reservoir are given velocitiesdrawn from a Maxwell-Boltzmann distribution,while a Langevin thermostat keeps the temper-ature constant. The method appears to be suc-cessful although the nonunique (ad hoc) choiceof force fields and Maxwell-Boltzmann distri-bution makes it not very theoretically pleas-ing. It is also not clear what the effect of theseforces are on the local fluid state (it is wellknown that even in a dilute gas [25] gravity-driven flow exhibits significant deviations fromNavier-Stokes behavior), but this effect is prob-



ably negligible since force fields are only actingin the reservoir region. Delgado-Buscalioni andCoveney [9] refined the above approach by us-ing an Usher algorithm to insert particles in theenergy landscape such that they have the de-sired specific energy, which is beneficial to im-posing a desired energy current while eliminat-ing the risk of particle overlap at some com-putational cost. This approach, however, usesa Maxwell–Boltzmann distribution, for the ini-tial velocities of the inserted particles. Temper-ature gradients are imposed by a small num-ber of thermostats placed in the direction of thegradient. Although no proof exists that the dis-turbance to the particle dynamics is small, itappears that this technique is successful at im-posing boundary conditions with moderate er-ror. Boundary conditions on MD simulationscan also be imposed through the method ofconstraint dynamics [26]. Although the ap-proach in [26] did not allow hydrodynamicfluxes across the matching interface, this fea-ture can be integrated into this approach witha suitable domain termination.

A method for terminating molecular dynam-ics simulations with small effect on particle dy-namics has been suggested and used in [16].This simply involves making the reservoir re-gion fully periodic. In this manner, the bound-ary conditions on ∂Ω also impose a boundaryvalue problem on R, where the inflow to Ω isthe outflow from R. As R becomes bigger, thegradients in R become smaller; thus, the flowfield in R will have a small effect on the solu-tion in Ω. The disadvantage of this method isthe number of particles that are needed to fillRis fairly large, especially in high dimensions.

We believe that significant contributions canstill be made by developing methods to imposeboundary conditions in hydrodynamically con-sistent and, most importantly, rigorous ap-proaches.

3. THE DIRECT SIMULATION MONTECARLO

The DSMC method was proposed by Bird [7]in the 1960s and has been used extensively to

model rarefied gas flows. A comprehensive dis-cussion of DSMC can be found in the reviewarticle by Alexander et al. [2]. The DSMC algo-rithm is based on the assumption that a smallnumber of representative “computational par-ticles” can accurately capture the hydrodynam-ics of a dilute gas as given by the Boltzmannequation. Air under standard conditions nar-rowly meets the dilute gas criterion. Empiri-cal results [7] show that a small number (≈ 20)of computational particles per cubic molecularmean-free path is sufficient to capture the rele-vant physics. This is approximately two ordersof magnitude smaller than the actual numberof gas atoms/molecules contained in the samevolume. This is one source of the DSMC’s sig-nificant computational advantage over a fullymolecular simulation.

The DSMC solves the Boltzmann equationusing a splitting approach: the time evolutionof the system is approximated by a sequenceof discrete timesteps ∆t in which particles un-dergo successively collisionless advection andcollisions. Collisions are performed betweenrandomly chosen particle pairs within smallcells of linear size ∆x. The flow solution isdetermined by averaging the individual parti-cle properties over space and time. This ap-proach has been shown to produce correct so-lutions of the Boltzmann equation in the limit∆x, ∆t → 0 [30]. The splitting approach elim-inates the computational cost associated withintegrating the equations of motion of all parti-cles, but most importantly allows the timestepto be significantly larger (see also below) than atypical timestep in a hard sphere molecular dy-namics simulation. This is another reason whythe DSMC is significantly more computation-ally efficient than “brute force” molecular dy-namics.

Recent studies [15, 18] have shown that forsteady flows or flows that are evolving attimescales that are long compared to the molec-ular relaxation times, a finite timestep leads toa truncation error that manifests itself in theform of timestep-dependent transport coeffi-cients; this error has been shown to be of theorder of 5% when the timestep is of the order



of a mean-free time and goes to zero as ∆t2.Quadratic dependence of transport coefficientson the collision cell size ∆x was shown in [3].

4. THE SCHWARZ METHOD FOR INCOM-PRESSIBLE FORMULATIONS

Although in some cases compressibility may beimportant, a large number of applications aretypically characterized by flows where use ofthe incompressible formulation results in a sig-nificantly more efficient approach [31]. As ex-plained in the introduction section, our defi-nition of incompressible formulation is basedon the flow physics and not on the numericalmethod used. Although we have used here a fi-nite element discretization based on the incom-pressible formulation, we believe that a precon-ditioned compressible formulation could alsobe used to solve the continuum subdomainproblem provided that it is matched to themolecular solution through a coupling methodwhich takes into account the flow physics asoutlined in Sec. 2.

Here, matching is achieved through an it-erative procedure based on the Schwarz alter-nating method for the treatment of steady-stateproblems. The Schwarz method was originallyproposed for molecular dynamics-continuummethods in [16] and extended in [17], but itis equally applicable to DSMC-continuum hy-brid methods [1, 33]. This approach was cho-sen because of its ability to couple different de-scriptions through Dirichlet boundary condi-tions (easier to impose on dense-system molec-ular simulations compared to flux conditionsbecause fluxes are nonlocal in dense systems),and its ability to reach the solution steady statein an implicit manner by using only steady so-lutions from each subdomain. The importanceof the latter characteristic cannot be overem-phasized; the implicit convergence in timethrough exchange of steady solutions guaran-tees timescale decoupling that is necessary forthe solution of macroscopic problems: whilethe integration of molecular trajectories at themolecular timestep for total times correspond-ing to macroscopic evolution times is, and will

for a long time be, infeasible, integration of themolecular region to its steady state is feasible.

Within the Schwarz coupling framework, anoverlap region facilitates information exchangebetween the continuum and atomistic subdo-mains in the form of Dirichlet boundary con-ditions. A steady-state continuum solution isfirst obtained using boundary conditions takenfrom the atomistic subdomain solution. For thefirst iteration, this latter solution can be a guess.A steady-state atomistic solution is then foundusing boundary conditions taken from the con-tinuum subdomain. This exchange of bound-ary conditions corresponds to a single Schwarziteration. Successive Schwarz iterations are re-peated until convergence, i.e., until the solu-tions in the two subdomains are identical in theoverlap region.

The Schwarz method was recently applied[1] to the simulation of flow-through micro-machined filters. These filters have passagesthat are sufficiently small that require a molec-ular description for the simulation of the flowthrough them. Depending on the geometry andnumber of filter stages, the authors have re-ported computational savings ranging from 2to 100. The approach in [1] used a Maxwellianvelocity distribution and a “control mecha-nism” to impose the flow field on the molec-ular simulation. This approach, although suc-cessful in quasi-one-dimensional flows, is notvery general; additionally, it is well known thatusing a Maxwellian distribution to impose hy-drodynamic boundary conditions, in general,if uncorrected will lead to slip (discrepancybetween the imposed and observed boundaryconditions). As discussed in Sec. 2.4 generalboundary condition imposition on dilute-gasmolecular simulations can be performed us-ing the Chapman-Enskog velocity distribution[8, 14]. This approach eliminates the need fora feedback correction since supplying the cor-rect local distribution function eliminates slip.A Chapman-Enskog procedure for the Schwarzmethod is described below.

Extensions of the Schwarz method to time-dependent problems is currently under investi-gation [32], although, as discussed in Sec. 2.3,



when the Mach number is low, the dispar-ity between the molecular and hydrodynamictimescales makes this a very stiff problem.

Before proceeding with an example, a sub-tle numerical issue associated with the incom-pressible formulation should be discussed. LetΓ1 be the portion of the continuum subdomainthat receives boundary data from the molecularsubdomain. Due to inherent statistical fluctu-ations in this data, the boundary condition onthe complete continuum subdomain boundary(φ ⊇ Γ1) may not conserve mass exactly. Al-though this phenomenon is an artifact of thefinite sampling of the atomistic solution (if asufficiently large —“infinite”— number of sam-ples are taken, the mean field obtained fromthe atomistic simulation should be appropri-ately incompressible), it is sufficient to cause anumerical instability in the continuum calcula-tion. The most common remedy is to apply acorrection to vΓ1 , the atomistic boundary datato be imposed on Γ1, namely

(vΓ1 .n)corrected = vΓ1 .n−∫φ vφ.ndS∫

Γ1dS

(3)

where n is the unit outward normal vectorto the boundary and dS is an element of theboundary. This correction essentially removesthe discrepancy in mass flux equally across allnormal velocity components of vΓ1 . Tests withvarious problems [16, 17, 33] indicate that it issuccessful at removing the numerical instabil-ity.

4.1. Driven Cavity Test Problem

In this section we discuss the Schwarz alternat-ing method in the context of the solution of thedriven cavity problem. We pay particular at-tention to the imposition of boundary condi-tions on the DSMC domain using a Chapman-Enskog distribution, which is arguably themost rigorous and general approach. For illus-tration and verification purposes we solve thesteady driven cavity problem (see Fig. 2), inwhich the continuum subdomain is describedby the Navier-Stokes equations solved by finite-element discretization. The hybrid solution is

FIGURE 2. Continuum and atomistic sub-domains for Schwarz coupling for the two-dimensional driven cavity problem.

expected to recover the fully continuum so-lution because the atomistic subdomain is farfrom solid boundaries and from regions of largevelocity gradients. This test, therefore, pro-vides a consistency check for the scheme.

Standard Dirichlet velocity boundary condi-tions for a driven cavity problem were appliedon the system boundaries; the horizontal ve-locity component on the left, right, and lowerwalls were held at zero while the upper-wallhorizontal velocity was set to 50 m/s. The ver-tical velocity component on all boundaries wasset to zero. Despite the relatively high veloc-ity, the flow is essentially incompressible andisothermal. The pressure is scaled by settingthe middle node on the lower boundary at at-mospheric pressure (1.013× 105 Pa).

Figure 3 shows the detailed structure of theapproach used for exchanging boundary condi-tions between the two subdomains. By center-ing DSMC cells on the finite element (FE) nodeswe can directly impose the molecular solutiononto the continuum calculation (after correctingfor mass conservation using Eq. 3). The impo-



FIGURE 3. Schematic of the boundary condi-tion imposition approach. Only the bottom leftcorner is shown.

sition of boundary conditions on the atomisticsubdomain is facilitated by the particle reser-voir shown in Fig. 3 which, in this implemen-tation acts also as part of the overlap region.Particles are created at locations x, y within thereservoir with velocities Cx, Cy, Cz drawn froma Chapman-Enskog velocity distribution f(C)given by [13],

f(C) = f0(C)Γ(C) (4)

where, C = C/(2kT/m)1/2 is the normalizedthermal velocity,

f0(C) =1

π3/2e−C2

(5)

and,

Γ(C) = 1 + (qxCx + qyCy + qzCz)(

25C2 − 1

)

− 2(τxyCxCy + τxzCxCz + τyzCyCz)− τxx(C2

x − C2z )− τyy(C2

y − C2z ) (6)

with,

qi = − κ

P

(2m

kT

)1/2 ∂T

∂xi(7)

τij =µ

P

(∂vi

∂xj+

∂vj

∂xi− 2

3∂vk

∂xkδij

)(8)

Here qi and τij are the dimensionless heatflux and stress tensor, respectively, with µ, κ, Pand v = (v1, v2, v3) being the viscosity, thermalconductivity, pressure, and mean fluid veloc-ity. The Chapman Enskog distribution of ve-locities can be generated using an “acceptance-rejection” scheme as detailed by Garcia andAlder [13].

The number and spatial distribution of par-ticles in the reservoir are chosen accordingto the overlying continuum cell mean densityand density gradients. After particles are cre-ated in the reservoir they move for a singleDSMC timestep. Particles that enter DSMCcells are incorporated into the standard convec-tion/collision routines of the DSMC algorithm.Particles that remain in the reservoir are dis-carded. Particles that leave the DSMC domainare also deleted from the computation. Therapid convergence of the Schwarz approach isdemonstrated in Fig. 4. The continuum nu-merical solution is reached to within ±10% atthe third Schwarz iteration and to within ±2%at the tenth Schwarz iteration. Our error es-timate, which includes the effects of statisti-cal noise [19] and discretization error due tofinite timestep and cell size, is approximately2.5%. Similar convergence of the vertical veloc-ity field is also observed.

The close agreement with the fully con-tinuum results indicates that the Chapman-Enskog procedure is not only theoretically ap-propriate, but also robust. Despite a Reynoldsnumber of Re ≈ 1, the Schwarz method(originally only shown to converge for ellipticproblems [23]) converges with negligible error.This is in agreement with the findings of Liu[24] who has recently shown that the Schwarzmethod is expected to converge for Re ∼ O(1).Extension of the the Schwarz method to flowswith higher Re ∼ O(100) has also been pos-



sible [12] provided suitable preconditioning isutilized.

5. ADAPTIVE MESH AND ALGORITHM RE-FINEMENT FOR COMPRESSIBLE FOR-MULATION

As discussed above, consideration of the com-pressible equations of motion leads to hybridmethods that differ significantly from their in-compressible counterparts. The hyperbolic na-ture of compressible flows means that steady-state formulations typically do not offer a sig-nificant computational advantage, and as a re-sult, explicit time integration is the preferredsolution method and flux matching is the pre-ferred coupling method. Given that the char-acteristic evolution time, τh, scales with thesystem size, the largest problem that can becaptured by a hybrid method is limited bythe separation of scales between the molecu-lar integration time and τh. Local mesh re-finement techniques [14, 34] minimize the re-gions of space that need to be integrated atsmall CFL timesteps (due to a fine mesh), suchas the regions adjoining the molecular subdo-main. Implicit timestepping methods [38] canalso be used to speed up the time integration ofthe continuum subdomain. Unfortunately, al-though both approaches enhance the computa-tional efficiency of the continuum subproblem,they do not alleviate the issues arising from thedisparity between the molecular timestep andthe total integration time.

As discussed in the introduction, over-whelming computational costs can be incurredwhen using a time-explicit flux-based couplingapproach to capture steady phenomena wherecompressibility effects are negligible, as is inmost cases, in dense fluids. In this case theintegration timestep of the continuum subdo-main also becomes of the order of the molecu-lar timescale, while the continuum subdomainis, presumably, much larger than the molecu-lar subdomain and evolves at a much longertimescale.This appears to not have been fullyappreciated by various groups that have at-tempted to develop dense-fluid hybrid meth-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

−12

−10

−8

−6

−4

−2

0

X (m)

U =

f (X

,Y=0

.425

× 10

−6m

)

Exact Iterate=1 Iterate=2 Iterate=3 Iterate=5 Iterate=10

FIGURE 4. Convergence of the horizontal ve-locity component along the y = 0.425 × 10−6mplane with successive Schwarz iterations.

ods based on the compressible continuum for-mulation and flux-based matching proceduresto solve steady and essentially incompressibleproblems.

On the other hand for compressible gas flow,locally refining the continuum solution cells tothe size of DSMC cells leads to a particularlyseamless compressible hybrid formulation inwhich DSMC cells differ from the neighboringcontinuum cells only by the fact that they areinherently fluctuating (the DSMC timestep re-quired for accurate solutions, see [3, 15, 18],is very similar to the CFL timestep of a com-pressible formulation). Thus a finite volumeformulation can be used to couple the two sub-domains quite naturally. In such a method[18, 19] the fluxes of mass, momentum, andenergy from DSMC to the continuum subdo-main given by particles leaving the DSMC re-gion and traveling toward the continuum sub-domain can be used directly for finite volumeintegration. Imposition of the continuum “in-terface conditions” onto DSMC requires the useof reservoirs similarly to the procedure outlinedin Sec. 4.1. The flux of mass, momentum, andenergy from the continuum to the atomistic do-main is provided by the particles that, upon ini-tialization in the reservoir at the continuum so-



lution conditions, travel into the DSMC region.In this paper we review recent developments[34, 35] that embed this methodology into anadaptive mesh refinement framework.

Another characteristic inherent to compress-ible formulations is the possibility of describingparts of the domain by the Euler equations ofmotion [34]. In that case, consistent couplingto the molecular formulation can be performedusing a Maxwell-Boltzmann distribution [14].

In a recent paper [4], Alexander et al. haveshown that explicit time-dependent flux-basedformulations preserve the fluctuating nature ofthe molecular description within the molecularregions, but the fluctuation amplitude decaysrapidly within the continuum regions; correctfluctuation spectra can be obtained in the en-tire domain by solving a fluctuating hydrody-namics formulation [21] in the continuum sub-domain.

5.1. Fully Adaptive Mesh and Algorithm Re-finement for a Dilute Gas

The compressible formulation of Garcia etal. [14], referred to as AMAR (Adaptive Meshand Algorithm Refinement), pioneered the useof mesh refinement as a natural frameworkfor the introduction of the molecular descrip-tion in a hybrid formulation. In AMAR, thetypical continuum mesh refinement capabilitiesare supplemented by an algorithmic refinement(continuum to atomistic) based on continuumbreakdown criteria. This seamless transition isboth theoretically and practically very appeal-ing.

In what follows we briefly discuss a re-cently developed [34, 35] fully adaptive AMARmethod. In this method, the DSMC providesan atomistic description of the flow while thecompressible two-fluid Euler equations serveas the continuum-scale model. The continuumand atomistic representations are coupled bymatching fluxes at the continuum-atomistic in-terfaces and by proper averaging and interpola-tion of data between scales. This is performedin three steps: a) the continuum solution val-ues are interpolated to create DSMC particles

FIGURE 5. Moving Mach 10 shock wavethough Argon. The AMAR algorithm tracks theshock by adaptively moving the DSMC regionwith the shock front.

in the reservoir region, here called buffer cells;b) the conserved quantities in each continuumcell overlaying the DSMC region are replacedby averages over particles in the same region;and c) fluxes recorded when particles cross theDSMC interface are used to correct the con-tinuum solution in cells adjacent to the DSMCregion. This coupling procedure makes theDSMC region appear as any other level in anAMAR grid hierarchy. Similarly to the over-lap region described for the Schwarz methodabove, the Euler solution information is passedto the particles via buffer cells surrounding theDSMC region. At the beginning of each DSMCintegration step, particles are created in thebuffer cells using the continuum hydrodynamicvalues.

The above algorithm allows grid and algo-rithm refinement based on any combination offlow variables and their gradients. Densitygradient-based refinement has been found to begenerally robust and reliable [34]. Concentra-tion gradients or concentration values withinsome interval are also effective refinement cri-teria especially for multispecies flows involv-ing concentration interfaces. In this particu-lar implementation, refinement is triggered byspatial gradients exceeding user defined toler-



90 100 110 120 130 140 150 160 170 180 1901

2

3

4

5

6

7

x 10−3

x/λ

Den

sity

(g

/cm

3 )

τm

= 9.44 τm

= 12.38 τm

= 15.33

AMARTheory

FIGURE 6. Moving Mach 10 shock wavethough Argon. The AMAR profile (red dots)is compared with the analytical time evolutionof the initial discontinuity (blue lines). τm is themean collision time.

ances. This approach follows from the contin-uum breakdown parameter method proposedby Bird [6].

Using the AMAR capabilities provided bythe Structured Adaptive Mesh Refinement Ap-plication Infrastructure (SAMRAI) developedat the Lawrence Livermore National Labora-tory [27], the above adaptive framework hasbeen implemented in a fully 3D, massively par-allel form in which multiple molecular (DSMC)patches can be introduced or removed asneeded.

Figure 5 shows the adaptive tracking of ashockwave of Mach number 10 used as a val-idation test for this method. Density gradient-based mesh refinement ensures the DSMC re-gion tracks the shock front accurately. Further-more, as shown in Fig. 6 the density profile ofthe shock wave remains smooth and is devoidof oscillations that are known to plague tradi-tional shock capturing schemes [5, 37].

6. DISCUSSION

One of the most important messages of thispaper is that boundary-condition impositionon molecular domains is quite independent of

the choice of the solution-coupling approach.As an example, consider the Schwarz method,which provides a recipe for making solutionsin various subdomains globally consistent sub-ject to exchange of Dirichlet conditions. Theimposition of these boundary conditions canbe achieved through any method, and no cer-tain method is favored by the coupling ap-proach. Flexibility in adopting appropriate el-ements from previous approaches and the im-portance of choosing the coupling method ac-cording to the flow physics are key steps tothe development of more sophisticated, next-generation hybrid methods.

Although hybrid methods provide signifi-cant savings by limiting molecular solutionsonly to the regions where they are needed, so-lution of time-evolving problems, which spana large range of timescales, is still not possibleif the molecular domain, however small, needsto be integrated for the total time of interest.New frameworks are, therefore, required thatallow timescale decoupling or coarse-grainedtime evolution of molecular simulations.

Significant computational savings can be ob-tained by using incompressible formulationwhen appropriate. Neglect of these simpli-fications can lead to a problem that is sim-ply intractable when the continuum subdo-main is appropriately large. It is interesting tonote that, when a hybrid method was used tosolve a problem of practical interest [1] whileproviding computational savings, the Schwarzmethod was preferred because it provided asteady solution framework with timescale de-coupling.

For dilute gases the Chapman-Enskog distri-bution provides a robust and accurate methodfor imposing boundary conditions. Furtherwork is required for the development of simi-lar frameworks for dense liquids.

ACKNOWLEDGMENTS

The authors wish to thank R. Hornung, A. L.Garcia, A. T. Patera, and B. J. Alder for help-ful comments and discussions. The work out-lined in Sec. 5.1 was performed jointly with R.



Hornung and A. L. Garcia. This work was sup-ported in part by the Center for ComputationalEngineering, and the Center for Advanced Sci-entific Computing, Lawrence Livermore Na-tional Laboratory, US Department of Energy,W-7405-ENG-48. The authors also acknowl-edge the financial support of the Singapore-MIT alliance.

REFERENCES

1. Aktas, O. and Aluru, N. R., A combined con-tinuum/DSMC technique for multiscale anal-ysis of microfluidic filters, Journal of Computa-tional Physics 178:342–372, 2002.

2. Alexander, F. J., and Garcia, A. L., The directsimulation Monte Carlo method, Computers inPhysics 11:588–593, 1997.

3. Alexander, F. J., Garcia, A. L., and Alder, B. J.,Cell size dependence of transport coefficientsin stochastic particle algorithms, Physics of Flu-ids 10:1540-1542, 1998; erratum, Physics of Flu-ids 12:731, 2000.

4. Alexander, F., Garcia, A. L. and TartakovskyD., Algorithm refinement for sochastic partialdiffential equations: I. Linear diffusion, Journalof Computational Physics 182(1):47–66, 2002.

5. Arora, M., and Roe, P. L., On postshock oscil-lations due to shock capturing schemes in un-steady flows, Journal of Computational Physics130:25, 1997.

6. Bird, G. A., Breakdown of translational androtational equilibrium in gaseous expansions,AIAA Journal 8:1998, 1970.

7. Bird, G. A., Molecular Gas Dynamics and the Di-rect Simulation of Gas Flows, Clarendon Press,Oxford, 1994.

8. Chapman, S. and Cowling, T. G., The Math-ematical Theory of Non-Uniform Gases, Cam-bridge University Press, Cambridge, 1970.

9. Delgado–Buscalioni, R., and Coveney, P. V.,Continuum-particle hybrid coupling for mass,momentum and energy transfers in unsteadyfluid flow, Physical Review E 67:046704, 2003.

10. Eggers, J., and Beylich, A., New algorithmsfor application in the direct simulation MonteCarlo method, Progress in Astronautics andAeronautics 159:166–173, 1994.

11. Flekkoy, E. G., Wagner, G., and Feder, J., Hy-brid model for combined particle and contin-uum dynamics, Europhysics Letters 52:271–276,2000.

12. Fortin, M. and Aboulaich, R. Schwarz’s de-

composition method for incompressible flowproblems, 1st Int. Symp. on Domain Decomposi-tion Methods for Partial Differential Equations, R.Glowinski, G. Golub, G. Meurant, and J. Peri-aux, Eds., SIAM, Philadelphia, 333–349, 1988.

13. Garcia, A .L., and Alder, B. J., Generation of theChapman-Enskog distribution, Journal of Com-putational Physics 140:66–70, 1998.

14. Garcia, A. L., Bell, J. B., Crutchfield, W. Y., andAlder, B. J. Adaptive mesh and algorithm re-finement using direct simulation Monte Carlo.Journal of Computational Physics 54 :134–155,1999.

15. Garcia, A. L., and Wagner, W. Time step trun-cation error in direct simulation Monte Carlo.Physics of Fluids 12:2621–2633, 2000.

16. Hadjiconstantinou, N. G., and Patera A. T.Heterogeneous atomistic-continuum represen-tations for dense fluid systems. InternationalJournal of Modern Physics C, 8:967–976, 1997.

17. Hadjiconstantinou, N. G., Hybrid atomistic-continuum formulations and the movingcontact-line problem. Journal of ComputationalPhysics 154:245–265 1999.

18. Hadjiconstantinou, N. G. Analysis of dis-cretization in the direct simulation MonteCarlo. Physics of Fluids 12:2634–2638, 2000.

19. Hadjiconstantinou, N. G., Garcia, A. L., Bazant,M. Z. and He, G., Statistical error in parti-cle simulations of hydrodynamic phenomena.Journal of Computational Physics 187:274–297,2003.

20. Hash, D., and Hassan, H. A hybridDSMC/Navier-Stokes solver. AIAA Paper95-0410, 1995.

21. Landau, L. D., and Lifshitz, E. M. Statistical Me-chanics Part 2, Pergamon Press, Oxford, 1980.

22. Li, J., Liao, D., and Yip S. Nearly exact solutionfor coupled continuum/MD fluid simulation.Journal of Comput-Aided Materials Design 6:95–102, 1999.

23. Lions, P. L., On the Schwarz alternatingmethod. I. 1st Int. Symp. on domain decomposi-tion methods for partial differential equations, R.Glowinski, G. Golub, G. Meurant, and J. Peri-aux, Eds., SIAM, Philadelphia, 1–42, 1988.

24. Liu, S. H. On Schwarz alternating methods forthe incompressible Navier-Stokes equations.Siam Journal of Scientific Computing 22:1974–1986, 2001.

25. Mansour, M. M., Baras, F. and Garcia, A. L.On the validity of hydrodynamics in planePoiseuille flows. Physica A 240:255–267, 1997.

26. O’Connell, S. T., and Thompson, P.A. Molec-



ular dynamics-continuum hybrid computa-tions: A tool for studying complex fluid flows.Physical Review E, 52:R5792–R5795, 1995.

27. Structured Adaptive Mesh RefinementApplication Infrastructure (SAMRAI),http://www.llnl.gov/CASC/

28. Wadsworth, D. C., and Erwin, D. A. One-dimensional hybrid continuum/particle simu-lation approach for rarefied hypersonic flows.AIAA Paper 90-1690, 1990.

29. Wadsworth, D. C. and Erwin, D. A., Two-dimensional hybrid continuum/particle simu-lation approach for rarefied hypersonic flows.AIAA Paper 92-2975, 1992.

30. Wagner, W. A convergence proof for Bird’sdirect simulation Monte Carlo method forthe Boltzmann equation. Journal of StatisticalPhysics, 66:1011, 1992.

31. Wesseling, P. Principles of Computational FluidDynamics, Springer, (2001).

32. Wijesinghe, H. S. Hybrid atomistic-continuumformulations for gaseous flows. PhD thesis,Dep. of Aeronautics and Astronautics, Mas-sachusetts Institute of Technology, 2003.

33. Wijesinghe, H. S., and Hadjiconstantinou, N.G. A hybrid continuum-atomistic scheme forviscous incmpressible flow. Proc. of 23th Int.Symp. on Rarefied Gas Dynamics, Whistler,British Columbia, 907–914, 2002.

34. Wijesinghe, H. S., Hornung, R., Garcia, A. L.and Hadjiconstantinou, N. G., “3–dimensionalhybrid continuum-atomistic simulations formultiscale hydrodynamics. Proc. of the 2003 Int.Mechanical Engineering Congress and Exposition,Vol. 2, Paper No. NANOT-41251.

35. Wijesinghe, H. S., Hornung, R., Garcia, A.L., and Hadjiconstantinou, N. G. Three–dimensional hybrid continuum-atomistic sim-ulations for multiscale hydrodynamics. Journalof Fluids Engineering (appear).

36. Wong, J. S., Darmofal, D. L. and Peraire, J. Thesolution of the compressible euler equations atlow mach numbers using a stabilized finite el-ement algorithm. Computer Methods in AppliedMechanics and Engineering 190:5719–5737, 2001.

37. Woodward, P. R., and Colella, P. The numericalsimulation of two-dimensional fluid flow withstrong shocks. Journal of Computational Physics54:115, 1984.

38. Yuan, X., and Daiguji, H. A specially com-bined lower-upper factored implicit schemefor three dimensional compressible Navier-Stokes equations. Computers and Fluids 30: 339-363, 2001


Documents

Discussion of Hybrid Atomistic-Continuum Methods for …web.mit.edu/ngh/www/Hettithanthrige.pdf · 2004. 8. 21. · pling methods nearly 50 years of continuum computational ﬂuid