A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with...

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Comput. Methods Appl. Mech. Engrg. 197 (2008) 821–833

A mesoscopic bridging scale method for fluids and couplingdissipative particle dynamics with continuum finite element method

Milos Kojic a,b,*, Nenad Filipovic a,b, Akira Tsuda a

a Harvard School of Public Health, 665 Huntington Avenue, Boston, MA 02115, USAb University of Kragujevac, 34000 Kragujevac, Serbia

Received 10 May 2007; received in revised form 6 August 2007; accepted 17 September 2007Available online 25 September 2007

Abstract

A multiscale procedure to couple a mesoscale discrete particle model and a macroscale continuum model of incompressible fluid flowis proposed in this study. We call this procedure the mesoscopic bridging scale (MBS) method since it is developed on the basis of thebridging scale method for coupling molecular dynamics and finite element models [G.J. Wagner, W.K. Liu, Coupling of atomistic andcontinuum simulations using a bridging scale decomposition, J. Comput. Phys. 190 (2003) 249–274]. We derive the governing equationsof the MBS method and show that the differential equations of motion of the mesoscale discrete particle model and finite element (FE)model are only coupled through the force terms. Based on this coupling, we express the finite element equations which rely on theNavier–Stokes and continuity equations, in a way that the internal nodal FE forces are evaluated using viscous stresses from the meso-scale model. The dissipative particle dynamics (DPD) method for the discrete particle mesoscale model is employed. The entire fluiddomain is divided into a local domain and a global domain. Fluid flow in the local domain is modeled with both DPD and FE method,while fluid flow in the global domain is modeled by the FE method only.

The MBS method is suitable for modeling complex (colloidal) fluid flows, where continuum methods are sufficiently accurate only inthe large fluid domain, while small, local regions of particular interest require detailed modeling by mesoscopic discrete particles. Solvedexamples – simple Poiseuille and driven cavity flows illustrate the applicability of the proposed MBS method.� 2007 Elsevier B.V. All rights reserved.

Keywords: Multiscale modeling of fluid flow; Mesoscopic bridging scale method; Dissipative particle dynamics method; Finite element method; CouplingNavier–Stokes and dissipative particle dynamics equations

1. Introduction

Although computer technology is continuously advanc-ing, still there are limitations facing the ever increasingdemands in computer modeling of scientific and engineer-ing problems. For example, we have such limitations inmolecular dynamics (MD) modeling of domains of onlyseveral millimeters in size and over a time span of seconds.An illustrative example of these limitations can be found in

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.09.011

* Corresponding author. Address: Center for Scientific Research ofSerbian Academy of Sciences and Arts and University of Kragujevac,34000 Kragujevac, Serbia.

E-mail address: [email protected] (M. Kojic).

the popularly written article [1], where significant achieve-ments in modeling protein conformation changes are pre-sented. However, the presented model is still very farfrom satisfying the needs in molecular biology of livingcells. It is cited that in a MD modeling of a small proteinin water, half a million sets of Cartesian coordinates aregenerated in a nanosecond time period for the positionsof 10,000 atoms. The handling of such large amounts ofdata is beyond the practical capabilities of computer hard-ware and software currently available.

During last decade, tremendous efforts in research incomputational physics and mechanics have been devotedtowards the development of methods to overcome limita-tions of the MD methods. One direction that has been

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822 M. Kojic et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 821–833

taken, is so-called coarse-graining, i.e., formulating meth-ods of discretizing continuum (fluids and solids) into meso-scale particles of micron length scale and micro-secondstime scale, considering these particles as clusters of atoms.This discretization relies on the Voronoi cell division (tes-sellation) of a continuum [2–4]. The origin of these discreteparticle (DP) methods comes from research in astro-phys-ics [5]. The Lagrangian description of motion employedin the DP methods assumes appropriate quantification ofinteraction forces, which include conservative, dissipativeand random forces (and moments). One of the mostadvanced methods in this field is the dissipative particledynamics (DPD) method for fluids, introduced by Hoo-gerbrugge and Koelman [6], further generalized theoreti-cally, particularly by Espanol and co-authors [7–16],Flekkoy and co-authors [2,3], and in [17–19], and appliedto various problems [20–24]. The DPD method will bedescribed here in some detail and used subsequently.

Another direction taken to overcome the MD limitationshas been in the development of multiscale methods, whichcouple MD and continuum methods. A review of the mul-tiscale methods is given by Curtin and Miller [25], Liu et al.[26] (see also [27,28]). One of these methods, which is usedas the basis for our developments, is the bridging scale(BS) method, (Liu and co-workers, e.g. Wagner and Liu[29], Tang et al. [30]) which was implemented in modelingvarious problems [31–35]. Development of the BS methodis inspired by other methods, such as the immerse boundarymethod in modeling of complex fluids [36,37]. The mainidea of this method is that the displacement of an atomcan be decomposed into a coarse scale displacement, calcu-lated by a continuum-based method, such as the finite ele-ment (FE) method, and a fine scale correction indisplacement calculated using the MD solution. The coarsescale displacement of an atom can be considered as a mean(averaged) displacement, so that the displacement correc-tion represents a fine scale fluctuation around the meanvalue [30]. The correction is obtained by introducing theappropriate projection operator, which provides theorthogonality of the fine scale displacement correction withrespect to the coarse scale (FE) interpolation functions.Moreover, what is most significant and fundamental, is thatthe decomposition in displacement allows the kinetic energyof the system to be represented as a sum of coarse scale andfine scale kinetic energies, which are uncoupled with respectto coarse scale and fine scale velocities. From this form ofkinetic energy follow differential equations of motion inboth scales which are coupled through the force terms only.Boundary conditions between the MD and FE domains arehandled by a history kernel matrix implemented using ana-lytical or numerical procedures [29,34,38].

Following the above cited properties of the BS method,we extend the formulation to couple the mesoscale DPDmodel with the macroscale continuum FE model. Namely,instead of coarse scale mean displacement and fine scaledisplacement fluctuation of an atom, we here introducethe coarse scale mean velocity and fine scale velocity fluctu-

ation of a mesoscopic particle. Bridging these two scales inmodeling of fluid flow offers a tool to efficiently solve prob-lems for which a detailed description of flow is needed inthe local regions only, while a usual FE method can beemployed for other domains of the flow field. The detaileddescription is in the mesoscale (hence an MD modelingwith the excessive computational demands avoided), suffi-ciently accurate in many applications, which is appropri-ately coupled with a cost-effective continuum description.

The proposed approach is particularly attractive formodeling a dilute mixture flow with a detailed insight intoflow in certain local regions, as in case of, for example,blood flow in a large artery with growing thrombus atthe wall. Development of the thrombus is dependent onboth the global hemodynamics within the artery, and localflow and interactions between blood constituents within asmall region around the thrombus. Continuum methodsare applicable for modeling global artery hemodynamics,but are inadequate for determination of local flows whichinvolve platelet activation, aggregation and adhesion. Byusing a discrete particle method in a small domain, eachplatelet can be modeled by a mesoscopic particle withinthe blood [39–45]. Blood can be considered as a Newtonianfluid in which plasma is the dominant medium due to lowhematocrit (volume fraction occupied by red blood cells)near the wall and small concentration of platelets. Sincethe mechanical events in the global and local regions arecoupled, a numerical procedure for this coupling, such asthe MBS method, is needed.

In practical applications of the introduced mesoscopicbridging scale (MBS) method we assume that the wholefluid domain is divided into: (a) global domain, where onlya continuum macroscopic model us used; and (b) localdomains where both macroscopic model and mesoscopicdiscrete particle model are used, with the appropriate cou-pling between these models.

Although the present method is applicable to dilute mix-tures with non-deformable particles, it has advantages withrespect to other multiscale methods, such as immersedboundary method [36,37], which allow deformation of par-ticles within a colloidal fluid. The advantage is, first of all,in simplicity of MBS. Also, the MBS provides an insightinto the interaction among particles and the evolution ofthe interaction forces during the flow, important in someapplications (mentioned above in modeling of platelet-mediated thrombosis).

The presentation is organized as follows: In the next sec-tion, we describe the proposed MBS method for fluid, uti-lizing the DPD method for the mesoscale model and theFE method for the macroscale continuum. In Section 3,we describe the coupling of the DPD and Navier–Stokesequations, as well as the boundary conditions at the inter-face between the global FE domain and the local domainswith two scale models coupled. In Section 4, we presenttwo examples, illustrating the applicability of the MBSmethod. Finally, we summarize the findings and concludeour study in Section 5.

M. Kojic et al. / Comput. Methods Appl. Mech. Engrg. 197 (2008) 821–833 823

2. The mesoscopic bridging scale (MBS) method for fluid

2.1. Decomposition of velocities

We discretize a fluid domain into the mesoscale discreteparticles, further called ‘‘particles’’, representing the finescale model. We also discretize the same fluid domain intofinite elements as the coarse scale model (Fig. 1). As we sta-ted in Introduction, the basic assumption is that the veloc-ity of a particle ‘‘i’’, vi, at any time, can be expressed as

vi ¼ �vi þ v0i; ð1Þ

where �vi is the coarse scale velocity, representing the meanparticle velocity, obtained by the FE method; and v0i is thevelocity correction, or fine scale velocity fluctuation, ob-tained from the fine scale solution. The coarse scale veloc-ity �vi can be expressed in terms of nodal velocities, V, as

�vi ¼ NiV; ð2Þwhere Niðni

1; ni2; n

i3Þ is the matrix of interpolation functions

for velocities within a finite element, and nik; k ¼ 1; 2; 3 are

the natural coordinates of particle ‘‘i’’. The nodal velocityvector is

VT ¼ fV 1x; V 1y ; V 1z; . . . ; V Nx; V Ny ; V NzgT; ð3Þ

where N is the number of FE nodes. The relations (1) and(2) can be written for all particles within the finite element,

v ¼ �vþ v0; ð4Þ�v ¼ NV: ð5Þ

The vector v and matrix N are defined by (we use vT and NT

for more compact writing)

vT ¼ fvT1 ; v

T2 . . . ; vT

nagT ð6Þ

and

NT ¼ ½NT1 ;N

T2 ; . . . ;NT

na�: ð7Þ

Here na is the number of particles within the finite element.Note that dimensions of the vector v (and v0) and matrix N

are 3na and 3na � 3N , respectively.The velocity vi of particles can be calculated using a

method of discrete particle dynamics, such as the dissipa-

Fig. 1. Discrete particle (DP) and finite element (FE) models of the samefluid domain (one finite element, 2D representation). Velocities andinteraction forces.

tive particle dynamics (DPD) method, see, e.g. [12,21].Details of the DPD method are given in Section 3.1.

We now introduce the projection operator Q to provideorthogonality condition for velocity v0 with respect to theinterpolation functions Nij. Following the idea introducedin [29,30], our aim is to minimize a residual R, which isdefined as

R ¼ ðv�NVÞTMAðv�NVÞ; ð8Þ

where MA is the discrete particle mass matrix,

MA ¼

m1I3

m2I3

. . .

. . .

mnaI3

2666666664

3777777775: ð9Þ

Here m1, m2, . . ., mna are masses of particles. The dimen-sion of the particle mass matrix MA is 3na � 3na. The min-imum of the residual corresponds to the followingrelationship:

MV ¼ NTMAv; ð10Þ

where M is the FE mass matrix, with dimensions 3N � 3N ,expressed as

M ¼ NTMAN: ð11Þ

Since the basic assumption of the mesoscale discretizationrelies on the Voronoi tesselation of space, the matrix M

represents the finite element consistent mass matrix �M, i.e.,

M ¼M ¼Z

Vq�NT �NdV ; ð12Þ

where q is the fluid density, V is the element volume, and�Nðn1; n2; n3Þ is the matrix of interpolation functions of or-der 3N � 3N , evaluated at material points continuouslydistributed within the finite element. From the relation(10) we obtain

V ¼M�1NTMAv: ð13Þ

Substituting (13) into (5) and then into (4), we obtain thefine scale velocity correction v0 as

v0 ¼ ðI�NM�1NTMAÞv ð14Þ

expressed in terms of the fine scale (mesoscale) velocity v.This equation can be written as

v0 ¼ Qv; ð15Þ

where

Q ¼ I� P ð16Þ

and

P ¼ NM�1NTMA ð17Þ


are the projection operators, and I is the identity matrix.We note that dimensions of the projection operators P

and Q are 3na � 3na.We list the following relations (see [29,30]) which are

important for further developments:

PP ¼ NM�1 NTMAN|fflfflfflfflffl{zfflfflfflfflffl}¼M

M�1NTMA ¼ P ð18Þ

and

QQ ¼ ðI� PÞðI� PÞ ¼ Q: ð19Þ

Also,

PTMA ¼MA NM�1NTMA|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}¼P

¼MAP ð20Þ

and consequently

QTMA ¼MAQ: ð21Þ

2.2. Decomposition of kinetic energy

We now calculate the kinetic energy Ek of particleswithin one finite element as

Ek ¼1

2vTMAv ¼ 1

2ð�vT þ v0TÞMAð�vþ v0Þ

¼ 1

2�vTMA�vþ 1

2v0TMAv0 þ 1

2�vTMAv0 þ 1

2v0TMA�v: ð22Þ

Using (15) and (5), we find that the last two terms are equalto zero, since we have that

v0TMA�v ¼ vTQTMANV

¼ vTðMAN�MANM�1 NTMAN|fflfflfflfflffl{zfflfflfflfflffl}¼M

ÞV ¼ 0 ð23Þ

This orthogonality relation results from the definition ofthe projection operators Q and P according to (16) and(17), and the relations (20) and (21). The last two termsare equal

v0TMA�v ¼ �vTMAv0 ¼ ðMAÞijv0i�vj ¼ ðMAÞij�viv0j

because MA is symmetric, ðMAÞij ¼ ðMAÞji.Therefore, the kinetic energy of a finite element, Ek, can

be expressed as the sum of two terms, kinetic energy of thecoarse (macroscale), �Ek, and kinetic energy of the velocitycorrections from the fine scale (mesoscale), E0k. Note thatthey are decoupled with respect to velocities:

Ek ¼ �Ek þ E0k; ð24Þ

where

�Ek ¼1

2�vTMA�v ¼ 1

2VTMV ¼ 1

2VT �MV ð25Þ

and

E0k ¼1

2v0TMAv0: ð26Þ

Decomposition of kinetic energy into a sum of coarse scalekinetic energy �Ek and fine scale kinetic energy E0 (Eq. (24))is the result of the fundamental importance for the meso-scopic bridging scale (MBS) method for fluids (as in theBS method). The relation (24) relies on the orthogonalityrelation (23).

2.3. Differential equations of motion

We use the principle of virtual power [46] to obtain dif-ferential equations of motion of fluid within one finite ele-ment in the local domain. Our mechanical system possesses3na þ 3N degrees of freedom, corresponding to particlefluctuation velocities v0 and FE nodal velocities V, respec-tively. The system is subjected to external and internalforces. The differential equations are

MA _v0 ¼ f 0ext þ f 0int; ð27Þ

where f 0ext and f 0int are the external force (such as gravity, orinertial forces due to motion of the reference coordinatesystem) and internal force (from action of surrounding par-ticles), respectively; and

M _V ¼ Fext þ Fint; ð28Þ

where the vectors Fext and Fint are the external and internalforces corresponding to the FE nodal velocity vector V (de-tails about these vectors are given below).

To find the relationships between the forces f 0ext and f 0int,and the interaction forces between particles, we proceed asfollows. First, from (15) we have

_v0 ¼ Q _v: ð29Þ

Substituting this relation into (27) and using the relation(21) we obtain

QTMA _v ¼ f 0ext þ f 0int: ð30Þ

On the other hand, if we consider motions of individualparticles, from the Newton law follow the differential equa-tions of motion of particles

MA _v ¼ fext þ f int: ð31Þ

The external force vector fext includes all external particleforces fext

i ; i ¼ 1; 2; . . . ; na. Regarding the internal forcesfor a particle ‘‘i’’, the internal force f int

i is given as

f inti ¼

Xj

f ij; ð32Þ

where f ij are the interaction forces exerted by the surround-ing particles ‘‘j’’ shown in Fig. 1. The summation goes overall particles ‘‘j’’ which are in the interaction domain of theparticle ‘‘i’’. We will describe details for the calculation ofthe interaction forces in Section 3.

From Eqs. (30) and (31) follows:

f 0ext þ f 0int ¼ QTðfext þ f intÞ: ð33Þ


This is the relationship between the fine scale particle forcescorresponding to the fluctuation velocities v0 and the inter-action forces corresponding to the particle velocities v.

With the relationship (33), differential equations ofmotion (27) can be written as,

MA _v0 ¼ QTðfext þ f intÞ: ð34Þ

We note that one of the differential equations of motion ofparticles (31) or (34), can be employed. The use of (31) iscomputationally more efficient because it does not requireevaluation of the projection operator Q. We emphasize,however, that the use of the projection operator Q servedto provide the theoretical background of the kinetic energydecomposition (24), as in the bridging scale method [29,30].

Details regarding the vectors Fext and Fint are as follows.Components of Fext are

F extKj ¼

ZV

NKf Vj dV ; ð35Þ

where K = 1,2, . . . ,N are the node numbers, N K are theinterpolation functions [49], and f V

j , j = 1,2,3, are the com-ponents of the volumetric force in a Cartesian coordinatesystem x, y, z. Components of the force Fint can be ex-pressed as

F intKl ¼

ZV

oNK

oxrl1 þ

oNK

oyrl2 þ

oNK

oxrl3

� �dV ; ð36Þ

where l ¼ 1; 2; 3 stands for x, y, z, and rl1; rl2; rl3 are thestress components in the Cartesian coordinate system.The stresses represent the internal forces (per unit area)and they can be calculated from the interaction forces[47]. Therefore, we can write the following functional rela-tionship in a vector form,

Fint ¼ Fintðf intÞ: ð37ÞIt is important to emphasize that the system of differentialEqs. (28) and (31) (or (34)), with the relationships (37), arecoupled through the forces acting on the particles.

Similarly to the coupling between the MD and contin-uum models according to the bridging scale (BS) method[29,30], the presented mesoscopic bridging scale method(MBS) for fluids enables us to couple the mesoscale modeland the continuum models with the following importantfeatures:

1. Differential equations of motion for the fine scale (meso-scale) and coarse scale (continuum) are coupled throughthe interaction forces between the mesoscale particles.

2. By calculating motions of the mesoscale particles withthe use of the discrete particle DPD method for instance,the corresponding nodal forces of a FE model can bedetermined. The coarse scale velocities, based on thecontinuum balance equations, can also be obtained.

3. The method enables us to divide a fluid domain into thedomains for a coarse scale modeling (only), and localdomains modeled by both coarse scale (FE) and detailedmesoscale DPD, with coupling between the governing

equations of the two (or multiple) different domains ina consistent manner. Appropriate boundary conditionsat the common boundary between the two domainsmust be imposed.

These results are fundamental for our MBS methodol-ogy and will be discussed in the next section in more detailwith respect to practical applications.

3. Coupling DPD and Navier–Stokes FE equations

3.1. Differential equations of motion according to DPD

method

As mentioned in Introduction, a number of discrete par-ticle methods for modeling of incompressible fluid flowhave been proposed and implemented [2–24,48]. One ofthese methods, the dissipative particle method (DPD),has gained particular attention and has been widely used[19–24]. For the sake of the completeness of the presenta-tion, we here describe the main features of the DPDmethod.

In the DPD methods, the fluid domain is discretized intothe mesoscopic particles (see Fig. 1) which in general havedifferent masses. These particles are represented by thepoints which are the centers of MD particle clusters. Theparticle clusters occupy the volumes which can be obtainedby Voronoi tessellation of the fluid space [2–4,9,12]. Differ-ential equations of motion have the form (31), where theinteraction force f ij (see Eq. (32)) is

f ij ¼ fCij þ fc

ij þ ~f ij; ð38Þ

where fCij , fc

ij and ~f ij are the conservative (repulsive), viscousand random forces, respectively. These forces can be ex-pressed as follows (see, for example, [3,10,12,21]):

fCij ¼

mia 1� rij

rmax

� �r0

ij; rij < rmax;

0; rij P rmax;

(ð39Þ

fcij ¼ �micwcðr0

ij � vijÞr0ij; ð40Þ

~f ij ¼ mir~wfijðDtÞ�1=2r0

ij; ð41Þ

where a is a material coefficient, ri and rj are the positionvectors of particles ‘‘i’’ and ‘‘j’’, rij ¼ ri � rj, rij ¼ kri � rjk,and r0

ij ¼ rij=rij; rmax is the radius of interaction domain be-tween particles; vij ¼ vi � vj is the relative velocity; c is thefriction coefficient or normal damping coefficient; wc and~w are the weight functions for viscous and random forces,with the relation wc ¼ ~w2 [8,21]; r is the random forceamplitude (for unit mass) r ¼ ð2ckBT Þ1=2 [8,21]; fij is a ran-dom number with zero mean and unit variance (chosenindependently for each pair of DPD particles and timestep); and Dt is a time step size used in the integration of dif-ferential equations of motion. For the sake of comparisonwith the MD interaction forces, we mention that the MDinteraction forces assume the conservative forces only,which can be calculated from an energy potential.


Differential equations of motion (31) can be solved incre-mentally by, for example, a Newmark incremental method[49], or a Verlet algorithm [21,29]. A review of various meth-ods used for integration of differential Eq. (31) is given in[19]. A Verlet-type algorithm is summarized in Table 1.

To find the DPD solution, appropriate boundary condi-tions must be satisfied. We implement boundary condi-tions, such as, for example, no-slip, Maxwellian orspecular boundary conditions at the walls, or periodicityconditions at surfaces where particles are entering or leav-ing the flow domain [50–55]. Similar boundary conditionshave been introduced in the MD, MD–FE multiscale mod-eling [26,38,56,57]. Boundary conditions at the commonFE–DPD boundary are described in Section 3.4.

3.2. Finite element Navier–Stokes equations

In this section we present the finite element (FE) equa-tions of balance in a form commonly used in the literature(e.g. [49,58–61]), and used here for the global fluid domain(modeled by finite elements only). This presentation alsoserves to clarify the modifications of these equations whichwill be introduced in Section 3.3. The Navier–Stokes equa-tions and continuity equation for incompressible fluid are

qovi

otþ ovi

oxjvj

� �¼ orij

oxjþ f V

i ; ð42Þ

ovj

oxj¼ 0; ð43Þ

where q is fluid density, rij are stresses at a fluid point, vi

are the velocity components, and f Vi is (external) body

force; summation is assumed for the repeated index ‘‘j’’,j = 1, 2, 3. The stresses and the constitutive equations forthe Newtonian fluid are

rij ¼ �pdij þ sij; ð44Þ

sij ¼ lovi

oxjþ ovj

oxi

� �; ð45Þ

where p is the pressure, sij are the viscous stresses, and l isthe fluid viscosity.

Table 1Computational steps according to a Verlet-type DPD integration scheme

Step Calculate

1 tf i ¼ tfCi þ tf

ci þ t~fi ¼

Pj

tfCi þ tf

ci þ t~fi

�þ tþDtfext

i2 ~vi ¼ tvi þ Dtðk=mÞtfi

3 tþDtri ¼ tri þ Dt~vi

4 tþDtf i ¼ tþDtfCi

tþDtri �

þ tþDtfci

tþDtri;~vi �

þ tþDt~fitþDtri �

þ tþDtfexti

5 tþDtvi ¼ ~vi þ Dtð1=2mÞtþDtf i

We note that the increment of velocity is calculated from the mean forcebetween the weighted forces at start and end of time step; increment ofdisplacement is obtained using the weighted force at start of time step; theconservative and stochastic forces at end of time step are evaluated usingparticle position t+Dtri at end of time step; while for the viscous force tþDtf

ci

the vector t+Dtri and a weighted velocity ~vi are used. Values of theparameter k are between 0 and 1 (we used k = 1/2) and time stepDt = DtDPD corresponds to the DPD integration.

By a standard Galerkin procedure [58,59], Eqs. (42) and(43), with use of (44) and (45), are transformed into a FEincremental-iterative form (for iteration ‘‘i’’)

1Dt Mþ tþDtK̂ði�1Þ Kvp

KTvp 0

" #DVðiÞ

DPðiÞ

( )¼

tþDtFextði�1Þ

0

( )

�1Dt Mþ tþDtK̂ði�1Þ Kvp

KTvp 0

" #tþDtVði�1Þ

tþDtPði�1Þ

( )þ

1Dt MtV

0

� �;

ð46Þ

where DVðiÞ and DPðiÞ are the vectors of nodal velocity andpressure increments (nodal pressure vector is P), andtþDtK̂ði�1Þ is the matrix

tþDtK̂ði�1Þ ¼ tþDtKði�1Þ þ Kl: ð47Þ

The matrices tþDtKði�1Þ and Kl are the convective and vis-cous matrices, respectively.

The terms in the above matrices are

MKJ½ �mm ¼Z

VqNKNJ dV ; no sum on m; ð48Þ

tþDtKði�1ÞKJ

h imn¼ q

ZV

NKN J ;jtþDtvði�1Þ

j dV

mm

þ qZ

VNK

tþDtvði�1Þm;n N J dV ;

no sum on m; ð49Þ

ðKlÞKJ

� �mm¼Z

VlN K;jNJ ;jdV ; no sum on m; ð50Þ

ðKvpÞKJ

� �mm¼ �

ZV

oN K

oxmN̂ J dV ; no sum on m; ð51Þ

where N̂ J are the interpolation functions for pressure; andm and n denote components corresponding to x; y; z coordi-nate directions. The components of the external force vec-tor tþDtFextðiÞ are

tþDtF extði�1ÞKj ¼

ZV

N KtþDtf V

j dV þZ

SN K

tþDtrði�1Þjl nldS; ð52Þ

where nl (l ¼ 1; 2; 3) are components of the normal n to theelement surface. Note that when these equations, corre-sponding to a finite element, are assembled, only surfaceforces on the external domain boundary remain. Also,the time step here is Dt which is usually larger than DDPD

in Table 1, Dt ¼ mDDtDPD where mD is usually of order 103.

3.3. Coupling the DPD and finite element Navier–Stokes

equations

In order to couple the finite element equations of bal-ance and the DPD Eq. (31), we keep the stress terms inthe form (44) and transform the differential equations ofbalance (42) and (43) into the FE equations as in Section3.2. Then we obtain the incremental-iterative FE equationsas


1Dt Mþ tþDtK̂ði�1Þ Kvp

KTvp 0

" #DVðiÞ

DPðiÞ

( )

¼tþDtFintði�1Þ

0

( )þ

tþDtFextði�1Þ

0

( )

�1Dt Mþ tþDtKði�1Þ Kvp

KTvp 0

" #tþDtVði�1Þ

tþDtPði�1Þ

( )

þ1Dt MtV

0

� �: ð53Þ

The internal nodal force vector is

tþDtF intði�1ÞKi ¼ �

ZV

NK;jtþDtsði�1Þ

ij dV ; ð54Þ

where tþDtsði�1Þij are the viscous stresses at end of time step.

The pressure part of the total stress tþDtrði�1Þij is taken into

account by the interpolation of pressure field over the finiteelements, as in Section 3.2. Note that we now have in (53)the internal force tþDtFintði�1Þ due to viscous stressestþDtsði�1Þ. This internal force replaces the viscous nodalforce equal to Kl

tþDtVði�1Þ on the right-hand side in Eq.(46).

A note about the coupling the DPD and Navier–Stokesdifferential equations should be added for the clarity of thephysical background of this coupling. Namely, the DPDequations are true Lagrangian, while the Navier–Stokesequations rely on the Eulerian description of motion. Inthe above derivation, the basis of the coupling relies onthe Lagrangian description expressed by Eqs. (27) and(28) and the interdependence of the force terms in thesetwo equations. The Navier–Stokes equations express thebalance of linear momentum of the fluid within a finite ele-ment in which the internal nodal forces are evaluated usingthe interaction forces among the mesoscale particles, there-fore the Lagrangian background of the coupling is pre-served in this formulation.

The formulation (53) is suitable for coupling with theDPD model because the shear stresses can be evaluatedby using the interaction forces tþDtf

intði�1Þi form the DPD

solution (see Eqs. (32) and (38), and Table 1). It is furtherassumed that the Eq. (53) written for one finite element isevaluated for all finite elements within the local domains(where both FE and DPD models are employed). Withthe appropriate assemblage procedure [44,49,58–60] weobtain the resulting system of equations which retains theform (53).

The system of incremental equations for the whole fluiddomain, which includes both global and local domains, isformed from the FE Eqs. (46) and (53). The unknown vari-ables in this system of equations are the increments ofnodal velocities DVðiÞ and increments of nodal pressuresDPðiÞ. The solution iterations continue until the conver-gence criteria are satisfied, such as kDVðiÞk=kVð1Þk 6 ev

and kDPðiÞk=kPð1Þk 6 ep, where Vð1Þ and Pð1Þ are the veloc-ities and pressures at start of iterations, and ev and ep are

the error tolerances for velocities and pressures. It shouldbe noted that the convergence is reached when the resultingvector on the right-hand side of assembled Eqs. (46) and(53) is small enough. The resulting vector represents theunbalanced force of the whole mechanical system.

In summary, it is important to note that in a local fluiddomain, common for the DPD and FE models, the coarsescale incremental Eq. (53) and the fine mesoscale incremen-tal equations are coupled through the internal forcestþDtFintði�1Þ. The computational schemes for this couplingare listed in Table 2 (iterations analogous to thosedescribed in [62]).

To obtain the DPD solution, the appropriate boundaryconditions for the common FE–DPD boundary need to beimposed, as we will discuss in the next section.

3.4. Boundary conditions between the FE and fine (meso)

scale domains

As we have stated in the Introduction, the fluid flow ismodeled by finite elements in the whole domain, and bythe DPD discrete particles within the local domain(Fig. 2 (2D representation)). The flow region outside thelocal domain is called the global domain. The boundarybetween the local and global domains is shown as the lineABCD. It is assumed that the boundary ABCD is fit alongthe lines of the FE mesh.

The following boundary conditions are generally used atthe common FE–DPD boundary:

1. Velocities of DPD particles at the common boundaryare equal to velocities in the coarse scale, at all coarsescale time steps,

tvDPD ¼ t�vFE;tþDtvDPD ¼ tþDt�vFE;

tþ2DtvDPD ¼ tþ2Dt�vFE; . . . ð55Þ

2. Number of DPD particles remains constant within thelocal domain by imposing periodic boundary condi-tions. Practical implementation of these conditions isspecific for each problem and must be appropriately for-mulated. Details on these implementations can be foundelsewhere [50–57].

To apply the velocity boundary conditions 1, we pro-ceed as follows. Since the time step of the fine scaleDtDPD is usually much smaller than the time step Dt ofthe coarse scale, the size of DtDPD can be expressed as

DtDPD ¼DtmD

; ð56Þ

where mD is a scaling factor. Our experience is that mD is oforder 103 (also in [21,54,63–66]). We impose the boundaryconditions for velocity (55) and iterate until the DPD con-vergence is reached,

jvðlÞ � vðl�1Þj 6 evDPDvref ; ð57Þ

Fig. 2. Two domains within a flow field and boundary conditions at thecommon boundary between fine (meso) and coarse (FE) models.

Table 2Computational steps to couple the DPD and FE solutions for the current time step, between time ‘t’ and time ‘t + Dt’

1. Initial parameters for the FE time step (between time ‘t’ and ‘t + Dt’).Set global (FE) iteration counter i = 0.Initial velocities and pressures for the coarse scale, and initial positions and velocities for DPD particles, correspond to the last time step of thecoarse scale (time ‘t’): V(0) = tV, P(0) = tP; (0)r = tr, (0)v = tv.

2. Global iteration loop: i = i + 1Loop over finite elements in the global domain.

3. DPD solution.Loop over local domains.

Iterations on the DPD solutions: Use the coarse scale solution for the boundary conditions at the DPD-FE boundary. Calculate the DPD solu-tions and interaction forces for time steps DtDPD (DPD integration scheme, e.g. as given in Table 1): ðlDDPDÞr; ðlDDPDÞv; ðlDDPDÞf ij;

ðlDDPDÞvboundary ¼tþDtv

ði�1ÞFE ; l ¼ 1; 2; . . . :; nD

DPD solution continues until the DPD convergence is reached (Eq. (58)). The number of time steps employed is nD, and this number can varyamong FE time steps.

4. FE nodal forces. Determine the FE nodal internal forces (54) for finite elements in the common DPD-FE domain, from the viscous stresses(obtained from the DPD interaction forces): t+DtFint(i�1) = Fint (t+Dts(i�1)), t+Dts(i�1) = s (v, fij)Evaluate finite element matrices using the coarse scale solution from iteration ‘i � 1’.End of loop over local domains.

5. FE solution. Solve finite element incremental equations for the assemblage of finite elements in the whole fluid domain (global and local domains).The FE solution for the whole domain is: t+DtV(i), t+DtP(i)

6. Convergence check. Convergence criteria:tþDtVðiÞ � tþDtVði�1Þ�� = tþDtVð1Þ

�� 6 ev and tþDtPðiÞ � tþDtPði�1Þ�� = tþDtPð1Þ�� 6 ep

If convergence criteria are not satisfied, repeat steps 2–5 (iterate on Eq. (53)).

The size of time step is Dt for the coarse scale (FE) and D tDPD = Dt/mD for the fine scale. The global iterations (FE) continue until the FE equations ofbalance are satisfied for the end of time step. Iterations on the DPD solutions are performed (for the current FE iteration) until the DPD convergence isreached. The initial solution (first iteration, i = 1) for the first time step is obtained using the FE model for the whole domain (global domain and localdomains).


where evDPD is the error tolerance, vref is a reference veloc-ity, and l is the iteration counter.

Other boundary conditions may also be applied at theFE–DPD boundary, such as periodic boundary conditions(see Examples in Section 4).

3.5. Stress evaluation using DPD method

One of the key steps in the presented multiscale algo-rithm is the evaluation of stresses from the DPD solution.The stress tensor from DPD is calculated using the Irving–Kirkwood model [42,43,47,63]

r ¼ �nX

i

miv̂i � v̂i þ1

2

Xi

Xj 6¼i

rij � f ij

" #* +; ð58Þ

where n is the number density of particles; the vector v̂i isdefined as v̂i ¼ vi � �vðxÞ; �vðxÞ is the stream velocity at theposition x, and h. . .i denotes the ensemble average.

In practical application of the expression (58) (see Sec-tion 4 – Examples), we used a very fine grid within theDPD domain, and the averaging in Eq. (56) was performedover the fine scale cells (bins), and over a number of DPDtime steps. We found that averaging over last 10% of DPDtime steps provides accurate results. Also, the fluid materialconstants which define the constitutive laws in the twoscales must provide matching of the fluid behavior in thetwo scales. The relations between the material constantsin the two scales have been reported in [21,50,54]. In ourexamples we considered Newtonian fluids and we deter-mined the DPD constants, which include the constants a,c and rmax in Eqs. (39) and (40), by fitting these values tomatch the FE and DPD solutions (see Section 4).

4. Examples

In this section, two simple examples are presented toillustrate the applicability of the proposed multiscale meth-odology and its solution accuracy.

4.1. Example 1: Poiseuille flow between two parallel plates

A steady flow of homogenous incompressible Newto-nian fluid between two parallel plates is considered(Fig. 3; the channel width and the length of the modelingdomain are shown in the figure). This simple example

Fig. 3. A 2D Poiseuille fluid flow between two parallel plates (dimensionsin lm).

Fig. 5. Finite element mesh with a DPD domain (box). The box issymmetrically positioned with respect to the axis of symmetry. The middleplane of the box is denoted as cross-section plane AB. Vertical coordinateis normalized.


was chosen because the analytical solution is available towhich we compare our numerical solutions (FE, DPDand multiscale solutions).

First, we obtain the FE solution with the velocity–pres-sure nodal variables in accordance with Eq. (46). We thenmodel the flow using the DPD method for the wholedomain to find the material constants of the DPD modelby matching the velocity field derived from the DPD closeto the analytical result. Finally, we implement the multi-scale approach by coupling the DPD model and FE model.

The Reynolds number and viscosity are set to beRe ¼ 10 and l ¼ 1:0 g=lms, respectively (viscosity corre-sponds to the constitutive law in Eq. (45)). The pressuredrop, which is op=ox ¼ 0:01 g=s2lm2, is used to set thepressure boundary conditions in our FE model. It isassumed that the initial velocities are equal to zero. TheFE solution calculated by using 9-node elements with pres-sures at the corner nodes [61,67] agrees with the analyticalsolution.

In the DPD model, the velocity of each DPD particlewas prescribed at the entrance of the channel to matchthe inlet velocity condition with a parabolic profile of theanalytical solution. The periodic boundary conditions wereimposed for the inlet and outlet boundaries (e.g. [19,20]).The bounce-back method is used for the no-slip rigid wallboundary conditions [54]. A good agreement with the ana-lytical solution was reached (Fig. 4) using a = 25, r ¼ 3,and c ¼ 4:5 (see Eqs. (39)–(41)). A time step used for the

Fig. 4. Analytical, DPD and FE solutions for velocity; 2D steadyPoiseuille flow. Coordinate Y is normalized with respect to the channelwidth 2h = 20 [lm].

DPD simulation was DtDPD ¼ 0:001, which correspondsto DtFE ¼ 0:1 s for the macroscale FE model [54,55].

Next, we solve the same problem using a multiscalemodel. The finite element mesh is the same as before (10elements in the lateral and 50 elements in the axial direc-tions) and a DPD domain is placed in the middle of thechannel, covering 6 · 6 finite elements, shown in Fig. 5.As the DPD domain is located symmetrically around theaxis of symmetry, the velocity profile is symmetric laterallywithin the domain and an interaction with the FE solutionis also symmetric. We used 2500 DPD particles for the box.

Two types of boundary conditions are implemented forthe DPD box:

(a) Given velocities at the top and bottom lines parallelto flow; and

(b) Periodic boundary conditions and given velocities atthe entering and outlet boundaries.

We implemented the Lees–Edwards [52–54] method forthe boundary conditions (a). According to this method, aparticle crossing the upper boundary of the DPD domainat the end of a DPD time step DtDPD is re-introduced atthe lower boundary with its xu-coordinate shifted by�ð�vxÞu � DtDPD and the x-velocity is decreased by ð�vxÞu.Here ð�vxÞu is the velocity obtained from the FE model. Fur-ther, a particle crossing the lower boundary, at a coordi-nate xl and having the velocity ð�vxÞl is re-introduced at

Fig. 6. The time evolution of velocity profiles in Poiseuille flow. FE andmultiscale FE-DPD solutions at AB plane cross-section (middle positionof the left DPD box); Y coordinate normalized.

Fig. 7. Velocity profile over DPD domain at AB cross-section for t = 200s(the DPD solutions correspond to averaging over the last 10,000 DPDtime steps of the total 100,000 DPD time steps per one FE time step). (a)Analytical and multiscale solutions using velocity averaging over (20·1)and (20·20) DPD bins; (b) Analytical and multiscale solutions usingvelocity averaging over axial layers 1, 6 and 20 inside the DPD domain,with bin division (20·20).

Fig. 8. Shear stress distribution along the AB line obtained by themultiscale approach vs. the FE (analytical) solution. The DPD shear stressvalues were obtained by averaging over (20·1) bins (Y coordinatenormalized).


the upper boundary of the box at x-coordinate shifted byð�vxÞl � DtDPD and the x-velocity is increased by ð�vxÞl.

Regarding the boundary conditions (b), we used theusual periodic boundary conditions [19,20]: if a particlelives the box at the outlet within time step DtDPD, a particle

Fig. 9. Driven cavity problem: multiscale FE-DPD model. Dimensions of thVelocities are prescribed (V=1lm/s) on the top boundary, from the right to theDPD domain. Boundary conditions for the DPD model are shown in the righ

is introduced at the same y-coordinate at the inlet bound-ary, with the boundary velocity obtained from the FEsolution.

Temporal evolution of the velocity profile at the middleof the DPD domain simulated by the newly developed FE–DPD multiscale bridging method is compared to the FE(analytical) solution in Fig. 6. The results demonstrate thatthe multiscale solutions agree very well with the analyticalsolutions.

Fig. 7a shows the ultimate velocity profile over the DPDdomain (A–B cross-section). The results are obtained usingtwo different bin divisions for velocity averaging. In thefirst one, shown by large circles, the averaging goes overthe bins obtained by 20 divisions of the DPD domain inthe Y direction and no division in X direction (division20 · 1). In the second division we have 20 divisions in bothY and X directions (division 20 · 20); the results are shownby small dots, where each dot corresponds to one bin. Itcan be noticed that bin division (20 · 1) gives results whichare closer to the analytical solution. In Fig. 7b, the DPD

e cavity are 30 · 30lm, and dimensions of the DPD box are 7.5·7.5lm.left. The flow domain is divided into the global FE domain and local FE-t figure.


solution is shown for several axial layers obtained by(20 · 20) bin division over the whole box. Note that thevelocity profiles have some deviations from the analyticalsolution.

Shear stress distribution along the AB line over thewhole cross-section, calculated by the multiscale approach,was compared to the analytical solution (Fig. 8). The devi-ations of the DPD values of the shear stress come from theaveraging over the DPD bins in Eq. (58).

Fig. 10. Velocity profiles along the verticle middle-line (shown by verticalline in the insets) passing through the local (FE-DPD) and global (FE)domains. The final time, t=10 sec, is reached after total 2,000,000 DPDtime steps (and 100 FE time steps). (a) Velocity component vx; (b) Velocitycomponent vy.

Fig. 11. Velocity vector field for the FE met

4.2. Example 2: Cavity flow

The driven cavity flow is chosen as a more complexproblem (note that there are fluid zones with the stress sin-gularity near the corners) which represents a recirculationflow with curve streamlines.

The flow in the cavity is induced by a horizontal motionof the upper wall with a constant velocity of V (Fig. 9). TheReynolds number, based on the size of the cavity and thevelocity of the upper wall movement, was set to 30. A localDPD domain (a small square region in Fig. 9) was selectedat the right upper corner of the cavity and the flow wassolved both by the FE method alone and by the FE–DPD multiscale bridging method. In the DPD simulation,900 mesoscale DPD particles were employed. The materialparameters were set the same as in Example 1. The bound-ary conditions for the DPD simulation are as follows: atthe no-slip rigid wall, the bounce-back condition wasemployed; at the moving wall it is assumed that particleshave the prescribed velocity V ¼ 1 lm=s; at the FE–DPDcommon boundaries, the velocities of particles are modeledas described in Section 3.4. The boundary conditions forthe FE model consist of the prescribed velocity at the mov-ing wall and zero velocity at the rigid walls. Also, the peri-odic boundary conditions are used for the entering (lowerhorizontal) and the outlet (left vertical) boundaries to keepnumber of particles within the DPD box constant. It isassumed that initial pressures and velocities in the wholedomain are equal to zero.

The velocity components ðvx; vyÞ along the vertical line,which goes through the middle of the local (FE + DPD)domain and extends through the global (FE) domain tothe bottom of the cavity (see the insets in Fig. 10), are cal-culated both by the FE method alone and by the FE–DPDmultiscale method. As can be seen from Fig. 10, the FE andmultiscale solutions agree very well. Velocity vector profilescalculated by the FE method (Fig. 11 left) and by the FE–DPD multiscale method (Fig. 11 right) are also comparedwith good agreement. Finally, it is noted that the pressurehas practically no effect on the solutions and that flow isgoverned by shear stresses.

hod (left) and multiscale method (right).


5. Conclusions

The proposed mesoscopic bridging scale (MBS) methodfor fluids provides the coupling between mesoscale dis-crete particle models and continuum-based finite elementmodels. It is an extension of the bridging scale methodfor coupling molecular dynamics and finite elementmodels.

This multiscale approach can be applied to modeling adilute mixture flow where in small local regions a detailedanalysis of particle motions is desirable, as in case of bloodflow in a large artery with platelet aggregation and adhe-sion to the vessel wall in a small domain of thrombosisdevelopment [42–45].

It can be seen from the presented examples that the pro-posed approach of coupling the mesoscopic DPD modeland continuum FE model is applicable to incompressiblefluid flow. Further, it is important to note that iterationson the DPD solution were performed until the equilibriumis reached, with the velocity boundary conditions imposedfrom FE solution for each FE time step. Also, we calcu-lated the stresses by averaging over the last 10% of DPDtime steps when the solution is close to the equilibriumstate. And we found that the random character of the totalinteraction DPD forces does not cause noticeable noiseoutside the DPD domain. The noise effects are small forsmall and moderate Reynolds numbers (in our examplesRe was around 30). Investigation of these effects will bethe subject of another study.

Acknowledgements

This work has been supported by National Institute ofHealth, USA, NHLBI Grant RO1 HL054885, HL070542,HL074022; and Ministry of Science of Serbia, Grant No.TR-6209A, OI-144028.

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A mesoscopic bridging scale method for fluids and coupling dissipative particle dynamics with...

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