Element-free simulation of dilute polymeric flows using Brownian … · 2004. 8. 24. · Element-free simulation of dilute polymeric flows using Brownian Configuration Fields Korea-Australia

Korea-Australia Rheology JournalVol. 16, No. 1, March 2004 pp. 1-15

lia

sis

l-,

ingons)lter-ing theo ky,

n-n-ullyeis-tered itsnts

lyfreeoidm

ow- belpandSTon-os-temb-

use

Element-free simulation of dilute polymeric flows using Brownian Configuration Fields

D. Tran-Canh and T. Tran-Cong*Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD 4350, Austra

(Received April 16, 2003; final revision received October 22, 2003)

Abstract

The computation of viscoelastic flow using neural networks and stochastic simulation (CVFNNSS) is devel-oped from the point of view of Eulerian CONNFFESSIT (calculation of non-Newtonian flows: finite ele-ments and stochastic simulation techniques). The present method is based on the combination of radial bafunction networks (RBFNs) and Brownian configuration fields (BCFs) where the stress is computed froman ensemble of continuous configuration fields instead of convecting discrete particles, and the velocityfield is determined by solving the conservation equations for mass and momentum with a finite pointmethod based on RBFNs. The method does not require any kind of element-type discretisation of the anaysis domain. The method is verified and its capability is demonstrated with the start-up planar Couette flowthe Poiseuille flow and the lid driven cavity flow of Hookean and FENE model materials.

Keywords: Brownian dynamics, RBFN, stochastic simulation, viscoelastic flow, Brownian ConfigurationFields, CONNFFESSIT, finite point method, element-free method.

1. Introduction

In recent time, several works concerned with hybrid sim-ulations using Brownian dynamics (Laso and Oettinger,1993; Feigl et al., 1995; Laso et al., 1997; Hulsen et al.,1997; Oettinger et al., 1997; Bonvin and Picasso, 1999;Laso et al., 1999; Somasi and Khomami, 2000; Cormen-zana et al., 2001; Suen et al., 2002, Tran-Canh and Tran-Cong, 2002; 2003) have been introduced to bypass theneed for closed-form constitutive equations (CE) which arerequired in the conventional macroscopic approaches. Theprincipal idea behind these schemes is to couple the con-tinuum problem with Brownian dynamics. In the tradi-tional CONNFFESSIT approach (Laso and Oettinger,1993), also called Lagrangian CONNFFESSIT (Suen etal., 2002), the polymer contribution to stress is calculatedfrom the configuration of a large ensemble of dumbbells.On the other hand, the Brownian Configuration Fields(BCF) method (Hulsen et al., 1997), also called EulerianCONNFFESSIT (Suen et al., 2002), uses an ensemble ofconfiguration fields which represent the internal degrees offreedom of the polymer molecules. The BCF methodavoids extra effort associated with the particle tracking pro-cess.

In the majority of the works relating to hybrid simula-tions, stochastic simulation techniques (SST for the cal-culation of the stress tensor) are coupled with element-

based methods (e.g. FEM for the solution of the governequations such as the continuity and momentum equatiin a micro-macroscopic approach. In general, as an anative to element-based discretisation of the governequations, various finite point methods can be used inso called meshless approach (Kansa, 1990; Belytschketal., 1996; Duarte and Oden, 1996; Randles and Libers1996; Dolbow and Belytschko, 1999; Mai-Duy and TraCong, 2001; Atluri and Shen, 2002). In particular, TraCanh and Tran-Cong, (2002; 2003) coupled successfthe RBFN-based finite point method with SST for thnumerical solution of the start-up Couette and 2-D vcoelastic fluid flows. In the macroscopic part, the discremodel is completely represented by a set of unstructudiscrete collocation nodes in the analysis domain and onboundary (i.e. there is no need to generate finite elemeor define any topological connectivity, which is commonreferred to as truly meshless or mesh-free or element-approach). In other words, the method can at least avthe extra effort of meshing and re-meshing (if the problerequires) associated with the element type methods. Hever, effective volumes for stress averages (EVSA) canflexibly generated around the collocation points to hedetermine the average polymer stress (Tran-Canh Tran-Cong, 2003). The resultant element-free RBFN-Smethod is Lagrangian as far as the microscopic part is ccerned, and particle tracking could be inconvenient. A psible disadvantage of the present method is that the sysmatrix is dense and may be ill conditioned for large prolems. However, this problem can be overcome with the

*Corresponding author: [email protected]© 2004 by The Korean Society of Rheology

Korea-Australia Rheology Journal March 2004 Vol. 16, No. 1 1

D. Tran-Canh and T. Tran-Cong

e ofnce

d on

to

icson

dE-Ptingtedat-asN-t isughextra

-he

d

i--nndticne

lltea

-

tting by

ec-.

of domain decomposition technique as will be reportedelsewhere.

In this paper, an Eulerian element-free RBFN-SSTmethod is developed following the Brownian Configura-tion Fields idea. In the present method, the polymer con-tribution to stress at all collocation points is calculatedusing the BCF technique and then the continuity andmomentum equations are solved using the RBFN-basedmethod for the velocity field and pressure. The paper isorganized as follows: Sections 2 is an outline of the schemein which the governing PDEs and SDEs for the elasticdumbbell models are briefly reviewed. In sections 3, theRBFN-based numerical method for solving the conserva-tion equations is briefly described, followed by numericalmethods of the solution of BCFs. The associated variancereduction techniques are described for the SDEs for theHookean and FENE dumbbell models. Section 4 presentsthe algorithm of the present scheme for viscoelastic flowproblems, highlighting the macroscopic-microscopic inter-faces of the method. Numerical examples are then dis-cussed in section 5, followed by a brief conclusion insection 6.

2. Governing equations

The present work is concerned with the flow of dilutepolymer solutions which are modelled as an incompress-ible suspension of non-interacting macromolecules in aNewtonian solvent. Under isothermal and steady state con-dition, an application of the penalty function method trans-forms the governing equations into

(1)

where the penalty equation is given by

(2)

subject to boundary conditions

,,

where u denotes the velocity field; n is the unit vector out-wardly normal to the boundary; L is the rate of strain ten-sor; ηηηηN is the Newtonian solvent viscosity; ττττ = ττττ s + ττττ p =2ηηηηNL + ττττ p is the extra stress; ρ is the fluid density; pe is asufficiently large penalty parameter. Although this methodproduces an error of O(pe

−1) (Baker, 1983) in approxi-mating ∇∇∇∇ · u = 0, it is considered as a good method whichallows the elimination of the incompressibility conditionand a corresponding reduction of the number of degrees offreedom of the problem in solving complex problems(Hughes et al., 1979; Crochet et al., 1984; Bernstein et al.,1994; Laso et al., 1997; 1999). Travis et al. (1990) havemade rigorous comparison between a number of numericalmethods and concluded that the methods based on penalty

function produce comparably accurate results. The valuthe penalty parameter can only be chosen from experieat this stage and the value chosen in this work is basethe results reported in the references cited above.

The system is closed by the specification of a methodcalculate the polymer contribution to the stress ττττ p. Here,the microscopic method employs the Brownian dynamsimulation (or SST) to determine the polymer contributito stress ττττ p via kinetic modelling (Bird et al., 1987; Oet-tinger, 1996; Halin et al., 1998). The kinetic theory-basemodels used here are the Hookean, FENE and FENdumbbell models. These models consist of non-interacelastic dumbbells having two Brownian beads connecby an entropic spring. The configuration of a dumbbell sisfies a certain stochastic differential equation (SDE) detailed in Laso and Oettinger (1993) where the CONFFESSIT idea was first proposed. In this approach, inecessary to convect a large number of molecules throthe domain under consideration, hence there are some effort associated with particle tracking (Hulsen et al., 1997;Laso, 1998).

Hulsen et al. (1997) proposed a modified CONNFFESSIT method which overcomes these drawbacks. Tmethod employs an ensemble of N continuous configu-ration fields Q(x,t) with respect to space and time insteaof convecting discrete connector vectors Qi's. The mainidea of this scheme is that after initiating N spatially unform configuration fields (N,Q) whose values are independently sampled from an equilibrium distributiofunction, the configuration fields are convected adeformed by the drift component (flow gradient, elasretraction) and by the diffusion component (Browniamotion). This evolution of a configuration field satisfies thfollowing SDE

(3)

where ζ is the friction coefficient between the dumbbeand the solvent; kB is Boltzmann constant; T is the absolutemperature; W(t) is a 3-component vector which is Wiener process with mean 〈Wi(t)〉 = 0 and covariance〈Wi(t)Wj(t')〉 = δij min(t,t') and accounts for the random displacement of the beads due to thermal motion; κκκκ = (∇∇∇∇u)T isthe velocity gradient; F is the spring connector forcebetween the two beads and depends on the model. LeH be the spring constant, the connector force is given

F = HQ, (4)

(5)

for the Hookean and the FENE dumbbell models, resptively, where Qo is the maximum possible spring lengthThe configuration fields (N,Q) are obtained by solving the

2ηηηηN∇ L⋅ ρρρρ u ∇⋅( )u pe∇ ∇ u⋅( )+ + ∇– ττττp,⋅=

p p– e ∇ u⋅( ),=

u uo= x Γu∈,n ∇u⋅ qo= x Γt∈,

dQ t( ) u– ∇Q⋅ κκκκ Q⋅ 2ζ---F Q( )–+ dt 4kBT

ζ-----------dW t( ),+=

F HQ

1QQo------

2–---------------------,=

2 Korea-Australia Rheology Journal


-

ed

F)

riv-

e

b-

ainPS-t of

nnduir-

SDE eqn. (3). The term u(x,t) · ∇∇∇∇Q(x,t) accounts for theconvection of the configuration fields by the flow. SincedW depends on time only, it affects the configuration fieldsin a spatially uniform way and hence the gradient of theconfiguration fields is well defined as smooth functions ofthe spatial coordinates (Hulsen et al., 1997). It can be seenthat the existence of the convective term in this Eulerianframework is completely equivalent to the particle trackingin the traditional Lagrangian CONNFFESSIT approach.Once the configuration fields are known, the stress can bedetermined as follows

(6)

where nd is the density of dumbbells; I is the identity tensorand F is the spring force. The configuration field Q is non-dimensionalised by and equation (3) becomes

(7)

where Q' = Q[H/(kBT)]1/2 is the dimensionless form of the

configuration field vector Q; λH = ζ /(4H) is the relaxation

time of dumbbells; is the square of the maximum

possible extension of the dimensionless configuration fieldQ' and F' is the dimensionless spring force given by

F' = Q', (8)

(9)

for the Hookean and FENE dumbbell models, respectively.For the sake of brevity, primes will be dropped in the fol-lowing discussion.

3. Computational schemes

In this section, computational techniques are describedfor the numerical solution of the conservation equations(momentum and continuity equations) and the Brownianconfiguration fields, respectively. For the stochastic pro-cesses, a variance reduction technique is described, fol-lowed by a presentation of the overall algorithm.

3.1. RBFN-based element-free method for solv-ing the momentum and continuity equations

An element-free method based on RBFNs for solvingPDEs was developed and reported elsewhere (e.g. Tran-Canh and Tran-Cong, 2003). Briefly, the method takesadvantage of the fact that a smooth function can be approx-imated by a RBFN such as (Haykin, 1999; Golberg et al.,1996)

(10)

(11)

where w j ∈ w (wT = [w1 w2...wm]) and are the synaptic weights; hj is the chosen radial basis function corresponding to the jth RBF-neuron; pk is the poly-nomial basis function corresponding to the kth PBF-neuron;m+ m is the total number of neurons. R and P are definedas follows

(12)

(13)

Let n be the number of collocation points is thcoordinate of the ith collocation point and is the desirevalue of function f at the collocation point xi. The RBF h

j

employed here is the Thin Plate Splines (TPS-RB(Duchon, 1976) which is given by

(14)

of which the corresponding first and second order deatives are given respectively by

(15)

χ(r) = 0 (16)

where r = x − cj and r = ||x − cj || is the Euclidean norm ofr; {c} m j=1, with m≤ n, is a set of RBF centers that can bchosen from among the training points; aj > 0 is the widthof the jth RBF (Haykin, 1999). Since the TPS-RBF is C2s-1-continuous, the power index s must be appropriately cho-sen for a given partial differential operator (Zerroukat etal., 1998). In the present work, the TPS-RBF with s= 2 ischosen to satisfy the continuity condition. For 2D prolems, the first order PBF is used as follows

(17)

It is interesting to note that the TPS-RBF does not contany adjustable parameter and in some situations the TRBFN methods can achieve an accuracy similar to thathe Multi-Quadric RBFN (MQ-RBFN) (Zerroukat et al.,1998; 2000; Tran-Canh and Tran-Cong, 2002).

3.1.1. RBF-centres, collocation points and RBFNtraining

The choice of the quantity and location of collocatiopoints (xi, i = 1,...,n) depends on the problem geometry adesired solution accuracy and is a major open issue req

ττττ p n– dkBTI nd Q F⋅〈 〉,+=

kBT H⁄ ,

dQ' u– x t,( ) ∇Q' x t,( )⋅ κκκκ x t,( ) Q⋅ ' x t,( ) 12λH---------F ' Q'( )–+ dt=

1λH------dW t( ),=

bHQ02

kBT-----------=

F' Q'

1 Q'2

b-------–

---------------=

f x( ) wjhjxj 1=

m

∑ λλλλkpk x( )k 1=

m

∑+ RT x( )w PT x( )λλλλ+= =

pk xi( )wi

i 1=

m

∑ 0= k 1 ..... m, ,=,

λk λ λT λ1λ2...λµ[ ]=( )∈

RT x( ) h1 x( )h2 x( )...hm x( )[ ],=

PT x( ) p1 x( )p2 x( )...pm x( )[ ].=

xi ŷi,( ); xiŷi

hj r( ) hj x cj –( ) r2s r( )log= = s 1 2 ...,, ,=,

∂hj

∂xi------- r2 s 1–( ) xi cij–( ) 2s r( )log 1+( )=

∂ 2hj∂ xi∂xl-------------- 2r2 s 2–( ) xi cij–( ) x1 x1j–( ) 2s s 1–( ) r( )log 2s 1–( )+[ ] χ+ r( ),=

χ r( ) r2 s 1–( )= 2s r( )log 1+( ) i∀ 1,=

i∀ 1≠

PT x( ) 1 x1 x2[ ].=



es-hts

ar

nd

d toi-

l-ttern-t-

i hefol-

esse

ing separate investigation (Fodoseyev et al., 2000; Orr,1999; Larsson and Fornberg, 2001). However, one canimagine an analogy between an adaptive discretisation inthe present finite point method and a finite elementmethod. In this respect, an advantage of the present finitepoint method is that points can be added or removed muchmore easily than a corresponding addition or removal offinite elements, since there is no topology to be concernedabout. In general, both collocation points and RBF centrescan be randomly and separately distributed in the analysisdomain. However, in the present work, collocation pointsare chosen to be the same as RBF centres, i.e. m= n, whichare uniformly distributed in the physical domain. Theunknown weights are found by minimizing an appropriatecost function given by

(18)

where Λ is a global regularization parameter; ,

Then the partial derivatives of f(x) can be calculated ana-lytically as follows

(19)

where Λ is a derivative operator.In particular, each variable in the momentum and con-

tinuity equations is approximated by an RBFN such as(10), and those equations are collocated at chosen pointsthroughout the analysis domain, yielding the followingsum square error without penalty method (planar flows indimensionless form)

(20)

and with penalty method

SSE =

(21)

where i denotes the ith collocation point; α = ηN/ηo; ηo =ηN + ηp; ηp is the polymer viscosity; Re= ρVa/ηo, V and aare characteristic velocity and length, respectively; Φ1i =

The stresses are

scaled by ηoV/a. Applying the general linear least squarprinciple to (20) or (21) (taking into account (11)), a sytem of linear algebraic equations of the unknown weigis obtained as follows

(22)

where B is the design matrix; w' is the vector of allweights; is the vector of known values. The non-lineconvective term (u · ∇∇∇∇)u in (20)-(21) is estimated using aPicard-type iterative procedure whose detail can be fouin Tran-Canh and Tran-Cong (2003).

3.2. Numerical solution of the configuration fieldsIn the present work, two numerical schemes are use

solve the SDE, namely the explicit Euler and the semimplicit predictor-corrector scheme. The former is reative simple and therefore not detailed here. The lafor the time discretisation of the elastic dumbbell cofiguration fields was described in Gardiner (1990) Oetinger (1996) Kloeden and Platen, (1995) and Somasetal. (2000) and therefore is presented only briefly for tFENE model. The technique consists of two steps as lows

(a) The predictor stepLet Qi = Q(ti), using a fixed time stepsize ∆∆∆∆t for the sto-

chastic process, the predicted BCF Q*n+1 at the time steptn+1 is explicitly determined as follows:

(23)

The updated configuration fields Q*n+1's are employed toestimate the polymer contribution to the predicted strττττ*n+1, according to (6), which is in turn used to get thsolutions of the predicted velocity at time tn+1 by solvingeqn. (1).

(b) The corrector step

C w Λ,( ) y i f xi( )–( )2

i 1=

n

∑ Λ W j( )2,j 1=

m

∑+= )

w w'∈w'T w1 w2 ... wm, , ,[ ]=( )j.

f x( ) wj hj x( )j 1=

m

∑ λk pk x( ),k 1=

m

∑+=

SSE∂u1∂x1--------

∂u2∂x2--------+

i

2

∑=

α ∂2u1

∂x12----------

∂ 2u1∂x22----------+ Re u1

∂u1∂x1-------- u2

∂u1∂x2--------+– ∂p∂x1

-------∂τ11p

∂x1---------

∂τ21p

∂x2---------++–

i

2

∑+

α ∂2u2

∂x12----------

∂ 2u2∂x22----------+ Re u1

∂u2∂x1-------- u2

∂u2∂x2--------+– ∂p∂x2

-------∂τ12p

∂x1---------

∂τ22p

∂x2---------++–

i

2

∑+

u1 uo–{ }2

i Γu∈∑ n1

∂u1∂x1-------- n2

∂u1∂x2--------+ qo–

i

2

i Γt∈∑+ +

α ∂2u1

∂x12----------

∂ 2u1∂x22----------+ Re u1

∂u1∂x1-------- u2

∂u1∂x2--------+– pe

∂ 2u1∂x12----------

∂ 2u1∂x1∂x2---------------+ Φ1i+ +

i

2

∑

α ∂2u2

∂x12----------

∂ 2u2∂x22----------+ Re u1

∂u2∂x1-------- u2

∂u2∂x2--------+– pe

∂ 2u2∂x22----------

∂ 2u1∂x1∂x2---------------+ Φ2i+ +

i

2

∑+

u1 uo–{ }2

i Γu∈∑ n1

∂u1∂x1-------- n2

∂u2∂x2--------+ qo–

i

2

i Γt∈∑+ +

∂ττττ11p

∂x1--------- xi( )

∂ττττ21p

∂x2--------- xi( );+ Φ2i

∂ττττ12p

∂x1--------- xi( )

∂ττττ22p

∂x2--------- xi( ).+=

BTB( )w' BTŷ=

ŷ

Q n 1+( )* Qn un ∇Qn⋅ κκκκn Qn⋅–Qn

2λH 1Qn2

Qo------–

----------------------------+

∆t ∆tλH------Wn,+–=

1 ∆t

4λH 1Q2n 1+

b-------------–

-----------------------------------+

Q n 1+( ) Qn12--- un– ∇Qn un 1+* ∇Qn 1+*⋅–⋅

+=



E

n-sws

the

g isrk,

ol-

or

d in as

ant

se

ce

yom-

(24)

Eqn. (24) leads to a unique cubic equation for |Qn+1 | ofwhich admissible solutions are those that satisfy 0≤ | Qi+1 |< (Oettinger, 1996). It is noted that in the present work,the gradients of the configuration and velocity on the RHSof (23)-(24) are determined by calculating directly thederivatives of their TPS-RBFN approximant as shown ineqn. (19).

The polymer stress tensor is then determined by the aver-age of the configuration fields evaluated at each collocationpoint and given by Kramers' expression as follows (Bird etal., 1987; Oettinger, 1996):

(25)

(26)

for the Hookean and FENE dumbbell models, respectively.

3.3. Variance reduction methodWithout increasing the number of dumbbells, in polymer

dynamics, a method is available to reduce the variance, butnot to change the average value of the parameters of inter-est (Melchior and Oettinger, 1996; Oettinger et al., 1997;Bonvin and Picasso, 1999). The variance reduction consistsof different techniques which are detailed in Oettinger etal. (1996); Kloeden and Platen (1995) and Kloeden et al.(1997). Owing to the Eulerian nature of the BCF scheme,the implementation of the variance reduction techniques isachieved easily in the present approach. In this work, thecontrol variate method is presented only for the FENEdumbbell model. Discussions on the efficiency of thescheme can be found in those references cited earlier (Mel-chior and Oettinger, 1996; Oettinger et al., 1997; Bonvinand Picasso, 1999) and are not repeated here.

3.3.1. Control variate method for the FENE dumbbellmodel

The method uses a control variate Xc which is correlatedwith a random variable X, to produce a better estimator of〈X〉. While 〈X〉 is unknown and needs to be estimated, 〈Xc〉can be calculated by a deterministic method. The methodhas been applied in other studies more recently (Jendrejacket al., 2000; Kroger et al., 2000 and Prabhakar andPrakash, 2002). In the Brownian Configuration Fieldsmethod, the control variate reduction technique is imple-mented as follows: at each collocation point, N dumbbellsare assigned and numbered from i = 1 ..N where dumb-bells having the same index in the whole analysis domainhave the same random number. Here, for illustrative pur-pose, this technique is presented for the numerical calcu-

lation of the polymer contribution to stress using the FENmodel (26) where the expectation of random variable

is required. At each time t and position x, let

(x, t) be the control variate corresponding to the cofiguration field Q(x, t). The variance reduction method icarried out by splitting the expectation above as follo(Bonvin and Picasso, 1999)

(27)

When = 0 there is no variance reduction. From (27), polymer stress tensor (26) is rewritten as follows

(28)

where

(29)

The first term of the RHS of (28) is calculated by usinBrownian dynamics simulations and the second termdetermined in a deterministic way. In the present wosince 's are estimated at equilibrium configuration ττττ pfeneis zero and the configuration vectors 's satisfy the flowing SDE

(30)

where F is determined by (9). The polymer stress tens(28) reduces to

(31)

4. Algorithm of the present method

In general, the overall approach can now be describea detailed algorithm (see Figs. 1 and 2 for flowcharts)follows:

a. Generate a set of collocation points and start withinitial velocity for the first iteration (zero in the presenwork) along with the boundary conditions of problem;

b. Assign N dumbbells to each collocation point. Thedumbbells are numbered from i = 1 to N. All dumbbellshaving the same index constitute a configuration. Henthere is an ensemble of N configuration fields Qi (i = 1 ..N). Initially, the polymer configuration fields are spatialluniform and their values are independently sampled frthe known equilibrium distribution function which is a 3

κκκκn Qn⋅ κκκκn 1+* Qn 1+*⋅Qn

2λH 1Qn2

Qo------–

----------------------------

∆tλH------Wn+–+ +

b

ττττ nd– kBT QQ〈 〉 I–( )=

ττττ nd– kBTQQ〈 〉

1 Q2

b------–

------------- I–

=

QQ1 Q2 b⁄–-------------------

Q

QQ

1 Q2

b------–

------------- Q Q

1 Q2

b------–

------------- QQ

1 Q2

b------–

------------- Q Q

1 Q2

b------–

-------------– .+=

Q

τ p ndkBTQQ

1 Q2

b------–

------------- Q Q

1 Q2

b------–

-------------– τfenep

,+=

τfenep

ndkBT–Q Q

1 Q2

b------–

-------------〈 〉 I–

.=

QQ

dQF

2λH---------dt– 1

λH------dW t( ),+=

τ p ndkBT–QQ

1 Q2

b------–

------------- Q Q

1 Q2

b------–

-------------– .=



BFss

on-heions

eder-cityc-

nt

l-

itart

D Gaussian distribution with zero mean and unit covari-ance (Bird et al., 1987; Oettinger, 1996). Since all thedumbbells having the same index receive the same randomnumbers, there is a strong correlation between dumbbellsin a configuration. The control variates 's associated withthe configuration fields Qi's are created as described in sec-tion 3.3;

c. Calculate velocity gradient fields directly by differ-entiating the RBFNs that approximate the velocity fields;

d. Calculate the polymer configuration fields using themethod described in section 3.2. To ensure strong corre-lation within a configuration field, all the dumbbells of thesame index have the same random numbers. For each con-figuration field Q, a corresponding control variate isdetermined according to the procedure described in section3.3. In this work, while the time discretisation of the BCFis carried out by a predictor-corrector scheme, the controlvariates which are governed by eqn. (30) is estimated byEuler method;

e. Determine the polymer contribution to stress by takingthe ensemble average of the polymer configurations at eachcollocation point, using (28) for the FENE dumbbell modelfor example. Impose the stress boundary conditions at thecollocation points located on the boundary;

f. The stress is then approximated globally by TPS-Rnetworks which are the ultimate description of the strefield;

g. With the stress field just obtained, solve the set of cservation equations for the new velocity field using tRBFN-based mesh-free method as described in sect3.1;

h. Terminate the simulation when either the desirtime or convergence is reached. The latter is detmined by a convergence measure for either the velofield or the stress field, which is defined for the veloity field by

(32)

where d is the number of dimension (2 in the presework); tol is a preset tolerance; ui is the i component of thevelocity at a collocation point; N is the total number of co

Q

Q

CM

uin uin 1––( )2

i 1=

d

∑1

N

∑

uin( )2

i 1=

d

∑1

N

∑-------------------------------------- tol


a-

toitysthear

ly at

ITnh

them- =t =

thes

m-

location points and n is the iteration number. Convergenceis also checked for the shear stress and the first normalstress difference;

i. Return to step (d) for the next time level of the micro-scopic process.

5. Numerical examples

The present method is verified with the simulation of thestart-up planar Couette and steady state planar Poiseuilleflows of Hookean and FENE model fluids. The capabilityof the method is then demonstrated with the simulation ofthe lid driven cavity flow of the Hookean model fluid. Forall examples, the criterion for convergence is tol = 10−4

applied to the velocity field.

5.1. Start-up planar Couette flowThis problem was earlier studied by Mochimaru (1983)

for the FENE-P model, by Laso and Oettinger (1993) andTran-Canh and Tran-Cong (2002) for the FENE andFENE-P models, and it is used here to verify the presentmethod. The problem is defined in Fig. 3. and the chosenphysical parameters are ηo = ηN + ηp = 1, ρ = 1.2757, λH =49.62, b = 50, ηN = 0.0521, ∆t = 10−2 (Mochimaru, 1983;Laso and Oettinger, 1993).

To ensure that the centre density is adequate, three levelsof discretisation are used, namely n = 17, n = 23 and n =25, and the results show that the solutions obtained do notdiffer significantly. Only the results corresponding to n =25 are presented here. The analysis is carried out for theFENE dumbbell model where the configuration fields areproduced with one thousand dumbbells at each collocationpoint and the velocity convergence is shown in Fig. 4. Thecontrol variate is calculated at the equilibrium state. Thesimulation is continued for t ≥ 0 until the flow reaches thesteady state.

Fig. 5 describes the evolution of the velocity at four loctions y = 0.2, y = 0.4, y = 0.6 and y = 0.8 and shows that thevelocity overshoot occurs sooner in fluid layers nearerthe moving wall. Fig. 6 depicts the evolution of the velocprofile with respect to the coordinate y, which confirmthat velocity undershoot is insignificant in comparison wiovershoot. Figs. 7 and 8 describe the evolution of the shstress and the first normal stress difference, respectivelocations y = 0.2, y = 0.4, y = 0.6 and y = 0.8. The presentresult is a close match with the results of CONNFFESS(Laso and Oettinger, 1993) and CVFNNSS (Tran-Ca

Fig. 3.The start-up planar Couette flow problem: the bottomplate moves with a constant velocity V = 1, the top plateis fixed; no-slip boundary conditions apply at the fluid-solid interfaces. The collocation point distribution is onlyschematic.

Fig. 4.The steady-state planar Couette flow problem using FENE model: the velocity convergence rate. The paraeters of the problem are number of collocation points25, the number of dumbbells at each collocation poin1000, λH = 49.62, b = 50, ηN = 0.0521 and ∆t = 10−2.

Fig. 5.The steady-state planar Couette flow problem using FENE model: the time evolution of velocity at locationy = 0.2, y = 0.4, y = 0.6 and y = 0.8. The parameters of theproblem are number of collocation points = 25, the nuber of dumbbells at each collocation point = 1000, λH =49.62, b = 50, ηN= 0.0521 and ∆t = 10−2.



b,. In are

theal

me

ett-

and Tran-Cong, 2002). It is notable that the quality of con-vergence is better than that achieved with the CVFNNSSmethod.

5.2. The steady state planar Poiseuille flowThe planar creeping Poiseuille problem and coordinate

system are described in Fig. 9a where only half of the fluiddomain needs to be considered, owing to symmetry. Forthis problem, the characteristic length is chosen to be a,half of the gap between the two parallel plates; the char-acteristic velocity V, the maximum velocity; the charac-teristic viscosity ηo= ηN + ηp; and the characteristic timeλH. The length of the domain under consideration is a.

Using two collocation densities, namely 15× 15 and25× 25, whose schematic distribution is shown in Fig. 91000 dumbbells are assigned at each collocation pointthis example, tow models, namely Hookean and FENE,

Fig. 6.The steady-state Couette flow problem using the FENEdumbbell model: the velocity profile with respect to loca-tion y at different times. The parameters are the same asin Fig. 5.

Fig. 7.The steady-state planar Couette flow problem using theFENE dumbbell model: the evolution of shear stress atlocation y = 0.2, y = 0.4, y = 0.6, y = 0.8 with respect totime. The parameters are the same as shown in Fig. 5.

Fig. 8.The steady-state planar Couette flow problem using FENE dumbbell model: the evolution of the first normstress difference at location y = 0.2, y = 0.4, y = 0.6,y = 0.8 with respect to time. The parameters are the saas shown in Fig. 5.

Fig. 9.a) The planar Poiseuille flow problem with parabolic inlvelocity profile; non-slip boundary conditions applied athe fluid-solid interfaces. b) The collocation point distribution is only schematic.



ssf the the isted.

first

ndthewsdif- 121,lu- inand

the

irst

eical

considered. The fluid parameters are as follows (Feigl etal., 1995)

ηN = 0.5; ηN/ηo = 0.5; λH = 1; b = 50.

5.2.1. Boundary conditions and analytical solutionThe macroscopic boundary conditions are given in

dimensionless form as follows:• On the wall (Γ4), there is no slip

u(x) = 0

• At the inlet section (Γ1), the flow is fully developedPoiseuille where the velocity profile is parabolic for theHookean model as follows

For the FENE model, this velocity profile is not parabolicand determined by using the periodic boundary conditionat the inlet and outlet. Although the planar Poiseuille flowcan be computed as a 1D-problem, the 2-D method usingthe model FENE is carried out as follows

− Initially, the inlet of the domain is given a parabolicprofile as described above for the Hookean model;

− The obtained outlet velocity profile at a step i is usedto update the inlet velocity profile of the next step (i + 1);

− The process is continued until there is no furtherchange in the outlet profile.

• At the outlet section ((Γ3)

u2(x) = 0,

• On the centreline (Γ2), the symmetry condition applies

For the Hookean dumbbell (Oldroyd-B) model, the creep-ing Poiseuille flow problem has the analytical solutiongiven by

(34)

where De= λH 〈u1〉/a = 2/3 λHV/a is the Deborah numberand stresses are scaled by ηoV/a. The above analytical solu-tion is used to judge the quality of the following numericalsimulation.

5.2.2. Sum square errorThe expression of sum square error (20) for the creeping

planar Poiseuille flows is given by

(35)

where

up

is the inlet velocity profile given in section (5.2.1).5.2.3. Results and discussion

The solutions obtained for the velocity field, shear streand the first normal stress difference are the averages olast 200 iterations after reaching the steady state. ForHookean dumbbell model, the parabolic velocity profileaccurately recovered in the downstream region as expecFigs. 10 and 11 show the polymer shear stress and thenormal stress difference on the middle plane x1 = 0.5 cor-responding to the two collocation densities 15x15 a25× 25. The results are in very good agreement with analytical solution given by eqn. (34). Fig 10 also shothe polymer shear stress and the first normal stress ference at several steps after convergence (steps 120,122) which depicts small oscillation in steady state sotions as iteration goes on. Such oscillation has its originstochastic nature of the microscopic stress calculation,

u1 x( ) up 1 x22–( ),= =u2 x( ) 0.=

u2 x( ) ∂ u1∂ x2-------- x( ) τ12 0.=, ,

τ11 3= 1 α–( )De∂ u1∂ x2--------

2; τ12 1 α–( )=∂ u1∂ x2--------; τ22 0,=

SSE∂u1∂x1-------- xi( )

∂u2∂x2-------- xi( )+

i

2

xi Ω∈∑=

α ∂2u1

∂x12

---------- xi( )∂ 2u1∂x2

2---------- xi( )+

∂p∂ x1-------– xi( ) Φ1 xi( )+

2

∑+

α ∂2u2

∂x12

---------- xi( )∂ 2u2∂x2

2---------- xi( )+ ∂p∂ x2

-------– xi( ) Φ2 xi( )+

i

2

∑+

u1 xi( ) up–{ }2

xi Γ1∈∑ u2

2 xi( )xi Γ1∈∑ u2

2 xi( )xi Γ2∈∑

∂u1∂x2-------- xi( )

2

xi Γ2∈∑+ + +

u22 xi( )

xi Γ3∈∑ u1

2 xi( )xi Γ4∈∑ u2

2 xi( ),xi Γ4∈∑+ + +

ΦΦΦΦ1i xi( )∂ττττ11p

∂x1--------- xi( )

∂ττττ21p

∂x2--------- xi( );+= ΦΦΦΦ2i xi( )

∂ττττ12p

∂x1--------- xi( )

∂ττττ22p

∂x2--------- xi( );+=

Fig. 10.The steady state planar Poiseuille flow problem using Hookean dumbbell (Oldroyd-B) model with 15× 15 col-location points: the polymer shear stress and the fnormal stress difference on the middle plane x1 = 0.5with respect to x2 are denoted by ‘x’ for the step 120, ‘o’for step 121, ‘*’ for step 122, solid line for the averagof the last 200 steps and dashed line for the analytsolution, respectively.



e

omeum

eat-

therefore the final result is obtained by averaging a largenumber of these `steady state' solutions.

5.3. Lid driven square cavityWhile this problem has attracted the interest of many

researchers in the case of viscous fluids, there are very fewnumerical results for viscoelastic fluids. Mendelson et al.(1982) and Grillet et al. (1999) use the FEM for the anal-ysis and Tran-Cong et al. (2002) employs a BEM and RBFapproach for the numerical solution for the Oldroyd-Bmodel. On the other hand, Pakdel et al. (1997) performedexperiments on an ideal Boger fluid. The results citedabove are used here for qualitative comparison with thepresent results since the fluids used in those studies are dif-ferent, except for the case of Tran-Cong et al. (2002), fromthe Hookean dumbbell model (Oldroyd-B model) usedhere. The flow is creeping, isothermal and in a steady state.

The geometry of the computational domain with the cho-sen coordinate system is shown in Fig. 12a. Let L and Hbe the width and height of cavity, respectively. Using sixdifferent sets of collocation points (11× 11 + 2); (15× 15+ 2); (17× 17 + 2), (19× 19 + 2), (21× 21 + 2) and (41×41 + 2) whose schematic distribution is described in Fig.12b, 1000 dumbbells are assigned at each collocationpoint. The fluid parameters are given by

λH = 1. (36)

Let V be the speed of the lid. The Deborah numbers isgiven by

,

Similar to the works of Tran-Cong et al. (2002) and Gril-let et al. (1999), the Dirichlet boundary conditions argiven, in dimensionless form, by (Fig. 12.a):

In order to reduce the number of the degrees of freedof the problem, the penalty function method for thmomentum equation (1) is employed and then the ssquare error (20) is rewritten as follows:

(37)αααα ηηηηNηηηηo------ 1 9;⁄= =

De λλλλHVH----⋅=

u1 x( ) 1=u2 x( ) 0=u x( ) 0=

x∀ ΓΓΓΓ1∈ , x∀ ΓΓΓΓ1∈ , x∀ ΓΓΓΓ2∈ .

SSE α ∂2u1

∂x12

---------- ∂2u1

∂x22

----------+xi

Peηo-----+ ∂

2u1∂x1

2---------- ∂

2u1∂x1∂x2---------------+

xi

Φ1i+ 2

xi Ω∈∑=

α ∂2u2

∂x12

---------- ∂2u2

∂x22

----------+xi

Peηo-----+ ∂

2u2∂x2

2---------- ∂

2u2∂x1∂x2---------------+

xi

Φ2i+ 2

xi Ω∈∑+

u1 1–( )xi2

xi Γ1∈∑ u2( )xi

2


2


2

xi Γ2∈∑+ + +

Fig. 11.The steady state planar Poiseuille flow problem using theHookean dumbbell (Oldroyd-B) model with 25× 25 col-location points: the polymer shear stress and the firstnormal stress difference (averaged of the last 300 steps)on the middle plane x1 = 0.5 with respect to x2 aredenoted by ‘∆’. The dashed line represents the analyticalsolution

Fig. 12.a) The lid driven square cavity problem: velocity of thupper lid is unity; non-slip boundary conditions apply the fluid-solid interfaces. b) The collocation point distribution is only schematic.



rgennce’

ted

e-i-

where are

known by virtue of the BCF simulation and approximated

using TPS-RBFN's and (.)xi denotes the value of (.) at xi.5.3.1. Results and discussionIn order to demonstrate that numerical solutions conve

to the correct solution, six different sets of collocatiopoints are used as described above and ‘mesh convergeis measured by the following criterion

(38)

where tp is the set of internal test points, uin-1 is the ith com-

ponent of the velocity at an internal test point associa

ΦΦΦΦ1i∂ττττ11p

∂x1--------- xi( )

∂ττττ21p

∂x2--------- xi( );+= ΦΦΦΦ2i

∂ττττ12p

∂x1--------- xi( )

∂ττττ22p

∂x2--------- xi( ).+=

CR

uin uin 1––( )2

i 1=

2

∑tp∑

uin( )2

i 1=

2

∑tp∑

--------------------------------------=

Table 1.The lid driven square cavity flow problem using theHookean dumbbell model: Trend of the `mesh conver-gence' measure, CR defined by with increasing numberof collocation points for De= 1. N: number of collo-cation points, tp: number of internal test points.

N tp CR

11× 11 + 2 81 1.000015× 15 + 2 81 0.044717× 17 + 2 169 0.012319× 19 + 2 225 0.011621× 21 + 2 289 0.009741× 41 + 2 443 0.0093

Fig. 13.The lid driven square cavity flow problem using theHookean dumbbell model: the velocity field (upper fig-ure); the zoomed velocity field around the primary vor-tex position (lower figure). The parameters are α = 1/9,λH = 1 (De= 1), (21× 21 + 2) collocation points.

Fig. 14.The lid driven square cavity flow problem using thHookean dumbbell model: the velocity field (upper figure); the zoomed velocity field around the vortex postions. The parameters are α = 1/9, λH = 1 (De= 1), (41×41 + 2) collocation points



lrterenp totion left be

ellet

m- thehe

e-r-me

with the coarser discretisation and uin is the corresponding

quantity associated with the finer one. Since the solution(velocity field) is the average of the results of a number ofiterations, ui

n-1 and uin are the average values at the internal

test points. Table 1 reports the trend of CR for the velocityfield with increasing collocation density for De= 1.0. Theprocess is deemed to have achieved `mesh convergence'when CR is O(10−2).

As in the previous examples, the numerical solutions arethe average of the results of the last 200 iterations afterconvergence. In the case of the Hookean dumbbell model,the result is in good agreement with the findings of Tran-Cong et al. (2002). Figs. 13 (collocation density 21× 21 +2) and 14 (collocation density 41× 41 + 2) depict thevelocity field for De= 1 and Figs. 15 describes the x1-com-

ponent velocity profile on the vertical central plane x1 = 0.5and the x2-component velocity profile on the horizontacentral plane x2 = 0.5. Fig. 16 depicts the velocity field foDe= 1.5. The results show that the primary vortex centends to shift upstream and towards the driving lid whDe increases. The primary vortex appears to extend uthe walls as shown in Fig. 14 where the size and locaof secondary vortices can also be observed in the lowerand right corners. Although the present results can onlycompared with Tran-Cong et al. (2002) as they used thesame model fluid (Oldroyd-B), it is generally in qualitativagreement on the typical flow features reported by Griet al. (1999), Mendelson et al. (1982), and Pakdel et al.(1997) in which the vortex shifts upstream as the De nuber increases. Since the discussion on the efficiency ofcontrol variate variance reduction is not the object of t

Fig. 15.The lid driven square cavity flow problem using theHookean dumbbell model: the profile of the velocitycomponent u2 on the horizontal central plane (upper fig-ure). The profile of the velocity component u1 on the ver-tical central plane (lower figure). The solid lines are forthe last several steps and ‘-o-’ denotes the average of theresults from the last 200 iterations. The parameters arethe same as in Fig. 13.

Fig. 16.The lid driven square cavity flow problem using thHookean dumbbell model: the velocity field (upper figure); the zoomed velocity field around the primary votex position (lower figure). The parameters are the saas in Fig. 13 except that λH = 1.5 (De= 1.5)



theer ofm-ac-redv-

s,illen-b-

eng-ip.

ikend

ler-

cs,

sl,op-

of-

87,,

for

free

m-

r-n-

ing

lesO

vedsth.

present work, only an observation of the effect of the num-ber of configuration fields on the velocity fields is given inFig. 17 for the cases of 600, 1000 and 1400 dumbbellsassigned at each collocation point and De= 1. The resultsshown in Fig. 17 demonstrate that the choice of 1000dumbbells is adequate.

6. Conclusions

This paper reports the development of a computationalmethod for viscoelastic flows using a combination of aRBFN-based element-free method and SST from the Eule-rian CONNFFESSIT point of view for 1-D and 2-D prob-lems.

The main advantages of the present method are that: par-

ticle tracking is not necessary; variance reduction of stochastic stress tensor is achieved for the same numbdumbbells used; the noise effect due to the Brownian coponent is reduced; the method is element-free in both mroscopic and microscopic parts and only an unstructuset of collocation points is required to discretise all goerning equations.

The method is verified with standard test problemnamely the start up Couette flow and the planar Poiseuflow problems. The potential of the method is demostrated with the successful solution of a non trivial prolem, namely the lid-driven square cavity problem.

Acknowledgements

This work is supported by a grant of computing timfrom the Australia Partnership for Advanced Computi(APAC) National Facility, grant number d72 to T. TranCong. D. Tran-Canh is supported by a USQ ScholarshThis support is gratefully acknowledged. The authors lto thank Prof H.C. Oettinger for his helpful discussions athe referees for their helpful comments.

References

Atluri, S.N. and S. Shen, 2002, The meshless local Petrov-Gakin (MLPG) method, Tech Science Press, Los-Angeles.

Baker, A.J., 1983, Finite element computational fluid mechaniMcGraw-Hill, New-York.

Belytschko, T., Y. Krongauz, D. Organ, M. Fleming and P. Kry1996, Meshless methods: An overview and recent develments, Comput, Methods Appl. Mech. Engrg. 139, 3-47.

Bernstein, B., K.A. Feigl and E.T. Olsen, 1993, Steady flowsviscoelastic fluids in axisymmetric abrupt contraction geometry: A comparison of numerical results, J. Rheol. 38(1), 53-71.

Bird, R.B, C.F. Curtiss, R.C. Armstrong and O. Hassager, 19Dynamics of polymeric liquids, Vol 2, John Wiley & SonsNew York.

Bonvin, J. and M. Picasso, 1999, Variance reduction methodsCONNFFESSIT-like simulations, J. Non-Newt. Fluid Mech.84, 191-215.

Cormenzana, J., A. Ledda and M. Laso, 2001, Calculation of surface flows using CONNFFESSIT, J. Rheol. 45(1), 237-259.

Crochet, M.J., A.R. Davies and K. Walters, 1984, Numerical siulation of non-Newtonian flow, Elsevier, Amsterdam.

Dolbow J., T. Belytschko, 1999. Numerical integration of Galekin weak form in meshfree methods, Computational Mechaics 23, 219-230.

Duarte C.A. and J.T. Oden, 1996, An h-p adaptive method usclouds. Comput, Methods Appl. Mech. Engrg. 139, 237-262.

Duchon, J., 1976, Interpolation des fonctions de deux variabsuivant le principle de la flexion des plaques minces, RAIRAnalyse Numeriques. 10, 5-12.

Fedoseyev, A.I, M.J. Friedman and E.J. Kansa, 2002. Impromultiquadric method for elliptic partial differential equationvia PDE collocation on the boundary, Compt. and Ma

Fig. 17.The lid driven square cavity flow problem using theHookean dumbbell model: the profile of the velocity com-ponent u2 on the horizontal central plane (lower figure).The profile of the velocity component u1 on the verticalcentral plane (upper figure). The average of the resultsfrom the last 200 iterations corresponding to the numberof dumbbells fixed at each collocation point 600, 1000 and1400. The parameters are the same as in Fig. 13, De= 1.



ernt,

ofnc-

m-

g,s-

ou-

et-

ids:s,

7,N-

itys

ly

foric

ro-om-

m-im-

ri-is

onry

is-on,

n of

Fsis

.H.ert,fi-

Applic., 43(3-5), 491-500.Feigl, K., M. Laso and H.C. Oettinger, 1995, CONNFFESSIT

approach for solving a two-dimensional viscoelastic fluid prob-lem, Macromolecules, 28, 3261-3274.

Gardiner, C.W, 1990, Handbook of stochastic methods for phys-ics, chemistry and the natural sciences, Springer-Verlag, Berlin.

Golberg, M.A., C.S. Chen and S.R. Karur, 1996, Improved mul-tiquadric approximation for partial differential equations, Eng.Anal. with Boundary Elements 18, 9-17.

Grillet, A.M., B. Yang, B. Khomami and E.S.G. Shaqfeh, 1999,Modelling of viscoelastic lid driven cavity flow using finite ele-ment simulations, J. Non-Newt. Fluid Mech. 88, 99-131.

Haykin, S., 1999, Neural networks: A comprehensive foundation,Prentice Hall, New Jersey.

Hughes, T.J.R., W.K. Liu and A. Brooks, 1979, Finite elementanalysis of incompressible viscous flows by the penalty func-tion formulation, J. Comp phys. 30, 1-60.

Hulsen, M.A., A.P.G. van Heel and B.H.A.A. van den Brule,1997, Simulation of viscoelastic flows using Brownian Con-figuration fields, J. Non-Newt. Fluid Mech. 70, 79-101.

Jendrejack, R.M., M.D. Graham and J.J. de Pablo, 2000, Hydro-dynamic interactions in long chain polymer: Application of theChebyshev polynomial approximation in stochastic simula-tions, J. Chem. Phys. 113, 2894-2900.

Kansa, E.J., 1990, Multiquadrics-a scattered data approximationscheme with applications to computational fluid dynamics-II:Solutions to parabolic, hyperbolic and elliptic partial differ-ential equations, Computers Math. Applic. 19(8-9), 147-161.

Kloeden, P.E. and E. Platen, 1995, Numerical solution of sto-chastic differential equations, Springer, Berlin.

Kloeden, P.E., E. Platen and H. Schurz, 1997, Numerical solutionof stochastic differential equations through computer exper-iments, Springer, Berlin.

Kroger, M., A. Alba-Perez, M. Laso and H.C. Oettinger, 2000,Variance reduced Brownian simulation of a bead-spring chainunder steady shear flow considering hydrodynamic interactioneffects, J. Chem. Phys. 113, 4767-4773.

Larsson, E. and B. Fornberg, 2001, A numerical study of someradial basis function based solution methods for elliptic PDEs,Tech. Rep. 470, Dept. of Applied Mathematics, Univ. of Col-orado at Boulder, CO.

Laso, M. 1998, Calculation of flows with large elongation com-ponents: CONNFFESSIT calculation of the flow of a FENEFluid in a planar 10:1 contraction. In: Adam, M.J., MashelkarR.A., Pearson J.R.A. and Rennie A.R., Dynamics of complexfluids: Proceedings of the second Royal Society Unilever Indo-UK forum in material science and engineering, Chapter 6, 73-87, Imperial College Press, London.

Laso, M. and H.C. Oettinger, 1993, Calculation of viscoelasticflow using molecular models: the CONNFFESSIT approach, J.Non-Newt. Fluid Mech. 47, 1-20.

Laso, M., M. Picasso and H.C. Oettinger, 1997, 2-D time-depen-dent viscoelastic flow calculations using CONNFFESSIT,AIChE Journal. 43(4), 877-892.

Laso, M., M. Picasso and H.C. Oettinger, 1999, Calculation offlows with large elongation components: CONNFFESSIT cal-culation of the flow of a FENE fluid in a planar 10:1 con-

traction. In: Nguyen, T.Q. and Kausch, H.H., Flexible polymchains dynamics in elongational flow: theory and experimeChapter 6, 101-136, Springer, Berlin.

Mai-Duy, N. and T. Tran-Cong, 2001, Numerical solution Navier-Stokes equations using multiquadric radial basis fution networks, Neural Networks. 14, 185-199.

Melchior, M. and H.C Oettinger, 1996, Variance reduced siulations of polymer dynamics, J. Chem. Phys. 105(8), 3316-3331.

Mendelson, M.A., P.W. Yeh, R.A. Brown and R.C. Armstron1982, Approximation error in finite element calculation of vicoelastic fluid flows, J. Non-Newt. Fluid Mech. 32, 197-224.

Mochimaru Y., 1983, Unsteady-state development of Plane Cette Flow for Viscoelastic Fluids, J. Non-Newt. Fluid Mech, 12,135-152.

Orr M.J.L., 1999. Recent advances in radial basis function nworks. http://www.cns.ed.ac.uk/$\sim$mjo.

Oettinger, H.C., 1996, Stochastic processes in polymeric flutools and examples for developing simulation algorithmSpringer, Berlin.

Oettinger, H.C., B.H.A.A. van den Brule and M.A. Hulsen, 199Brownian configuration fields and variance reduced CONFFESSIT, J. Non-Newt. Fluid Mech. 70, 255-261.

Pakdel, P., S.H. Spiegelberg and G.H. McKinley, 1997, Cavflows of elastic liquids: Two-dimensional flows, Phys. Fluid9(11), 3123-3140.

Phelan, F.R, M.F. Malone and H.H. Winter, 1989, A purehyperbolic model for unsteady viscoelastic flow, J. Non-Newt.Fluid Mech. 32, 197-224.

Prabhakar, R. and J.R. Prakash, 2002, Viscometric functionsHookean dumbbells with excluded volume and hydrodynaminteraction, J. Rheol. 46, 1191-1220.

Randles P.W. and L.D. Libersky, 1996, Smoothed particle hyddynamics: some recent improvements and applications, Cput, Methods Appl. Mech. Engrg. 139, 375-408.

Somasi M. and B. Khomami, 2000, Linear stability and dynaics of viscoelastic flows using time-dependent stochastic sulation techniques, J. Non-Newt. Fluid Mech. 93, 339-362.

Suen J.K.C, Y.L. Joo and R.C. Armstrong, 2002, Molecular oentation effects in viscoelasticity In: J. L. Lumley, S.H. Davand P. Moin, Annual Review of Fluid Mechanics, V34 , 417-444.

Tran-Canh, D. and T. Tran-Cong, 2002, BEM-NN Computatiof generalized Newtonian flows, Eng. Anal. With BoundaElement 26, 15-28.

Tran-Canh, D. and T. Tran-Cong, 2002, Computation of vcoelastic flow using neural networks and stochastic simulatiKorea-Australia Rheology Journal 14(4), 161-174.

Tran-Canh, D. and T. Tran-Cong, 2003, Meshless computatio2D viscoelastic flows, Korea-Australia Rheology Journal, sub-mitted.

Tran-Cong, T., N. Mai-Duy and N. Phan-Thien, 2002, BEM-RBapproach for visco-elastic flow analysis, Engineering Analywith Boundary Element, 26, 757-762.

Travis, B.J., C. Anderson, J. Baumgardner, C.W. Gable, BHager, R.J. O'connell, P. Olson, A. Raefsky and G. Schub1990, A benchmark comparison of numerical methods for in



pe

nite Prandrl number thermal convection in two-dimensionalCartesian geometry, Geophys. Astrophys. Fluid Dynamics, 55,137-160.

Zerroukat, M., H. Power and C.S. Chen, 1998, A numericalmethod for heat transfer problems using collocation and radial

basis functions, Int. J. Numer. Meth. Engng. 42, 1263-1278.Zerroukat, M., K. Djidjeli and A. Charafi, 2000, Explicit and

implicit meshless methods for linear advection diffusion-typartial differential equations, Int. J. Numer. Meth. Engng. 48,19-35.


Element-free simulation of dilute polymeric flows using Brownian Configuration FieldsD. Tran-Canh and T. Tran-Cong*Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD 4350, Aus...(Received April 16, 2003; final revision received October 22, 2003)

AbstractThe computation of viscoelastic flow using neural networks and stochastic simulation (CVFNNSS) is...Keywords�:�Brownian dynamics, RBFN, stochastic simulation, viscoelastic flow, Brownian Configurat...

Element-free simulation of dilute polymeric flows using Brownian … · 2004. 8. 24. · Element-free simulation of dilute polymeric flows using Brownian Configuration Fields Korea-Australia

Documents