an explicit scheme for tracking the ﬁlling front during...

An explicit scheme for tracking the ®lling front during polymermold ®lling

J.A. Luoma, V.R. Voller *

St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis,

MN 55414-2196, USA

Received 13 July 1998; received in revised form 12 November 1999; accepted 22 December 1999

Abstract

Tracking the ®lling front during polymer mold ®lling can be achieved by solving the so-called ``volume of ¯uid''

(VOF) equation. In this paper, a method that uni®es many of the existing VOF-based schemes is introduced. Each

variant of this uni®ed method requires, at each time step, the solution of a system of equations which can be nonlinear.

The introduction of pseudo compressibility transforms the basic VOF equation into an equation that matches the well-

known enthalpy formulation for solid/liquid phase change. This leads to a fully explicit scheme of the governing ®lling

equations that does not require the solution of a system of equations. If the pseudo compressibility is small enough, the

explicit solution is close to the solution obtained with a direct solution of the incompressible VOF equation. Further, in

some cases, the explicit scheme is computationally e�cient in terms of CPU time and accuracy. Ó 2000 Elsevier

Science Inc. All rights reserved.

Keywords: Explicit; Implicit; Molding; Polymer; Free surface

www.elsevier.nl/locate/apm

Applied Mathematical Modelling 24 (2000) 575±590

Nomenclatureanb coe�cient for support nodes of node iai coe�cient for node ia�i forced coe�cient for node ib 1/2 height of an injection mold cavityc psuedo compressiblity factorCp speci®c heatF ®ll fractiong liquid fractionG total amount of ¯uidH enthalpyk e�ective permeabilityK thermal conductivityL latent heatm0 pre-exponent coe�cient of power law viscosity model

* Corresponding author. Tel.: +1-612-625-5522; +1-612-626-7750.

E-mail address: [email protected] (V.R. Voller).

0307-904X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.

PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 0 4 - 4

1. Introduction

The success of the use of synthetic polymers for consumer products is based, in no small part,on the widespread availability of reliable simulation tools and methodologies. Currently, part andprocess design engineers routinely use computer-based tools to simulate the molding process [1,2].This has greatly reduced the trial-and-error process and has increased the reliability of the partsproduced.

One of the critical areas in simulating injection molding processes is tracking the poly-mer ®lling front. A knowledge of the ®lling front movement helps the mold designer avoidsuch defects as short shots in injection molding (IM) [3] and dry spots in resin transfermolding (RTM) [4]. A common means of determining the movement of the polymer front isto solve a model based on the so-called ``volume of ¯uid'' (VOF) method introduced by Hirtand Nichols [5]. The basic feature in this model is to track the evolution of a ®ll factor, F,which takes a value of F � 1 in ®lled regions and a value of F � 0 in empty regions. Ex-amples of the use of this approach in numerical methods of polymer molding operations canbe found in the work of Liang and Tucker [6], Young [7] and Voller and Peng [8]. A hallmark of these various models is that, at each time step, a solution of a set of equations isrequired to advance the polymer front. In some cases, depending on the nature of the timeintegration and the problem at hand, this system can be nonlinear requiring an iterativesolution.

In the context of polymer mold ®lling, two contributions are made in the paper:1. A common scheme that uni®es the previous VOF solution methodologies is presented.2. On the introduction of pseudo compressibility, the basic VOF equation is transformed into an

equation that matches the well-known enthalpy formulation for solid/liquid phase change. Thisleads to a fully explicit scheme of the governing ®lling equations.

n outward pointing normal on C(t)n exponent of power law viscosity modelP pressure ®eldP0�t� prescribed gate pressure w.r.t. timeq�t� prescribed gate ¯ux w.r.t. timeS ¯uidity tensort timeT temperatureu velocity vectorX �t� one-dimensional front position w.r.t. timeXH�t� Neumann analytical solution w.r.t. timeDt time step

Greek lettersC�t� ®ll front location w.r.t. timed one-dimensional grid spacingl absolute ¯uid viscosityk auxiliary variable in Neumann equations shear stressX entire mold region, ®lled and emptyXe empty region of moldXf ®lled region of mold

576 J.A. Luoma, V.R. Voller / Appl. Math. Modelling 24 (2000) 575±590

The introduction of an explicit scheme has a number of advantages over previous ®lling al-gorithms. At the top of the list, there is no need to solve a system of equations at each time stepand the code implementation is straightforward. This results in an easy-to-use approach that has ahigh degree of modeling ¯exibility, e.g., complex situations involving nonlinear temperature-dependent properties can be readily handled. Further, this technique simpli®es the iterative so-lution method of existing schemes thus making the algorithm compatible with the next generationarchitecture of supercomputer, i.e., multiple parallel processors. The downside of using theproposed explicit scheme is that, in order to match results from a conventional VOF treatment, asmall pseudo compressibility factor is required. This, in turn, requires a small simulation time stepto meet the stability requirement and can result in excessive computation times.

2. The VOF model

The ®lling stage of a polymer molding operation is shown schematically in Fig. 1. With ref-erence to this ®gure, the task of the ®lling simulation is to track the movement of the ®lling frontC�t�. The basic VOF equation is written in the conserved form as [5]

oFot�r � uF� � � 0; �1�

where u is the velocity of the ¯uid. Eq. (1) is valid in the entire domain.Appropriate boundary conditions are: (1) a prescribed pressure, P0(t), or prescribed ¯ux

q�t� � u � n at the gates (where n is the outward pointing normal), and (2) no normal ¯ow, i.e.,u � n � 0, elsewhere.

A simpli®cation of Eq. (1) can be made on noting that in many polymer molding operationsthe following relationship holds between the pressure, P, and velocity, u:

u � ÿSrP ; �2�where S is a ¯uidity tensor. The de®nition of S depends on the speci®c process being modeled.Examples include:1. Isotropic porous mats in RTM where, by Darcy's law [4],

S � kel

I; �3�

k is the e�ective permeability of the porous medium, l the ¯uid viscosity, and I is the identitymatrix.

Fig. 1. Generic molding process in the ®lling stage.

J.A. Luoma, V.R. Voller / Appl. Math. Modelling 24 (2000) 575±590 577

2. A power law ¯uid in a thin section IM process where, using the Hele±Shaw approximation [3]

S � m0� �ÿ1=n� � b 2�1=n� �

2� 1=n

� �rP � rP� � �1ÿn�=2n� �

I; �4�

b is half of the thickness of the mold cavity and m0 and n are coe�cients in the power law shearmodel

s � m0

ouoy

� �n

: �5�

In addition to the relationship in Eq. (1), it is also reasonable to assume that the gradient ofpressure in the un®lled regions, where F � 0, is 0, i.e.,

rP � 0 if F � 0: �6�Then, using Eq. (2) in the ®lled region and Eq. (6) in the empty region a suitable VOF equation

for polymer molding is obtained

oFot� r � �SrP �: �7�

The VOF model in Eq. (7) is completed on noting that appropriate boundary conditions are

rP � n � 0 �8�on the nongated part of the boundary of X and

SrP � q�t� or P � P0�t� �9�on the gated parts. The central attraction of using Eq. (7) in the simulation of a ®lling process isthat it is valid for the entire region (Xf and Xe).

3. A uni®ed VOF method

A number of numerical methods for solving the VOF method have been presented in [3±8]. Thebasic elements in these solutions are essentially the same. A discrete form of Eq. (7), subject to theboundary conditions in Eqs. (8) and (9), is solved for a nodal pressure ®eld. With this pressure®eld a combination of Eqs. (2) and (7) can be used to update the nodal ®ll factor ®eld, F. A basicdemarcation between the methods is the way in which the time stepping in the discretization ofEq. (7) is handled. One group of methods [3,4] uses an explicit time stepping. An alternative andmore recent approach [8,9] uses an implicit time stepping. The explicit time stepping schemes aresubject to a Courant criterion which restricts the time step. The implicit time scheme is free of thiscriterion. The aim of this section is to show how both these schemes can be combined into auni®ed scheme through the choice of the update step for the ®ll factor ®eld, F.

On a grid of nodes the discrete VOF equation is

aiPi �X

anbPnb � F oldi ÿ Fi

Dt; �10�

where a�'s are coe�cients (generated by ®nite element [4], control volume ®nite elements [9] or®nite di�erence schemes [10]), Pi is the pressure at node i, Fi is the ®ll factor at node i, and thesubscript nb stands for nodes in the region of support of node i. Since the absolute viscosity of thedisplaced ¯uid (air) is generally at least three orders of magnitude less than that of the injected


polymer, the supplanted air can be assumed to remain at a gauge pressure of 0 unless trapped in abubble. Thus the conditions of the nodal regions containing the ®lling front C(t) or in un®lledregions are zero pressure, i.e.,

Pi � 0 if Fi < 1:0: �11�A time step solution of Eq. (10) is achieved with the following procedure.

1. At the start of the time step the old ®eld values Piold and Fi

old are known.2. Initial guesses for the new pressure and ®ll factor ®elds are arrived on setting

Pi � P oldi and Fi � F old

i : �12�

3. A search of the ®ll factor ®eld is made and at every point where Fi < 1:0 the coe�cient ai, in Eq.(10) is replaced by a�i � large number (1016 is used in the current work).

4. The modi®ed system of equations

a�i Pi �X

anbPnb � F oldi ÿ Fi

Dt�13�

is solved.5. Due to the large value of a�i in empty (Fi � 0) and ®lling cells (Fi < 1), a nodal value of Pi� 0 is

returned for all empty and partially ®lled cells. This solution meets the requirement of Eq. (11).6. The next step is to update the ®ll factor. The update can be obtained on returning to the normal

values of ai and rearranging Eq. (10) to give

Fi � F oldi � Dt

XanbPnb

�ÿ aiPi

�: �14�

Two alternative implementations of Eq. (14) can be used.6a. In the implicit time integration scheme of Voller and Peng [8] the ®ll factor from Eq. (14) is

treated as an unknown value and in an iterative fashion, steps 3±6 are repeated until con-vergence. At each application of Eq. (14), over®lling of a cell is avoided by resetting Fi � 1at nodes where Eq. (14) predicts a value of Fi > 1.

6b. Iterations can be avoided if the explicit time integration scheme of Bruschke and Advani [4]is used. In this scheme, the time required to ®ll each node's control volume is calculatedfrom a rearrangement of Eq. (10), i.e.,

Dti �1ÿ F old

i

ÿ �PanbPnb ÿ aiPi

: �15�

Then selection of the smallest positive number from this set ensures that a single applicationof Eq. (14) will not over-®ll a computational cell.

The main advantage of the Bruschke and Advani [4] time explicit approach for updating the ®llfactor (step 6b above) is that no iteration in a time step is required. The over®lling of a com-putational cell by Eq. (14) is avoided on choosing a time step that exactly ®lls the next availablecell. The downside is that if the ®lling front has a high morphology then many small time stepsmay be required to move the front forward. This situation can be remedied, at the experience ofsome mass loss, on ¯agging cells with Fi > 0.99 as ®lled. Unlike the time explicit method, thealternative time implicit approach (step 6a above) requires iteration in the time step. This dis-advantage is o�-set on noting that, in situations where material properties are only spatiallydependent, the implicit solution is independent of the time step [11]. That is, after convergence of


the iterations, the ®ll factor nodal ®eld predicted with one large time step of n� Dt is identical tothe ®ll factor ®eld predicted with n times steps of Dt.

4. An enthalpy formulation

The central di�culty in solving the VOF equation (Eq. (7)) is that there is no explicit rela-tionship between the pressure and ®ll factor ®elds. A situation that necessitates the two-step al-gorithm of a complete pressure solution followed by a ®ll factor update. In some respects the lackof an explicit relationship between the pressure and ®ll factor is analogous to the lack of anexplicit relationship between the pressure and velocity in solving the incompressible Navier±Stokes equations. If the material being injected was compressible, then such a relationship wouldexist. A well-known means of solving the incompressible Navier±Stokes equations is to assumeslight compressibility, thereby introducing an equation of state that leads to an explicit pressurevelocity coupling in the continuity equation (see the work of Chorin [12]). This approach shouldalso work for the case of mold ®lling. By introduction of a pseudo compressibility factor, c� (Kg/Pa m3), the density at any point is given by

q � c�P � q0F ; �16�where q0 is the reference density of the material at P � 0, gauge. The appropriate de®nition of F inthis equation is a Heaviside step function, which has a value of unity if the point is full of ¯uid and0 otherwise. Dividing through by the reference density, q0, we get a new dimensionless variable Gthat explicitly couples the ®ll factor F and pressure P

G � cP � F ; �17�where

c � c�

q0

�18�

thus

G � qq0

: �19�

With this de®nition, we can now generate the governing equation for the problem. Referring toFig. 2, we have

Fig. 2. Control volume undergoing ®lling.


d

dt

ZXf�Xe

GdV �Z

Rf�Re

Gu � ndA: �20�

That is, the dimensionless mass in the control volume is determined by the dimensionless mass¯ux through the surface. Rf and Re are the portions of the control surface covered by injectionmaterial and still empty, respectively. By utilizing Eqs. (2) and (17) we get

d

dt

ZXf�Xe

GdV �Z

Rf�Re

FSrP � ndA�Z

Rf�Re

�cP�SrP � ndA: �21�

At this point an observation is made, in the limit of c! 0, Eq. (21) can be reduced to the basicVOF equation. This suggests that, if c is small enough, the solution obtained from Eq. (21) will beclose to the solution obtained from the basic VOF equation. We are interested in simulating theincompressible ¯ow where c � 0, thus as long as c is small we can neglect the ®nal term in Eq. (21)and retain the e�ects of compressibility only where necessary to couple the equation.

Further, examining the condition on the front boundary, following Chen et al. [13], we notethat

rP � 0 if F < 1 and F � 1 if P > 0; �22�thus, we can further simplify Eq. (21) and get

d

dt

ZXf�Xe

GdV �Z

Rf�Re

SrP � ndA: �23�

Application of Leibnitz's and Gauss' theorems yields

oGot� r � �SrP �; �24�

which in the limit of c! 0 will reduce to standard VOF form, Eq. (7). In addition to the aboveanalysis, it is also noted that rearrangement of Eq. (17) provides an explicit relationship betweenthe pressure, P, and ®ll factor, F

P �0 if G < 1;

Gÿ 1

cotherwise

8<: �25�

and

F � min�G; 1�: �26�In essence, Eq. (25) states that at points in the domain which are empty or ®lling, P is ®xed at

the gauge pressure of 0.Eq. (24) along with the condition in Eqs. (17) and (25) is identical in form to the enthalpy

formulation of a one-phase melting problem [14,15], i.e.,

oHot� r � �KrT �; �27�

where the enthalpy H is related to the temperature, T, through the relationship

H � CpT � gL; �28�Cp is the constant speci®c heat, g the liquid fraction and L is the latent heat. K in Eq. (27) isthermal conductivity The analogy between Eqs. (27) and (24) can be made exact by replacingvariables between the two equations as listed in Table 1.


An extensive range of numerical techniques have been developed to solve the enthalpy for-mulation, see [16]; this suggests that adaptation of these melting schemes could be used for ®llingproblems. Indeed, the implicit VOF method of Voller and Peng [8], outlined above, is simply themelting enthalpy method proposed by Swaminathan and Voller [17] implemented in the limitc! 0.

5. An explicit enthalpy ®lling scheme

The simplest enthalpy melting scheme is an explicit time integration scheme. This can beapplied to the enthalpy ®lling equations, Eqs. (17) and (24), using the following steps:1. At each time step the old pressure, ®ll factor and G ®eld are known.2. The new G ®eld is established from an explicit time integration of Eq. (24)

Gi � Goldi � Dt

XanbP old

nb

�ÿ aiP old

i

�: �29�

3. Following the solution of Eq. (29) new values of P are obtained from the discretized relation-ship in Eq. (17), i.e.,

Pi �0 if Gi < 1;

Gi ÿ 1

cotherwise:

8<: �30�

4. This completes the calculation in the time step.The major advantage of this explicit scheme is that no system of equations needs to be solved

and secondly, due to its explicit nature, nonlinearity can be readily handled. The major disad-vantage is that the simulation time step may need to be small, e.g., in a one-dimensional problem,solved on a uniform grid, the time step needs to satisfy the condition

SDt

cd2<

1

2; �31�

where d is the grid spacing. The e�ect of this restriction is reduced if the value of c is large. Inorder, however, to obtain ®lling predictions close to those obtained with the conventional in-compressible VOF method one might expect to use relatively small values of c. Hence, in using theexplicit enthalpy ®lling scheme, the critical adjustment is to make c large enough to maintainreasonable time steps and yet small enough to retain accuracy. In the next section the e�ect of thechoice of c on the e�ciency and accuracy of ®lling calculations is investigated. In a subsequentsection it is shown how the enthalpy ®lling algorithm can be modi®ed to allow for large values ofc, fast solution times, and accurate predictions.

Table 1

Comparison between phase change and ®lling variables

Enthalpy phase change Explicit ®lling

T P

K S

H G

Cp c

g F

L 1


6. Performance

6.1. A one-dimensional problem

An investigation of the e�ect of the value of c on ®lling predictions can be conducted using aone-dimensional problem of ®lling a porous tube (length 1 unit), with unit properties and anapplied pressure of P � 1 at x � 0, see Fig. 3. For this case the analytical ®lling solution for the®lling front movement is

X �t� ��2tp

: �32�Using various values for c, front movement predictions from the enthalpy ®lling algorithm are

compared with the analytical solution in Fig. 4. In these calculations the numerical solution used10 equally spaced nodes and in each case a time step was set at 99% of the stability limit. Theresults in Fig. 4 suggest that a value of c on the order of 0.1 is required to determine the frontposition to within a few percent error.

In the case of a non-zero c, the predictions from the enthalpy algorithm should match theNeumann analytical solution for a one-phase melting of a pure material characterized by a Stefannumber St � c [14], i.e.,

Fig. 3. One-dimensional ®lling of a tube.

Fig. 4. Analytic solution of the one-dimensional ®lling problem compared against a 10-node enthalpy scheme using

various values of the pseudo compressibility factor c.


XH�t� � 2k

��tc

r; �33�

where

kek2

erf�k� � cp1=2

: �34�

This provides for a more exact way of accounting for the error e�ect of c, viz.,

Error � 100%�1ÿ XH�0:5��: �35�Recognize that at time t � 0:5, by Eq. (33), X � 1. Fig. 5 is a log±log plot of this error against

the pseudo compressibility factor. As a reference point note that at a value of c � 0:1 the fronterror (at t � 0:5) is less than 2.0%.

6.2. Comparison with experimental measurements

One use of the ®lling simulation model, as mentioned previously, is predicting the ®lling frontin RTM. The RTM process can be described brie¯y as follows [4]: (1) A hollow mold is con-structed. (2) The mold is ®lled with reinforcing mats, e.g., ®berglass or carbon ®ber. These matsmust be cut on the edge to match the mold wall. (3) The mold is injected with polymer. (4) Thepolymer cures in the mold and the part is ejected.

Due to the cutting of the ®ber mats at the edge of the mold, the injected polymer tends to ¯owpreferentially along the mold edge. This phenomenon is often called `race tracking' because thepolymer runs on the outside edge of the molded part. Numerically, the race track phenomena canbe modeled by adjusting the permeability of computational cells adjacent to the mold edge.

In order to con®rm its physical validity, the ®lling front predictions from the proposed enth-alpy ®lling method are compared with experimental race track data reported by Chen [11]. These

Fig. 5. Percent error in the prediction of the ®lling front against the pseudo compressibility factor c.


experiments are described in detail by Sheard et al. [18]. The experiment was designed to simulateRTM ®lling and to capture the ®lling front in real time. Fig. 6 shows the boundary conditions andthe geometry as well as the plan dimensions of the mold used. This ®gure also shows the threenoded linear triangular control volume ®nite element grid used in the proposed explicit scheme.Following Chen [11] (who had performed an appropriate inverse analysis for calculating thepermeability of the race tracked zone) the nominal permeability (S � 1:05� 10ÿ9) of all elementswith one or more nodes on the mold wall were enhanced by a factor of 27.5. The boundaryconditions are no normal ¯ux on the top, bottom, and right edge of the model and a constantpressure of 250 kPa along the entire left edge.

Fig. 7 compares the numerical prediction (with the compressibility factor, c � 0:1) with theexperimental times of t � 5:0, 14.0, and 21.0 s (no attempt has been made to smooth the ex-perimental data). The comparison between the experimental and numerical data is close to thesame correlation achieved by the implicit method reported by Chen [11].

Fig. 7. Comparison of ®lling front predictions using the enthalpy scheme against experimental data at 5, 14, and 21 s.

Fig. 6. Boundary conditions and numerical grid for the two-dimensional simulation comparison to experimental data.


6.3. General testing

To test the general performance of the proposed enthalpy ®lling scheme, the test geometry,conditions, and grid (linear control volume ®nite element) shown in Fig. 8 was used. In this ar-rangement, the upper, right, and lower boundaries are no ¯ow boundaries. The left boundary, the®lling gate, is held a ®xed, dimensionless pressure of P � 1, the nominal dimensionless perme-ability is S � 1; but, in the upper half of the domain this value is enhanced by a factor of 30. This50/50 permeability split is a practical and stringent test condition. In all simulations using theenthalpy ®lling scheme, a time step at 99% of the stability limit is used. In terms of accuracy, theperformance of the proposed method will be compared with a benchmark obtained from the timestep independent implicit method proposed by Voller and Peng [8] and developed by Chen et al.[19]. In these simulations a single unit time step (1.75 s) is chosen, the inner iterations on thepressure solver, Eq. (13), are converged to a tolerance of 10ÿ3 maximum nodal change, and theouter iterations on the ®ll factor are converged to a tolerance of 10ÿ3 total error in the sum of allthe computational cells. In comparing the explicit enthalpy solution to this benchmark, the focuswill be on the e�ect of the pseudo compressibility factor, c, on the predicted shape and movementof the ®lling front.

Fig. 9 compares the implicit predictions with three enthalpy simulations (c � 0:05, 1.0, and5.0). The enthalpy scheme with a pseudo compressibility factor c � 0:05 gives a solution nearlyindistinguishable from the benchmark solution. The simulation with c � 1:0 noticeably divergesfrom the benchmark and the simulation with a c � 5:0 is quite far from the implicit solution. TheCPU requirements for these runs are shown in the ®rst four lines of Table 2. From these resultsthe indication is that the enthalpy method with pseudo compressibility can lead to reliable resultsbut at the expense of high computation times. In the next section a modi®cation of the enthalpy®lling algorithm is suggested that maintains accuracy and e�ciency.

6.4. Modi®ed enthalpy scheme numerical performance

Each of the enthalpy ®lling simulations in Fig. 9 is terminated when t � 1 unit. This is thesimulation time used in the implicit solution. It is noticed that as the value of c is increased, thefront predicted by the enthalpy scheme falls further behind the benchmark solution. This is due to

Fig. 8. Boundary conditions and numerical grid for two-dimensional general testing.


the fact that a larger value of c will have a larger level of apparent mass loss due to compress-ibility. Two modi®cations are suggested to improve this performance:1. If the focus of interest is only on the shape and movement of the ®lling front and not the actual

®lling time, a comparison at di�erent fractions of ®ll (as opposed to time) may provide a better®t. This is typically the goal when trying to determine the last point to ®ll.

2. In addition to the error in the prediction of the front position the enthalpy method also leads toerrors in the pressure ®eld. Fig. 10 schematically indicates the nature of this error in a one-di-mensional test case. The analytical VOF solution is linear where the enthalpy solution is con-cave downward, the nonlinearity arising from the need to provide material to feed thecompressibility. The smaller the value of c the closer to a linear pressure pro®le. In a generalcase, within a time step, once the enthalpy equations (Eqs. (29) and (30)) have been solvedfor new G, F, and P ®elds a pressure ®eld of the correct form can be obtained by solving thepressure equation

r � SrP� � � 0 �36�over the ®lled computational cells imposing a boundary condition of P � 0 in all partially ®lledand empty cells. In practice, a reasonable pressure solution of Eq. (36) can be obtained using aJacobi one-iterative pass. That is, after obtaining explicit ®rst estimates of the G and P ®eldsfrom Eqs. (29) and (30), calculate, in a single pass, a modi®ed pressure ®led as

Fig. 9. Comparison of the enthalpy method at various pseudo compressibility factors against the implicit method.

Table 2

Comparison of CPU times for various simulationsa

Algorithm type Compressibility factor, c CPU seconds SGI-octane 200

MHz-R10000

Simulation ®ll time

prediction (s)

Implicit N/A 6.6 1.75

Enthalpy 0.05 271.0 1.75

Enthalpy 1.00 14.4 1.75

Enthalpy 5.00 3.7 1.75

Modi®ed enthalpy 1.00 28.1 1.85

Modi®ed enthalpy 5.00 7.7 2.21

Time explicit N/A 42.0 1.75

aNote: all enthalpy schemes use explicit time stepping.


P modi �

PanbPnb� �ai

�37�

and then obtain a modi®ed ®ll fraction Gi on all ®lled cells

Gi � cP modi � 1:0 �38�

to complete the time step calculations. It should be noted that this single pass update does notguarantee convergence of the pressure ®eld, it is merely an attempt to obtain a better pressure®eld for large values of c.Fig. 11 shows the predicted pro®le of the fully implicit method, and the modi®ed enthalpy

method obtained with c values of 1 and 5, respectively. In these results the comparison with thebench mark solution is made when the ®ll fractions are the same value. The position and shape ofthe enthalpy predicted front, for both the c � 1 and c � 5 case, is close to the benchmark. Inaddition CPU times are reasonable, approaching the implicit CPU times when c � 5, line 6 inTable 2. As expected, however, the actual ®ll times are slower than the bench mark, see the lastcolumn in Table 2.

As an additional reference point, the time explicit version of the uni®ed VOF method (based onthe method originally presented by Advani and co-workers [4,20]) is applied to the test problem.The grid in Fig. 8 is used, the following settings are made: (1) if a computational cell has a ®ll valuegreater than 0.99, it is considered ®lled, and (2) the implicit solver for the pressure ®eld allows anerror of up to 10ÿ4 for any given node. Convergence limits much beyond these added unacceptableerror to the solution of the explicit VOF method. The front prediction from this method is in closeagreement with the time and position of the implicit benchmark, see Table 2 and Fig. 9. The CPUusage is relatively high, however, around a factor of 6 times more than the implicit benchmarkscheme and the modi®ed enthalpy scheme using a pseudo compressibility of c � 5.

Fig. 10. Comparison of nondimensional pressure at the point of ®lling at tube of unit length using various values of the

pseudo compressiblity factor c.


7. Conclusions

A popular way of tracking the ®lling front in polymer molding is to use a numerical schemebased on the VOF equation. In this paper, a uni®ed VOF ®lling approach has been introducedthat incorporates many previous methods, in particular implicit and explicit time integrationschemes for updating the pressure ®eld. A drawback in this uni®ed schemes is the need, at eachtime step, to solve a system of equations in the pressure ®eld.

On introducing the concept of a pseudo compressibility, c, it has been shown how a VOFtreatment can be made to match an enthalpy description of the melting of a one-phase purematerial. This opens the door to a range of phase change solution algorithms that in the limit ofc! 0 will match the VOF solution. In particular an explicit enthalpy solution has been proposedthat avoids the solution of a system of equations for the pressure. Although the proposed explicitscheme is easy to apply and in the limit of small c produces accurate results; its CPU usage, due tostability requirements, can be excessive. This problem is overcome by the introduction of anadditional single iterative sweep that improves the pressure prediction from the enthalpy ®llingmethod. With this extra step, accurate predictions of front shape can be obtained with CPU timecomparable to the most e�cient VOF schemes.

The proposed enthalpy ®lling scheme will really come into its own, when directed at problemswith high degrees of nonlinearity, in which the simulation step for accuracy is below that requiredfor stability of the scheme. In such situations the enthalpy scheme will be robust and have a highdegree of modeling ¯exibility. Examples of these types of problems would include polymerizationreactions during ®lling of large RTM parts [21] and severe temperature dependence of the vis-cosity model in IM [22]. Recently, Luoma [23] has demonstrated this feature by using the pro-posed explicit scheme to model an IM process with a temperature-dependent viscocity. Resultscompared well with short-shot experiements and were very close to those obtained with com-mercial IM ®lling codes.

Acknowledgements

This work has been partly supported by an NSF project on the product and process engi-neering for starch-based polymers, Grant DMI 97-00126. The authors also acknowledge the

Fig. 11. Comparison of the implicit, time explicit, and improve enthalpy ®lling front predictions.


support of the Minnesota Supercomputer Institute and thank Profs. M. Bhattacharya andK. Stelson for their helpful discussions.

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