An algorithm for analysis of polymer filling of molds

An Algorithm for Analysis of Polymer Filling of Molds

V. R. VOLLER and S. PENG

Department oJCiuiZ Engineering University of Minnesota

Minneapolis, Minnesota 55455-0220

A numerical implementation of the Volume of Fluid (VOF) free surface technique, applied to a filling process governed by a potential flow, is introduced. The implementation is based on a finite clement control volume space discretization and uses an implicit time integration. This allows for the accurate tracking of the filling front using large simulation time steps in molds with complex geometries. The approach is validated on comparison with the analytical solution for filling a one-dimensional tube and with two-dimensional results obtained with previously-presented filling algorithms. Examples of the application of the a p proach in the analysis of structural resin transfer molding (SRTM) are presented. The capability of the method is demonstrated on predicting “race tracking” and “dry spotting” phenomena. The CPU requirement for a typical analysis is on the order of 1 to 20 s on personal computers.

INTRODUCTION

nalysis of mold filling operations cuts across a A wide number of manufacturing processes from the filling of sand molds with molten metals through to structural reaction injection molding (SRIM) of polymers. The typical filling process involves the feed- ing of a material into a gas filled cavity (i.e., a mold). The central issue in the analysis of a filling process is the tracking of the free surface between the filling material and the escaping gas (usually air) which was present in the initially empty mold. In general terms a filling analysis requires the coupled solution of the flow equations (i.e., the Navier-Stokes equations) with some means of tracking the free surface. From the literature, three alternative tracking approaches can be identified ( 1):

1. Lagrangian In this approach, the computational space grid deforms as the flow field calculation proceeds such that a line of node points (or grid cell boundaries) always coincides with the free surface, Fig. Ib. An alternative, but equivalent approach, is to use a domain transformation and perform calculations in a fixed domain with a simple geometry.

In this technique, the progress of the free surface is followed through a fixed Eulerian space grid by tracking the movement of imaginary tracer particles, see Fig. Ic. A most straightforward implementation is based on an explicit solution of the Navier-Stokes equations

2. Marker and Cell: (MAC)

followed by an updating of the free surface based on the calculated velocities in the surface grid cells.

3. Volume of Fluid (VOF) This technique, first p r e posed by Hirt and Nichols (2), is widely used for modeling free surface flows. In this approach, the free surface is tracked on solving a transport equation of the form

- d F + -7. ( UF) = 0

d t (1)

where a constant density in the filling fluid has been assumed, u is the velocity and F is the volume fraction of the liquid. In a solution based on a fixed grid of computational cells, a cell value of F = 0 would indicate a n empty cell, a cell value of F = 1 would indicate a completely filled cell and a cell value of 0 < F < 1 would indicate a partially filled cell, Fig. Id.

All three of the above approaches, or variations, have been applied to polymer molding processes. Ex- amples of applications in structured resin transfer molding (SRTM) include: Aoyagi et al (3 ) who use a Lagrange approach based on a domain transformation, Um and Lee (4) who use a variation of the marker and cell updating the front position based on the current velocity field, Hamill et aL (5) who use a basic VOF approach based on explicit time integration of Eq I and most recently W-B Young (6) who combines a n explicit VOF with a simplified pressure

1758 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22

A n Algorithm for Analys is of Polymer Filling of Molds

calculation. Examples in injection molding include Wang et aL (7) who implements a 3-D explicit VOF approach, and Osswald and Tucker (8) who imple- ment an explicit VOF approach on a control volume finite element mesh.

A major simplification in the filling analysis of a polymer molding processes is that a potential flow approximation is valid. The objective of this paper is to introduce a new variation on the VOF free surface algorithm which is suitable for tracking the liquid/gas front in filling processes governed by potential flow. There are three key characteristics that distinguishes this algorithm.

1. The flow field and position of the free surface are effectively determined from a single governing equation.

2. The time integration in the resulting numerical solution is implicit and as such large simulation time steps can be used with little or no loss in accuracy.

3. The algorithm is independent of the space discretization used, in particular a discretization based on unstructured meshes (e.g. finite elements or control volumes) can be used.

AN SRTM PROCESS As an example system we will consider an SRTM

process, involving the injection molding of polymer into a cavity that contains pre-placed fiber reinforce- ment mats. Appropriate assumptions are, Lee (9):

1. The polymer is incompressible. 2. The Reynolds number is small so that inertia terms

in the equation of motion can be neglected. 3. Body forces (e.g., due to gravity) are neglected.

Under these assumptions, at the macroscopic scale of the process, the flow is a potential flow and the velocity field is governed by Darcy's Law, i.e.,

k , d P k , d P k , d P u = u = - - _ ; w =- - - , p dx' P d y P d z

(2 )

where u, u, and w are the velocity components, k,, etc. are the directional permeabilities associated with the mats, P is pressure, and p is the viscosity. Note that with appropriate definitions of the permeabilities Eq 2 will be suitable for describing any situation involving a potential flow. As such the subsequent development of the filling algorithm is applicable to any system in which a potential flow assumption is valid.

THE GOVERNING EQUATION

The first step in the development of the filling alge rithm focuses on the development of a single governing equation that can determine both the flow field in the polymer and the position of the polymer/air free surface. Consider a small representative fixed control volume through which the polymer/air surface passes, Fig. 2. A mass balance, assuming a uniform polymer density, on this volume can be written in the integral form

(3)

where 4 is the local liquid fraction at a point in the control volume V, taking a value of 1 or 0, S is the

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Yo/. 35, No. 22 1759

V. R. Voller and S. Peng

IQg. 2. Representative Eulerian control volume.

surface of V , and n is the unit outward normal to S. On defining the volume averaged liquid fraction in V (i.e., the fraction of filling) to be

(4)

Eq 3 can be rewritten as

( 5 )

and on using the Darcy law, Eq 2, this equation becomes

(6)

It is reasonable to assume that pressure gradients are negligible in the empty cavity ahead of the filling front. As such the appearance of 4 on the right hand side of Eq 6 is redundant (i.e., the pressure gradient is zero at every point where d = 0) and Eq 6 can be simplified to

k= CL d z d p 1 V ~ = / s ( ~ ~ n x + - - n y + - - n z k , d P dS (7)

Equation 7 will be the governing filling equation used in this work. Essentially on providing a suitable numerical grid the surface S in Eq 7 can be associated with the surface of a numerical cell. The subsequent solution in the nodal F field will then provide the fractions of fill in each cell.

p dx CL dLJ

THE NUMERICAL ALGORITHM

To focus directly on the numerical solution a p proach for Eq 7, the following simplifications will be made:

1. The flow is Newtonian, i.e., shear rate effects on the viscosity are ignored.

2. An isothermal non-reacting filling situation is assumed so that the thermal properties, in particular the viscosity, can be taken as constant.

3. Only two dimensional processes are considered.

The last two of these simplifications would not, in general, be valid in a given SRTM process. It is

stressed, however, that the extension to three dimen- sions is trivial (but not necessarily easy) and the incorporation of temperature/curing dependent viscosity can be achieved on coupling with a solution of the thermal and curing fields. Note that Um and Lee (4) also make these assumptions in the investigation of their filling algorithm.

Figure 3 shows a section of an unstructured control volume finite element mesh. On assuming that the value of F at the node point is constant over the control volume and using an implicit time integration the governing equation, Eq 7, on this mesh can be written as

where V, is the cell volume, the summation on the right hand side is over the control volume interfaces, A t is the time step, and A xi and A yj are the length components (measured in a counterclockwise sense around node I) along the j th volume interface. Use of a suitable approximation of the pressure gradients at the control volume interfaces, in terms of neighboring nodal values, will result in a system of non-linear equations in pressure P with the general point form

(9)

where the summation is over the neighboring nodes to node I. The non-linearity in these equations can be identified on noting that in control volumes that are filling or empty, i.e., volumes where FI < 1, the pressure P = 0. In the specific case of a uniform grid of square control volumes, Fig. 4 , the coefficients in Eq 9 are, Patankar (10)

(10)

where the upper case subscripts indicate evaluation at a node point and the lower case subscripts indicate evaluation at control volume interfaces.

Fig. 3. Section of an unstructured control volume mesh.


An Algorithmfor Analysis of Polymer Filling of Molds

5. The above update is applied at every node, followed by the under/over shoot correction

Fig. 4. A section of a square control uolume mesh.

There is a close resemblance between the general discrete filling equation, Ey 9, and the discrete numerical equations used to describe a liquid to solid thermal phase transformation, e.g., see Eq 17 in Swaminathan and Voller (1 1). In effect, as described in Crank (12). a potential filling process can be thought of as a liquid to solid phase change with zero specific heat, with the pressure P taking the role of temperature and the variable F taking the role of an enthalpy. This analogy allows us to use the extensive numerical “technologies” that have been developed for solving phase change problems to solve filling problems. Note that this proposed approach has been successfully applied to track filling fronts in flows governed by the general Navier-Stokes equations, Swaminathan and Voller (1). The objective of the current work is to develop a specialized and more simple treatment for the analysis of potential filling processes.

Adapting the phase change technique proposed by Voller (13) the solution of Ey 9, at a given time step, is based on an iterative updating of the nodal liquid fraction filed, i.e., the F,’s. This iterative process is as follows:

1. At the start of the time step the nodal liquid fraction field is initialized on setting it equal to the old liquid fraction values, i.e., at each node point Frm =

F,’ = where the superscript m indicates the iteration level.

2. Then, at each node where F, < 1, the nodal coeffi- cient aI is set to a large value 4 (10 l6 is currently used).

3. The set of equations, defined in Ey 9 with the above modification is then solved; currently a line- by-line iterative technique is employed. In this solution the effect of the modification to the ar coeffi- cient is to return a pressure value of P = 0 in all the filling or empty control volumes. These would be the correct values if the filling cells had been correctly identified from the current iterative value of the nodal liquid fraction field.

4. To account for the fact that the current iterative nodal liquid fraction may have been incorrect, after calculation of the nodal pressure field, the liquid fraction field is updated on manipulation of the un-modified Eq 9, i.e.,

F , = F p ‘ d + Can,Pn, (1 1)

where it is assumed that P, = 0.

to account for fully filled/fully empty control volumes or control volumes that complete filling in the given time step.

6. The iteration in steps 2 to 5 continues until changes in the total liquid volume (sum of the nodal liquid fractions X nodal volumes), between iterations, falls below a given small value (Domain Volume x in the current work).

VALIDATION

Comparison With An Analytical Solution

The proposed approach is validated on solution of a one-dimensional isothermal filling problem under a constant pressure boundary condition. Consider a one-dimensional tube of fiber mat, 0 < x < L, initially filled with air and open at both ends. Starting at time t= 0 the tube is filled with a polymer (from left to right) on fixing the pressure at x = 0 to a value PO. Under these conditions there is an analytical solution for the front movement X( t ) given by

(13)

The following data k,= 2 x lo-’ m2, p = 10 Pa.s, P = 5 x lo5 Pa (gauge) and L = 0.2 m, consistent with values used in the recent study by Um and Lee (4). is chosen as a test case. The performance of the proposed model and numerical scheme to track the front in the tube mold is compared with the analytical solution in Fig. 5. In the numerical scheme a fxed space grid with 50 steps of A x = 0.01 is used. Three

X l t )

Analytical Numerical step 1s

fdd&%tNumerical step 10s 0000[3 Numerical step 25s

0.12 4 Af--

O . O 4 ~ 0.00 0 20 40 Time s . 60 80 1

Fig. 5. Movement of one-dimensional$lling front.

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22 1761


solutions are obtained using constant time steps of A t = 1, 10, and 25, s respectively. The results in Fig. 5 indicate that

1. The proposed model provides accurate predictions. 2. The front can move through more than one cell in

a time step with no loss of accuracy, (e.g., with a time step of A t = 25 s the front moves through seven cells in the first time step).

3. The accuracy of the proposed model is not reduced when a larger time step is used. In effect at each time step the problem is equivalent to solving a steady state free surface problem with a sink condition at the free surface.

As will be demonstrated, the above behavior will also be present in multi-dimensional problems.

Comparison With A Previous M e t h o d

As a further validation exercise the predictive performance of the proposed method is compared with the performance of the experimentally validated Boundary Element Method (BEM) presented by Um and Lee (4). Towards this end, the progress of filling in a diverging-converging mold (0.1 m x 0.1 m), investigated by Um and Lee, is analyzed, Fig. 6. This mold, containing a fiber mat with uniform permeability k =

2 X lo-' m2, is filled by a polymer with viscosity of p = 10 Pa. s across a 10 mm wide gate at a constant velocity of 4.878 mm/s. The proposed filling method is applied on a 20 x 20 grid of square control volumes (5.036 mm X 5.036 mm), see Fig. 6, and a time step of A t = 25 s (to match Um and Lee, Ref. 4). Results, predicted positions of the front are shown in Fig. 7. The agreement with the results obtained by Um and Lee (Fig. 8 in Ref. 4) is seen to be close. The CPU requirement for this calculation, using Fortran 77 code running on an Intel 486 DX2/50 processor, was 20.22 s.

With reference to Fig. 7 the following comments are made:

1. The position of the front is obtained on interpolat- ing for the 0.5 F-isoline in the filling fraction field.

Fig. 6. Diuerging mold (after Urn and Lee. Re$ 4) showing control uolume discretization.

Fig. 7. Prediction oJJlling-front in diverging mold with time step of 25 s. Tht. symbols are taken from the results presented by Um and Lee (41.

Since only one line of control volumes will take F values between 1 and 0 the interpolation for the 0.5 isoline is subject to small oscillations. Use of a finer grid (a 30 x 30 grid has been employed) will predict the fronts in the same locations and will reduce, but not remove, the oscillations.

2. In the test case investigated by Um and Lee (4), the nominal gate filling velocity was 5 mm/s. The method proposed by Um and Lee (41, however, suffers from a small mass imbalance, e.g., at the end of the filling the percent mass imbalance (i.e., [(Predicted Volume Filled)/(Volume Across Ingate) - 11 100) is - 2.5%. The percent mass imbalance in the proposed method is small ( - lo-'%). As such, to match the experiments the gate velocity of 4.878 mm/s obtained on dividing the mold area (0.1 141 m2) by the filling time reported by Um and Lee (234 s), was used.

RACE TRACKING A common problem in SRTM is so-called "race

tracking." This phenomenon is caused by the mal- distribution of permeability, in particular the ten- dency for the permeability of the reinforced mats to increase near to the mold boundary. As an example, consider the case of filling a thin rectangular cavity of dimension 0.1 m x 0.2 m, from a central gate of width 0.02 m, Fig. 8. Filling is accomplished on applying a constant gauge pressure at the gate of Po = 5 x lo5 Pa. The nominal permeability of the reinforced mats is k , = k , = 2 x lo-'. It is assumed that during filling the air, initially in the cavity, can escape across the top at y = 0.2 m. The region of interest is covered by a uniform array of square control volumes, 10 x 20 in dimension, and the filling simulation is terminated



4

0.2 m

t

Fg. 8. Geometry and grid used in race tracking example.

when the control volume in the center of the top line reaches a filling value of F > 0.5. In the first instance a uniform permeability of the nominal value is assumed. The results using a time step At = 4 s are shown in Fig. 9. The CPU requirement for this calculation using a Fortran 77 code running on an Intel

128 s. _ _ _ _ _ _ - - - - - -_____.

- - - - - _ _ 64 st - - - .

, , _,16-9._ ‘ \ \ /

1 /’ \ \

/ / ‘ \

I / \ / \

/ \ I \ ’+

Fig. 9. Movement ofJillingfront wi th unqorm permeability.

486 DX2/50 processor was 6.26 s. In the second case the permeability in the layer of control volumes adja- cent to the wall, in the direction parallel to the wall, is raised to a value 10 times larger than the nominal value used in the remainder of the mold. In this case, using the same time step, it is seen that a dry spot forms, Fig. 10. The CPU requirement on the same hardware for this calculation was 1 1.48 s. In keeping with the observation that the isothermal problem is essentially a steady state problem an identical result is obtained when the numerical run is repeated with only four variable time steps, to arrive at simulations times of t = 16, 32, 64, and 128 s respectively. The CPU requirement in this case was reduced to 2.91 s. The same final “dry spot” at t = 128 s can also be predicted using a single time step of At = 128, with a CPU requirement of only 1.48 s.

A CONTROL VOLUME FINITE ELEMENT EXAMPLE

All the test examples to date have been carried out on structured uniform square grids. The method, however, as indicated by Figs. 2 and 3, can be a p plied on a control volume finite element mesh: a mesh that can be unstructured. As an example consider the flow of polymer ( p = 10 Pa’s) into an annu- lus mold, containing mat with a permeability k = 2 X

lo-” m2, under a constant pressure of Po = 5 X lo5 Pa acting on r = 0.1 m. This problem will be solved using the mesh of 200 linear triangular elements shown in Fig. 1 1 . In cylindrical coordinates this prob-

- - - - , . I \

\ I

\ \ 128 S;/’ \

\ - *’ \ / \ I

I I I I \ I \ I \ I \ I \ I \ /

I I \ I \ / \ / \ /

\ \ 32 s. ,I ,

\ 16 s. I

\ . \ - _ _ - - _ - I I I

\ I . J ._- _ _ - - ‘._.

II Fig. 10. Movement of.filling front with non-unqorm permeability.

POLYMER ENGtNEERlNG AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22 1763


f i g . 1 1. Finite element grid.

lem has an analytical solution for the movement of the filling front with time, viz.,

1 ~ Analytical OoOoo Finite Element

where t is time, r = a (= 0.1 m) is the inner radius and r = s( t ) is the position of the filling surface. The mesh shown in Fig. 1 I , however, is a Cartesian mesh and on this mesh the problem is two dimensional and a good test of the finite element method. Figure 12 shows the predicted position of the filling front at time t = 32 s. Note that the expected semi-circular shape is well maintained. On calculating the area filled, AJill, at each time step a prediction of the radial position with time can be obtained from

s ( t ) = 2- + a' [ *:" r2 (15)

Predictions obtained from Eq 15 are compared with the analytical solution ( E q 14) in Fig. 13. This result, along with Fig. 12, confirms that the proposed a p proach can be applied on a finite element mesh.

CONCLUSIONS

A new numerical algorithm for predicting the filling, under the assumption of potential flow, of polymer into a mold has been presented. The operation of the algorithm is very similar to previous numerical algorithms used for solving metal filling and solidification problems, Swaminathan and Voller (1, 11).

Fig. 12. Position of31ling front at time t = 32 s.

0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f

0.00 10.00 20.00 30.00 4c Time s .

Fig. 13. Comparison of analytical and finite element s o h tions.

The approach has been validated on comparing predictions with a one-dimensional analytical solution and with a previously proposed filling algorithm. The main attributes of the proposed algorithm are:

1. The solution of only one equation is required to fully determine the fluid flow field and position of the filling front.

2. The predictions are mass conserving. 3. In the numerical solution implicit time stepping

can be used, leading to accurate and efficient solution times. In some cases an appropriate analysis can be carried out in a matter of seconds on rela- tively low performance hardware.

4. Based on the solidification work of Swaminathan and Voller (1 1) it is known that the algorithm is independent of the space discretization used. In particular, a discretization based on unstructured meshes (e.g. finite elements or control volumes) can be used. In addition, three-dimensional prob lems should pose no new conceptual barriers.

As the algorithm is presented it requires that the pressure ahead of the filling front is atmospheric (i.e., the air can easily exit the mold). The algorithm can, however, be readily extended to cases where the air is trapped on simply compressing the air and forcing the pressure solution in front cells to take a value equal to the compressed pressure.

The essence of the proposed algorithm is that the analysis of the filling alone places no restriction on the size of the simulation time step. The numerical results presented in this work assume a constant viscosity. More realistic molding processes can be modeled on solving for the thermal and curing fields and using an appropriate constitutive relationship for the viscosity. Such a comprehensive viscosity treatment, however, could place restrictions on the size of the simulation time step.



Additional work will include more testing of the filling algorithm, including,

1. comprehensive comparisons with alternative

2. analysis of problems which form trapped air pock-

3. applications to problems with variable viscosity.

methods,

ets, and

NOMENCLATURE

U

u. u, and w V X ( t )

=Volume averaged liquid fraction. = Permeability. = Directional permeabilities. =Domain length. =Unit outward normal to S. = Directional components of n =Pressure. =Boundary pressure. =Surface of control volume V. =Velocity vector. =Velocity components. =Volume of control volume. =Position of one-dimensional filling

front.

Subscripts

I, W, E, S, N nb

Superscripts

old =Old time value. m =Iteration counter.

=Node points. =Neighbor node points to node I.

Greek

kJ. 44,

A t =Time step. P =Dynamic viscosity. 9 =Local liquid fraction.

=Length components along volume interface.

REFERENCES

1. C. R. Swaminathan and V. R. Voller, Appl Math Model,

2. C . W. Hirt and B. D. Nichols, J. Comput. Phys.. 23, 276

3. H. Aoyagi, M. Uenoyama, and S. I. Guceri, Int. Polym

4. M-K. Um and W. 1. Lee, Polym Eng. Sci., 31. 765 (1991). 5. I . S. Hamill, L. Jun, and N. P. Waterson, in Mathematical

Modelling fo r Materials Processing. M. Cross, J. F. T. Pittman, and R. D. Wood, Eds., Clarendon F’ress, Oxford. England ( 1993).

18, 101 (1994).

(1981).

Proc., 7, 7 1 ( 1992).

6. W.-B. Young, J. Adu. Mater., 25, 60 (1994). 7. V. W. Wang, C. A. Hieber, and K. K. Wang, SPE ANTEC

8. T. A. Osswald and C. L. Tucker, rnt. Polym Proc.. 5, 79

9. C-C. Lee, PoLym Eng. Sci., 30. 1607 (1990). 10. S. V. Patankar, Numerical Heat Transfer and Fluid Flow.

1 1 . C. R. Swaminathan and V. R. Voller, Int. J . Num M e t h

12. J. Crank, Free and Moving Boundary Problems, Claren-

13. V. R. Voller. Num Heat Transfer B, 17, 155 (1990).

Tech Papers, 32, 97 (1986).

( 1990).

Hemisphere, Washington, D.C. (1980).

Heat Fluid Flow, 3. 233 (1993).

don Press, Oxford, England ( 1984).

Revised June 1994

POLYMER ENGINEERING AND SCIENCE, NOVEMBER 1995, Vol. 35, No. 22 1765

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An algorithm for analysis of polymer filling of molds