optimizing flow in plastic injection molding

Journal of Materials Processing Technology 72 (1997) 333–341

Optimizing flow in plastic injection molding

L.W. Seow, Y.C. Lam *Department of Mechanical Engineering, Monash Uni6ersity, Wellington Road, Clayton 3168, Victoria, Australia

Received 31 July 1996

Abstract

The mold and part design of plastic parts for injection molding is a complicated process, considerations for producing a partranging from cost and speed of production to structural, ergonomics and aesthetic requirements. One of the routines faced by adesigner when designing quality into a part is the process of cavity balancing. This entails controlling the plastic flow in the fillingphase such that the melt front reaches the boundaries of the mold at the same time. This is done by adjusting the thicknesses ofvarious sections and can be a tedious trial and error process. In this paper, a method is described whereby the thickness-adjust-ment process can be automated. An optimization routine is used to generate the thicknesses necessary to balance the mold cavity.The method is implemented on a PC through interfacing of the Fortran code with the commercial software, Moldflow©. Usingthe method, good results have been obtained for several basic geometric models. © 1997 Elsevier Science S.A.

Keywords: Injection molding; Flow optimisation; Cavity balancing

1. Introduction

The injection molding process involves the injectionof a polymer melt into a mold where the melt cools andsolidifies to form a plastic product. It is generally athree phase process comprising filling, packing andcooling phases. Its popularity is typified by the numer-ous products produced in this way at the present time.

The introduction of simulation software has made asignificant impact in the industry where in the past,much was unknown about the injection process itself.Indeed, it was considered by some to be a ‘black art’known by only a handful of experts. However, with theincreasing use of computers in design engineering, theamount of commercially available software on the mar-ket has also increased [1]. To the versatile user, simula-tions can produce a variety of results on all aspects ofthe injection process. Traditional trial runs on thefactory floor can be replaced by less costly computersimulations.

Recently, research on plastic injection molding hasincluded a growing number of papers on optimizationalgorithms, the focus being in generating routines to

assist the designer in the work of mold and part design.Lee and Kim [2] used a modified complex method toreduce warpage by optimizing the thicknesses of differ-ent surfaces, the warpage being further reduced byobtaining the optimum processing conditions. Optimalinjection gate locations were studied by Pandelidis andZou [3] who defined the optimum location with aquality function consisting of temperature differences,overpack and frictional heating terms. A combinationof simulated annealing and the hill-climbing methodwas then used to find the optimal node. The processvariables were also optimized in a later paper [4]. Leeand Kim [5] also investigated optimal gate locationsusing evaluation criteria of warpage, weld and meldlines and izod impact strength. An intensive searchroutine was avoided by allowing the designer to firstselect some sets of gate locations. A local search wasthen conducted on the nodes using the criteria todetermine the quality of the node. Jong and Wang [6]described the optimal design of runner systems (i.e. thesystems that supply the melt to the injection gates in amulti-part mold).

In this paper, the method presented focuses on cavitybalancing to reduce distortion. By balancing the flow,over-packing and residual stresses are decreased. Thepresence of weld lines and air-traps may also be elimi-

* Corresponding author. Tel.: +61 3 99053521; fax: +61 399051825; e-mail: [email protected]

0924-0136/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

PII S 0 924 -0136 (97 )00188 -X

L.W. Seow, Y.C. Lam / Journal of Materials Processing Technology 72 (1997) 333–341334

Fig. 1. Fill pattern of a rectangular mold.Fig. 3. Flow leaders to optimize flow.

The flow to the corners and the shorter edges (i.e. theedges further away from the gate) has to be accelerated.This is currently carried out by the introduction of flowleaders or flow deflectors. A flow leader encouragesflow in a particular direction by using local increases inthe mold thickness. Conversely, a flow deflector is alocal reduction in mold thickness to resist the flow in aparticular direction. A possible solution to the flow inFig. 1 would be the use of flow leaders to the corners,as shown in Fig. 3.

Clearly the process of cavity balancing can be aninvolved process. To the less experienced designer, it isvery much a trial and error approach. Much dependson the part designer in recognizing where to place flowleaders or deflectors. The physical dimensions such aswidth, length and the changes in thicknesses are alsorequired. Use of commercial software such asMoldflow© provides invaluable assistance to the de-signer in determining the effects of such changes with-out the need for costly trials. However, human expertiseis still required in cavity balancing, as existing softwarecannot as yet determine the necessary modificationsrequired to optimize the flow.

3. Review of flow governing equations

Plastic flow in the filling phase is like flow betweentwo plates separated by a small distance. This is wellmodelled by the Hele-Shaw approximation in general.Assuming an incompressible, generalized, non-Newto-nian fluid, the equations for the filling phase can bewritten as:

nated, in addition to material savings and the reductionof cycle time.

1.1. Ca6ity balancing

Cavity balancing is still one area that depends heavilyon human interaction and input. The primary aim ofcavity balancing is to fulfil the design criteria wherebythe flow front of the plastic melt reaches the boundaryor extremities of the mold at about the same time, withequal pressure.

Balanced flow is critical to the quality of the finalproduct, as unbalanced flow during filling often leads towarping. Consider a centre gated rectangular cavitywith uniform thicknesses as shown in Fig. 1.

During the filling phase when the molten plastic isinjected into the mold, the expanding circular flowfront will fill the top and bottom edges first. Theseedges will continue to fill and pack whilst the materialfills the corner and side edges. As a result, over-packingoccurs and high pressure differentials occur between theedges, which leads to distortion of the end productwhich is undesirable.

2. Current approach

To achieve balanced flow and thus satisfy the designcriteria, the circular flow front in Fig. 1 would have tobe changed to a rectangular flow front (Fig. 2).

Fig. 2. Rectangular flow front. Fig. 4. Cross-sectional view of the flow front.

L.W. Seow, Y.C. Lam / Journal of Materials Processing Technology 72 (1997) 333–341 335

Fig. 5. Flow paths in unbalanced flow.

Fig. 7. Plot of fill times at different thicknesses.

Continuity equation:

(u(x

+(6

(y+(w(z

=0 (1)

Momentum equations:

(p(x

=(

(z�

h(u(z�

(2)

(p(y

=(

(z�

h(6

(z�

(3)

(p(z

=0 (4)

Energy equation:

rCp�(T(t

+u(T(x

+6(T(y

�=hg2+k

(2T(z2 (5)

where (x, y, z) are the Cartesian coordinates and(u, 6, w) are the velocity components, respectively. T isthe temperature, p the pressure, r is the density, Cp isthe specific heat and k is the thermal conductivity of thematerial whilst h is the shear viscosity where the shearrate g is:

g='�(u(z�2

+�(6(z�2

(6)

Following the treatment by Kennedy [7], the continuityand momentum equations can be combined to yield:

(

(x�

S(p(x�

+(

(y�

S(p(y�

=0 (7)

where

S=& h

0

�z2

h

�dz (8)

is the flow conductance, often called the fluidity termand h denotes the half-gap thickness.

Eqs. (5)–(8) represent the simplified governing equa-tions applicable to the modelling of the filling phase.The solution of these equations by Hieber and Shen[8,9] used a finite element/finite difference approachcombined with a control volume method for automaticflow front advancement: Most commercial softwareuses this method. Since then, post-filling stages havealso been added, for instance by Chiang et al. [10,11].

3.1. Effect of thickness 6ariations on the flow-rate

During filling, two assumptions can be made aboutthe flow of the melt. Firstly, there is a frozen layeralong the mold wall where the velocities u, 6 and w arezero. Secondly, the flow field is assumed to be symmet-rical about the cavity centre line (Fig. 4). Following on,the applicable boundary conditions are as follows:

(u, 6, w)=0 at z=h (9)

Fig. 8. Elements associated with a flow path.Fig. 6. Flow paths in balanced flow.


Fig. 9. The structure of the optimization routine.

Fig. 10. Fill patterns for an ellipsoidal shaped cavity with an interiorgate: (a) Before balancing; (b) after balancing.

my=2& h

0

6 dz (16)

then combining with Eqs. (13) and (14) will give:

mx= −2S(p(x

(17)

my= −2S(p(y

(18)

It is clear that the fluidity, S, is an important term indictating the flow and filling characteristics: high valuesimply low resistance to flow and similarly, low valuessuggest high resistance to flow. It is important to notethat the fluidity is not a material characteristic althoughit is related to viscosity which in turn depends ontemperature and shear rate. From Eq. (8), the fluidityalso depends on a geometrical property, the thickness.By varying the thickness, the fluidity and hence theflow-rate during filling, can be changed.

As the solution of the governing equations is solvedusing the finite element method, it is here that a charac-teristic of the method can be exploited. Since the entiremodel is discretized into elements, thicknesses need notbe constrained to surfaces and regions, but instead eachelement can have a separate thickness value. The fluid-ity across the entire mesh can be controlled by thethicknesses of the elements. As such, the task would beto ascertain at the elemental level whether the flow-ratewithin each element is satisfactory to achieve optimalflow. The routine will then generate a thickness distri-bution that will satisfy the criteria.

(u(z

=(6

(z=0 at z=0 (10)

Integrating Eqs. (2) and (3) in the z direction with theboundary conditions will result in:

(p(x

z=h(u(z

(11)

(p(y

z=h(6

(z(12)

From the integration of Eq. (11) and Eq. (12), andtaking average velocities, it follows that:

u= −Sh(p(x

(13)

6= −Sh(p(y

(14)

If the mass flow rate per unit length in the x and ydirection are defined to be:

mx=2& h

0

u dz (15)Fig. 11. Selected distribution of thicknesses.


Fig. 12. Fill patterns of a quarter model of a centre gated squarecavity: (a) Before balancing; (b) after balancing.

4.2. Updating parameter

To adjust the thicknesses along the flow paths, adirect linear relationship is assumed between the flow-rates and the thicknesses. This is based on the plotbelow showing the fill times at various thicknessesobtained from a centre gated, disc shaped cavity (Fig.7). The analyses were performed using the materialpolypropylene (PP) under constant injection pressure.

Clearly, a degree of linearity is observed with the filltime of the cavity varying inversely proportional to thethickness: As the thickness increases, the fill time de-creases and vice versa. Hence, using polypropylene (PP)in the filling analyses, the updating equation can bestated as follows:

Znew=�tnode

tshort

�Zold (19)

where Znew is the updated thickness, Zold is the currentthickness, tnode is the time when the melt front fills thenode and tshort is the time when the melt front fills thereference boundary node.

The fill time of the shortest flow path, tshort, is used asa reference time to which all other fill times can becompared. Conversely, the fill time for the longest flowpath can be used or, where applicable, some desired filltime can act as the reference time instead.

4.3. Correlating elements to the flow path

To effect the thickness changes of Eq. (19), all of theelements within the discretized cavity are associatedwith the flow paths via their nodes. This is achieved byassigning each node to the closest flow path, whereinthe node assumes the thickness of the flow path, theelemental thicknesses then being the average of itsnodal thickness (Fig. 8).

4.4. Optimality criterion

One characteristic of flow during injection molding isthat the pressure at the advancing melt front is equal tozero. In unbalanced flow, pressure builds up on theedges because the melt fills these edges first and con-tinue to pack them as depicted in Fig. 1. However, inbalanced flow this is not the case. As the flow frontreaches the boundaries at the same time, no over-pack-ing occurs. In addition, the pressure at the boundarynodes is the pressure on the flow front. Thus, thepressure of the boundary nodes in a balanced cavityshould be equal to zero.

As such, the above condition can be used as acriterion to terminate the optimization routine. Obvi-ously, in reality it is impossible to generate a flow frontthat will reach each boundary node at exactly the sametime and some over-packing will occur. In this case, a

4. Optimization routine

4.1. Flow path concept

A flow path is simply the path traced by, say, aparticle when injected through the gate until the moldhas been filled. For simple geometries without inserts,unbalanced flow would indicate that the flow pathchanges direction once a boundary is encountered. Theflow paths may look something like those shown in Fig.5.

For balanced flow, the melt front reaches theboundary at the same time for all paths. A possiblesolution to achieve this is to have a constant flowdirection during filling and to adjust the flow rates byadjusting the thicknesses of the part. As such, the flowrates along these paths are not constant, but insteadvary, depending on the distance travelled by the melt.Thus, any flow path traced from the injection nodewould be a straight line to the boundary (Fig. 6).

In reality, this may not be the actual flow path, but agood approximation. The melt can then be assumed toflow along these straight flow paths to the boundariesin balanced flow. By changing the thicknesses alongthese flow paths and therefore their flow-rates, cavitybalancing can be achieved.

Fig. 13. Thickness distribution.


Fig. 14. Fill patterns of a non-centrally gated square cavity: (a) Before balancing; (b) after balancing.

tolerance is set whereby if the pressure at the boundarynode is less than 1% of the maximum pressure of thecavity then the pressure at the boundary node is as-sumed to be zero. The condition for convergence is100% optimality where the optimality is given by,

Optimally=� N

Ntotal

�100 (20)

where N is the number of zero pressure boundary nodesand Ntotal is the total number of boundary nodes.

4.5. Assumptions

In the interests of determining a starting point, someassumptions have to be applied. These assumptions arenot seen to be necessary conditions and their form maybe modified wherever suitable. The assumptions for thecurrent studies are: (1) The initial thickness of allelements are 1 mm; (2) the reference node is the short-est flow path; (3) the same reference node is usedthroughout the optimization process; (4) nodes alongthe same flow path have a thickness equal to that ofboundary node thickness; (5) the thickness of the injec-tion node is constant; and (6) the thickness of eachelement is the average of its nodal thicknesses.

4.6. Methodology

The method has been implemented on a computerand acts as an outer loop to the main filling analysis of

Moldflow©. The structure of the routine is shown inFig. 9.

5. Results

The following are some models optimized using theabove method. The fill patterns are shown before andafter balancing. The fill patterns are simply lines ofiso-fill times at constant time increments and depict theadvancing melt front of the plastic during injection.

5.1. Model 1

The first model is a simple ellipsoidal shaped cavitywith an interior injection node depicted by the blacksquare. Fig. 10(a) shows the fill pattern of the cavitywith uniform thickness distribution; this being the ini-tial condition before optimization, whilst Fig. 10(b) isthe final pattern achieved after optimization.

Fig. 11 shows the thickness distribution after opti-mization. For the sake of clarity, only the higher rangeof thicknesses are displayed using a grey level scale. Inthis model, only the thicknesses of more than 64% ofthe maximum thickness are shown. The darker sectionsindicate thicker sections.

5.2. Model 2

This model is a quarter plate model of a centrallygated square cavity. The injection node is at the lowerleft-hand corner of the model. Fig. 12(a) shows the fillpattern with uniform thickness distribution whilst Fig.

Fig. 15. Thickness distribution.

Table 1Number of iterations to achieve optimality

Increase in vol-No. of Itera-Modelume (%)tions

61. Ellipsoidal shaped cavity 58.724.3142. Quarter plate model of

square plate (centre-gated)40.13. Rectangular cavity 6


Fig. 16. Fill pattern of a cavity with an insert using uniform thicknessdistribution.

Fig. 18. Flow paths around an insert for part of the model in Fig. 16.

12(b) is the pattern of the balanced cavity after opti-mization.

The same criteria for Fig. 11 are used to display thefinal thickness distribution in Fig. 13. As can be seen,the thicker elements form a flow leader to the corner ofthe model, which is an expected solution to a problemof this kind.

5.3. Model 3

The third model is a rectangular cavity gated at thebottom edge. The initial fill pattern is depicted in Fig.14(a) whilst the final optimized fill pattern is shown inFig. 14(b).

In Fig. 15, the thickness distribution is shown. Theflow leaders are clearly evident in the model, extendingfrom the injection node to the two corners, as would beexpected to balance the cavity.

The results of the three models indicate the effective-ness of the method in obtaining an optimal solution.Since the reference flow path was constrained to be theshortest, Eq. (19) results in an overall thickening of theelements. This essentially leads to the formation of flowleaders. The number of iterations taken to achieveoptimality is tabulated in Table 1.

6. Observations

When an insert is present, it is not clear if thealgorithm could achieve balanced flow. The reason isthat an insert is not considered explicitly in the originalformulation of the problem. To illustrate this and themethod used to overcome it, Fig. 16 depicts the fillpattern of a cavity with an insert. The pattern shown isthe initial fill pattern resulting from a uniform thicknessdistribution. The injection node is indicated by theblack square.

From Fig. 16, point A is the last node to fill. In thesame cavity but without the insert, point A would alsohave been the last point to fill due to being the furthestboundary node from the injection node. However, thepresence of the insert exacerbates the situation becauseof two consequences, one being that the distance trav-elled by the melt is increased, and the second that thedirection of the flow is altered completely. The inserteffectively forces the melt to diverge and flow around it.In this case, the assumptions of straight flow paths willnot hold.

Fig. 17 shows the result of an optimization run usingthe straight flow paths assumption. An optimality of87.8% was achieved after 37 iterations. Further itera-tions did not improve the optimality and the method

Fig. 17. Fill pattern obtained using the straight flow paths assump-tion. Fig. 19. Fill pattern obtained using the modification.


Table 2Comparsions with the basic method without modifications

No. of iterations Optimality (%)Method Increase in volume (%)

87.8 154.7521. Basic straight flow path assumption49.5100.02. Modified straight flow path assumption 24

failed to achieve the desired optimality of 100%. The fillpattern shown is the result obtained at the end of 52iterations with an optimality of 87.8%.

The solution is unsatisfactory and clearly some im-provement to the method is required. One possiblemodification to overcome this shortcoming is to assumethat the flow paths travel around the insert but subse-quently branch out to become straight flow paths again.This is shown in Fig. 18 for the part with an insert.Flow path b is shared jointly by the straight flow paths1 and 2. Flow paths a and c belong to the straight flowpath 1 which curves around the insert.

In this instance, for the portions of the flow sharedby two or more flow paths, the assumption is made thatthe thickness of the shared portions is simply the aver-age of the flow paths. In Fig. 18, the thickness of flowpath b will be the average thickness of paths 1 and 2.Other criteria can be used, but it was considered suffi-cient for this preliminary investigation.

Fig. 19 is the result obtained for the same cavity butimplementing the modification to the straight flowpaths discussed above. Comparisons with the basicmethod without modifications is tabulated in Table 2.

As observed in Table 2, the modified method per-formed considerably better, overcoming the complexityintroduced by the insert and converging to 100% opti-mality in 24 iterations. The final thickness distributionis shown in Fig. 20 using a grey level scale as before. Aswith the earlier models, flow leaders can be seen toform and as expected, the flow is encouraged to flow inthe direction of the insert.

Despite the improvement obtained with the modifica-tion, it can be expected that as the complexity of the

model increases, this limitation may become more sig-nificant. The simple assumptions used may or may notsuffice, research currently being focused on this aspect.

7. Conclusions

Balancing the flow in a cavity is an essential designstep in product development as it can improve thequality of the final product. The optimization routinedescribed in this paper has shown its effectiveness inoptimizing the thickness distribution to achieve bal-anced flow. Although simple, its robustness is indicatedby its ability to overcome the presence of an insert. Themethod can be implemented easily and adapted tocommercial software. Further improvements to themethod are currently being investigated.

Acknowledgements

The authors would like to acknowledge the supportof Moldflow© Pty. Limited, in particular that of MrPeter Kennedy, Director, who provided valuable assis-tance and stimulating discussions. This project is sup-ported by an ARC Collaborative Research Grant andthe first author gratefully acknowledges the financialsupport of Monash University in the form of a MonashGraduate Scholarship and the Australian Governmentin the form of the Overseas Postgraduate ResearchScholarship.

References

[1] L.T. Manzione (Ed.), Applications of Computer Aided Engi-neering in Injection Molding, Hanser, New York, 1987.

[2] B.H. Lee, B.H. Kim, Optimization of part wall thickness toreduce warpage of injection-molded parts based on the modifiedcomplex method, Polym.-Plast. Technol. Eng. 34 (1995) 793–811.

[3] I. Pandelidis, Q. Zou, Optimization of injection molding design.Part I: Gate location optimization, Polym. Eng. Sci. 30 (1990)873–882.

[4] I. Pandelidis, Q. Zou, Optimization of injection molding design.Part II: Molding conditions optimization, Polym. Eng. Sci. 30(1990) 883–892.

[5] B.H. Lee, B.H. Kim, Automated selection of gate location basedon desired quality of injection molded part, SPE Annual Techni-cal Conference, ANTEC 1995, pp. 554–560.

Fig. 20. Thickness distribution of the cavity using the modifiedmethod.


[6] W.R. Jong, K.K. Wang, Automatic and optimal design ofrunner systems in injection molding based on flow simulation,SPE Tech. Papers 36 (1990) 385–389.

[7] P. Kennedy, Flow Analysis of Injection Molds, Hanser, Ver-lag, 1995.

[8] C.A. Hieber, S.F. Shen, Flow analysis of the non-isothermaltwo-dimensional filling process in injection molding, Israel J.Technol. 16 (1978) 248–254.

[9] C.A. Hieber, S.F. Shen, A finite element/finite difference simu-

lation of the injection molding filling process, J. Non-Newto-nian Fluid Mech. 7 (1980) 1–32.

[10] H.H. Chiang, C.A. Hieber, K.K. Wang, A unified simulationof the filling and postfilling stages in injection molding. Part I:Formulation, Polym. Eng. Sci. 31 (1991) 116–124.

[11] H.H. Chiang, C.A. Hieber, K.K. Wang, A unified simulationof the filling and postfilling stages in injection molding. Part2: Experimental verification, Polym. Eng. Sci. 31 (1991) 125–139.

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optimizing flow in plastic injection molding

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