Algorithm to optimise number of gates andlocations in an injection mould

C. Shen and M. Zhai*

Determination of the number of gates and their locations is an important issue in injection mould

design. An optimisation strategy based on a genetic algorithm was developed to deal with this

problem. The objective function was expressed as the sum of the injection pressure at the end of

filling stage and a penalty term, and the constraint was to limit the injection pressure to remain

below a reference value. The coordinates of the gates were chosen as design variables. The

penalty term of the objective function was to reduce the number of gates. The number of gates

was modified according to their distance when a mould filling process was simulated to evaluate

the required injection pressure and the objective function. Through optimisation, the minimum

number of gates which satisfies the limit of injection pressure and their optimum locations leading

to balanced flow could be obtained simultaneously. Examples are given to demonstrate the

effectiveness of the proposed scheme.

Keywords: Injection moulding, Optimisation, Gate location, Number of gates

IntroductionInjection moulding is a cyclical process of formingplastic into a desired shape by forcing the material underpressure into a cavity. With a proper mould design andprocess control, multiple parts with tight tolerance andcomplex shapes can be produced in a single cycle.Within many design parameters, gate design is a criticalfactor affecting part quality. Optimisation of thenumber of gates and their locations is of great interestto mould designers.

Several studies have investigated the optimisation ofthe injection moulding process. Pandelidis and Zou1

presented the optimisation of gate location using thecombined scheme of simulated annealing and hill-climbing. The quality of the design was quantified asan additive function of a temperature differential term,an overpacking term and a frictional overheating term,with appropriate weighting factors. Young2 employed agenetic algorithm to search for the optimum gatelocations by minimising the mould filling pressure,uneven filling pattern, and temperature difference duringthe mould filling process. Lee and Kim3 determined theinitial gate design based on the human designer’sintuition, and located the optimum gate by the adjacentnode evaluation method. Irani et al.4 developed a systemthat performed the gate design in two stages: a globalsearch followed by a local search. Gokce et al.5 adopteda branch and bound search to find the optimum gatelocation.

The related studies described above do not auto-matically determine the optimum number of gates andtheir locations simultaneously. In injection moulding,the injection pressure is often limited by the capacity ofinjection machines. When the injection pressure exceedsthe pressure limit, it must be lowered. Changing the gatelocation to reduce the maximum flow length in the partis good practice to lower the injection pressure. If thepressure is still too high for the optimum single gateconfiguration, multiple gates should be considered.However, multiple gates always create additionalunwanted weld lines. Design optimisation should beemployed to find the minimum number of gatessatisfying the injection pressure limit and their optimumlocations.

An optimisation strategy based on a geneticalgorithm is developed in this investigation to searchfor the optimum number of gates and their locations.The objective function is expressed as the sum of theinjection pressure at the end of filling stage and a penaltyterm. The penalty term is used to reduce the number ofgates. The constraint is to limit the injection pressure toremain below a reference value. The coordinates of gatesare chosen as design variables. The number of gates iscorrected according to their distance when the requiredinjection pressure was evaluated. The minimum numberof gates satisfying the limit of injection pressure andtheir optimum locations leading to balanced flow can beobtained simultaneously through optimisation.

Definition of optimisation problemTo apply optimisation theory to the injection mouldingprocess, quantitative measures of the part quality firstneed to be developed since the ultimate goal in

National Engineering Research Center for Polymer Processing,Zhengzhou University, Zhengzhou 450002, P. R. China

*Corresponding author, email mmzhai@ntu.edu.sg

330

� 2004 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 8 March 2004; accepted 20 December 2004DOI 10.1179/174328904X22323 Plastics, Rubbers and Composites 2004 VOL 33 NO 8

optimising the injection moulding design is to improvepart quality. There are two types of part qualitymeasures: direct and indirect. Direct measures candetermine the measurable quantities that characterise aproduct. In contrast, an indirect measure of the qualityis a quantity that is correlated but does not produce adirect estimate of that quality. The determination ofindirect part quality measures will now be discussed.

The indirect quality measures used in this investiga-tion are those related to part warpage and weld lines,which are critical quality issues for most injectionmoulded parts. Part warpage is a dimensional distortionwhich causes structural unfitness and aesthetic pro-blems. It occurs because of non-uniform shrinkage,which is induced by an unbalanced flow pattern. Moreuniform polymer flow leads to smaller warpage. Weldlines can reduce the strength and spoil the appearance ofthe moulded part. The number of weld lines is directlyrelated to the number of gates. Multiple injection gatesalways create additional weld lines. So the aim of gateoptimisation in this investigation is to achieve balancedflow and the minimum number of gates satisfying theinjection pressure limit of an injection machine.

Balanced flow can be related to injection pressure.The injection pressure for a design with an unbalancedflow pattern will be higher than that for a uniform flowpattern, as an unbalanced flow will lead to overpackingand thus higher injection pressure. With a set of fixedprocessing conditions, minimising the injection pressurewill lead to more balanced flow.

Though fewer gates are always more favourable toreduce the number of weld lines, such designs alwaysrequire higher injection pressure. The optimised gatedesign must satisfy the capacity of the injection machine.According to Moldflow design principles,6 the injectionpressure required during the mould filling processshould be lower than 80% of the maximum pressurethe injection machine can provide. If not, the part maycontain voids or have other quality problems. Thus, thispressure limit will be employed as a constraint.

In this investigation, the coordinates of gate locationsare chosen as design variables. From these discussions,the gate optimisation problem can be stated as

Minimise

F (X)~PIzn|b

Subject to

PI{80%|Pmax¡0

X li¡Xi¡Xu

i , i~1,2,:::2|N

where F(X) is the objective function; PI is the injectionpressure at the end of the filling stage; n6b is thepenalty term, which was to reduce the number of gates; nis the modified number of gates which are used tosimulate the mould filling process to evaluate theinjection pressure; b is the penalty factor; Pmax is themaximum pressure the injection machine can provide; Nis the number of initial gates; and X5[X1, … Xi,…X26N]are the design variables representing the coordinates ofgates, and their upper and lower bounds (i.e. Xu and Xl,respectively) constrain the gate location to a specifiedregion of the mould.

Through solving this optimisation model, the mini-mum number of gates which satisfy the limit of injection

pressure and their optimum locations leading tobalanced flow can be obtained simultaneously.

In the present study, the gate locations are defined ascontinuous functions of the design variables. During theoptimisation routine, when the gate location for a designdoes not coincide with a pre-assigned node location inthe finite element mesh, the pre-assigned node nearestthe gate location is moved to the gate location. Theadjacent nodes are moved in the same way so that thegeometry of the elements connected to the gate noderemains unchanged and distortion of the mesh near thegate is avoided. Because the pressure field at a pointsource is singular, such movement of the mesh nodeshelps to neutralise the effect of the errors introduced bythe point source as the design variable.

Optimisation schemeA genetic algorithm is based on the mechanism ofnatural selection and natural genetics. It encodes apotential solution to a specific problem on a simplechromosome like data structure and applies recombina-tion operators to these structures so as to preservecritical information.7,8 Genetic algorithms have beenapplied to a wide range of optimisation problems inengineering.9–13 These algorithms do not use any processknowledge or information while creating new designparameter values during an iteration, hence they can beinherently inefficient.

An optimisation strategy based on a genetic algorithmis proposed to solve the optimisation problem in thispaper. The proposed scheme is described next.

Initial populationTo begin with, it is assumed that N gates are required.Every coordinate of the N gates will be represented by abinary string. These binary strings are connected in ahead-to-tail way to form an L-bit binary chromosome torepresent a set of N gates. Then, an initial population ofa fixed number (m) of chromosomes is created to startthe genetic algorithm. Each chromosome is a binaryvector of L bits. All L bits for each chromosome areinitialised randomly.

Fitness evaluationThis procedure is to evaluate the objective functionindicating the ‘goodness’ (fitness) of a member in ageneration. It consists of the following steps:

(i) One chromosome in the generation is decodedand the coordinates of this set of candidate gatesare obtained.

(ii) A correction process is performed to modify thenumber of gates according to the distancebetween them. If the distance between two gatesis less than a given value, they will be identifiedby one gate located at the midpoint betweenthem.

(iii) The mould filling process is simulated tocalculate the required injection pressure for themodified gate design. Then, the injection pres-sure and the modified number of gates will beused to evaluate the objective function.

These three steps are repeated until all the chromosomesin the generation are evaluated. The calculated objectivefunction values are, to a large extent, determined by the

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Plastics, Rubbers and Composites 2004 VOL 33 NO 8 331

penalty term, which is related to the modified number ofgates. Fewer gates (modified) will lead to a lower valueof the penalty term and a lower objection function value,and thus exhibit higher fitness and have a higherprobability of being selected into the next populationduring the evolution process. On the other hand,multiple gate design will be penalised and have a lowerprobability to survive. As the genetic algorithm pro-ceeded generation by generation, the penalised chromo-somes will die. Thus, the minimum number of gatessatisfying the constraint will be obtained.

Genetic operatorsAfter fitness evaluation, a mating pool with the samesize as the population is formed by selecting the fitterchromosomes. Then, members of the pool are randomlyselected for crossover and mutation transformations.

Crossover combines the features of two parentchromosomes to form two similar offspring by swappingcorresponding segments of the parents. It is performedby randomly selecting two parent strings from themating pool and interchanging the binary bits of arandomly chosen section of the mating parent strings. Acrossover probability is defined to determine whethercrossover should be performed. Usually, a random numberbetween 0 and 1 is generated. If the number is less thanthe crossover probability, crossover is performed.

Mutation alters one randomly chosen gene of aselected chromosome to prevent the search process frompremature loss of important genetic information.Operation of the mutation is determined by themutation probability. If a random number generatedbetween 0 and 1 is less than the mutation probability,mutation will be performed.

One cycle of the genetic algorithm is thus completedand an improved population is generated. The wholeprocedure, i.e. evaluate fitness of current population,select chromosomes, perform crossover and mutation, isrepeated until the maximum number of generations ismet.

Application exampleThe performance of the proposed optimisation strategyis demonstrated by two examples.

Example 1. Square plateA square plate was chosen as the first example forverification purposes since the optimum gate locationscan be easily identified by symmetry. The dimensions ofthe plate was 0.2 m60.2 m. Its finite element meshmodel consisted of 155 nodes and 266 triangularelements. This part was moulded with material PS(Styron 685, Dow Chemical). The moulding conditionsemployed were mould temperature537uC, melttemperature5200uC, and injection rate51E–04 m3 s21.A fixed population size of 50 was used in conjunction

with crossover probability50.6 and mutation probabil-ity50.001 for the genetic search. The penalty factor bwas set as 1Ez08. Because the number of gates ismainly determined by the capacity of an injectionmachine, two cases with different injection pressurelimits will be discussed.

Case 1

The injection pressure limit was 9Ez07 Pa for the firstcase. This limit was rather high, so both single gate andtwo gates can satisfy the constraints. In order to reducethe number of weld lines, single gate design was morereasonable. For single gate configuration, the gateshould be located at the centre of the square plate, i.e.(0.1000 m60.1000 m).

The proposed optimisation strategy was applied withtwo initial gates. Then 50 candidate pairs of gates weregenerated for the first generation. For each pair of gates,if the distance between them was less than a given value,they would be identified by one gate located at themidpoint between them. Subsequently, the mould fillingprocess was simulated and the objective function wasevaluated based on the corrected number of gates andtheir locations. Then, the fitness of each candidate gatedesign would be obtained. According to the fitness,those pairs of gates representing better solutions weregiven more chances to reproduce in the next generation.Through the natural selection and reproduction, thepopulation was improved. After nine generations with atotal of 450 mould filling simulations, the optimumresults (Table 1) were attained.

The locations of the two gates are shown as points Aand B in Fig. 1. The two gates were very close and theyhad been identified by one gate automatically by theprogram when mould filling process was performed toevaluate the objective function. That is to say, a singlegate design was obtained after the optimisation process,and the optimum gate was located at the midpointbetween the two gates, i.e. (0.1005 m60.1006 m), which

Table 1 Optimum results for case 1

Gate location Lower bound, m Upper bound, m Optimal value, m

Gate 1 X1 0.04 0.19 0.1076Y1 0.04 0.19 0.0940

Gate 2 X2 0.04 0.19 0.0934Y2 0.04 0.19 0.1071

1 Schematic diagram of optimisation result

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332 Plastics, Rubbers and Composites 2004 VOL 33 NO 8

was close to the expect value (0.1000 m, 0.1000 m).Figure 2 shows the melt front advancements for theoptimum design. As expected, the melt reached theextremities of the mould almost simultaneously.

Case 2

In this case, the injection pressure limit was3.8Ez07 Pa. The maximum injection pressure is ratherlow and could not meet the requirement of a single-gate mould. So two gates are required and theiroptimal location should be at (0.1500 m, 0.1000 m),(0.0500 m, 0.1000 m); or (0.1000 m, 0.0500 m),(0.1000 m, 0.1500 m) by symmetry.

As in case 1, two initial gates were supposed. After 12generations with 600 filling simulations, the optimisationresults were obtained (Table 2). The locations of the twogates are shown as points C and D in Fig. 1. Thedistance between the two gates was rather long and theyshould not been identified as a single gate automaticallyby the program when the mould filling process isperformed to evaluate the objective function. That isto say, two gates were obtained after the optimisationand their optimum locations (0.1498 m, 0.0995 m),(0.0497 m, 0.1010 m) were close to one pair of expectedgates (0.1500 m, 0.1000 m), (0.0500 m, 0.1000 m).Figure 3 indicated the melt front advancements for theoptimum design. It can be seen that the melt flow wasbalanced in the cavity.

Example 2. L-plateThe second example considered an L-plate, as shownin Fig. 4. Its finite element model consisted of 360nodes and 642 triangular elements. The resinmaterial, process conditions and parameters associatedwith the genetic algorithm were the same as those forexample 1.

The injection pressure limit was 230 MPa in thisexample. Three initial gates were used for optimisation.After 15 generations with 750 filling simulations, theoptimum results were obtained (Table 3). The locationsof the three gates are shown as points A, B and C in Fig. 4.The three gates were very close and they have beenidentified by a single gate to evaluate the objective function.That is to say, a single gate was favourable in this example,which should be located at [(0.5606z0.6534z0.5886)/3.0 m, (0.4123z0.3515z0.3374)/3.0 m], i.e. (0.6009 m,0.3671 m). Figure 5 shows the melt front advancementsin the cavity for the optimum design. It can be seen that abalanced flow was achieved.

2 Melt front advancement of optimum design for case 1

Table 2 Optimum results for case 2

Gate location Lower bound, m Upper bound, m Optimal value, m

Gate 1 X1 0.04 0.19 0.1498Y1 0.04 0.19 0.0995

Gate 2 X2 0.04 0.19 0.0497Y2 0.04 0.19 0.1010

3 Melt front advancement of optimum design for case 2

4 Schematic diagram of L-plate; dimensions in m

Table 3 Optimum results for example 2

Gate 1 Gate 2

Gate location X1 Y1 X2 Y2

Lower bound, m 0.0782 0.0782 0.0782 0.00782Upper bound, m 0.6830 0.6330 0.6830 0.6330Optimum, m 0.5606 0.4123 0.6534 0.3515

5 Melt front advancement of optimum design for L-plate

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Plastics, Rubbers and Composites 2004 VOL 33 NO 8 333

ConclusionIn this paper, an optimisation strategy based on agenetic algorithm was developed to optimise the numberof gates and their locations in an injection mould. Toavoid the difficulties induced by the integer program-ming and the change of the number of design variables,only the coordinates of the gate locations were chosen asdesign variables. The number of gates was correctedaccording to the distance between them to perform themould filling simulation to evaluate the objectivefunction. The validity of the present methodology wasdemonstrated by two examples.

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