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Int J Adv Manuf Technol (2001) 17:2973042001 Springer-Verlag London LimitedOptimum Gate Design of FreeForm Injection Mould using theAbductive NetworkJ. C. LinDepartment of Mechanical Design Engineering, National Hu-Wei Institute of Technology, Yunlin, TaiwanThis study uses the injection position and size of the gate asthe major control parameters for a simulated injection-mould.Once the injection parameters (gate size and gate position) aregiven, the product performance (deformation) can be accuratelypredicted by the abductive network developed.To avoid the numerous influencing factors, first the part-lineof the parameter equation is created by an abductive networkto limit the range of the gate. The optimal injection parameterscan be searched for by a simulation annealing (SA) optimisa-tion algorithm, with a performance index, to obtain a perfectpart. The major purpose is searching for the optimal gatelocation on the part-surface, and minimising the air-trap anddeformation after part formation. This study also uses a prac-tical example which has been and proved by experiment toachieve a satisfactory result.Keywords: Abductive network; Injection mould; Simulationannealing (SA)1. IntroductionOwing to the rapid development of industry and commerce inrecent years, there is a need for rapid and high volumeproduction of goods. The products are manufactured usingmoulds in order to save the time and cost. Plastic products arethe majority. Owing to these products not requiring complicatedprocesses it is possible to cope with market demand speedilyand conveniently.In traditional plastic production, the designs of the portionsof the mould are determined by humans. However, because ofthe increased performance requirements, the complexity ofplastic products has increased. First, the geometric shapes ofthe plastic products are difficult to draw, and the internalshape is often complex which also affects the production ofthe product.Injection processing can be divided into three stages:Correspondence and offprint requests to: Dr. J. C. Lin, Departmentof Mechanical Design Engineering, National Hu-Wei Institute ofTechnology, Yunlin 632, Taiwan. E-mail: linrcK.tw1. Heat the plastic material to a molten state. Then, by highpressure, force the material to the runner, and then into themould cavity.2. When the filling of the mould cavity is completed, moremolten plastic should be delivered into the cavity at highpressure to compensate for the shrinkage of the plastic. Thisensures complete filling of the mould cavity.3. Take out the product after cooling.Though the filling process is only a small proportion of thecomplete formation cycle, it is very important. If filling inincomplete, there is no pressure holding and cooling is required.Thus, the flow of the plastic fluid should be controlledthoroughly to ensure the quality of the product.The isothermal filling model of a Newtonian fluid is thesimplest injection mould-flow filling model. Richardson 1produced a complete and detailed concept. The major conceptis based on the application of lubrication theory, and hesimplified the complex 3D flow theory to 2D Hele-Shaw flow.The Hele-Shaw flow was used to simulate the potential flowand was furthermore used in the plasticity flow of the plastic.He assumed the plasticity flow on an extremely thin plate andfully developed the flow by ignoring the speed change throughthe thickness. Kamal et al. used similar methods to obtain thefilling condition for a rectangular mould cavity, and the analyti-cal result obtained was almost identical to the experimentalresult.Plastic material follows the Newtonian fluid model for flowin a mould cavity, and Bird et al. 24 derived mould-flowtheory based on this. When the shape of a mould is complicatedand there is variation in thickness, then the equilibrium equa-tions changes to nonlinear and has no analytical solution.Thus, it can be solved only by finite-difference or numericalsolutions 2,5Of course, as the polymer is a visco-elastic fluid, it is bestto solve the flow problem by using visco-elasticity equations.In 1998, Goyal et al. used the White Metzner visco-elasticitymodel to simulate the disk mould-flow model for centralpouring. Metzner, using a finite-difference method to solve thegoverning equation, fould the visco-elasticity effect would notchange the distribution of speed and temperature. However, itaffects the stress field very much. If it is a pure visco-elastic298 J. C. Linflow model, the popular GNF model is generally used toperform numerical simulation.Currently, finite-element methods are mostly used for thesolution of mould-flow problems. Other methods are purevisco-elastic models, such as C-FOLW and MOLD-FLOWsoftware. We used this method as well. Some software employsthe visco-elastic WhiteMetzner model, but it is limited to 2Dmould-flow analysis. Simple mould flow analysis is limited byCPU time. For the complicated mould shapes, Papthanasionet al. used UCM fluid for filling analysis, using a finite-difference method and BFCC coordination system applicationfor the solution of the more complicated mould-shape andfilling problem, but it was not commercialised 6.Many factors affect plastic material injection. The fillingspeed, injection pressure and molten temperature, holding press-ure 712, cooling tube 13,14 and injection gate affect theaccuracy of the plastic product, because, when the injectionprocessing is completed, the flow of material in the mouldcavity results in uneven temperature and pressure, and inducesresidual stress and deformation of the workpiece after cooling.It is difficult to decide on the mould part-surface and gatepositions. Generally, the mould part-surface is located at thewidest plane of the mould. Searching for the optimal gateposition depends on experience. Minimal modification to themould is required if you are lucky. However, the time andcost required for the modification of most injection-mouldsexceeds the original cost, if the choice of the part-line is poor.For the mould part surface, many workers used various methodsto search for the optimal mould part-line, such as geometricshape and feature-based design 1517. Some workers usedfinite-element methods and abductive networks to look for theoptimal gate design for a die-casting mould 18.This study used an abductive network to establish the para-meter relationship of the mould part-line, and used this formulafor searching for 22 points on the injection mould part-line toserve as the location for an injection gate. Abductive networksare used to match injection pressure and pressure holding timeto perform injection formation analysis, and to establish arelationship between these parameters, and the output result ofthe injection process.It has been shown that prediction accuracy in abductivenetworks is much higher than that in other networks 19.Abductive networks based on the abductive modelling tech-nique are able to represent complex and uncertain relationshipsbetween mould-flow analysis results and injection parameters.It has beeen shown that the injection-strain and injection-stressin a product can be predicted, with reasonable accuracy, basedon the developed networks. The abductive network has beenconstructed once the relationships of gate location that areinput and simulated have been determined; an appropriateoptimisation algorithm with a performance index is then usedto search for the optimal location parameters.In this paper, an optimisation method for simulated annealing20 is presented. The simulated annealing algorithm is asimulation of the annealing process for minimising the perform-ance index. It has been successfully applied to filtering inimage processing 21, VLSI layout generation 22, discretetolerance design 23, wire electrical discharge machining 24,deep-draw clearance 25, and casting-die runner design 26,etc. It provides an experimental foundation based on theoryfor the development and application of the technologies.2. Mould-Flow TheoryThe mould flow analysis include four major parts:1. Filling stage.2. Pressure holding stage.3. Cooling and solidification stage.4. Shrinkage and warp, i.e. stress residue stage.Thus, the major mould flow equations are divided into fourgroups. In the filling stage, the mould cavity is filled withmolten plastic fluid at high presssure. Thus, the governingequations include:1. Continuity equation. The plastic deformation or shapechange accompany the flow during the filling process(mass conservation):rt+ = (=V) = 0 (1)r = plastic density; V = vector velocity2. Momentum equation. Newtons second law is used to derivethe momentum (acceleration condition) or force balancegenerated by plastic flow:rFVt+(V =)VG=- VP + = t + rf (2)P = flow pressure; f = body force; t = stress tensor.3. Energy equation. The energy conservation of system andlaws of conservation of flow material, if the fluid is incom-pressible:rCP FTt+(V =)TG=- = q + t: =V (3)T = temperature; CP= specific heat of constant pressure; q= heat flux4. Rheology equationt = fn(g, T, P, %) (4)g= =V +(=V)T(5)=V = deform tensor; (=V)T= transport vector.Holding pressure analysis. The holding pressure process isto maintain the pressure after the mould cavity is filled inorder to inject more plastic, to compensate for the shrinkagein cooling.rV1t=-Px1+Ft11x1+t21x2+t31x3G(6)rV2t=-Px2+Ft12x1+t22x2+t32x3G(7)rV3t=-Px1+Ft13x1+t23x2+t33x3G(8)Optimum Gate Design of FreeForm Injection Mould 299Cooling analysis. The analysis of the cooling process con-siders the relationship of the plastic flow distribution and heattransmission. The homogenous mould temperature and thesequence of filling will be affected by the shrinkage of theproduct formed. If the temperature is distributed non-uniformly,it tends to produce warp. This is mainly due to heat-transferand crystallisation heat of the plastic.rCPTt= kF2Tx21+2Tx22+2Tx33G+ rCPrDH (9)r = crystallisation rate; DH = crystallisation heat3. Abductive Network Synthesis andEvaluationMiller 22 observed that human behaviour limits the amountof information considered at a time. The input data are summar-ised and then the summarised information is passed to a higherreasoning level.In an abductive network, a complex system can be decom-posed into smaller, simpler subsystems grouped into severallayers using polynomial function nodes. These nodes evaluatethe limited number of inputs by a polynomial function andgenerate an output to serve as an input to subsequent nodesof the next layer. These polynomial functional nodes arespecified as follows:1. NormaliserA normaliser transforms the original input variables into arelatively common region.a1= q0+ q1x1(10)Where a1is the normalised input, q0, q1are the coefficientsof the normaliser, and x1is the original input.2. White nodeA white node consists of linear weighted sums of all theoutputs of the previous layer.b1= r0+ r1y1+ r2y2+ r3y3+ % + rnyn(11)Where y1, y2, y3, ynare the input of the previous layer, b1isthe output of the node, and the r0, r1, r2, r3, %, rnare thecoefficients of the triple node.3. Single, double, and triple nodesThese names are based on the number of input variables. Thealgebraic form of each of these nodes is shown in thefollowing:single: c1= s0+ s1z1+ s2z21+ s3z31(12)double: d1= t0+(t1n1+ t2n21+ t3n31)+(t4n2+ t5n22+ t6n32)+(t7n1n2)(13)triple: e1= u0+(u1o1+ u2o21+ u3o31)+(u4o2+ u5o22+ u6o32)+(u7o3+ u8o23+ u9o33)+ u10o1o2+ u11o2o3+ u12o1o3+ u13o1o2o3(14)where z1, z2, z3, %, zn, n1, n2, n3, %, nn, o1, o2, o3, %, onare the input of the previous layer, c1, d1, and e1are theoutput of the node, and the s0, s1, s2, s3, %, sn, t0, s1, t2, t3,%, tn, u0, u1, u2, u3, %, unare the coefficients of the single,double, and triple nodes.These nodes are third-degree polynomial Eq. and doublesand triples have cross-terms, allowing interaction among thenode input variables.4. UnitiserOn the other hand, a unitiser converts the output to a realoutput.f1= v0+ v1i1(15)Where i1is the output of the network, f1is the real output,and v0and v1are the coefficients of the unitiser.4. Part-Surface ModelThis study uses an actual industrial product as a sample, Fig.1. The mould part surface is located at the maximum projectionarea. As shown in Fig. 1, the bottom is the widest plane andis chosen as the mould part surface. However, most importantis the searching of gate position on the part surface.This study establishes the parameter equation by using anabductive neuron network, in order to establish the simulatedannealing method (SA) to find the optimal gate path position.The parameter equation of a part-surface is expressed byF(Y) = X. First, use a CMM system to measure the XYZcoordinate values (in this study z = 0) of 22 points on themould part-line on the mould part-surface as illustrated inTable 1, and the gate position is completely on the curve inthis space.Prior to developing a space-curve model, a database has tobe trained, and a good relationship msut exist between thecontrol point and abductive network system. A correct andFig. 1. Injection-mould product.300 J. C. LinTable 1. X, Y coordinate.Set number X-coordinate Y-coordinate1 0.02 - 4.62 - 1.63 - 4.333 - 3.28 - 3.54 - 5.29 - 2.045 - 7.31 - 0.566 - 9.34 0.97 - 11.33 2.358 - 12.98 3.949 - 13.85 5.5710 - 14.12 7.3411 - 13.69 9.6712 - 12.96 11.913 - 10.00 21.0314 - 9.33 23.1615 - 8.64 25.2816 - 7.98 27.3917 - 7.87 28.3118 - 7.80 29.2919 - 7.83 30.3420 - 7.60 31.3021 - 7.07 32.1522 - 6.11 32.49precise curve Eq. is helpful for finding the optimal gatelocation.To build a complete abductive network, the first requirementis to train the database. The information given by the inputand output parameters must be sufficient. A predicted squareerror (PSE) criterion is then used to determine automaticallyan optimal structure 23. The PSE criterion is used to selectthe least complex but still accurate network.The PSE is composed of two terms:PSE = FSE + KP(16)Where FSE is the average square error of the network forfitting the training data and KPis the complex penalty of thenetwork, shown by the following equation:KP= CPM2s2pKN(17)Where CPM is the complex penalty multiplier, KPis a coef-ficient of the network; N is the number of training data to beused and s2pis a prior estimate of the model error variance.Based on the development of the database and the predictionof the accuracy of the part-surface, a three-layer abductivenetwork, which comprised design factors (input: various Ycoordinate) and output factors (X coordinate) is synthesisedautomatically. It is capable of predicting accurately the spacecurve at any point under various control parameters. All poly-nomial equations used in this network are listed in AppendixA. (PSE = 5.8 10- 3).Table 2 compares the error predicted by the abductive modeland CMM measurement data. The measurement daa is excludedfrom the 22 sets of CMM measurement data for establishingthe model. This set of data is used to test the appropriatenessof the model established above. We can see from Table 2 thatthe error is approximately 2.13%, which shows that the modelis suitable for this space curve.Table 2. CMMS-coordinate and neural network predict compared (itis not included in any original 22 sets database).Items CMMS neural network Error valuescoordinate predict (CMMS-predict)/coordinate CMMSCoordinate (- 11.25, 16.0) (- 11.01, 16.0) 2.13%5. Create the Injection-Mould ModelSimilarly, the relationship is established between input para-meters (gate location and gate size) and the output parameter(deformation) during the injection process. To build a completeabductive network, the first requirement is to train the database.The information given by the input and the output data mustbe sufficient. Thus, the training factor (gate location) forabductive network training should be satisfactory for makingdefect-free products. Figure 2 shows the simulation of FEMmould-flow. Table 3 shows the position of the gate and themaximum deformation of the product obtained from mould-flow analysis.Based on the development of the injection-mould model,three-layer abductive networks, which are comprised of injec-tion-mould conditions and the injection-results (deformation),are synthesised automatically. They are capable of predictingaccurately the product strain (the result of injection-mouldedproduct) under various control parameters. All polynomialequations used in this network are listed in Appendix B (PSE= 2.3 10- 5).Table 4 compares the error predicted by the abductive modeland the simulation case. The simulation case is excluded fromthe 22 sets of simulation cases for establishing the model. Thisset of data is used to test the appropriateness of the modelestablished above. We can see from Table 4 that the error isFig. 2. The deformation of FEM mould-flow.Optimum Gate Design of FreeForm Injection Mould 301Table 3. Gate location and the maximum strain relationship.Set number X-coordinate Y-coordinate Gate width Gate length Produce max. strain1 0.02 - 4.6 0.525 1.1475 0.3482 - 1.63 - 4.33 0.7 1.53 0.31533 - 3.28 - 3.5 0.875 1.9125 0.27104 - 5.29 - 2.04 1.05 2.295 0.28585 - 7.31 - 0.56 0.525 1.1475 0.30176 - 9.34 0.9 0.7 1.53 0.5267 - 11.33 2.35 0.875 1.9125 0.23698 - 12.98 3.94 1.05 2.295 0.25179 - 13.85 5.57 0.525 1.1475 0.278810 - 14.12 7.34 0.7 1.53 0.277311 - 13.69 9.67 0.875 1.9125 0.298812 - 12.96 11.9 1.05 2.295 0.299713 - 10.00 21.03 0.525 1.1475 0.257614 - 9.33 23.16 0.7 1.53 0.262415 - 8.64 25.28 0.875 1.9125 0.254216 - 7.98 27.39 1.05 2.295 0.249517 - 7.87 28.31 0.525 1.1475 0.250318 - 7.80 29.29 0.7 1.53 0.245619 - 7.83 30.34 0.875 1.9125 0.259620 - 7.60 31.30 1.05 2.295 0.245721 - 7.07 32.15 0.525 1.1475 0.249922 - 6.11 32.49 0.7 1.53 0.2511Table 4. Mould-flow simulated and neural network predict compared(it is not included in any original 22 set database).Items FEM mould-flow Neural networksimulation predictX-coordinate - 1