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Int J Adv Manuf Technol (2001) 17:297304 2001 Springer-Verlag London Limited Optimum Gate Design of FreeForm Injection Mould using the Abductive Network J. C. Lin Department of Mechanical Design Engineering, National Hu-Wei Institute of Technology, Yunlin, Taiwan This study uses the injection position and size of the gate as the major control parameters for a simulated injection-mould. Once the injection parameters (gate size and gate position) are given, the product performance (deformation) can be accurately predicted by the abductive network developed. To avoid the numerous influencing factors, first the part-line of the parameter equation is created by an abductive network to limit the range of the gate. The optimal injection parameters can be searched for by a simulation annealing (SA) optimisa- tion algorithm, with a performance index, to obtain a perfect part. The major purpose is searching for the optimal gate location on the part-surface, and minimising the air-trap and deformation after part formation. This study also uses a prac- tical example which has been and proved by experiment to achieve a satisfactory result. Keywords: Abductive network; Injection mould; Simulation annealing (SA) 1. Introduction Owing to the rapid development of industry and commerce in recent years, there is a need for rapid and high volume production of goods. The products are manufactured using moulds in order to save the time and cost. Plastic products are the majority. Owing to these products not requiring complicated processes it is possible to cope with market demand speedily and conveniently. In traditional plastic production, the designs of the portions of the mould are determined by humans. However, because of the increased performance requirements, the complexity of plastic products has increased. First, the geometric shapes of the plastic products are difficult to draw, and the internal shape is often complex which also affects the production of the product. Injection processing can be divided into three stages: Correspondence and offprint requests to: Dr. J. C. Lin, Department of Mechanical Design Engineering, National Hu-Wei Institute of Technology, Yunlin 632, Taiwan. E-mail: linrcKsunws.nhit.edu.tw 1. Heat the plastic material to a molten state. Then, by high pressure, force the material to the runner, and then into the mould cavity. 2. When the filling of the mould cavity is completed, more molten plastic should be delivered into the cavity at high pressure to compensate for the shrinkage of the plastic. This ensures complete filling of the mould cavity. 3. Take out the product after cooling. Though the filling process is only a small proportion of the complete formation cycle, it is very important. If filling in incomplete, there is no pressure holding and cooling is required. Thus, the flow of the plastic fluid should be controlled thoroughly to ensure the quality of the product. The isothermal filling model of a Newtonian fluid is the simplest injection mould-flow filling model. Richardson 1 produced a complete and detailed concept. The major concept is based on the application of lubrication theory, and he simplified the complex 3D flow theory to 2D Hele-Shaw flow. The Hele-Shaw flow was used to simulate the potential flow and was furthermore used in the plasticity flow of the plastic. He assumed the plasticity flow on an extremely thin plate and fully developed the flow by ignoring the speed change through the thickness. Kamal et al. used similar methods to obtain the filling condition for a rectangular mould cavity, and the analyti- cal result obtained was almost identical to the experimental result. Plastic material follows the Newtonian fluid model for flow in a mould cavity, and Bird et al. 24 derived mould-flow theory based on this. When the shape of a mould is complicated and 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 numerical solutions 2,5 Of course, as the polymer is a visco-elastic fluid, it is best to solve the flow problem by using visco-elasticity equations. In 1998, Goyal et al. used the White Metzner visco-elasticity model to simulate the disk mould-flow model for central pouring. Metzner, using a finite-difference method to solve the governing equation, fould the visco-elasticity effect would not change the distribution of speed and temperature. However, it affects the stress field very much. If it is a pure visco-elastic 298 J. C. Lin flow model, the popular GNF model is generally used to perform numerical simulation. Currently, finite-element methods are mostly used for the solution of mould-flow problems. Other methods are pure visco-elastic models, such as C-FOLW and MOLD-FLOW software. We used this method as well. Some software employs the visco-elastic WhiteMetzner model, but it is limited to 2D mould-flow analysis. Simple mould flow analysis is limited by CPU time. For the complicated mould shapes, Papthanasion et al. used UCM fluid for filling analysis, using a finite- difference method and BFCC coordination system application for the solution of the more complicated mould-shape and filling problem, but it was not commercialised 6. Many factors affect plastic material injection. The filling speed, injection pressure and molten temperature, holding press- ure 712, cooling tube 13,14 and injection gate affect the accuracy of the plastic product, because, when the injection processing is completed, the flow of material in the mould cavity results in uneven temperature and pressure, and induces residual stress and deformation of the workpiece after cooling. It is difficult to decide on the mould part-surface and gate positions. Generally, the mould part-surface is located at the widest plane of the mould. Searching for the optimal gate position depends on experience. Minimal modification to the mould is required if you are lucky. However, the time and cost required for the modification of most injection-moulds exceeds the original cost, if the choice of the part-line is poor. For the mould part surface, many workers used various methods to search for the optimal mould part-line, such as geometric shape and feature-based design 1517. Some workers used finite-element methods and abductive networks to look for the optimal 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 formula for searching for 22 points on the injection mould part-line to serve as the location for an injection gate. Abductive networks are used to match injection pressure and pressure holding time to perform injection formation analysis, and to establish a relationship between these parameters, and the output result of the injection process. It has been shown that prediction accuracy in abductive networks 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 relationships between mould-flow analysis results and injection parameters. It has beeen shown that the injection-strain and injection-stress in a product can be predicted, with reasonable accuracy, based on the developed networks. The abductive network has been constructed once the relationships of gate location that are input and simulated have been determined; an appropriate optimisation algorithm with a performance index is then used to search for the optimal location parameters. In this paper, an optimisation method for simulated annealing 20 is presented. The simulated annealing algorithm is a simulation of the annealing process for minimising the perform- ance index. It has been successfully applied to filtering in image processing 21, VLSI layout generation 22, discrete tolerance 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 theory for the development and application of the technologies. 2. Mould-Flow Theory The 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 four groups. In the filling stage, the mould cavity is filled with molten plastic fluid at high presssure. Thus, the governing equations include: 1. Continuity equation. The plastic deformation or shape change accompany the flow during the filling process (mass conservation): r t + = (=V) = 0 (1) r = plastic density; V = vector velocity 2. Momentum equation. Newtons second law is used to derive the momentum (acceleration condition) or force balance generated by plastic flow: r F V t +(V =)V G =- VP + = t + rf (2) P = flow pressure; f = body force; t = stress tensor. 3. Energy equation. The energy conservation of system and laws of conservation of flow material, if the fluid is incom- pressible: rC P F T t +(V =)T G =- = q + t: =V (3) T = temperature; C P = specific heat of constant pressure; q = heat flux 4. Rheology equation t = f n (g , T, P, %) (4) g = =V +(=V) T (5) =V = deform tensor; (=V) T = transport vector. Holding pressure analysis. The holding pressure process is to maintain the pressure after the mould cavity is filled in order to inject more plastic, to compensate for the shrinkage in cooling. r V 1 t =- P x 1 + F t 11 x 1 + t 21 x 2 + t 31 x 3 G (6) r V 2 t =- P x 2 + F t 12 x 1 + t 22 x 2 + t 32 x 3 G (7) r V 3 t =- P x 1 + F t 13 x 1 + t 23 x 2 + t 33 x 3 G (8) Optimum Gate Design of FreeForm Injection Mould 299 Cooling analysis. The analysis of the cooling process con- siders the relationship of the plastic flow distribution and heat transmission. The homogenous mould temperature and the sequence of filling will be affected by the shrinkage of the product formed. If the temperature is distributed non-uniformly, it tends to produce warp. This is mainly due to heat-transfer and crystallisation heat of the plastic. rC P T t = k F 2 T x 2 1 + 2 T x 2 2 + 2 T x 3 3 G + rC P rDH (9) r = crystallisation rate; DH = crystallisation heat 3. Abductive Network Synthesis and Evaluation Miller 22 observed that human behaviour limits the amount of information considered at a time. The input data are summar- ised and then the summarised information is passed to a higher reasoning level. In an abductive network, a complex system can be decom- posed into smaller, simpler subsystems grouped into several layers using polynomial function nodes. These nodes evaluate the limited number of inputs by a polynomial function and generate an output to serve as an input to subsequent nodes of the next layer. These polynomial functional nodes are specified as follows: 1. Normaliser A normaliser transforms the original input variables into a relatively common region. a 1 = q 0 + q 1 x 1 (10) Where a 1 is the normalised input, q 0 , q 1 are the coefficients of the normaliser, and x 1 is the original input. 2. White node A white node consists of linear weighted sums of all the outputs of the previous layer. b 1 = r 0 + r 1 y 1 + r 2 y 2 + r 3 y 3 + % + r n y n (11) Where y 1 , y 2 , y 3 , y n are the input of the previous layer, b 1 is the output of the node, and the r 0 , r 1 , r 2 , r 3 , %, r n are the coefficients of the triple node. 3. Single, double, and triple nodes These names are based on the number of input variables. The algebraic form of each of these nodes is shown in the following: single: c 1 = s 0 + s 1 z 1 + s 2 z 2 1 + s 3 z 3 1 (12) double: d 1 = t 0 +(t 1 n 1 + t 2 n 2 1 + t 3 n 3 1 ) +(t 4 n 2 + t 5 n 2 2 + t 6 n 3 2 )+(t 7 n 1 n 2 )(13) triple: e 1 = u 0 +(u 1 o 1 + u 2 o 2 1 + u 3 o 3 1 ) +(u 4 o 2 + u 5 o 2 2 + u 6 o 3 2 )+(u 7 o 3 + u 8 o 2 3 + u 9 o 3 3 ) + u 10 o 1 o 2 + u 11 o 2 o 3 + u 12 o 1 o 3 + u 13 o 1 o 2 o 3 (14) where z 1 , z 2 , z 3 , %, z n , n 1 , n 2 , n 3 , %, n n , o 1 , o 2 , o 3 , %, o n are the input of the previous layer, c 1 , d 1 , and e 1 are the output of the node, and the s 0 , s 1 , s 2 , s 3 , %, s n , t 0 , s 1 , t 2 , t 3 , %, t n , u 0 , u 1 , u 2 , u 3 , %, u n are the coefficients of the single, double, and triple nodes. These nodes are third-degree polynomial Eq. and doubles and triples have cross-terms, allowing interaction among the node input variables. 4. Unitiser On the other hand, a unitiser converts the output to a real output. f 1 = v 0 + v 1 i 1 (15) Where i 1 is the output of the network, f 1 is the real output, and v 0 and v 1 are the coefficients of the unitiser. 4. Part-Surface Model This study uses an actual industrial product as a sample, Fig. 1. The mould part surface is located at the maximum projection area. As shown in Fig. 1, the bottom is the widest plane and is chosen as the mould part surface. However, most important is the searching of gate position on the part surface. This study establishes the parameter equation by using an abductive neuron network, in order to establish the simulated annealing method (SA) to find the optimal gate path position. The parameter equation of a part-surface is expressed by F(Y) = X. First, use a CMM system to measure the XYZ coordinate values (in this study z = 0) of 22 points on the mould part-line on the mould part-surface as illustrated in Table 1, and the gate position is completely on the curve in this space. Prior to developing a space-curve model, a database has to be trained, and a good relationship msut exist between the control point and abductive network system. A correct and Fig. 1. Injection-mould product. 300 J. C. Lin Table 1. X, Y coordinate. Set number X-coordinate Y-coordinate 1 0.02 - 4.6 2 - 1.63 - 4.33 3 - 3.28 - 3.5 4 - 5.29 - 2.04 5 - 7.31 - 0.56 6 - 9.34 0.9 7 - 11.33 2.35 8 - 12.98 3.94 9 - 13.85 5.57 10 - 14.12 7.34 11 - 13.69 9.67 12 - 12.96 11.9 13 - 10.00 21.03 14 - 9.33 23.16 15 - 8.64 25.28 16 - 7.98 27.39 17 - 7.87 28.31 18 - 7.80 29.29 19 - 7.83 30.34 20 - 7.60 31.30 21 - 7.07 32.15 22 - 6.11 32.49 precise curve Eq. is helpful for finding the optimal gate location. To build a complete abductive network, the first requirement is to train the database. The information given by the input and output parameters must be sufficient. A predicted square error (PSE) criterion is then used to determine automatically an optimal structure 23. The PSE criterion is used to select the least complex but still accurate network. The PSE is composed of two terms: PSE = FSE + K P (16) Where FSE is the average square error of the network for fitting the training data and K P is the complex penalty of the network, shown by the following equation: K P = CPM 2s 2 p K N (17) Where CPM is the complex penalty multiplier, K P is a coef- ficient of the network; N is the number of training data to be used and s 2 p is a prior estimate of the model error variance. Based on the development of the database and the prediction of the accuracy of the part-surface, a three-layer abductive network, which comprised design factors (input: various Y coordinate) and output factors (X coordinate) is synthesised automatically. It is capable of predicting accurately the space curve at any point under various control parameters. All poly- nomial equations used in this network are listed in Appendix A. (PSE = 5.8 10 - 3 ). Table 2 compares the error predicted by the abductive model and CMM measurement data. The measurement daa is excluded from the 22 sets of CMM measurement data for establishing the model. This set of data is used to test the appropriateness of the model established above. We can see from Table 2 that the error is approximately 2.13%, which shows that the model is suitable for this space curve. Table 2. CMMS-coordinate and neural network predict compared (it is not included in any original 22 sets database). Items CMMS neural network Error values coordinate predict (CMMS-predict)/ coordinate CMMS Coordinate (- 11.25, 16.0) (- 11.01, 16.0) 2.13% 5. Create the Injection-Mould Model Similarly, 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 complete abductive network, the first requirement is to train the database. The information given by the input and the output data must be sufficient. Thus, the training factor (gate location) for abductive network training should be satisfactory for making defect-free products. Figure 2 shows the simulation of FEM mould-flow. Table 3 shows the position of the gate and the maximum 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 predicting accurately the product strain (the result of injection-moulded product) under various control parameters. All polynomial equations used in this network are listed in Appendix B (PSE = 2.3 10 - 5 ). Table 4 compares the error predicted by the abductive model and the simulation case. The simulation case is excluded from the 22 sets of simulation cases for establishing the model. This set of data is used to test the appropriateness of the model established above. We can see from Table 4 that the error is Fig. 2. The deformation of FEM mould-flow. Optimum Gate Design of FreeForm Injection Mould 301 Table 3. Gate location and the maximum strain relationship. Set number X-coordinate Y-coordinate Gate width Gate length Produce max. strain 1 0.02 - 4.6 0.525 1.1475 0.348 2 - 1.63 - 4.33 0.7 1.53 0.3153 3 - 3.28 - 3.5 0.875 1.9125 0.2710 4 - 5.29 - 2.04 1.05 2.295 0.2858 5 - 7.31 - 0.56 0.525 1.1475 0.3017 6 - 9.34 0.9 0.7 1.53 0.526 7 - 11.33 2.35 0.875 1.9125 0.2369 8 - 12.98 3.94 1.05 2.295 0.2517 9 - 13.85 5.57 0.525 1.1475 0.2788 10 - 14.12 7.34 0.7 1.53 0.2773 11 - 13.69 9.67 0.875 1.9125 0.2988 12 - 12.96 11.9 1.05 2.295 0.2997 13 - 10.00 21.03 0.525 1.1475 0.2576 14 - 9.33 23.16 0.7 1.53 0.2624 15 - 8.64 25.28 0.875 1.9125 0.2542 16 - 7.98 27.39 1.05 2.295 0.2495 17 - 7.87 28.31 0.525 1.1475 0.2503 18 - 7.80 29.29 0.7 1.53 0.2456 19 - 7.83 30.34 0.875 1.9125 0.2596 20 - 7.60 31.30 1.05 2.295 0.2457 21 - 7.07 32.15 0.525 1.1475 0.2499 22 - 6.11 32.49 0.7 1.53 0.2511 Table 4. Mould-flow simulated and neural network predict compared (it is not included in any original 22 set database). Items FEM mould-flow Neural network simulation predict X-coordinate - 11.01 - 11.01 Y-coordinate 16.0 16.0 Gate width 1.8 1.8 Gate height 0.9 0.9 Produce max. deformation 0.3178 0.3325 Error values 4.62% (FEM-predict)/FEM approximately 4.62%, which shows that the model is suitable for this model requirement. 6. Simulation Annealing Theory In 1983, a theory that was capable of solving the global optimisation problem was developed for the optimised problem. The concept was a powerful optimisation algorithm based on the annealing of a solid which solved the combinatorial optimisation problem of multiple variables. When the tempera- ture is T and energy E, the thermal equilibrium of the system is a Boltzman distribution: P r = 1 Z(T) exp S - E K B T D (18) Z(T) = normalisation factor; K B = Boltzman constant; Exp(- E/K B T) = Boltzman factor Metropolis 24 proposed a criterion for simulating the cool- ing of a solid to a new state of energy balance. The basic criterion used by Metropolis is an optimisation algorithm called “simulated annealing”. The algorithm was developed by Kirk- patrick et al. 20. In this paper, the simulation-annealing algorithm is used to search for the optimal control parameters for gate location. Figure 3 shows the flowchart of the simulated annealing search. First, the algorithm is given an initial temperature T s and a final temperature T e , and a set of i