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G Journal of Mechanical Science and Technology 21 (2007) 789798 Journal of Mechanical Science and Technology Micro Genetic Algorithm Based Optimal Gate Positioning in Injection Molding Design Jongsoo Lee * , Jonghun Kim School of Mechanical EngineeringYonsei University, Seoul 120-749 Korea (Manuscript Received December 12, 2006; Revised March 26, 2007; Accepted March 26, 2007) - Abstract The paper deals with the optimization of runner system in injection molding design. The design objective is to locate gate positions by minimizing both maximum injection pressure at the injection port and maximum pressure difference among all the gates on a product with constraints on shear stress and/or weld-line. The analysis of filling process is conducted by a finite element based program for polymer flow. Micro genetic algorithm (mGA) is used as a global optimization tool due to the nature of inherent nonlinearlity in flow analysis. Four different design applications in injection molds are explored to examine the proposed design strategies. The paper shows the effectiveness of mGA in the context of optimization of runner system in injection molding design.G Keywords: Micro genetic algorithm; Design optimization; Filling injection mold - 1. Introduction Injection molding process has been recognized as one of the most efficient manufacturing technologies since high performance polymer materials can be utilized to accurately manufacture a product with complicated shape (Chiang, et al., 1991; Chang and Yang, 2001; Himasekhar, et al., 1992; Kwon and Park, 2004). Also, the demand on injection molded products such as from conventional plastic goods to micro optical devices is being dramatically increased over the recent years (Piotter, et al., 2001; Kang, et al., 2000). In general, the injection mold process is initiated by the filling stage where the polymer materials fill into a cavity under the injection temperature. After the cavity is completely filled, the post-filling stage, that is, the packing stage is conducted to be additionally filled with the high pressure polymer, thereby resulting in the avoidance of material shrinkage. Subsequently, the cooling stage is required for a molded product to be ejected without any deformation. It is important to accommodate the molding conditions in the filling stage since it is the first stage in the overall injection molding design (Zhou and D. Li, 2001). After that, one can success- fully expect more improved molding conditions during post-filling stages such as packing, cooling stages. The paper deals with optimal conditions of the filling injection molding design in which the flow pattern and pressure for the polymer materials to be filled through gates of a runner are of significant. That is, one of design requirements are such that when the polymer comes into a cavity through a number of gates located at different positions, pressure levels on the surface of a product should be as uniform as possible. Such design can be performed through the intelligent gate positioning to generate the more * Corresponding author. Tel.: +82 2 2123 4474; Fax.: +82 2 362 2736 E-mail address: jleejyonsei.ac.kr 790 Jongsoo Lee and Jonghun Kim / Journal of Mechanical Science and Technology 21(2007) 740749 uniform distribution of injection pressure over the product surface. There have been a number of studies of optimal gate location in the context of CAE filling injection molding design problems where various kinds of optimizer have been employed to conduct design optimization (Kim et al., 1996; Young, 1994; Pan- delidis and Zou, 2004; Lin, 2001; Li and Shen, 1995). The paper explores the design of injection mold system using micro genetic algorithm (mGA). Ge- netic algorithm (conventional GA) is based on the Darwins theory of the survival of the fittest, and adopts the concept of natural evolution; the competitive designs with more fit are survived by selection, and the new designs are created by crossover and mutation (Lee, 1996; Lee and Hajela, 1996). A conventional GA works with a multiple number of designs in a population. Handling with such designs results in increasing a higher probability of locating a global optimum as well as multiple local optima. GA is also advantageous when the design problem is represented by a mixture of integer/dis- crete and continuous design variables. Nevertheless, it requires expensive computational costs especially when combining with finite element based CAE analysis tools. A conventional GA determines the population size depending upon the stringlength of a chromosome that is a coded value of a set of design variables. The main difference between a conven- tional GA and mGA resides on the population size. The population size in mGA is based on Goldbergs concept such that Evolution process is possible with small populations to reduce the cost of fitness function evaluation (Goldberg, 1988). This implies that mGA employs a few number of populations for GA evolution regardless of the number of design variables and the complexity of design parameters (Krishnakumar, 1989; Dennis and Dulikravich, 2001). The paper discusses the design requirements of filling injection mold optimization to construct the proper objective functions and design constraints. Four different design applications in injection molds are explored to examine the proposed design strategies. The paper shows the effectiveness of mGA in the context of optimization of runner system in injection molding design. 2. Mold flow analysis The flow of a polymer in injection molding process obeys the following governing equations: 22 ()()0 pp SS xxyy wwww wwww (1) 2 2 2 () pxy TTT T Ck txy z UQQKJ www w www w (2) where, 2 2 0 h z Sdz K . In the above equations, p is a flow pressure, T is a temperature of polymer, and t is denoted as time. Parameters K, J, and k are viscosity, shear rate and thermal conductivity, respectively (Lee, 2003). It is assumed that polymer is a non-compaction substance in the filling analysis. The flow analysis in the present study is conducted by Computer Aided Plastics Application (CAPA) (Koo, 2003), a finite element based commercial code for polymer flow of injection molding. The runner system in injection mold covers the passage of molten polymer from injection port to gates. The present study develops two different runner systems where a cold system requires the change in polymer temperature, and a hot system keep it unchanged while the flow passes through the runner. For the hot runner system has a geometrically consistent thickness due to the constant temperature as shown in Fig. 1a. However, the CAE result of a cold runner system depends on the thickness and shape Table 1. Ten-bar truss design results. micro GA conventional GA Case 1 Case 2 Case 3 Case 1 Case 2 Case 3 Reference 20 X 1 7.86 8.15 7.85 8.15 7.30 7.81 7.90 X 2 0.41 0.18 0.19 0.10 0.83 0.45 0.10 X 3 8.38 7.99 8.15 8.20 8.77 8.37 8.10 X 4 5.05 3.83 3.89 3.97 3.27 4.16 3.90 X 5 0.12 0.96 0.15 1.10 0.75 0.55 0.10 X 6 0.41 0.25 0.25 0.10 0.82 0.30 0.10 X 7 6.41 5.67 5.87 5.84 6.74 6.30 5.80 X 8 5.23 6.29 5.52 5.68 5.06 5.26 5.51 X 9 3.83 3.85 5.05 5.07 2.89 3.86 3.68 Optimal area X 10 0.50 0.25 0.25 0.40 1.16 0.42 0.14 Optimal weight 1599 1587 1588 1593 1590 1585 1499 # of function evaluations 57540 54230 25335 78894 69497 73533 Jongsoo Lee and Jonghun Kim / Journal of Mechanical Science and Technology 21(2007) 789798 791 (a) Hot runner system (b) Cold runner system Fig. 1. Modeling of runner system. shape of a runner. The typical illustration of the geometric model in a cold runner system is shown in Fig.1b where the runner thickness is changed according to the temperature gradient. 3. Molding design requirements 3.1 Objective functions One of the most significant factors considered in the injection molding design is a flow pattern, which implies that a balanced flow should be maintained while a polymer arrives at each part of a design product. Once the improvement on flow balance is obtained, the flow of molten polymer smoothes and the maximum injection pressure is decreased with the same or at least evenly distributed injection pressure level at each gate. In a case where the certain part of a product within the mold is filled up earlier than other parts, each part would fall into over-packing and under-packing situations during the filling process of a polymer into mold. Such problem further evokes a malformation like twisting and bending, resulting from the difference in contraction rate during the course of cooling-off. The difference in pressure triggers the flow of polymer during the filling process, in which the maximum injection pressure is detected at the injection port of polymer. The polymer always flows from high-pressure region to low-pressure one. When a flow pattern improves, the flow of polymer gets smoother with the maximum injection pressure decreased. However, the flow instability sometimes happens, thereby requiring a higher pressure to fill up. That is, the maximum injection pressure needs to be reduced in order to improve the flow instability. The pressure gap (i.e., the highest and lowest pressure values) among all of gates is also taken as another objective function to determine whether the whole mold is being filled at once. Most commonly accepted design strategy to improve the flow pattern is the adjustment of gate location. The present study controls the flow pattern by developing the optimal gate positioning problems with proper objective function(s) and design cons- traints. Objective functions for injection molding design are considered as both maximum injection pressure (MIP) and maximum pressure difference (MPD). It should be noted that the maximum injection pressure is calculated at the injection port and the maximum pressure difference is a numerical difference between the highest and lowest values of pressure among all of gates. The aforementioned statements could be interpreted as a multiobjective design problem, hence the present study simply employs a weighting method as follows: * () () () MIP x MPD x Fx MIP MPD DE (3) where, D and E are weighting factors as D+E=1, and x is a set of design variables which are Cartesian coordinates of gates on a product. Each component in the above equation is normalized by optimal single- objective function value, (i.e., MIP*, MPD*). It is mentioned that the number of gates is considered as a problem parameter in the study. 3.1 Constraints Weld-lines are easily detected when more than two flow fronts having different temperature values meet during the filling process. The weld-line is one of the weakest points in molded product; it is very 792 Jongsoo Lee and Jonghun Kim / Journal of Mechanical Science and Technology 21(2007) 740749 vulnerable to a shock and subsequently causes external defects of a very glossy polymer. The weld- line should be moved into a less weak region by adjusting the width of a product, the size and/or shape of gates and runners, and the position of gates, etc. The present study considers the position of a weld- line as a constraint in optimal gate positioning of mold design. Once a designer specifies areas where weld-lines should not be generated, all of the finite element nodes in such areas are constrained not to form the weld-lines. Shear stress is defined as a shear force imposed on the wall of a mold by the shear flow of a polymer. The magnitude of shear stress is proportional to the pressure gradient of each position. In general, the shear stress is zero at the center of a molded product, and reaches a maximum value on the wall. High shear stress triggers the molecule cultivation on the surface of a molded product. Flow instability such as melt fracture has a close relationship with the shear stress. The clear surface of a molded product can be obtained by reducing the magnitude of shear stress. That is, shear stress should be minimized during the mold filling process in order to improve the quality of a molded product, particularly on its surface. Maximum allowable shear stress depends on the kinds of polymer, and is generally taken as 1% of tensile strength of a polymer. Shear stress affecting the quality of end product is considered as another constraint. 3.3 Formulation of optimization problem The statement of a mold design optimization problem can be written as follows: Find 12 (, , ) (, , ), (, , ),., (, , ) N x ijk xijk x ijk x ijk (4) to minimize * () () () MIP x MPD x Fx MIP MPD DE (5) subject to shear stress(i, j, k) shear stress allowable (6) weld-line(i, j, k) = designated area(s) only (7) where, lower upper xxxdd A set of design variables, x are Cartesian coordi- nates (i, j, k) of gates on the surface of a molded product, where N is the number of gates. A traditional weighted-sum method in the context of multiob- jective optimization is employed by using two wei- Fig. 2. Micro GA process. ghting factors of D and E, where D+E=1. Multi- objective functions considered in the present study are maximum injection pressure (MIP) measured at the injection port and maximum pressure difference (PD) among all of gates. The constants, MIP* and MPD* are optimal objective function values obtained via single-objective optimization. The permission of weld-lines to designated areas only and the upper limits on shear stress are imposed as design cons- traints. The flow pattern analysis is performed by CAPA as mentioned in the earlier section, and the optimization is conducted through mGA. It should be noted that Cartesian coordinates (i, j, k) is recognized as nodal points when a molded product is discretized by finite elements in CAPA. 4. Micro GA The overall process of mGA in the present study is depicted in Fig. 2, and a stepwise procedure can be explained as follows: Step-1) Generate an initial population at random. The recommended population size is 3, 5, or 7. Step-2) Perform a conventional GA evolution until the nominal convergence is satisfied. In the present study, the population size is selected as 5, and a tournament selection operator is used. The crossover probability in mGA is 1.0 due to the small size in population, while a conventional GA is preferred to use it less than 1.0. The nominal convergence means that the difference of 1s and/or 0s among string positions is within 5% out of the stringlength, thereby resulting in the convergence to a local solution. Step-3) During the user-specified number of ge- nerations, a new population is updated; one individual is selected by elitism; the remaining individuals in a Jongsoo Lee and Jonghun Kim / Journal of Mechanical Science and Technology 21(2007) 789798 793 new population are generated at random. It should be noted that the selection operation adopts tournament for activating the diversity and elitism for keeping the best solution. Since the updated populations except for the elitism are generated at random, mGA seldom considers the mutation. bfaa biaa cbaa ceaa chaa daaa a baaaa caaaa daaaa eaaaa faaaa gaaaa haaaa iaaaa N_function Objective (a) A conventional GA bfaa biaa cbaa ceaa chaa daaa a baaaa caaaa daaaa eaaaa faaaa gaaaa haaaa iaaaa N_function O b jective (b) Micro GA Fig. 3. Convergence histories of ten-bar truss problem. G Fig. 4. Seven discrete design spaces for vehicle dashboard problem. Fig. 5. Initial gate location of vehicle dashboard. In summary, mGA enables to locate an optimal solution thanks to the small size of populations, tournament and elitism operations in selection, and the full participation in crossover. However, mGA has a drawback upon finding one of multiple local optima only due to the small size of populations and the nominal convergence strategy. A conventional GA is superior to maintaining the diversity while mGA is advantageous of savings in computational resource requirements. 4.1 Truss design The proposed mGA is verified by a typical ten-bar planar truss optimization problem. The objective is to find optimal cross-sectional areas by minimizing the structural weight subjected to stress constraints (Haftka and Gurdal, 1993). Optimal solutions are obtained via mGA and a conventional GA to compare with each other. The population size in mGA is 5, while a conventional GA requires 250 individuals in a population since the stringlength in this problem is 100. Crossover and mutation probabilities in a con- ventional GA used are 0.8 and 0.01, respectively. After two genetic search methods are conducted ten times by changing randomly generated initial popul- ations, the most fit design results are demonstrated in Table 1. The convergence history for each optimizer demonstrates that mGA produces the better design and locates the near-optimal solution at the earlier stage of evolution in Fig. 3. 5. Results of design applications 5.1 Vehicle dashboard A passenger car in-panel has been first examined. This model is supposed to have 7 gates, and design spaces for use in genetic evolution are shown in Fig. 4. Objective functions of MIP and MPD are taken into account, but no constraints are imposed in this model. The initial design is shown in Fig. 5; this design has been obtained through experience and trial-and-errors in an automotive part molding company. Optimized results by mGA are shown in Figs. 6 to 8, whose objective functions were considered as MIP only, MPD only and both MIP and MPD, respectively. Design results for each case are summarized in Table 2 as well. It is noted that both MIP and MPD is calculated with D changing from 0.0 to 1.0 with an increment of 0.1 while keeping DE .0. 794 Jongsoo Lee and Jonghun Kim / Journal of Mechanical Science and Technology 21(2007) 740749 Fig. 6. Optimized design of vehicle dashboard (MIP only). Fig. 7. Optimized design of vehicle dashboard (MPD only). Fig. 8. Optimized design of vehicle dashboard (both MIP and MPD). In case of MIP only in Fig. 6, the maximum injection pressure value has an improvement of 23.9% compared with an initial model, but the pressure distribution on the product becomes worse, resulting in over-packing on the left region. When a case of MPD only is considered, the design performance in Fig. 7 is achieved in terms of not only maximum pressure difference but also maximum injection pressure as shown. It is expected that the flow gets smoother during the improvement of pressure distribution, and the maximum injection pressure is decreased as well. In case of both MIP and MPD in Fig. 8, its result is quite similar to a case Table 2. Optimization results of vehicle dashboard. maximum pressure MPa maximum difference MPa Initial design 242.69 20.26 MIP only 184.73 35.08 MPD only 231.22 12.44 objective both MIP and MPD 229.92 12.58 Table 3. Optimization results of TV monitor. maximum pressure MPa maximum difference MPa shear stress 0.5 MPa Initial design 80.55 13.71 0.45 MIP only 68.46 4.06 0.43 MPD only 72.27 3.04 0.45 objective both MIP and MPD 68.46 4.06 0.43 of MPD only in terms of gate locations from Figs. 7 and 8 and the percentile improvement in Table 2. A weighted-sum method is used to obtain the mul- tiobjective optimal solutions by changing D and Esimultaneously, but yields the same results out of a total of 11 weighting factor based trials. The reason why a few number of Pareto solutions are detected is such that the maximum pressure is not counter to pressure distribution in the filling injection molding. In other words, when the overall pressure distribution is improved thanks to the enhancement of flow balance and the smoothness of polymer flow, the maximum pressure is consequently decreased. As far as the pressure distribution of a modeled product is concerned, the change in gate position is noticeable; Gate_5 of optimized models moves from right to left region compared with an initial model. 5.2 TV monitor The model of a TV monitor equipped with 4 gates is now optimized using objective functions and the upper limit on shear stress constraint, where the shear st