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UNIVERSIDAD AUTNOMA DE NUEVO LEN FACULTAD DE INGENIERA MECNICA Y ELCTRICA Divisin de Posgrado en Ingeniera de Sistemas Serie de Reportes Tcnicos Reporte Tcnico PISIS-2004-04 Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding Carlos E. Castro 1 Mauricio Cabrera Ros 2 Blaine Lilly 1,3 Jos M. Castro 3 (1) Department of Mechanical Engineering Ohio State University Columbus, Ohio, EUA (2) Programa de Posgrado en Ingeniera de Sistemas FIME, UANL E-mail: mcabrerauanl.mx (3) Department of Industrial, Welding, and Systems Enginering Ohio State University Columbus, Ohio, EUA 08 / Septiembre / 2004 2004 by Divisin de Posgrado en Ingeniera de Sistemas Facultad de Ingeniera Mecnica y Elctrica Universidad Autnoma de Nuevo Len Pedro de Alba S/N, Cd. Universitaria San Nicols de los Garza, NL 66450 Mxico Tel/fax: +52 (81) 1052-3321 E-mail: pisisyalma.fime.uanl.mx Pgina: http:/yalma.fime.uanl.mx/pisis/ Simultaneous Optimization of Mold Design and Processing Conditions in Injection Molding Carlos E. Castro 1 , Mauricio Cabrera Ros 3 , Blaine Lilly 1,2 , and Jos M. Castro 2 1 Department of Mechanical Engineering and 2 Departement of Industrial, Welding however it might prove a challenge to determine if this solution lies in the efficient frontier, especially in the case where the PMs show nonlinear behavior. In addition, this solution is dependent on the bias of the user defining the weights. In engineering practice it is often times impossible to define one optimal solution to all criteria. Instead, it is both feasible and attractive to determine the best compromises between PMs: that is the combinations of PMs that cannot be improved in one single dimension without harming another. Data Envelopment Analysis (DEA) provides an unbiased way to find these efficient compromises. It is the purpose of this paper to demonstrate the determination of efficient solutions (best compromises) in an IM context through a series of case studies comprising several potential industrial applications. These solutions prescribe the settings for IM process and design variables. Additionally, the identification of robust solutions is discussed. The Optimization Strategy Proposed by Cabrera-Rios, et al 2, 3 the general strategy to find the best compromises between several PMs consists of five steps: Step 1) Define the physical system. Determine the phenomena of interest, the performance measures, the controllable and non-controllable variables, the experimental region, and the responses that will be included in the study. Step 2) Build physics-based models to represent the phenomena of interest in the system. Define models that relate the controllable variables to the responses of interest. If this is not feasible, skip this step. 3 Step 3) Run experimental designs. Create data sets by either systematically running the models from the previous step, or by performing an actual experiment in the physical system when a mathematical model is not possible. Step 4) Fit metamodels to the results of the experiments. Create empirical expressions (metamodels) to mimic the functionality in the data sets. Step 5) Optimize the physical system. Use the metamodels to obtain predictions of the phenomena of interest, and to find the best compromises among the PMs for the original system. The best compromises are identified here through DEA. In the method outlined here, the metamodels are empirical approximations of the functionality between the controllable (independent) variables, and the responses (dependent variables). These metamodels are used either for convenience or for necessity. Because DEA as it is used here requires that many response predictions be made, it is more convenient to obtain these predictions from metamodels rather than more complicated physics-based models. In addition, when physics-based models are not available to represent the phenomena of interest, the use of metamodels becomes essential. Data Envelopment Analysis (DEA) Cabrera-Rios et al 2,3 have demonstrated the use of DEA to solve multiple criteria optimization problems in polymer processing. DEA, a technique created by Charnes, Cooper, and Rhodes 4, provides a way to measure the efficiency of a given combination of PMs relative to a finite set of combinations of similar nature. The efficiency of each combination is computed through the use of two linearized versions of the following mathematical programming problem in ratio form: 4 free nj T T T T j T j T T T 0 min 0 min 0 min 0 max min 0 0 max 0 0 ,.,11 , 1 Y 1 Y Y Y s.t. Y Y Maximize toFind = + + (1) (2) (3) (4) (5) where, and are vectors containing the values of those PMs currently under analysis to be maximized and minimized respectively, is a vector of multipliers for the PMs to be maximized, is a vector of multipliers for the PMs to be minimized, max 0 Y min 0 Y 0 is a scalar variable, n is the number of total combinations in the set, and is a very small constant usually set to a value of 1x10 -6 . The solutions deemed efficient by the two linearized versions of the model shown above represent the best compromises in the (finite) set of combinations of PMs. A complete description of the linearization procedure as well as the application of this model can be found in any of the references 1 through 5. Determination of settings of process variables and injection point Consider the part shown in Figure 1. This part, which we introduced in previous works 5,6,7, represents a case where the location of the weld lines is critical, and the part flatness plays a major role. The part is to be injection molded using a Sumitomo IM machine using PET with a fixed flow rate of 9cc/s. Nine PMs were included in this 5 study: (1) maximum injection pressure, P I , (2) time to freeze, t f , (3) maximum shear stress at the wall, S W , (4) deflection range in the z-direction, R Z , (5) time at which the flow front touches hole A, t A , (6) time at which the flow front touches hole B, t B , (7) time at which the flow front touches the outer edge of the part, t oe , (8) the vertical distance from edge 1 to the weld line, d 1 , and (9) the horizontal distance from edge 2 to the weld line, d 2 . For production purposes it is desirable to minimize P I , t f , S W , and R Z : P I to keep the machine capacity unchallenged, t f to reduce the total cycle time, S W to minimize plastic degradation, and R Z to control the part dimensions. It is desirable to maximize t A , t B , t oe , d 1 , and d 2 : t A , t B , t oe in order to minimize the potential for leakage, and d 1 and d 2 to keep the weld lines away from corners which were assumed to be areas of stress concentration. Figure 1: Part of constant thickness with cutouts. 6 Five controllable variables were varied at the levels shown in Table 1 in a full factorial design. These controllable variables include: (a) the melt temperature, T m , (b) the mold temperature, T w , (c) the ejection temperature, T e , (d) the horizontal coordinate of the injection point, x, and (e) the vertical coordinate of the injection point, y. T e was only varied at two levels because a preliminary study showed that a third level did not add any meaningful variation. The injection point location is constrained to be in the region shown in Figure 1, due to limitation of the IM machine. This point will be characterized by the variables x and y in a Cartesian coordinate system with its origin at the lower left corner of the part. Table 1: Levels of each of the controllable variables for the initial dataset T m T w T e x y Label C C C cm cm -1 260 120 149 15 10 0 275 130 159 20 17.5 1 290 140 25 25 A finite element mesh of the part was created in Moldflow TM in order to obtain estimates for the performance measures. An initial dataset was obtained from the full factorial design. Following with the general optimization strategy, this initial dataset was used to create metamodels to mimic the behavior of each the performance measures. In general, it is favorable to fit a simple model to the data. In this study, second order linear regressions were initially considered as models for the performance measures. When simple models do not suffice, then more complicated models, in this case ANNs, become necessary. In general the ANNs outperformed the second order linear regression for every performance measure in terms of approximation quality and prediction capability, and were therefore used to obtain predictions for each PM at previously untried 7 combinations of controllable variables. The results for the performance of the regression models and the ANNs obtained can be found Table 2. Table 2: Summary of performance and results from residual analysis results for the regression metamodels The complete multiple criteria optimization problem originally posed for this case contained all nine performance measures. To solve the optimization problem, it was necessary to generate a large number of feasible level combinations of the controllable variables. This was achieved by varying T m and T w at five levels, and the rest of the variables at three levels within the experimental region of interest (see Table 1) in a full factorial enumeration. This experimental design resulted in a total of 675 combinations. The results after applying DEA were that over 400 of the 675 combinations were found to be efficient. Such a large number of efficient combinations can be explained by examining Table 3, which summarizes the results of the analysis of variance of each PM in regression form. Notice that the last five PMs are only dependent on the injection point position determined by variables x and y. Any specific combination of values (x * ,y * ) will 8 give the same result on all of these five PMs regardless of the values that the rest of the other controllable variables T m , T w , and T e take. Having used a full factorial enumeration with x and y at three levels, it follows that we can obtain only nine different values for these five PMs, but each of the nine specific combinations (x,y) have in fact 75 combinations of the rest of the controllable variables. In the high dimensionality of the problem, this elevated amount of repetition results in a large number of efficient solutions. In order to increase the discrimination power i.e. obtain fewer efficient solutions, one can solve the DEA model shown in Eqs. 1 through 5 by setting 0 equal to zero. The resulting model is similar to the Charnes-Cooper-Rhodes (CCR) DEA model 8. Table 3: The significant sources of variation (linear, quadratic and second order interaction terms in the linear regression metamodel) to each performance measure. 9 Using the simple modification described above, the number of efficient combinations comes down to 149. It can be shown that these combinations are a subset of those 400 plus found previously. These efficient combinations are shown in terms of the PMs in Figure 2. Figure 2: Levels of the PMs that corresponded to the efficient solutions when all nine were included It is important to notice that we can exploit the information our methods gave us about the functionality of the PMs in order to tailor the optimization problem. To illustrate, five sub cases were defined for practical applications of the conceptual part shown in Figure 1: (i) an excess capacity injection machine application, (ii) a dimensional quality and economics critical application, (iii) a structural part application, 10 (iv) a part quality critical application, and (v) a case including PMs that are only dependent on the injection location 7. Excess Capacity Injection Molding Machine: For a case in which the injection-molding machine has excess capacity, it would be possible to not consider the maximum injection pressure in the optimization problem. For simplicity, in this case S W , t A , t B , and t oe were also dropped from the optimization, leaving four performance measures. The DEA model was again solved here by setting the constant 0 equal to zero in order to improve the discrimination power of DEA. The functionality shown in Table 3 called for inclusion of all variables, and the factorial enumeration with 675 combinations was used. In this case, fourteen combinations were found to be efficient. Figure 3 shows the levels of the PMs for the efficient solutions. The compromise between the locations of the weld lines is evident. A noticeable compromise also arises between t f and R z . This is an understandable compromise, because the two depend oppositely on the ejection temperature. 11 Figure 3: Efficient solutions for the excess machine capacity application in terms of the levels of the PMs considered. Figure 4 shows the locations of the injection gate for the efficient solution. The positions in this case help to define attractive areas to locate the injection port, since they tend to cluster in specific sections. In this case the efficient injection locations clustered along right and bottom edges. The three PMs that are affected by the location of the injection gate are the weld line positions and the deflection in the z-direction. The additional PM here is the time to freeze, which is not affected by the injection location according to the analysis of variance. 12 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X Y Figure 4: Injection Locations of the efficient solutions to the excess machine capacity application transformed to fall between -1 and 1. Table 4 shows the values for all of the controllable variables at the efficient solutions. Notice that T w and T m were at 120 and 260 degrees Celsius respectively for all of the efficient solutions. In industrial practice, if the PMs involved in this case were the only ones of interest, this would be a good indication that T m and T w should be set at these temperatures. Also notice that the ejection temperature values of the efficient solutions vary over the entire range. According to the analysis of variance, d 1 and d 2 do not depend on the ejection temperature, so this fact must be due to the compromise between R z and t f previously mentioned. 13 Table 4: Efficient Solutions for the excess machine capacity application xyT w T m T e t f R z d 1 d 2 cm cm C C C s mm mm mm 25 25 120 260 149 20.89 0.0005 37.9 130.9 25 25 120 260 154 18.99 0.002 37.9 130.9 25 25 120 260 159 17.27 0.007 37.9 130.9 25 25 120 260 149 22.71 0.000 37.9 131.4 25 17.5 120 260 149 20.90 0.001 94.9 107.8 25 17.5 120 260 154 19.00 0.005 94.9 107.8 25 17.5 120 260 159 17.28 0.010 94.9 107.8 25 17.5 120 260 149 22.72 0.001 94.9 108.9 15 10 120 260 149 20.92 0.001 124.6 67.4 15 10 120 260 154 19.02 0.005 124.6 67.4 15 10 120 260 159 17.30 0.011 124.6 67.4 25 10 120 260 149 20.92 0.001 124.7 82.1 25 10 120 260 154 19.02 0.006 124.7 82.1 25 10 120 260 159 17.30 0.012 124.7 82.1 Controllable Variables Performance Measures Dimensional Quality and Economics Critical Application: In this case it was assumed that the economic concerns included minimizing the cycle time and keeping the machine capacity untested in order to have long machine life and smaller power consumption. These two concerns are defined by t f and P I respectively. R z defines the dimensional quality. The analysis of variance shows that all of the controllable variables affect at least one of these PMs, so the enumeration with 675 combinations again was applied. Twenty-five efficient solutions were found. Since the problem is three-dimensional the efficient frontier can be visualized. The efficient points are shown in Figure 5 with respect to the rest of the data set. 14 Figure 5: A Visualization of the efficient frontier of the economics critical and dimensional Application Figure 6 shows the efficient solutions in terms of the levels of the PMs. The direct compromise between the time to freeze and deflection is confirmed here. Notice that they follow opposite trends while it is favorable to minimize both. 15 Figure 6: Efficient solutions for the dimensional quality and economic application in terms of the levels of the PMs considered. Figure 7 shows the locations of the injection gate for the efficient solutions. This case contradicts the first case. In the large machine capacity case, the attractive areas for the injection gate were found at the bottom and right edges of the feasible area, but in this case, the top edge and bottom left corner proved to be the efficient locations. This is due to the fact that the positions of the weld lines were not considered in this case. From these results we can conclude that d 1 and d 2 are the main drivers for keeping the injection location on the right or bottom edge. They are the only PMs affected by x and y that were included in the first case and not in this case. 16 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X Y Figure 7: Injection Locations of the efficient solutions to the dimensional quality and economics critical application transformed to fall between -1 and 1. Table 5 shows the twenty-five combinations of the controllable variables that proved to be efficient for the dimensional quality and economics critical application. Eighteen out of the twenty-five efficient solutions had the injection gate located at the upper left corner of the feasible region, which is close to the center of the part. This is the most robust injection location for this application. According to the analysis of variance, P I is affected by the location of the injection gate. Locating the injection gate towards the center would favorably decrease P I . Since d 1 and d 2 were not included in this case there were no negative effects of moving the injection gate towards the center. 17 Table 5: Efficient Solutions for the dimensional quality and economics critical application xyT w T m T e P I t f R z cm cm C C C MPa s mm 15 25 140 290 159 9.35 26.7 0.0098 15 25 140 290 149 9.35 37.9 0.0015 15 25 140 282.5 159 9.55 23.8 0.0099 15 25 140 282.5 149 9.55 32.0 0.0009 15 25 140 275 149 9.75 27.4 0.0006 15 25 140 275 159 9.75 21.9 0.0101 15 25 140 267.5 159 9.96 20.4 0.0102 15 25 140 267.5 149 9.96 24.5 0.0006 15 25 140 260 154 10.17 20.9 0.0045 15 25 140 260 159 10.17 19.1 0.0101 15 25 140 260 149 10.17 22.7 0.0006 15 25 130 275 149 12.25 26.5 0.0005 15 25 125 260 149 14.69 21.4 0.0005 15 25 125 260 159 14.69 17.8 0.0095 15 25 120 275 149 16.00 25.5 0.0004 15 25 120 260 154 16.61 19.0 0.0032 15 25 120 260 149 16.61 20.9 0.0005 15 25 120 260 159 16.61 17.3 0.0090 25 25 140 260 149 17.46 22.7 0.0004 25 25 135 260 149 19.43 22.4 0.0004 20 25 120 260 159 26.92 17.3 0.0071 25 25 120 260 159 27.16 17.3 0.0065 25 25 120 260 154 27.16 19.0 0.0020 15 10 120 282.5 149 28.50 29.8 0.0003 15 10 120 275 149 29.21 25.5 0.0003 Controllable Variables Performance Measures A Structural Application In this application, the PMs included were the vertical distance from edge 1 to the weld line, d 1 , and the horizontal distance from edge 2 to the weld line, d 2 . The location of weld lines is considered critical to design a structurally sound part. From the analysis of variance, it was known that these PMs depended only on the position of the injection gate, characterized by variables x and y. In order to avoid the repetition described in the full set, a new dataset was created by varying x and y at nine levels creating a finer sampling grid for the injection location. The rest of the variables were set to a value in the 18 middle of their respective ranges. The levels of the controllable variables for this dataset are shown in Table 6. The total number of combinations of controllable variables in this dataset was 81. Table 6: Levels of controllable variables used for the dataset for x,y dependent PMs he efficient frontier for this two-dimensional case is shown here in Figure 8. T m T w T e xy CCC cm cm 130 275 154 15 10 16.25 11.875 17.5 13.75 18.75 15.625 20 17.5 21.25 19.375 22.5 21.25 23