注塑模具外文文献翻译-注塑成型模具设计和加工条件的同步优化
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UNIVERSIDAD AUTNOMA DE NUEVO LENFACULTAD DE INGENIERA MECNICA Y ELCTRICADivisin de Posgrado en Ingeniera de SistemasSerie de Reportes TcnicosReporte Tcnico PISIS-2004-04Simultaneous Optimization of Mold Design and Processing Conditions in Injection MoldingCarlos E. Castro1 Mauricio Cabrera Ros2Blaine Lilly1,3 Jos M. Castro3(1) Department of Mechanical EngineeringOhio State University Columbus, Ohio, EUA(2) Programa de Posgrado en Ingeniera de SistemasFIME, UANLE-mail: mcabrerauanl.mx(3) Department of Industrial, Welding, and Systems EngineringOhio State University Columbus, Ohio, EUA08 / 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. UniversitariaSan Nicols de los Garza, NL 66450MxicoTel/fax: +52 (81) 1052-3321E-mail: pisisyalma.fime.uanl.mx Pgina: http:/yalma.fime.uanl.mx/pisis/Simultaneous Optimization of Mold Design and Processing Conditions in Injection MoldingCarlos E. Castro1, Mauricio Cabrera Ros3, Blaine Lilly1,2, and Jos M. Castro21 Department of Mechanical Engineering and 2Departement of Industrial, Welding & Systems EngineeringThe Ohio State University Columbus, Ohio, 432103 Graduate Program in Systems Engineering Universidad Autnoma de Nuevo LenSan Nicols de los Garza, Nuevo Len, Mxico, 66450AbstractInjection molding (IM) is the most prominent process for mass-producing plastic products. One of the biggest challenges facing injection molders today is to determine the proper settings for the IM process variables. Selecting the proper settings for an IM process is crucial because the behavior of the polymeric material during shaping is highly influenced by the process variables. Consequently, the process variables govern the quality of the part produced. The difficulty of optimizing an IM process is that the performance measures (PMs), such as surface quality or cycle time, that characterize the adequacy of part, process, or machine to intended purposes usually show conflicting behavior. Therefore, a compromise must be found between all of the PMs of interest. In this paper, we present a method comprised of Computer Aided Engineering, Artificial Neural Networks, and Data Envelopment Analysis (DEA) that can be used to find the best compromises between several performance measures. The approach discussed here also allows for the identification of robust variable settings that might help to define a starting point for negotiation between multiple decision makers.29IntroductionInjection Molding (IM) is the most prominent process for mass-producing plastic parts. According to the Society of the Plastics Industry, over 75% of all plastics processing machines are IM machines, and close to 60% of all plastics processing facilities are injection molders 1. Selecting the proper IM process settings is crucial because the behavior of the polymeric material during shaping is highly influenced by the process variables. Consequently, the process variables govern the quality of the part produced. A substantial amount of research has been directed towards determining the process settings for the IM process as well as the optimal location of the injection gate.The challenge of optimizing an IM process is that the performance measures often show conflicting behavior when they are functions of process or design variables in common. For example, the cycle time and the part warpage will both be affected by the ejection temperature. Increasing the ejection temperature would be favorable for minimizing cycle time. However, letting the part cool to a lower ejection temperature before demolding would decrease the part warpage. Therefore, a compromise must be found between these two performance measures to set the ejection temperature. For this reason, when optimizing an IM process it is nearly impossible to find one best solution. However, it is feasible to determine a set of best compromises between multiple PMs.The problem of considering several PMs simultaneously, i.e. finding the best compromises, is referred to as a multiple criteria optimization. Conventional methods of multiple criteria optimization involve combining individual weighted PMs into one objective function and optimizing that function. These methods will converge to asolution; 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 StrategyProposed 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.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:Find, m0toMaximizeT Ymax + m 00Y(1)s.t.j0T Ymax + mTmin 0YTminj 1j = 1,., n(2)TYTmin 0TYTmin 0 e 1 e 1(3)(4)m0free(5)where,Ymax and Yminare vectors containing the values of those PMs currently under00analysis to be maximized and minimized respectively, m is a vector of multipliers for the PMs to be maximized, n is a vector of multipliers for the PMs to be minimized, m0 is ascalar variable, n is the number of total combinations in the set, and e is a very smallconstant 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 pointConsider 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 thisstudy: (1) maximum injection pressure, PI , (2) time to freeze, tf, (3) maximum shear stress at the wall, SW, (4) deflection range in the z-direction, RZ, (5) time at which the flow front touches hole A, tA, (6) time at which the flow front touches hole B, tB, (7) time at which the flow front touches the outer edge of the part, toe, (8) the vertical distance from edge 1 to the weld line, d1, and (9) the horizontal distance from edge 2 to the weld line, d2. For production purposes it is desirable to minimize PI , tf, SW, and RZ : PI to keep the machine capacity unchallenged, tf to reduce the total cycle time, SW to minimize plastic degradation, and RZ to control the part dimensions. It is desirable to maximize tA, tB, toe, d1, and d2 : tA, tB, toe in order to minimize the potential for leakage, and d1 and d2 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.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, Tm, (b) the mold temperature, Tw, (c) the ejection temperature, Te, (d) the horizontal coordinate of the injection point, x, and (e) the vertical coordinate of the injection point, y. Te 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 datasetTmTwTexyLabelCCCcmcm-1260120149151002751301592017.512901402525A finite element mesh of the part was created in MoldflowTM 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 untriedcombinations 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 regressionmetamodelsThe 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 Tm and Tw 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 PMin 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*) willgive the same result on all of these five PMs regardless of the values that the rest of the other controllable variables Tm, Tw, and Te 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 m0 equal tozero. The resulting model is similar to the Charnes-Cooper-Rhodes (CCR) DEA model8.Table 3: The significant sources of variation (linear, quadratic and second order interaction terms in the linear regression metamodel) to each performance measure.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 includedIt 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,(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 SW, tA, tB, and toe were also dropped from the optimization, leaving four performance measures. The DEA model was again solved here by setting the constant m0 equal to zero in order to improve the discrimination power of DEA. Thefunctionality shown in Table 3 called for inclusion of all variables, and the factorialenumeration 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 tf and Rz. This is an understandable compromise, because the two depend oppositely on the ejection temperature.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.1.00.2Y0.0-0.2-0.4-0.6-0.8-1.0-1.0 -0.8 -0.6 -0.4 -0.2 0.00.81.0XFigure 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 Tw and Tm 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 Tm and Tw 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, d1 and d2 do not depend on the ejection temperature, so this fact must be due to the compromise between Rz and tf previously mentioned.Table 4: Efficient Solutions for the excess machine capacity applicationControllable VariablesPerformanceMeasuresxyT wT mT et fRzd 1d 2cmcmCCCsmmmmmm252512026014920.890.000537.9130.9252512026015418.990.00237.9130.9252512026015917.270.00737.9130.9252512026014922.710.00037.9131.42517.512026014920.900.00194.9107.82517.512026015419.000.00594.9107.82517.512026015917.280.01094.9107.82517.512026014922.720.00194.9108.9151012026014920.920.001124.667.4151012026015419.020.005124.667.4151012026015917.300.011124.667.4251012026014920.920.001124.782.1251012026015419.020.006124.782.1251012026015917.300.012124.782.1Dimensional 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 tf and PI respectively. Rz 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.Figure 5: A Visualization of the efficient frontier of the economics critical and dimensional ApplicationFigure 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.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 d1 and d2 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.1.00.2Y0.0-0.2-0.4-0.6-0.8-1.0-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0XFigure 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, PI is affected by the location of the injection gate. Locating the injection gate towards the center would favorably decrease PI. Since d1 and d2 were not included in this case there were no negative effects of moving the injection gate towards the center.Table 5: Efficient Solutions for the dimensional quality and economics critical applicationControllable VariablesPerformance MeasuresxyT wT mT eP It fR zcmcmCCCMPasmm15251402901599.3526.70.009815251402901499.3537.90.00151525140282.51599.5523.80.00991525140282.51499.5532.00.000915251402751499.7527.40.000615251402751599.7521.90.01011525140267.51599.9620.40.01021525140267.51499.9624.50.0006152514026015410.1720.90.0045152514026015910.1719.10.0101152514026014910.1722.70.0006152513027514912.2526.50.0005152512526014914.6921.40.0005152512526015914.6917.80.0095152512027514916.0025.50.0004152512026015416.6119.00.0032152512026014916.6120.90.0005152512026015916.6117.30.0090252514026014917.4622.70.0004252513526014919.4322.40.0004202512026015926.9217.30.0071252512026015927.1617.30.0065252512026015427.1619.00.00201510120282.514928.5029.80.0003151012027514929.2125.50.0003A Structural ApplicationIn this application, the PMs included were the vertical distance from edge 1 to the weld line, d1, and the horizontal distance from edge 2 to the weld line, d2. 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 themiddle 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 PMsT mT wT exyCCCcmcm130275154151016.2511.87517.513.7518.7515.6252017.521.2519.37522.521.2523.7523.1252525The efficient frontier for this two-dimensional case is shown here in Figure 8.140130Weld Position 2 (mm)1201101009080706050402030405060708090100110120130Weld Position 1 (mm)Figure 8: Visualization of the efficient frontier in the structural applicationThe seven efficient solutions for a structural part are shown in Figure 9 in terms of the levels of the two PMs in increasing order of d1. The compromise between the positions of the weld lines is confirmed. We want to maximize both of the weld line positions, but where one of them is at a maximum, the other is at a minimum.Figure 9: Efficient Solutions for the structural application in terms of the weld line positions d1 and d2.Figure 10 shows the positions of the injection gate corresponding to the seven best compromises. The entire area shown is the feasible injection region. In this case the attractive clusters occur at the bottom right corner of the feasible injection area and along the right edge of the feasible injection region. These results tend to agree with the large machine capacity case. Since the locations of the weld lines are independent of the other controllable variables, any of these x,y pairs would obtain the same results for d1 and d2 regardless of the temperature levels. In this case the efficient solutions are defined1.00.20.0-0.2-0.4-0.6-0.8-1.0-1.0 -0.8 -0.6 -0.4 -0.20.00.81.0XYby the injection location, so temperature levels are not shown. In other words, had the temperatures been left at the maximum or minimum of their respective feasible ranges, we would have arrived at the same results for the locations of the weld lines.Figure 10: Injection Locations of the seven efficient solutions to the structural application transformed to fall between -1 and 1.A Quality Critical ApplicationAppearance in our test part we defined as related to the position of the weld lines and the flatness of the part, i.e. d1, d2, and RZ. From the analysis of variance in Table 3 it is known that d1 and d2 depend only on the x and y position of the injection gate. However, the temperatures cannot be disregarded in this case, because RZ depends on all three of them. Therefore, we used the factorial enumeration already created for all the variables (x, y, Tm, Tw, and Te) with 675 combinations. The resulting sixteen efficient solutions are shown in Figure 11 with respect to the rest of the dataset. Notice that the data isorganized into columns. The different columns illustrate the repetitions that were referred to earlier. Each of the columns corresponds to one x,y pair, and the variation in height of the data points in these columns is determined by the controllable temperatures. Since only one other PM was involved, Rz, this repetition did not cause a problem.Figure 11: Visualization of the efficient frontier for the part quality applicationFigure 12 shows the efficient solutions with respect to the values of the PMs in increasing order of d1. The compromise between d1 and d2 is again evident.Figure 12: Efficient solutions for the part quality application in terms of the position of the weld lines and deflection range.Figure 13 analyzes the clusters of the design variables x and y. Again, the entire space shown is the feasible area for the injection gate. This case did not use the same fine grid for the injection location that was used in the structural application, so the attractive clusters are not as well defined. However, it is evident that the right and bottom edges would be the best areas to locate the injection gate. This case agrees with the previous cases of the large machine capacity, and the structural application.Figure 13: Injection Locations of the seven efficient solutions to the part quality application transformed to fall between -1 and 1.Table 7 shows the levels of the controllable variables that correspond to the efficient combinations of PMs. Notice that for all of the efficient solutions, the value of Te was 149 degrees C. Allowing the part to cool to a lower ejection temperature favorably affects the part deflection in the z-direction. In this case, the time to freeze was not considered. Allowing the part to cool longer did not introduce any negative effects, so the efficient ejection temperature was always at the minimum of the range.Table 7: Efficient solutions for the quality critical applicationControllable VariablesPerformance MeasuresXYT mT wT eRzd 1d 2cmcmCCCmmmmmm15101302601490.0003109.764.415101352601490.0003109.764.720101202601490.0007124.667.420101302601490.0008124.677.825101202601490.0009124.782.125101252601490.0009124.782.52517.51252601490.000894.9108.925251252601490.000437.9131.425101302601490.0011124.782.92517.51302601490.000994.9110.02525130267.51490.000537.9131.625251302751490.000637.9131.625101352601490.0016124.783.32517.51352601490.001394.9110.925101402601490.0023124.783.62517.51402601490.002094.9111.8Injection Location Dependent Performance Measures:From the results of the analysis of variance we can see that there are some PMs that are dependent only on the location of the injection gate. These PMs, tA, tB, toe, d1, and d2, were considered in a separate case that is only concerned with determining the location of the injection gate. For this case the factorial enumeration of the levels of the controllable variables shown in Table 6 was used. Again because of the high dimensionality of this case, a DEA model with m0 equal to zero was used. The resulting fourteen efficientsolutions are shown in Figure 14. Notice the compromises between the time to touchhole A and the time to touch the outer edge. The peaks of these two PMs always contrast each other. On the other hand, the trend of the time to touch hole B follows a similar path as the time to touch hole A. As observed before the compromise between the weld line positions is evident.Position Weld Line 1time to touch hole A time to touch outer edgePosition Weld Line 2time to touch hole B1403.01202.5Weld Position 1 and 2 (mm)1002.0time to touch (s)801.5601.0400.520002468101214Efficient Compromises0.0Figure 14: Efficient solutions for the case of determining the injection location in terms of the levels of the PMs considered.The efficient gate locations for this case are shown in Figure 15. This case agreed with some of the earlier cases. The attractive clusters for the injection gate occurred along the bottom and right edges. Only a few new injection locations resulted from introducing the flow times into this case on top of the weld line locations, which were previously considered by themselves in the structural applications. Additionally, these new injection locations are still in the same general area. This implies that generally, the flow times as a group do not introduce definite compromises with respect to the location of the injection gate with the locations of the weld lines. Here the efficient solutions are only dependent on x and y, so the levels of the temperatures are not shown.1.00.2Y0.0-0.2-0.4-0.6-0.8-1.0-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0XFigure 15: Injection Locations of the efficient solutions for the application considering only x,y dependent PMsAnalysis of Robust SolutionsThe discussion of the different cases in the previous section leads to an additional analysis: finding robust efficient solutions. Robust solutions can be found within the individual cases, and some of those were discussed previously. As it can be inferred, a robust efficient solution is a combination of controllable variable settings that remains efficient when analyzing different subsets of performance measures. It is also beneficial to determine which solutions were robust on a large scale, i.e.which combinations of process variables were deemed efficient in several subsets of optimization. Indeed forthis case it was possible to identify that the combination of (x, y, Tm, Tw, Te) = (20 cm, 10 cm, 120 oC, 260 oC, 149 oC) is a robust efficient solution.Determining a suitable location for the inje
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