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For the example shown in Fig. 2, given a preliminary design composed by four parallel channels, layout design 1 and layout design 2 can be obtained by the preliminary design with two and three operations of Number-oriented mutations, layout design 3 can be obtained by layout design 1 and layout design 2 with one operation of Number-oriented crossover, the other three layout designs can be obtained by similar methods. It illustrates that different designs in solution space can be obtained from the same preliminary design through this evolutionary process. IV. GA PARAMETERS GA parameters include: population size, initial population, crossover rate, mutation rate, fitness function and convergence condition. Population size is an important parameter in genetic algorithm, which determines the number of chromosomes in one generation. If the population size is too small, there will be fewer possibilities to perform crossover and only a small part of the search space is explored. If it is too large, the algorithm will slow down [12]. We set 100 as population size first, then adjust the value according to the result of optimization. The first generation is initialized with designs that have only one channel with different attributes and different geometric positions. All kinds of crossover rates and mutation rates are set by the size of inpidual search space and adjusted according to the optimal results. The algorithm will stop if it doesn’t make an improvement in the last 100 generations. At this point the best found solution so far, is used as output of the GA. Fig.3 (b) Mold Wall Temperature Fig.3 (c) Temperature difference Fig.3 (a) Part Shape The fitness value of every chromosome in each generation is obtained by a fuzzy evaluation method developed previously[3]: ) ( ) ( ) ( s s m m c c R I w I w I w I ⊗ ⊕ ⊗ ⊕ ⊗ = This is a fuzzy weighted average [13, 14] of the cooling performance index Ic, the manufacturability index Im, and the structural strength index Is. The operators ⊕ and ⊗ represent fuzzy extended addition and multiplication. The weighting factors w’s are fuzzy variables defined with five levels of importance: very high (VH), high (HI), medium (MI), low (LO), and very low (VL). The user specifies the importance level for each weighting, so that the evaluation can be biased towards a specific design requirement. For example, the user may choose a VH for wc, a VL for wm, and MI for ws when designing an expensive precision mould, and may choose a MI for wc, VH for wm, and VH for ws when designing a low-cost mould for a high production volume. The indices Ic and Im are fuzzy variables, which are obtained by fuzzy weighted averages of multiple factors. The fitness value is calculated by defuzzifying IR. V. CASE STUDY To verify the feasibility of the proposed evolutionary method, an experimental system has been developed using C++ and is interfaced to the Unigraphics II CAD/CAM system.
TABLE I. DOFS OF EXPAMPLE DESIGN Fig.3 (a) shows the part shape to be tested. The layout design of the cooling system can be found in Fig.3 (b) and (c). For illustration purpose, only the design in the core side of the mould is optimized. TABLE I. lists the optimized geometric position of the design denoted by the DOFs of the cooling system. C-Mold analysis has been applied to confirm that the layout design generated by the proposed method is able to provide an efficient cooling performance. The analysis results for the layout design and Layout-1 are shown in Figures 3(b) and (c). With a cooling time of 14 seconds, it can be seen from the two figures that the maximum mould-wall temperature is about 60oC and the maximum temperature differences is less than 6oC, which indicates that the cooling function provided by this layout design is satisfactory. VI. CONCLUSION An evolutionary approach with ad hoc operators to concurrently optimize the topological connection and geometric position of a cooling system is proposed in this research. A mixed encoding scheme is developed to encode a candidate solution, which is represented in the form of a variant-length chromosome. Ad hoc evolutionary operators and parameters adapting to the characteristics of cooling system design are devised. An experimental system is implemented to verify the feasibility of the approach, and the results of case study illustrates the validity of this approach. Degree X1 X2 X3 X4 X5 19.64 21.04 134.23 110.08 199.99 Degree X6 X7 X8 X9 X 10 129.60 134.96 133.86 110.44 221.90 Degree X11 X12 X13 X14 X 15 134.44 109.97 200.61 299.94 134.51 Degree X16 110.58 ACKNOWLEDGMENT The work described in this paper was supported by Education of ZheJiang Province (No. Y200907365) and NSF of China (No. 50875069). REFERENCES [1] C.L. Li, “ A feature-based approach to injection mould cooling system design”, Computer-Aided Design, Vol33 no. 14, pp 1073-1090, Dec 2001. [2] C.L. Li, “Automatic synthesis of cooling system design for plastic injection mould”, ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 27-th Design Automation Conference, Pittsburgh, Pennsylvania, September 9-12, 2001. [3] C.L. Li, CG Li and ACK MOK, “Automatic layout design of plastic injection mould cooling system”, Computer-Aided Design, , Jun 2005, Pages 645-662. [4] C.L. Li, C.G. Li, “Manufacturable and Functional layout design of cooling system for plastic injection mould – an automatic approach” ICMA 2004. [5] CL Li, C.G Li, “Automating the design of injection mould cooling system by genetic algorithm”, ICMA2007, Singapore May 2007, Pages 25-28. [6] CG. Li and CL. Li, “A new configuration space method for the design of plastic injection mould cooling system”, Computer-Aided Design, 2009. [7] Y.C. LAM, L.Y. ZHAI, K. TAI and S.C. FOK, “An evolutionary approach for cooling system optimization in plastic injection moulding”, International Journal product research, 15 May 2004, vol.42, No. 10, 2047-2061. [8] http://www.moldflow.com/ [9] XuanGuangNan and ChengRunWei, Genetic Algorithms and Engineering Optimization Tsinghua publisher, 2004. [10] Stefano Rizzi. “Genetic operators for hierarchical graph clustering”. Pattern Recognition Letters 1998 1293-1300. [11] Hironobu Katagiri, Kotaro Hirasawa, Jinglu Hu and Junichi Murata, “Comparing some graph crossover in genetic network programming”, SICE2002 Aug. 5-7, 2002, Osaka. [12] Mitsuo Gen, Runwei Cheng, “Genetic algorithms and engineering optimization”, EISBN: 0-471-31531-1, Copyright © 2000 by John Wiley & Sons, Inc. [13] Dong WM,Wong FS. “Fuzzy weighted averages and implementation of the extension principle”, Fuzzy Sets Syst, 21, 1987; Pages 183-199. [14] Guh Y-Y, Hon C-C, Lee ES. “Fuzzy weighted average: the linear programming approach via Charnes and Cooper’s rule”, Fuzzy Sets Syst, 117, 2001; Pages 157-160.