基于kriging代理模型的优化设计方法及其在注塑成型中的应用
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摘要
信息产品和电子产品的迅速发展,对注塑制品的质量提出了更高的要求,如轻便、小巧等,这些就促使了高精密塑料制品的出现。由于制件的壁很薄(一般为1-2mm),所以更容易出现收缩、翘曲、残余应力等缺陷。对于这些薄壳类精密制品而言,翘曲变形严重影响制品的质量。注塑成型过程是一个高度非线性时变性的多参数作用过程,所以质量与整个过程中的各个参数相关。随着一些大型的注塑成型分析软件的发展和成熟,基于CAE技术的优化设计在注塑成型工艺中的应用极大地提高了制品成型的质量和效率。然而,基于这些大型的分析软件,实施有效的优化设计仍然是一个有挑战性的课题。
本文首先对kriging代理模型的数学机理进行了详细阐述,它是一种基于统计理论的插值技术,没有假定的具体形式,应用灵活。随后通过工程优化实例将基于kriging代理模型的优化方法与序列线性规划算法进行了比较,结果显示出基于kriging代理模型优化方法的诸多优点。
鉴于kriging代理模型近似优化的优点,将其引入到注塑成型领域,同时发展了带有随机移动的网格取样方法,并结合二者设计基于最小化响应面和最大化期望提高加点准则的序列黑箱优化程序。这种序列优化设计对所建立的代理模型进行序列更新,能有效地提高最优解精度。应用该程序对注塑制品成型工艺进行了有效的优化设计,并基于优化结果分析了注塑工艺参数对制件翘曲变形的影响。
提出一种同时考虑预测值及其不确定性的kriging代理模型序列近似优化方法。这种优化方法在每次更新模型时增加当前最优设计和预测不确定性较大的点,提高模型的全局预测精度,同时进一步提高当前最优解处的预测精度。其中加点准则在每次加点迭代过程中所增加的新点一般为多个,称之为多点加点准则。通过数学函数对所提出的序列近似优化方法进行了较详细的探讨,并且与最大化期望提高加点准则进行了比较,结果显示出本文提出的优化方法能有效地逼近全局最优解。在此基础上,将多点加点准则引入到考虑不确定性的稳健优化设计中,实现了序列优化过程,并且引入了双重kriging代理模型的构建,不仅减小优化过程中的函数运行次数而且减小稳健优化过程中不确定性分析的计算量,从而有效地提高计算效率。
注塑制件的翘曲变形产生机理复杂,而且与注塑工艺、制件尺寸等因素的关系式高度非线性的。针对减小注塑制品翘曲的要求,建立了注塑工艺和制件几何壁厚的优化模型,并结合多点加点准则实施了减小翘曲的序列近似优化设计,有效地减小了制品的翘曲变形,并分析了工艺参数以及材料参数(黏度和PVT关系)对制件翘曲变形的影响,为提高注塑制品的质量提供了依据。考虑到注塑工艺中某些参数的不稳定性,建立了相应的稳健优化模型,结合基于kriging代理模型的序列稳健优化设计方法对注塑工艺和制件壁厚进行了稳健优化设计,优化结果表明所提出的序列近似稳健优化设计方法是有效的。
本文工作得到国家自然科学基金重大项目《高聚物成型加工与模具设计中的关键力学和工程问题》(No.10590354)的资助。
The developments of communication and consumer electronic products require higher for the qualities of high precise injection molded parts,especially as the components continue to move toward being lighter,thinner,shorter,and smaller.Shrinkage,warpage and residual stress are important defects for the thin-wall components(1～2mm thick),and among them the warpage affects greatly the quality of the components.Injection molding is a highly non-linear process with interactions of parameters,thus qualities of components are related with all parameters through the total process.With the developments of some commercial analysis software for injection molding,the application of optimization based on computer aided engineering(CAE) technologies greatly improves the qualities of the components and efficiency of molding.However,it is still a challenge problem to implement effective optimization based on the computational time-consumed analysis software.
First,theory of kriging surrogate model is expatiated,which is a kind of special interpolated technology based on statistics with no specific formula assumed.Through an engineering problem,a compare between sequential linear programming and kriging-based optimization method is done,and it shows kriging-based optimization method has a lot of merits.
With the merits of the kriging-based optimization method,the kriging surrogate model is introduced in injection molding field.At the same time,a modified rectangular grid sampling developed.Then the sequential approximation optimization procedure,with minimizing response surface and maximizing expected improvement,is proposed to find the optimal process parameters and good result is obtained.Furthermore,an analysis about the effects of process parameters on warpage is performed at the optimal process setting.
A sequential approximation optimization method,considering simultaneously the predictor and its uncertainty,is represented.This sequential method updates the kriging surrogate model by adding current optimal point for the accuracy of the interest region and the points with larger predicted uncertainty for the accuracy of the global approximation. Using this optimization method,more than one new point is selected to improve the surrogate, so it is called "multi-point addition" criterion.Through two mathematic functions,an investigation is done for the proposed optimization method,and a compare is implemented between the expected improvement sampling criterion and the multi-point addition criterion. The compare results show the proposed optimization method here effectively improves the optimum.Subsequently,the multi-point addition criterion is used for the robust optimization under uncertainty.It makes adaptive iteration come true through the consideration of predictor and its uncertainty and reduce the function evaluation.In addition,the use of the second-layer kriging model saves computational complexity and improves the computational efficiency.
The mechanism of warpage is rather complicated,and it has high nonlinear relationship with part geometry and process parameters.For the reduction of warpage,the optimization model is proposed with the process parameters and part geometry as variables.The sequential optimization is performed through multi-point addition criterion and the warpage is reduced effectively.Meanwhile,the effects of process and material properties(viscosity and PVT relationship) on warpage are analyzed,and the results provide some advice for improvement of the part quality.In addition,considering the instability of some process parameters,the robust optimization model with variables process parameters and part geometry is constructed and the optimal robust design is obtained by the sequential robust optimization method based on kriging surrogate model.The optimization result shows the proposed sequential approximation optimization method is effective.
The authors gratefully acknowledge financial support for this work from the Major program(No.10590354) of the National Natural Science Foundation of China.
本文首先对kriging代理模型的数学机理进行了详细阐述,它是一种基于统计理论的插值技术,没有假定的具体形式,应用灵活。随后通过工程优化实例将基于kriging代理模型的优化方法与序列线性规划算法进行了比较,结果显示出基于kriging代理模型优化方法的诸多优点。
鉴于kriging代理模型近似优化的优点,将其引入到注塑成型领域,同时发展了带有随机移动的网格取样方法,并结合二者设计基于最小化响应面和最大化期望提高加点准则的序列黑箱优化程序。这种序列优化设计对所建立的代理模型进行序列更新,能有效地提高最优解精度。应用该程序对注塑制品成型工艺进行了有效的优化设计,并基于优化结果分析了注塑工艺参数对制件翘曲变形的影响。
提出一种同时考虑预测值及其不确定性的kriging代理模型序列近似优化方法。这种优化方法在每次更新模型时增加当前最优设计和预测不确定性较大的点,提高模型的全局预测精度,同时进一步提高当前最优解处的预测精度。其中加点准则在每次加点迭代过程中所增加的新点一般为多个,称之为多点加点准则。通过数学函数对所提出的序列近似优化方法进行了较详细的探讨,并且与最大化期望提高加点准则进行了比较,结果显示出本文提出的优化方法能有效地逼近全局最优解。在此基础上,将多点加点准则引入到考虑不确定性的稳健优化设计中,实现了序列优化过程,并且引入了双重kriging代理模型的构建,不仅减小优化过程中的函数运行次数而且减小稳健优化过程中不确定性分析的计算量,从而有效地提高计算效率。
注塑制件的翘曲变形产生机理复杂,而且与注塑工艺、制件尺寸等因素的关系式高度非线性的。针对减小注塑制品翘曲的要求,建立了注塑工艺和制件几何壁厚的优化模型,并结合多点加点准则实施了减小翘曲的序列近似优化设计,有效地减小了制品的翘曲变形,并分析了工艺参数以及材料参数(黏度和PVT关系)对制件翘曲变形的影响,为提高注塑制品的质量提供了依据。考虑到注塑工艺中某些参数的不稳定性,建立了相应的稳健优化模型,结合基于kriging代理模型的序列稳健优化设计方法对注塑工艺和制件壁厚进行了稳健优化设计,优化结果表明所提出的序列近似稳健优化设计方法是有效的。
本文工作得到国家自然科学基金重大项目《高聚物成型加工与模具设计中的关键力学和工程问题》(No.10590354)的资助。
The developments of communication and consumer electronic products require higher for the qualities of high precise injection molded parts,especially as the components continue to move toward being lighter,thinner,shorter,and smaller.Shrinkage,warpage and residual stress are important defects for the thin-wall components(1～2mm thick),and among them the warpage affects greatly the quality of the components.Injection molding is a highly non-linear process with interactions of parameters,thus qualities of components are related with all parameters through the total process.With the developments of some commercial analysis software for injection molding,the application of optimization based on computer aided engineering(CAE) technologies greatly improves the qualities of the components and efficiency of molding.However,it is still a challenge problem to implement effective optimization based on the computational time-consumed analysis software.
First,theory of kriging surrogate model is expatiated,which is a kind of special interpolated technology based on statistics with no specific formula assumed.Through an engineering problem,a compare between sequential linear programming and kriging-based optimization method is done,and it shows kriging-based optimization method has a lot of merits.
With the merits of the kriging-based optimization method,the kriging surrogate model is introduced in injection molding field.At the same time,a modified rectangular grid sampling developed.Then the sequential approximation optimization procedure,with minimizing response surface and maximizing expected improvement,is proposed to find the optimal process parameters and good result is obtained.Furthermore,an analysis about the effects of process parameters on warpage is performed at the optimal process setting.
A sequential approximation optimization method,considering simultaneously the predictor and its uncertainty,is represented.This sequential method updates the kriging surrogate model by adding current optimal point for the accuracy of the interest region and the points with larger predicted uncertainty for the accuracy of the global approximation. Using this optimization method,more than one new point is selected to improve the surrogate, so it is called "multi-point addition" criterion.Through two mathematic functions,an investigation is done for the proposed optimization method,and a compare is implemented between the expected improvement sampling criterion and the multi-point addition criterion. The compare results show the proposed optimization method here effectively improves the optimum.Subsequently,the multi-point addition criterion is used for the robust optimization under uncertainty.It makes adaptive iteration come true through the consideration of predictor and its uncertainty and reduce the function evaluation.In addition,the use of the second-layer kriging model saves computational complexity and improves the computational efficiency.
The mechanism of warpage is rather complicated,and it has high nonlinear relationship with part geometry and process parameters.For the reduction of warpage,the optimization model is proposed with the process parameters and part geometry as variables.The sequential optimization is performed through multi-point addition criterion and the warpage is reduced effectively.Meanwhile,the effects of process and material properties(viscosity and PVT relationship) on warpage are analyzed,and the results provide some advice for improvement of the part quality.In addition,considering the instability of some process parameters,the robust optimization model with variables process parameters and part geometry is constructed and the optimal robust design is obtained by the sequential robust optimization method based on kriging surrogate model.The optimization result shows the proposed sequential approximation optimization method is effective.
The authors gratefully acknowledge financial support for this work from the Major program(No.10590354) of the National Natural Science Foundation of China.
引文
1.廖正品.中国塑料制品行业发展趋势与市场分析.国外塑料,2008,26(4):22-30.
2.瞿金平,黄汉雄,吴舜英.塑料工业手册-注塑、模压工艺与设备.北京:化学工业出版社,2001.
3.廖正品.中国塑料工业(2005).中国塑料,2006,20(4):6-12.
4.刘来英.注塑成型工艺.北京:机械工业出版社,2005.
5.Zoetelief WF.Multi-component Injection Molding:(PhD thesis).Eindhoven:Eindhoven University of Technology,1995.
6.吴崇峰.实用注塑模CAD/CAE/CAM技术.北京:中国轻工业出版社,2000.
7.李海梅,申长雨.注塑成型及模具设计.北京:化学工业出版社,2002.
8.申长雨,陈静波,刘春太等.塑料成型加工的计算机模拟技术.工程塑料应用,2000,28(4):31-34.
9.Bushko WC,Stokes VK.Solidification of thermoviscoelastic melts.Part Ⅱ:Effects of processing conditions on shrinkage and residual stresses.Polymer Engineering and Science,1995,35(4):365-383.
10.Jacques MS.An analysis of thermal warpage in injection molded flat parts due to unbalanced cooling.Polymer Engineering and Science,1982,22(4):241-247.
11.Li D,Zhou H.Modelling and simulation of residual stress and warpage in injection moulding.Proceedings of the Institution of Mechanical Engineerings,Part C:J.Mechanical Engineering Science.2004:521-530.
12.Mlekusch B.The warpage of corners in the injection moulding of short-fibre-reinforced thermoplastics.Composites Science and Technology,1999,59:1923-1931.
13.李海梅,申长雨,顾元宪.不均匀冷却对注塑残余应力的影响.中国塑料,1999,13(6):106-111.
14.李海梅,谢英,余晓容,申长雨.注射压力对塑件残余应力和翘曲变形的影响.模具工业,1999(7):27-30.
15.Wang TJ,Yoon CK.Shrinkage and warpageanalysis of injection-molded parts.SPE ANTEC Tech.2000:687-692.
16.Huang MC,Tai CC.The effective factors in warpage problem ofan injection-molded part with a thin shell feature.Journal of Materials Processing Technology,2001,110:1-9.
17.董斌斌,申长雨,李倩.注塑成型薄壁制品收缩与翘曲因素.化工学报,2005,56(4):727-731.
18.董斌斌,申长雨,刘春太.注射工艺参数对PC/ABS材料制品收缩与翘曲的影响.高分子材料科学与工程,2005,21(4):233-235.
19.Liao SJ,Chang DY,Chen HJ et al.Optimal process conditions of shrinkage and warpage of thin-wall parts.Polymer Engineering and Science,2004,44(5):917-928.
20.Wang TH,YoungWB,Wang J.Process design for reducing the warpage in thin-walled injection.International Polymer Processing ⅩⅦ,2002(2):146-152.
21.Oktema H,Erzurumlub T,Uzman I.Application of Taguchi optimization technique in determining plastic injection molding process parameters for a thin-shell part.Materials and Design,2007,28:1271-1278.
22.申长雨,余晓容,王利霞等.塑料注塑成型浇口位置优化.化工学报,2004,55(3):445-449.
23.余晓容,申长雨,李倩等.基于流动平衡的注塑模浇口位置优化设计.高分子材料科学与工程2004,20(3):159-162.
24.翟明,顾元宪,申长雨.注射模浇口数目和位置的优化设计.化工学报,2003,54(8):1141-1145.
25.翟明,顾元宪,申长雨.注塑成型充填过程中注射速率的优化设计.高分子学报,2003(1):35-38.
26.翟明,顾元宪,申长雨.注射模浇注系统位置优化设计.机械工程学报,2002,38(2):40-43.
27.刘春太,肖长江,申长雨.基于遗传算法的注塑成型充模过程优化.郑州大学学报,2002,23(4):4-8.
28.郑占明,刘春太,申长雨.基于遗传算法的注塑模注射速率优化.工程塑料应用,2002,30(7):25-27
29.Pandelidis I,Zou Q.Optimization of injection molding design,part Ⅰ:gate location optimization.Polymer Engineering and Science,1990,30(15):873-882.
30.王德翔,刘来英,王振宝等.基于人工神经网络技术的注塑成型工艺参数优化.模具技术,2001(6):1-4.
31.郑生荣,辛勇,杨国泰等.基于混合神经网络与遗传算法的注塑参数优化.计算机应用,2004,24(2):91-94.
32.吴建军,马泽恩.以边界元法为基础的注塑模冷却系统优化设计.西北工业大学学报,1998,16((4)):554-559.
33.Lee BH,Kim BH.Optimization of part wall thickness to reduce warpage of injection molded part based on the modified complex method.Polymer-Plastics Technology and Engineering,1995,34(5):793-811.
34.申长雨,王利霞,张勤星.神经网络与混合遗传算法结合的注塑成型工艺优化.高分子材料科学与工程,2005,21(5):23-27.
35.Shen CY,Wang LX,Li Q.Optimization of injection molding process parameters using combination of artificial neural network and genetic algorithm method.Journal of Materials Processing Technology,2007,183:412-418.
36.Kurtaran H,Ozcelik B,Erzurumlu T.Warpage optimization of a bus ceiling lamp base using neural network model and genetic algorithm.Journal of Materials Processing Technology,2005,169:314-319.
37.Kurtaran H,Ozcelik B,Erzurumlu T.Effective warpage optimization of thin shell plastic parts using response surface methodology and genetic algorithm.International Journal of Advanced Manufacturing Technology,2006,27(5):468-472.
38.Ozcelik B,Erzurumlu T.Determination of effecting dimensional parameters on warpage of thin shell plastic parts using integrated response surface method and genetic algorithm.International Communications in Heat and MassTransfer,2005,32:1085-1094.
39.Ozcelik B,Erzurumlu T.Comparison of the warpage optimization in the plastic injection molding using ANOVA,neural network model and genetic algorithm.Journal of Materials Processing Technology,2006,171:437-445.
40.Chang SH,Hwang JI,Doong JL.Optimization of the injection molding process of short glass fiber reinforcedp polycarbonate composites using grey relational analysis.Journal of Material Processing Technology,2000,97:186-193.
41.姚涛,王鹏程.基于灰色关联分析的注塑成型工艺参数优化设计.内蒙古工业大学学报,2004,23(4):273-276.
42.Ye H,Wu Y,Wang KK CAE and Intelligent Processing of Polymeric Materials.ASME International Mechanical Engineering Congress and Exposition Texas,USA.1997:139-149.
43.Yen C,Lin JC,Li W et al.An abductive neural network approach to the design of runner dimensions for the minimization of wrappage in injection mouldings.Journal of Materials Processing Technology,2006,178:1-8.
44.Goldberg DE.Genetic algorithms in search,optimization and machine learning.Massachusetts:Addison-Wesley,1989.
45.Kirkpartick S,Gelatt Jr CD,Vecchi MP.Optimization by simulated annealing.Science,1983,220(4598):671-680.
46.Myers RH,Khuri AI,Carter Jr WH.Response Surface Methodology:1966-1988.Technometrics,1989,31(2):137-157.
47.Zeidenberg,Matthew.Neural networks in atificial intelligence.New York:Ellis Horwood Limited,1990.
48.Shie JR.Optimization of injection molding process for contour distortions of polypropylene composite components by a radial basis neural network.International Journal of Advanced Manufacturing Thechnology Polymer Processing,2008,36:1091-1103.
49.Vapnik VN.An overview of statistical learning theory.IEEE Transactions on Neural Networks,1999,10(3):988-999.
50.邓乃扬,田英杰.数据挖掘中的新方法-支持向量机.北京:科学出版社,2004.
51.Zhou J,Turng LS,Kramschuster A.Single and multi objective optimization for injection molding using numerical simulation with surrogate models and genetic algorithms.International Polymer Processing 2006,21(5):509-520.
52.Zhou J,Turng LS.Process optimization of injection molding using an adaptive surrogate model with Gaussian process approach.Polymer Engineering and Science,2007,47:684-694.
53.Zhou J,Turng LS.Adaptive multiobjective optimization of process conditions for injection molding using a Gaussian process approach.Advances in Polymer Technology,2007,26:71-85.
54.AlamK,Kamal MR.A robust optimization of injection molding runner balance.Computers and Chemical Engineering,2005,29:1934-1944.
55.Matherton.Principles of geo-statistics.Economic Geology,1963,58:1246-1266.
56.张崎.基于kriging方法的机构可靠性分析及优化设计:(博士学位).大连:大连理工大学,2005.
57.Giunta AA,Watson LT.A comparison of approximation modeling techniques:polynomial vs.interpolating models.AIAA.1998.
58.Costa JP,Pronzato L,Thierry E.A comparison between kriging and radial basis function networks for nonlinear prediction.Dana Proc.NSIP' 99,Antalya.1999.
59.Simpson TW,Lin DKJ,Chen W.Sampling strategies for computer experiments:design and analysis.International Journal of Reliability and Safety,2001,2(3):209-240.
60.Martin JD,Simpson TW.A study on the use of kriging models to approximate deterministic computer models.Design Engineering Technical Conferences and Computers and Information in Engineering Conference,Chicago.2003.
61.Simpson TW,Mauery TM,Korte JJ,Mistree F.Comparison of response surface and kriging models for multidisciplinary design optimization.7th Symposium on Multidisciplinary Analysis and Optimization.1998:AIAA-98-4755.
62.王晓峰,席光,王尚锦.Kriging与响应面方法在气动优化设计中的应用.工程热物理学报,2005,26(3):423-425.
63.Sakata S,Ashida F,Zako M.Structural optimization using kriging approximation.Computer Methods in Applied Mechanics and Engineering,2003,192:923-939.
64.Sakata S,Ashida F,Zako M.An efficient algorithm for kriging approximation and optimization with large scale sampling data.Computer Methods in Applied Mechanics and Engineering,2004,193:385-404.
65.Booker AJ,Corm AR,Dennis JE et al.Global modeling for optimization:Boeing/IBM/Rice collaborative project.ISSTECH-95-032,The Boeing Company,Seattle.1995.
66.Lee KH,Park GJ A global robust optimization using the kriging based Approximation model.JSME International Journal,2006,49(3):779-788.
67.张柱国,姚卫星,刘克龙.基于进化Kriging模型的金属加筋板结构布局优化方法.南京航空航天大学学报,2008,40(4):892-901.
68.曹鸿钧,段宝岩.基于kriging模型的后优化研究.机械设计与研究,2004,20(5):10-13.
69.Koch PN,Wujek B,Golovidov O et al.Facilitating probabilistic multidisciplinary design optimization using kriging approximation models.9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization.Atlanta,Georgia.2002.
70.Welch WJ,Buck RJ,Sacks J.Predictingand computer experiments.Technometrics,1992,34(1):15-25.
71.Sacks J,Welch WJ,Mitchell TJ et al.Design andanalysis of computer experiments.Statistics Science,1989,4(4):409-435.
72.Lucifredi A,Mazzieri C and Rossi M.Application of multi-regressive linear models,Dynamic kriging models and neural network models to predictive maintenance of hydroelectric power systems.Mechanical Systems and Signal Processing,2000,14(3):471-494.
73.Lee SH,Kim HY,Oh SI.Cyindrical tube optimization using response surface method based on stochastic process.Journal of Material Processing Technology,2002,130:490-496.
74.Jones DR,Schonlau M,Welch WJ.Efficient global optimization of expensive black-box functions.Journal of Global Optimization,1998,13:445-492.
75.Lophaven SN,Nielsen HB,Sondergaard J DACE-A Matlab Kriging Toolbox,Informatics and mathematical modelling,Denmark.2002.
76.Simpson TW,Peplinski JD,Koch PN et al.Metamodels for computer-based engineering design:survay and recommendations.Engineering with Computers,2001,17:129-150.
77.Koehler JR,Owen AB.Computer experiments,in Ghosh S Rao CR.Handbook of Statistics.New York:Elsevier Science.1996:261-308.
78.Welch WJ,Yu TK,Kang SM,Sacks J.Computer experiments for quality control by parameter design.Journal of Quality Technology,1990,22(1):15-22.
79.Sacks J,Schiller SB,Welch WJ.Designs for computer experiments.Technometrics,1989,31(1):41-47.
80.Jones DR.A texonomy of global optimization methods based on response surfaces.Journal of Global Optimization,2001,23:345-383.
81.Mckay MD,Beckman RJ,Conover WJ.A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics 1979,21(2):239-245.
82.Ye Q,Li W,Sudjianto A.Algorithmic construction of optimal symmetric Latin Hypercube designs.Journal of Statistical Planning and Inference,2003,90:145-159.
83.Lindley DV.On a measure of the information provided by an experiment.Ann.Math.Statist.,1956,27:986-1005.
84.Shewry MC,Wynn HP.Maximum entropy sampling.Journal of Applied Statistics,1987,14:165-170.
85.Park JS.Optimal Latin-hypercube designs for computer experiments.Journal of Statistical Planning and Inference,1994,39:95-111.
86.Mockus J,Tiesis V,Zilinskas A.The application of Bayesian methods for seking the extremum,in GP.Dixon LCW and Szego.Towards Global Optimization.North Holland:North-Holland Publishing Company.1978.
87.Schonlau M,Welch WJ,Jones DR.Global versus local search in constrained optimization of computer models,National Institute of Statistical Sciences,NC.1998.
88.Sasena MJ,Papalambros PY,Goovaerts P.Metamodeling sampling criteria in a global optimization framework.AIAA,Long beach.2000.
89.Sasena MJ,Papalambros PY,Goovaerts P.Exploration of metamodeling sampling criteria for constrained global optimization.Engineering Optimization,2002,34:263-278.
90.Sobester A,Leary SJ,Keane AJ.On the design of optimization strategies based on global response surface approximation models.Journal of Global Optimization,2005,33:31-59.
91.Huang D,Allen TT,Notz WI et al.Global optimization of stochastic black-box systems via sequential kriging meta-models.Journal of Global Optimization,2005.
92.Huang D,Allen TT,Notz WI et al.Sequential kriging optimization using multiple fidelity evaluations.Structral and multidisciplinary optimization,2006,32(5):369-382.
93.宋旭军.汽轮机基础框架结构动力响应及灵敏度近似重分析与多目标优化设计:(博士论文).大连:大连理工大学,2002.
94.Subremanian NR,Lin T,Seng YA.Optimizing warpage analysis for an optical housing.Mechatronics,2005,15:111-127.
95.Sahu R,Yao DG,Kim B.Optimal Molds Design Methodology to Minimize Warpage in Injection Molded Parts.Annual Technical Conference-ANTEC.1997:3308-3312.
96.Taguchi G.Taguchi on robust technology development:bringing quality engineering upst ream.New York:ASME Press,1993.
97.GuL,Yang R.Optimizationand robustness for crashworthiness of side impact.International Journal of Vehicle Design(Special Issue) 2001,26(3):348-360.
98.Varadarajan S,Chen W,Pelka J C..Robust concept exploration of propulsion systems with enhanced model approximation capabilities Engineering Optimization,2000,32(3):309-334.
99.Jin R,Chert W,Simpson TW.Comparative studies of metamodelling techniques under multiple modeling criteria.Structral and Multidisciplinary Optimization,2001,23(1):1-13.
100.许焕卫,孙伟,张旭.基于混合响应面的多目标稳健设计.机械科学与技术,2008,27(5):628-632.
101.崔庆安.基于支持向量机的稳健性参数设计与实现.设计与研究,2008(8):5-8.
102.Park GJ,Lee TH,Lee KHet al.A review of robust design methodologies.Transaction of the KSMEA(in Korean),2004,28(9):1368-1383.
103.Law A M,Kelton W D.Simulation modeling and analysis.New York McGraw - Hill Company,1982.
104.Hammersley JM,Handscomb DC.Monte Carlo methods.London:Chapman and Hall,1964.
105.Lee HL,Park GJ A global robust optimization using the kriging based Approximation model.JSME International Journal,2006,49(3):779-788.
106.ViningGG,Myers RH.Combining Taguchi and response surface philosophies:a dual response approach Journal of quality technology,1990,22(1):38-45.
107.Myers RH,Carter WH.Response surface techniques for dual response systems.Technometrics,1973,15(3):301-317.
108.Yeniay O,Unalb R,Lepsch RA.Using dual response surfaces to reduce variability in launch vehicle design:a case study.Reliability Engineering & System Safety,2006,91(4):407-412.
109.Gao YH,Wang XC.An effective warpage optimization method in injection molding based on kriging model.International Journal of Advanced Manufacturing Technology,2008,37(9):953-960.
110.Gao YH,TurngLS,Wang XC.Adaptive geometry and process optimization for injection molding using kriging surrogate model trained by numerical simulation.Advances in Polymer Technology,2008,26(5):1-16.
111.Gao YH,Wang XC.Surrogate-based process optimization for reducing warpage in injection molding.Journal of Materials Processing Technology,2009,209:1302-1309.
112.高月华,王希诚.注塑制品的翘曲优化及影响因素分析.化工学报,2007,58(6):1575-1580.
2.瞿金平,黄汉雄,吴舜英.塑料工业手册-注塑、模压工艺与设备.北京:化学工业出版社,2001.
3.廖正品.中国塑料工业(2005).中国塑料,2006,20(4):6-12.
4.刘来英.注塑成型工艺.北京:机械工业出版社,2005.
5.Zoetelief WF.Multi-component Injection Molding:(PhD thesis).Eindhoven:Eindhoven University of Technology,1995.
6.吴崇峰.实用注塑模CAD/CAE/CAM技术.北京:中国轻工业出版社,2000.
7.李海梅,申长雨.注塑成型及模具设计.北京:化学工业出版社,2002.
8.申长雨,陈静波,刘春太等.塑料成型加工的计算机模拟技术.工程塑料应用,2000,28(4):31-34.
9.Bushko WC,Stokes VK.Solidification of thermoviscoelastic melts.Part Ⅱ:Effects of processing conditions on shrinkage and residual stresses.Polymer Engineering and Science,1995,35(4):365-383.
10.Jacques MS.An analysis of thermal warpage in injection molded flat parts due to unbalanced cooling.Polymer Engineering and Science,1982,22(4):241-247.
11.Li D,Zhou H.Modelling and simulation of residual stress and warpage in injection moulding.Proceedings of the Institution of Mechanical Engineerings,Part C:J.Mechanical Engineering Science.2004:521-530.
12.Mlekusch B.The warpage of corners in the injection moulding of short-fibre-reinforced thermoplastics.Composites Science and Technology,1999,59:1923-1931.
13.李海梅,申长雨,顾元宪.不均匀冷却对注塑残余应力的影响.中国塑料,1999,13(6):106-111.
14.李海梅,谢英,余晓容,申长雨.注射压力对塑件残余应力和翘曲变形的影响.模具工业,1999(7):27-30.
15.Wang TJ,Yoon CK.Shrinkage and warpageanalysis of injection-molded parts.SPE ANTEC Tech.2000:687-692.
16.Huang MC,Tai CC.The effective factors in warpage problem ofan injection-molded part with a thin shell feature.Journal of Materials Processing Technology,2001,110:1-9.
17.董斌斌,申长雨,李倩.注塑成型薄壁制品收缩与翘曲因素.化工学报,2005,56(4):727-731.
18.董斌斌,申长雨,刘春太.注射工艺参数对PC/ABS材料制品收缩与翘曲的影响.高分子材料科学与工程,2005,21(4):233-235.
19.Liao SJ,Chang DY,Chen HJ et al.Optimal process conditions of shrinkage and warpage of thin-wall parts.Polymer Engineering and Science,2004,44(5):917-928.
20.Wang TH,YoungWB,Wang J.Process design for reducing the warpage in thin-walled injection.International Polymer Processing ⅩⅦ,2002(2):146-152.
21.Oktema H,Erzurumlub T,Uzman I.Application of Taguchi optimization technique in determining plastic injection molding process parameters for a thin-shell part.Materials and Design,2007,28:1271-1278.
22.申长雨,余晓容,王利霞等.塑料注塑成型浇口位置优化.化工学报,2004,55(3):445-449.
23.余晓容,申长雨,李倩等.基于流动平衡的注塑模浇口位置优化设计.高分子材料科学与工程2004,20(3):159-162.
24.翟明,顾元宪,申长雨.注射模浇口数目和位置的优化设计.化工学报,2003,54(8):1141-1145.
25.翟明,顾元宪,申长雨.注塑成型充填过程中注射速率的优化设计.高分子学报,2003(1):35-38.
26.翟明,顾元宪,申长雨.注射模浇注系统位置优化设计.机械工程学报,2002,38(2):40-43.
27.刘春太,肖长江,申长雨.基于遗传算法的注塑成型充模过程优化.郑州大学学报,2002,23(4):4-8.
28.郑占明,刘春太,申长雨.基于遗传算法的注塑模注射速率优化.工程塑料应用,2002,30(7):25-27
29.Pandelidis I,Zou Q.Optimization of injection molding design,part Ⅰ:gate location optimization.Polymer Engineering and Science,1990,30(15):873-882.
30.王德翔,刘来英,王振宝等.基于人工神经网络技术的注塑成型工艺参数优化.模具技术,2001(6):1-4.
31.郑生荣,辛勇,杨国泰等.基于混合神经网络与遗传算法的注塑参数优化.计算机应用,2004,24(2):91-94.
32.吴建军,马泽恩.以边界元法为基础的注塑模冷却系统优化设计.西北工业大学学报,1998,16((4)):554-559.
33.Lee BH,Kim BH.Optimization of part wall thickness to reduce warpage of injection molded part based on the modified complex method.Polymer-Plastics Technology and Engineering,1995,34(5):793-811.
34.申长雨,王利霞,张勤星.神经网络与混合遗传算法结合的注塑成型工艺优化.高分子材料科学与工程,2005,21(5):23-27.
35.Shen CY,Wang LX,Li Q.Optimization of injection molding process parameters using combination of artificial neural network and genetic algorithm method.Journal of Materials Processing Technology,2007,183:412-418.
36.Kurtaran H,Ozcelik B,Erzurumlu T.Warpage optimization of a bus ceiling lamp base using neural network model and genetic algorithm.Journal of Materials Processing Technology,2005,169:314-319.
37.Kurtaran H,Ozcelik B,Erzurumlu T.Effective warpage optimization of thin shell plastic parts using response surface methodology and genetic algorithm.International Journal of Advanced Manufacturing Technology,2006,27(5):468-472.
38.Ozcelik B,Erzurumlu T.Determination of effecting dimensional parameters on warpage of thin shell plastic parts using integrated response surface method and genetic algorithm.International Communications in Heat and MassTransfer,2005,32:1085-1094.
39.Ozcelik B,Erzurumlu T.Comparison of the warpage optimization in the plastic injection molding using ANOVA,neural network model and genetic algorithm.Journal of Materials Processing Technology,2006,171:437-445.
40.Chang SH,Hwang JI,Doong JL.Optimization of the injection molding process of short glass fiber reinforcedp polycarbonate composites using grey relational analysis.Journal of Material Processing Technology,2000,97:186-193.
41.姚涛,王鹏程.基于灰色关联分析的注塑成型工艺参数优化设计.内蒙古工业大学学报,2004,23(4):273-276.
42.Ye H,Wu Y,Wang KK CAE and Intelligent Processing of Polymeric Materials.ASME International Mechanical Engineering Congress and Exposition Texas,USA.1997:139-149.
43.Yen C,Lin JC,Li W et al.An abductive neural network approach to the design of runner dimensions for the minimization of wrappage in injection mouldings.Journal of Materials Processing Technology,2006,178:1-8.
44.Goldberg DE.Genetic algorithms in search,optimization and machine learning.Massachusetts:Addison-Wesley,1989.
45.Kirkpartick S,Gelatt Jr CD,Vecchi MP.Optimization by simulated annealing.Science,1983,220(4598):671-680.
46.Myers RH,Khuri AI,Carter Jr WH.Response Surface Methodology:1966-1988.Technometrics,1989,31(2):137-157.
47.Zeidenberg,Matthew.Neural networks in atificial intelligence.New York:Ellis Horwood Limited,1990.
48.Shie JR.Optimization of injection molding process for contour distortions of polypropylene composite components by a radial basis neural network.International Journal of Advanced Manufacturing Thechnology Polymer Processing,2008,36:1091-1103.
49.Vapnik VN.An overview of statistical learning theory.IEEE Transactions on Neural Networks,1999,10(3):988-999.
50.邓乃扬,田英杰.数据挖掘中的新方法-支持向量机.北京:科学出版社,2004.
51.Zhou J,Turng LS,Kramschuster A.Single and multi objective optimization for injection molding using numerical simulation with surrogate models and genetic algorithms.International Polymer Processing 2006,21(5):509-520.
52.Zhou J,Turng LS.Process optimization of injection molding using an adaptive surrogate model with Gaussian process approach.Polymer Engineering and Science,2007,47:684-694.
53.Zhou J,Turng LS.Adaptive multiobjective optimization of process conditions for injection molding using a Gaussian process approach.Advances in Polymer Technology,2007,26:71-85.
54.AlamK,Kamal MR.A robust optimization of injection molding runner balance.Computers and Chemical Engineering,2005,29:1934-1944.
55.Matherton.Principles of geo-statistics.Economic Geology,1963,58:1246-1266.
56.张崎.基于kriging方法的机构可靠性分析及优化设计:(博士学位).大连:大连理工大学,2005.
57.Giunta AA,Watson LT.A comparison of approximation modeling techniques:polynomial vs.interpolating models.AIAA.1998.
58.Costa JP,Pronzato L,Thierry E.A comparison between kriging and radial basis function networks for nonlinear prediction.Dana Proc.NSIP' 99,Antalya.1999.
59.Simpson TW,Lin DKJ,Chen W.Sampling strategies for computer experiments:design and analysis.International Journal of Reliability and Safety,2001,2(3):209-240.
60.Martin JD,Simpson TW.A study on the use of kriging models to approximate deterministic computer models.Design Engineering Technical Conferences and Computers and Information in Engineering Conference,Chicago.2003.
61.Simpson TW,Mauery TM,Korte JJ,Mistree F.Comparison of response surface and kriging models for multidisciplinary design optimization.7th Symposium on Multidisciplinary Analysis and Optimization.1998:AIAA-98-4755.
62.王晓峰,席光,王尚锦.Kriging与响应面方法在气动优化设计中的应用.工程热物理学报,2005,26(3):423-425.
63.Sakata S,Ashida F,Zako M.Structural optimization using kriging approximation.Computer Methods in Applied Mechanics and Engineering,2003,192:923-939.
64.Sakata S,Ashida F,Zako M.An efficient algorithm for kriging approximation and optimization with large scale sampling data.Computer Methods in Applied Mechanics and Engineering,2004,193:385-404.
65.Booker AJ,Corm AR,Dennis JE et al.Global modeling for optimization:Boeing/IBM/Rice collaborative project.ISSTECH-95-032,The Boeing Company,Seattle.1995.
66.Lee KH,Park GJ A global robust optimization using the kriging based Approximation model.JSME International Journal,2006,49(3):779-788.
67.张柱国,姚卫星,刘克龙.基于进化Kriging模型的金属加筋板结构布局优化方法.南京航空航天大学学报,2008,40(4):892-901.
68.曹鸿钧,段宝岩.基于kriging模型的后优化研究.机械设计与研究,2004,20(5):10-13.
69.Koch PN,Wujek B,Golovidov O et al.Facilitating probabilistic multidisciplinary design optimization using kriging approximation models.9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization.Atlanta,Georgia.2002.
70.Welch WJ,Buck RJ,Sacks J.Predictingand computer experiments.Technometrics,1992,34(1):15-25.
71.Sacks J,Welch WJ,Mitchell TJ et al.Design andanalysis of computer experiments.Statistics Science,1989,4(4):409-435.
72.Lucifredi A,Mazzieri C and Rossi M.Application of multi-regressive linear models,Dynamic kriging models and neural network models to predictive maintenance of hydroelectric power systems.Mechanical Systems and Signal Processing,2000,14(3):471-494.
73.Lee SH,Kim HY,Oh SI.Cyindrical tube optimization using response surface method based on stochastic process.Journal of Material Processing Technology,2002,130:490-496.
74.Jones DR,Schonlau M,Welch WJ.Efficient global optimization of expensive black-box functions.Journal of Global Optimization,1998,13:445-492.
75.Lophaven SN,Nielsen HB,Sondergaard J DACE-A Matlab Kriging Toolbox,Informatics and mathematical modelling,Denmark.2002.
76.Simpson TW,Peplinski JD,Koch PN et al.Metamodels for computer-based engineering design:survay and recommendations.Engineering with Computers,2001,17:129-150.
77.Koehler JR,Owen AB.Computer experiments,in Ghosh S Rao CR.Handbook of Statistics.New York:Elsevier Science.1996:261-308.
78.Welch WJ,Yu TK,Kang SM,Sacks J.Computer experiments for quality control by parameter design.Journal of Quality Technology,1990,22(1):15-22.
79.Sacks J,Schiller SB,Welch WJ.Designs for computer experiments.Technometrics,1989,31(1):41-47.
80.Jones DR.A texonomy of global optimization methods based on response surfaces.Journal of Global Optimization,2001,23:345-383.
81.Mckay MD,Beckman RJ,Conover WJ.A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics 1979,21(2):239-245.
82.Ye Q,Li W,Sudjianto A.Algorithmic construction of optimal symmetric Latin Hypercube designs.Journal of Statistical Planning and Inference,2003,90:145-159.
83.Lindley DV.On a measure of the information provided by an experiment.Ann.Math.Statist.,1956,27:986-1005.
84.Shewry MC,Wynn HP.Maximum entropy sampling.Journal of Applied Statistics,1987,14:165-170.
85.Park JS.Optimal Latin-hypercube designs for computer experiments.Journal of Statistical Planning and Inference,1994,39:95-111.
86.Mockus J,Tiesis V,Zilinskas A.The application of Bayesian methods for seking the extremum,in GP.Dixon LCW and Szego.Towards Global Optimization.North Holland:North-Holland Publishing Company.1978.
87.Schonlau M,Welch WJ,Jones DR.Global versus local search in constrained optimization of computer models,National Institute of Statistical Sciences,NC.1998.
88.Sasena MJ,Papalambros PY,Goovaerts P.Metamodeling sampling criteria in a global optimization framework.AIAA,Long beach.2000.
89.Sasena MJ,Papalambros PY,Goovaerts P.Exploration of metamodeling sampling criteria for constrained global optimization.Engineering Optimization,2002,34:263-278.
90.Sobester A,Leary SJ,Keane AJ.On the design of optimization strategies based on global response surface approximation models.Journal of Global Optimization,2005,33:31-59.
91.Huang D,Allen TT,Notz WI et al.Global optimization of stochastic black-box systems via sequential kriging meta-models.Journal of Global Optimization,2005.
92.Huang D,Allen TT,Notz WI et al.Sequential kriging optimization using multiple fidelity evaluations.Structral and multidisciplinary optimization,2006,32(5):369-382.
93.宋旭军.汽轮机基础框架结构动力响应及灵敏度近似重分析与多目标优化设计:(博士论文).大连:大连理工大学,2002.
94.Subremanian NR,Lin T,Seng YA.Optimizing warpage analysis for an optical housing.Mechatronics,2005,15:111-127.
95.Sahu R,Yao DG,Kim B.Optimal Molds Design Methodology to Minimize Warpage in Injection Molded Parts.Annual Technical Conference-ANTEC.1997:3308-3312.
96.Taguchi G.Taguchi on robust technology development:bringing quality engineering upst ream.New York:ASME Press,1993.
97.GuL,Yang R.Optimizationand robustness for crashworthiness of side impact.International Journal of Vehicle Design(Special Issue) 2001,26(3):348-360.
98.Varadarajan S,Chen W,Pelka J C..Robust concept exploration of propulsion systems with enhanced model approximation capabilities Engineering Optimization,2000,32(3):309-334.
99.Jin R,Chert W,Simpson TW.Comparative studies of metamodelling techniques under multiple modeling criteria.Structral and Multidisciplinary Optimization,2001,23(1):1-13.
100.许焕卫,孙伟,张旭.基于混合响应面的多目标稳健设计.机械科学与技术,2008,27(5):628-632.
101.崔庆安.基于支持向量机的稳健性参数设计与实现.设计与研究,2008(8):5-8.
102.Park GJ,Lee TH,Lee KHet al.A review of robust design methodologies.Transaction of the KSMEA(in Korean),2004,28(9):1368-1383.
103.Law A M,Kelton W D.Simulation modeling and analysis.New York McGraw - Hill Company,1982.
104.Hammersley JM,Handscomb DC.Monte Carlo methods.London:Chapman and Hall,1964.
105.Lee HL,Park GJ A global robust optimization using the kriging based Approximation model.JSME International Journal,2006,49(3):779-788.
106.ViningGG,Myers RH.Combining Taguchi and response surface philosophies:a dual response approach Journal of quality technology,1990,22(1):38-45.
107.Myers RH,Carter WH.Response surface techniques for dual response systems.Technometrics,1973,15(3):301-317.
108.Yeniay O,Unalb R,Lepsch RA.Using dual response surfaces to reduce variability in launch vehicle design:a case study.Reliability Engineering & System Safety,2006,91(4):407-412.
109.Gao YH,Wang XC.An effective warpage optimization method in injection molding based on kriging model.International Journal of Advanced Manufacturing Technology,2008,37(9):953-960.
110.Gao YH,TurngLS,Wang XC.Adaptive geometry and process optimization for injection molding using kriging surrogate model trained by numerical simulation.Advances in Polymer Technology,2008,26(5):1-16.
111.Gao YH,Wang XC.Surrogate-based process optimization for reducing warpage in injection molding.Journal of Materials Processing Technology,2009,209:1302-1309.
112.高月华,王希诚.注塑制品的翘曲优化及影响因素分析.化工学报,2007,58(6):1575-1580.