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编号： 毕业设计毕业设计(论文论文)任务书任务书题 目： 支承管注射模具设计 学院： 国防生学院 专 业：机械设计制造及其自动化学生姓名： 谭宇宙 学 号： 1000110111 指导教师单位： 机电工程学院 姓 名： 何玉林 职 称： 讲师 题目类型 ：理论研究 实验研究 工程设计 工程技术研究 软件开发 2013 年 12 月 9 日一、一、毕业设计（论文）的内容毕业设计（论文）的内容1、塑件的分析 2、塑件材料的选用与性能分析（特性及成型工艺参数） 3、拟定模具的结构形式塑（塑件要有侧凹或侧凸或螺纹） 4、浇注系统的设计 5、分流道的设计 6、浇口的设计 7、冷料穴和拉料杆的设计 8、成型零件的设计 9、导向机构的设计10、脱模推出机构的设计 11、侧向分型与抽心机构设计 12、排气系统的设计 13、冷却系统的设计二、二、毕业设计（论文）的要求与数据毕业设计（论文）的要求与数据 1、外型尺寸及精度 2、使用环境-1040 3、收缩率 4、外观要求 5、塑料壁厚 6、ABS 主要技术指标及工艺参数 7、设计中的计算 8、安装尺寸的校核3、毕业设计（论文）应完成的工作毕业设计（论文）应完成的工作 1、完成二万字左右的毕业设计说明书（论文） ；在毕业设计说明书（论文）中必须包括详细的 300-500 个单词的英文摘要；2、独立完成与课题相关，不少于四万字符的指定英文资料翻译（附英文原文） ；3、对于纯机械类课题，绘图工作量折合 A0 图纸 3 张以上，其中必须包含两张 A3 以上的计算机绘图图纸；四、应收集的资料及主要参考文献四、应收集的资料及主要参考文献1 李学峰.塑料模设计及制造M. 北京：机械工业出版社,2001.2 翁其金.塑料模塑成型技术M. 北京：机械工业出版社,2002.3 钱泉森.塑料成型工艺及模具设计M. 济南：山东科学技术出版社，2004.4 塑料模设计手册编著组.塑料设计手册.北京：机械工业出版 社,2002.5 陈剑鹤.模具设计基础M. 北京：机械工业出版社，2004.6 王文广等.塑料注塑模具设计技巧与实例M. 北京：化学工业出版社，20047 章飞.型腔模具设计与制造M. 北京：化学工业出版社，2003.8 谭雪松，林晓新，温利编。新编塑料模设计手册M.北京：人民邮电出版社，2007.1.9 朱光.塑料注塑模中小型模架及其技术条件M. 北京：清华大学出版社，2003,1.10 2006HERBERT W.YANKEE MANUFACTURING PROCESSES Prentice-Hall,Inc.五、试验、测试、试制加工所需主要仪器设备及条件五、试验、测试、试制加工所需主要仪器设备及条件计算机(autoCAD,及 pro/E,protel 软件)任务下达时间：2013 年 12 月 9 日毕业设计开始与完成时间：2013 年 12 月 17 日至 2014 年 05 月 8 日组织实施单位：教研室主任意见：签字： 2013 年 12 月 14 日院领导小组意见：签字： 2013 年 12 月 16 日2014 年机电工程学院毕业设计(论文)进度计划表学生姓名：学生姓名： 学号：学号：序号起止日期计划完成内容实际完成内容检查日期检查人签名12013.12.912.1522013.12.1612.2232013.12.2312.2942013.12.30-2014.1.552014.1.6-2014.1.1262014.1.13-2014.1.1972014.2.24-2014.3.282014.3.3-2014.3.9(本表同时作为指导教师对学生的 16 次考勤记录)2014 年机电工程学院毕业设计进度计划表（续）学生姓名：学生姓名： 学号：学号：序号起止日期计划完成内容实际完成内容检查日期检查人签名92014.3.10-2014.3.16102014.3.17-2014.3.23112014.3.24-2014.3.30122014.3.31-2014.4.6132014.4.7-2014.4.13142014.4.14-2014.4.20152014.4.21-2014.4.27162014.4.28-2014.5.4任务下达时间：2013 年 12 月 9 日(本表同时作为指导教师对学生的 16 次考勤记录)编号： 毕业设计（论文）外文翻毕业设计（论文）外文翻译译 （原文）（原文） 学 院： 国防生学院 专 业：机械设计制造及其自动化学生姓名： 谭宇宙 学 号： 1000110111 指导教师单位： 机电工程学院 姓 名： 何玉林 职 称： 讲师 2014 年 1 月 18 日编号： 毕业设计毕业设计( (论文论文) )开题报告开题报告题 目： 支承管注射模具设计 院 （系）： 国防生学院 专 业： 机械设计制造及其自动化 学生姓名： 谭宇宙 学 号： 1000110111 指导教师单位： 机电工程学院 姓 名： 何玉林 职 称： 讲师 题目类型 ：理论研究 实验研究 工程设计 工程技术研究 软件开发 2013 年 12 月 23 日- 0 -1毕业设计的主要内容、重点和难点等 注射成型的基本原理、冷流道注射模具浇注系统、凸模凹模以及成型零件的计算、排气系统的设计、侧向分型与抽芯机构的设计、温度调节系统和顶出系统。并介绍 OUTCAD 的应用以及对导柱和导套进行参数化设计。1、塑件的分析 2、塑件材料的选用与性能分析（特性及成型工艺参数）3、拟定模具的结构形式塑（塑件要有侧凹或侧凸或螺纹） 4、浇注系统的设计 5、分流道的设计 6、浇口的设计 7、冷料穴和拉料杆的设计 8、成型零件的设计 9、导向机构的设计 10、脱模推出机构的设计 11、侧向分型与抽心机构设计 12、排气系统的设计 13、冷却系统的设计2准备情况（查阅过的文献资料及调研情况、现有设备、实验条件等） 1、模具技术的现状模具是机械、汽车、电子、通讯、家电等工业产品的基础工艺装备之一。作为工业基础，模具的质量、精度、寿命对其他工业的发展起着十分重要的作用，在国际上被称为“工业之母” ，对国民经济发展起着不容质疑的作用。 2、注塑模简介 注塑成型又称注塑模具，是热塑性塑料制件的一种主要成型方法，并且能够成功地将某些热固性塑料注塑成型。注塑成型可成型各种形状的塑料制品，其优点包括成型周期短，能一次成型外形复杂、尺寸精密、带有嵌件的制品，且生产效率高易于实现自动化，因而广泛应用在塑料制品生产当中。 3、注塑成型原理及特点 塑料的注塑成型过程，就是借助螺杆或柱塞的推力，将已塑化的塑料熔体以一定的压力和速度注入模具型腔内，经过冷却固化定型后开模而获得制品。因此，可以说注塑成型在塑料装配生产中具有重要地位。- 1 - 4、注塑成型原理注塑成型所用的模具即为注塑模（也称为注射模） ，注塑成型的原理（以螺杆式注射机为例） 。首先将颗粒或粉状的塑料加入料斗，然后输送到侧装有电加热的料筒中塑化。螺杆在料筒前端原地转动，使被加热预塑的塑料在螺杆的转动作用下通过螺旋槽输送至料筒前端的喷嘴附近。螺杆的转动使塑料进一步化，料温在剪切摩擦热的作用下进一步提高并得以均匀化。当料筒前端堆积的体对螺杆产生一定的压力时（称为螺杆的背压） ，螺杆将转动后退，直至整好的行程开关接触，从而使螺母与螺杆锁紧。具有模具一次注射量的塑料预塑和储过程结束。这时，马达带动气缸前进，与液压缸活塞相连接的螺杆以一定的速度和压力将熔料通过料筒前端的喷嘴注入温度较低的闭合模具型腔中。熔体通过喷嘴注入闭合模具腔后，必须经过一定时间的保压，熔融塑料才能冷却固化，保持模具型腔所赋予形状和尺寸。当合模机构打开时，在推出机构的作用下，即可顶出注塑成型的塑料制品。5、支承管注射模具设计的流程：(1)思考与创新：绘制草图，确定晾衣叉的外观形式；(2)实践操作：通过 Pro-e 软件画出晾衣叉外壳的三维模型；(3)用 Pro-e 做出内部的结构，实现外观要求；(4)将 Pro-e 做的图导入 AutoCAD 中； (5)修改结构图。6、注射模具的设计过程(1)对塑料零件的材料、形状和功能进行分析(2)确定型腔的数目确定型腔的数目条件有：最大注射量、锁模力、产品的精度要求和经济性等。(3)选择分型面分型面的选择应以模具结构简单、分型容易，且不破坏已成型的塑件为原则。(4)型腔的布置方案型腔的布置应采用平衡式排列，以保证各型腔平衡进料。型腔的布置还要注意与冷却管道、推杆布置的协调问题。(5)确定浇注系统浇注系统包括主流道、分流道、浇口和冷料穴。浇注系统的设计应根据模具的类型、型腔的数目及布置方式、塑件的原料及尺寸等确定。(6)确定脱模方式脱模方式的设计应根据塑件留在模具的部分而同。由于注射机的推出顶杆在动模部分，所以，脱模推出机构一般都设计在模具的动模部分。因此，应设计成使塑件能留在动模部分。设计中，除了将较长的型芯安排在动模部分以外，还常设计拉料杆，强制塑件留在动模部分。但也有些塑件的结构要求塑件在分型时，留在定模部分，在定模一侧设计出推出装置。推出机构的设计也应根据塑件的不同结构设计- 2 -出不同的形式，有推杆、推管和推板等结构。(7)确定调温系统结构模具的调温系统主要由塑料种类决定。模具的大小、塑件的物理性能、外观和尺寸精度都对模具的调温系统有影响。(8)确定凹模和型心的固定方式当凹模或型心采用镶块结构时，应合理地划分铁块并同时考虑镶块的强度、可加工性及安装固定。(9)确定排气尺寸一般注射模的排气可以利用模具分型面和推杆与模具的间隙；而对于大型和高速成型的注射模，必须设计相应的排气装置。(10)确定注射模的主要尺寸根据相应的公式，计算成型零件的工作尺寸，以及决定模具型腔的侧壁厚度、动模板的厚度、拼块式型腔的型腔板的厚度及注射模的闭合高度。(11)选用标准模架根据设计、计算的注射模的主要尺寸，来选用注视模的标准模架，并尽量选择标准模具零件。(12)绘制模具的结构草图在以上工作的基础上，绘制注射模的完整的结构草图，绘制模具结构图是模具设计十分重要的工作，其步骤为先画俯视图（顺序为：画模架、型腔、冷却管道、支撑柱、推出机构） ，再画出主视图。(13)校核模具与注射机有关尺寸对所使用的注射机的参数进行校核：包括最大注射量、注射压力、锁模力及模具的安装部分的尺寸、开模行程和推出机构的校核。(14)注射模结构设计的审查对根据上述有关注视模结构设计的各项要求设计出来的注射模，应进行注射模结构设计的初步审查，同时，也有必要对提出的要求加以确认和修改。(15)绘制模具的装配图装配图是模具装配的主要依据，因此应清楚地表明注视模的各个零件的装配关系、必要的尺寸（如外形尺寸、定位圈直径、安装尺寸、活动零件的极限尺寸等） 、序号、明细表、标题栏及技术要求。(16)绘制模具的零件图由模具装配图拆绘零件图的顺序为：先内后外，先复杂后简单，先成型零件后结构零件。(17)复核设计图样注射模具设计的最后是审核所设计的注射模，应多关注零件的加工、性能。已查阅的文献资料- 3 -1 李学峰.塑料模设计及制造M. 北京：机械工业出版社,2001.2 翁其金.塑料模塑成型技术M. 北京：机械工业出版社,2002.3 钱泉森.塑料成型工艺及模具设计M. 济南：山东科学技术出版社，2004.4 塑料模设计手册编著组.塑料设计手册.北京：机械工业出版社,2002.5 陈剑鹤.模具设计基础M. 北京：机械工业出版社，2004.6 王文广等.塑料注塑模具设计技巧与实例M. 北京：化学工业出版社，20047 章飞.型腔模具设计与制造M. 北京：化学工业出版社，2003.8 谭雪松，林晓新，温利编。新编塑料模设计手册.北京：人民邮电出版社，2007.1.9 朱光.塑料注塑模中小型模架及其技术条件M. 北京：清华大学出版社， 2003,1.10 cunha,L,et.al.,performance of chromium nitride and titanium nitride coatings during platics injection moulding. Surface and coating Technology, 2002. 153(2-3):p.160-165. 现有设备及实验条件：计算机一台，使用软件为 Pro/Engineer5.0 及 Auto CAD2008、Moldflow insight，以上实验条件可满足本次毕业设计的要求.3、实施方案、进度实施计划及预期提交的毕业设计资料 1、2013 年 12 月 17 日至 2013 年 12 月 30 日，理解消化毕设任务书要求并收集、分析、消化资料文献，根据毕设内容完成并交开题报告； 2、2013 年 1 月 6 日至 2014 年 1 月 13 日，开展调研，了解塑件结构，对原材料进行分析，考虑塑件的成型工艺性、模具的总体结构的形式，并完成部分英文摘要翻译。 3、2014 年 3 月 4 日至 2013 年 3 月 31 日，查阅资料，熟悉注射模的结构及有关计算，拟定模具的方案设计、总体设计及主要零件设计，拟定成型工艺过程，查阅有关手册确定适宜的工艺参数，注射机的选择及确定注射设备及型号规格； 4、2014 年 4 月 1 日至 2014 年 4 月 21 日，完成设计计算任务，总体结构的设计和完成总装配图及零件图的设计； 5、2014 年 4 月 22 日至 2014 年 5 月 1 日，完成设计，图纸绘制任务，工艺规程说明书的编写； 6、2014 年 5 月 1 日至 2014 年 5 月 4 日，完善设计并完成论文的撰写；- 4 -7、2014 年 5 月 4 日至 2014 年 5 月 8 日，修改并打印毕业论文及整理相关资料，交指导老师评阅，准备论文答辩。指导教师意见指导教师（签字）： 2013 年 12 月 日开题小组意见开题小组组长（签字）：2014 年 1 月 日 院（系、部）意见 主管院长（系、部主任）签字： 2014 年 1 月日毕业设计（论文）中期检查表（指导教师）毕业设计（论文）中期检查表（指导教师）指导教师姓名：何玉林填表日期： 2014 年 4 月 20 日学生学号1000110111学生姓名谭宇宙题目名称支承管注射模具设计已完成内容开题并做调研，进行翻译；确定其方案设计；完成结构设计；绘制结构草图；完成相关计算；完成英文翻译；绘制装配图；绘制零件图；撰写论文；完成毕业设计。 检查日期：2014-4-20完成情况全部完成按进度完成滞后进度安排存在困难解决办法查阅相关资料，并且与指导老师和同学们一起讨论解决方案。预期成绩优 秀良 好中 等及 格不及格建议 教师签名：教师签名： 教务处实践教学科制表教务处实践教学科制表说明：说明：1、本表由检查毕业设计的指导教师如实填写；2、此表要放入毕业设计（论文）档案袋中；3、各院(系)分类汇总后报教务处实践教学科备案Computer-Aided Design 40 (2008) 334349/locate/cadPlastic injection mould cooling system design by theconfiguration space methodC.G. Li, C.L. LiDepartment of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong KongReceived 3 May 2007; accepted 18 November 2007AbstractThe cooling system of an injection mould is very important to the productivity of the injection moulding process and the quality of themoulded part. Despite the various research efforts that have been directed towards the analysis, optimization, and fabrication of cooling systems,support for the layout design of the cooling system has not been well developed. In the layout design phase, a major concern is the feasibilityof building the cooling system inside the mould insert without interfering with the other mould components. This paper reports a configurationspace (C-space) method to address this important issue. While a high-dimensional C-space is generally required to deal with a complex systemsuch as a cooling system, the special characteristics of cooling system design are exploited in the present study, and special techniques that allowC-space computation and storage in three-dimensional or lower dimension are developed. This new method is an improvement on the heuristicmethod developed previously by the authors, because the C-space representation enables an automatic layout design system to conduct a moresystematic search among all of the feasible designs. A simple genetic algorithm is implemented and integrated with the C-space representation toautomatically generate candidate layout designs. Design examples generated by the genetic algorithm are given to demonstrate the feasibility ofthe method.c ? 2007 Elsevier Ltd. All rights reserved.Keywords: Cooling system design; Plastic injection mould; Configuration space method1. IntroductionThe cooling system of an injection mould is very importantto the productivity of the injection moulding process andthe quality of the moulded part. Extensive research has beenconducted into the analysis of cooling systems 1,2, andcommercial CAE systems such as MOLDFLOW 3 andMoldex3D 4 are widely used in the industry. Researchinto techniques to optimize a given cooling system has alsobeen reported 58. Recently, methods to build better coolingsystems by using new forms of fabrication technology havebeen reported. Xu et al. 9 reported the design and fabricationof conformal cooling channels that maintain a constant distancefrom the mould impression. Sun et al. 10,11 used CNCmilling to produce U-shaped milled grooves for coolingchannels and Yu 12 proposed a scaffolding structure for thedesign of conformal cooling.Corresponding author.E-mail address: meclli.hk (C.L. Li).Despite the various research efforts that have focused mainlyon the preliminary design phase of the cooling system designprocess in which the major concern is the performance ofthe cooling function of the system, support for the layoutdesign phase in which the feasibility and manufacturability ofthe cooling system design are addressed has not been welldeveloped. A major concern in the layout design phase is thefeasibility of building the cooling system inside the mouldinsert without interfering with the other mould components.Consider the example shown in Fig. 1. It can be seen thatmany different components of the various subsystems of theinjection mould, such as ejector pins, slides, sub-inserts, andso forth, have to be packed into the mould insert. Finding thebest location for each channel of the cooling circuit to optimizethe cooling performance of the cooling system and to avoidinterference with the other components is not a simple task.Another issue that further complicates the layout designproblem is that the individual cooling channels need to beconnected to form a path that connects between the inlet andthe outlet. Therefore, changing the location of a channel may0010-4485/$ - see front matter c ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.11.010C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349335Fig.1. Thecoolingsysteminsideamouldinsertpackedwithmanyothermouldcomponents.require changing the other channels as well. Consider theexample shown in Fig. 2. The ideal location of each channelto optimize the cooling performance of the system is shownin Fig. 2(a). Assume that when the cooling system and theother mould components are built into the mould insert, amould component O1is found to interfere with channel C1.As C1cannot be moved to a nearby location due to the possibleinterference with other components, it is shortened. As a result,C2is moved and C3is elongated accordingly to maintain theconnectivity, as shown in Fig. 2(b). Owing to its new length,C3is found to interfere with another mould component, O2,and further modification is needed, which results in the finaldesign shown in Fig. 2(c). Given that a typical injection mouldmay have more than ten cooling channels, with each channelpotentially interfering with a few other mould components,finding an optimal layout design manually is very tedious.This paper reports a new technique that supports theautomation of the layout design process. In this new technique,a configuration space (C-space) method is used to provide aconcise representation of all of the feasible layout designs. TheC-space representation is constructed by an efficient methodthat exploits the special characteristics of the layout designproblem. Instead of using heuristic rules to generate layoutdesigns, as in the automatic layout design system developedpreviously by the authors 13,14, this new C-space methodenables an automatic layout design system to conduct a moresystematic search among all of the feasible layout designs.2. The configuration space methodIn general, the C-space of a system is the space thatresults when each degree of freedom of that system is treatedas a dimension of the space. Regions in the configurationspace are labeled as blocked region or free region. Pointsin the free regions correspond to valid configurations of thesystem where there is no interference between the componentsof the system. Points in the blocked regions correspond toinvalid configurations where the components of the systeminterfere with one another. C-space was initially formalizedby Lozano-Perez 15 to solve robot path planning problemsand a survey in this area of research has been reported byWise and Bowyer 16. The C-space method has also beenused to solve problems in qualitative reasoning (e.g., 17,18)(a) Interference occurs between cooling channel C1and mould component O1at the ideal location ofC1.(b) Channel C1is shortened, C2is moved, and C3iselongated.(c) C3is moved and C2is shortened to give the finaldesign.Fig. 2. An example showing the tediousness of the layout design process.336C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 3. An example showing the degrees of freedom of a cooling system.and the analysis and design automation of kinematic devices(e.g., 1921). The author investigated a C-space method in theautomatic design synthesis of multiple-state mechanisms 22,23 in previous research.2.1. C-space of a cooling systemA high-dimensional C-space can be used to represent all ofthe feasible layout designs of a given preliminary design ofa cooling system. Fig. 3 gives an example. The preliminarydesign of this cooling system consists of four cooling channels.To generate a layout design from the preliminary design, thecenters and lengths of the channels are adjusted. As shown inFig. 3, the center of channel C1can be moved along the X1and X2directions, and its length can be adjusted along the X3direction. Similarly, the length of C2can be adjusted along theX4direction, while its center adjustment is described by X1and X3and thus must be the same as the adjustment of C1tomaintain the connectivity. By applying similar arguments to theother channels, it can be seen that the cooling system has 5degreesoffreedom,andtheyaredenotedas Xi,i = 1,2,.,5.In principle, the C-space is a five-dimensional space and anypoint in the free region of this space gives a set of coordinatevalues on the Xiaxes that can be used to define the geometry ofthe channels without causing interference with the other mouldcomponents.Todeterminethefreeregioninahigh-dimensionalC-spaceofacoolingsystem,thefirststepistoconstructthefreeregions in the C-spaces of the individual channels.2.2. C-space construction of individual cooling channelsWhen an individual channel Ciis considered alone, it hasthree degrees of freedom, say X1and X2for its center locationand X3for its length. As the ideal center location and lengthhave already been specified in the preliminary design, it isreasonable to assume a fixed maximum allowable variation cfor X1, X2, and X3. The initial free region in the C-spaceof channel Ciis thus a three-dimensional cube Biwith thedimensions c c c.To avoid any possible interference with a mould componentOiwhen channel Ciis built into the mould insert by drilling,a drilling diameter D and drilling depth along X3have to beconsidered. To account for the diameter D, Oiis first offsetby D/2 + M to give O0i, where M is the minimum allowabledistance between the channel wall and the face of a component.This growing of Oiin effect reduces channel Cito a line Li.Consider the example illustrated in Fig. 4. Fig. 4(a) shows achannel Ciand three mould components, O1, O2,and O3,thatmay interfere with Ci. Fig. 4(b) shows the offsets O01, O02,and O03of the mould components, and the reduction of Citoa line segment Lithat is coincident with the axis of Ci. Ifthere is no intersection between Liand the offsets of the mouldcomponents, then the original channel Ciwill not intersect with(a) Channel Ciand three mouldcomponents inside the mould insert.(b) Offsets of the mould components andCirepresented by line segment Li.(c) Sweeping the offsets of the mouldcomponents and Cirepresented by point Pi.(d) The initial free region of Ci.(e) Subtracting O00ifrom B0i.(f) The free region FRiof Ci.Fig. 4. The major steps in the construction of the free region FRiof a channel Ci.C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349337the mould components. This growing or offset of an obstacle isa standard technique in the C-space method 15.A channel is formed by drilling from a face of the mouldinsert, and any obstacle Oiwithin the drilling depth will affectthe construction of the channel. To account for the drillingdepth, the offset O0iof Oiis swept along the drilling directionuntil the opposite face of the mould insert is reached to generateO00i. This sweeping of O0iin effect reduces line Lito a point Pilocated at the end of Li. As shown in Fig. 4(c), if the point Piis outside O00i, the drilling along Lito produce Ciis feasible.The free region FRiof channel Ciis obtained as follows.First, the initial free region Biis constructed with its centerat Pias shown in Fig. 4(d). Bithen intersects with the mouldinsert to obtain B0i. B0irepresents all of the possible variationsof Ciwhen only the geometric shape of the mould insert isconsidered. Then, FRiis obtained by subtracting from B0itheO00iof all of the obstacles. Fig. 4(e) and (f) show the subtractionand the resulting FRiof the example.2.3. Basic approach to the construction of the C-space ofcooling systemTo determine the free region FRFin the C-space of acooling system, the free regions of each cooling channel haveto be “intersected” in a proper manner so that the effect ofthe obstacles to all of the channels are properly representedby FRF. However, the standard Boolean intersection betweenthe free regions of two different channels cannot be performedbecause their C-spaces are in general spanned by different setsof axes. Referring to the example in Fig. 3, the C-spaces ofC1and C2are spanned by X1, X2, X3 and X1, X3, X4,respectively. To facilitate the intersection between free regionsin different C-spaces, the projection of a region from the C-space of one channel to that of another channel is needed. Thefollowing notations are first introduced and will be used inthe subsequent discussions on projections and the rest of thepaper.Notations used in describing high-dimensional spacesSndenotes an n-dimensional space spanned by the set of axesXn= X1, X2,., Xn.Smdenotes an m-dimensional space spanned by the set of axesXm= X01, X02,., X0m.pndenotes a point in Snand pn= (x1,x2,.,xn), where xidenotes a coordinate on the ith axis Xi.Rndenotes a region in Sn(Rn Sn). Rnis a set of points in Sn.PROJSm(pn) denotes the projection of a point pnfrom SntoSm.PROJSm(Rn) denotes the projection of a region Rnfrom SntoSm.Notations used in describing a cooling systemnCdenotes the number of channels in the cooling system.nFdenotes the total degrees of freedom of the cooling system.Cidenotes the ith channel of the cooling system.Sidenotes the C-space of Ci.FRidenotes the free region in Si. That is, it is the free region ofan individual channel Ci.SFdenotes the C-space of the cooling system.FRFdenotes the free region in SF. That is, it is the free regionof the cooling system.Consider the projection of a point pnin Snto a point pminSm. Fig. 5(a) illustrates examples of projection using spaces ofone dimension to three dimensions. Projections are illustratedfor three cases: (i)XmXn; (ii)XmXn; and (iii)Xm6Xn,Xn6Xm, andXnXm6= . For (i), each coordinate ofpmis equal to a corresponding coordinate of pnthat is on thesame axis. For (ii) and (iii), the projection of pnis a region Rm.For each point pmin Rm, a coordinate of pmis equal to thatof pnif that coordinate is on a common axis of Snand Sm.For the other coordinates of pm, any value can be assigned.The reason for this specific definition of the projections, inparticular, for cases (ii) and (iii), is as follows. Consider twoadjacent channels Cnand Cm. As they are adjacent, they mustbe connected and thus their C-spacesSnand Smshare somecommon axes. Assume that a configuration that correspondsto a point pnin Snhas been selected for Cn. To maintainthe connectivity, the configuration for Cmmust be selectedsuch that the corresponding point pmin Smshares the samecoordinates with pnon their common axes. This implies thatpmcan be any point within the projection of pnon Sm, wherethe method of projection is defined above. The projections of aregion Rnin Snto Smare simply the projections of every pointin Rnto Sm. Fig. 5(b) illustrates the region projections. Theformal definition of projection is given below.Definition 1 (Projection).1.1. IfXmXn, PROJSm(pn) is a pointpm=(x01,x02,.,x0m), where for X0i= Xj, x0i= xjfor all i 1,m. To simplify the notations in subsequent discussion,this projection is regarded as a region that consists of thesingle point pm. That is, PROJSm(pn) = pm.1.2. IfXmXn, PROJSm(pn) is a regionRm=pm|PROJSn(pm) = pn.1.3. IfXm6Xn,Xn6Xm, andXnXm6= , PROJSm(pn)isa region Rm= pm|PROJSI(pm) = PROJSI(pn), whereSIis the space spanned byXnXm. IfXnXm= ,PROJSm(pn) is defined as Sm.1.4. PROJSm(Rn) is defined as the region Rm= pm|pmPROJSm(pn), pn Rn.As discussed in Section 2.1, any point pFin FRFgives avalue for each degree of freedom of the cooling system so thatthe geometry of the channels is free from interference with theother mould components. In other words, the projection of pFto each Siis in the free region FRiof each Ci. Thus, FRFisdefined as follows.Definition 2 (Free Region in the C-space of a Cooling System).FRF= pF|PROJSi(pF) FRi,i 1,nC338C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 5. The projections of points and regions in Snto Sm.Note that according to Definition 1.1, the projection of pFto Sialways contains only a single point because the set of axesthat span Siis always a subset of the axes that span Sn.The construction of the free region FRiof each Cihasalready been explained in Section 2.2. To find FRFfrom FRi,the following theorem is useful.Theorem 1.FRF=nCi=1PROJSF(FRi).Intuitively, this theorem says that to find FRF, all of the FRiarefirst projected to the C-space of the cooling system SF. FRFcan then be obtained by performing the Boolean intersectionsamong the projections. The proof of Theorem 1 and the lemmasused in the proof are given in the Appendix.2.4. Representation and computation of the C-spaceTo represent the free region FRFand to facilitate thecomputation of the Boolean intersections between the regionsin a high-dimensional space, we can use a kind of cellenumeration method similar to the one used in 21,24. Thebasic idea is to approximate a high-dimensional region RFinSFby a set of high-dimensional boxes. Each box is defined byspecifying an interval on each axis of SF. The intersection oftwo regions is achieved by the intersection of the two sets ofboxes. The intersection between two high-dimensional boxesis simply the intersection between the intervals of each of theboxes in each axis.Assuming that each FRiis approximated by m three-dimensional boxes, the projection PROJSF(FRi) can then beapproximated by mnF-dimensional boxes. The constructionof FRFthat uses Theorem 1 then requires mnCintersectionsC.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349339(a) A simple cooling system with four channels and four degrees of freedom.(b) The free region FRiof each channel in its configuration space Si.Fig. 6. A simplified example of a cooling system design.between nF-dimensional boxes, and FRFis represented by amaximum of mnCnF-dimensional boxes. Although the numberof boxes used to represent the intermediate results of theintersections and FRFcan be reduced by special techniques, itis anticipated that the memory and computational requirementsare still major problems of this method. In the next section, animproved method is developed.3. An efficient technique for C-space constructionTo avoid the high memory and computational requirementsfor the representation and construction of FRF, we choose notto represent and not to compute FRFexplicitly. Instead, wefocus on a technique that enables the computational process towork on the C-spaces of each individual channel.First, consider the simplified design example shown inFig. 6. For the purpose of illustration, it is assumed in thisexample that there is no variation in FRialong the Z directionof the mould insert and thus the cooling system has four degreesof freedom as shown in Fig. 6(a). The Siof each channel Ciaretwo dimensional and the assumed FRiare shown in Fig. 6(b).Consider a simple method for designing channel C1. First, apoint p1can be selected from within FR1so that C1is freefrom interference with any obstacle. However, S1is spanned340C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(c) Free regions in S1after “intersection” withFR2.(d) A valid point p1for the designs of C1and C2results in an invalid design for C4.Fig. 6. (continued)by X1and X2, and X2is shared by S2. Hence, the constraintsimposed by those obstacles in S2must also be considered. Inan attempt to find all of the feasible points for designing C1,FR1is “intersected” with FR2. The result of this “intersection”is shown in Fig. 6(c), which is obtained by removing the regionin FR1where x2 j,CRj,i= PROJSj(CRj+1,i) FRj.If i j,CRj,i= PROJSj(CRj1,i) FRj.If i = j,CRj,i= FRi.As an example, Fig. 8 shows the sequence of compositionsthat leads to the construction of CR1,4. The first step is toconstruct CR3,4, which is given by CR3,4= PROJS3(FR4) FR3, as shown in Fig. 8(a). Then, CR2,4is constructed byCR2,4=PROJS2(CR3,4) FR2, as shown in Fig. 8(b).Finally, CR1,4is given by CR1,4= PROJS1(CR2,4) FR1, asshown in Fig. 8(c). It is obvious from Fig. 8(c) that the resultingCR1,4takes into account the effects of the free regions of all ofthe channels that make up the cooling system. Therefore, forany point in CR1,4, it is guaranteed that a valid design for thecooling system can be constructed.By applying the composition operations, a valid design canbe obtained by selecting points in each Siafter the free regionsof all of the other channels have been “composited” into Si.However, we would also like to ensure that no valid designis being excluded from the free region after the compositionoperations are applied. Otherwise, some valid designs that maygive better cooling performance can never be obtained by thismethod. Taking the design of C1as an example, it is importantthat CR1,4in Fig. 8(c) not only represent a part of the validdesign for C1, but also represent all of the valid designs forC1. To address this issue, we introduce the following theoremthat applies to a cooling system that consists of a sequence ofchannels Ci, i 1,nc.Theorem 2.PRi= CRi,1 CRi,nC.Theorem 2 states that PRi, which represents all of thevalid designs for channel Ci, can be obtained by a Booleanintersection between CRi,1and CRi,nC. An important featureof this theorem is that PRican be obtained by computationsin three-dimensional spaces, because both CRi,1and CRi,nCare regions in Siand thus the intersection is performed in Si.Moreover, CRi,1and CRi,nCare obtained by the intersectionof regions in Sj. That is, PRiis obtained by a sequenceof operations in three-dimensional spaces. If the assumption342C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(a) Constructing CR3,4by PROJS3(FR4) FR3.Fig. 8. The sequence of operations by which CR1,4is constructed.stated in Section 2.4 is used again, that is, if each FRiisapproximated by m three-dimensional boxes, then both CRi,jand PRican also be represented by m 3D boxes. Therefore,a total of ncm three-dimensional boxes is needed to representall of the PRi. It can be shown that O(ncm2) intersectionsbetween three-dimensional boxes are needed to generate all ofthe PRi. Therefore, the use of Theorem 2 prevents the need tostore regions in a high-dimensional space, and avoids the highmemory and computational requirements of the method givenin Theorem 1.The following gives the proof of Theorem 2. It consists oftwo parts: the proof of CRi,1 CRi,nC PRiand the proofof CRi,1 CRi,nC PRi. The lemmas used in the proof arestated in the Appendix.3.1. Proof of Theorem 2(1) To prove: CRi,1 CRi,nC PRipi CRi,1 CRi,nC pi CRi,1,pi CRi,nC.(i) Inducing from pi CRi,1pi CRi,1CRi,1= PROJSi(CRi1,1) FRi(By Definition 4) pi PROJSi(CRi1,1)pi FRipi PROJSi(CRi1,1) pi1 CRi1,1such that pi PROJSi(pi1)(By Definition 1.4)pi PROJSi(pi1) piand pi1have the same coordinates in the common axesof Siand Si1.(By Definitions 1.11.3)pi1 CRi1,1CRi1,1= PROJSi1(CRi2,1) FRi1 pi1 FRi1.Using the same method, we can determine a point pi2FRi2such that pi1and pi2have the same coordinates inthe common axes Si1and Si2.Repeatedly using this method, we can determine a series ofpoints pk,k 1,i 1, such that pk FRk, and pkand pk+1have the same coordinates in the common axes of Skand Sk+1.(ii) Inducing from pi CRi,nCpi CRi,nC.C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349343(b) Constructing CR2,4by PROJS2(CR3,4) FR2.Fig. 8. (continued)Using a similar method, we can determine another series ofpoints pk,k i + 1,nC, such that pk FRk, and pkandpk1have the same coordinates in the common axes of SkandSk1.By (i) and (ii), we obtain a series of points pk,k 1,nC,such that pk FRk, and any two adjacent points in this serieshave the same coordinates in their common axes.For a cooling system that consists of a sequence of coolingchannels Ci, the C-spaces Siand Si+1of two adjacentchannels Ciand Ci+1always share some common axes becauseof the physical connection between the channels. Furthermore,if there is a common axis Xcin the C-spaces of channels Ciand Cj, Xcmust also appear in all of the C-spaces of thechannels between Ciand Cj. Therefore, the series of pointspk,k 1,nC constructed by the above method will give aunique coordinate for each axis of SF. Let pFbe the pointconstructed from this set of coordinates. It is obvious thatpk = PROJSk(pF),k 1,nCpk FRk,k 1,nC PROJSk(pF) FRk,k 1,nC pF PROJSF(FRk),k 1,nC(By Lemma 8)i.e. pFnCk=1PROJSF(FRk) PROJSi(pF) PROJSi nCk=1PROJSF(FRk)!(By Lemma 1) pi PROJSi nCk=1PROJSF(FRk)! CSi,1 CSi,nC PROJSi nCk=1PROJSF(FRk)! CSi,1 CSi,nC PROJSi(FRF)(By Theorem 1) CSi,1 CSi,nC PRi(By Definition 3).(2) To prove: CRi,1 CRi,nC PRiPROJSF(CRi,1) = PROJSF?PROJSi(CRi1,1) FRi?= PROJSF(PROJSi(CRi1,1) PROJSF(FRi)(By Lemma 4) PROJSF(CRi1,1) PROJSF(FRi)(By Lemma 6)= PROJSF(PROJSi1(CRi2,1) FRi1) PROJSF(FRi)= PROJSF(PROJSi1(CRi2,1) PROJSF(FRi1) PROJSF(FRi)(By Lemma 4) PROJSF(CRi2,1) PROJSF(FRi1) PROJSF(FRi)344C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(c) Constructing CR1,4by PROJS1(CR2,4) FR1.Fig. 8. (continued)(By Lemma 6). PROJSF(FR1) PROJSF(FR2) PROJSF(FRi)=ik=1PROJSF(FRk).Using a similar method, we can obtain:PROJSF(CRi,nC) nCk=iPROJSF(FRk)PROJSF(CRi,1) PROJSF(CRi,nC) nCk=1PROJSF(FRk)PROJSF(CRi,1 CRi,nC) = PROJSF(CRi,1) PROJSF(CRi,nC)(By Lemma 4) PROJSF(CRi,1 CRi,nC) nCk=1PROJSF(FRk) PROJSi(PROJSF(CRi,1 CRi,nC) PROJSi nCk=1PROJSF(FRk)!(By Lemma 2) CRi,1 CRi,nC PROJSi nCk=1PROJSF(FRk)!(By Lemma 5) CRi,1 CRi,nC PROJSi(FRF)(By Theorem 1) CRi,1 CRi,nC PRi(By Definition 3).By (1) and (2):PRi= CRi,1 CRi,n.4. Generation of candidate designsGiven a preliminary design of a cooling system that specifiesa sequence of channels and their ideal geometry, the first stepis to construct an FRifor each channel. Then, the PRiforeach channel is obtained by applying the composition operationas specified in Theorem 2. One way to generate a candidatedesign for a cooling system is to select a set of coordinatesfrom the set of PRias follows. To simplify the explanation,assume that each channel Cihas degrees of freedom XiandXi+1, and Xi+1is shared by the adjacent channel Ci+1. Togenerate a design, a point (x1,x2) in PR1is chosen. Then, anx3is chosen such that (x2,x3) is within PR2. This selectionC.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349345process is then repeated for the next coordinate in the PR ofthe next channel until the coordinates of all of the degrees offreedom are determined. An important feature of the methodis that whatever value is selected for a coordinate in one step,there always exist valid values that can be selected for the nextcoordinate in a subsequent step.5. The automation of the design process using a geneticalgorithmTo test the feasibility of the C-space method in supportingthe automation of the layout design process, a simple geneticalgorithm (GA) 25 is implemented and integrated with the C-space construction program. A simple chromosome structure isused in the implementation of the GA. It consists of a stringof nFreal values g1g2.gnF, for which each gihas a realvalue between 0 and 1, and nFis the number of the degrees offreedom of the cooling system. To generate a design from thechromosome, the approach described in the previous sectionis used, with the giused as a percentage value to select acoordinate. For example, if the valid values for a coordinate xiin PRiexist in the intervals x1i,x2i and x3i,x4i, where x1ix2i x3i,x4i, then the selected value for xiwill be x1i+gi(x2ix1i)+(x4ix3i) if gi (x2ix1i)/(x2ix1i+x4ix3i) (i.e., xilies in the first interval). Otherwise, xiwill be set to x3i+ (gi1)(x2i x1i) + gi(x4i x3i) (i.e., xilies in the second interval).A standard one-point crossover operation, a mutationoperation,andtheroulettewheelselectionmethod26areusedin the GA process. The fuzzy evaluation method developed inour previous research 13,14 is used to perform fast evaluationof the fitness of the candidate design that corresponds to achromosome. Note also that before the GA process starts,the PRifor each channel is constructed. The constructionof the PRiis done only once and thus it will not affectthe computational time for the evolution process of the GA.Examples of the layout designs generated by the GA processare given in the next section.6. Case studyTwo views of an example part are shown in Fig. 9(a).Fig.9(b)illustratesthepreliminarydesignofthecoolingsystemthat specifies the ideal location of each cooling channel whenonly the cooling performance of the system is considered (forthe purpose of illustration, only the cooling system in thecavity half is shown). In the ideal location, interference occursbetween channel C5and a mould component O1. Using theproposed method to automate the layout design, the FRiandthen the PRiof each channel are constructed. As an example,Fig. 9(g) and (h) show FR4and PR4for channel C4. It isnoted that PR4is obtained from FR4by composition with otherFRi, and thus PR4is a subset of FR4, as is evident from thefigures. After all of the PRiare computed, the GA process isinvoked, and the maximum fitness value among the candidatedesigns generated in each generation during the evolutionaryprocess is shown in Fig. 9(j). The maximum fitness value startsto converge after approximately 600 generations. As shown inFig. 9(c), the cooling system consists of 15 degrees of freedomXiand their values are listed in Table 1. In the table, therow labeled “Preliminary design” shows the xivalues for thepreliminary design. The next row lists the values for Design 1,which is the best design generated by the GA process after 1000generations. As highlighted in the table, Design 1 is obtainedby reducing x6by 1.21 mm. Fig. 9(d), which shows Design 1,this adjustment corresponds to the lowering of C5along the Zdirection to clear the interference between C5and O1. Due tothe connectivity among the channels, the same adjustment alsoapplies to channels C4and C6to C13. Table 1 also shows thatall of the other xiin Design 1 are maintained to within 0.2 mmfrom values specified in the preliminary design.To demonstrate further the capability of the C-space method,the mould component O2is moved along the Y direction sothat it interferes with C13, as shown in Fig. 9(e). This newobstacle imposes a new constraint in the free region of C13sothat the feasible adjustment along X6is further limited. Thiseffect is demonstrated in the updated PR4shown in Fig. 9(i) inwhich only the upper portion of the PR4shown in Fig. 9(h)is retained. The GA process is invoked again with the newPRiof all of the channels to generate Design 2. The fitnessvalue is shown in Fig. 9(k). Note that the best fitness valueattained is lower than that of Design 1. This is justified becausewith the imposition of more constraints, larger deviation fromthe ideal design is expected. The values of the xiobtainedfrom the GA process are shown in the last row of Table 1. Ashighlighted in the table, x6is adjusted for 5 mm to clear theinterference with O2. This corresponds to moving channels C4to C13along the Z direction. Now, the interference between C5and O1can no longer be cleared by adjusting x6. Instead, x4and x5are adjusted, which corresponds to moving C52.94 mmalong the Y direction, and moving C46.22 mm along theX direction, as shown in Fig. 9(f). Channels C2and C3arealso adjustedaccordingly to maintain theconnectivity.Design2demonstrates that when the constraint in one channel (e.g., C13)is changed, the proposed C-space method properly propagatesthis effect to the other channels (e.g., C4and C5) so that theset of all of the feasible designs for these channels is adjustedaccordingly.Cooling analysis with C-Mold has been used to analyzethe layout designs generated. It can be seen from Fig. 10(a)through (d) that the maximum mould-wall temperature is about46C with a cooling time of 20 s for both designs, and thattheir maximum temperature differences are less than 8C,which indicate that the proposed method is able to generatesatisfactory layout designs for both cases. It is also observedfrom Fig. 10(c) and (d) that a much larger portion of the part inDesign 1 is not colored when Design 1 is compared to Design2. This indicates that the temperature difference in most of thepart in Design 1 is less than 5.5C. This is because in Design2 as the channels in the cavity half are moved by 5 mm towardsthe mould impression, the cooling effect becomes less uniform,which demonstrates that when more constraints are imposed,maintaining the ideal cooling effect of the preliminary designbecomes more difficult. It also explains why the maximumfitness value for Design 2 is slightly lower than that of Design 1.346C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349(a) An example part.(b) Preliminary design of the cooling system.(c) The 15 degrees of freedom of the cooling system.(d) Design 1.(e) O2moved to interfere with C13.(f) Design 2.Fig. 9. The layout design examples generated by the proposed method.Table 1Degrees of freedom of the cooling systemDegree offreedomX1X2X3X4X5X6X7X8X9X10X11X12X13X14X15Preliminarydesign32.516955212.5139.52932.5110212.580.532.5212.55121.532.5Design 132.41168.9154.95212.34139.4927.7932.36109.97212.5180.6032.30212.5251.1021.5632.52Design 232.51169.0455.13206.28136.5634.0032.51109.84212.4980.4532.50212.5250.9321.6632.64C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349347(g) FR4.(h) PR4.(i)UpdatedPR4.(j) Fitness value against the generation for Design 1.(k) Fitness value against the generation for Design 2.Fig. 9. (continued)7. Discussion and conclusionIn the implementation of the C-space method, a cellenumeration scheme is used to simplify the implementation.In the current implementation, the resolution in the C-spacerepresentation is 0.15 mm in each dimension, which shouldbe adequate for the cooling system design because for a veryfine adjustment, say 0.01 mm, the change in the coolingperformance may not be noticeable. However, the methodologyand theorems developed in this research are not limited to aparticular scheme for representation. In fact, for the methodbased on Theorem 2, all C-space computation and storage aredone in three dimension, and thus standard geometric modelingtechniques can be used.A major contribution of this research is the developmentof a specific C-space method that supports the layout designprocess. Using the C-space method, all of the feasible layoutdesigns are properly represented. We have demonstrated thatthe C-space method can be used to support design generationin which not only designs that are optimal in terms ofcooling performance are generated, but also, the designs aremanufacturable. This new method overcomes the limitation ofrelying on specific heuristics to generate the layout design,as in our previous method 13,14. This C-space method canalso be used as a stand-alone system to support interactivelayout design. It allows a designer to explore design alternativesinteractively without having to check for interference betweenthe cooling system and the other mould insert components.The focus of this research is on the geometric aspect of thecooling system design. It is understood that other parameters,such as the coolant flow rate, cooling time, packing time,ejection time, etc. need to be considered as well when designinga cooling system. One possible approach to take all theseparameters into account is to integrate the C-space methodwith a more sophisticated GA such as the one reported in 8.Further investigation on this approach is needed and otherfurther research directions include the extension of the C-spacemethod to deal with topology changes in the cooling systemand specific design constraints, such as various geometry andtopology constraints specified between selected channels in thepreliminary design.AcknowledgementThe work described in this paper was fully supported bya Strategic Research Grant from the City University of HongKong (Project No. 7001775).AppendixLemma 1. Given a region Rnand a point pnin space Sn. Ifpn Rn, thenPROJSm(pn) PROJSm(Rn).348C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349Fig. 10. A comparison of the two layout designs using CAE mould cooling analysis.Lemma 2. Given two regions Rnand R0nin spa