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用配置空间的方法对注塑模冷却系统进行设计c.g.李， c.l.李* 香港城市大学制造工程及工程管理部，香港2007年5月3日收到; 2007年11月18日接纳摘要 注塑模的冷却系统对注射模具的成型过程和塑料零件质量影响是非常重要的。尽管已有各种针对冷却系统的分析、优化和制作的研究，但冷却系统的布局设计方面并没有得到很好的发展。在规划设计阶段，我们主要关注的是冷却系统的可行性和其他模具组件插入是否发生干预。本文介绍了利用配置空间（C空间）的方法来解决这一重要问题。然而高维配置空间方法一般需要处理一个如冷却系统般复杂的系统，冷却系统的特殊特点设计目前正在探索研究中，利用C空间在三维空间或更低维空间计算和存储的特别技术也在发展中。这种新方法是由作者对以前启发式方法的改善，因为C空间的代表性能使自动布局设计系统在所有可行的设计中进行更系统的搜索。自动生成候选布局设计的一个简单的遗传算法是C空间代表性的实施和综合。遗传算法所产生的设计实例，给这种方法提供了可行性证明。 c 2007 Elsevier公司有限公司，保留所有权利。关键词: 冷却系统设计；注塑模具；配置空间的方法1.导言 注塑模的冷却系统对注射模具的成型过程和塑料零件质量影响是非常重要的。大量涉及对冷却系统分析 1,2 ，及商业CAE系统，如Moldflow 3 和moldex3d 4 的研究被广泛应用于工业。以优化某一特定的冷却系统的研究技术亦已报道 5-8 。最近，通过使用新形式的制造技术以建立更好的冷却系统的研究已被报告。徐等人 9 报道了他们的模具意念：保持一定距离的冷却水道的设计和制作。孙等人 10,11 用数控铣床铣削生产U形槽冷却渠道和俞 12 提出了一个棚架形冷却结构的设计。尽管各种研究的重点主要集中在冷却系统的初步设计过程中冷却系统的功能实现问题，布局设计阶段过程中没有得到很好发展的冷却系统的可行性和可制造性设计问题。关注的重点主要是：在初步设计阶段冷却系统的可行性且与其他的模具部件是否干预。如图1所示 。从中可以看到注塑模的各子系统许多不同的组成部分，如喷射器的管脚，滑块等等，都必须装入模具中。为每个回路冷却水道寻找最佳位置以优化冷却性能并避免与其他组件干扰不是一项简单的任务。另一个让规划布局设计更复杂的问题是，单独的冷却水道需要和出水道和进水道连接而形成一条环形水道。因此，改变一条水道的位置，其他水道可能也需要改变。 在图 2所示 。优化冷却系统的每个水道的理想位置都如图2（a）所示 。假设当冷却系统及其他模具组件都装入模具内部时，模具组件O1和水道C1是干扰的。因为C1与其他组件可能的干扰而无法移到附近的一个位置，它必须被缩短长度。因此， 通过移动C2和延长C3使他们保持连接，如图2（b）所示。基于其新的长度， C3又与其他模具组件O2发生干扰，进一步修改是必要的，最后的设计结果如图2（c）所示 。鉴于一个典型的注塑模具可能有10条以上的冷却水道，每个水道与其他模具组件都可能存在着潜在的干扰，手工找出一个优化布置设计是非常繁琐的。 本文介绍了一种在设计过程中支持自动布局的新技术。对于这种新技术，配置空间（C空间）的方法是用来在所有可行的设计中提供一个简洁的有代表性的布图设计。C空间的代表性是通过利用解决布局设计问题这个特殊特点的有效方法构建的，而不是采用启发式规则来生成的布局设计，这就就好比以前作者开发的自动布局设计系统 13,14 ，这个新的C空间方法能使自动布局设计系统在所有可行的布图设计中进行更系统的搜索。2.配置空间的方法 一般来说， 一个系统的C空间是当该系统的每个自由度被视为一个层面的结果而导致的空间。配置空间中的区域被标记为堵塞区域或自由区域。在自由地区的点对应于组件间没有相互干扰的系统的有效配置。在被堵塞区域的点对应于组件间相互干扰的系统的无效配置。 C空间最初被洛萨诺-佩雷斯定形 15 以解决机器人路径规划的问题和关于这方面的研究一项调查已被明智和鲍耶 16 报道 。C空间的方法也被用来解决定性推理方面的问题（例如， 17,18 ）和运动装置的自动化分析与设计（例如， 19-21 ） 。作者在由多个国家组成的自动设计机构做研究时 22 ， 23日研究了一种C空间的方法。 （a） 冷却水道C1 和模具组件 （b）c1截短，c2移动，c3延长O1干扰发生在理想的位置 （c）c3移动，c2截短从而效果最佳图3冷却系统的自由度2.1一个冷却系统的C空间 一个高维C空间可以用来表示给定的某一冷却系统的初步设计中所有可行的布图设计。图3给出了一个例子。冷却系统的初步设计由4冷却水道组成。从初步设计中生成一个布局设计，渠道的中心和长度需要被调整。正如图3所示，该水道c 1的中心可沿着X1 和X 2方向移动，其长度可以沿X 3 方向调整。同样地，C2长度的可以沿X 4方向调整，而其中心可以按X1 和X 3所描述的调整 ，因此必须与调整C 1保持连接性的情况相同。通过运用类似的观点对其他水道，可以看出，冷却系统有5个自由度，它们都是标注为Xi，i= 1 ， 2 ， 5 。原则上， C空间是一个五维空间而这个空间的自由区域中的任何一点都给定了一个对应的坐标值在X i轴上，可以用来界定渠道的几何位置且没有与其他模具组件造成干扰。在一个冷却系统的高维C空间中确定一个自由区域，第一步是在独立水道的C空间中构建自由区域。2.2 独立水道的c空间构造 当一个独立的水道c1被确定为单独时，它有三个自由度，则X 1和X 2为其中心位置而X 3是它的长度。因为理想的中心位置和长度已经在初步设计中指明，因此假定一个固定的允许最大变化量 C为X1 ，X2 ，X3是合理的。c1水道的C空间中最初确定的自由区域，是一个尺寸为ccC的三维立方体。为避免与模具组件oi发生任何可能的干扰当水道通过钻孔插入模具内部时，钻头直径D和沿X3的钻孔深度必须考虑。假设直径D ，Oi开始时用D/2 +M对于O 抵销，其中M是水道内壁和附近的一个组件间所允许的最短距离。Oi的增长有效的减少了水道Ci的长度对于直线Li来说 。以图4为例子 。图4（a）表明了水道Ci和三模具组件O1、O2、O3可能会与Ci发生干扰。图4（b）显示了模具组件O , O , O 和O 的偏移及 Ci相对于线段Li的减少量与Ci的x值相符情况。如果Li和模具其他组成部分没有交汇点，那么，原来的水道Ci将不会与模具组件相交。 （a） 水道Ci和模具的 （b）模具组件和Ci相对Li的偏移三个组件 （c） 模具组件和Ci相对Pi点的补偿 （d）Ci的自由度 （e）Oi相对Pi的减少量 （f）Ci的自由点Fri 图4在一个通道CI自由区FRi施工的主要步骤水道是通过钻孔从对模具的表面插入的，任何如Oi的障碍以及钻孔深度将会影响水道的构建。钻孔深度及Oi的补偿O沿钻孔的方向延伸，直到钻到模具对应的另一面生成水道为止。Oi相对 Pi沿直线Li的减少至Li的终点。如图4（c）所示，如果点Pi位于Oi之外 ，沿Li钻孔产生水道Ci是可行的。水道Ci的自由区域Fri用如下方法取得。首先，初始自由地区Bi是用如图4（d）所示的Pi点作为中心构建的。然后插入与模具交叉取得B 0 。 B 0代表Ci所有可能的变化当仅考虑插入的模具几何形状时。然后Fri是从所有障碍的Oi中减去Bi获得。图4（e）和（f）显示了这种减法以及这种例子的结果FRi。2.3 基本接近法构建冷却系统的C空间 在一个冷却系统的C空间中确定自由区域FRF，每个冷却水道的自由区域必须以一个适当的方式“交叉”，以使障碍的效果能恰当的通过FRF描绘对于所有水道来说。然而在两个不同水道之间的自由区域的标准布尔交叉口无法执行，因为他们的C空间在一般跨距于不同的轴线。以图3为例子 ，C1和C2的C空间分别为（ X1 ， X2 ，X 3 ）和（X 1 ，X3 ，X4 ）。为了更方便在不同的C 空间中的自由区域之间确定交叉口，从一个渠道和另一个渠道的C空间中推算一个地区是必要的。以下批注首先介绍了并将用于随后的讨论和其余的文件。标记法用于描述高维空间S n是指一个通过坐标定义的n维空间 = X 1, X 2, . . . , X n.Sn是指一个通过坐标定义的m维空间= X , X , . . . , X . Pn 是指在Sn 的一个点 p n = (x 1, x 2, . . . , x n) Rn属于区间S n(R n S n) 标记法用于描述冷却系统n c指在冷却系统中水道的数目。 n f指冷却系统总的自由度。 ci指冷却系统第i个水道。 s i指Ci的C空间。FRi是指在Si中的自由地区。也就是说，它是独立水道Ci的自由区域。 SF指冷却系统的C空间。 FRF是指SF中的自由区域。也就是说，它是冷却系统的自由区域。假设Pn在Sn中，Pm在Sm中，图5（a）用一唯和三唯的的空间点明了突出的例子(i) (ii) ；而(iii) ， 且对(i)Pn 和Pm的坐标是一样的如果Sn和Sm在同一区间时。对(ii)和(iii)Pn在区间Rm中。因为Pm在Rm中，当点位于Sn和Sm中时Pn等于Pm。而对另一坐标Pm其可以是任意值；特别对(ii)和(iii)，假设水道Cn和Cm，因为它们相近所以必须连接。这样它们的C空间Sn、Sm有相同的坐标值。假设那是一个结论？对应到在S n中一个点P n已选定为Cn。保持连通性，结论呢？ Cm必须被选择在以使Sm中的相应点Pm与P n共用相同坐标在共同的轴线。这意味着Pm和PN可以是任何点在区间Sm中，该方法已经在前面予以定义。在区域Sn和Sm中的一区域Rn是Rn和Sm中每一点的简化。图5（ b ）说明了相应的区域。投影的正式定义如下面所示。定义1 （投影）1.1.如果X mX n, PROJ Sm ( pn )是一个点=(x,x,x),因为X = Xj, x = xj因为i 1,m。为了在随后的讨论中简化符号，这一投影是被视为单独点Pm的区间。也即是PROJ Sm ( pn )=Pm。. 1.2.如果X m X n，PROJ Sm ( pn )是一个区间 pm |PROJ Sn ( pm ) = pn .1.3.如果X m Xn , X n Xm ,并且 X n X m , PROJ Sm( pn )是一个区间Rm = pm|PROJ SI( pm ) = PROJ SI( pn ),其中Si位于区间X n X m ，如果 n X m =，PROJ Sm( pn )则定义为Sm。1.4.ROJSm(Rn) 定义在区间Rm=Pm|PmPROJ(Pn),PnRn.正如在2.1节所讨论的，在FR中的任意点P为冷却系统的每个自由度给定了一个值，使水道与其他模具组件在几何空间是不会发生任何干涉。另一方面， P相对每个点s i的投影是，在Ci的每个自由区域FR中。因此，FR定义如下。定义2 （一个冷却系统C空间的自由区域）FRF = pF | P R O JSi ( pF ) FRi , i 1, nC 图 5 点和区间在Sn至Sm区间中的投影。根据定义1.1知道， 从到的区间投影始终只包含一个单一的点，因为跨距s i始终是s n一个子轴线. Ci的每一个自由区域FR的构造，已经在第2.2节中解释。从FR中找出FRF，下面的定理是很有用的。 定理1 . 这定理很直观表明为找出，所有的FR首先投影到冷却系统的C空间. 可以从投影的布尔交叉口得到。定理1的证明和所用的引理，都已在附录中标出。2.4.C -空间的表示和计算为了表示自由区域和便于在一个高维空间的区域布尔交叉口之间的计算，我们可以利用类似 21,24 中的一种细胞枚举法。基本思路是用一高维立方体在中逐渐靠近一高维区间。每个立方体是通过对每个轴指定间隔来确定的。两个区间的交汇点是通过两个立方块交汇点所取得的。两个高维立方体的交叉点只不过是在每个轴的立方体之间间隔的普通交叉点。 假设每个FR是近似由m个三维立方体组成，投影PROJ S（FR）便可近似由维立方体组成。使用定理1对的构建，需要在n-三维立方体中交叉，是用一个n-三维立方体只中的最大值表示。虽然用来代表交叉点中间结果的立方体的数量和 可通过特殊技术减少，可以预料到记忆和计算的要求仍然是这种方法的主要问题。在下一节中将介绍一种更先进的方法。（二）在配置空间Si中每个水道的自由区域。（一）一个拥有四个水道和四个自由度的简单冷却系统3.C空间构建的一种有效率技术对的表示和构建时为了避免高的内存和计算的要求，我们选择不表示和不计算。相反，我们专注于对每一独立水道的C -空间计算过程是否有效的技术。首先，我们看显示在图6的简化设计例子 。假设在这个例子中模具沿z方向插入时在FR中不存在变异，那么冷却系统有四个如图6（ a ）所示的自由度。每个水道的Si是两维和假设的FR如图6（ b ）所示。为水道考虑一个简单的设计方法。首先，点可以从FR中选择，以使不会和任何障碍发生干涉。然而，由X 1和X 2确定 ，而X2在S 2中 。因此那些在S 2中的障碍所施加的约束，还必须考虑。为了找出设计的所有可行点，是与 “交叉”。这个“交叉点”结果如图6（ c ）所示，这是通过移动区间x 2 6得到的 ，因为该自由区域， 2 6 ， 10 。现在，如图6（ c ）所以示给定一个与任何障碍不发生干涉的水道，并在其自由区间的任何一点的选定，始终为C 2存在着这样一种设计：例如，它可以连接到（他们都有一个共同的 2值）并和任何障碍不发生干涉。然而，这个简单方法的一个主要问题是在为C 1和C 2进行有效的设计时并不保证冷却系统其他水道存在有效的设计。例如，如果一个点选定如图6（d）所示，则 2 8 ,10 ,那么由， 3 6 ，8 ，在并没有有效点和在这个区间。 上述例证表明，在为水道设计时，只考虑与相邻并有一个共同轴的的自由区域和是不恰当的。事实上，其他所有的都必须加以考虑，尽管他们的C 空间并没有共同轴和（且他们也不和C 1相邻 ），因为组成冷却系统的冷却水道是相接的。一个自由度的选择会影响冷却系统另一自由度的选择。 为每一个独立水道的C空间发展一个设计的过程，主要关注的是：在一个水道C的空间选择一个点后，必须始终存在和所有其他s i相应的点，以使所有的水道可以连接到一起形成一个有效的冷却系统。为解决这一问题，到每个量s i的投影是必要的。 （c）在与相交以后的自由区间 （d）为C1和C2设计的一个有效点P1使C4成为无效的设计。 图6定义3 。定义为到投影 = PRO ()显然，对在选定的任何点，始终存在着相应的点在中 ，因为和都是点在的投影，在中选中的任何点，很明显总是有一些相应的设计对应其他所有的渠道以使这些水道可以连接在一起形成一个有效的冷却系统。因此，为了保证冷却系统能有效的设计，的构建是很重要的。根据定理3，为到投影。然而，如在第2.4节所讨论的，我们并不想构建基于大容量空间和繁琐计算要求。另一种可供选择的更有效的方法是直接构建。而不是作用在高维空间，这个方法通过一个工作在空间三维或更少维数的序列运行来建构。该方法正式介绍之前，在图6所举的例子再次被使用来说明这种方法的基本概念。为了开始一个设计过程，在的点P 1 =（ 1， 2 ）首先被选择如图7所示 。因为 有一点在中 ，必须有一个值，以使我们可以找到=（ 2 ， 3 ）在。又有一个坐标在，坐标必须有一个值，以使我们可以找到=（ 3 ， 4 ）在 。此外，因为在有和 ，=（，）必须在。图7显示了为水道构建一个有效设计的点、和的顺序。 上述例子显示，为了在代表所有的有效设计的中确定有效的区间，自由区域应首先考虑。的影响应该可以 “促使”以确定有效的区间在中，然后是，最后是。在的有效区域产生的结果包括、的所有影响。为达到这一目的，组合的运作正式被界定。定义4 （组成）对于在一个冷却系统里的两个相邻水道和，他们从到的自由区域的组合，标注为，而他们从到自由区域的组合，标注为，定义如下： (b)FRi每个通道的自由地区Si的配置空间 图6冷却系统设计的一个简化的例子对于冷却系统一个水道Ci序列的构成， 从到自由区域的组成，标注为，定义如下文。如果如果如果图8显示了促使 构建的组合序列。第一步是要构建，就像图8（a）所示这已被给定在=PROJ(FR)FR， 。然后如在图8（b）所示CR的构建由公式CR=PROJ()FR得。最后，CR，由CR=PROJ( CR)FR。如图8（c）所示。从图8（c）很明显的得出，CR对组成冷却系统的所有水道的自由区域存在着影响。因此，对于CR中的任意一点，可以保证冷却系统的一个有效设计可以被构造。 通过组合序列的运用，一个有效的设计可以通过在每个中选择点获得。在其他所有水道的自由区域已经组合到中时。不过，我们也想确保没有将有效的设计从自由区域中排除，当组合序列被应用以后。否则，有些可能提供更佳的冷却性能的有效设计将不能用这个方法得到。以C的设计为例，图8（c）的CR不仅仅代表着C一部份有效设计，而且代表着C所有的有效设计，这对C来说尤为重要。为了解决这一问题，我们提出以下定理：应用水道C的一个序列C,i1,到冷却系统。定理2 定理2说明代表水道C所有有效的设计PR，可以通过和之间的一个布尔交点得到。这定理的一个重要特点是PR可以在三维立体空间中计算得到，因和都在S中，所以交点在S中。此外和也可以通过在中的区间相交得到。这样，PR可以通过在三维立体空间的序列得到。如果在第2.4节中的假设说明再次被使用，即是说如果每个通过M个三维立方体近似得到，那么和PR也可以用M个三维立方体表示。所以，nm所有的三维立方体需要代表所有的PR。因此可以证明三维立方体之间的交点O需要产生所有的PR。因此，使用定理2可以防止在高维空间存储区域的需要，并可以避免高容量和繁琐计算的要求如在定理1所证明的。图8 CR构建所用的序列以下给出了定理2的证明 。它由两部分组成：该引理中所使用的证明如附录所示。 3.1定理2证明（1） 为了证明：（i） 由p因为p 和 有相同的坐标在和用同样的方法，我们可以确定一点以使和具有相同的坐标在和。 使用这种方法，我们也可以确定一系列点，k1,i -1,以使，那么和具有相同的坐标在轴线和。（ii） （b）由PROJ()构建 用类似的方法，我们可以确定另一系列点，ki+1,以使，那么和具有相同的坐标在轴线和。由（i）及（ii）知，我们确定了一系列的点，k1,以使，在连续的任何两个相邻的点具有相同的坐标在他们的共同轴线。对于由一系列冷却水道构成的冷却系统，在两相邻水道和的C空间和总是存在着一些共同的轴线由于它们之间的空间联系。此外，如果在和的C空间有一个公共轴，也必须存在于和间所有水道的C 空间。所以，由上述方法构建的一系列点，k1,将为的每个轴提供唯一的坐标。令为由坐标构建的点。很明显： （c）由PROJ()构建 用类似的方法，可以得到： 初始设计给定一个为冷却系统指定一系列水道和他们理想几何尺寸的初步设计，第一步是为每个水道建构一个。然后，每个水道的可以通过应用定理2的组合操作得到。为冷却系统产生初始设计的一个方法是，是要从中选出一套坐标。为了简化解释，假设每个水道词拥有自由度和，而和相邻的水道有着相同的坐标。为了生成一个设计，在的点（，）必须被选择。然后，点被选择为了让（，）在中。此选择4候选设计产生由于冷却系统初始设计对水道系列和它们的理想几何结构进行了具体化，第一步要做的是为每个水道建立FRi，然后通过将复合应用应用到定理2中得到每个水道的PRi。一个产生冷却系统候选设计的方法是从如后PRi系列中选出坐标系。为简化阐述，假设每个水道C的自由度为和，被邻近水道共用。为得到一个设计，选择了PR1中的一个点（X1，X2），然后，选择一个X3使（X3，X2）在PR2内。这个选择过程在下一个水道PR坐标中重复，直到确定所有的自由度时停止。此方法的一个重要的特点是在一个步进中无论坐标值如何选取，后续步骤中总存在一个下一坐标可选有效值。5应用源运算法则的自动化设计过程 为测试C-空间方法在支持自动化布局设计过程时的可行性，在C-空间建立项目中插入与应用了一个简单源运算法则（GA）25。在实施GA时候用到了一个简单的染色体结构，它由一系列nF真值g1g2gnF组成，其中gi的真值在01之间，nF冷却系统的自由度。为得带一个形状设计，用到了前面部分提到的方法和应用g作为一个百分比值来选择坐标。例如，中坐标的有效值的在区间和，其中，就得的选取值为，（也就是在第一区间）否则就设置为（也就是在第二区间内）一个单点交叉操作，一个转化操作和转迹线轮选择方法26被用于GA过程中。之前研究中提到的模糊记值方法13,14对相对于机构的候选设计的适合性进行快速评定。必须注意的是在在GA过程开始之前，建立起每个水道的，经过一次建立得到，因此不会影响GA演变过程的计算时间。下一部分给出了一些由GA过程得到的布局设计实例。6.实例研究 图9（a）显示出了实例部分的2个观察结果。图9（b）显示了当只考虑系统冷却效果时，具体给出每个冷却水道的理想位置的冷却系统的初始设计。（为了便于表征，只给出了行腔部分冷却系统的图示）。在理想位置上，水稻C5和模具组成发生干涉现象。用提出的方法进行布局设计，自动化，就建立起了每个水道的。例如，图9（g）和（h）显示了水道的和。值得注意的是是通过将和其他复合得到，因此是亚设置，如数据明显指出。在所有的计算完成之后，GA过程开始调用，图9（j）显示了演变过程中得到的初始设计最大适合值。最大适合值在产生值接近600时开始收敛。如图9（c）所示，冷却系统由15个自由度组成，他们的值在表1中列出。叫“初始设计”的行显示初始设计的值。下一行显示设计1的值，它是GA过程在1000生产后得到最好的设计。如表中明显之处，涉及1通过减小1.21mm得到。图9（d）显示设计1，这个调整对应于沿着Z方向减小以消除和之间的干涉。这个调整对水道和到也适用。表1也显示设计1中所有其它的值都保持在规定初始至0.2mm误差以内。为更好的表征C-空间方法，模成分沿着Y方向移动同相截，如图9（e）所示。这个新障碍增加了自由区域的约束以至于方向体移动性受到很大限制。这个效应在更新中显示出来，如图9（i）所示，其中只有的上部分在图9（h）中显示出来。以所有水道新的再次调用GA过程以获得设计2。适合值在图9（k）中显示。值得注意的是最佳适合值比设计1中获得的要小。这很合理，因为约束的增加，偏移量与真实值的差距很大。又GA过程获得的值在表1的最后一行中显示出来。如表中所示，调整5mm以清除同的干涉。这同沿Z方向移动水道到相对应。现在和截面不能通过调整使其光亮。而调整和，相应地将沿-Y方向移动2.94mm，沿-X方向移动6.22mm如图9（e）所示。为保持连结性，和也作相应的调整。设计2显示，当一个水道的约束数（如）变化时，提出C-空间方法很好的将这个效应传播到其它水道（如和）中去，以至于所有这些水道的可行设计组得到相应的调整。C-模型冷却分析用于分析设计得到的布局图。从图10（a）到（d）可见，两个设计中，冷却时间为20s时，最高模-壁温度在以上。它们的最大温度偏差小于，这表明两种情形下，提出的方法能够得到满意的设计布局。从图10（c）和（d）观察得到，同设计2比较，涉及1中工件大部分没有产生变色。这表明在设计1中很多工件的温度偏差在以内。这是因为在设计2中，随着空腔中的水道向模压移动了5mm，冷却效果变得不均匀，这表示当施加很多约束时，保持初始理想冷却效果很困难。它也解释了为什么设计2的最大适切性稍微小于设计1的最大适切性。 （a）示例零件（b）冷却系统的初始设计（a）冷却系统的15个自由度（b）设计1（b）移动和相交（b） 设计2图9 分层设计表1.冷却系统的自由度7讨论与结论在执行C-空间方法中，一个单元列举方案被用于简化这个方法的执行，在目前的执行中，C-空间一维分辨率为0.15mm。对冷却系统设这个分辨率是足够的，因为对一个好的调整，如0.01mm，冷却系统的功能变化是很难发现的，然而，该研究中所发展的理论与方法并不局限于相应的表现项目。实际上，基于理论2的方法，所有C-空间计算和存储都在3维空间内完成，因此标准校核模型技巧可以应用。 该研究的一个主要贡献是发展了一个特别的支持布局设计的C-空间方法。应用这个C-空间方法，所有的可行布局设计很好的被显示出来。同时我们得出了该方法不仅可以用于冷却系统设计的优化设计支持，还可以用于生产制造。该方法克服特殊启发产生布局设计的局限，如前面的方法13，14。这个C-空间方法能够独立作为一个系统去支持互动布局设计。它使设计者在不用检查冷却系统截面和其它模型插件能够开发出设计方法。该研究主要目的集中在冷却系统设计的几何形状构成方面。在设计冷却系统时，其它参数如冷流率，冷却时间，包装时间，挤出时间都需要被考虑进来。一个可行的方法就是将这些所有参数进行考虑插入配备更复杂的GA的C-空间方法，如8报道所示。需要对该方法进一步研究，其他研究方向包括C-空间方法处理冷却系统拓扑变化和具体设计约束，如初始设计选择水道之间的变化几何形状和拓扑约束扽。鸣谢该文章中所完成的工作得到香港城市大学战略研究部（项目No.7001775）的大力支持。（a）设计1的模具温度 （b）设计2的模具温度（c）设计1零件的不同温度 （d）设计2零件的不同温度图10。用CAE模具冷却分析系统比较这两个布图设计引理2在中给定两个区间和。如果，那么引理3 在中给定，那么引理4 在中给定任意两个和。如果，则它们对的投影满足：引理5 给定两个区间和满足。则在中的区间满足：引理6 给定三个区间、和满足和。则在中的区间满足：引理7 引理8 给定两个区间和满足，其中点在中，点在中，如果 那么： 定理1参考文献： 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 thepr