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Computer-Aided Design 40 (2008) space C.L. producti moulded 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 feasibility of building the cooling system inside the mould insert without interfering with the other mould components. This paper reports a configuration space (C-space) method to address this important issue. While a high-dimensional C-space is generally required to deal with a complex system such as a cooling system, the special characteristics of cooling system design are exploited in the present study, and special techniques that allow C-space computation and storage in three-dimensional or lower dimension are developed. This new method is an improvement on the heuristic method developed previously by the authors, because the C-space representation enables an automatic layout design system to conduct a more systematic search among all of the feasible designs. A simple genetic algorithm is implemented and integrated with the C-space representation to automatically generate candidate layout designs. Design examples generated by the genetic algorithm are given to demonstrate the feasibility of the method. c? 2007 Elsevier Ltd. All rights reserved. Keywords: Cooling system design; Plastic injection mould; Configuration space method 1. Introduction The cooling system of an injection mould is very important to the productivity of the injection moulding process and the quality of the moulded part. Extensive research has been conducted into the analysis of cooling systems [1,2], and commercial CAE systems such as MOLDFLOW [3] and Moldex3D [4] are widely used in the industry. Research into techniques to optimize a given cooling system has also been reported [5–8]. Recently, methods to build better cooling systems by using new forms of fabrication technology have been reported. Xu et al. [9] reported the design and fabrication of conformal cooling channels that maintain a constant distance from the mould impression. Sun et al. [10,11] used CNC Despitethevariousresearcheffortsthathavefocusedmainly on the preliminary design phase of the cooling system design process in which the major concern is the performance of the cooling function of the system, support for the layout design phase in which the feasibility and manufacturability of the cooling system design are addressed has not been well developed. A major concern in the layout design phase is the feasibility of building the cooling system inside the mould insert without interfering with the other mould components. Consider the example shown in Fig. 1. It can be seen that many different components of the various subsystems of the injection mould, such as ejector pins, slides, sub-inserts, and so forth, have to be packed into the mould insert. Finding the best location for each channel of the cooling circuit to optimize Plastic injection mould cooling configuration C.G. Li, Department of Manufacturing Engineering and Engineering Received 3 May 2007; accepted Abstract The cooling system of an injection mould is very important to the milling to produce U-shaped milled grooves for cooling channels and Yu [12] proposed a scaffolding structure for the design of conformal cooling. ? Corresponding author. E-mail address: meclli@cityu.edu.hk (C.L. Li). 0010-4485/$ - see front matter c? 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2007.11.010 334–349 system design by the method Li? Management, City University of Hong Kong, Hong Kong 18 November 2007 vity of the injection moulding process and the quality of the the cooling performance of the cooling system and to avoid interference with the other components is not a simple task. Another issue that further complicates the layout design problem is that the individual cooling channels need to be connected to form a path that connects between the inlet and the outlet. Therefore, changing the location of a channel may 335 Fig.1. Thecoolingsystem components. require changing the example shown in to optimize the cooling in Fig. 2(a). Assume other mould components mould component As C1 cannot be mo interference with other C2 is moved and C connectivity, as sho C3 is found to interfere mould components, is very tedious. that supports the this new technique, used to provide a layout designs. The an efficient method the layout design to generate layout system developed w C-space method to conduct a more layout designs. is the space that system is treated the configuration free region. Points of the the components correspond to of the system initially formalized planning problems shortened and further modification is needed, which results in the final design shown in Fig. 2(c). Given that a typical injection mould may have more than ten cooling channels, with each channel (a) Interference occurs between cooling channel C1 and mould component O1 at the ideal location of C1. (c) C3 is moved and C2 is design. Fig. 2. An example showing the tediousness and a survey in this area of research has been reported by Wise and Bowyer [16]. The C-space method has also been used to solve problems in qualitative reasoning (e.g., [17,18]) (b) Channel C1 is shortened, C2 is moved, and C3 is elongated. to give the final C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334–349 insideamouldinsertpackedwithmanyothermould other channels as well. Consider the Fig. 2. The ideal location of each channel performance of the system is shown that when the cooling system and the are built into the mould insert, a O1 is found to interfere with channel C1. ved to a nearby location due to the possible components, it is shortened. As a result, 3 is elongated accordingly to maintain the wn in Fig. 2(b). Owing to its new length, with another mould component, O2, potentially interfering with a few other finding an optimal layout design manually This paper reports a new technique automation of the layout design process. In a configuration space (C-space) method is concise representation of all of the feasible C-space representation is constructed by that exploits the special characteristics of problem. Instead of using heuristic rules designs, as in the automatic layout design previously by the authors [13,14], this ne enables an automatic layout design system systematic search among all of the feasible 2. The configuration space method In general, the C-space of a system results when each degree of freedom of that as a dimension of the space. Regions in space are labeled as blocked region or in the free regions correspond to valid configurations system where there is no interference between of the system. Points in the blocked regions invalid configurations where the components interfere with one another. C-space was by Lozano-Perez [15] to solve robot path of the layout design process. 336 and (e.g., automatic 23 2.1. the y c 3 se (e) a cooling system. Fig. 3 gives an example. The preliminary design of this cooling system consists of four cooling channels. To generate a layout design from the preliminary design, the centers and lengths of the channels are adjusted. As shown in Fig. 3, the center of channel C1 can be moved along the X1 and X2 directions, and its length can be adjusted along the X3 direction. Similarly, the length of C2 can be adjusted along the X4 direction, while its center adjustment is described by X1 and X3 and thus must be the same as the adjustment of C1 to maintain the connectivity. By applying similar arguments to the other channels, it can be seen that the cooling system has 5 (a) Channel Ci and three mould components inside the mould insert. (b) Offsets of the mould Ci represented by line (d) The initial free region of Ci. Fig. 4. The major steps in the construction considered. To account for the diameter D, Oi is first offset by D/2 + M to give Oprimei, where M is the minimum allowable distance between the channel wall and the face of a component. This growing of Oi in effect reduces channel Ci to a line Li. Consider the example illustrated in Fig. 4. Fig. 4(a) shows a channel Ci and three mould components, O1, O2, and O3, that may interfere with Ci. Fig. 4(b) shows the offsets Oprime1, Oprime2, and Oprime3 of the mould components, and the reduction of Ci to a line segment Li that is coincident with the axis of Ci. If there is no intersection between Li and the offsets of the mould components, then the original channel Ci will not intersect with components and gment Li. (c) Sweeping the offsets of the mould components and Ci represented by point Pi. Subtracting Oprimeprimei from Bprimei. (f) The free region FRi of Ci. C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334–349 Fig. 3. An example showing the degrees of freedom of a cooling system. the analysis and design automation of kinematic devices [19–21]).TheauthorinvestigatedaC-spacemethodinthe design synthesis of multiple-state mechanisms [22, ] in previous research. C-space of a cooling system A high-dimensional C-space can be used to represent all of feasible layout designs of a given preliminary design of degreesoffreedom,andtheyaredenotedas Xi,i = 1,2,...,5. In principle, the C-space is a five-dimensional space and an point in the free region of this space gives a set of coordinate values on the Xi axes that can be used to define the geometry of the channels without causing interference with the other mould components.Todeterminethefreeregioninahigh-dimensional C-spaceofacoolingsystem,thefirststepistoconstructthefree regions in the C-spaces of the individual channels. 2.2. C-space construction of individual cooling channels When an individual channel Ci is considered alone, it has three degrees of freedom, say X1 and X2 for its center location and X3 for its length. As the ideal center location and length have already been specified in the preliminary design, it is reasonable to assume a fixed maximum allowable variation? for X1, X2, and X3. The initial free region in the C-space of channel Ci is thus a three-dimensional cube Bi with the dimensions?c ×?c ×?c. To avoid any possible interference with a mould component Oi when channel Ci is built into the mould insert by drilling, a drilling diameter D and drilling depth along X have to be of the free region FRi of a channel Ci. C.G. Li, C.L. Li / Computer-Aided the mould components. This growing or offset of an obstacle is a standard technique in the C-space method [15]. A channel is formed by drilling from a face of the mould insert, and any obstacle Oi within the drilling depth will affect the construction of the channel. To account for the drilling depth, the offset Oprimei of Oi is swept along the drilling direction until the opposite face of the mould insert is reached to generate Oprimeprimei . This sweeping of Oprimei in effect reduces line Li to a point Pi located at the end of Li. As shown in Fig. 4(c), if the point Pi is outside Oprimeprimei , the drilling along Li to produce Ci is feasible. The free region FRi of channel Ci is obtained as follows. First, the initial free region Bi is constructed with its center at Pi as shown in Fig. 4(d). Bi then intersects with the mould insert to obtain Bprimei. Bprimei represents all of the possible variations of Ci when only the geometric shape of the mould insert is considered. Then, FRi is obtained by subtracting from Bprimei the Oprimeprimei of all of the obstacles. Fig. 4(e) and (f) show the subtraction and the resulting FRi of the example. 2.3. Basic approach to the construction of the C-space of cooling system To determine the free region FRF in the C-space of a cooling system, the free regions of each cooling channel have to be “intersected” in a proper manner so that the effect of the obstacles to all of the channels are properly represented by FRF. However, the standard Boolean intersection between the free regions of two different channels cannot be performed because their C-spaces are in general spanned by different sets of axes. Referring to the example in Fig. 3, the C-spaces of C1 and C2 are spanned by {X1, X2, X3} and {X1, X3, X4}, respectively. To facilitate the intersection between free regions in different C-spaces, the projection of a region from the C- space of one channel to that of another channel is needed. The following notations are first introduced and will be used in the subsequent discussions on projections and the rest of the paper. Notations used in describing high-dimensional spaces Sn denotes an n-dimensional space spanned by the set of axes ˉXn = {X1, X2,..., Xn}. Sm denotes an m-dimensional space spanned by the set of axes ˉXm = {Xprime1, Xprime2,..., Xprimem}. pn denotes a point in Sn and pn = (x1,x2,...,xn), where xi denotes a coordinate on the ith axis Xi. Rn denotes a region in Sn(Rn ? Sn). Rn is a set of points in Sn. PROJSm(pn) denotes the projection of a point pn from Sn to Sm. PROJSm(Rn) denotes the projection of a region Rn from Sn to Sm. Notations used in describing a cooling system nC denotes the number of channels in the cooling system. nF denotes the total degrees of freedom of the cooling system. Ci denotes the ith channel of the cooling system. Si denotes the C-space of Ci. Design 40 (2008) 334–349 337 FRi denotes the free region in Si. That is, it is the free region of an individual channel Ci. SF denotes the C-space of the cooling system. FRF denotes the free region in SF. That is, it is the free region of the cooling system. Consider the projection of a point pn in Sn to a point pm in Sm. Fig. 5(a) illustrates examples of projection using spaces of one dimension to three dimensions. Projections are illustrated forthreecases:(i) ˉXm ? ˉXn;(ii) ˉXm ? ˉXn;and(iii) ˉXm negationslash? ˉXn, ˉXn negationslash? ˉXm, and ˉXn ∩ ˉXm negationslash= ?. For (i), each coordinate of pm is equal to a corresponding coordinate of pn that is on the same axis. For (ii) and (iii), the projection of pn is a region Rm. For each point pm in Rm, a coordinate of pm is equal to that of pn if that coordinate is on a common axis of Sn and Sm. For the other coordinates of pm, any value can be assigned. The reason for this specific definition of the projections, in particular, for cases (ii) and (iii), is as follows. Consider two adjacent channels Cn and Cm. As they are adjacent, they must be connected and thus their C-spacesSn and Sm share some common axes. Assume that a configuration that corresponds to a point pn in Sn has been selected for Cn. To maintain the connectivity, the configuration for Cm must be selected such that the corresponding point pm in Sm shares the same coordinates with pn on their common axes. This implies that pm can be any point within the projection of pn on Sm, where the method of projection is defined above. The projections of a region Rn in Sn to Sm are simply the projections of every point in Rn to Sm. Fig. 5(b) illustrates the region projections. The formal definition of projection is given below. Definition 1 (Projection). 1.1. If ˉXm ? ˉXn, PROJSm(pn) is a point pm = (xprime1,xprime2,...,xprimem), where for Xprimei = X j, xprimei = xj for all i ∈ [1,m]. To simplify the notations in subsequent discussion, this projection is regarded as a region that consists of the single point pm. That is, PROJSm(pn) = {pm}. 1.2. If ˉXm ? ˉXn, PROJSm(pn) is a region Rm = {pm|PROJSn(pm) = {pn}}. 1.3. If ˉXm negationslash? ˉXn, ˉXn negationslash? ˉXm, and ˉXn∩ ˉXm negationslash= ?, PROJSm(pn)is a region Rm = {pm|PROJSI (pm) = PROJSI (pn)}, where SI is the space spanned by ˉXn ∩ ˉXm. If ˉXn ∩ ˉXm = ?, PROJSm(pn) is defined as Sm. 1.4. PROJSm(Rn) is defined as the region Rm = {pm|pm ∈ PROJSm(pn), pn ∈ Rn}. As discussed in Section 2.1, any point pF in FRF gives a value for each degree of freedom of the cooling system so that the geometry of the channels is free from interference with the other mould components. In other words, the projection of pF to each Si is in the free region FRi of each Ci. Thus, FRF is defined as follows. Definition 2 (Free Region in the C-space of a Cooling System). FRF = {pF|PROJSi (pF) ? FRi,i ∈ [1,nC]} -Aided Note that according to to Si always contains only that span Si is always a subset The construction of the already been explained in the following theorem is useful. Theorem 1. FRF = nCintersectiondisplay i=1 PROJSF(FRi). Intuitively, this theorem says first projected to the C-space can then be obtained by performing among the projections. The used in the proof are given of the C-space F and to facilitate the between the regions can use a kind of cell used in [21,24]. The region RF in Each box is defined by SF. The intersection of of the two sets of high-dimensional boxes intervals of each of the by m three- OJSF(FRi) can then be boxes. The construction Fig. 5. The projections of points and regions in Sn to Sm. Definition 1.1, the projection of pF a single point because the set of axes of the axes that span Sn. free region FRi of each Ci has Section 2.2. To find FRF from FRi, that to find FRF, all of the FRi are of the cooling system SF. FRF the Boolean intersections proof of Theorem 1 and the lemmas 2.4. Representation and computation To represent the free region FR computation of the Boolean intersections in a high-dimensional space, we enumeration method similar to the one basic idea is to approximate a high-dimensional SF by a set of high-dimensional boxes. specifying an interval on each axis of two regions is achieved by the intersection boxes. The intersection between two is simply the intersection between the boxes in each axis. Assuming that each FRi is approximated dimensional boxes, the projection PR approximated by mnF-dimensional 338 C.G. Li, C.L. Li / Computer in the Appendix. Design 40 (2008) 334–349 of FRF that uses Theorem 1 then requires mnC intersections between nF-dimensional maximum of mnCnF of boxes used to represent intersections and FR is anticipated that the are still major problems improved method is 3. An efficient technique To avoid the high for the representation . Instead, we process to example shown in is assumed in this along the Z direction hasfourdegrees each channel Ci are shown in Fig. 6(b). channel C1. First, a (a) A simple cooling system with four channels and four degrees of freedom. (b) The free region FRi of each channel in its configuration space Si. Fig. 6. A simplified example of a cooling system design. boxes, and FRF is represented by a -dimensional boxes. Although the number the intermediate results of the F can be reduced by special techniques, it memory and computational requirements of this method. In the next section, an developed. for C-space construction to represent and not to compute FRF explicitly focus on a technique that enables the computational work on the C-spaces of each individual channel. First, consider the simplified design Fig. 6. For the purpose of illustration, it example that there is no variation in FRi ofthemouldinsertandthusthecoolingsystem of freedom as shown in Fig. 6(a). The Si of two dimensional and the assumed FRi are Consider a simple method for designing C.G. Li, C.L. Li / Computer-Aided memory and computational requirements and construction of FRF, we choose not Design 40 (2008) 334–349 339 point p1 can be selected from within FR1 so that C1 is free from interference with any obstacle. However, S1 is spanned -Aided continued even though their C-spaces of C1, (i.e., they are as well, because the system are connected. have an effect in the cooling system. To develop a design of each individual channels, selection of a point always exist a corresponding that all of the channels system. To address this Si is needed. Definition 3. PRi is PRi = PROJSi (FRF Obviously, for an always a correspondi FR2. Again, as p2 x3 must have a value FR3. Also, as must also be inside p1, p2, p3, and p4 C1. determine the valid designs