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Computer-Aided Design 40 (2008) 334349 Plastic injection mould cooling system design by the confi guration space method C.G. Li, C.L. Li Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong Received 3 May 2007; accepted 18 November 2007 Abstract 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. 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 confi guration 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; Confi guration 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 58. 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 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: .hk (C.L. Li). Despite the various research efforts that have focused mainly 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 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 0010-4485/$ - see front matter c ? 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2007.11.010 C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349335 Fig.1. Thecoolingsysteminsideamouldinsertpackedwithmanyothermould components. require changing the other channels as well. Consider the example shown in Fig. 2. The ideal location of each channel to optimize the cooling performance of the system is shown in Fig. 2(a). Assume that when the cooling system and the other mould components are built into the mould insert, a mould component O1is found to interfere with channel C1. As C1cannot be moved to a nearby location due to the possible interference with other components, it is shortened. As a result, C2is moved and C3is elongated accordingly to maintain the connectivity, as shown in Fig. 2(b). Owing to its new length, C3is found to interfere with another mould component, O2, and further modifi cation is needed, which results in the fi nal design shown in Fig. 2(c). Given that a typical injection mould may have more than ten cooling channels, with each channel potentially interfering with a few other mould components, fi nding an optimal layout design manually is very tedious. This paper reports a new technique that supports the automation of the layout design process. In this new technique, a confi guration space (C-space) method is used to provide a concise representation of all of the feasible layout designs. The C-space representation is constructed by an effi cient method that exploits the special characteristics of the layout design problem. Instead of using heuristic rules to generate layout designs, as in the automatic layout design system developed previously by the authors 13,14, this new C-space method enables an automatic layout design system to conduct a more systematic search among all of the feasible layout designs. 2. The confi guration space method In general, the C-space of a system is the space that results when each degree of freedom of that system is treated as a dimension of the space. Regions in the confi guration space are labeled as blocked region or free region. Points in the free regions correspond to valid confi gurations of the system where there is no interference between the components of the system. Points in the blocked regions correspond to invalid confi gurations where the components of the system interfere with one another. C-space was initially formalized by Lozano-Perez 15 to solve robot path planning problems 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) (a) Interference occurs between cooling channel C1 and mould component O1at the ideal location of C1. (b) Channel C1is shortened, C2is moved, and C3is elongated. (c) C3is moved and C2 is shortened to give the fi nal design. Fig. 2. An example showing the tediousness of the layout design process. 336C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349 Fig. 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 the automatic design synthesis of multiple-state mechanisms 22, 23 in previous research. 2.1. C-space of a cooling system A high-dimensional C-space can be used to represent all of the feasible layout designs of a given preliminary design of 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 C1can be moved along the X1 and X2directions, and its length can be adjusted along the X3 direction. Similarly, the length of C2can be adjusted along the X4direction, while its center adjustment is described by X1 and X3and thus must be the same as the adjustment of C1to maintain the connectivity. By applying similar arguments to the other channels, it can be seen that the cooling system has 5 degreesoffreedom,andtheyaredenotedas Xi,i = 1,2,.,5. In principle, the C-space is a fi ve-dimensional space and any point in the free region of this space gives a set of coordinate values on the Xi axes that can be used to defi ne the geometry of the channels without causing interference with the other mould components.Todeterminethefreeregioninahigh-dimensional C-spaceofacoolingsystem,thefi rststepistoconstructthefree regions in the C-spaces of the individual channels. 2.2. C-space construction of individual cooling channels When an individual channel Ciis considered alone, it has three degrees of freedom, say X1and X2for its center location and X3for its length. As the ideal center location and length have already been specifi ed in the preliminary design, it is reasonable to assume a fi xed maximum allowable variation c for X1, X2, and X3. The initial free region in the C-space of channel Ciis thus a three-dimensional cube Biwith the dimensions c c c. To avoid any possible interference with a mould component Oiwhen channel Ciis built into the mould insert by drilling, a drilling diameter D and drilling depth along X3have to be considered. To account for the diameter D, Oi is fi rst offset by D/2 + M to give O0 i, where M is the minimum allowable distance 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 a channel Ciand three mould components, O1, O2,and O3,that may interfere with Ci. Fig. 4(b) shows the offsets O0 1, O 0 2, and O0 3 of the mould components, and the reduction of Cito a line segment Lithat is coincident with the axis of Ci. If there is no intersection between Liand the offsets of the mould components, then the original channel Ciwill not intersect with (a) Channel Ciand three mould components inside the mould insert. (b) Offsets of the mould components and Cirepresented by line segment Li. (c) Sweeping the offsets of the mould components and Cirepresented by point Pi. (d) The initial free region of Ci.(e) Subtracting O00 i from B0 i. (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) 334349337 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 Oiwithin the drilling depth will affect the construction of the channel. To account for the drilling depth, the offset O0 i of Oiis swept along the drilling direction until the opposite face of the mould insert is reached to generate O00 i . This sweeping of O0 i in effect reduces line Lito a point Pi located at the end of Li. As shown in Fig. 4(c), if the point Pi is outside O00 i , 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 center at Pias shown in Fig. 4(d). Bithen intersects with the mould insert to obtain B0 i. B 0 i represents all of the possible variations of Ciwhen only the geometric shape of the mould insert is considered. Then, FRiis obtained by subtracting from B0 i the O00 i of all of the obstacles. Fig. 4(e) and (f) show the subtraction and the resulting FRiof the example. 2.3. Basic approach to the construction of the C-space of cooling system To determine the free region FRFin 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 C1and C2are 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 fi rst introduced and will be used in the subsequent discussions on projections and the rest of the paper. Notations used in describing high-dimensional spaces Sndenotes an n-dimensional space spanned by the set of axes Xn= X1, X2,., Xn. Smdenotes an m-dimensional space spanned by the set of axes Xm= X0 1, X 0 2,., X 0 m. pndenotes a point in Snand pn= (x1,x2,.,xn), where xi denotes 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 Snto Sm. PROJSm(Rn) denotes the projection of a region Rnfrom Snto Sm. Notations used in describing a cooling system nCdenotes 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 of an individual channel Ci. SFdenotes the C-space of the cooling system. FRFdenotes the free region in SF. That is, it is the free region of the cooling system. Consider the projection of a point pnin Snto a point pmin Sm. Fig. 5(a) illustrates examples of projection using spaces of one dimension to three dimensions. Projections are illustrated for three cases: (i) Xm Xn; (ii) Xm Xn; and (iii) Xm6 Xn, Xn6 Xm, and Xn Xm6= . For (i), each coordinate of pmis equal to a corresponding coordinate of pnthat is on the same axis. For (ii) and (iii), the projection of pnis a region Rm. For each point pmin Rm, a coordinate of pmis equal to that of 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 specifi c defi nition of the projections, in particular, for cases (ii) and (iii), is as follows. Consider two adjacent channels Cnand Cm. As they are adjacent, they must be connected and thus their C-spacesSnand Smshare some common axes. Assume that a confi guration that corresponds to a point pnin Snhas been selected for Cn. To maintain the connectivity, the confi guration for Cmmust be selected such that the corresponding point pmin Smshares the same coordinates with pnon their common axes. This implies that pmcan be any point within the projection of pnon Sm, where the method of projection is defi ned above. The projections of a region Rnin Snto Smare simply the projections of every point in Rnto Sm. Fig. 5(b) illustrates the region projections. The formal defi nition of projection is given below. Defi nition 1 (Projection). 1.1. If Xm Xn, PROJSm(pn) is a pointpm= (x0 1,x 0 2,.,x 0 m), where for X0i = Xj, x0 i = xjfor 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 Xm Xn, PROJSm(pn) is a regionRm= pm|PROJSn(pm) = pn. 1.3. If Xm6 Xn, Xn6 Xm, and XnXm6= , PROJSm(pn)is a region Rm= pm|PROJSI(pm) = PROJSI(pn), where SIis the space spanned by Xn Xm. If Xn Xm= , PROJSm(pn ) is defi ned as Sm. 1.4. PROJSm(Rn ) is defi ned as the region Rm= pm|pm PROJSm(pn), pn Rn. As discussed in Section 2.1, any point pFin FRFgives 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 Siis in the free region FRiof each Ci. Thus, FRFis defi ned as follows. Defi nition 2 (Free Region in the C-space of a Cooling System). FRF= pF|PROJSi(pF) FRi,i 1,nC 338C.G. Li, C.L. Li / Computer-Aided Design 40 (2008) 334349 Fig. 5. The projections of points and regions in Snto Sm. Note that according to Defi nition 1.1, the projection of pF to Sialways contains only a single point because the set of axes that span Siis always a subset of the axes that span Sn. The construction of the free region FRiof each Cihas already been explained in Section 2.2. To fi nd FRFfrom FRi, the following theorem is useful. Theorem 1. FRF= nC i=1 PROJSF(FRi). Intuitively, this theorem says that to fi nd FRF, all of the FRiare fi rst projected to the C-space of the cooling system SF. FRF can then be obtained by performing the Boolean intersections among the projections. The proof of Theorem 1 and the lemmas used in the proof are given in the Appendix. 2.4. Representation and computation of the C-space To represent the free region FRFand to facilitate the computation of the Boolean intersections between the regions in a high-dimensional space, we can use a kind of cell enumeration method similar to the one used in 21,24. The basic idea is to approximate a high-dimensional region RFin SF by a set of high-dimensional boxes. Each box is defi ned by specifying an interval on each axis of SF. The intersection of two regions is achieved by the intersection of the two sets of boxes. The intersection between two high-dimensional boxes is simply the intersection between the intervals of each of the boxes in each axis. Assuming that each FRiis approximated by m three- dimensional boxes, the projection PROJSF(FRi) can then be approximated by mnF-dimensional boxes. The construction of FRFthat uses Theorem 1 then requires mnCintersections C.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 FRi of each channel in its confi guration space Si. Fig. 6. A simplifi ed example of a cooling system design. between nF-dimensional boxes, and FRFis represented b