- Email: [email protected]
Part decomposition for die pattern machining
Journal of Materials Processing Technology 130±131 (2002) 599±607
Part decomposition for die pattern machining Dongwoo Kia,*, Kunwoo Leeb a
Technical and Research Center, CubicTek Co., AceTechno-Tower 1101, 684-1 Deungchon-dong, Gangseo-gu, Seoul 157-721, South Korea b School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1, Shilim-dong, Kwanak-gu, Seoul 151-742, South Korea
Abstract Casting patterns are usually made of weak materials that can be machined easily, and thus they are good candidates to be fabricated by hybrid RP machines that involve both deposition and 3-axis machining processes. However, the setup and machining time, and the number of required workpiece in the sequential processes are proportional to the number of the decomposed components. Hence, reducing the number of decomposed components at the pre-processing stage is an essential requirement to reduce the production time and the number of required workpiece. Existing hybrid RP machines, however, consider only one machining direction for all layers in decomposing a part into layers. This usually produces more layers than a method allowing different machining directions for each layer. This paper proposes a novel decomposition method for pattern fabrication that can reduce the number of components at the pre-processing stage. To this end, we propose a new three-step decomposition method, which comprises: (1) the determination of a set of machining directions, (2) the grouping of a part's outer faces, and (3) the decomposition of a part's inner region. The proposed approach in this paper could reduce the number of decomposed components, therefore would decrease the production time. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Part decomposition; Die manufacturing; Pattern manufacture; Multi-direction machining; Hybrid rapid prototyping
1. Introduction Generally, a die such as a stamping die for a car body panel, is fabricated using a casting process with a lost form pattern, which is also called die pattern or pattern. In recent years, NC machining has been considered as a good alternative for making die patterns. The fabrication of a die pattern with NC machining in general consists of three steps: (i) part decomposition, (ii) machining, and (iii) joining. These steps are analogous to the hybrid rapid prototyping process [1] using both machining and deposition. Since both of the above processes need machining processes for each of the individually decomposed elements, the production time increases in proportion to the number of the decomposed components. And, each of the decomposed components is machined by two-setup machining consisting of a front-face machining and a back-face machining. Hence, to reduce the production time in the process, it is essential to decompose a part into the minimum number of components. The suitability of each component for machining should also be *
Corresponding author. E-mail addresses: [email protected] (D. Ki), [email protected] (K. Lee).
guaranteed in the decomposition step so that the decomposed components can be machined by a 3-axis NC machine. So as to minimize the number of decomposed elements, various directions for machining are considered in this proposed method. And, the machining easiness and the machining ef®ciency are considered in decomposing the part as well. A new proposed approach will produce fewer elements than those that can be generated when only one machining direction is considered for each element. The part decomposition method proposed in this paper consists of three sequential steps: (i) the determination of a set of machining directions in Section 2, (ii) the grouping of a part's outer faces in Sections 3 and 4, and (iii) the decomposition of part's inner region in Section 5 are explained. Section 6 presents an example of part decomposition, and ®nally, some conclusions are presented in Section 7. 2. Related work Ramaswami et al. [2±4] presented an algorithm of part decomposition for SDM [5]. In that method, for part decomposition, silhouette curves that serve as the boundary
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 7 5 8 - 6
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D. Ki, K. Lee / Journal of Materials Processing Technology 130±131 (2002) 599±607
between the undercut and non-undercut portions of the surface are used. Ilinkin et al. [6] proposed a decomposition method that decomposes polyhedral models to minimize the support requirements in RP. This method is based on the plane-sweep paradigm that generates expressions for the support volume and the contact area as a function of the height of the sweep plane, and optimizes these while moving in a xy-plane along the z-direction. Rapid prototyping systems with an ability to deposit along multiple directions can improve the surface quality and reduce the support volume. Singh and Dutta [7] presented a multi-directional slicing method applicable to these systems. To determine a set of build directions, they introduced the concept of a build map, which is similar to a visibility map [8,9]. In choosing a build direction among a set of build directions, they used surface quality and collision-free deposition. Using a build map, they identi®ed surfaces that are non-buildable along a selected build direction. To improve the ef®ciency in the LOM process, Karunakaran et al. [10] presented an ef®cient method of cutting the remaining stock to extract the object. In their method, they segmented the remaining stock into two stock halves that open about a parting surface of minimum area. In searching the parting direction and parting surface to create stock halves, Karunakaran et al. used a visibility map. These methods are not applicable for decomposing a part into the minimum components for two-step machining. This paper proposes a new part decomposition method for pattern, where NC machining is performed in a pair of opposite directions for each decomposed component. 3. Determination of a set of machining directions 3.1. Machining direction Before decomposing a part, the machining directions that will be used for NC machining of each decomposed component need to be determined ®rst. To select such machining directions, we considered two factors: simplicity and stability of setup, and quality of machined surfaces as follows. For the stability of workpieces during machining, it is desirable to machine a component in a direction that has a simple and stable setup. If workpieces are not held by simple supports using planar surfaces of the part, the stability may not be guaranteed and thus supplement supports or complexly shaped supports would be needed, which will cause more setup time and fabrication cost. Hence, noting that we use 3-axis NC machines, the normal directions of planar surfaces of the part could be good candidates for machining directions. Therefore, the normal vectors of planar surfaces in a part are used to select machining directions, which is explained below. First, each group of planar surfaces with the same normal direction is searched for among all the faces in a part. Then the total area of each face group is summed up and stored
Fig. 1. Machining directions for the 2D example part.
together with its normal direction. The normal directions of the face groups are the candidates for machining directions. An example for a 2D part is illustrated in Fig. 1, and three 1, D 2 , and D 3 can be selected by machining directions of D the foregoing selection procedure. As mentioned above, i and D i for each there is two setups corresponding to D machining direction. 3.2. Face split Once in a while, a surface may have two regions, one of which can be machined from one direction and the other can be machined from the opposite direction. This means this surface cannot be machined in one setup. Thus, such surfaces need to be split into two or more regions so that each region can be accessed from one of two directions. Face split is a pre-processing routine to ®nd a set of connected faces that can be machined at a pair of opposite directions (D), and to decide a machining direction out of one of the two directions to machine the member faces. For each machining direction, all silhouette curves are calculated, and the faces in a part are split by using them. 4. Grouping of the part's outer faces 4.1. Definition of a set For grouping of a part's faces, let Setj Di be the jth set of i or connected faces that can be machined from either D i direction. Fig. 2 shows an example that the faces indicatD ed by `' can be machined in one setup, while the faces indicated by ` ' can be machined by the opposite setup. Faces indicated by `0' can be machined in either of two setups. i , a sequential grouping of the faces For a direction D indicated by ` ' and the faces indicated by `' forms a set i setups. And, there can be that can be machined in D i , and they should include all several sets for each direction D faces in part. After creating Setj Di for all machining direc i , these sets are represented by a Venn diagram. Fig. 3 tions D
D. Ki, K. Lee / Journal of Materials Processing Technology 130±131 (2002) 599±607
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Fig. 2. Definition of a set.
shows a Venn diagram for the part in Fig. 1, where all the face sets for the three machining directions are illustrated. 4.2. Subset removal When all Setj Di are represented by a Venn diagram, there may be a face set that is completely enclosed by another face set. This means that the faces of the inside face set can be machined by the machining direction corresponding to the outer face set in the Venn diagram, and thus the inside face set can be removed from the Venn diagram. The face sets enclosed by another face set are indicated by the dotted boundary in Fig. 3. Fig. 4 shows a Venn diagram after eliminating the face sets enclosed by other face sets. On the other hand, there may be a situation where two or more face sets with different machining directions have exactly the same face members. Since the faces of such face sets can be machined using any direction of them, such face sets are kept until a speci®c direction among them is assigned to that at a later stage.
Fig. 4. Venn diagram after subset removal.
that belong only to its own face set and the face members can be machined in only the direction represented by the face set. Hence, the face set should be included in the minimum number of face sets. The second type of face sets is called a dependent set. All the face elements of such face sets belong to at least two face sets, and thus they can be machined in different directions. The following equation can be used to determine whether Seti Dj is an independent set or a dependent set 6 f : independent set; Seti Dj U Seti Dj f : dependent set
4.3. Determination of minimum face sets After the subsets are removed from the face sets as described previously, we can classify the remaining face sets into two types. The ®rst type of face sets is called an independent sets, and means that these can be face members
Fig. 3. Venn diagram of the 2D example part.
Fig. 5. Process to find a combination of minimum sets: (a) Venn diagram after selection of independent sets; (b) tree of selected combination of sets.
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where U is the union of the remaining sets in the Venn diagram. Since the faces in a dependent set can be machined in several different directions, we can obtain several combinations of face sets of which the union contains all the faces of a part. Among those face sets, we need to decide the group of minimum face sets. A procedure to ®nd minimum face sets is described below in detail with an example. Fig. 5(a) shows an example of dependent face sets, where it can be seen that every element is shared by two or more sets. We assume in the example that independent sets are already selected and separated. To select a combination of the minimum face sets among the dependent sets in the Venn diagram, the following procedure is applied. First, a face set that has the most intersections with other face sets will have the highest priority, and set difference operation is applied such that the face set and its face elements are separated from the Venn diagram. If two or more sets have the same priority, each of them is selected and tried in turn to consider all the possible cases, which will result in a tree-structure as shown in Fig. 5(b). The foregoing procedure continues until all the elements in the Venn diagram are removed. From Fig. 5(b), it can be concluded that the minimum face sets is realized by Set2, Set4, and Set7. And, returning to the example in Fig. 4, it can be recognized that the group of minimum face sets is Set1 D3 Set1 D2 or Set1 D3 Set1 D1. 5. Decomposition of a part's inner region 5.1. Cover surface generation As mentioned earlier in Section 4.1, because each set of outer faces is not a solid, it is required to create covers to form a solid containing each face set. To generate such cover faces, the present approach introduced a concept of machining direction aligned min/max-box (MAMB). The MAMB is de®ned as the smallest bounding box that encloses all the faces in a face set. Notice that MAMB is oriented when calculating it such that one of its three orthogonal axes is parallel to the machining direction of the face set. First, the top surfaces of a cover are generated as follows. Each face in the face set is then projected onto the MAMBs top or bottom face according to the angle between the face's i . If the angle is less than p/2, the face normal vector and D is projected on the MAMBs top face whose normal vector is i , and if the angle is greater than p/2, the face is projected D i . If on the MAMBs bottom face whose normal vector is D the angle is equal to p/2, a face is not projected. After that all the faces are projected on either the top or the bottom face of the MAMB, two planar surfaces whose boundaries are made up of silhouette curves of the face set is created at the i , respectively. MAMBs highest and lowest positions in D And, the top surfaces are created through Boolean difference operation that subtracts the projected surfaces from the planar surface. If a top surface resulting from Boolean
Fig. 6. Process of making a cover for a set: (a) top surface of a cover; (b) top/side surfaces of a cover.
difference operation on anMAMBs top or bottom is null, a cover is not required in the direction. Fig. 6(a) shows a top surface of cover created on an MAMBs face whose normal and a cover is not required in D. vector is D Note that the top surfaces are complete, the side surfaces of the cover need to be made next. The side surfaces of the cover can be produced by sweeping the top surface's boundary curve along the opposite direction of the top surface's normal. The intersection curves between the swept surfaces and the faces in the face set constitute the swept surfaces' boundary. Fig. 6(b) shows an example of a cover (Cover1) produced by the above procedure. 5.2. Generation of seed component Using the covers generated from each face set as parting surfaces, a part is ®rst decomposed into seed components. Each of the seed components is associated with its machining direction. In generating seed components, because of the intersection of face sets, a component that has two or more machining directions may be produced. In general, such a component can be merged into an adjacent component that has the same machining direction. To reduce the number of
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Fig. 7. Creating the seed component for example part1: (a) MAMB for example part1; (b) covers for example part1: (c) covers for example part1.
the decomposed components, it is desirable that the component is merged with one among candidates. With reference to Fig. 7, the decomposition procedure will be explained in detail for the example part in Figs. 1 and 4. Fig. 7(a) shows the two MAMBs of Set1D2 and Set1D3 of the example part, Fig. 7(b) displays the two covers (Cover1 for Set1D3 and Cover2 for Set1D2) created using the MAMBs in Fig. 7(a). Three components decomposed using the covers in Fig. 7(b) is presented in Fig. 7(c). C1 is originated from Set1D2, C3 from Set1D3, and C2 from both Set1D2 and Set1D3, since the faces 110 and 12 of C2 are common face elements of Set1D2 and Set1D3. Notice here that C2 partitioned by Cover1 and Cover2 can be regarded as a common component of both Set1D2 and Set1D3. Thus, it is desirable in the light of the number of decomposed components to be
Fig. 8. Example of a second decomposition: (a) example part and machining directions; (b) decomposed components after the first decomposition; (c) traces and their expanded direction; (d) split faces; (e) decomposed components.
machined that C2 should be merged to one of the C1 and C3. To decide which one is a better candidate for mergence, a metric called aggregation ef®ciency (AE) was developed. It is composed of the machining easiness and the machining ef®ciency, and helps decide which combination will yield a
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more desirable component for machining. The detailed description will be given later. By measuring the suitability of merged components, we could ®nd that the resulting component produced by merging C2 with C3 will have a larger contact area with the work table, and thus will have higher AE. Hence, the example part will ®nally be decomposed into two components of C1 and C2 C3 . 5.3. Core region's decomposition In the example shown in Fig. 7, an additional decomposition was not necessary after partitioning the part with the covers of each face set. However, there are some cases where the part's core will not be decomposed by the covers. To deal with such undecomposed core region, we decompose it further into smaller components and merge them with proper seed components obtained previously. The decomposition of a core region uses the traces on it generated in the creating of seed components. Since the boundary faces of a core region were created from the covers of each face set during the generation of the seed components, they are planar top surfaces of cover or sweep surfaces. To create parting surfaces for the core region, the boundary curves of top surfaces of a cover are swept in®nitely along the opposite direction to the normal vector of the surface. After the parting surfaces are obtained, they are used to decompose the core region. Fig. 8(a) shows an example where a core region's decomposition is required. Note that the machining directions are also shown in the ®gure. Fig. 8(b) displays the decomposed components after the ®rst decomposition for generating seed components. Each decomposed component is displayed with an arrow indicating its machining direction, shown in Fig. 8(a). Note that the core of the part is not associated with a machining direction. Actually, the core region will be further decomposed into several components and the newly decomposed components will be associated with their machining directions to which they are merged. The core region is then decomposed by the traces on it generated in the creating of the seed components. Fig. 8(c) illustrates the planar surfaces
Fig. 10. Decomposed components after the second decomposition: (a) decomposed components after the second decomposition; (b) combinations of components can be joined with a seed component; (c) components after combining.
Fig. 9. Estimation of ease of machining of a combination of components.
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Fig. 11. Experimental example: (a) pattern model in top view; (b) pattern model in bottom view; (c) minimum face sets; (d) components.
on the part's core, and their sweep direction. The parting surfaces and decomposed components are displayed in Fig. 8(d) and (e), respectively. To minimize the total number of the decomposed components created in the decomposition process, it is desirable to merge the components with one another, if possible. The components created in decomposing the core region are considered as candidates to be merged with the seed components. As mentioned earlier, a metric called AE has been developed to decide which combination of mergence will produce the most suitable components for machining. To
estimate the AE between components, the machining easiness and the machining ef®ciency are measured. As each component should be machined in two setups corresponding to two machining directions opposite to each other, it should be checked that all faces of the component to be merged could be accessible from the two machining directions before estimating the machining easiness and the machining ef®ciency. The machining easiness is the ratio of contact areas to the projection area of a new component that will be produced by the combination of components. The contact areas mean the sum of areas on which the new component
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contacts with the work table, and the projection area is the area of projection of the new component onto the work table. The machining easiness is calculated as follows: X Contact Areai MEa machining easiness wa1 Projection Area As an example for the calculation of the machining easiness, consider the combination of components in Fig. 9. The machining easiness is calculated by the formula wa1 Ac1 Ac2 =Ap . Since a component is machined in two opposite directions, the machining easiness needs to be calculated twice for each direction. The smaller value of two values is regarded as the value of the machining easiness of the combined component. When two or more components are combined, the machining time will be decreased if the total area to be machined becomes smaller. To search for such a combination, a concept of the machining ef®ciency is introduced into the AE. The machining ef®ciency is the ratio of individual areas to merged area for machining. The individual areas mean the sum of machining areas on candidate components for merging. The merged area is the machining area of a new merged component. It can be denoted by an equation as follows: MEf machining efficiency
Area machinedbefore join Area machinedafter join
Once the machining easiness MEa and the machining ef®ciency MEf are calculated, the AE is derived by a weighted sum of the two factors as below AE aggregation efficiency Wa MEa Wf MEf For all possible combinations, AE is calculated and the combination having the highest AE value is joined. This process is terminated when all components have an own machining direction. Fig. 10(a) shows all the components resulting from the decomposition process, and Fig. 10(b) illustrates all the combinations for each of the seed components and the combinations selected only based on the MEa values obtained. Fig. 10(c) shows the ®nal components that are produced by merging the components obtained in the core region's decomposition with the seed components. Notice that all of the ®nal components are associated with their own machining directions. 6. Case study To show the capability of the proposed method, we demonstrated it with a practical example. Fig. 11(a) and (b) shows a test part, pattern model considered in case study. The test part has several holes along different axes and so many undercut regions. First, though the process of determination of a set of machining directions using planar faces in model, three
1 (1, 0, 0), D 2 directions for machining are obtained: D (0, 1, 0), D3 (0, 0, 1). Then, the part's faces are grouped into several face sets of connecting faces corresponding to each machining direction. To ®nd a group of minimum face sets include all faces in model, a subset removal and a determination of dependent sets are carried out, and four face sets are obtained as shown Fig. 11(c). In Fig. 11(c), the arrows represent the machining directions corresponding to each face set. Then, the process of creating cover surfaces is performed to form a solid for each face set. After a solid component is created, and it is adjusted considering the machining easiness and machining ef®ciency. Fig. 11(d) shows four ®nal components corresponding to the face set. When only z-direction for machining is considered, six layers are generated, and when only x- or y-direction is considered, a lot of layers are generated because of free-form surfaces and undercuts. In contrast, the proposed method considering several machining directions generates only four components as demonstrated previously. 7. Conclusions The proposed method allows various machining directions to deduce the number of decomposed components. Thereby, it could produce smaller number of components than considering only one machining direction. In addition to the feasibility of two-setup machining, it also considers the machining easiness and the machining ef®ciency during the decomposition. Therefore, if the proposed decomposition method is applied to pattern fabrication, it is anticipated to shorten the time required in mold manufacturing. Further research is required on the following issues. First, consideration on the removal volume of material is necessary. Secondly, improvement in ®nding candidate combinations for merging after the core region's decomposition is required. The present process relies on a brute-force search, and thus an optimal search using some geometric properties of the candidate components will help improve the computation ef®ciency. References [1] J. Hur, K. Lee, Z. Hu, J. Kim, Hybrid rapid prototyping system using machining and deposition, Comput.-Aid. Des., in press. [2] K. Ramaswami, Y. Yamaguchi, F.B. Prinz, Spatial partitioning of solids for solid freeform fabrication, in: Proceedings of the Fourth ACM/SIGGRAPH Symposium on Solid Modeling and Applications, May 1997. [3] K. Ramaswami, Process planning for shape deposition manufacturing, Doctoral Thesis, Stanford University, January 1997. [4] Y.C. Chang, J.M. Pinilla, J.H. Kao, J. Dong, K. Ramaswami, F.B. Prinz, Automated layer decomposition for additive/subtractive solid freeform fabrication, in: Proceedings of the Solid Freeform Fabrication Symposium, The University of Texas at Austin, August 1999.
D. Ki, K. Lee / Journal of Materials Processing Technology 130±131 (2002) 599±607 [5] R. Merz, F.B. Prinz, K. Ramaswami, M. Terk, L.E. Weiss, Shape deposition manufacturing, in: Proceedings of the Solid Freeform Fabrication Symposium, The University of Texas at Austin, August 1994. [6] I. Ilinkin, R. Janardan, J. Majhi, J. Schwerdt, M. Smid, R. Sriram, A decomposition-based approach to layered manufacturing, Comput. Geom., in press. [7] P. Singh, D. Dutta, Multi-direction slicing for layered manufacturing, J. Comput. Inform. Sci. Eng. 1 (June 2001).
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[8] P. Gupta, R. Janardan, J. Majhi, T.C. Woo, Efficient geometric algorithms for workpiece orientation in 4- and 5-axis NC machining, Comput.-Aid. Des. 28 (8) (1996) 577±587. [9] T.C. Woo, Visibility maps and spherical algorithms, Comput.-Aid. Des. 28 (1) (1994) 6±16. [10] K.P. Karunakaran, S. Dibbi, P. ViveKananda Shanmuganathan, Efficient stock cutting for laminated manufacturing, Comput.-Aid. Des. 34 (4) (2002) 281±298.
Part decomposition for die pattern machining Dongwoo Kia,*, Kunwoo Leeb a
Technical and Research Center, CubicTek Co., AceTechno-Tower 1101, 684-1 Deungchon-dong, Gangseo-gu, Seoul 157-721, South Korea b School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1, Shilim-dong, Kwanak-gu, Seoul 151-742, South Korea
Abstract Casting patterns are usually made of weak materials that can be machined easily, and thus they are good candidates to be fabricated by hybrid RP machines that involve both deposition and 3-axis machining processes. However, the setup and machining time, and the number of required workpiece in the sequential processes are proportional to the number of the decomposed components. Hence, reducing the number of decomposed components at the pre-processing stage is an essential requirement to reduce the production time and the number of required workpiece. Existing hybrid RP machines, however, consider only one machining direction for all layers in decomposing a part into layers. This usually produces more layers than a method allowing different machining directions for each layer. This paper proposes a novel decomposition method for pattern fabrication that can reduce the number of components at the pre-processing stage. To this end, we propose a new three-step decomposition method, which comprises: (1) the determination of a set of machining directions, (2) the grouping of a part's outer faces, and (3) the decomposition of a part's inner region. The proposed approach in this paper could reduce the number of decomposed components, therefore would decrease the production time. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Part decomposition; Die manufacturing; Pattern manufacture; Multi-direction machining; Hybrid rapid prototyping
1. Introduction Generally, a die such as a stamping die for a car body panel, is fabricated using a casting process with a lost form pattern, which is also called die pattern or pattern. In recent years, NC machining has been considered as a good alternative for making die patterns. The fabrication of a die pattern with NC machining in general consists of three steps: (i) part decomposition, (ii) machining, and (iii) joining. These steps are analogous to the hybrid rapid prototyping process [1] using both machining and deposition. Since both of the above processes need machining processes for each of the individually decomposed elements, the production time increases in proportion to the number of the decomposed components. And, each of the decomposed components is machined by two-setup machining consisting of a front-face machining and a back-face machining. Hence, to reduce the production time in the process, it is essential to decompose a part into the minimum number of components. The suitability of each component for machining should also be *
Corresponding author. E-mail addresses: [email protected] (D. Ki), [email protected] (K. Lee).
guaranteed in the decomposition step so that the decomposed components can be machined by a 3-axis NC machine. So as to minimize the number of decomposed elements, various directions for machining are considered in this proposed method. And, the machining easiness and the machining ef®ciency are considered in decomposing the part as well. A new proposed approach will produce fewer elements than those that can be generated when only one machining direction is considered for each element. The part decomposition method proposed in this paper consists of three sequential steps: (i) the determination of a set of machining directions in Section 2, (ii) the grouping of a part's outer faces in Sections 3 and 4, and (iii) the decomposition of part's inner region in Section 5 are explained. Section 6 presents an example of part decomposition, and ®nally, some conclusions are presented in Section 7. 2. Related work Ramaswami et al. [2±4] presented an algorithm of part decomposition for SDM [5]. In that method, for part decomposition, silhouette curves that serve as the boundary
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 7 5 8 - 6
600
D. Ki, K. Lee / Journal of Materials Processing Technology 130±131 (2002) 599±607
between the undercut and non-undercut portions of the surface are used. Ilinkin et al. [6] proposed a decomposition method that decomposes polyhedral models to minimize the support requirements in RP. This method is based on the plane-sweep paradigm that generates expressions for the support volume and the contact area as a function of the height of the sweep plane, and optimizes these while moving in a xy-plane along the z-direction. Rapid prototyping systems with an ability to deposit along multiple directions can improve the surface quality and reduce the support volume. Singh and Dutta [7] presented a multi-directional slicing method applicable to these systems. To determine a set of build directions, they introduced the concept of a build map, which is similar to a visibility map [8,9]. In choosing a build direction among a set of build directions, they used surface quality and collision-free deposition. Using a build map, they identi®ed surfaces that are non-buildable along a selected build direction. To improve the ef®ciency in the LOM process, Karunakaran et al. [10] presented an ef®cient method of cutting the remaining stock to extract the object. In their method, they segmented the remaining stock into two stock halves that open about a parting surface of minimum area. In searching the parting direction and parting surface to create stock halves, Karunakaran et al. used a visibility map. These methods are not applicable for decomposing a part into the minimum components for two-step machining. This paper proposes a new part decomposition method for pattern, where NC machining is performed in a pair of opposite directions for each decomposed component. 3. Determination of a set of machining directions 3.1. Machining direction Before decomposing a part, the machining directions that will be used for NC machining of each decomposed component need to be determined ®rst. To select such machining directions, we considered two factors: simplicity and stability of setup, and quality of machined surfaces as follows. For the stability of workpieces during machining, it is desirable to machine a component in a direction that has a simple and stable setup. If workpieces are not held by simple supports using planar surfaces of the part, the stability may not be guaranteed and thus supplement supports or complexly shaped supports would be needed, which will cause more setup time and fabrication cost. Hence, noting that we use 3-axis NC machines, the normal directions of planar surfaces of the part could be good candidates for machining directions. Therefore, the normal vectors of planar surfaces in a part are used to select machining directions, which is explained below. First, each group of planar surfaces with the same normal direction is searched for among all the faces in a part. Then the total area of each face group is summed up and stored
Fig. 1. Machining directions for the 2D example part.
together with its normal direction. The normal directions of the face groups are the candidates for machining directions. An example for a 2D part is illustrated in Fig. 1, and three 1, D 2 , and D 3 can be selected by machining directions of D the foregoing selection procedure. As mentioned above, i and D i for each there is two setups corresponding to D machining direction. 3.2. Face split Once in a while, a surface may have two regions, one of which can be machined from one direction and the other can be machined from the opposite direction. This means this surface cannot be machined in one setup. Thus, such surfaces need to be split into two or more regions so that each region can be accessed from one of two directions. Face split is a pre-processing routine to ®nd a set of connected faces that can be machined at a pair of opposite directions (D), and to decide a machining direction out of one of the two directions to machine the member faces. For each machining direction, all silhouette curves are calculated, and the faces in a part are split by using them. 4. Grouping of the part's outer faces 4.1. Definition of a set For grouping of a part's faces, let Setj Di be the jth set of i or connected faces that can be machined from either D i direction. Fig. 2 shows an example that the faces indicatD ed by `' can be machined in one setup, while the faces indicated by ` ' can be machined by the opposite setup. Faces indicated by `0' can be machined in either of two setups. i , a sequential grouping of the faces For a direction D indicated by ` ' and the faces indicated by `' forms a set i setups. And, there can be that can be machined in D i , and they should include all several sets for each direction D faces in part. After creating Setj Di for all machining direc i , these sets are represented by a Venn diagram. Fig. 3 tions D
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Fig. 2. Definition of a set.
shows a Venn diagram for the part in Fig. 1, where all the face sets for the three machining directions are illustrated. 4.2. Subset removal When all Setj Di are represented by a Venn diagram, there may be a face set that is completely enclosed by another face set. This means that the faces of the inside face set can be machined by the machining direction corresponding to the outer face set in the Venn diagram, and thus the inside face set can be removed from the Venn diagram. The face sets enclosed by another face set are indicated by the dotted boundary in Fig. 3. Fig. 4 shows a Venn diagram after eliminating the face sets enclosed by other face sets. On the other hand, there may be a situation where two or more face sets with different machining directions have exactly the same face members. Since the faces of such face sets can be machined using any direction of them, such face sets are kept until a speci®c direction among them is assigned to that at a later stage.
Fig. 4. Venn diagram after subset removal.
that belong only to its own face set and the face members can be machined in only the direction represented by the face set. Hence, the face set should be included in the minimum number of face sets. The second type of face sets is called a dependent set. All the face elements of such face sets belong to at least two face sets, and thus they can be machined in different directions. The following equation can be used to determine whether Seti Dj is an independent set or a dependent set 6 f : independent set; Seti Dj U Seti Dj f : dependent set
4.3. Determination of minimum face sets After the subsets are removed from the face sets as described previously, we can classify the remaining face sets into two types. The ®rst type of face sets is called an independent sets, and means that these can be face members
Fig. 3. Venn diagram of the 2D example part.
Fig. 5. Process to find a combination of minimum sets: (a) Venn diagram after selection of independent sets; (b) tree of selected combination of sets.
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where U is the union of the remaining sets in the Venn diagram. Since the faces in a dependent set can be machined in several different directions, we can obtain several combinations of face sets of which the union contains all the faces of a part. Among those face sets, we need to decide the group of minimum face sets. A procedure to ®nd minimum face sets is described below in detail with an example. Fig. 5(a) shows an example of dependent face sets, where it can be seen that every element is shared by two or more sets. We assume in the example that independent sets are already selected and separated. To select a combination of the minimum face sets among the dependent sets in the Venn diagram, the following procedure is applied. First, a face set that has the most intersections with other face sets will have the highest priority, and set difference operation is applied such that the face set and its face elements are separated from the Venn diagram. If two or more sets have the same priority, each of them is selected and tried in turn to consider all the possible cases, which will result in a tree-structure as shown in Fig. 5(b). The foregoing procedure continues until all the elements in the Venn diagram are removed. From Fig. 5(b), it can be concluded that the minimum face sets is realized by Set2, Set4, and Set7. And, returning to the example in Fig. 4, it can be recognized that the group of minimum face sets is Set1 D3 Set1 D2 or Set1 D3 Set1 D1. 5. Decomposition of a part's inner region 5.1. Cover surface generation As mentioned earlier in Section 4.1, because each set of outer faces is not a solid, it is required to create covers to form a solid containing each face set. To generate such cover faces, the present approach introduced a concept of machining direction aligned min/max-box (MAMB). The MAMB is de®ned as the smallest bounding box that encloses all the faces in a face set. Notice that MAMB is oriented when calculating it such that one of its three orthogonal axes is parallel to the machining direction of the face set. First, the top surfaces of a cover are generated as follows. Each face in the face set is then projected onto the MAMBs top or bottom face according to the angle between the face's i . If the angle is less than p/2, the face normal vector and D is projected on the MAMBs top face whose normal vector is i , and if the angle is greater than p/2, the face is projected D i . If on the MAMBs bottom face whose normal vector is D the angle is equal to p/2, a face is not projected. After that all the faces are projected on either the top or the bottom face of the MAMB, two planar surfaces whose boundaries are made up of silhouette curves of the face set is created at the i , respectively. MAMBs highest and lowest positions in D And, the top surfaces are created through Boolean difference operation that subtracts the projected surfaces from the planar surface. If a top surface resulting from Boolean
Fig. 6. Process of making a cover for a set: (a) top surface of a cover; (b) top/side surfaces of a cover.
difference operation on anMAMBs top or bottom is null, a cover is not required in the direction. Fig. 6(a) shows a top surface of cover created on an MAMBs face whose normal and a cover is not required in D. vector is D Note that the top surfaces are complete, the side surfaces of the cover need to be made next. The side surfaces of the cover can be produced by sweeping the top surface's boundary curve along the opposite direction of the top surface's normal. The intersection curves between the swept surfaces and the faces in the face set constitute the swept surfaces' boundary. Fig. 6(b) shows an example of a cover (Cover1) produced by the above procedure. 5.2. Generation of seed component Using the covers generated from each face set as parting surfaces, a part is ®rst decomposed into seed components. Each of the seed components is associated with its machining direction. In generating seed components, because of the intersection of face sets, a component that has two or more machining directions may be produced. In general, such a component can be merged into an adjacent component that has the same machining direction. To reduce the number of
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Fig. 7. Creating the seed component for example part1: (a) MAMB for example part1; (b) covers for example part1: (c) covers for example part1.
the decomposed components, it is desirable that the component is merged with one among candidates. With reference to Fig. 7, the decomposition procedure will be explained in detail for the example part in Figs. 1 and 4. Fig. 7(a) shows the two MAMBs of Set1D2 and Set1D3 of the example part, Fig. 7(b) displays the two covers (Cover1 for Set1D3 and Cover2 for Set1D2) created using the MAMBs in Fig. 7(a). Three components decomposed using the covers in Fig. 7(b) is presented in Fig. 7(c). C1 is originated from Set1D2, C3 from Set1D3, and C2 from both Set1D2 and Set1D3, since the faces 110 and 12 of C2 are common face elements of Set1D2 and Set1D3. Notice here that C2 partitioned by Cover1 and Cover2 can be regarded as a common component of both Set1D2 and Set1D3. Thus, it is desirable in the light of the number of decomposed components to be
Fig. 8. Example of a second decomposition: (a) example part and machining directions; (b) decomposed components after the first decomposition; (c) traces and their expanded direction; (d) split faces; (e) decomposed components.
machined that C2 should be merged to one of the C1 and C3. To decide which one is a better candidate for mergence, a metric called aggregation ef®ciency (AE) was developed. It is composed of the machining easiness and the machining ef®ciency, and helps decide which combination will yield a
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more desirable component for machining. The detailed description will be given later. By measuring the suitability of merged components, we could ®nd that the resulting component produced by merging C2 with C3 will have a larger contact area with the work table, and thus will have higher AE. Hence, the example part will ®nally be decomposed into two components of C1 and C2 C3 . 5.3. Core region's decomposition In the example shown in Fig. 7, an additional decomposition was not necessary after partitioning the part with the covers of each face set. However, there are some cases where the part's core will not be decomposed by the covers. To deal with such undecomposed core region, we decompose it further into smaller components and merge them with proper seed components obtained previously. The decomposition of a core region uses the traces on it generated in the creating of seed components. Since the boundary faces of a core region were created from the covers of each face set during the generation of the seed components, they are planar top surfaces of cover or sweep surfaces. To create parting surfaces for the core region, the boundary curves of top surfaces of a cover are swept in®nitely along the opposite direction to the normal vector of the surface. After the parting surfaces are obtained, they are used to decompose the core region. Fig. 8(a) shows an example where a core region's decomposition is required. Note that the machining directions are also shown in the ®gure. Fig. 8(b) displays the decomposed components after the ®rst decomposition for generating seed components. Each decomposed component is displayed with an arrow indicating its machining direction, shown in Fig. 8(a). Note that the core of the part is not associated with a machining direction. Actually, the core region will be further decomposed into several components and the newly decomposed components will be associated with their machining directions to which they are merged. The core region is then decomposed by the traces on it generated in the creating of the seed components. Fig. 8(c) illustrates the planar surfaces
Fig. 10. Decomposed components after the second decomposition: (a) decomposed components after the second decomposition; (b) combinations of components can be joined with a seed component; (c) components after combining.
Fig. 9. Estimation of ease of machining of a combination of components.
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Fig. 11. Experimental example: (a) pattern model in top view; (b) pattern model in bottom view; (c) minimum face sets; (d) components.
on the part's core, and their sweep direction. The parting surfaces and decomposed components are displayed in Fig. 8(d) and (e), respectively. To minimize the total number of the decomposed components created in the decomposition process, it is desirable to merge the components with one another, if possible. The components created in decomposing the core region are considered as candidates to be merged with the seed components. As mentioned earlier, a metric called AE has been developed to decide which combination of mergence will produce the most suitable components for machining. To
estimate the AE between components, the machining easiness and the machining ef®ciency are measured. As each component should be machined in two setups corresponding to two machining directions opposite to each other, it should be checked that all faces of the component to be merged could be accessible from the two machining directions before estimating the machining easiness and the machining ef®ciency. The machining easiness is the ratio of contact areas to the projection area of a new component that will be produced by the combination of components. The contact areas mean the sum of areas on which the new component
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contacts with the work table, and the projection area is the area of projection of the new component onto the work table. The machining easiness is calculated as follows: X Contact Areai MEa machining easiness wa1 Projection Area As an example for the calculation of the machining easiness, consider the combination of components in Fig. 9. The machining easiness is calculated by the formula wa1 Ac1 Ac2 =Ap . Since a component is machined in two opposite directions, the machining easiness needs to be calculated twice for each direction. The smaller value of two values is regarded as the value of the machining easiness of the combined component. When two or more components are combined, the machining time will be decreased if the total area to be machined becomes smaller. To search for such a combination, a concept of the machining ef®ciency is introduced into the AE. The machining ef®ciency is the ratio of individual areas to merged area for machining. The individual areas mean the sum of machining areas on candidate components for merging. The merged area is the machining area of a new merged component. It can be denoted by an equation as follows: MEf machining efficiency
Area machinedbefore join Area machinedafter join
Once the machining easiness MEa and the machining ef®ciency MEf are calculated, the AE is derived by a weighted sum of the two factors as below AE aggregation efficiency Wa MEa Wf MEf For all possible combinations, AE is calculated and the combination having the highest AE value is joined. This process is terminated when all components have an own machining direction. Fig. 10(a) shows all the components resulting from the decomposition process, and Fig. 10(b) illustrates all the combinations for each of the seed components and the combinations selected only based on the MEa values obtained. Fig. 10(c) shows the ®nal components that are produced by merging the components obtained in the core region's decomposition with the seed components. Notice that all of the ®nal components are associated with their own machining directions. 6. Case study To show the capability of the proposed method, we demonstrated it with a practical example. Fig. 11(a) and (b) shows a test part, pattern model considered in case study. The test part has several holes along different axes and so many undercut regions. First, though the process of determination of a set of machining directions using planar faces in model, three
1 (1, 0, 0), D 2 directions for machining are obtained: D (0, 1, 0), D3 (0, 0, 1). Then, the part's faces are grouped into several face sets of connecting faces corresponding to each machining direction. To ®nd a group of minimum face sets include all faces in model, a subset removal and a determination of dependent sets are carried out, and four face sets are obtained as shown Fig. 11(c). In Fig. 11(c), the arrows represent the machining directions corresponding to each face set. Then, the process of creating cover surfaces is performed to form a solid for each face set. After a solid component is created, and it is adjusted considering the machining easiness and machining ef®ciency. Fig. 11(d) shows four ®nal components corresponding to the face set. When only z-direction for machining is considered, six layers are generated, and when only x- or y-direction is considered, a lot of layers are generated because of free-form surfaces and undercuts. In contrast, the proposed method considering several machining directions generates only four components as demonstrated previously. 7. Conclusions The proposed method allows various machining directions to deduce the number of decomposed components. Thereby, it could produce smaller number of components than considering only one machining direction. In addition to the feasibility of two-setup machining, it also considers the machining easiness and the machining ef®ciency during the decomposition. Therefore, if the proposed decomposition method is applied to pattern fabrication, it is anticipated to shorten the time required in mold manufacturing. Further research is required on the following issues. First, consideration on the removal volume of material is necessary. Secondly, improvement in ®nding candidate combinations for merging after the core region's decomposition is required. The present process relies on a brute-force search, and thus an optimal search using some geometric properties of the candidate components will help improve the computation ef®ciency. References [1] J. Hur, K. Lee, Z. Hu, J. Kim, Hybrid rapid prototyping system using machining and deposition, Comput.-Aid. Des., in press. [2] K. Ramaswami, Y. Yamaguchi, F.B. Prinz, Spatial partitioning of solids for solid freeform fabrication, in: Proceedings of the Fourth ACM/SIGGRAPH Symposium on Solid Modeling and Applications, May 1997. [3] K. Ramaswami, Process planning for shape deposition manufacturing, Doctoral Thesis, Stanford University, January 1997. [4] Y.C. Chang, J.M. Pinilla, J.H. Kao, J. Dong, K. Ramaswami, F.B. Prinz, Automated layer decomposition for additive/subtractive solid freeform fabrication, in: Proceedings of the Solid Freeform Fabrication Symposium, The University of Texas at Austin, August 1999.
D. Ki, K. Lee / Journal of Materials Processing Technology 130±131 (2002) 599±607 [5] R. Merz, F.B. Prinz, K. Ramaswami, M. Terk, L.E. Weiss, Shape deposition manufacturing, in: Proceedings of the Solid Freeform Fabrication Symposium, The University of Texas at Austin, August 1994. [6] I. Ilinkin, R. Janardan, J. Majhi, J. Schwerdt, M. Smid, R. Sriram, A decomposition-based approach to layered manufacturing, Comput. Geom., in press. [7] P. Singh, D. Dutta, Multi-direction slicing for layered manufacturing, J. Comput. Inform. Sci. Eng. 1 (June 2001).
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[8] P. Gupta, R. Janardan, J. Majhi, T.C. Woo, Efficient geometric algorithms for workpiece orientation in 4- and 5-axis NC machining, Comput.-Aid. Des. 28 (8) (1996) 577±587. [9] T.C. Woo, Visibility maps and spherical algorithms, Comput.-Aid. Des. 28 (1) (1994) 6±16. [10] K.P. Karunakaran, S. Dibbi, P. ViveKananda Shanmuganathan, Efficient stock cutting for laminated manufacturing, Comput.-Aid. Des. 34 (4) (2002) 281±298.