Computer-Aided Design 57 (2014) 15–28

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Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Automatic generation of mold-piece regions and parting curves forcomplex CAD models in multi-piece mold design✩

Alan C. Lin a,∗, Nguyen Huu Quang b

a Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 106, Taiwanb Department of Mechanical Engineering, University of Economic and Technical Industries, 456 Minh Khai Street, Hai Ba Trung District, Hanoi, Vietnam

a r t i c l e i n f o

Article history:Received 7 February 2013Accepted 25 June 2014

Keywords:Mold designParting curvesMulti-piece molds

a b s t r a c t

Multi-piece molding technology is an important tool in producing complex-shaped parts that cannot bemade by traditional two-piece molds. However, designing multi-piece molds is also a time-consumingtask. This paper proposes an approach for automatic recognition ofmold-piece regions and parting curvesfor free-form CAD models. Based on the geometric properties of objects and mathematical conditions ofmoldability, a collection of feasible parting directions is formed from which the sets of visible-moldablesurfaces are identified for each parting direction.Moldable surfaces that are partially visible to a particularparting direction are recognized and divided into fragments by silhouette detection and edge extrusion. Aset of criteria is proposed to arrange tentative fragments, which can be simultaneously visible to severalparting directions, into appropriate regions formold pieces. Finally, the parting curves formold pieces areextracted from the corresponding mold-piece regions. The proposed algorithm overcomes the problemsfound in previous multi-piece molds and at the same time achieves high accuracy and high performance.Examples of industrially complex models are used to demonstrate the performance and robustness ofthe proposed algorithm. The approach is generic in nature, allowing its application to be extended to anycomplex geometry in 3-D mold design.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Injection molding is one of the most common manufacturingprocesses extensively used today; it is able to produce parts withgood quality and accuracy. From a geometric perspective, moldscan be either two-piece or multi-piece. Conventional two-piecemolds have only one primary parting direction, which constrainsthe two mold pieces to one axis of motion. However, undercutscan often be found in complex parts and, as a result, a numberof side-cores is required to form the shapes of these undercuts.The higher the number of side-cores required, the higher thetooling costs and time consumed in the machine. Some complexparts with multiple undercuts may even be impossible to producewith two-piece molds. On the contrary, the use of multi-piecemolds can overcome the aforementioned restrictions of traditionaltwo-piece molds. With many different parting directions allowingfree movement of mold pieces, undercuts can be eliminated sothat no actual side-cores are required, thus significantly reducing

✩ This paper has been recommended for acceptance by Kin-Chuen Hui.∗ Corresponding author.

E-mail addresses: alin@mail.ntust.edu.tw, alanclin0313@gmail.com (A.C. Lin).

http://dx.doi.org/10.1016/j.cad.2014.06.0140010-4485/© 2014 Elsevier Ltd. All rights reserved.

total manufacturing costs. Consequently, multi-piece molding hasbecome an important technology in handling complex geometriesthat cannot be made solely by using two-piece molds. Fig. 1 showsexamples of typical molded parts, two-piece molds, and multi-piece molds provided by Protoform GmbH.

The main challenge in multi-piece mold design lies in iden-tifying feasible parting directions, mold-piece regions, and thecorresponding parting curves and parting surfaces. In this paper,a systematic approach to automating the determination of theserequisite data for constructing mold pieces is proposed. The ba-sic problem of multi-piece molds can be described thus: given afree-form CAD model, how do we determine the feasible partingdirections D, mold-piece regions Si for each element di of D, andthe corresponding parting curves?

The remainder of the paper has been organized in the followingmanner: in Section 2, related works are reviewed and an overviewof the proposed algorithm is presented. Background informationof surface moldability and ray testing for surface visibility are de-scribed in Section 3. Section 4 discusses the algorithm for au-tomatic recognition of mold-piece regions and parting curves.Several proposed criteria are also included. The algorithm imple-mentation and examples are discussed in Section 5. Conclusionsare given in Section 6.

16 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

Fig. 1. Examples of molded parts, two-piece molds, and multi-piece molds.

2. Related works and overview of the algorithm

2.1. Related works

Automated processes for mold design have been developed inmany publications. Most work focuses on traditional two-piecemolds, including the determination of parting direction, undercuts,parting curves, and parting surfaces. Fu et al. [1] introducedan algorithm related to the determination, classification, andrecognition of feature parameters for detecting undercuts. Ismailet al. [2] proposed amethod to recognize cylindrical-based featuresbased on an edge-boundary classification technique. Kharderkaret al. [3] described an algorithm to identify and display undercutfeatures by implementing the Gauss map. Building upon theserelated techniques, Fu et al. [4] proposed an algorithm thatdetermined the optimal parting direction in injection-moldedparts by the number of undercut features and their correspondingvolumes. Furthermore, Chen et al. [5] posed a method in whichthree possible parting directions were defined by surface normalvectors of a bounding box. Feasible parting directions were thenestimated based on the dexel model and fuzzy decision-making.

In order to determine parting curves and parting surfaces,Fu et al. [6] described a technique employing the maximum ex-ternal edge loops between the core- and cavity-molded surfacegroups. Chakraborty et al. [7] presented a method to determinethe parting curve and parting surface for a two-piece permanentmold based on a combination of the surface area of the undercut,the flatness of the parting surface, and the draw depth. In addi-tion, Wong et al. [8] proposed an uneven slicing approach to find-ing the feasible parting curves of a CADmodel. The optimal partingcurve was evaluated based on the criteria described by Ravi andSrinivasan [9]. Furthermore, Paramio et al. [10] evaluated the de-moldability of injection-molded parts through the slicing of theirCAD models, which could then be used to identify feasible partingcurves. For the generation of parting surfaces, Li et al. [11,12] posedan approach by evaluating the extrudability of parting curves; asubdivision technique was employed to generate parting surfaceregions for the portions of the parting curves that were not extrud-able.

Side cores or pins can be generated along with the recognitionof undercuts. Banerjee et al. [13] used the retraction space ofeach undercut surface to identify the shapes of the side cores. Theundercut surfaces were grouped into undercut regions accordingto a discrete set of feasible and non-dominated retractions, afterwhich the geometry of individual side cores could be obtained.Fu [14] used the concepts of surface visibility, demoldability,and moldability to identify the surfaces molded by side cores.Furthermore, Ran and Fu [15] described an algorithm for automaticdesign of internal pins after identifying the inner undercuts andextracting the related surfaces.

The development of CAD research formold design has also beenconnected to manufacturability and manufacturing costs. Bidkaret al. [16] presented a feature recognition method based on the el-emental cubes to assess the critical manufacturability information

of injection-molded parts. Denkena et al. [17] introduced amethodin which a CAD-based application of the calculation tool ‘visualform calculator’ was used to generate and analyze CAD models ofmold cavities in order to compute tool accessibility and manufac-turing costs.

In the area of multi-piece molding, few articles have been pub-lished. Dhaliwal et al. [18] presented a feature-based approachto automatic design of multi-piece sacrificial molds. In their ap-proach, the desired gross mold shape is decomposed into simplershapes for manufacturability and assemblability purposes. By thesame rationale, Huang et al. [19] described an algorithm for gen-erating multi-piece sacrificial molds with an accessibility drivenspatial partitioning scheme. Chen and Rosen [20,21] introduced aregion-basedmethod for partitioning parting surfaces into regionsand combining them into mold pieces. The basic elements in theirapproach are concave regions and convex surfaces. A reverse glueoperation is then proposed to automate the construction of multi-piece molds based on the generation of parting surfaces. In addi-tion, Priyadarshi et al. [22] described a geometric algorithm forautomatic design of multi-piece permanent molds. The moldpieces are constructed based on the results of a global accessibilityanalysis of the part.

2.2. Problems of multi-piece mold design

As summarized previously, the most significant advantage ofmulti-piece molds over conventional two-piece molds is that theycan be used to handle complex-shaped parts. However, current ap-proaches to multi-piece mold design have two limitations. First,they require simple polyhedral parts or approximate complexparts by facets [18–22], which may not be acceptable in the indus-try. Second, some algorithmshave limited application domains. Forinstance, the algorithms proposed byDhaliwal [18] andHuang [19]do not handle general partitioning cases such as partitioning alongnon-planar faces. Moreover, in Huang’s approach, manufacturabil-ity is measured only by the number of cuts involved in the parti-tioning. In reality, manufacturability is more directly related to thenumber of components and their geometric complexity. In Chenand Rosen’s method [20,21], only local disassemblability evalua-tion is performedwhen generating parting surfaces. As a result, thecurrent paper proposes a systematic approach to automatic recog-nition of mold-piece regions and parting curves for constructingmold pieces. In the proposed approach, several criteria and tech-niques, such as visibility-ray testing and silhouette detection, aredeveloped to determinemoldable surfaces and appropriate surfaceregions for mold pieces. The algorithm is sufficiently generic to beapplied in commercial CAD systems.

2.3. Overview of the proposed algorithm

All curved/free-form surfaces of the input CAD model are in-serted into the proposed system. Information of the geometric en-tities of the model (vertices, edges, and surfaces) is also extractedand used as input for the algorithm. Equations describing edgesand surfaces are formed based on such information. The followingsteps have been developed to generate suitable parting directionsand parting curves for the piece.

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 17

sisi

a b c

Fig. 2. (a) Moldable planar surface, (b) moldable quadric surface, and (c) moldable free-form surface.

(1) A collection D of tentative parting directions is first formedby analyzing the geometric features of the input part. This isdescribed in Section 4.1.

(2) Using these tentative parting directions, as well as thesurfaces of the CAD model, surface visibility and moldability areexamined for each surface of the model. The set Si of visible-moldable surfaces for each direction di of the collection D is thendetermined. This is described in Section 4.2.

(3) If there exists a surface that is not in any visible-moldablesurface sets, a further accessibility analysis is performed to deter-mine a feasible parting direction for such a surface. This parting di-rection is then inserted into the collection D. This step is describedin Section 4.3.

(4) All visible-moldable surfaces of sets Si are used to determinethe different regions for mold pieces. In the case where a surfacesimultaneously belongs to two or more surface sets, the surfaceis rearranged into the most appropriate mold-piece region. This isdescribed in Section 4.4.

(5) Finally, the algorithm for locating parting curves of mold-piece regions is implemented. All outer and inner loops of the part-ing curves are determined for further generation of mold-pieceparting surfaces and structures. This process is detailed in Sec-tion 4.5.

3. Background information on moldable surfaces and raytesting for visibility

In order to describe the proposed algorithm, the definitionsof moldable surfaces, which are extended from those of surfacevisibility and moldability mentioned in [6], are first presented.This section focuses on the following three types of surfaces:planar surfaces (first-order surfaces), quadric surfaces, and free-form surfaces (third-order surfaces or higher). It is assumed that siis a surface of model M and ni is the normal vector of an arbitrarypoint on si. Let di be one of the parting directions of model M .Surface si is moldable in direction di if the following condition ismet:

ni · di ≥ 0. (1)

The physical meaning of Eq. (1) is that rays from infinity that areparallel to the parting direction di cannot cast any shadow in thisdirection onto the surface; in other words, the surface is visible inthese rays. Equality occurs when the surface is a vertical wall.

For planar and quadric surfaces, the normal vectors of points onthese two types of surfaces possess exact directions, thus allowingthe fulfillment of Eq. (1) to be easily confirmed (refer to Fig. 2(a) and(b)). However, for a free-form surface, a sub-division methodmustbe employed: the surface is divided into small regions with equaldistances in parametric coordinates u and v, and the normal vectorni,j at a corresponding node (i, j) of the surface is determined. Theentire set of normal vectors is then used to confirm the fulfillmentof Eq. (1). Only if the equation is met for all nodes can the surfacebe defined as moldable. Fig. 2(c) shows an example of a moldablefree-form surface along parting direction di.

For surfaces of revolution, a substantially different methodmust be applied to determine the moldable surfaces: the axis

Fig. 3. Moldable and unmoldable conical surfaces.

of revolution is used, instead of the normal vector, to ascertainwhether a revolved surface can be molded. In general, cylindersand cones are present in the design of industrial parts. When thesurface is a cylinder, the condition for moldability isax · di = 1 (2)where ax is the axis of the cylinder. Eq. (2) asserts that a cylindri-cal surface is moldable if its axial vector is parallel to the partingdirection di. As for conical surfaces, the following equation is used:

ax · di ≥ cosα

2

(3)

where ax is the axis of the cone and α is the angle at the apex.Eq. (3) shows that a conical surface ismoldable if the angle betweenits axis and parting direction di is smaller than α/2. Moreover, thissituation allows removal of the part from themoldwithout any dif-ficulty caused by these conical surfaces. Fig. 3 shows an example ofmoldable and unmoldable conical surfaces.

Once a surface is defined as moldable in a certain direction, itsvisibility must be examined. Accordingly, all moldable surfaces ineach parting direction di are collected and stored in a so-called ‘S-table’. Each moldable surface si is then tested in sequence by raysto determine whether it is obscured by other surfaces included inthe S-table. Each ray is originated from the end point or themiddlepoint pi of each edge ei of the test surface si along direction di. Ifthe rays do not intersect with other surfaces in the S-table exceptthe points pi themselves, the test surface si is identified as a visibleone; otherwise, si is identified as an invisible surface. Examples ofvisible and invisible moldable surfaces in a direction di of a modelare shown in Fig. 4.

4. Algorithm for automatic parting curve generation of multi-piece molds

4.1. Collection of tentative parting directions

Inmulti-piecemold design, the parting direction is the directionalong which a mold piece can be separated without any obstacles,and thus the determination of feasible parting directions is a prior-ity issue that needs to be addressed. In our approach, a collection Dof tentative parting directions is formed based on the geometricalproperties of the part’s features by considering the following threetypes of directions: relative coordinate axes, axes of features of rev-olution, and normal directions of planar surfaces (which follows

18 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

a b

Fig. 4. (a) Visible-moldable surface and (b) invisible-moldable surface.

Fig. 5. Tentative parting directions. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

that of [22]). These are the directions from which most surfaces ofthe part can be accessed. Fig. 5 shows a typical part to be analyzedduring the design process ofmulti-piecemolds. All tentative direc-tions for accessing the surfaces of the part are foundusing the prop-erties of its geometric features; these are presented with differentcolored arrows in the figure. Note that for each tentative partingdirection di in D, −di is also included in the collection. The partshown in the figure will be used as a running example for describ-ing steps of the proposed algorithm.

4.2. Formation of visible-moldable surfaces

The accessibility of all surfaces of a part is analyzed accordingto the tentative parting directions in collectionD. Visible-moldablesurfaces for each direction di in D can then be found. The processis performed in the following steps:

Step 1: Identify moldable surfaces for direction di based onEqs. (1)–(3). All moldable surfaces related to each partingdirection di are grouped into a set Si.

Step 2: For eachmoldable surfaces of set Si, a ray test, as presentedin Section 3, is performed to determine whether thissurface is visible in direction−di. Invisible surfaces, whichare obscured by other surfaces in Si, are removed from Si.

As shown in Fig. 6, both the red and blue surfaces of the partare identified as moldable surfaces along parting direction d1.However, the red surface is completely obscured by other surfaceswhen viewed from infinity along direction −d1. Thus, the redsurface is removed from visible-moldable surface set S1.

During the process of identifying visible-moldable surfaces,theremay be two caseswhere the surfaces are partially visible. Thefirst is called the ‘dual moldable surface’ where Eqs. (1), (2), and/or(3) are satisfied only by a group of points on these surfaces (see thegreen surfaces in Fig. 7(a)). The second is the ‘partially obscuredsurface’ inwhich only a part of its area is obscured by other surfaces

Fig. 6. Visible and invisiblemoldable surfaces. (For interpretation of the referencesto color in this figure legend, the reader is referred to theweb version of this article.)

(see the red surface in Fig. 7(b), which is partially obscured by theblue surfaces when viewed from infinity along −d3).(a) Dual moldable surface

A silhouette-detecting algorithm is employed to identify theexact boundary between the visible and invisible regions of a dualmoldable surface. First, a dualmoldable surface S(u, v) is describedby a NURBS equation, which is generally in the following form:

S(u, v) =

x(u, v)y(u, v)z(u, v)

=

n1i=0

n2j=0

Bi,k(u) · Bj,l(v) · wi,j · Ci,j

n1i=0

n2j=0

Bi,k(u) · Bj,l(v) · wi,j

wi,j > 0 (4)

Bi,k =u − ui

ui+k − uiBi,k−1(u) +

ui+k+1 − uui+k+1 − ui+1

Bi+1,k−1(u)

Bi,0 =

1, ui ≤ u ≤ ui+10, otherwise.

Bj,l =v − vj

vj+l − vjBj,l−1(v) +

vj+l+1 − v

vj+l+1 − vj+1Bj+1,l−1(v)

Bj,0 =

1, vj ≤ v ≤ vj+10, otherwise

where surface S(u, v) has degree k in parameter u and degree lin parameter v, Ci,j is the control point, wi,j is its correspondingweight, n1 and n2 are the number of control points in directionsu and v, and Bi,k and Bj,l are the B-spline basis functions defined inu and v. The silhouette of a free-form object is typically defined asthe set of points on the object’s surface where the surface normalvector is perpendicular to the vector from the viewpoint. A point onsurface S(u, v) with corresponding surface normal vector N(u, v)is a silhouette point if the angle between the parting direction diand N(u, v) is 90°. This means that the following constraint mustbe met:

di · N(u, v) = 0 (5)

in which N(u, v) is computed by the following rational equation:

N(u, v) =

Nx(u, v)Ny(u, v)Nz(u, v)

=

∂S(u, v)

∂u×

∂S(u, v)

∂v. (6)

Let Nm(u, v) =n1

i=0n2

j=0 Bi,k(u) · Bj,l(v) · wi,j · Ci,j; its partialderivative with respect to parameter u is Nm′

u(u, v) =n1

i=0n2

j=0

Bui,k(u) · Bj,l(v) · wi,j · Ci,j. Also, let Dn(u, v) =

n1i=0n2

j=0 Bi,k(u) ·

Bj,l(v)·wi,j so that its partial derivativewith respect to parameter u

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 19

a

b

Fig. 7. (a) Dual moldable surfaces and (b) partially obscured surface. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Fig. 8. Surface regions divided from dual moldable surfaces.

is Dn′u(u, v) =

n1i=0n2

j=0 Bui,k(u) ·Bj,l(v) ·wi,j ·Bu

i,k is the derivativeof a B-spline basis function and can be expressed by the followingequation:

Bui,k(u) =

k − 1ui+k − ui

Bi,k−1(u) +k − 1

ui+k+1 − ui+1Bi+1,k−1(u). (7)

Therefore, the partial derivative of S(u, v) with respect to u is:

∂S(u, v)

∂u=

Nm(u, v)

Dn(u, v)

′

u

=Nm′

u(u, v) · Dn(u, v) − Dn′u(u, v) · Nm(u, v)

Dn2(u, v)

=Nm′

u(u, v) · Dn(u, v) − Dn′u(u, v) · Dn(u, v) · S(u, v)

Dn2(u, v)

=Nm′

u(u, v) − Dn′u(u, v) · S(u, v)

Dn(u, v). (8)

Similarly, the partial derivative of S(u, v) with respect to v is:

∂S(u, v)

∂v=

Nm′u(u, v) − Dn′

u(u, v) · S(u, v)

Dn(u, v). (9)

A silhouette curve can be found by substituting Eqs. (6)–(9) intoEq. (5). In fact, Eq. (5) is a polynomial equation of two variables uand v. In this scenario, there are fewer equality constraints thanthere are variables. Its zero-solution set can be computed basedon the convex hull and subdivision properties of rational splinefunctions. A set of discrete points approximating the zero set isgenerated by recursive subdivision based on the Newton–Raphsonmethod. The method to solve a general system of m polynomialequations of n variables is described by G. Elber and M.S. Kim [23].Fig. 8 shows an example of surface regions divided fromdualmold-able surfaces of the pedal part in Fig. 5.(b) Partially obscured surface

This type of surface will be divided into regions that arecompletely visible or invisible to the current parting directionthrough an edge-extrusion algorithm, as illustrated in Fig. 9. In the

Fig. 9. Division of a partially obscured surface s1 into two surface regions.

figure, surface s1 is partially obscured by surface s2. Each boundaryedge of s2 is used to build an extrusion surface s3 along the currentparting direction. The intersection between surface s1 and surfaces3 can then be used to divide s1 into an un-obscured region and anobscured region.

Fig. 10 shows an example of the surface regions divided froma partially obscured surface of the part in Fig. 5. The red surfaceis partially obscured by others when viewed from infinity alongdirections −d3, −d5, and −d6. The edge-extrusion algorithm isthus applied, resulting in the eleven surface regions divided fromthe red surface as shown.

4.3. Determination of additional parting directions

Assuming that there are some ‘inaccessible’ surfaces invisiblefrom the directions in D, an analysis is performed to identifyadditional parting directions (APDs) along which such surfaces arevisible. APDs found are inserted into the collection D.

The determination of APDs is implemented via the use of G-mapand V-map introduced in [24]. A G-map is a map of surface normalsonto a unit sphere, where each point on themap represents the in-tersection of the transferred surface normal vectorwith the surfaceof the unit sphere. On the other hand, a V-map of a surface is a setof points in the unit sphere whereby every point in the V-map de-viates from the corresponding point in the G-map by an angle less

20 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

Fig. 10. Surface regions divided from partially obscured surfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

a

b

c

Fig. 11. (a) Inaccessible surfaces of a pocket, (b) V-map of the pocket, and (c) APD found from the V-map.

than 90°. To identify the G-map and V-map of each free-form sur-face of the CAD part, we employ Eqs. (6)–(9) in computing the nor-mal vectors ni,j at the corresponding nodes (i, j) of each surface.

Fig. 11 illustrates the determination of an APD using a V-map.The pocket feature of the example part in Fig. 11(a) has fiveinaccessible surfaces, each of whose V-maps is a hemisphere(Fig. 11(b)). The intersection of these hemispherical V-maps formsthe V-map of the pocket. The vector passing through the centralpoint of the V-map and its origin located at the unit sphere is theAPD of the pocket (Fig. 11(c)).

In the proposed algorithm, each of APDs is only identified forinaccessible surfaces which are in the same undercut feature.Of course, if the determination process of an APD is applied toinaccessible surfaces belonging to different undercut features, it ispossible that no feasible parting direction can be found. Therefore,all inaccessible surfaces must be classified into different groupsbefore the V -map and G-map methods can be used. Each groupconsists of adjacent inaccessible surfaces connected together toform an undercut feature (Fig. 12).

To classify inaccessible surfaces into different groups, all inac-cessible surfaces are collected andnumbered from is1 to isn (where:n is the total number of inaccessible surfaces). Each of inacces-sible surface isi will be examined to find its adjacent surfaces. Inthe process, if two surfaces have at least one common edge, theyare considered as adjacent ones. Once inaccessible surface isj is

Fig. 12. Example of groups of adjacent inaccessible surfaces.

determined as the adjacent surface of isi, isj will be inserted intothe same group of isi, or in other words, they are in the same un-dercut feature.

Importantly, an APD may be a possible parting direction forthe visible-moldable surface set Si identified previously. As shownin Fig. 13, groups s1, s2, and s3 are identified as visible-moldablesurfaces associated with parting direction d1 while all surfaces ofthe pocket are visible-moldable with the APD d7 determined bythe V-maps. However, with d7, the surface group (s1, s2, s3) andthe surfaces of the pocket are all visible-moldable. In such a case,the surfaces are re-classified into the same set associated with APDd7. Therefore, once an APD of an undercut feature is identified,

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 21

Fig. 13. Possible parting directions for one surfaces set and the pocket.

its accessibility to the surfaces of each set Si must be analyzed. Ifall surfaces of Si are accessible by the APD, the surfaces of Si aregrouped with the surfaces of the undercut feature to create a newsurface set associated with the APD. This process ensures that thenumber of mold pieces is reduced to a minimum.

4.4. Formation of regions for mold pieces

The regions for mold pieces are determined using all visible-moldable surfaces of each set Si associated with direction di. Theseregions are groups of adjacent surfaces connected together to forma united region. In the proposed algorithm, a region of a moldpiece is called ‘mold-piece region’, and is denoted by Ri. Individualvisible-moldable surfaces and surface regions divided from dualmoldable surfaces or divided from partially obscured surfaces aregenerally called ‘fragments’. In reality, a fragment of surface setsSi may either belong to exactly one mold-piece region associatedwith oneparting directiondi (inwhich case it is an ‘exact fragment’,denoted by f e), or to several mold-piece regions associated withseveral parting directions (inwhich case it is a ‘tentative fragment’,denoted by f t ). Rearrangement of tentative fragments in surfacesets Si into appropriate regions is the most important process inthe formation of mold-piece regions. Three criteria are proposedfor this work.

Criterion 1. If a tentative fragment f t is passed through whenwithdrawing the mold piece of an exact fragment f e along its

corresponding parting direction, the fragment f t is rearranged tothe same mold piece region as the fragment f e.

As shown in Fig. 14, the fragment f e is obscured by othersurfaces when viewed from infinity along direction −d3. Thus, it isan exact fragment that only belongs to the mold piece associatedwith parting directiond5. Conversely, f t is a tentative fragment thatis visible in both directions −d3 and −d5; hence, it can belong toeither mold pieces associated with d3 or d5. However, the moldpiece containing f e will pass through f t if it is withdrawn alongdirection d5. Therefore, f t is rearranged into the same region as f e.

To check whether a tentative fragment f t is passed throughby the mold piece of the exact fragment f e, we create extrusionsurfaces from edges of f e along the parting direction. If the interiorof f t intersects with the extrusion surfaces, f t is considered to bepassed through by the mold piece of f e.

Criterion 2. When a tentative fragment f t is adjacent to severalexact fragments f e of several mold-piece regions, the tentativefragment is rearranged into themold-piece region having themostexact fragments adjacent to f t .

As shown in Fig. 15, fragments f e1 and f e2 are exact fragmentsof the mold-piece region associated with parting direction d5.Fragment f t1 is a tentative fragment adjacent to both f e1 andf e2 . Moreover, there are no exact fragments of other mold-pieceregions adjacent to f t1 . Thus, f

t1 is rearranged into the same mold-

piece region as f e1 and f e2 .

Criterion 3. A tentative fragment f t is rearranged into the mold-piece region associated with the parting direction along which thewithdraw distance of f t is minimal.

Thewithdrawdistance of a tentative fragment f t ismeasured bythe distance along theparting direction from the farthest point of f tto the surface of the bounding box of themolded part. Fig. 16 showstwo withdraw distances of a tentative fragment; these correspondto two parting directions d3 and d5, and are measured from thefarthest points p2 and p1, respectively.

In Fig. 17, the tentative fragment f t can belong to either of themold-piece regions associated with parting directions d3 and d5.However, the withdraw distance of f t corresponding to direction

Fig. 14. Tentative fragment passed through by the mold piece of an exact fragment.

Fig. 15. Tentative fragment adjacent to exact fragments.

22 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

Fig. 16. Withdraw distances of a tentative fragment along the parting directions.

d3 is less than that corresponding to d5. Therefore, f t is rearrangedinto the mold-piece region associated with direction d3.

The order of applying the three criteria must be determined torearrange tentative fragments. Among the three criteria, Crite-rion 1 must be applied first. It is used until no more tentative frag-ment passed through by the mold-pieces of the exact fragmentshas been found. That is because if a tentative fragment f t is rear-ranged into amold-piece region basedwithout first checking Crite-rion 1, the corresponding mold-piece region created may not validas a united region. As shown in Fig. 18, the tentative fragment f t1is passed through by the exact fragment f e1 when withdrawing themold-piece of f e1 along the direction d5. If f t1 is rearranged into othermold-piece region different from that of f e1 (according to the Crite-rion 2 or 3), the mold-piece region of f e1 will have a gap in betweenand it cannot create a valid mold-piece structure.

Next, Criterion 2 should be applied before Criterion 3. In reality,if a tentative fragment is processed by first considering Criterion 3,it can create a very complicated or an invalidmold-piece structure.As shown in Fig. 19, tentative fragments f t4 ∼ f t8 are onlyadjacent to exact fragments f te1 , f e1 , f e3 , f e4 of the mold-piece regionR5 associated with the direction d5. Conversely, the withdrawdistances of f t4 ∼ f t8 along the direction d3 are shorter than thatof f t4 ∼ f t8 along the direction d5. If f t4 ∼ f t8 are rearranged into themold-piece region R3 due to the consideration of Criterion 3, boththe mold-piece region R3 and R5 cannot create reasonable mold-piece structures which cause difficulties in industrial machining.

Fig. 19. Unreasonable mold-piece regions.

Moreover, the mold-piece created from R3 may not be withdrawnalong the direction d3.

Besides the proposed criteria, the order in which the setsSi of visible-moldable surfaces are processed in computing thecorresponding mold-piece regions affects the whole geometries ofmulti-piece molds. As mentioned in Section 4.1, parting directionsare selected from a set of different tentative directions. Therefore,the sets Si are divided into three groups: a group of Si associatedwith coordinate axes, a group associated with axes of revolutionfeatures and normal directions of planar surfaces, and a groupassociatedwith APDs. In the proposed algorithm, the following twodirectives are utilized for these groups: (1) The group associatedwith the coordinate axes is processed before the group associatedwith axes of revolution features or normal directions of planarsurfaces. The group associated with APDs is processed last. (2) For

Fig. 17. Tentative fragment with its minimal withdraw distance.

Fig. 18. Invalid mold-piece region due to the existence of a gap.

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 23

Fig. 20. Flow diagram of generation of mold-piece regions.

sets Si in each group, the processing orders are based on the totalarea of their included surfaces. The set Si with the largest totalsurface area is processed first and the one with the smallest isprocessed last. After these directives, all sets Si are rank orderedand indexed from i =1 to i = imax (where imax is equal to the totalnumber of sets Si).

All tentative fragments are then rearranged into appropriatemold-piece regions Ri with the processed order of sets Si andthe criteria proposed above. The complete mold-piece region

algorithm (MPR algorithm) is presented in Fig. 20. After the MPRalgorithm is performed, mold-piece regions associated with allparting directions di are identified to further generate partingcurves and construct mold-piece structures.

The application of the MPR algorithm is illustrated for allfragments shown in Fig. 21. In the figure, f e1 , f e2 , f e3 , and f e4 are exactfragments of S5, and can only belong to the mold-piece regionassociated with parting direction d5; f e5 is an exact fragment ofS3, and can only belong to the mold-piece region associated withparting direction d3. In processing set S5, the tentative fragmentf t1 is adjacent to f e1 and is passed through by the mold piece off e1 associated with direction d5. Thus, f t1 is rearranged into thesame mold-piece region as f e1 (following Criterion 1). Fragment f t1is immediately updated to be a new exact fragment for furtherprocessing. Next, fragment f t3 is adjacent only to exact fragments ofthemold-piece region associatedwithd5; it is therefore rearrangedinto this mold-piece region (adhering to Criterion 2). Similarly, f t3is immediately updated to be a new exact fragment. Criterion 2 isalso repeated for tentative fragments f t4 ∼ f t8 .

In processing S3, tentative fragments f t2 and f t11 are rearrangedinto the same mold-piece region as the exact fragment f e5associated with parting direction d3 (again following Criterion 1).Meanwhile, f t9 and f t10 are tentative fragments that can belong toboth mold-piece regions associated with d3 and d5. However, thewithdraw distance of f t9 and f t10 corresponding to d3 is less thanthat corresponding to d5. Hence, f t9 and f t10 are rearranged intothe mold-piece region associated with direction d3 (as stipulatedby Criterion 3). This analysis continues for other fragments untiltentative fragments are exhausted.

4.5. Location of parting curves for mold pieces

Finally, all mold-piece regions Ri are used to obtain partingcurves for mold pieces. In general, a parting curve is a closed loopthat identifies surfaces at which the mold is split into differentpieces. Hence, parting curves are the boundaries of mold-pieceregions. There may exist one or several closed loops depending onthe existence of ‘hole-features’. If there is more than one closedloop, the following technique is employed to identify the outer andinner loops: all loops are projected onto a plane perpendicular tothe parting direction associated with the mold-piece region under

Fig. 21. Fragments for applying theMPR algorithm.

24 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

a

b

Fig. 22. (a) Two different views of a mold-piece region, and (b) parting curves generated for the mold-piece region.

consideration. The loop that contains all other loops is identified asthe outer loop and those remaining are inner loops.

The process for determining the boundary of each mold-pieceregion involves the removal of all common edges. First, all edgesof the surfaces of a mold-piece region are checked to identifycommon edges. In particular, an edge e is determined to be acommon edge (eCM ) of two adjacent surfaces if the followingequation is met:

e ∈ (CB(sj) ∧ CB(sk)) j, k = 1, 2, 3 . . . n (10)

where sj and sk are surfaces of themold-piece regionwhose partingcurve is to be determined, n is the total number of surfaces inthis region, and CB(sj) and CB(sk) are the corresponding closedboundaries of surfaces sj and sk, respectively. In Eq. (10), the term(CB(sj) ∧ CB(sk)) refers to the edges simultaneously belonging tothe closed boundary loops of both surfaces sj and sk. Now, letΣ1 bethe set of all edges of all surfaces in the mold-piece region and Σ2be the set of all common edges eCM ; then the parting curve epc of amold-piece region is determined as follows:

epc = (Σ1 − Σ2) . (11)

The physical meaning of Eq. (11) is that the parting curves of amold-piece region are the remaining external edges after subtract-ing all the common edges between adjacent surfaces included inthe region. Fig. 22(a) shows a mold-piece region associated withone parting direction of the molded part mentioned in Figs. 5, and22(b) is its parting curve. As seen from the figure, the boundaryedges form one outer loop and three inner loops from which theparting surfaces and mold piece may be suitably constructed.

5. Implementation

The proposed algorithm has been implemented using theAPI-based Pro/Toolkit for Pro/Engineer Wildfire 5.0. Geometricinformation of the CAD parts (vertices, edges, and surfaces) is

extracted and used as input for the algorithm. The part featuredin this research is a pedal; it has been used as a running examplein the previous sections to illustrate our algorithm. The proposedsystem analyzed all surfaces of the part, identifying the feasibleparting directions and corresponding visible-moldable surfaces.Six directions d1 ∼ d6 were found relative to the coordinateaxes; these formed a sufficient collection of parting directions. Byusing the MPR algorithm discussed in Section 4.4, six mold-pieceregions were determined based on the surface sets’ visibility andmoldability, as shown in Fig. 23. The final parting curves for eachmold piece were generated by connecting the external edges ofouter surfaces after subtracting all common edges. As seen fromthe figure, these parting curves can be used directly to generateparting surfaces for mold design.

To demonstrate in more details the ability of the proposedsystem, another complex CADpart is used. The part is a plugmodel.As shown in Fig. 24, six possible directions denoted from d1 to d6are found to form a sufficient collection of parting directions.

When determining visible-moldable surfaces sets for eachparting direction di, there are some tentative fragments which canbelong to several mold-piece regions (Fig. 25). They are actuallyfragments divided from partially obscured surfaces when viewedfrom infinity along the direction −d3, −d4.

In Fig. 25, fragments from f e21 to f e26 are exact fragments of S5,and can only belong to the mold-piece region R5 associated withparting direction d5.

When processing S5, tentative fragments f t26 is adjacent to theexact fragment f e24 and is passed through by the mold-piece off e24 along the parting direction d5. Thus, f t26 is rearranged into thesame mold-piece region R5 with f e24 (following to Criterion 1). Itis immediately updated to be a new exact fragment for furtherprocess. It is similar to other tentative fragments f t21, f

t27. Besides,

tentative fragments f t22, ft23 are adjacent tomost of exact fragments

of S5, therefore they are also rearranged into mold-piece region R5

(adhering to Criterion 2). Meanwhile, f t28 is the tentative fragment

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 25

a b

c d

e f

g h

Fig. 23. Mold-piece regions and parting curves generated for the implemented example.

that can belong to several mold-piece regions associated withd1, d2, d3, and d5. However, the withdraw distance of f t28 along tod5 is shortest. Hence, f t28 is rearranged into the mold-piece regionR5 associated with direction d5 (as stipulated by Criterion 3). Theanalysis continues for other fragments of S5 until nomore tentativefragment is found.

The argument is similar for tentative fragments f t11, ft12, and f t13

of surfaces set S1. After the implementation of the MPR algorithm,all tentative fragments are rearranged into appropriatemold-pieceregions associated with parting directions d1 ∼ d6. Fig. 26 showssix mold-piece regions and their corresponding parting curves for

the example part. They are then used to extract successfully sixmold-pieces as shown in Fig. 27.

For efficient implementation of the algorithm, a filteredprogram was employed to prune unnecessary information. Afterthe CAD model was loaded, the working mode of the system didnot require any adjustment from the user. Through the runningexample—the pedal part that we used to present the capacity ofthe algorithms, the computation time was acceptable althoughit was not a simple part. It found all tentative parting directionsof the set D (following the rules described in Section 4.1) inabout two seconds. Based on the three proposed criteria, the MPR

26 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

i j

k l

m

Fig. 23. (continued)

algorithm successfully created six valid mold-piece regions andcorresponding parting curves from the sets of visible-moldablesurfaceswithin 5min. For the plug part, the total computation timeis about 4 min and 12 s. On other complex parts it is capable of

finding feasible solution. However, it takes the computation timemore than 5 min. The reason is that the optimality may not beguaranteed in our programming. This will be updated in the futurework.

A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28 27

Fig. 24. Sufficient collection of parting directions for the plug model.

Fig. 25. Rearranging tentative fragments into appropriate mold-piece regions.

The numerical experiments show that for high-order interpo-lation (fourth order or higher) it can be more convenient to makedistributions of nodes of curves/surfaces to reduce the conditionnumbers of the resulting elemental matrices. Moreover, using agood distribution of nodes, the number of computational matricescan be reduced two or three times, in comparisonwith the originaldistribution of nodes.

6. Limitations

There are some limitations and future works can be addressedas follows. First, the set D of tentative parting directions isdetermined based on three types of directions: relative coordinateaxes, axes of features of revolution, and normal directions ofplanar surfaces. These directions are always good candidates fora parting direction of a mold-piece. If there are surfaces thatcannot be accessible from any of the directions in D, the APD’salgorithm will be applied to detect a feasible direction for theseinaccessible surfaces. However, it may be possible that even afterapplying the APD’s algorithm, these surfaces are still inaccessibleas shown in Fig. 28. The pocket A is located inside the pocket B.The APD (denoted by the blue vector) of inaccessible surfaces ofthe pocket A intersect with a surface of the pocket B. Therefore,the corresponding mold-piece created from inaccessible surfacesof the pocket A cannot be freely withdrawn along only the APDdirection. In the proposed algorithm, such part is rejected as non-moldable. Although for industrial parts, such cases are rare, theimprovements for the proposed algorithm to create split-cores forsuch parts must be required in the future works.

Second, using the visibility map to identify the APD direction ofundercut features may not be satisfied in some cases. As shown inFig. 29, the V -map of each planar surface is a hemisphere but theV -map of the entire part is empty. Indeed, there is not a feasibledirection from which the interior of the model is visible from the

(a) Mold-piece region 1 and its corresponding parting curves.

(b) Mold-piece region 2 and its corresponding parting curves.

(c) Mold-piece region 3 and its corresponding parting curves.

(d) Mold-piece region 4 and its corresponding partingcurves.

(e) Mold-piece region 5 and its corresponding partingcurves.

(f) Mold-piece region 6 and its corresponding parting curves.

(g) Parting surfaces extended from parting curves.

Fig. 26. Mold-piece regions and their corresponding parting curves.

exterior. The main reason is that the visibility map of an objectis constructed using mainly the local accessibility information in-stead of the global accessibility. Although such kinds of structuresare not common in industrial molded parts, the consideration ofglobal accessibility is still required. Global accessibility also en-

28 A.C. Lin, N.H. Quang / Computer-Aided Design 57 (2014) 15–28

Fig. 27. Mold-pieces of the plug model.

Fig. 28. Example of inappropriate APD. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

Fig. 29. Example of the part with an empty V -map.

sures the problem that the tentative parting directions set is com-plete and the mold-pieces can be disassembled in the design ofmulti-piece multi-cavity molds.

7. Conclusions

This paper presents a practical approach to determine mold-piece regions and parting curves of the complex CAD parts in

the field of multi-piece mold design. There are two significantimprovements with respect to the following characteristics:• The approach can automatically recognize both mold-piece

regions and parting curves for complex CAD models. Basedon the three criteria and the MPR algorithm proposed in thepaper, visible-moldable surface regions for eachmold-piece areidentified to generate reasonable mold-piece structures.

• Current approaches to multi-piece mold design normallyrequire polyhedral parts or approximate complex parts byfacets whichmay not be acceptable in industry. In the proposedapproach, the system can handle surface models containingfree-form surfaces by dividing surfaces into small fragments forrearranging into appropriate mold-piece regions. The capacityof the proposed system has been shown through examplesdescribed in the Implementation section.

Although there exist some limitations, we expect that the algo-rithms described in this paper will provide feasible foundations forautomating themold-piece regions and parting curves. Through il-lustrative example parts, the approach is proven robust and adapt-able for use with current CAD/CAM systems and applications inindustry. It will help in reducing the working time in the field ofmulti-piece mold design and manufacturing.

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