Automatic recognition of intersecting features for side core design in two-piece permanent molds

ORIGINAL ARTICLE

Automatic recognition of intersecting features for side coredesign in two-piece permanent molds

Rajnish Bassi & N. Venkata Reddy & Sanjeev Bedi

Received: 17 September 2008 /Accepted: 7 January 2010 /Published online: 6 February 2010# Springer-Verlag London Limited 2010

Abstract The automatic recognition of molding features(protrusions, depressions, and their intersections), core, andcavity surfaces plays an important role in shortening the leadtime in mold design and manufacturing as well as aiding theside core design. Consequently, in this paper, an automaticmold feature recognition system to recognize protrusion,depression as well as intersecting depression features isproposed and implemented. The recognition of various typesof intersecting features is a significant contribution to theliterature. The output generated by the accessibility analysis(without discretizing the part) is used as the input to thefeature recognition module. The newly developed system isassessed by comparing its results with those of earliersystems. A comprehensive case study is presented that candemonstrate the additional capabilities of the proposedsystem to those of the present in the published literature.

Keywords Feature recognition . Accessibility analysis .

Intersecting depression features . Die/Mold design .

Mold undercuts

1 Introduction

The design and fabrication of two-piece permanent moldsand their additional components are crucial to near net-

shape molding/casting processes. In fact, more than 70% ofnon-standard components in the consumer product marketsare manufactured by these processes [1]. In addition, molddesign and fabrication are on the critical path of any newproduct development due to the lead time. Computer-aidedengineering design (CAED) systems built to assist engi-neers in tasks such as the analysis, simulation, design, andmanufacturing of a product, are relied on to design tools toreduce the lead time and to optimize the design. In the lasttwo decades, many researchers have attempted to automatethe process of finding the optimal parting direction, lines,and surfaces. In spite of efforts to automate the featurerecognition for side core/cavity as well as the core andcavity design, their success is limited. In the subsequentparagraphs, a review of the literature, pertaining to variousaspects of mold design with an emphasis on automating theside core design, is presented.

Most of the methodologies developed for automatingmold design rely on visibility and accessibility analysis fordetermining the demoldability of the given part along theparting direction. Hui and Tan [2] have used semi-infiniterays, originating from surface grid points toward the chosenparting direction, to classify the surfaces as obstructed orunobstructed. They have defined a factor (i.e., a blockingfactor) to determine the extent of the blockage along thechosen parting direction. Woo [3] has described theimportance of visibility maps in various manufacturingapplications, including die and mold design. Chen et al. [4]have used the concept of visibility maps developed by Woo[3], to determine the best parting direction. The best partingdirection is determined by locating a pair of antipodalpoints that is contained by maximum number of V-maps.However, the approach cannot be used for parts with non-planar surfaces. Later, Chen et al. [5] have identified twolevels of visibility, i.e., complete and partial, to determine

R. Bassi : S. Bedi (*)Department of Mechanical Engineering, University of Waterloo,Waterloo, ON, Canadae-mail: [email protected]

N. V. ReddyDepartment of Mechanical Engineering,Indian Institute of Technology,Kanpur, India

Int J Adv Manuf Technol (2010) 50:421–439DOI 10.1007/s00170-010-2524-7

the demoldability of polyhedral surfaces. Furthermore, theauthors have divided the partially visible surfaces into thosethat can be molded with and without side cores. Dhaliwal etal. [6] have determined global accessibility cones for allfacets of a polyhedral part. The exact inaccessible region fora facet is computed to obtain the global accessibility conesfor all facets of the part. Priydarshi et al. [7] have developeda methodology to automate the design of multi-piecepermanent molds. The researchers have determined theaccessibility of each facet along the chosen parting direction,by checking the obstruction of each facet with rest of thefacets of the part. For near-vertical facets, they have slightlyrotated the viewing direction in such way that a near-verticalfacet becomes a front-facing facet. Such a methodology isapplicable to polyhedral objects since free-form surfaces areapproximated by using a number of linear surfaces.

There have also been attempts to determine accessibilitywith the aid of computer graphics hardware. Khardekar etal. [8] have developed a user routine to identify theundercut faces of a part by using the depth texturecapabilities of graphics hardware, i.e., by using the depthvalues at each pixel location. The authors have used thetessellated CAD model as an input. Later, Priyadarshi andGupta [9] have used the graphics hardware capabilities toidentify various mold-piece regions of a part by assumingthat each facet belongs to one mold-piece region.

Ravi and Srinivasan [10] have proposed nine criteria fordetermining the optimal parting surface for a given partingdirection. To determine the optimal parting lines forinjection molding and die casting, Weinstein and Manoo-chehri [11] have presented an approach considering thecombined effect of the parting line complexity, draw depth,number of undercuts, number of unique side cores, andmachining complexity but is limited to polyhedral partsonly. Later, Fu et al. [12] have classified the surfaces intothree groups (core, cavity, and local tool-molded surfaces)based on their geometrical and topological characteristicsand then determined the parting lines by using the externaledges of the core or cavity molded surfaces. Wong et al.[13] have proposed an algorithm to determine the partinglines by using uneven slicing planes that are normal tochosen draw direction. Although this method can work onall kinds of surfaces, it does not consider all possibledirections; hence, global optimality is not guaranteed.Rubio et al. [14] have also proposed an approach basedon slicing a part CAD model for moldability analysis. Inaddition, a 2D intersection profile analysis has been carriedout to obtain the parting lines and demoldability zones.Madan et al. [15] have determined the optimal partingdirection and parting lines for die-cast parts. They haverecognized interacting and nested die-cast features byapplying geometric reasoning. However, the study islimited to polyhedral parts only. Chakraborty and Reddy

[16] have developed a system that can take an STL(stereolithography) file of the part as the input anddetermine the best pair of parting directions, parting line/surfaces, and side core/cavity directions for two-piecepermanent molds. A weighted sum of the minimization ofthe surface area of the undercuts, the flatness of the partingline, and the minimization of the draw depth is conductedto determine the best pair of parting directions. However,the system cannot handle components requiring split cores.

To automate the side core design process, it is importantto recognize the mold features. In the molding context, afeature is either a depression or a protrusion. For automaticfeature recognition, Ganter and Skoglund [17] havepresented an approach to extract three types of side corefeatures: internal voids, single and multi-surface holes, andboundary perturbations. However, the approach can recog-nize depression undercuts only. Nee et al. [18] havedeveloped an algorithm, based on part geometry andtopology, to determine optimal parting direction and toextract undercut features. The undercuts are classifiedaccording to the characteristics of the edge loops thatconnect target and adjacent surfaces, and the releasedirections are determined. Their method is appropriate forcylindrical, conical, spherical, revolved, and B-splinesurfaces. However, B-spline surfaces are analyzed bydetermining the surface characteristics at various points.Fu et al. [1] have improved upon the work of Nee et al. [18]by considering the blending surfaces (fillets) as part of theundercut. An optimal parting direction is attained byminimizing the volume of the undercuts. But, interactingfeatures cannot be recognized by the authors' methodology.

Yin et al. [19] have used the concept of volumedecomposition for determining the optimal parting directionand recognizing the interactive depression undercut features.To identify the undercuts, the maximal connected compo-nents are determined by the convex hull difference anddecomposed into convex cells. Subsequently, the potentialfeatures are reconstructed by systematically connecting theseconvex cells by using the non-directional blocking graph ofthe cells. Yin et al.'s work is limited to polyhedral parts only.Bidkar and McAdams [20] have decomposed the part intosmall elements (cubes), and their accessibility and relation-ship is used to recognize the features and to assess the partmoldability. The mesh size is determined by the smallestfeature that can be produced by that manufacturing process.However, an appropriate and variable mesh size is necessaryto make the approach robust.

Joshi and Chang [21] have introduced a graph isomor-phism based approach for recognizing polyhedral machin-ing features by using an attributed adjacency graph. Later,Ye et al. [22] have developed a hybrid method (graph-basedcombined with hint-based) to recognize the interactingmold undercut features by searching the cut-set of the

422 Int J Adv Manuf Technol (2010) 50:421–439

undercut sub-graph. The face and edge properties are usedas hints to recognize the interacting features. However, thehybrid method cannot determine the feature parameters,namely, the volume and release direction. Therefore, sidecores cannot be generated with this approach. Niraj et al.[23] have developed a polyhedron face adjacency graph torecognize the undercut features of the part with planar andBezier surfaces. To obtain such a graph, the authors havedivided the Bezier surfaces into planer polyhedron surfaces.

Hui and Tan [2] have used a solid sweep and regularizedboolean operations to generate the core and cavity of aninjection-molded part, but are limited to parts with nothrough holes. For side core extraction, Shin and Lee [24]have developed a procedure to identify interference facesby applying polygon overlap tests. To generate the sidecores, the Euler operation is used which is inherentlyefficient. However, Shin and Lee's method can generateambiguous results, when two or more than two undercutsoverlap. Ye et al. [25] have extended their own approach[22] of using attributed adjacency graphs to recognizeinteractive undercuts and to generate side cores. To form aside core, a bounding box was created to enclose theblocking portion of the undercut feature. The box wastrimmed with all the faces of the feature and then sweptalong the release direction to obtain a linkage portion. Theunion of the linkage portion and blockage portion formedthe side core for the undercut feature. However, their workhas not considered intersecting features [16] which areusually present in the molded parts. Recently, Banerjee andGupta [26] have developed an algorithm to automaticallygenerate the shapes of the side actions by using the STLformat. The side actions are secondary mold pieces whichare removed from the part by using translation directionsother than the mold opening directions. Given a moldopening direction, a side action translation motion isgenerated by determining a collision-free 2D translationspace for horizontal and vertical accessibility to eachundercut facet. Here, the side action is considered to beretracted in a plane perpendicular to the mold openingdirection only. However, the approach gives an optimalsolution only if each connected undercut region of the partrequires three or fewer side actions.

The previous literature review indicates that manyattempts have been made to automate the determination ofthe parting line and surface, compared with the research onmold feature recognition for automating the design of sidecores/cavities, cores, and cavities. The techniques torecognize features are either limited to polyhedral parts(including STL) or parts containing, at best, Beziersurfaces. In addition, most of the research, related tomolding feature recognition, is limited to non-complexfeatures with no tolerance for arbitrary interaction betweenthe features. Lately, Ye et al. [25] have classified and

recognized some interacting features by combining graph-and hint-based approaches. However, their approach canlead to ambiguous results, if more than one candidate edgeexists for a cut-set at some nodes, i.e., if more than oneclosed loop exists as a cut-set for the undercut.

The intention of this paper is to derive an automaticfeature recognition methodology for tool design by takingthe part geometry as the input in the B-Rep format and itsspecified parting direction. In practice, most of the moldingparts are designed by keeping the parting direction in mind[26]. Therefore, the parting direction is taken as the inputalong with the part geometry from the user.

2 Problem statement and solution methodology

2.1 Problem statement

The objective of the work, described in this paper, is todevelop an algorithm for the automatic recognition of core,cavity, and side core/cavity surfaces (their release directionas well) of two-piece permanent molds, ensuring thedemoldability of the given part along the given partingdirection.

Input:

& A 3D part in the B-rep format& A parting direction for demoldability

Output:

& The recognition of the core, cavity, and side core/cavitysurfaces

2.2 Solution methodology

The part faces are first classified according to theaccessibility in the given parting direction. The accessibilityanalysis is carried out by a continuous approach rather thana discretized approach, and the procedure is discussed inSection 3. After the accessibility-based face classification,the feature recognition process begins. The mold featuresare classified based on geometry, topology, and accessibil-ity of the part faces into simple protrusion and intersectingdepression features in Section 4. The part faces thatconstitute a simple protrusion are identified, and theprocedure to recognize protrusion features is discussed inSection 5. Next, the intersecting depression features arerecognized in Section 6 by using the release direction ofdepression features. To maintain the flow, in Section 6, theprocedure for determining the release direction of the moldfutures is detailed in Appendix B. The face classificationand feature recognition algorithms have been tested onmany parts; four of these are described in Section 7 to

Int J Adv Manuf Technol (2010) 50:421–439 423

demonstrate the working of the proposed algorithms. Ablock diagram of the approach, developed in the presentwork, is shown in Fig. 1.

3 Accessibility analysis

The objective of the accessibility analysis is to classify eachface of the solid part/object (O) into different categoriesaccording to its accessibility in the given parting directiond!� �

. The output generated by accessibility analysis is usedas the input to the feature recognition module.

A face/surface S is classified into the followingcategories, as shown in Fig. 2, in relation to the results oftranslating S to infinity in the direction d

!:

& Fully-accessible: if the translation does not cause anintersection with the interior of the object O.

& Fully-inaccessible: if the translation results in anintersection of all the points on S with the interior ofthe object O, at least once.

& Partially-accessible: if the translation results in anintersection of some points on S with the interior ofthe object O.

& Outer-boundary-inaccessible: if the translation resultsin an intersection of some inside points and all theboundary points of S with the interior of the object O.

& Partial-outer-boundary-accessible: if the translationresults in an intersection of some inside points and someboundary points of S with the interior of the object O.

3.1 Methodology overview

The part faces are classified in terms of their orientation, andthis classification is used as the input for the accessibility

analysis. The faces are classified by using the cosine of angle(θ) between the surface normal and the positive partingdirection þ d

!� �, resulting in four categories, namely,

positive, negative, perpendicular, and dual, as listed inTable 1. The idea is to minimize the checks during theaccessibility analysis, e.g., a positively oriented face cannotbe accessible from a negative parting direction. Therefore, anaccessibility analysis of the positive and negative faces iscarried out in the positive þ d

!� �and the negative � d

!� �

parting direction, respectively. However, a perpendicular facecan be accessible in both the positive and negative partingdirections; therefore, the perpendicular faces are analyzed inboth directions. Most of the parts have dual faces, i.e., faceswith positive and negative orientations, as shown in Fig. 3a.Those surfaces are divided into positive, negative, andperpendicular surfaces by generating the silhouettes in theparting direction, as illustrated in Fig. 3.

The accessibility analysis of the part is performedalong a given parting direction to classify each surface(S) as Fully-accessible, Fully-inaccessible, and Partially-accessible. The positive and negative Partially-accessiblesurfaces are further divided into Partial-outer-boundary-accessible and Outer-boundary-inaccessible in relation tothe accessibility of the outer edge boundary of the consi-dered surface.

To identify a Fully-inaccessible surface, it is furthercategorized into two types: type1 and type2. If the inacces-sible surface is shadowed by a connected set of interferingsurfaces, it is classified as type1; otherwise, it is classifiedas type2, denoted in Fig. 4.

The methodology adopted for accessibility analysis ofpositive and negative surfaces is presented first and isfollowed by the methodology adopted for perpendicularsurfaces.

3.2 Methodology adopted for positive and negative surfaces

Each part surface undergoes a series of sweeping andregularized boolean operations and is classified at dif-ferent steps of the process by applying various geo-metrical and topological checks, as described in thefollowing sub-sections.

3.2.1 Fully-accessible surfaces

Given a B-rep part (Boriginal) and parting direction � d!� �

,each one of the positive/negative surfaces (Sconsidered) isselected and swept along the þ d

!=� d

!direction, respec-

tively, by a distance greater than the maximum dimension(say twice the maximum dimension) of the part in theparting direction to generate a new solid body Bswept1. Forinstance, surface Sconsidered, shown in Fig. 5a, is swept togenerate a new body Bswept1, in Fig. 5b. Here, Bswept1

Protrusion Features Recognition

Accessibility Analysis

Edge Classification

Release Directions Determination

A part in B-Rep Format along with Parting Direction

Depression Features Recognition

Determining Various Intersections in RecognizedDepression Features

Fig. 1 Mold feature recognition procedure


represents the space swept by Sconsidered, as it translatesalong the parting direction during demolding.

If Bswept1 does not interfere with Boriginal, then Sconsideredshould be accessible from parting direction without anyobstruction. To check the interference of original body(Boriginal) with Bswept1, a regularized boolean subtraction,shown in Fig. 5c, is performed between the bodies asfollows:

Bresult ¼ Boriginal � »Bswept1 ð1ÞThe interference is checked by comparing the volumes

of Boriginal and Bresult. If the volume of the resulting body(Bresult) is equal to that of Boriginal, then the surface underconsideration is classified as Fully-accessible in thatdirection and no further test is required for Sconsidered.

3.2.2 Fully-inaccessible (type1) surfaces

If the surface is not classified as Fully-accessible, then thefollowing regularized boolean operation is considered to

evaluate the surface for partial-accessibility and fully-inaccessibility:

Bresult1 ¼ Bswept1 � »Boriginal ð2ÞIf Bswept1 interferes with Boriginal, then the above regular-

ized boolean subtraction results in the interfering space beingsubtracted from Bswept1, depicted in Fig. 5d. If the resultingbody (Bresult1) has two or more disjoined bodies, then itmeans that the interfering surfaces of the original bodyare connected, and they shadow the considered surfacecompletely, in which case, Sconsidered is classified as Fully-inaccessible (type1) along the considered direction. Other-wise, a further analysis is carried out to determine theinaccessibility of type2 and the partial accessibility of theconsidered surface.

3.2.3 Partially-accessible and fully-inaccessible (type2)surfaces

IfBresult1 is a single body, as shown in Fig. 5d, then Sconsideredcan be either partially covered by the shadow of theinterfering surfaces or fully covered by the shadow of thedisconnected interfering surfaces, e.g., surface S3 in Fig. 2.For further analysis, the surfaces of Bresult1 are classified interms of their orientation with respect to the parting directionof the part to be molded, as represented in Fig. 5e.

If the surface Sconsidered is positive, then the negativesurfaces (one at a time) of the resulting body Bresult1 areswept in the+direction by a distance greater than themaximum dimension of Bresult1 in that direction, to generatea new solid body Bswept2, in Fig. 5f. These negative surfaces

Table 1 Orientation-based classification

Cosine of angle (θ) Surface type

Cosine (θ)>0 Positive

Cosine (θ)<0 Negative

Cosine (θ)=0 Perpendicular

Cosine (θ)≤0 and cosine (θ)>0 Dual

d+

d−

1S

2S

3S

4S

5S

Fig. 2 Accessibility classification, considering surface S translatesalong þ d

!: (S1) “Fully-Accessible”, (S2 and S3) “Fully-Inaccessible”,

(S4) “Outer-Boundary Inaccessible” (“Partially-Accessible”), and (S5)“Partial-Outer-Boundary-Accessible” (“Partially-Accessible”)

Fig. 2 Accessibility classification, considering surface S translates along þ d!: (S1) “Fully-Accessible”, (S2 and S3) “Fully-Inaccessible”, (S4)

“Outer-Boundary Inaccessible” (“Partially-Accessible”), and (S5) “Partial-Outer-Boundary-Accessible” (“Partially-Accessible”)


of Bresult1 are swept because they correspond to the positivesurfaces of Boriginal that cast their shadows on Sconsidered inthe parting direction. To determine if any portion ofSconsidered is not under the shadow of the interferingsurfaces, the following regularized boolean operation iscarried out:

Bresult2 ¼ Bresult1 � »Bswept2 ð3Þ

If any negative surface (except the surface correspondingto Sconsidered) is identified on Bresult2 after the boolean opera-tion, then Bresult2 is renamed as Bresult1, and the booleanoperation is carried out again. This process is continued untilall the negative surfaces on Bresult2, except the surfacecorresponding to Sconsidered, are considered. If the resultingbody Bresult2, as shown in Fig. 5g, has the same height (h) asthat of Bswept1, then it is concluded that the surface underconsideration (Sconsidered) is Partially-accessible; otherwise, itis Fully-inaccessible surface (type2).

3.2.4 Boundary-based classification

To further classify the Partially-accessible surfaces intoPartial-outer-boundary-accessible and Outer-boundary-inaccessible, the outer edge boundary of the positive surfacesat the uppermost level of Bresult2, shown by the bold lines in

Fig. 5g, is compared with the outer edge boundary of theoriginal surface Sconsidered. If there is a full or partialoverlapping of the edges when projected on a plane inthe � d

!direction, then Sconsidered is classified as Partial-

outer-boundary-accessible; otherwise, as Outer-boundary-inaccessible.

The procedure in Fig. 5 for a Partial-outer-boundary-accessible surface demonstrates all the steps of theaccessibility analysis and is explained here. The surfaceSconsidered, in Fig. 5a, is swept in the given parting direction(assume the Y-axis) to generate the new body Bswept1, inFig. 5b. After the regularized boolean operation isperformed as per Eq. 1, the volume of the resulting body,Bresult, in Fig. 5c is compared with that of Boriginal and isfound to be different. Therefore, this surface is notclassified as Fully-accessible. To analyze further, a regular-ized boolean operation is performed between Bswept1 andBoriginal, as per Eq. 2, to generate the new body Bresult1,reflected in Fig. 5d. Since Bresult1 has only one solid body,the surface is not classified as Fully-inaccessible (type1).Next, the negative surfaces of Bresult1, Fig. 5e, except thesurface corresponding to Sconsidered, are swept in thepositive parting direction, as depicted in Fig. 5f. The sweptbody Bswept2 is subtracted from the Bresult1 as per Eq. 3 togenerate the new body Bresult2, in Fig. 5g. Since the height(h) of Bresult2 is the same as that of Bswept1, the surface is not

(a) (b) (c)type1 type2 type2

d+

d−

Fig. 4 Inaccessible surfacesclassification: a “type1”, b, andc “type2” inaccessible

(a) Before splitting (b) After splitting

Perpendicular Dual

Negative

Positive Positive

Perpendicular

Negative

d+

d−

Fig. 3 Orientation-basedclassification


classified as Fully-inaccessible (type2). Next, the outeredge boundary of the uppermost positive surface of Bresult2

(the bold lines in Fig. 5g) is compared with that ofSconsidered. Since their outer edge boundaries partiallyoverlap when projected on a plane in the parting direction,the considered surface is classified as Partial-outer-boundary-accessible, not as Outer-boundary-inaccessible.

3.3 Methodology adopted for perpendicular surfaces

A perpendicular surface differs from a positive/negativesurface since it can be accessible from both � d

!directions,

if it is not obstructed by any other surfaces of the part.Moreover, a perpendicular surface might not be visible in theparting direction, but can be accessible. To determine theaccessibility of the perpendicular surfaces, Priyadarshi andGupta [7] slightly rotated the near-vertical facets of a STLpart such that the near-vertical facet becomes a front-facet inthe parting direction. Later, Priydarshi et al. [9] determinedthe accessibility of near-vertical facets by thickening the

concave contour edges. To determine the accessibility of theperpendicular faces in the B-rep format, a new accessibilityevaluation methodology is developed and presented next.

The perpendicular surface under consideration, Sconsidered,illustrated in Fig. 6a, is offset by a small distance (say0.5 mm) towards the outward drawn normal of the samesurface. Thereafter, a new solid body (Boffset) is generatedbetween offset surface and Sconsidered, in Fig. 6b. The acces-sibility of Boffset is evaluated to determine the accessibilityof Sconsidered.

The accessibility of Sconsidered is evaluated by assessingall the positive /negative surfaces of Boffset from theþ d!=� d

!direction, respectively, by the procedure in

Section 3.2. It is observed that if all the positive surfacesare fully accessible, then Sconsidered is Fully-accessiblefrom the þ d

!parting direction. If all of the positive

surfaces are inaccessible, then Sconsidered is consideredas Fully-inaccessible from þ d

!. The following operations

are carried out on Boffset to classify the perpendicularsurfaces based on accessibility.

Fig. 5 Accessibility analysis procedure for “Partial-outer-boundary-accessible” surfaces: a Boriginal, Sconsidered, and the parting direction� d!

along Y-axis, b Boriginal and Bswept1, c Bresult as per Eq. 1, d Bresult1

as per Eq. 2, e negative face of Bresult1, f Bresult1 and Bswept2, g Bresult2

as per Eq. 3


To evaluate Sconsidered from þ d!, all the positive surfaces

of Boffset are swept towards þ d!

by a distance greater thanthe maximum dimension (say twice the maximum dimen-sion) of the part in the parting direction and generating anumber of solid bodies (Bswept_1, Bswept_2...... Bswept_i,where i is the total number of swept bodies), shown inFig. 6c. A similar procedure is adopted for evaluating theaccessibility from the � d

!direction by sweeping the

negative surfaces toward � d!. Next, the following regular-

ized boolean union operation is performed on Boffset andBswept_i bodies, depicted in Fig. 6d, resulting in a singlebody Bcombined:

Bcombined ¼ Boffset þ »Bswept 1 þ »Bswept 2::::::

þ »Bswept i ð4ÞFinally, the accessibility is evaluated by performing the

boolean operations similar to those represented in Eqs. 1and 2 by replacing Bswept1 with Bcombined. After performingthe operations given in Eqs. 1 and 2, the surface underconsideration is classified either as Fully-accessible, Fully-inaccessible, or Partially-accessible. Partially-accessibleperpendicular surfaces are not further classified as is donefor the positive/negative surfaces because such surfacescannot be Outer-boundary-inaccessible.

4 Classification of the molding features

Molding features/undercuts hinder the part demoldabilityand require additional elements, other than the core andcavity, to ensure demoldability. Each feature is recognizedby using the geometric and topologic information, embed-ded inside the B-Rep format of the given solid part.

In the feature recognition module, the output of the faceclassification module (discussed in Section 3) is used as aninput. In addition to the face classification, each edge isclassified into three categories (convex, concave, andtangent) to recognize the features. In a valid solid model,an edge is shared by two faces only and is categorized byconsidering the angle (φ) between the two adjacent faces onthe material side. The criteria for classifying an edge aregiven in Table 2.

BoriB iginal

SconsiS ideredd

(a)

(d)

BcombB

B

bined

oriB

+

d−

iginal

Bd

l

or

B

iginal

swepB

offB

B

1_pt

ffset

swB

_wept

B

2

offseB

(b)

(c)

t

B

B

B

sweB

swepB

origB

3_ept

4_ept

ginal

3

4

Fig. 6 Accessibility analysisprocedure for perpendicular sur-faces: a Boriginal, Sconsidered, andthe parting direction � d

!along

Y-axis, b Boriginal and Boffset, cBoriginal, Boffset, and Bswept_i, dBoriginal and Bcombined

Table 2 Edge classification criteria

Edge type Angle (φ)

Concave >180°

Convex <180°

Tangent =180°


4.1 Molding features classification

The literature and observation of many molded componentsindicate that molding features can be broadly classified intoprotrusions, depressions, and their intersections. Twofeatures are considered to be intersecting if their face-setintersect to alter each other's shape and need differentmolding elements to ensure demoldability. Intersectingfeatures are further classified depending on the elementsused during demolding. To facilitate the recognition, themold features are defined next.

“Protrusion” is the restriction region of the part duringdemolding and signifies material addition over the baseface (feature “FT6”, Fig. 7); whereas, a “depression”signifies material removal from the base face (feature“FT8”, Fig. 7). A depression undercut can be furtherclassified as through (feature “FT7”, Fig. 7) or blind(feature “FT1”, Fig. 7).

The intersecting depression features are classified intotwo main categories: Type-I intersection and Type-IIintersection. A Type-I intersection results in connectingthe core and cavity features either to each other or tofeatures moldable by using side cores and requires shut-offsurfaces to ensure demoldability (feature “FT5”, Fig. 7). A“Type-II intersection” is the result of the intersection of twodepression features that are moldable by using only the sidecores (feature “FT8”, Fig. 7).

Furthermore, “Type-II” intersecting features are classifiedinto two categories, depending on the edge boundary at theintersection of the two depression features. If one depressionfeature exists inside another depression feature, is bounded byclosed convex edge loops and requires side/split-core fordemoldability, then the features are called as a child depressionfeature (feature “FT3” and “FT4”, Fig. 7) and a parentdepression feature (feature “FT1”, Fig. 7), respectively, andboth are called “Bounded-Type-II Intersecting” features.Otherwise, if two depression features intersect with eachother such that they are moldable by using the side cores andthe intersection boundary is not definable by a convex edgeloop, then the two features are called “Unbounded-Type-IIIntersecting” features (feature “FT8”, Fig. 7).

In the feature recognition procedure, the protrusions arerecognized first, described in Section 5, as their presencesometimes affects the characteristics of the face(s) fromwhich they originate, e.g., base face f2 is partially accessibleas shown in Fig. 8. Then, depression features arerecognized, detailed in Section 6, by suppressing theinformation of protrusion features from the part. Next,Type-I intersecting features are recognized, in Section 6.2,prior to recognizing the Type-II intersecting features, inSection 6.3. At the end, the remaining faces that are not partof any of the recognized feature are grouped into individualfeatures (core, cavity, and additional) according to theiradjacency and accessibility.

Fig. 7 Nomenclature of mold-ing features: a moldable part, bpart in different view, and cfeature nomenclature (“FT1” isa “Blind Depression Parent”feature, “FT2” is a “ProtrusionChild” feature, “FT3” is a“Blind Depression Child”feature, “FT4” is a “ThroughDepression Child” feature,“FT5” is a “Type-I Intersecting”feature, “FT6” is a “SimpleProtrusion” feature, “FT7” is a“Simple Depression” feature,and “FT8” is a “Unbounded-Type-II Intersecting” feature)


5 Recognition of protrusion features

Simple protrusions have concave and/or tangent boundingedges along the base faces (e.g., f2 and f3 are the base facesfor the simple protrusion in Fig. 8) and consist of a convexface-set. A convex face-set of the simple protrusion isdenoted in green in Fig. 8. However, the protrusion featurescan have concave faces (signified in red in Fig. 8) due to theintersection with the depression(s), as well as protrusion(s).The recognition of such intersecting protrusions is beyondthe scope of this paper.

5.1 Procedure for protrusion features recognition

A positive/negative face with at least one concave edge ischosen as the seed face to initiate the recognition process,e.g., face f1 is chosen as the seed face for the simpleprotrusion in Fig. 8. All the adjacent faces of the seed face,sharing one or more convex edges with it, are grouped in anarray (“Protrusion_Face_Set”) along with the seed face.Next, the faces adjacent to these grouped faces that sharethe convex edge with them are also added to the same array.This process is continued until all the faces, grouped in thearray, are processed (the grouped faces for the seed face f1are shown in green in Fig. 8). However, if the facesgrouped in the “Protrusion_Face_Set” are all positive or allnegative, then the grouped faces cannot form a restrictionregion and are not classified as protrusion faces (e.g., groupof f4 and f5 faces in Fig. 8). These faces are not processedfurther. Next, all the concave edges of the faces in the“Protrusion_Face_Set” array are identified, stored in anarray (“Protrusion_Bounding_Edges”) and arranged basedon their connectivity. If the arranged concave edges formone closed loop (such a loop is shown in Fig. 8 in bold),then the faces grouped in the “Protrusion_Face_Set” arrayare classified as the face-set of a simple protrusion.

6 Recognition of depression features

Depression undercuts have “Fully-inaccessible/Partially-accessible” faces and need side/split-cores to be molded.

In the feature recognition procedure, the existence of adepression undercut is indicated by the presence of “Fully-inaccessible” or “Partially-accessible” faces (e.g., face f1 inFig. 9) that are identified during the accessibilityanalysis, as discussed in Section 3. The “Partial-outer-boundary-accessible” faces are split manually into “Fully-accessible” and “Fully-inaccessible” faces. Any of the“Fully-inaccessible/Partially-accessible” faces can be usedas a seed face for starting the recognition process. Thefaces that are identified as member of the protrusion face-set are not considered as the seed face. In addition, theseed face should not be obstructed by only the face-set ofidentified protrusions.

6.1 Procedure for identifying and grouping depression faces

Once the seed face is identified, the recognition process ofthe depression feature is initiated. “Fully-inaccessible” or“Partially-accessible” faces, adjacent to the seed face, aregrouped in an array (“Depression_Face_Set”) that alsoincludes the seed face. Next, “Fully-inaccessible/Partially-accessible” faces, adjacent to the faces in the array, are alsoadded to the same array. This process is continued until allthe faces in the array have been processed. However, thefaces, obstructed only by the faces forming the identifiedprotrusions, are not considered as part of undercut (e.g.,face f1 in Fig. 7) and are not added to the array. (The faces,obstructed by identified protrusion only, are grouped in thecategory of the “Additional Faces” at the end of the featurerecognition process. These “Additional Faces” should beconsidered while determining the parting line and mightneed additional elements to ensure demoldability).

Once the “Depression_Face_Set” array is populated, thebounding edges of the grouped faces are stored in an arraycalled “Depression_Bounding_Edges”, and these edges arearranged according to their connectivity. In Fig. 9, the facesof the “Depression_Face_Set” array, for seed face f1, are

Fig. 9 Bounding edge loops of depression undercut feature “FT1” ofFig. 7

Fig. 8 Simple protrusion face-set, considering parting direction alongthe Y-axis


shown in yellow, and the edge loops of “Depression_Bounding_Edges” array are shown in bold lines. A depres-sion feature can have more than one closed loop, based on thedepression type.

It should be noted that, in Fig. 9, face f2 is not included inthe “Depression_Face_Set” array because it is fully visiblethrough the loop formed by the Type-I intersection. Thus, theaforementioned procedure does not include “Fully-Accessi-ble” faces that are part of the undercut in the given partingdirection into “Depression_Face_Set” array of the consideredundercut. The identification of such missing faces and theiraddition to the “Depression_Face_Set” array requires furtherprocessing.

6.1.1 Procedure for adding missing faces to the depressionfaces

The convex hull faces of the part are determined by usingthe approach given in Appendix A. Then, the part faces aretemporarily categorized into three types: the faces includedin the “Depression_Face_Set” array, convex hull faces, andremaining faces. The remaining faces are further sub-grouped based on their adjacency, and then, each sub-group is checked to see if it is enclosed by the faces of the“Depression_Face_Set” array. If the sub-group is enclosed,then the sub-grouped faces (e.g., face f2 in Fig. 9) are addedto the “Depression_Face_Set” array.

Next, the “Type-I Intersecting” features (e.g., featureFT5 in Fig. 7) which are accessible in the parting directionand, at the same time, interact with the faces of thedepression features that require the side core, must beidentified. These features form an interacting zone thatneeds to be shut-off to ensure demoldability. Moreover,the zone can be interpreted as an undercut opening,instead of an intersection, for calculating the releasedirection. Such intersections must be identified, and thecorresponding closed loops need to be removed from“Depression_Bounding_Edges” array. In this paper, thefollowing method is adopted to recognize such intersect-ing features.

6.2 Procedure for Type-I intersecting features recognition

To recognize the Type-I intersecting features accessiblefrom þ d

!direction, as shown in Fig. 10a, a negative face

of the “Depression_Face_Set” array that is nearer fromoutside the part in þ d

!direction is identified, and the face

is used as a seed face, in Fig. 10b. All the negative andperpendicular faces adjacent to the negatively oriented seedface are selected from the “Depression_Face_Set” array andare sub-grouped into an array (“TypeI_Bounding_Faces”).Negative and perpendicular faces, adjacent to faces of“TypeI_Bounding_Faces” array, of “Depression_Face_Set”

array are also grouped to “Type I_Bounding_Faces” array,as shown in Fig. 10b. If the “Type-I” intersection is present,then there is a closed convex edge boundary within theouter edge boundary of the grouped faces, shown usingbold lines in Fig. 10c. A similar approach is adopted forrecognizing “Type-I Intersecting” features those are acces-sible from � d

!direction. The convex edge loop needs to be

shut-off to ensure demoldability, and the loop edges arestored in an array called “TypeI_Bounding_Edges”. Oncethe “Type-I Intersecting” features are identified, the “Type-II Intersecting” features are detected by the followingprocedure.

6.3 Procedure for identifying the presence of Type-IIintersecting features

To determine the type of depression feature (blind, through,or intersecting), the faces in the “Depression_Face_Set”array are further processed by determining the releasedirection and performing the accessibility analysis withrespect to the release direction. It is evident from the earlierdiscussions that the presence of more than one loop in the“Depression_Bounding_Edges” array indicates that thedepression under consideration can be through, intersecting,or a combination of these two; otherwise, it is blind with orwithout a child.

Let the depression under consideration be formed by thefaces belonging to setU and stored in a “Depression_Face_Set”array, as stated in Section 6.1. Corresponding “Depression_-Bounding_Edges” array contains the information of number ofedge loops and related information, as stated in Section 6.1.The methodologies adopted to classify the depressions are asfollows.

& The possible range of the release direction(s) of eachundercut is determined, first, to classify the depression.The release direction is determined for each edge looppresent in “Depression_Bounding_Edges” array. Thedetailed procedure for determining the release directionis presented in Appendix B to maintain continuity.

& Let Ui be the set indicating depression faces fullyaccessible along the release direction determined by usingloop li, and i varies from 1 to N, where N is the totalnumber of loops in each “Depression_Bounding_Edges”array. Note that U contains all the faces forming theundercut, including the protrusion face-sets that arepresent in that undercut. The undercut type is determinedby using the set theory as follows.

Let N be the total number of loops present in“Depression_Bounding_Edges” array.

U={f: f is a face of the depression undercut underevaluation}


Ui={f∈U: f is visible in the release direction deter-mined by using loop li and 1≤ i≤N}

i. If N=1 and U1=U, then the undercut under evaluationis simple and blind (FT1, Fig. 11).

ii. If N=2 and U1=U2=U, then the undercut is throughand accessible from both the directions (FT2, Fig. 11).

iii. If N=2 and U1 or U2=U and the other one is {∅},then the undercut is through but accessible from oneside only (FT3, Fig. 11).

iv. If N≥2 and U1∪U2∪....∪UN=U and Ui≠U, then theundercut is a “Type-II” intersecting feature. However,it can be a bounded (“FT4”, Fig. 7) or unbounded(FT8, Fig. 7) “Type-II” intersecting feature. Furtheranalysis is required to classify the undercut feature andto evaluate the convex edge boundary in the case of the“Bounded-Type-II” intersecting feature. The detailedprocedure is presented in Section 6.4.

v. For any N and U1∪U2∪....∪UN≠U and non-visiblesurfaces are obstructed only by the protrusion presentin that undercut (“FT4”, Fig. 11), then the undercutunder evaluation needs a split core to be molded.

vi. For any N and U1∪U2∪....∪UN≠U, then the undercutunder evaluation has inaccessible child depressionfeatures (FT3, Fig. 7). Once the presence of child isidentified, the edge boundary between the parent andchild is determined. The details are presented later inthe following Section 6.4.

Once the presence of intersecting depression features isidentified by using these conditions, the following proce-dure is followed to determine the type of intersection and toextract the edge boundary between the parent and child forthe “Unbounded-Type-II” intersecting features.

6.4 Procedure for Type-II intersecting features recognition

All the possible loops, consisting of convex edges otherthan the ones present in “Depression_Bounding_Edges”array or “TypeI_Bounding_Edges” array, are determined, asportrayed in Fig. 12. One edge can exist in multiple loops,e.g., edge e1 in Fig. 12a is present in two convex edge loopsas shown the by bold lines. If there is no convex edge loopand condition “iv” mentioned in Section 6.3 is satisfied,then all the requirements of the “Unbounded-Type-II”intersection are fulfilled, and the feature is identified. Thistype of feature must be divided into more than one featurefor demoldability, e.g., feature “FT8” in Fig. 7. Theirautomatic division is outside the scope of this work.

Next, the undercut faces, inaccessible from the releasedirection, are grouped based on adjacency. If any undercutface shares concave edges with the group of inaccessiblefaces, then that face is also added to the group. Now, theconvex edge loops are evaluated for each inaccessible facegroup to determine the boundary between the parent andthe child. The loop, shown in bold in Fig. 12c, separates thefaces that are accessible and inaccessible from the releasedirection of the undercut under evaluation, is identified. Toidentify the faces of the child undercut, the undercut faceson the inaccessible side of the loop are grouped. Todetermine if the child undercut is through or blind, thebounding edges of the group of faces, forming the childundercut, are extracted from the part geometry. If theseedges are present in “Depression_Bounding_Edges” array,

Fig. 11 a Different depression features and b faces obstructing thefeatures are hidden to show cross-sections

Fig. 10 “Type-I Intersecting”feature recognition procedure: apart with “Type-I Intersecting”feature, b Seed and groupedfaces (to avoid clutter,face obstructing the feature arehidden in figure), and cconvex edge loop of “Type-IIntersecting” feature


then the child undercut is a through depression. Otherwise,the child undercut is a blind depression.

If the child feature is not a through depression, then itsrelease direction is determined by using the edge loopseparating the parent and child features. However, theretraction space availability for releasing the split-coreduring demolding is not checked.

If the child undercut is classified as a through depres-sion, its accessibility is checked along the release directionthat is calculated by using the corresponding edge loop ofthe “Depression_Bounding_Edges” array.

Once the features are identified, including the “Type-I”and “Type-II” intersecting features, the remaining facesof the part are grouped as core, cavity, and additionalfeatures by using the accessibility analysis results and theiradjacency.

7 Implementation

To recognize mold features, including intersecting ones, thenovel mold feature recognition methodology presented hereis implemented by using Visual Basics 6.0 and Solidworks2007. Solidworks has a large set of user routines that can beused to obtain geometrical and topological data from a B-rep model. The newly developed feature recognition systemis executed on an Intel Pentium IV CPU 1.75GHz, 1 GBRAM with Windows XP operating system. The moldfeature recognition system requires the user to input thepart information in the B-rep format and specify the partingdirection. The system then recognizes the depression andprotrusion features, core, and cavity surfaces and deter-mines the release direction of the undercut features. Anefficient data structure is developed to store and retrieve thenecessary information. The developed system is tested onvarious parts with a variety of surfaces, including free-formsurfaces, to validate its effectiveness.

Four case studies are presented in this section. The firstcase study, in Fig. 13a, is geometrically similar to thebenchmark part considered by Ye et al. [22, 25]. Figure 13billustrates the features that are recognized by the newlydeveloped system. It can be clearly seen that the proposedalgorithm successfully recognizes five depression features

and six protrusion features, along with their releasedirections (shown by the dotted arrows). Similar featureinformation is reported in [22, 25]. The faces, other thanthose of the undercut features, are classified into core,cavity, and core/cavity, as observed in Fig. 13d. However,similar to the reported work, there is a need to split the facef (Fig. 13a) into base face and undercut faces to recognizethe protrusion features, in Fig. 13c.

The second case study, represented in Fig. 14a, isgeometrically similar to an automobile part (the switchhousing) [27] used in luxury cars. This injection-moldedpart has three obliquely positioned holes. The proposedmethodology successfully recognizes three depressionfeatures with their release directions, as denoted inFig. 14c. Of three identical depression features, the releasedirection for one of the depression features is shown inFig. 14c as a dotted arrow. The core, core/cavity, and cavitysurfaces are shown in Fig. 14d, e, and f, respectively.

The third case study, in Fig. 15a is geometrically similarto an industrial part (the throttle knob). This part has onedepression feature and one through hole in the partingdirection. The face f, in Fig. 15a, of the depression featureis partially accessible (i.e., the face is partially outside thedepression undercut and is successfully identified duringthe accessibility analysis in the present work as a “partial-outer-boundary-accessible” surface). The novel systemprompts the user to split such faces; however, the splittingis done interactively. Once face f is split into the two faces,in Fig. 15c, the algorithm has successfully recognized thedepression features, along with their release direction, inFig. 15d. The core, core/cavity and cavity surfaces areshown in Fig. 15e, f, and g, respectively.

The geometry of the fourth case study is selected toexplain the relative merits of the present system withrespect to the work reported in the literature. The part inFig. 16a has free-form surfaces as well as protrusion,depression, and intersecting depression features. Thefeature FT1, in Fig. 16b, is a depression feature (commonlyknown as a slot in machining) with one blind depression,one through depression, one protrusion child, and one“Type-I” intersecting feature. In this depression feature,face f1 in Fig. 16b is fully accessible from the partingdirection. This face has been successfully recognized as

Fig. 12 Convex edge loops forseed edge e1 are shown usingbold lines in b and c


Fig. 13 Test part 1from the work of Ye et al. [22, 25]: a part model, b recognized features and release directions, c splitting required to recognizeprotrusions, and d core, cavity, and core/cavity surfaces

Fig. 14 Test part 2, an injection-molded part from [27]: a partmodel, b part model in anotherview, c recognized features andrelease direction, d core surfaces,e core/cavity surfaces, and fcavity surfaces


part of the depression undercut. The faces of the “Type-I” intersecting features are accessible from the partingdirection and are successfully identified as part of coresurfaces. The feature FT2 is an “intersecting” depressionfeature and is intersecting with two features. One ofthese is a “Type-I Intersection” and the other one is a“Bounded-Type-II Intersection”. All three features arerecognized, and their release directions are determined.The feature FT3 is a simple depression undercut. Thefeature FT4 is a depression feature with an opening inthe parting direction, and its release direction is deter-mined along the cylinder axis. The feature FT5 is a resultof the intersection of two depression features and doesnot have any convex edge loop that can separate the twointersecting features. This feature has been identified asan “Unbounded-Type-II” intersection, and the releasedirections are determined for both features involved. Thesolid arrows in Fig. 16c show the release directions of theundercut features. Figure 16d conveys the surfaces thoseform the core, cavity, and core/cavity.

8 Conclusion

The automated recognition of molding features is animportant step to automate the mold design process. Asystem to recognize protrusion as well as intersecting

depression features is developed, implemented, and testedsuccessfully. The system analyzes the accessibility of thepart faces and identifies the faces that cannot be molded byusing the core and cavity. This information, along with thegeometric and topologic information stored in the B-repformat of the solid model, is used to recognize the complexintersecting features.

The methodology, adopted for the accessibility analysis,classifies each face without discretizing it. Moreover, theanalysis involves sweeping and regularized boolean oper-ations to analyze all types of surfaces, including free-formones. A robust approach is presented to analyze theaccessibility of perpendicular surfaces by generating anoffset body. The mold features (protrusions, blind, andthrough depressions) are identified successfully. Further-more, the features involving the intersection betweendepression features have also been classified and identified.Case studies are presented to demonstrate the additionalfeature recognition capabilities of the present systemcompared with those of Ye et al. [22, 25]. The novelfeature recognition algorithm is capable of handling alltypes of surfaces, unlike in earlier attempts [15, 22]. Inaddition, a procedure to determine the feasible uniquerelease direction is presented. However, the recognition ofthe protrusion features with concave faces due to theintersection with depression(s) as well as protrusion(s) isbeyond the scope of this paper.

Fig. 15 Test part 3, an injection-molded industrial part: a part model and parting direction along the Y-axis, b part model in different view, c face fsplit, d recognized feature and its release direction shown by dotted arrow, e core surfaces, f core/cavity surfaces, and g cavity surfaces


The proposed methodology can be further extended todevelop a system for the automatic generation of moldelements to achieve the objective of design-manufacturingintegration.

9 Limitations and scope for future work

The system developed in this work can be applied torecognize simple as well intersecting depression features;however, the new system cannot recognize protrusion

features with concave edges due to their intersection withother features. Further work is in progress to recognize suchfeatures as well.

To generate the side cores of the recognized intersectingfeatures, a simple approach is developed by assuming thatthe undercut will have ruled property in the release directionbut is not very robust. Moreover, it is necessary to split the“Unbounded-Type-II” intersecting feature to generate itsmold elements since it does not have any defined boundary.Research is being conducted to automatically generate theside cores/cavities for the intersecting features.

Fig. 16 Test part 4, an injection-molded part with intersecting features: a part model, b recognized features, c intersecting features along withtheir release direction shown by solid arrow, and d core, core/cavity, and cavity surfaces


The authors declare that they have no conflict of interest.

Appendix A

Determination of part faces shared by its convex hull

A 3D convex hull envelops the part and each face of theconvex hull subdivides space such that all the entities(faces, edges, and vertices) of the part lie wholly on or toone side of that face, as illustrated in Fig. 17. The purposeof determining the part faces that are shared by its convexhull is to identify “Fully-accessible” missing faces of adepression undercut, as discussed in Section 6.1.1.

Methodology for determining the faces that are part ofconvex hull

Weinstein and Manoochehri [14] have proposed a method-ology to identify the surfaces of the polyhedral part so thatall the other surfaces are located on the material side ofthese surfaces. Their methodology has been extended toinclude the ruled, as well as free-form surfaces and ispresented next.

Given a surface, each edge of its outer edge loop isparameterized and divided into a number of equal seg-ments. The dot product of the surface normal at eachsegment’s node with the vectors from the node to everyvertex of other faces of the part is taken. If all the dotproducts are negative or zero, then the considered surface isdetermined as part of the convex hull.

Appendix B

Determination of the undercut release direction

The release direction for the protrusion and depression featuresis required to ensure demoldability of the side core and cavity.For the protrusion undercuts, if the face-set is fully accessiblealong the � d

!direction, the undercut is considered moldable

by using core and cavity; otherwise, a release direction for thefeature is generated. The release direction for the depressionundercuts is determined with the help of the information

present in “Depression_Face_Set” and “Depression_-Bounding_Edges” arrays (whereas, “Protrusion_Face_Set”and “Protrusion_Bounding_Edges” array are used for deter-mining the release direction of protrusion undercuts) by usingthe procedure presented next.

Procedure for determining undercut release direction

Each edge loop present in the “Depression_Bounding_Edges”array of the considered undercut is processed in turn todetermine its unique release direction. First, the adjacentfaces of the edges in the loop are selected (shown in gray inFig. 18b), grouped in an array and arranged based onadjacency between them. The normal vectors of thesefaces are determined on the vertices of the edge present inthe considered loop, as denoted in Fig. 18b. The cross-product of the normal vectors of the adjacent faces of thegroup is determined. The cross-product calculation startsfrom two adjacent edges of the loop and traverses in onedirection until the same pair is reached. However, if thenormal vectors of a pair are parallel or opposite to eachother, then that pair is not considered, e.g., edge pair 1 and4 in Fig. 18b.

The cross-product vector indicates a direction alongwhich both the adjoining faces can slide. The purpose offurther work is to determine a unique center vector of allthe cross-product vectors in 3D. The objective is simplifiedby solving it in 2D, and cross-products are analyzed on twoplanes that are perpendicular to each other. The detailedprocedure is presented in the following section and aunique center vector in 3D is determined.

These cross-product vectors are translated to a commonpoint (Fig. 18c) and projected on to a plane (p1)perpendicular to the parting direction � d

!� �, as shown

Fig. 18d. If the number of projected cross-product vectors isfewer than four, the two outer most vectors are selected andtheir center vector (v1) is calculated. Otherwise, the twoinnermost vectors are selected to determine v1, as depictedin Fig. 18d.

Now, the cross-product vectors are rotated about theparting direction on to a plane p2, which is perpendicular top1, in such a way that all the rotating vectors are pointingtowards one hemisphere, in Fig. 18e. A center vector (v2) isdetermined from the rotated vectors by using a similarprocedure that is followed for v1.

After the center vectors v1 and v2 are normalized, inFig. 18f, they are used to determine the release directionvector (v) in 3D. It can be perceived that if viewed in theparting direction, the release direction vector v wouldoverlap the vector v1. However, angle between plane p1and vector v is not known. Also, it can be visualized that in3D that vectors v and v2 would have the same angle withrespect to plane p1.Fig. 17 a A non-convex part and b its convex hull [4]


Finally, the release direction vector v is determined byrotating vector v2 about the parting direction by an angleequivalent to the minimum angle between v1 and p2, shownin Fig. 18f.

Such a procedure is graphically illustrated in Fig. 18with the help of a depression undercut (Fig. 18a), and theparting direction is chosen to be parallel to Y-axis.Figure 18b, c show the normal vectors, and theircorresponding cross-product vectors, respectively. The

cross-product vectors are then projected onto plane ZX(plane p1), and their center vector v1 is determined, asillustrated in Fig. 18d. Next, the cross-product vectors arerotated about the parting direction, i.e., Y-axis, on to XYplane (plane p2) in such a way that all the vectors pointtowards the +X direction, and then their center vector v2 isdetermined, as depicted in Fig. 18e. Finally, the releasedirection is determined by rotating vector v2 about Y-axis byangle α equivalent to that of the minimum angle between

Fig. 18 Procedure for release direction determination (LHS coordinate system is used)


vector v1 and plane p2, as shown in Fig. 18f. It can be seenin Fig. 18g that all the undercut faces are fully visible fromthe release direction.

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Automatic recognition of intersecting features for side core design in two-piece permanent molds

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