Automatic determination of undercut regions and their releasing directions in plastic mold design

To discover the surfaces belonging to undercut regions, attributes are then assigned to the surfaces of the part model based on the topological relationship of adjacent surfaces of each

D9803803

Automatic Determination of Undercut

Regions and Their Releasing Directions

in Plastic Mold Design

Tran Anh Son

103 1 20

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ABSTRACT

The determination of undercut regions and their releasing directions plays an important role in injection mold design Most of the approaches in the literature face difficulties in recognizing the undercut regions of real-life parts This thesis proposes an approach to automating the determination of undercut regions and their releasing directions for complex parts with free-form surfaces In order to delineate the border of undercut regions, orthogonal cutting planes are firstly employed to automatically find the inner loops

of a part model using the concept of “shared vertices” and “adjacent points” The inner loops

are classified into two groups: closed inner loops and open inner loops In order to determine undercut regions, open loops are further converted to closed ones through the introduction of additional line segments To discover the surfaces belonging to undercut regions, attributes are then assigned to the surfaces of the part model based on the topological relationship of adjacent surfaces of each inner loop After that, the concept of “target facets” is proposed to

separate the undercut regions from other surfaces in the model Through the recognized surfaces of the undercut regions, the concept of “visibility map” is further applied to

determine feasible releasing directions for each of the undercut regions Delaunay triangulation is adopted here to represent a set of releasing directions The undercut regions having the same releasing direction are finally grouped to form a slider in the injection mold

In addition to proposing the methodologies to find undercut regions and their releasing directions for free-form surfaces, this thesis also uses commercial software packages Pro/Engineer Wildfire 5.0 and Matlab 9.0 to implement the algorithms developed for the proposed methodologies Several real-life parts, such as cell phone cover and bike helmet, are used as testing examples to demonstrate the applicability of the implemented system While these example parts contain a large number of complicated free-form surfaces,

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ACKNOWLEDGEMENTS

This dissertation would never have been to finish without the guidance and support of

my advisors, suggestions of committee members, the facilitation of both National Taiwan University of Science and Technology (NTUST) and Ho Chi Minh City University of Technology (HCMUT) during my study period, helps from friends, and great supports from

my family

First and foremost, I would like to express my deepest gratitude and respect to my advisor, Prof Alan C Lin for his excellent guidance, caring, patience, and providing me with both material and spiritual support for doing research Besides my advisor, I would also like

to thank the members of defense committee for the valuable comments and suggestions so that my dissertation can be modified perfectly

I want to thank the President and Professors of National Taiwan University of Science and Technology (NTUST) who have created good opportunities and kindly environment for

me to access and complete my Ph.D degree

During my study period in NTUST, I was indebted to many of my lab mates to provide me a happy and peaceful environment Specially, I would also like to express my appreciation and thank to Dr Tzu-Kuan Lin and my dear friend, Dr Luu Quoc Dat, who always share and help me in my research and life for the past years

Finally and profoundly, I gratefully acknowledge the silent self-abnegation of my beloved father and my wife for supporting me in every aspect of my life I would also like to thank my brothers, my sisters, my uncles and my children for everything they have done for

me Without my family love, this thesis would not be finished

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TABLE OF CONTENTS

ABSTRACT iv

ACKNOWLEDGEMENTS vi

TABLE OF CONTENTS vii

LIST OF FIGURES x

LIST OF TABLES xvi

NOTATIONS xvii

Chapter One INTRODUCTION 1

1.1 Research background and motivation 1

1.2 Research objectives 4

1.3 Thesis structure 4

Chapter Two LITERATURE REVIEW 6

2.1 Definition and classification of undercut regions 6

2.1.1 Definition of undercut regions 7

2.1.2 Classification of undercut regions 7

2.1.3 Geometric factors of an undercut region 10

2.1.3.1 Surfaces of undercut regions 10

2.1.3.2 Releasing directions of undercut regions 11

2.2 Recognition of undercut regions 12

2.3 Determination of parting directions 27

2.4 Slicing methods for mold design 30

2.5 Concept of visibility map 32

2.6 Delaunay triangulation representation 36

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2.7 Comments on the past literatures 38

Chapter Three AUTOMATIC DETERMINATION OF INNER LOOPS OF UNDERCUT REGIONS 44

3.1 Classification of inner loops 44

3.2 Workflow of finding inner loops 46

3.3 Selection of parting directions 48

3.4 Formation of 3 sets of orthogonal cutting planes 48

3.5 Extraction of intersection points 50

3.6 Collection of candidate points belonging to inner loops 51

3.7 Formulation of inner loops 58

3.8 Conversion of open inner loops to closed loops 61

3.9 Discussions 64

Chapter Four AUTOMATIC DETERMINATION OF SURFACES OF UNDERCUT REGIONS 66

4.1 Determination of surfaces of undercut regions using B-rep data 66

4.1.1 Formation of three sets of cutting planes 68

4.1.2 Generation of intersection curves of cutting planes and part surfaces 69

4.1.3 Generation of projected curve of inner loops onto the current cutting plane 70

4.1.4 Analysis of loops in the current cutting plane 70

4.1.5 Assignment of surface attributes 71

4.2 Determination of undercut surfaces using STL file 74

4.2.1 Formation of three sets of cutting planes 75

4.2.2 Determination of intersection line-segments between cutting planes and STL model 76 4.2.3 Projection of inner loops 80

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4.2.4 Assignment of facet attributes 85

4.3 Discussions 87

Chapter Five AUTOMATIC DETERMINATION OF RELEASING DIRECTIONS OF UNDERCUT REGIONS 88

5.1 Calculation of releasing directions of undercut regions 88

5.2 Grouping of undercut regions 94

5.2.1 Grouping of undercut regions into a side-core region 94

5.2.2 Grouping of undercut regions into core or cavity 98

5.3 Discussions 100

Chapter Six SYSTEM IMPLEMENTATIONS 101

6.1 Implementation Example 1 Lamp cover 107

6.2 Implementation Example 2 – Plastic cover of a hair dryer 114

6.3 Implementation Example 3 – Component of cell phone 117

6.4 Implementation Example 4 – Bike helmet model 121

Chapter Seven CONCLUSIONS AND DISCUSSIONS 146

7.1 Conclusions 146

7.2 Future works 147

REFERENCES 150

BRIEF INTRODUTION OF THE AUTHOR 155

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LIST OF FIGURES

Figure 1-1 Solid models of some real-life parts 3

Figure 2-1 Main mold components 6

Figure 2-2 Undercut regions and mold components 8

Figure 2-3 Target surface and inner loops of external undercut regions [1] 8

Figure 2-4 Illustration of target surfaces and inner loops 9

Figure 2-5 Completely and partial visible regions 10

Figure 2-6 Surfaces of an undercut region 11

Figure 2-7 Releasing directions of undercut regions 12

Figure 2-8 Surface visibility 13

Figure 2-9 Definition of complete and partial visibility 13

Figure 2-10 Recognition of undercut regions using the concept of sealed pocket 15

Figure 2-11 Modelling of core and cavity geometry by solid-sweep operation 16

Figure 2-12 Concept of blockage 17

Figure 2-13 External undercut region 18

Figure 2-14 Potential undercut Surface using the concept of blockage 18

Figure 2-15 Internal undercut region 19

Figure 2-16 Connectivity cases of target surface with first adjacent surfaces 20

Figure 2-17 An undercut region composed of planar and curve surfaces 21

Figure 2-18 Candidate undercut faces 22

Figure 2-19 An undercut region and its graph-based 23

Figure 2-20 An undercut region with free-form surface 23

Figure 2-21 Cylindrical-based EBC patterns 24

Figure 2-22 A countersink through hole 25

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Figure 2-23 Mold design navigating process [12] 26

Figure 2-24 Multi-piece mold and two-piece mold 27

Figure 2-25 Extreme points of product model on each slicing plane 30

Figure 2-26 Missing information of parting lines 31

Figure 2-27 The grid line test on each slicing plane 32

Figure 2-28 Recognizing undercut regions using slicing method 32

Figure 2-29 Visibility map 33

Figure 2-30 A tessellated part presentation and its undercut region 34

Figure 2-31 Illustration of V-map calculation 36

Figure 2-32 Delaunay triangulation in 2-D 37

Figure 2-33 Non-Delaunay triangulation 37

Figure 2-34 Delaunay triangulation in 3-D 38

Figure 2-35 V-map and Delaunay triangulation 38

Figure 2-36 Real-life parts having complex free-form surfaces 40

Figure 2-37 Recognition of surfaces molded by core and cavity 41

Figure 2-38 Case study in [44] 42

Figure 2-39 Missing surface 42

Figure 3-1 Inner loop and target surfaces 45

Figure 3-2 Illustration of inner loops 45

Figure 3-3 Closed inner loops 46

Figure 3-4 An open inner loop 46

Figure 3-5 Workflow for determination of inner loops 47

Figure 3-6 Main parting direction 48

Figure 3-7 Three sets of cutting planes 49

Figure 3-8 Intersection points on one cutting plane of main set 51

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Figure 3-9 Shared vertices and adjacent points 52

Figure 3-10 Shared vertex and adjacent points at the inner loop 53

Figure 3-11 Collected points of inner loops 55

Figure 3- 12 Candidate points at several cutting planes of the main set 56

Figure 3-13 Inner undercut region inside the main undercut region 56

Figure 3-14 Main set of cutting planes PLz 57

Figure 3-15 Candidate points on inner loops 57

Figure 3-16 Formation of inner loops using shared vertices and candidate points 59

Figure 3-17 Inner loop created by line segments 60

Figure 3-18 Two inner loops created by serial line segments 60

Figure 3-19 An open inner loop and its end points 61

Figure 3-20 Inner loops of main region and inner region 61

Figure 3-21 Cases of additional line segments of an open inner loop 62

Figure 3-22 Creation of an additional line segment of an open inner loop 63

Figure 3-23 Adjacent facets of an end point 64

Figure 3-24 Boundary curve and closed inner loop 64

Figure 4-1 Workflow of determination of surface attributes 67

Figure 4-2 Target surfaces and inner loops 68

Figure 4-3 Three sets of cutting planes 69

Figure 4-4 Non-watertight structure of B-rep data and intersection curves in a cutting plane 70

Figure 4-5 Projected curve in the current cutting plane 71

Figure 4-6 Analysis of intersection curves and projected curve in the current cutting plane 71 Figure 4-7 Surface numbers and surface attributes 72

Figure 4-8 Illustration of checking points on each checking loop 73

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Figure 4-9 Surfaces of undercut regions 74

Figure 4-10 Cutting planes 76

Figure 4-11 Target facet 77

Figure 4-12 Determination of target facets 77

Figure 4-13 Intersection line-segments between the current cutting plane and part facets 78

Figure 4-14 Intersections in current cutting planes PLx and PLy 78

Figure 4-15 Intersection in current cutting plane PLz of a model with two holes 79

Figure 4-16 Intersection line-segments in cutting plane PLy of the model with an open loop 80

Figure 4-17 Projected curve of the inner loop of the model with a closed loop 81

Figure 4-18 Projected curves of loop 1 of the model with two holes 81

Figure 4-19 Projected curves of loop 2 of the model with two holes 82

Figure 4-20 Projected curves of the open loop in cutting planes PLx, PLy, and PLz 83

Figure 4-21 Connected group of the model with one closed inner loop 84

Figure 4-22 Connected groups of the model with two closed inner loops 84

Figure 4-23 Connected group of the model with one open inner loop 85

Figure 4-24 Facet attributes in current cutting plane PLz 85

Figure 4-25 Facet numbers for connected groups of a part model with two inner loops 86

Figure 5-1 Examples of surfaces and their corresponding V-maps 88

Figure 5-2 The analytic results of V-map 89

Figure 5-3 Normal vectors at several points on a free-form surface 90

Figure 5-4 Determination of V-map of region F 91

Figure 5-5 Numbers of facets of undercut region F 92

Figure 5-6 Result of V-map and potential releasing directions 92

Figure 5-7 Center point CF and random results of releasing directions RDw 94

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Figure 5-8 Grouping of undercut regions 95

Figure 5-9 Two close undercut regions 96

Figure 5-10 V-maps of undercut regions 96

Figure 5-11 Intersection of V-maps and possible releasing directions 96

Figure 5-12 Feasible shared releasing directions using V-map 97

Figure 5-13 Using Delaunay Triangulation to represent V-maps 97

Figure 5-14 Intersection of Delaunay triangulations 98

Figure 5-15 Shared releasing directions using Delaunay triangulation 98

Figure 5-16 Releasing directions of undercut regions 99

Figure 6-1 Workflow of the implemented system 102

Figure 6-2 Structure of STL file in ASCII format 103

Figure 6-3 Flowchart of determination of inner loops 104

Figure 6-4 Flowchart of determination of undercut facets 106

Figure 6-5 Flowchart of determination of releasing directions 107

Figure 6-6 Inner loops of lamp cover model 109

Figure 6-7 Connected group in the current cutting plane PLy 109

Figure 6-8 Surfaces of undercut regions of lamp cover model 110

Figure 6-9 Undercut regions of the lamp cover 112

Figure 6-10 Verification of results of the lamp cover model 113

Figure 6-11 Inner loops of the plastic cover of a hair dryer 115

Figure 6-12 Potential undercut regions of the plastic cover of hair dryer 116

Figure 6-13 Inner loops of component of cell phone 117

Figure 6-14 Surfaces of undercut regions of cell phone model 119

Figure 6-15 Undercut regions of the component of cell phone 120

Figure 6-16 Verifying results of the component of cell phone 121

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Figure 6-17 Bike helmet model 122

Figure 6-18 Half helmet with eleven holes 122

Figure 6-19 STL file of the half helmet 123

Figure 6-20 Inner loops 123

Figure 6-21 Surfaces of undercut regions of the half-helmet 124

Figure 6-22 Final result of surfaces of undercut regions for the half-helmet 130

Figure 6-23 Potential groups of undercut regions 132

Figure 6-24 Shared releasing directions of each group of undercut regions 134

Figure 6-25 Procedure to choose a shared releasing direction 136

Figure 6-26 Result of releasing directions of each group of undercut regions 137

Figure 6-27 Checking of interferences for the helmet 139

Figure 6-28 Checking of interference volumes of slider 2 142

Figure 6-29 Final result of the releasing direction of the helmet model 144

Figure 7-1 Releasing direction of a zig-zag hole 148

Figure 7-2 Partition of the undercut region and feasible releasing directions 148

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LIST OF TABLES

Table 1 Surface attributes of points 1, 2, and 3 on inner loop 1 73

Table 2 Surface attributes of points 1 and 2 on inner loop 2 74

Table 3 The results of surface attributes 74

Table 4 Attribute values FAwv for facets of the part with two inner loops 87

Table 5 Normal vectors ni of facets of undercut region F 93

Table 6 Center points and releasing directions of potential undercut regions of the lamp cover model 112

Table 7 Input information of the lamp cover model for interference checking 113

Table 8 Center points and releasing directions of potential undercut regions of the plastic cover of hair dryer 116

Table 9 Center points and releasing directions of potential undercut regions of the component of cell phone 118

Table 10 Center points and releasing directions of feasible undercut regions of the component of cell phone 120

Table 11 Center points and releasing directions of undercut regions of the partial helmet models 138

Table 12 Input information of the helmet model for interference checking 139

Table 13 Interference volume of slider 2 and the molding part along the releasing direction RD2 145

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NOTATIONS

VMap(F) : Visible map of a region F

VMap(Si) : Visible map of a patch surface Si

VMap(fi) : Visible map of a triangular facets fi

x, y, z : Coordinate value corresponding to the x-, y-, and z-axis, respectively

v : Indexes of axes of Cartesian coordinate system

Sx, Sy, Sz : Projection area on a plane taken perpendicular to the x-, y-, and

q : Number of triangular facets of an undercut region

RD : Releasing direction of an undercut region

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Chapter One INTRODUCTION

This thesis focuses on automatic determination of undercut regions and their releasing directions in injection mold design The background and motivation, the objectives, and the structure of the thesis are presented in this chapter

Injection molding is an important manufacturing process Different plastic products have been produced using injection molds such as aircraft and automobile parts, personal utensils, electrical devices, kid toys, etc In the field of injection molding, efficiency is achieved by transmuting a product from the conceptual design stage to a real part quickly and inexpensively The high demand for shorter design and manufacturing lead times, good dimensional and overall quality, and rapid design changes has become the bottlenecks in mold industries [1] Hence we need efficient algorithms to generate injection molds automatically in a short period of time However, the existing algorithms have not satisfied all the demands of automation because of the diversified form of plastic products The existing approaches use the expertise of experienced mold designers Since the entire process will be done requiring a lot of manual interventions, it will take weeks to complete the design Moreover, the manual design related to the human intervention may not be the optimum design In addition, the effective communication between the part designers and the mold designers is very important Usually, the final part and mold designs require multiple iterations of those designers Without this communication, it becomes very difficult even for

a highly skilled designer to visualize the mold geometries for complex parts consisting of multiple undercut regions that need to be recognized and molded by local mold components

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For instance, it is very hard for mold designers to recognize the symmetrical geometries, large curvature surfaces, small and hidden recesses, etc of undercut regions Keeping in view all these issues, automatic recognition of undercut regions is a necessity

Moreover, the constitution of side actions is an important part of the mold system Along with the recognition of undercut regions, the determination of releasing directions for the undercut regions is also a necessary task For an undercut region, it may have one or many possible releasing directions that make the undercut region feasible for mold opening The determination of feasible releasing directions helps reduce the complexity of mold structure Furthermore, it helps group several undercut regions into the same region and decrease the number of mold components

Figure 1-1 shows some examples of part models in real life These models are formed

by many complex free-form surfaces and consist of many cavities This makes it difficult to recognize undercut regions In addition, the determination of releasing directions and the grouping of undercut regions for these parts are not trivial, even for experienced mold designers Furthermore, based on the recognized undercut regions, mold designers need to consider the constitution of side actions for the creation of sliders and lifters These works not only rely on mankind’s experiences but also features of surfaces and regions of part models

The variety of surfaces and their connection, and the appearance and interweaving of many depression and protrusion regions are always hard challenges for injection mold design

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This thesis focuses on the recognition of undercut regions and the determination of their releasing directions in injection mold design for real-life parts which may consist of large quantities of free-form surfaces and are full of many depression and protrusion regions Part models in the form of Stereolithography (STL) are used as the input data In the first place, orthogonal cutting planes are created and used to automatically determine the inner loops by following the subsequent steps: (1) extract the intersection points of cutting planes and part surfaces, (2) collect points of interest using a so-called “shared vertices and common

edges” concept, and (3) connect relevant points to form each individual inner loop Inner

loops are then classified into closed inner loops and open inner loops In order to determine undercut regions, open loops must be further converted to closed ones through the generation

of additional line segments Once all inner loops are found, attributes are assigned to all surfaces of the part model based on the topological relationship of adjacent surfaces of each inner loop Through the recognized surfaces of the undercut regions, the concept of V-map is used to find all possible releasing directions Delaunay triangulation is adopted here to represent a set of releasing directions Several undercut regions which have the same releasing direction are then grouped into one slider region

This thesis is composed of the following seven chapters: Introduction, Literature review, Automatic determination of inner loops of the undercut regions, Automatic determination of surfaces of the undercut regions, Automatic determination of releasing directions of undercut regions, System implementations, and Conclusions and Discussions

Chapter three discusses the method to recognize inner loops of undercut regions using orthogonal cutting planes Next, the inner loops are classified into closed and open inner loops based on the geometrical characteristics of loops Open loops are then converted to closed loops

Chapter four addresses the issue of determining surfaces of undercut regions Based

on the topological relationship of part surfaces, undercut regions are recognized using the attributes of the surfaces

Chapter five focuses on the discussion of determination of releasing directions for each of the undercut regions Then, the undercut regions having the same releasing direction are grouped to reduce the number of the local mold components

Chapter six presents system implementations Several industrial parts are used to demonstrate the effectiveness and advantages of the proposed approach

Chapter seven concludes the contribution of this thesis Moreover, some future works are addressed

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Chapter Two LITERATURE REVIEW

This chapter reviews the definition and classification of undercut regions Then, the existing methods for the recognition of undercut regions and their releasing directions are discussed

2.1 Definition and classification of undercut regions

In the design of two-piece molds, plastic mold design process generally utilizes primarily two mold halves, namely, core and cavity as shown in Figure 2-1 Normally, the cavity is the stationary half and the core is the movable half Surfaces on the core create internal surfaces of molded parts, while surfaces on the cavity create major external surfaces Molten materials are filled into the space between the core and cavity to solidify the shape of

a part After that, the core and the cavity are separated and the molded part can be removed

Parting direction

Part

Undercut regionSlider

Undercut releasing direction

Cavity

Core

Figure 2-1 Main mold components

A direction along the opening path of core and cavity is called the parting direction The parting direction is a pair of opposite directions and if one of them is defined, the other

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one is also defined In the following sections, the definition and classification of the undercut regions will be discussed in detail

2.1.1 Definition of undercut regions

Undercut regions can be defined as convex and concave portions of molding parts In the molding part, undercut regions are multiform such as local recesses, holes, slots or bosses, cylinders, cones and spheres [1] If undercut regions cannot be added in by core or cavity, they will require the incorporation of side-cores, side-cavities, or other local mold components in the mold structure [2] In the approach proposed by Ran and Fu [3], undercut regions were defined as surface regions which prevent the typical two-piece molds from opening by their local molding tools after the plastic solidified, as shown in Figure 2-1(b)

2.1.2 Classification of undercut regions

Based on the mold structure and the requirement of local tools for a plastic mold, undercut regions could be classified into the following two types: external undercut regions (EU) and internal undercut regions (IU), as shown in Figure 2-2 The external undercut regions are the restriction regions, which prevent the withdraw of molding from the cavity, while the internal undercut regions prevent the ejection of the molding from the core [4] The external undercut regions are molded by side-cores or side-cavities and the internal undercut regions are molded by pins or split-cores inside the core and cavity Side-cores and side-cavities are outwards withdrawn from the molded part before injection, while pins or split-cores are inwards withdrawn from the molded part [1] In Figure 2-2, there are three undercut regions in the molded part and the given parting direction Undercut region 1 can be molded

in by core, but undercut regions 2 and 3 cannot be molded by core/cavity and would need side-cores and side-cavities for the molding

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Undercut region 2 Part

Side-cavity Cavity

Undercut region 3 Core

Side-core

Figure 2-2 Undercut regions and mold components

According to the geometric characteristics, external undercut regions are further classified into inside external undercut regions (IEU) and outside external undercut regions (OEU) [1, 2, 4] The concept of target surfaces was used in these approaches to determine the inside external undercut regions In Figure 2-3(a), the surface that contains undercut regions

is identified as the target surface In Figure 2-3(b), the external edge-loop 1 is the outermost edge-loop Two other edge-loops, e.g internal edge-loops 2 and 3, are the internal edge-loops [1, 5] The internal edge-loops are called the inner loops Figure 2-4 illustrates the plastic model and its target surfaces

External edge-loop 1

(a) A molding (b) Its top view

Figure 2-3 Target surface and inner loops of external undercut regions [1]

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Parting direction

Target surfaces

Inner loops

(a) Plastic model (b) Target surfaces and inner loops

Figure 2-4 Illustration of target surfaces and inner loops

From the moldability/formability point of view, undercut regions are categorized into side-core and split-core regions The concept of “completely and partially visible” in a

viewing direction d has been used for the classification [6~9], as shown in Figure 2-5(a) A completely visible region is a region which is completely visible when seen from outside In Figure 2-5(b), the side-core having the same shape of the completely visible region is used to create the surfaces of the undercut region On the contrary, a partially visible region is a region which is not completely visible when seen from outside [8] In Figure 2-5(c) and (d), the split-core is used to solve the partially visible region The split-core is released from the undercut regions by first moving along the parting direction and then along the direction of split-core

Based on observation of the number of releasing paths, undercut regions are classified into slider and lifter regions [10~12] Before mold opening, the slider simply clears out of the slider region with only one moving path, as shown in Figure 2-5(b), while the lifter needs

at least two moving paths for its releasing, as shown in Figure 2-5(c) and (d) The releasing path of the lifter is a combination of two moving paths including primary moving out the undercut region and secondary moving along the parting direction

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Completely

visible region

Partially visible region

Side-core

(a) A molding (b) Side-core moving

Split-core

(c) First moving of split-core (d) Second moving of split-core

Figure 2-5 Completely and partial visible regions

In summary, the undercut regions are classified into the following three main types depending on different points of view: (1) external and internal undercut regions (based on the mold structure and the requirement of local tools for a molding), (2) completely and partially visible regions (based on the moldability/formability for undercut surfaces in a given releasing direction), and (3) slider and lifter regions (based on the number of releasing paths)

2.1.3 Geometric factors of an undercut region

Geometric factors of undercut regions are surfaces and releasing directions, details being discussed below

2.1.3.1 Surfaces of undercut regions

A given part model has two groups of surfaces including surfaces of undercut regions and other ones that do not belong to undercut regions Surfaces of an undercut region are also surfaces of the molding which are constituted by side actions in plastic molding processes

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For example, the part depicted in Figure 2-6 has four surfaces, f1 ~ f4, and one undercut region

as the blind hole Hence, two surfaces, f3 and f4, are the undercut surfaces

Inner loop

Target surface

Undercut surfaces

f 2

f 1

f 3

f 4

(a) Plastic part (b) Undercut surfaces

Figure 2-6 Surfaces of an undercut region

2.1.3.2 Releasing directions of undercut regions

For a given undercut region, its releasing directions can be defined as directions in which the local mold component is most efficiently withdrawn from the undercut region to avoid the warpage of the molding [1, 9] If the undercut region is composed of a conical surface, a cylindrical surface, or a spherical surface, the releasing direction can be determined

by the axial direction of those surfaces In the case that the undercut region is made up of several surfaces, its releasing direction can be determined from all releasing directions of each surface using the concept of V-map Figure 2-7(a) displays the part model with two undercut regions which are formed by several surfaces Figure 2-7(b) shows the releasing directions of undercut regions In the figure, the releasing directions are represented by vectors passing through the center points of undercut regions The x-, y- and z-coordinates of the center points of undercut regions can be calculated as equation (1) [3, 9]:

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Center points

Releasing directions (a) Plastic part (b) Releasing directions and center points

Figure 2-7 Releasing directions of undercut regions

A number of studies have been proposed to recognize undercut regions based on taxonomy, characteristics, and geometric entities of the undercut regions [1-4, 6-26] Chen et

al [6] used the concept of surface visibility (V-map), developed by Woo [27], to determine the best parting direction The basic concept is: let Fi be a surface of part surfaces S, i.e S = {F1, F2,…, Fn}, where n is the total number of surfaces In order for Fi being visible along direction d, the following two conditions must hold: (1) any line of sight along direction d intersects Fi no more than once, and (2) any line of sight along direction intersects no other surfaces of before it intersects Fi Figure 2-8 shows an example with three surfaces

F1  F3 and three cases of checking lines R1  R3 For line R1, the first condition does not hold, the surface F1 thus is not visible in direction d Similarly, surface F3 is not visible in direction d because both the first and the second conditions do not hold Only surface F2 is visible in direction d

d S

Figure 2-8 Surface visibility

Chen et al [7] identified two levels of visibility, i.e., complete and partial, to determine the demoldability of undercut regions If a surface is not completely visible and hence cannot be separated from the mold assembly along any direction, it can be decomposed into two portions: those that are separable and those that form undercuts Figure 2-9 shows the definition of complete and partial visibility

(a) Complete visibility and demoldability

(b) Partial visibility and use of cores

Figure 2-9 Definition of complete and partial visibility

From the solid modeling perspective, Chen et al [7] generated a convex hull CH(P) for a given 3D polyhedral part P and found potential undercuts (or sealed pockets)

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 1, 2, , m

S  S S S between the convex hull CH(P) and the model P, viz., SCH P  PThen, V-map is applied to determine the undercut regions from the potential undercut

regions The recognition of undercut regions using the concept of “sealed pocket” is

illustrated in Figure 2-10 Let Lid S i and Pocket S i denote the lid faces and the pocket faces of Si, respectively, and Si Pocket S i If Si is completely visible along certain direction d’, Si can be separated from the mold along direction d’ Since the sealed pockets

are also a 3D solid with several single undercut regions linked together (see Figure 2-10(b)), decomposing them into many single undercut regions is equivalent to the recognition and extraction of undercut regions from 3D solid models It thus becomes another non-trivial regions recognition issue In addition, the approach suffers from another limitation that the input model is given with polyhedral surfaces

Hui and Tan [26] proposed a method to design the core and cavity inserts of a mold with sweeping operations The steps for the modelling of a core or cavity by the use of a solid-sweep operation can be summarized as follows: (1) Generate a solid by sweeping the molded part along the parting direction of the mold and determine the core and cavity sides of the swept solid (see Figure 2-11(a)), (2) Construct a cavity mold block with the required parting surfaces and subtract it with the swept solid at the parting line location (see Figure 2-11(b)), (3) Generate the second mold block and subtract it with the swept solid from the core side at the parting line location (see Figure 2-11(c)), and (4) Subtract the result of step 2 from that of step 3 with the molded plates in the closed position to obtain the core block (see Figure 2-11(d)) Then, the concept of blocking factor was used to indicate the amount of undercuts by measurement of the relative amount of obstruction in a certain direction

(d) Normal vectors of S2 (e) V-map and undercut direction of S2

Figure 2-10 Recognition of undercut regions using the concept of sealed pocket

(d) Modelling of mold-cavity plate P 2

Figure 2-11 Modelling of core and cavity geometry by solid-sweep operation

is studied based on the concept of “blockage” in a given direction A point p on part S is

blocked in direction d if vector (p+d) intersects the interior of S, where (0,) Figure

2-12 demonstrates the concept of blockage

d

pp+1d

p+2d Potential

undercut

Figure 2-12 Concept of blockage

External undercut region is defined based on the concept of blockage In Figure 13(a), external undercut region QE is a subset of the cavity solid such that QE is blocked along d or –d, but is cleared (not blocked) in certain direction d , where d  d and d  –d

2-However, the concept of blockage is similar to the concept of complete visibility as stated by Chen et al [6] In addition, this approach has been only applied to given polyhedral models

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with many flat faces and straight edges In special cases, several surfaces are not recognized

as the potential undercut surfaces using the concept of blockage, as shown in Figure 2-14 Surface f4 is not included as the potential undercut

Parting direction d

Q E

(a) A part with external undercut (b) Resolving external undercut with side core

Figure 2-13 External undercut region

Parting direction

f3

f2

f4

f5

Figure 2-14 Potential undercut Surface using the concept of blockage

Fu et al [1, 2] proposed an approach to recognizing surfaces of undercut regions and categorize undercut regions for part models described by B-rep data Further, the calculation

of releasing direction of each undercut region was presented In particular, by employing the

concept of “surface visibility” along a parting direction, surfaces of the part model molded by

the core and cavity are identified Then, the remaining surfaces should be molded by cores These surfaces are further recognized as surfaces of undercut regions After finishing the identification of surfaces of undercut regions, Fu et al [1, 2] developed a methodology to categorize undercut regions and determine undercut directions Potential undercut regions are

side-19

first classified into external and internal undercut regions Then, undercut directions are defined as the directions whereby the local tools can be withdrawn easily and most effectively to avoid the warpage of the molding In particular, for the determination of internal undercut regions, a ray was used to check the surface visibility The rule of ray checking is similar to the concept of the blockage discussed in Hui [15] The rays in the undercut direction and its opposite direction will have more than one intersection point with the molding beside the point in which the rays are cast In Figure 2-15, if the checking ray is

in the undercut direction, there are two intersection points P1 and P2; and in its opposite direction, there is one intersection point P3 between the ray and molding excluding the casting point Therefore, the potential undercut region is considered as the internal undercut region

Internal undercut region

Figure 2-15 Internal undercut region

For the determination of external undercut regions, Fu et al [1, 2] have cataloged individual cases in great details and come up with specific formulae based on the topological relationships of target surface Fi with its adjacent surfaces, i.e., first adjacent surfaces SAF and secondary adjacent surfaces SAS Figure 2-16 shows three types of connectivity cases of Fi

with SAF, namely, three-edge-surface, four-edge-surface, and more than four-edge-surface Let Li be the normal vector of Fi at the surface center, Lji or Lji, Lhi be the normal vectors of

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the first adjacent surfaces in the cases of three-surface-edge and four-surface-edge undercut regions, respectively, where j = 1, 2, 3 or j = j1, j2; h = h1, h2, the first subscript refers to the adjacent surface and the second subscript refers to the coordinate axis

(b) Four-edge-surface undercut region and its directions

Figure 2-16 Connectivity cases of target surface Fi with first adjacent surfaces SAF

on the normal vectors of the target surfaces and the adjacent surfaces Both kinds of surfaces are planar surfaces and each surface only has one normal vector For the case that these surfaces are sculptured surfaces having many normal vectors (refer to Figure 2-17), their method for determination of undercut direction becomes infeasible

Normal vectors

Target surface

First adjacent surfaces

Figure 2-17 An undercut region composed of planar and curve surfaces

A few approaches of recognizing undercut regions in B-rep data using graph-based method was put forward by Ye et al [20, 21] They proposed a hybrid method for recognition

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of undercut regions from molded parts with planar, quadric and free-form surfaces The

undercut regions are defined using “extended attributed face-edge graphs” (EAFFEG) and

recognized by finding cut-sets of sub-graphs instead of commonly-used graph matching techniques The procedure of their method consists of several main phases: (1) pre-processing: determining the face property, inputting the parting lines, and constructing EAFEG of molded part, (2) identifying candidate undercut faces (see Figure 2-18), and (3) recognizing undercut regions with inner cut-sets In order to identify candidate undercut faces, a definition of face property was presented The face property of face f is its orientation with respect to parting direction d Let  be the angle between d and normal vector r of the

face The face property is positive or negative corresponding to the value cos  0 or cos  0, respectively Then, all candidate undercut faces are identified based

on a simple observation of edge eij shared by adjacent faces and the face property Negative face fi and positive fj are a pair of candidates of undercut faces, if shared edge eij is not a parting line In Figure 2-18, faces fi and fj are the potential undercut faces

f p

f i

f j

Parting direction

Figure 2-18 Candidate undercut faces

The above research work did not indicate the method to construct plane fp in order to determine the face property Moreover, the identification of candidate undercut faces using

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the face property is ambiguous in several cases In the first case, Figure 2-19(a) shows an undercut region as a leaning hole According to the identification of candidate undercut faces using the face property in [20, 21], face f4 is not denoted as a candidate undercut face Thus, the undercut region cannot be recognized using the concept of cut-set Ac, as shown in Figure 2-19(b) The corrected cut-set Ac should be the graph-base shown in Figure 2-19(c), if the face is considered as the potential undercut face In another case, Figure 2-20 shows an undercut region with a free-form surface Potential undercut face f2 cannot be identified based

on the surface property proposed in [20, 21]

(a) Undercut region (b) Graph-base (c) Graph-based and undercut region

Figure 2-19 An undercut region and its graph-based

f 1

f 2

f 3

Parting direction

e 12

r 1

r 2

r 2

Figure 2-20 An undercut region with free-form surface

Ismail et al [5] accomplished a technique based on edge boundary classification (EBC) patterns for recognition of simple and interacting cylindrical-based regions Regions