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1Pr e c is io n M ach in e r y D e s ign

Chapter 8 Geometric Modeling

L. H. Hsu

Department of MechanicalEngineering

National Cheng Kung University

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Abstract

Pr e c is io n M ach in e r y D e s ign

The fundamental principles of geometric models, including

curves, surfaces, solid models and feature-based modeling

method, are introduced in this chapter.

Design model construction and assembly modeling, finite-

element modeling and analysis, rapid prototyping and

manufacturing, CAD and CAM Integration and numericalcontrol, and various case studies based on the principles of

geometric modeling are discussed and demonstrated.

In addition, the higher-level integrated CAX systems such as

CATIA and I-DEAS will be introduced to match the principles

and their applications of geometric models.

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Contents


8.1 Introduction to CAD/CAE/CAM/PLM Systems 8.2 Curves

8.3 Surfaces 8.4 Solids 8.5 Applications

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8.1 Introduction toCAD/CAE/CAM/PLM Systems

8.1-1 Components of CAD/CAM/CAE/PLM Systems

8.1-2 Basic Concepts of Graphics Programming and

Drafting System

8.1-3 Geometric Modeling Systems

8.1-4 Geometric Models and Their EngineeringApplications

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8.1-1 Components of CAD/CAM/CAE/PLM

SystemsAn Overview of CAX Systems

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SystemsThe CAD/CAE Tools

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Systems

The CAM Tools

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The PLM Environment


Systems

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Systems

The Evolution of CAX/PLM

1965 1970 1975 1980 1985 1990

Wireframe modellers

Surface modellers

Solid modellers

Feature modellers

Modelle

r

1995 2000 2005

Product LifecycleManagement

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8.1-2 Basic Concepts of Graphics

Programming and Drafting System Hardware Components

Computing Machine

Graphical Environments

Graphics Devices

Software Components

Drafting

Geometric Kernel Modeling

Design Analysis, Manufacturing and

Management

Windows-based CAX Systems

OS / Intranet / Internet

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- Graphics Interface

- Textual I/O- Textual Menus + Dialogs

- Iconized Menus + Interactive Tools

- Drafting System:- Lines / Curves

- Annotation, i.e. dimensions and notes, etc.

- Programming Environment Tools- OPENGL

- MFC

- PRO/Program

8.1-2 Basic Concepts of Graphics

Programming and Drafting System

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8.1-3 Geometric Modeling Systems

- Wireframe Modeling Sys tem: mainly for drafting

- Surface Model ing System :mainly for styling andNC code generation

- Sol id Model ing Sys tem: mainly for 3Dconstruction and design function definition

- Nonmani fo ld Model ing System: for discreteobjects definition

- Assembly Model ing Capabi l i ty: for sophisticated

objects and production engineering- Web-based Modeling: for company-wide

integration and collaboration

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8.1-4 Geometric Models and

Their Engineering Applications

- Design Model Construction and AssemblyModeling

- Finite-Element Modeling and Analysis

- Rapid Prototyping and Manufacturing

- CAD and CAM Integration and Numerical Control

- Various Case Studies, such as designautomation for prosthetic socket

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8.2-1 Hermite Curves 8.2-2 Bezier Curves

8.2-3 B-Spline Curves 8.2-4 NURBS Curves

8.2 Curves

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Equation:

: the position vectors of the two end points of thecurve

:the tangent vectors at the end points

8.2-1 Hermite Curves( ) [ ]

+++=

'

1

'

0

1

0

32323232 223231

p

p

p

p

uuuuuuuuuuP

K

K

K

K

K

0P

G

1P

G

'

0PG

'

1PG

( )10 u

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8.2-1 Hermite Curves

Effect of and on curve shape'

0

PG

'

1P

G

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Equation:

where the basis function are

and where

is the position vector of the ith vertex

8.2-2 Bezier Curves( ) ( )

=

=n

i

nii uBPuP0

,

KK

( )10 u

( ) ( ) inini uui

nuB

= 1,

( )!!

!

ini

n

i

n

=

iPK

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8.2-2 Bezier CurvesCubic Bezier curves

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Equation:

where

are the B-Spline functions

are the knot values


8.2-3 B-Spline Curves( ) ( )

=

=n

i

kii uNPuP0

,

KK

( )11 + nk tut

( ) ( ) ( ) ( ) ( )1

1,1

1

1,

,

++

++

+

+

=

iki

kiki

iki

kii

kitt

uNut

tt

uNtuuN

( ) =

+

otherwisetutuN

ii

i01 1

1,

it

( )uN ki ,

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20

Nonuniform B-Spline curves:n=5,k=3

Uniform B-Spline curves:n=5,k=3 andn=5,k=4


8.2-3 B-Spline Curves

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Equation:

where are the weights

is a position vector composed of (xi, yi, zi)

8.2-4 NURBS Curves

( )( )

( )

=

==n

i

kii

n

i

kiii

uNh

uNPhuP

0

,

0

,

K

K

ih

iPK

Non-Uniform Rational B-Spline

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Example of cubic NURBS

8.2-4 NURBS Curves

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8.3-1 The Bicubic Hermite Surface

8.3-2 Bezier Surfaces

8.3-3 B-Spline Surfaces

8.3-4 NURBS Surfcaes

8.3 Surfaces

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Equation:

where are algebraic vector coefficients with

x,y,z components


( ) = =

=3

0

3

0

,i i

ji

ij vuavuPK

( )10,10 vu

ija

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Elements of a Bicubic Hermite surface


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Equation:

where are the control points at the vertices ofthe control polyhedron

and are the blending functions usedin Bezier curves


( ) ( ) ( )= =

=n

i

m

j

mjniji vBuBPvuP

0 0

,,,,KK

( )10,10 vu

jiP,K

niB , mjB ,

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A Bezier Surface and its control polyhedron


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Equation:

where are the control points located at thevertices of the control polyhedron

and are the blendingfunctions used in B-Spline curves


( ) ( ) ( )= =

=n

i

m

j

ljjiji vNuNPvuP0 0

,,,,KK

( )1111 , ++ mlnk tvtsus

jip ,K

( )uN ki, ( )vN lj ,

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4 x 5 B-Spline surface patches

(a)patch approximates data points (b) patch interpolates datapoints



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Equation:

Parameters:

where are the x,y,z coordinates

are the homogeneous coordinates of

the control points and the weights

8.3-4 NURBS Surfaces

( )( ) ( )

( ) ( )

= =

= ==n

i

m

j

ljkiji

n

i

m

j

ljkijiji

vNuNh

vNuNPh

vuP

0 0

,,,

0 0

,,,,

,

K

K

( )1111 , ++ mlnk tvtsus

jip ,K

jih ,

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8.4-1 Fundamentals of Solid Modeling

8.4-2 Half-spaces 8.4-3 Boundary Representation (B-rep) 8.4-4 Constructive Solid Geometry (CSG)

8.4-5 Other Representations 8.4-6 Solid Manipulations

8.4 Solids

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- Set theory

- Regularization of set operations

- Set membership classification

- Neighborhood concept, for boundary evaluation

- Euler operators, for constructing B-rep solid model


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- a set of objects: a collection or aggregation of objects

- requirements of the elements of a set

- definitive object, no fuzzy one- no identical element appears twice, involving datatype

defined in a system- the order of elements immaterial

- Set algebra (or set operations) including (Venn diagram)

- Laws of set algebra

- commutative / associative / distributive /idemoptence / involution

Set theory


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Set algebra


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- regularized set

- geometrically closed to avoid nonsense objects (or

geometric closure)- a regular set including

- interior subset, and

- boundary subset, i.e. skin wrapped around the interior

S = k iS i.e. a set with closure (k) and interior(i)

- preserve spatial dimensionality, i.e. no lower dimension

- maintain homogenous, i.e. no dangling parts

- regularized union (U*), regularized intersection ( ), and

regularized difference (-*)


Regu lar ized set operators (or Boolean Operators )

*


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Set regu lari ty


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(a)Non-regu larized sets


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(b )Regu lar sets


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- A process to study the behavior of the candidate set X

relative to the reference set S; S: a given object, (classified

as three subsets, iS bS cS i.e. interior, boundary, and

complement of S); X: a geometric entity used to classifyagainst S; the relationship denoted as

M[X,S] = (X in S, X on S, X out S)

- for solving geometric intersection problem, i.e. therelationship between point / line / surface / solid with

respect to another solid

- boundary evaluation

- for shading (ray tracing algorithm) and mass properties

calculation

- solid / solid interference checking

- the example of line/polygonPr e c is io n M ach in e r y D e s ign

Set membersh ip c lass i f icat ion


8 4 1 F d t l f S lid M d li

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Line/polygon set membership classification


Set membership classi f icat ion



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- An example, line (L)/polygon (R) relationship, to describe

the algorithm of the process

- find boundary crossings, P1, P2, using line/edgeintersection technique if reference set described in B-rep

- sort the boundary crossings, [P0, P1, P2 , P3] in this

case

- classfy L wrt R

- odd boundary crossings -> in segment

- even boundary crossings -> out segment

- the result of the case:

[P0, P1] L out R

[P1, P2] L in R

[P2, P3] L out RPr e c is io n M ach in e r y D e s ign




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Line/polygon set membership for B-

rep





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Another example (line/polygon) for reference set in CSG rep


Set membersh ip c lass i f icat ion


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- this algorithm is used to make up the shortage of

set membership- classification to evaluate the boundary portion

when candidate set on the boundary of

reference set

- discussed in CSG section


Neighborhood con cept , for boundary evaluat ion

Euler operators , for cons truc t ing B -rep sol id

mode l

- discussed in B-rep section


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- A basic representation scheme, or lower level, to

construct bounded solids

- Unbounded geometric entities; dividing therepresentation

- space into two infinite portions, one filled with material

and the other empty

- A half-space is defined as

H = {P: P and f(P) < 0}

- A half-space boundary defined by f(P) = 0, a

unbounded surface

8.4-2 Half-spaces

B i l t

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- Most widely used half-spaces including planar / cylindrical/ spherical / conical / toroidal, each one of them depictedwith their local coordinate systems

- These basic elements described by a set of ordered triples(x,y,z) as follows:

Planar half-space: H={(x,y,z):z

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Unbounded half-spaces

Bu i ld ing operat ions

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- Building procedure

- Transform (i.e. rotation, translation of rigid motion)

each defined half-space to a proper position

- Use Boolean operations, intersection or subtraction

of the transformed half-space to form closed objects

- Union of the closed objects (if not using predefinedclosed objects, an unbounded object may be

defined), to form the required object

- Two examples to be demonstrated



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Half-space representation of solid S

g


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Solid fillet and its half-space representation

8 4-3 Boundary Representation (B-rep)

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- A well -known representat ionto describe solid object

using basic elements and Euler operators under the

rule of Euler Law

- Euler [Poincare] law

F - E + V - L = 2(B - G)

where F: number of faces

E: number of edges

V: number of vertices

L: number of of inner loops within faces

B: number of exterior & interior bodies, or shell

G: number of handles (through holes), or called genus

8.4 3 Boundary Representation (B rep)

8.4-3 Boundary Representation (B-rep)

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- Bas ic e lements :

- used to construct

solid object step bystep

- faces / edges / vertices

[loops, genus, andbodies] and surfacenormal

- data structu refor

boundary modeling

shown at the right

8.4 3 Boundary Representation (B rep)

8 4 3 B d R t ti (B )

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Euler operators :- rules used to build effective object

- defined by specific system, e.g.


8 4 3 Boundary Representation (B rep)

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- Euler operator under Euler Laws



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- Anexampleof buildingan object

8.4-4 Constructive Solid Geometry (CSG)

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- One of the most popular scheme, i.e.- Succinct

- Easy to create and store

- Easy to check for validation

- Provide conceptual means for material removal (i.e.

difference) by solid primitives and interference

checking (e.g. robot path planning), etc...

- Building operators (Boolean Operators)

- Union (U*)

- Difference ( )

- Intersection (-*)

y ( )

*


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CSG Primitives

y ( )


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Building processrecorded by a tree graph

- nodes: Booleanoperator

- terminals: primitive

y ( )


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Another example

8 4-5 Other Representations

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Sweep Representation

- For creating 2-1/2 D objects- Moving points or curves to create closed curves

- A process to gain extruded solids via linear or

revolution (or translation or rotation) of closed

curves

- Applications mainly on

- material removal and

- interference detection of a moving object

8.4-5 Other Representations

Sweep Representat ion

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- Types of Sweep

Analyt ic Sol id Model ing (ASM)

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- Subdivided intoindividualhyperpatches --extensions of

parametric solidsbounded by analyticor parametric surface

patches

- Based on design andanalysis application,incapable to storeinformation for

manufacturing suchas surface normal

- ASM model of solid S

Organizat ion of Sol id Modelers

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- Multiple internal representations

--> hybrid system

- Four major parts

- Input system

- Geometric modeling system

- Application system

- Output system

g

Organizat ion of Sol id Modelers

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- Architecture of a typical solid modeler

g

- Types of so l id modelers

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-Types of systems

- Single representation:CSG or B-rep -->

CSG / B-rep- Dual representation:primary CSG -->transform to

B-rep if needed- Hybrid representation:

B-rep and CSG input

--> consistant internalrepresentations

8.4-6 Solid Manipulations

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- Displaying

- Evaluating Points, Curves, and Surfaces on Solids

- Segmentation

- Trimming and Intersection

- Transformation

- Editing

8 6 So d a pu at o s

8.5 Applications

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8.5-1 Design Model Construction and Assembly Modeling 8.5-2 Finite-Element Modeling and Analysis 8.5-3 Rapid Prototyping and Manufacturing 8.5-4 CAD and CAM Integration and Numerical Control

8.5-5 Various Case Studies

8.5 Applications

8.5-1 Design Model Construction and

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g

Assembly Modeling

- Initial Physical Model Created by Stylist


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- CAD Model Created by Designer

Press for jaw

construction

Press for cover

construction

Assembly Modeling


Assembly Modeling

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Assembly Modeling


Assembly Modeling

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Assembly Modeling

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- Screw Driver

Press for

simulation

Assembly Modeling


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- Screw DriverAssembly Modeling


Assembly Modeling

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- Screw Driver with Moment Handle

Press for

simulation

Press for

simulation

y g

Screw Driver with Moment Handle


Assembly Modeling

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Assembly Modeling

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- General Steps of Finite Element Modeling

8.5-2 Finite-Element Modeling and Analysis

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p g

- Example Knuckle 8.5-2 Finite-Element Modeling and Analysis

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Use Auto create Sections

and take all defaults

Use Remove Loop

to suppress holes

(use RMB to turnAnchor Nodes on)

Remove and Add

Curves as you like

The 4 volumes

in the center

will be definedas mapped

Fillet creates difficult

section needs clean up

Narrow area canbe cleaned up with

Replace Curve

Press for function description

Function

8.5-2 Finite-Element Modeling and Analysis

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Press for

mesh generation

Press for

boundary setup

Press forFEM results

simulation

8.5-2 Finite-Element Modeling and AnalysisMesh generation

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8.5-2 Finite-Element Modeling and AnalysisBoundary Setup

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8.5-2 Finite-Element Modeling and AnalysisFEM results

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8.5-3 Rapid Prototyping and

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Manufacturing

Press for

upper part

Press for

lower part

8.5-3 Rapid Prototyping and Manufacturing

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8.5-3 Rapid Prototyping and Manufacturing

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8.5-4 CAD and CAM Integration and

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Numerical Control

- Two examples of

NC machining after

CAD models havebeen created

8.5-5 Various Case Studies

- Design Automation for Prosthetic Socket

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Socket

Press for machining

socket mold

Machining for Prosthetic Socket Mold

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References and Suggestion Readings

R f

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References:1. Anand, V B, Computer Graphics and Geometric Modeling for

Engineers, John Wiley and Sons, 1993 .

2. Lee, K, Principles of CAAD/CAE/CAM Systems, Addison Wesley,

1999.3. McMahon, C and Browne, J, CADCAM from Principles to Practice,

Addison-Wesley, 1993.

4. Zeid, I, CAD/CAM Theory and Practice, McGraw-Hill Inc, 1991.

Suggestion Readings:1. Farin, G, Curves and Surfaces for Computer-Aided Geometric

Design, Academic Press, 1988.

2. Mortenson, M E, Geometric Modeling, John Wiley and Sons Inc,1997.

3. Piegl L, and Tiller W, The NURBS Book. Springer. 1997.

4. Rogers, D and Adams, A, Mathematical Elements of Computer

Graphics, 2nd Ed, McGraw-Hill Inc, 1990. Pr e c is io n M ach in e r y D e s ign

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