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Chapter 3Geometric Modeling

Agenda

- Introduction to geometric modeling
- Wireframe modeling
- Surface Modeling
- Solid Modeling
- Parametric and Variational Design
- Computer-Aided Engineering Analysis (CAE)
- CAD/CAM data exchange

Introduction to Geometric Modeling

- The geometric information about an object

essentially includes types of surfaces and edges

and their dimensions and tolerances - Geometric modeling refers to a set of techniques

concerned mainly with developing efficient

representations of geometric aspects of a design - The requirements of geometric modeling trade-off

between storage and fast access - The completeness of the part representation

topological and geometric data - Easy to use by designers
- Rendering capability
- Why is knowledge of geometric modeling necessary?
- Object oriented nature and limited database
- The knowledge of structure and technique of the

software to fully understand software manual - High level of understanding the CAD process
- Geometric modeling approaches
- 2D view drawing
- 3D models wireframe, surface and solid modeling

Wireframe Modeling

- Wireframe modeling uses points, curves and so

forth to define objects - Characteristics of wireframe modeling
- Simple and straightforward in concept
- Contain only low-level information
- The virtual edges are not usually provided
- Ambiguous representations of real objects may be

created - require more user effort to input necessary

information than that of solid models. Provide

limited information - Wireframe entities points and lines
- Analytic wireframe entities
- Synthetic wireframe entities

Analytic Curves

- Analytic entities points, lines, arcs, circles,

ellipses, parabolas, hyperbolas - Example Create the wireframe model of the

following object by utilizing a CAD/CAM system - Solution To create the wireframe of this part
- Create the 2D profile of the part
- Sweep the profile across a space distance in the

direction perpendicular to the profile plane 3D

model of the base of the part - Repeat the process to create the pocket
- Create the two holes create one hole first then

use of some edit commands to duplicate the holes

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Synthetic Curves

- Synthetic curves curves that are constructed by

many curve segments - Requirements of a good representation of

engineering objects - Easy to control of the continuity of the curves

to be designed - Requires less storage to represent a curve
- Less computation time and no computational

problems - Easy to input by user
- Continuity the smoothness of the connection of

two curves or surfaces at the connection points

or edges - C0 continuity connecting two curves
- C1 continuity the gradients at the joining point

are the same - C2 continuity (curvature continuity) the

gradients and the center of curvature are the

same - Types of synthetic curves provided by major

CAD/CAM systems - Hermite cubic spline
- Bezier curves
- B-spline curves
- Rational B-splines
- Nonuniform rational B-splines

Type of Continuities

Hermite Cubic Spline

- Each segment is approximated by a parametric

cubic function (interpolation techniques) - Why?
- A cubic polynomial is the minimum-order

polynomial function that generates C0, C1, C2

continuity curves - A cubic polynomial is the lowest-degree

polynomial that permit inflection within a curve

segment and allows representation of non planar

space curves - Higherorder polynomials have some drawbacks,

such as oscillation about control points, and are

uneconomical in terms of storing information and

computation - The general form of a cubic function
- r V(t) a0 a1t a2t2 a3t3 0 t 1

Bezier Curves

- The shape of a Bezier curve is controlled by

control points. The Bezier curves do not pass

through all the given data points except the

first and the last control point ( approximation

techniques) - The curves pass through the first and last

control points - The tangents at the first and last points are in

the directions of the first and last segments of

the characteristic polygon. - The Bezier curve has the convex hull property

the entire curve lies within the characteristic

polygon

Bezier Curves

Other Synthetic Curves

- The B-spline is considered a generalization of

the Bezier curve local control is an interesting

feature of B-spline curves - Rational B-splines (RBSs) are generalizations of

B-splines. Each control point has an associated

weight to control the behavior of the curve - The nonuniform rational B-spline (NURBS) is a

class of RBS. Using a NURBS, a designer can model

free-form surfaces by defining a mesh of control

points. NURBS is now used in many software

packages.

Surface Modeling

- Surface modeling define the objects by their

bounding faces. Surface modeling systems contain

definitions of surfaces, edges, and vertices. - Advantages
- Contain more information wireframe information,

connection of two surfaces, etc. - Can be used to determine the curtter path
- Offer better graphic interaction
- Disadvantages
- Do not provide the topology of the entities (can

not distinguish the interior and exterior of an

object) - The collection of surfaces may not define a

physical part

Surface Entities

- Plane Surface
- Ruled (lofted) Surface
- Surface of Revolution
- Tabulated Cylinder
- Bezier Surface and B-spline Surface

Surface Representations

- Implicit Equation F(x,y,z)0
- Explicit Equation V x,y,zT x,y,f(x,y)T

Surface Representations

- Parametric equation of a surface
- V(s,t) x,y,zT X(s,t), Y(s,t), Z(s,t)T,
- smin s smax, tmin t tmax
- Parametric representation of synthetic surfaces
- Hermite bicubic surface patch
- Bezier surface patches patch
- Uniform cubic B-spline surfaces control points
- Surfaces are normally defined in patches each

patch corresponds to a rectangular domain in s-t

space

Solid Modeling

- In a solid modeling system, objects are defined

directly by primitive shapes called building

blocks. - Representation schemes for solid modeling
- Boundary representation (BREP) for complex

designs - Constructive solid geometry (CSG) easy to

create, simple objects - Sweep representation
- Primitive instancing
- Cell decomposition
- Analytical solid modeling

Boundary Representation

- Describe the geometry of an object in terms of

its boundaries, namely vertices, edges, and

(orientable) surfaces - Basic entities for BREP face, edge, and vertex
- Validation of BREP model using Eulers law to

ensure that a real object if formed or bounded - polyhedron F E V 2
- polyhedral with passageways and holes
- F E V L 2(B G)
- where F faces, E edges, V vertices, L

faces inner loops, B bodies, G genera (

torus, through hole) - Database the object-body-genus-face

loop-edge-vertex

Example of Boundary Representation

Example

- Verify Eulers law for the two parts in the

following figures - Solution
- F E V 5 9 6 2
- Assume that 2 cylinders are approximated by 2

cubic holes (12 edges for each cubic hole, 4 end

loops, 2 genuses) - F E V L 21 54 36 5 -2
- 2(B G) 2(1 - 2) -2

Constructive Solid Geometry (CSG)

- A solid object is constructed by simple solid

objects and Boolean operators under tree

structure - Advantages easy, the structure is concise and

less storage - Disadvantages slow in displaying the objects
- Solid entities (primitives) block, cylinder,

cone, sphere, etc. - Half-spaces is considered the basic elements of

primitives H V f(V)lt0, V ? E3, - The point set V f(V) 0 surface, f(V)lt0

solid, f(V)gt0 empty. - Common half-spaces planar, cylindrical,

spherical, conical . - E.g. a block 6 planar half-spaces using AND

operators - Regularized set operations UNION, INTERSECTION,

DIFFERENCE

Constructive Solid Geometry (CSG)

CSG Illustrative Example

Sweep Representation

- A solid is defined in terms of volumes swept out

by two-or-three-dimensional laminae as they move

along a curve ( path) - Translational sweep a planar 2D laminae is moved

a distance in space in a perpendicular direction

to the plane of the laminae - Rotational sweep rotating the laminae about an

axis at a given angle

Primitive Instancing Method

- Construct an object that has the same topology as

a potential primitive but different geometry - E.g. a bolt can be define by BOLT primitive and

their basic parameters ( number of sides,

length, pitch, diameter) specified by the designer

Cell Decomposition Scheme

- Represent a solid object by dividing its volume

into smaller volumes or cells. - The cuboid cells are often chosen and all cells

are identical - Three types of cell empty, full, partial. The

partial cells may be further decomposed into

empty, full or partial ? partial cell size

resolution - Decomposition schemes
- Simple regular grid slicing the 3D space into an

array of equal-sized and regularly spaced cells - Vfull Vobject Vfull Vpartial
- Octree adaptive grid a hierarchical subdivision

Analytical Solid Modeling (ASM)

- A parametric representation of a solid by the

tensor product formulation of parametric solid or

hyperpatch - The variable point of the solid
- V(s,t,u) x,y,z x(s,t,u), y(s,t,u),

z(s,t,u) - Where smin ssmax, tminttmax, uminuumax
- A general solid can be represented by the

following polynomial

Parametric and Variational Design

- Capability of a CAD/CAM system to support the

modifications in the geometric models and

dimensions

Computer-Aided Engineering Analysis

- Engineering analysis is concerned with analysis

and evaluation of engineering product designs - Finite-element analysis (FEA) is used to analyze

and study functional performance of an object by

dividing it into a number of small building

blocks (finite elements). E.g. the object

structures stresses and deflection are predicted

by FEA. - Divide the object into a grid of elements

(square, cube, etc.) - The FE program has information of the elements to

write the governing equations in the form of a

stiffness matrix - The unknowns for each element are the

displacement at the node points - The FE program assembles the stiffness matrices

for these simple elements to form the global

stiffness matrix for the entire model - This stiffness matrix is solved for the unknown

displacements given the known forces and boundary

conditions

Computer-Aided Engineering Analysis

- Steps in applying FEA
- Discretization of the given continuum (object)
- Selection of the solution approximation
- Development of element matrices and equations
- Assembly of the element equations
- Solution for the unknown at the nodes
- Interpretation of the result
- Static, dynamic, and natural frequency analysis

determine stress, deflections, strains of the

structure caused by - Fixed load (Static analysis)
- Changing load (Dynamic analysis)
- Vibrations (Natural frequency analysis)
- Heat transfer analysis determine the temperature

distribution - Plastic analysis analyze the elements of the

plastic injection molding process such as the

plastic part, runner geometry, material

properties, mold gate and vent locations, cooling

system and molding temperatures and pressures to

identify potential problems and obtain optimum

part. Mold and process design early in the

development process

Computer-Aided Engineering Analysis

- Fluid flow analysis analyze various

characteristics of fluid flow such as flow rate,

diffusion, dispersion and consolidation for the

purpose of piping system design - Motion analysis (kinematic analysis) is the

analysis of geometric properties ( displacement,

velocity acceleration) of a mechanism to produce

a desired motion by 3D simulation of an object in

motion. - Tolerance analysis is the process of determining

the proper assignment of tolerances - Design optimization the analysis process for

achieving the some specific design objectives

(goals)

CAD/CAM Data Exchange

- Using a neutral format file is the best solution

to establishing communication between dissimilar

CAD/CAM systems - IGES ( Initial Graphics Exchange Specification)
- The basic elements of IGES are entities

geometric entities (shape, curve, surface) and

nongeometric entities ( such as relation between

various entities) - Each entity is assigned a number
- 1 599 and 700-5000 specific assignments
- 600-699 and 10,000-99,999 user-defined entities
- Two Cartesian Coordinate systems MCS and WCS
- IGES reserve numbers 100-199 to define geometric

entities. Each entity has two main types of data

directory data (type of entity) and parameter

data (parameter of entities) and some other

related data are provided - In case of entities are not covered by IGES

approximate conversions

CAD/CAM Data Exchange

- IGES file structure
- Flag section standards name, version, and

conversion errors - Start section name of source and target CAD/Cam

systems - Global section global information of the stored

entities - Directory entry section entities name
- Parameter data define entities
- Terminate section a checking record

CAD/CAM Data Exchange

- PDES (Product Data Exchange Standard or Product

Data Exchange using STEP) - PDES is done in terms of applications, IGES

utilizes entities as basic elements - 3 layers
- application layer application model, description

and information are expressed - logical layer provide a consistent and computer

independent description - physical layer structure and format of the

exchange file itself to keep efficiency in file

size - DXF (Drawing exchange file, autoCAD) interchange

AutoCAD drawing and other programs. DXF file

structure - Header section general information
- Tables section definitions of named items
- Block section block definition entities
- Entities section the drawing entities, including

any block reference - END OF FILE

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