Solid Model Design

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GIRIJANANDA CHOWDHURY INSTITUTE OF

MANAGEMENT AND TECHNOLOGY

HATHKHOWAPARA, AZARA, GUWAHATI-17

DEPARTMENT OF MECHANICAL ENGINEERING

GENERAL PROFICIENCYPROJECT ON

SOLID MODEL DESIGN

Group Members: 5th

Sem Mechanical

NAME ROLL NO.

PULAKESH CHETIA 152

TRINAYAN SARMAH 067

PRIYANSHU SHARMA PATHOK 039

DIGANTA SARMA 155

PRITOM BARMAN 008

DHARINDOM SONOWAL 129

SEUJ PALLAV KHATANIAR 084

SIMANTA JEET BORAH 010


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CONTENTS

Acknowledgement 3

Introduction 4

Overview 5

Solid Representation Schemes 6

Constructive Solid Geometry (CSG) 8

Application of CSG 13

Advantages & Disadvantages Of CSG 14

Boundary Representation (B-Rep) 15

Advantages & Disadvantages Of B-Rep 20

Spatial Enumeration 21

Parameterized Primitive Instancing 26

Cell Decomposition 26Sweeping 27

Implicit Representation 27

Parametric & Feature Based Modeling 28

Advantages & Disadvantages Of

Solid Model Design 29


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ACKNOWLEDGEMENT

We would like to express our profound gratitude

to Mukulesh Baruah, Principal, Girijananda

Chowdhury Institute of Management and

Technology, Guwahati, for permitting us to carry

out our project work.

We are grateful to Prof. P.C.Baruah, Head of the

Department, Mechanical Engineering, G.I.M.T.,

Guwahati, for his cheerful encouragement and

valuable suggestions.

We thank all the teachers of Mechanical

Engineering department, G.I.M.T., from the bottom

of our heart for their sincere guidance throughout

the project.

Last but not the least; we thank our friends for

their support who helped us doing this project.


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INTRODUCTION

Solid Modeling:

'Solid Modeling' is a method used to design parts by

combining various 'solid objects' into a single three-dimensional

(3D) part design. It is a combination of Geometric modeling and

computer graphics which together forms the foundation for

computer-aided design (CAD)and in general support the

creation, exchange, visualization, animation, interrogation, and

annotation of digital models of physical objects. Solid modeling

is distinguished from related areas of Geometric modeling and

Computer graphics by its emphasis on physical fidelity. As a

field, solid modeling spans several disciplines, including

mathematics, computer science, and engineering.


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OVERVIEW

The use of solid modeling techniques allows for the automation

of several difficult engineering calculations that are carried out

as a part of the design process. Simulation, planning, and

verification of processes such as machining and assembly were

one of the main catalysts for the development of solid

modeling. More recently, the range of supportedmanufacturing applications has been greatly expanded to

include sheet metal manufacturing, injection molding, welding,

pipe routing etc. Beyond traditional manufacturing, solid

modeling techniques serve as the foundation for rapid

prototyping, digital data archival and reverse engineering by

reconstructing solids from sampled points on physical objects,mechanical analysis using finite elements, motion planning and

NC path verification, kinematic and dynamic analysis of

mechanisms, and so on. A central problem in all these

applications is the ability to effectively represent and

manipulate three dimensional geometry in a fashion that is

consistent with the physical behavior of real artifacts. Solidmodeling research and development has effectively addressed

many of these issues, and continues to be a central focus of

computer aided engineering.


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SOLID REPRESENTATION SCHEMES

Based on assumed mathematical properties, any scheme of

representing solids is a method for capturing information about

the class of semi-analytic subsets of Euclidean space. This

means all representations are different ways of organizing the

same geometric and topological data in the form of a data

structure. All representation schemes are organized in terms of

a finite number of operations on a set of primitives. Thereforethe modeling space of any particular representation is finite,

and any single representation scheme may not completely

suffice to represent all types of solids.

For example, solids defined via combinations of regularized

Boolean operations cannot necessarily be represented as the

sweep of a primitive moving according to a space trajectory,except in very simple cases. This forces modern geometric

modeling systems to maintain several representation schemes

of solids and also facilitate efficient conversion between

representation schemes.


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Below is a list of common techniques used to create or

represent solid models. Modern modeling software may use a

combination of these schemes to represent a solid:-

(1) Constructive Solid Geometry (CSG)(2) Boundary Representation (B-rep)(3) Spatial Enumeration(4) Instantiation or Parameterized Primitive

Instancing

(5) Cell Decomposition(6) Sweeping(7) Implicit Representation(8) Parametric & Feature Based Modeling


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CONSTRUCTIVE SOLID GEOMETRY (CSG)

Constructive Solid Geometry (CSG) denotes a family of schemes for

representing rigid solids as Boolean constructions or combinations ofprimitives via the regularized set operations discussed above. CSG is

currently the most important representation schemes for solids. CSG

representations take the form of ordered binary trees where non-

terminal nodes represent either rigid transformations (orientation

preserving isometrics) or regularized set operations. Terminal nodes are

primitive leaves that represent closed regular sets. The semantics of

CSG representations is clear. Each sub tree represents a set resultingfrom applying the indicated transformations/regularized set operations

on the set represented by the primitive leaves of the sub tree. CSG

representations are particularly useful for capturing design intent in the

form of features corresponding to material addition or removal (bosses,

holes, pockets etc). The attractive properties of CSG include

conciseness, guaranteed validity of solids, computationally convenient

Boolean algebraic properties, and natural control of a solid's shape in

terms of high level parameters defining the solid's primitives and their

positions and orientations. The relatively simple data structure and

elegant recursive algorithms have further contributed to the popularity

of CSG.

The simplest solid objects used for the representation are called

Primitives. Typically they are the objects of simple shape: cuboids,

cylinders, prisms, pyramids, spheres, cones. The set of allowable


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primitives is limited by each software package. Some software packages

allow CSG on curved objects while other packages do not.

BOOLEAN OPERATION

It is said that an object is constructed from primitives by means of

allowable operations, which are typically Boolean operations on sets:

Union, Intersection and Difference.

A primitive can typically be described by a procedure which accepts

some number of parameters; for example, a sphere may be described

by the coordinates of its center point, along with a radius value. These

primitives can be combined into compound objects using operationslike these:

UNION:-o The sum of all points in each of two defined sets. (logical

OR)

o Also referred to as Add, Combine,Join, Merge.


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DIFFERENCE :-o The points in a source set minus the points common to a

second set. (logical NOT)o Set must share common volumeo Also referred to as subtraction, remove, cut


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INTERSECTION :-o Those points common to each of two defined sets (logical

AND)

o Set must share common volumeo Also referred to as common, conjoin

When using Boolean operation, we should be careful to avoid situation

that do not result in a valid solid.


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DATA STRUCTURE

Data structure does not define model shape explicitly but ratherimplies the geometric shape through a procedural description

o E.g: object is not defined as a set of edges & faces but by theinstruction : union primitive1 with primitive 2

This procedural data is stored in a data structure referred to as aCSG tree

The data structure is simple and stores compact data easy tomanage

CSG TREE

CSG tree stores the history of applying Boolean operations onthe primitives.

It stores in a binary tree format. The outer leaf nodes of tree

represent the primitives. The interior nodes represent the Booleanoperations performed.More than one procedure (and hence database)can be used to arrive at the same geometry.


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APPLICATIONS OF CSG

Constructive solid geometry has a number of practical uses. It is used in

cases where simple geometric objects are desired, or where

mathematical accuracy is important.CSG is popular because a modeler

can use a set of relatively simple objects to create very complicated

geometry. When CSG is procedural or parametric, the user can revise

their complex geometry by changing the position of objects or by

changing the Boolean operation used to combine those objects.

One of the applications of CSG is that it can easily assure that objects

are "solid" or water-tight if all of the primitive shapes are water-tight.

This can be important for some manufacturing or engineering

computation applications. By comparison, when creating geometry

based upon boundary representations, additional topological data is

required, or consistency checks must be performed to assure that the

given boundary description specifies a valid solid object.

A convenient property of CSG shapes is that it is easy to classify

arbitrary points as being either inside or outside the shape created by

CSG. The point is simply classified against all the underlying primitivesand the resulting Boolean expression is evaluated. This is a desirable

quality for some applications such as collision detection.


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ADVANTAGES OF CSG

CSG is powerful with high level command. Easy to construct a solid model minimum step. CSG modeling techniques lead to a concise database less

storage.

Complete history of model is retained and can be altered atany point.

Can be converted to the corresponding boundary representation.DISAVANTAGES OF CSG

Only Boolean operations are allowed in the modeling processwith Boolean operation alone, the range of shapes to be modeled

is severely restricted not possible to construct unusual shape.

Requires a great deal of computation to derive the information onthe boundary, faces and edges which is important for theinteractive display/ manipulation of solid.

CSG representation is unevaluated --Faces, edges, vertices notdefined in explicit


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Boundary Representation (B-Rep)

Solid model is defined by their enclosing surfaces or boundaries.

This technique consists of the geometric information about thefaces, edges and vertices of an object with the topological data on

how these are connected. Since the boundaries of solids have the

distinguishing property that they separate space into regions

defined by the interior of the solid and the complementary

exterior .Every point in space can unambiguously be tested

against the solid by testing the point against the boundary of the

solid. Recalling that ability to test every point in the solid providesa guarantee of solidity. Using ray casting, it is possible to count

the number of intersections of a cast ray against the boundary of

the solid. Even number of intersections corresponds to exterior

points, and odd number of intersections corresponds to interior

points. The assumption of boundaries as manifold cell complexes

forces any boundary representation to obey disjointedness of

distinct primitives, i.e. there are no self intersections that cause

non-manifold points. In particular, the manifoldness condition

implies all pairs of vertices are disjoint, pairs of edges are either

disjoint or intersect at one vertex, and pairs of faces are disjoint or

intersect at a common edge. Several data structures that are

combinatorial maps have been developed to store boundary

representations of solids. Boundary representations have evolved

into a ubiquitous representation scheme of solids in most

commercial geometric modelers because of their flexibility in

representing solids exhibiting a high level of geometric

complexity.


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Why B-Rep includes such topological information?

- A solid is represented as a closed space in 3D space (surfaceconnect without gaps)

- The boundary of a solid separates points inside from pointsoutside solid.

B-Rep v/s Surface Modeling-

Surface model A collection of surface entities which simply enclose a

volume lacks the connective data to define a solid (i.e.

topology).

B- Rep model Technique guarantees that surfaces definitively divide model

space into solid and void, even after model modification

commands.


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B-Rep Data Structure

B-Rep graph store face, edge and vertices as nodes, with pointers, or

branches between the nodes to indicate connectivity.

Boundary representation- Validity

System must validate topology of created solid. B-Rep has to fulfill certain conditions to disallow self-intersecting

and open objects

This condition include- Each edge should adjoin exactly two faces and have a vertex

at each end.

- Vertices are geometrically described by point coordinates


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At least three edges must meet at each vertex. Faces are described by surface equations

The set of faces forms a complete skin of the solid with nomissing parts.

Each face is bordered by an ordered set of edges forming aclosed loop.

Faces must only intersect at common edges or vertices. The boundaries of faces do not intersect themselves

Validity also checked through mathematical evaluation Evaluation is based upon Eulers Law (valid for simple

polyhedra no hole)

(V E + F )= 2 ; where V=vertices; E=edges ;F=face loops

V = 5, E = 8, F = 5

5 8 + 5 = 2


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Expanded Eulers law for complex polyhedrons (with holes) Euler-Poincare Law:

(V-E+F-H)=2(B-P) Where H= number of holes in face; P=number of passages or

through holes; B=number of separate bodies; V=vertices;

E=edges ;F=face loops

V = 24, E=36, F=15, H=3, P=1,B=1

Boundary Representation- Ambiguity & Uniqueness

Valid B-Reps are unambiguous Not fully unique, but much more so than CSG Potential difference exists in division of

Surfaces into faces. Curves into edges


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ADVANTAGES OF B-REP

Capability to construct unusual shapes that would not be possiblewith the available CSG aircraft fuselages, swing shapes

Less computational time to reconstruct the image

DISADVANTAGES OF B-REP

Requires more storage More prone to validity failure than CSG Model display limited to planar faces and linear edges

- complex curve and surfaces only approximated


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SPATIAL ENUMERATION

This scheme is essentially a list of spatial cells occupied by the solid. The

cells, also called Voxels are cubes of a fixed size and are arranged in afixed spatial grid (other polyhedral arrangements are also possible but

cubes are the simplest). Each cell may be represented by the

coordinates of a single point, such as the cell's centroid. Usually a

specific scanning order is imposed and the corresponding ordered set

of coordinates is called a spatial array. Spatial arrays are unambiguous

and unique solid representations but are too verbose for use as

'master' or definitional representations. They can, however, representcoarse approximations of parts and can be used to improve the

performance of geometric algorithms, especially when used in

conjunction with other representations such as Constructive Solid

Geometry.

Spatial Enumeration Variation : Representation scheme that alleviates some of the storage

requirements of spatial enumeration.

Most common is the Octree representation

Fundamental idea is that of divide andconquer.

More easily understood by examiningthe 2D variant first (quadtrees)

QUADTREES

Quadtree is a tree data structure of storage used for 2dimensional cellular decomposition.

Quadtree is derived by successively sub-dividing a 2D plane in

quadrants

Each quadrant fully occupied, partially.


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Graphic representation of quadtree data structure similar to CSG

tree except:

not binary, but quartic (four branches at node)

nodes represent last level of decomposition

nodes indicate occupancy level

full

partial

empty

A partially occupied quadrant is recursively subdivided into

subquadrants.

This continues until either:

no partially occupied quadrants remain

a prescribed level of accuracy for the model has been reached.

Quadtrees are often used as an image representation format (screen

display)

a) 2D spatial enumeration of shape shown.

b) Quadtree representation of the same.


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OCTREES

Octrees represent the 3 dimensional extension of the quadtree

concept.

expand quadtree (2D) to 3D space Spatial volumes are sub-divided into a set of eight cells or octants.

Storage tree now has eight branches at each node.

Octree Representation


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Quadtree Storage Example

Spatial Enumeration : Domain and expressive power

Representation is approximate

Domain can be considered unlimited providing one excepts the

following conditions

that it is an approximation and,

high storage requirements exist

Spatial Enumeration : Validity

creates valid models if minimal connectivity required

each filled cell must have a neighbor

if connectivity required,validity check is straightforward


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PARAMETERIZED PRIMITIVE INSTANCING

This scheme is based on the notion of families of objects, each memberof a family distinguishable from the other by a few parameters. Each

object family is called a generic primitive, and individual objects within

a family are calledprimitive instances. For example a family of bolts is a

generic primitive, and a single bolt specified by a particular set of

parameters is a primitive instance. The distinguishing characteristic of

pure parameterized instancing schemes is the lack of means for

combining instances to create new structures which represent new and

more complex objects. The other main drawback of this scheme is thedifficulty of writing algorithms for computing properties of represented

solids. A considerable amount of family-specific information must be

built into the algorithms and therefore each generic primitive must be

treated as a special case, allowing no uniform overall treatment.

CELL DECOMPOSITION

This scheme follows from the combinatoric (algebraic topological)

descriptions of solids detailed above. A solid can be represented by its

decomposition into several cells. Spatial occupancy enumeration

schemes are a particular case of cell decompositions where all the cells

are cubical and lie in a regular grid. Cell decompositions provide

convenient ways for computing certain topological properties of solids

such as its connectedness (number of pieces) and genus (number of

holes). Cell decompositions in the form of triangulations are the

representations used in 3d finite elements for the numerical solution of

partial differential equations.


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SWEEPING

The basic notion embodied in sweeping schemes is simple. A set

moving through space may trace or sweep out volume ( a solid) thatmay be represented by the moving set and its trajectory. Such a

representation is important in the context of applications such as

detecting the material removed from a cutter as it moves along a

specified trajectory, computing dynamic interference of two solids

undergoing relative motion, motion planning, and even in computer

graphics applications such as tracing the motions of a brush moved

on a canvas. Most commercial CAD systems provide (limited)

functionality for constructing swept solids mostly in the form of atwo dimensional cross section moving on a space trajectory

transversal to the section. However, current research has shown

several approximations of three dimensional shapes moving across

one parameter, and even multi-parameter motions.

IMPLICIT REPRESENTATIONA very general method of defining a set of pointsXis to specify

a predicate that can be evaluated at any point in space. In

other words,Xis defined implicitlyto consist of all the points

that satisfy the condition specified by the predicate. The

simplest form of a predicate is the condition on the sign of a

real valued function resulting in the familiar representation of

sets by equalities and inequalities. For example iff= ax+ by+cz + dthe conditionsf(p) = 0 ,f(p) > 0, andf(p) < 0 represent

respectively a plane and two open linear half spaces. More

complex functional primitives may be defined by Boolean

combinations of simpler predicates. Furthermore, the theories


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of R-functions allow conversions of such representations into a

single function inequality for any closed semi analytic set. Such

a representation can be converted to a boundary

representation using polygonization algorithms, for example,

the marching cubes algorithm.

PARAMETRIC AND FEATURE BASED MODELINGFeatures are defined to be parametric shapes associated with attributes

such as intrinsic geometric parameters (length, width, depth etc),

position and orientation, geometric tolerances, material properties, andreferences to other features. Features also provide access to related

production processes and resource models. Thus, features have a

semantically higher level than primitive closed regular sets. Features

are generally expected to form a basis for linking CAD with downstream

manufacturing applications, and also for organizing databases for

design data reuse.


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ADVANTAGES OF SOLID MODEL DESIGN

(i) Better visualization of the design.(ii) CAD models can be brought into sophisticated analytical

programs.

(iii) Weights, CG's, and other geometric data can be had directly

from the CAD system, for more accuracy in less time.

(iv) Some design changes can be quicker and easier.

DISADVANTAGES OF SOLID MODEL DESIGN

(i) Some design changes can be slower and harder.

(ii) Requires a lot of computer knowledge.

Solid Model Design

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