Solid Modeling 471

7/28/2019 Solid Modeling 471

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5/2/2013 I/C: Regalla Srinivasa Prakash 1

INTRO

SOLID MODELING


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CHARACTERISTICS SOLID MODELING

Solids models are known to be complete, valid,and unambiguous representations of objects.

A complete solid is one which enables a point inspace to be classified relative to the object, if it isinside, outside oron the object.

This classification is called as spatialaddressabilityorset membership classification.

A valid solid should not have dangling edges orfaces, then only it will allow interference

analysis, mass property calculations, finiteelement modeling and analysis, CAPP, machinevision, and NC part programming.


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SOLID MODELING APPROACHES IN CAD PACKAGES

All commercial CAD packages offer one orboth of two different solid modeling

approaches:1) Primitives based

2) Feature based

UNIGRAPHICS (EDS Technologies), CATIA(Dassault Systems), I-DEAS (StructuralDynamics Research Corporation) offer both

approaches.SolidWorks (Dassault Systems), Pro/Engineer

(Parametric Technology Corporation).


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SOLID ENTITIES

APPROACH ENTITIES

Primitives based

approach

Solid primitives (block,

cylinder, cone, sphere,wedge and torus)

Feature based approach Sketches


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PRIMITIVE BASED SOLID MODELING

This approach allows designers to use

predefined shapes (primitives) as buildingblocks to create complex solids.

Designers must use Boolean operations to

combine the primitives This approach is limited by the restricted

shapes of the primitives.

A

B

C

A, B and C are primitive solids.

A = Block

B = Cylinder

C = Cylinder

A B C = D :Boolean operation; Create block A and

subtract two cylinders from it using primitives approach.

D = Final solid


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FEATURE BASED SOLID MODELING This method is more flexible because it allows the construction of more

complicated objects and more elaborate solids more readily than theprimitive based modeling.

Feature based modeling is in fact a generalization of primitives approach.

Boolean operations are still used, but are hidden from the user. Forexample, creating a protrusion on the face of a cube is a Boolean unionand creating a cut in the cube is a Boolean subtraction. These operationsare must for creation of the final solid.

* Create a rectangle

* Subtract two circles

* Extrude the resulting feature

* The required solid is obtained

Alternatively,

* Create a rectangle

* Extrude the rectangle to create the block* Selecting the top face of the block as

sketching plane, draw two circles

* Create through cuts by extrusion to

obtain the final solid


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SOLID MODELING

Geometry and topology

Solid entities

Fundamentals of solid modeling

Half-spaces

Boundary representation (B-Rep)

Constructive Solid Geometry (CSG)

Sweeps Solid Manipulations


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Geometry and topology

Geometry is the actual dimensions that define

the entities of the object. It is also sometimescalled as metric information.

Topology (sometimes called as combinatorial

structure) is the connectivity and associativity of

the object entities.


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Solid primitives


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Desirable properties of solid models:

1) Rigidity: Shape of the solid model is invariant

2) Homogeneous 3-Dimensionality: No danglingportions, no isolated portions, solid boundariesare in contact with the interiors

3) Finiteness and finite describability: The two aredifferent; a (P, R, H) set describe a finitecylinder but may have infinite faces to describe

4) Closure under rigid motion and Booleanoperations: Should produce valid solids

5) Boundary determinism: Boundary must clearlydetermine the solid


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Most commonly used representation schemes:

1) Half-Spaces

2) B-Rep (boundary representation)

3) CSG (Constructive Solid Geometry)

4) Sweeping

5) Analytic Solid Modeling

6) Cell decomposition

7) Octree Encoding8) Spatial Enumeration

9) Primitive instancing


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HALF SPACE FORMAL DEFINITION

A half-space is that portion of

an n-dimensional space

obtained by removing that

part lying on one side of an

(n-1)-dimensional hyperplane.

For example, half a Euclideanspace is given by the three-

dimensional region satisfying

x>0, ;

while a half-plane is given bythe two-dimensional region

satisfyingx>0 ,
http://mathworld.wolfram.com/Space.htmlhttp://mathworld.wolfram.com/Half-Plane.htmlhttp://mathworld.wolfram.com/Half-Plane.htmlhttp://mathworld.wolfram.com/Half-Plane.htmlhttp://mathworld.wolfram.com/Half-Plane.htmlhttp://mathworld.wolfram.com/Space.html


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BOUNDARY REPRESENTATION (B-Rep)

One of the two most popular and widely usedschemes (the other being CSG)

Based on the concept that a solid is made of aset of faces, which are subsets of closed andorientable surfaces

A closed surface is one that is continuouswithout breaks.

An orientable surface is one where it ispossible to distinguish two sides by using thedirection of the surface normal to point inside oroutside the solid model.

Each face is bounded by edges and each edgeis bounded by vertices


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Euler Operations and Euclidean

Calculations:

Topology is created by Euler operations Euler operations can be used to create, manipulate,

edit the faces, edges, and vertices of a boundarymodel

Euler operations, similar to Boolean operations,ensure the validity (closedness, no dangling faces oredges etc.) of B-rep models

Geometry is created by the Euclidean

calculations Geometry includes coordinates of vertices, rigid

motion and transformation


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Elements of B-Rep models: Faces: Face is a closed, orientable and bounded

(by edges) surface. Edges: It is finite, non- self intersecting directed

space curve bounded by two vertices

Vertices: Vertex is a point in space.

Loops: It is an ordered alternating sequence ofvertices and edges

Boundary Hole: A blind hole

Interior Hole: A hole lying inside and having no

boundary on the surface of the solid Handles: Handle is a through hole in the solid. Itmay be termed as a 3-D hole. The number ofhandles in a solid is called as genus.


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POLYHEDRAL OBJECTS

Four different classes:

1. Simple polyhedra

2. Polyhedra having loops

3. Polyhedra having boundary (blind) holes

and interior holes

4. Polyhedra having through holes or handles


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A DISJOINT SOLID

A solid having more than one body is

called as disjoint solid. Thus a hollow

sphere, a cuboid with internal hole, a solid

having two pieces that are completelydisconnected etc. are examples of disjoint

solids.

Can you create a disjoint solid inPro/Engineer?


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EULER OPERATIONS Euler in 1752 proved that polyhedra that are

homomorphic to a sphere, that is their faces arenon self-intersecting and belong to closedorientable surfacse, are topologically valid if theysatisfy the following Euler-Poincare Lawequation:

F E + V L= 2(B G)F= Number of faces

E= Number of edges

V= Number of vertices

L = Inner loops on facesB= bodies

G = genus (handles)


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SIMPLE POLYHEDRA

When L=B=G=0, then the solid satisfies

the following equation and is called as

simple polyhedron.

F E + V = 2


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A tetrahedron is the simplest:

F= 4

E= 6

V= 4

In this case F+ V- E= 2.

A cuboid is a simple solid:

F= 6

E= 12

V= 8In this case F+ V- E= 2.

The given solid is simple:F= 8

E= 18

V= 12

In this case F+ V- E= 2.


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SOLIDS THAT ARE NON-HOMOMORPHIC

TO A SPHERE (OPEN SOLIDS)

Open solids satisfy the following version of

Euler law:

F E + V L = B G

In this equation B refers to an open body

which can be a wire, an area or a volume.


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Open solids

WIRE OPEN POLYDRALAMINA OPEN POLYDRA

SHELL OPEN POLYDRA OPEN POLYDRA (OBJECTS)

HAVING NO TOP FACE


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CURVED POLYHEDRA

Simplest curbed polyhedra are cylinder

and sphere.

F = 3; E = 3; V = 2

F = 1; E = 0; V = 1


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CURVED POLYHEDRA If the curved objects are represented by storing

the equations of curves and surfaces of edgesand faces, the resulting boundary scheme iscalled as exact B-Rep scheme.

Alternatively, one may use faceted B-Rep (also

called as tesselated representation), in whicheach curved face is divided intoplanar facets.Increasing the number of facets increasesaccuracy of display but takes more time.

Faceted representation is not good for CNCmachining because the machine hardware willdo one more level of interpolation resulting inerrors.


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DATA STRUCTURE FOR B-Rep SOLIDS

TOPOLOGY GEOMETRY

ModelBody

Genus

Face Underlying surface equation

Loop

Edge Underlying curve equation

Vertex

CONSTRUCTIVE SOLID GEOMETRY (CSG)


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CONSTRUCTIVE SOLID GEOMETRY (CSG)

Principle: A physical object can be divided into a

set of primitives that can be combined in a

certain order following a set of rules (Booleanoperations) to form the object.

Primitives themselves are valid CSG models.

Each primitive is also a solid considered to have

been built by a B-Rep process of combiningfaces from edges, edges from vertices.

Database contains both topology and geometry

Validity check for CSG solids is much simplerthan B-Rep solids because each primitive is

already a valid solid.


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Data structures of CSG

representation

Graph

Diagraph

Tree

Binary tree

Inverted Binary tree


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Data Structure for CSG Solids:

CSG Trees

D t St t f CSG S lid CSG T


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Data Structure for CSG Solids: CSG Trees

How to divide a given solids into primitives?OP7

OP7

OP3

P1

P4

OP1

P2

P3

OP7

OP3

P1

P5

OP1

P2

P3

nL + nR = 2n 2

Perfect Tree:

nL = nR = n 1

n = Total nodes


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SWEEPING

A point set is swept along a directrix.

1. Translational sweep: Along a straightline

directrix

2. Rotational sweep: axi-symmetric rotation

3. Non-linear sweep: along a curve directrix

4. Hybrid sweep: More than one directrix5. Invalid Sweep: Produces dangling faces

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Solid Modeling 471