Copy of Solid-Modeling 12

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

CH- 88 SOLID MODELING

INTRO


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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 or on the object.

This classification is called as spatial addressability or set membership classification .

A valid solid should not have dangling edges or faces, 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 or both of two different solid modelingapproaches:

1) Primitives based2) 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 = BlockB = Cylinder C = Cylinder

A B C = D :Boolean operation; Create block A andsubtract 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. For example, 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 assketching plane, draw two circles* Create through cuts by extrusion toobtain 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 invariant2) Homogeneous 3-Dimensionality: No dangling

portions, 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) Sweeping5) Analytic Solid Modeling6) Cell decomposition

7) Octree Encoding8) Spatial Enumeration9) Primitive instancing


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HALF SPACE FORMAL DEFINITION A half-space is that portion of an n-dimensional space obtained by removing thatpart 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 regionsatisfying x >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 used

schemes (the other being CSG) Based on the concept that a solid is made of a

set 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 or outside the solid model.

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


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Euler Operations and EuclideanCalculations: 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 or edges 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: Types of Objects

Two types of objects:1) Polyhedral objects

Consist of plane faces and straight edges

2) Curved objects

Consist of curvilinear general surfaces andgeneral curvilinear edges


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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 directedspace curve bounded by two vertices

Vertices: Vertex is a point in space.

Loops: It is an ordered alternating sequence of vertices 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 of handles in a solid is called as genus .


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POLYHEDRAL OBJECTS

Four different classes:1. Simple polyhedra2. Polyhedra having loops3. Polyhedra having boundary (blind) holes

and interior holes4. Polyhedra having through holes or handles


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A DISJOINT SOLID

A solid having more than one body iscalled as disjoint solid. Thus a hollowsphere, a cuboid with internal hole, a solidhaving two pieces that are completelydisconnected etc. are examples of disjointsolids.

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 facesE= Number of edgesV= Number of vertices

L = Inner loops on facesB= bodiesG = genus (handles)


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SIMPLE POLYHEDRA

When L=B=G=0, then the solid satisfiesthe following equation and is called assimple polyhedron.

F E + V = 2


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A tetrahedron is the simplest:F = 4E = 6V = 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 = 8E = 18V = 12

In this case F + V - E = 2.


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SOLIDS THAT ARE NON-HOMOMORPHICTO A SPHERE (OPEN SOLIDS)

Open solids satisfy the following version of Euler law:

F E + V L = B GIn this equation B refers to an o p en b o d y

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 into planar 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 SOLIDSTOPOLOGY GEOMETRY

ModelBody

Genus

Face Underlying surface equationLoop

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 havebeen 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 simpler than B-Rep solids because each primitive isalready a valid solid.


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Data structures of CSGrepresentation

GraphDiagraph

TreeBinary treeInverted Binary tree


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Data Structure for CSG Solids:CSG Trees

S f CSG S lid CSG


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Data Structure for CSG Solids: CSG TreesHow to divide a given solids into primitives?

OP7

OP7

OP3

P1

P4

OP1

P2

P3

OP7

OP3

P1

P5

OP1

P2

P3nL + n R = 2n 2

Perfect Tree:nL = n R = n 1

n = Total nodes


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SWEEPING

A point set is swept along a directrix. 1. Translational sweep: Along a straightline

directrix2. Rotational sweep: axi-symmetric rotation3. Non-linear sweep: along a curve directrix

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

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