solid modeling 471
TRANSCRIPT
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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