ie 590 j cecil nmsu 1 ie 590 integrated manufacturing systems lecture 4 cad & geometric modeling
TRANSCRIPT
1 IE 590 J Cecil NMSU
IE 590 Integrated Manufacturing Systems
Lecture 4
CAD & Geometric Modeling
2 IE 590 J Cecil NMSU
Geometric Modeling
• Technique for providing complete/compatible description of the geometry of the part
• Studies computer based representation of geometry and related information needed for various applications such as engineering design, manufacturing, planning, inspection, etc.
• Involves the study of data structures, algorithms and file formats for creating, representing and communicating geometric information of parts and processes
3 IE 590 J Cecil NMSU
Terms&Concepts
• Geometric Model: the representation of a 3D shape
• Geometric Modeling: the technique of constructing 3D shape
• 2 Broad categories: - Solid modeling&curved surface modeling• Solid Modeling Focus: - Two widely used representations, Constructive
Solid Geometry CSG representations and Boundary Representations Brep
4 IE 590 J Cecil NMSU
Solid Model
• They represent complete shape of object as a closed space in 3D
• Only in a solid model, is it possible to check if a point in space is included in the solid or not
5 IE 590 J Cecil NMSU
Applications of Solid Modeling
• Interference checks:
- design of assembly or design of assembled machine
- interference can be checked automatically
- can be computed and displayed
• Collision detection:
- examples?
- How?
6 IE 590 J Cecil NMSU
• Computation of volume and area:
- decomposition of solid into cells
- count of cells yields the volume
- accuracy is det. by size of cells
Applications of Solid Modeling
7 IE 590 J Cecil NMSU
Applications
• Cutter Path Generation and Visualization
- Cutter Path: what is it?
- Leads to automatic verification of NC code possible
- Detect interferences and collisions
• Finite Element Analysis
- to generate meshes of parts, solid models are required
- meshes can be generated automatically
8 IE 590 J Cecil NMSU
Constructive Solid Geometry (CSG)
• Widely used representation method
• CSG uses PRIMITIVE shapes as building blocks AND BOOLEAN OPERATORS to build parts or objects
• Boolean Operators:?
• Union, Subtraction and Difference
• Drawbacks:
- Limited operations
- Time to display is too long
9 IE 590 J Cecil NMSU
Example of CSG based Part Construction
• CSG Models are rep. In a CSG Tree
• Primitives form the leaf and the interior nodes correspond to Boolean operations
10 IE 590 J Cecil NMSU
un
a. Part
b. CSG Tree
dif.
Box 1 Box 2
Hole
Hole = CYL(…)AT(…)Box1 = BLO(…)AT(…)Box2 = BLO(…)AT(…)Box = Box1 UN Box2Part = Box DIF Hole
c. Instructions to construct part
CSG example.
11 IE 590 J Cecil NMSU
Boundary Representations
• Objects are rep. By a collection of bounding faces plus topological information, which defines relationship:
- between faces, edges and vertices
- Hierarchy: Faces are composed of edges
>>Edges are composed of vertices
• BReps are difficult to create but provide easy graphics interaction and display
12 IE 590 J Cecil NMSU
Boundary Representation
A solid composed of faces, edges and vertices
E1F3
E2
E3
E4
E5E6
E7E8
V1
V2
V3V4
F1
F2
F4
F5
13 IE 590 J Cecil NMSU
BRep
Face table Edge table Vertex table
Face edges edge vertices vertex coordinate
F1 E4, E3, E2, E1 E1 V1, V2 V1 x1, y1,z1F2 E2, E7, E6 E2 V3, V2 V2 x2, y2, z2F3 E1, E6, E5 E3 V3, V4 V3 x3, y3, z3F4 E4, E5, E8 E4 V1, V4 V4 x4, y4, z4F5 E3, E7, E8 E5 V1, V5 V5 x5, y5, z5 E6 V2, V5
14 IE 590 J Cecil NMSU
CSG Vs BReps
• CSG Advantages:
• Data Structure(viz Tree based) is simple, internal management is easy
• CSG operations always result in a physically valid solid(see figure)
• Easy to modify a solid shape(corr. to a CSG rep)(see figure)
15 IE 590 J Cecil NMSU
(Taken from Solid Modeling by H. Chiyokura)
16 IE 590 J Cecil NMSU
CSG Vs BRep
CSG Drawbacks:
• Operations available are limited(to boolean type) - no local operations
• Display of complex parts requires longer time
Brep Advantages:
• Fast display and graphical interaction. Why?• No restriction on the availability of operations
- wide variety of operations supported
17 IE 590 J Cecil NMSU
CSG Vs BRep
Brep Drawbacks:
• Data structure is complex
- requires large memory space
- internal management is complex
• Do not always correspond to a valid solid (see figure)
18 IE 590 J Cecil NMSU
Mistak
es in B
oolean O
peration
sM
istakes in
Eu
ler Op
erations
(Taken from
Solid Modeling by H
. Chiyokura)
19 IE 590 J Cecil NMSU
• Important: In any system, you need a recovery facility
- Option 1: store all data in an external file (prev. Designed solid state can be retrieved)
- Option 2: store all commands performed (backtrack and undo)
20 IE 590 J Cecil NMSU
Validity of an engineering part or object
• Polyhedron: a part which has flat or planar polygonal surfaces only
• For the validity test of solids, Euler’s formula can be used
• For Polyhedrons without holes:
(# of faces)+(# of vertices)+# of edges +2
F+V = E+2,
where F, E and V are number of faces, edges and vertices
21 IE 590 J Cecil NMSU
• For Polyhedrons with through – holes:
F+V = E+2+R-2H,
where R is the # of disconnected interior edge rings in faces,
H is the number of holes in the body
22 IE 590 J Cecil NMSU
Example: Euler’s formula
Consider sample parts: F = 6, V = 8, E = 12
6 + 8 = 12 + 2
14 = = 14 (valid object)
F = 10(6 plus additional 4)
V = 16, E = 24
R = 2 (as its through hole)
H = 1
10 + 16 = 24 +2 +2 –2(1)
26 = = 26
23 IE 590 J Cecil NMSU
Example: Part with blind hole
If this part contained a blind hole, then?
Formula check: F+V = E+2+R
F = 6+5 = 11
V = 16, E = 24
R = 1(as its blind hole)
H = 0
11 +16 + 24 +2 +1 – 2(0)
27 = = 27
24 IE 590 J Cecil NMSU
Example: Part with Projection
F + V = E +2 +R-2H
F =11(6 + 4 +1)
V = 16, E = 24, H = 0
R = 1 (at base of projection)
F + V = E + 2 +R – 2H
11 +16 = 24 +2 +1-2(0)
27 = = 27
For 2 projections on a part,
F=16, V=24, E=36, R=2, H=0
16+24 = 36 +2+2
40 = = 40
25 IE 590 J Cecil NMSU
Example: Projection and Blind Hole
F + V = E + 2 +R –2H
F=5+11 (from prev. slide) =16
V=8+16=24
E=12+24=36
R=1+1 (at base of projection and top of hole)
F+V = E+2+R-2H
16+24 = 36+2+2-2(0)
40 = = 40
26 IE 590 J Cecil NMSU
Example: Projection and Through HoleF + V = E + 2 +R –2H
F=4+11 (from prev. slide) =15
V=8+16=24
E=12+24=36
R=1+2 (at base of projection and top of hole)
F+V = E+2+R-2H
15+24 = 36+2+3-2(1)
39 = = 39
27 IE 590 J Cecil NMSU
Euler Operators• As these operators follow Euler’s formula for solid
objects, they are called Euler Operations (EO)• Some Operators include: (consider solid A)• Make an Edge and a Loop (MEL)• Kill and Edge and a Loop (KEL)• Make a Vertex and an Edge (MVE)• Kill a Vertex and an Edge (KVE)• Make and Edge and a Vertex (MEV)• Make an Edge, a Vertex, a Vertex and a Loop (MEVVL)• Kill an Edge, a Vertex, a Vertex and a Loop (KEVVL)
28 IE 590 J Cecil NMSU
Figure E1
MEL (Make an Edge and a Loop) MEL(A, E1, L2, L1, V1, V2)
Edge E1 is generated between vertices V1 and V2 in loop L1 of solid A, as shown in Figure E1. At the same time, Loop L1 is separated into two loops L1 and L2.
29 IE 590 J Cecil NMSU
KEL (Kill an Edge and a Loop) KEL(A, E1, L2, L1, V1, V2)
Edge E1 of solid A is deleted, as shown in Figure E1. At the same time, two loops L1 and L2 are combined, and a new loop L2 is created. KEL is the inverse operation of MEL.
30 IE 590 J Cecil NMSU
MVE (Make a Vertex and an Edge) MVE(A, V1, E1, E2, x, y,z)
Vertex V1 of solid A is generated at a point (x,y,z) on edge E2, , as shown in Figure E2. As a result, edge E2 is separated into two edges E1 and E2.
Figure E2
31 IE 590 J Cecil NMSU
KVE (Make a Vertex and an Edge) KVE(A, V1, E1, E2, x, y,z)
Vertex V1 is deleted, as shown in Figure E2. As a result, two edges E1 and E2 are combined, and a new edge E2 is generated. KVE is the inverse operation of MVE.
32 IE 590 J Cecil NMSU
MEV (Make an Edge and a Vertex) MEV(A, E1,V1,V2, L1, x, y, z)
Edge E1 is generated between vertex V2 in loop L1 and a point(x,y,z), as shown in Figure E3.At the same time, vertex V1 is generated at the same point(x,y,z).
Figure E3
33 IE 590 J Cecil NMSU
KEV (Kill an Edge and a Vertex)
Edge E1 and vertex V1 are deleted, as shown in Figure E3. KEV is the inverse operation of MEV.
34 IE 590 J Cecil NMSU
MEVVL (Make an Edge, a Vertex, a Vertex and a Loop) MEVVL(A,E1,V1,V2, L1,x1,y1,z1,x2,y2,z2)
Edge E1 is generated between a point(x1, y1, z1) and a point(x2, y2, z2), as shown in Figure E4. At the same time, vertices V1 ,V2 and Loop L1 are generated.
Figure E4
35 IE 590 J Cecil NMSU
KEVVL (Kill an Edge, a Vertex, a Vertex and a Loop) KEVVL(A, E1,V1,V2,L1,x1,y1,z1,x2,y2,z2)
Edge E1 is deleted, as shown in Figure E4, and vertices V1 ,V2 and Loop L1 are also deleted. KEVVL is the inverse operation of MEVVL.