ie 590 integrated manufacturing systems lecture 4

1 IE 590 J Cecil NMSU

IE 590 Integrated Manufacturing Systems

Lecture 4

CAD & Geometric Modeling


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


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


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


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?


• 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


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


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


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


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.


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


Boundary Representation

A solid composed of faces, edges and vertices

E1F3

E2

E3

E4

E5E6

E7E8

V1

V2

V3V4

F1

F2

F4

F5


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


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)


(Taken from Solid Modeling by H. Chiyokura)


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


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)


Mistak

es in B

oolean O

peration

sM

istakes in

Eu

ler Op

erations

(Taken from

Solid Modeling by H

. Chiyokura)


• 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)


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


• 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


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


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


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


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


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


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)


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.


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.


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


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.


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


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.


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


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.

ie 590 integrated manufacturing systems lecture 4

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