IE 590 Integrated Manufacturing Systems Lecture 4

1. IE 590 Integrated Manufacturing Systems Lecture 4 CAD & Geometric Modeling

2. 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. 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. 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. 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. Applications of Solid Modeling • Computation of volume and area: - decomposition of solid into cells - count of cells yields the volume - accuracy is det. by size of cells

7. 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. 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. 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. dif. Hole Box 1 Box 2 a. Part b. CSG Tree un Hole = CYL(…)AT(…) Box1 = BLO(…)AT(…) Box2 = BLO(…)AT(…) Box = Box1 UN Box2 Part = Box DIF Hole c. Instructions to construct part CSG example.

11. 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. Boundary Representation F5 F4 E8 E7 E5 F2 E6 E3 V4 V3 E4 V1 E2 E1 F3 V2 F1 A solid composed of faces, edges and vertices

13. BRep Face table Edge table Vertex table Face edges edge vertices vertex coordinate F1 E4, E3, E2, E1 E1 V1, V2 V1 x1, y1,z1 F2 E2, E7, E6 E2 V3, V2 V2 x2, y2, z2 F3 E1, E6, E5 E3 V3, V4 V3 x3, y3, z3 F4 E4, E5, E8 E4 V1, V4 V4 x4, y4, z4 F5 E3, E7, E8 E5 V1, V5 V5 x5, y5, z5 E6 V2, V5

14. 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. (Taken from Solid Modeling by H. Chiyokura)

16. 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. 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. Mistakes in Boolean Operations Mistakes in Euler Operations (Taken from Solid Modeling by H. Chiyokura)

19. 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. 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. 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. 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. 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. 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. 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. Example: Projection and Through Hole F + 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. 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. 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. 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. 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. 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. Figure E3 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).

33. 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. Figure E4 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.

35. 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.