IE 590 J Cecil NMSU 1 IE 590 Integrated Manufacturing Systems Lecture 4 CAD & Geometric Modeling

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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?

IE 590 J Cecil NMSU 6 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 10 un a. Part b. CSG Tree dif. Box 1 Box 2 Hole Hole = CYL(…)AT(…) Box1 = BLO(…)AT(…) Box2 = BLO(…)AT(…) Box = Box1 UN Box2 Part = Box DIF Hole c. Instructions to construct part CSG example.

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 12 Boundary Representation A solid composed of faces, edges and vertices E1 F3 E2 E3 E4 E5 E6 E7 E8 V1 V2 V3 V4 F1 F2 F4 F5

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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)

IE 590 J Cecil NMSU 15 (Taken from Solid Modeling by H. Chiyokura)

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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)

IE 590 J Cecil NMSU 18 Mistakes in Boolean Operations Mistakes in Euler Operations (Taken from Solid Modeling by H. Chiyokura)

IE 590 J Cecil NMSU 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)

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 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

IE 590 J Cecil NMSU 22 Example: Euler’s formula Consider sample parts: F = 6, V = 8, E = = = = 14 (valid object) F = 10(6 plus additional 4) V = 16, E = 24 R = 2 (as its through hole) H = = –2(1) 26 = = 26

IE 590 J Cecil NMSU 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 = – 2(0) 27 = = 27

IE 590 J Cecil NMSU 24 Example: Part with Projection F + V = E +2 +R-2H F =11( ) V = 16, E = 24, H = 0 R = 1 (at base of projection) F + V = E + 2 +R – 2H = (0) 27 = = 27 For 2 projections on a part, F=16, V=24, E=36, R=2, H= = = = 40

IE 590 J Cecil NMSU 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 = (0) 40 = = 40

IE 590 J Cecil NMSU 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 = (1) 39 = = 39

IE 590 J Cecil NMSU 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)

IE 590 J Cecil NMSU 28 Figure E1 MEL (Make an Edge and a Loop)       MEL(A, E 1, L 2, L 1, V 1, V 2 ) Edge E 1 is generated between vertices V 1 and V 2 in loop L 1 of solid A, as shown in Figure E1. At the same time, Loop L 1 is separated into two loops L 1 and L 2.

IE 590 J Cecil NMSU 29 KEL (Kill an Edge and a Loop)       KEL(A, E 1, L 2, L 1, V 1, V 2 ) Edge E 1 of solid A is deleted, as shown in Figure E1. At the same time, two loops L 1 and L 2 are combined, and a new loop L 2 is created. KEL is the inverse operation of MEL.

IE 590 J Cecil NMSU 30 MVE (Make a Vertex and an Edge)        MVE(A, V 1, E 1, E 2, x, y,z) Vertex V 1 of solid A is generated at a point (x,y,z) on edge E 2,, as shown in Figure E2. As a result, edge E 2 is separated into two edges E 1 and E 2. Figure E2

IE 590 J Cecil NMSU 31 KVE (Make a Vertex and an Edge)        KVE(A, V 1, E 1, E 2, x, y,z) Vertex V 1 is deleted, as shown in Figure E2. As a result, two edges E 1 and E 2 are combined, and a new edge E 2 is generated. KVE is the inverse operation of MVE.

IE 590 J Cecil NMSU 32 MEV (Make an Edge and a Vertex)         MEV(A, E 1,V 1,V 2, L 1, x, y, z) Edge E 1 is generated between vertex V 2 in loop L 1 and a point(x,y,z), as shown in Figure E3.At the same time, vertex V 1 is generated at the same point(x,y,z). Figure E3

IE 590 J Cecil NMSU 33 KEV (Kill an Edge and a Vertex) Edge E 1 and vertex V 1 are deleted, as shown in Figure E3. KEV is the inverse operation of MEV.

IE 590 J Cecil NMSU 34 MEVVL (Make an Edge, a Vertex, a Vertex and a Loop)            MEVVL(A,E 1,V 1,V 2, L 1,x 1,y 1,z 1,x 2,y 2,z 2 ) Edge E 1 is generated between a point(x 1, y 1, z 1 ) and a point(x 2, y 2, z 2 ), as shown in Figure E4. At the same time, vertices V 1,V 2 and Loop L 1 are generated. Figure E4

IE 590 J Cecil NMSU 35 KEVVL (Kill an Edge, a Vertex, a Vertex and a Loop)            KEVVL(A, E 1,V 1,V 2,L 1,x 1,y 1,z 1, x 2,y 2,z 2 ) Edge E 1 is deleted, as shown in Figure E4, and vertices V 1,V 2 and Loop L 1 are also deleted. KEVVL is the inverse operation of MEVVL.