4.2 GEOMETRIC MODELS
• There are many ways to model a part, the major categories are,
- Elemental (using lines and points like drafting)
- Surfaces (such as polygons used in ES 206)
- Solids - Swept
- Solids - BRep
- Solids - CSG
- Features
• If describing a block with a hole in it, each of the methods above will result in different descriptions
• Which is best???? all of them in the right situations.
• Each method has its particular advantages, and disadvantages.
• The best software and hardware supports a combination of all methods.
• It is assumed that other information is used to describe the geometries above, like,
- Position
- Orientation
- Dimensions
• The geometries can also be used to associate other information,
- Materials
- Tolerances
- Finishes
- Inspection information
4.2.1 Elemental Depiction:
• Depicted with the simplest of details (lines, points arcs, etc.)
• Advantages,
- very easy to store and alter
- well suited to line based problems
- does not require a powerful computer
- easy to perform traditional drafting
• Disadvantages,
- not capable of carrying complex information
- ambiguous
- complex items require long time to model
- requires repetition
- very hard to connect to programs for FEM, etc.
• Typically used in older CAD systems like AUTOCAD, CADKEY, etc.
• A classic demonstration of the arbitrary nature is shown below,
4.2.2 Surface Description
• The geometry is described with polygons which should represent an entire surface of an object.
• Generally these polygons do not indicate which side a volume lies on, but inside/outside is defined with tricks like defining polygon vertices so that counter-clockwise is out.
• STL is a good example of an engineering use of this surface representation.
• This method is also used in computer games where speed is important, and the overhead of the full solid information is not desired.
• Advantages,
- gives appearance of solids.
- well known, and fast software and hardware for drawing.
• Disadvantages,
- because objects are not solid, they may be subject to ambiguities
- hard to pass data to other systems, like FEA
- not well suited to CAD
- polygon selection is problematic
• Commonly used in graphics packages like HOOPS, PHIGS, CORE, etc. Also acts as the basis for the SGI computer graphics.
• An example of the polygon meshes is given below.
• We can also define these geometries using edge meshes.
4.2.3 Solid - Swept
• A profile is created in 2D, and then swept along a path to create a volume, or to cut a volume.
• The path may be straight, rotating about an axis, rotation along a helix, following a curved twisting path.
• Advantages,
- Can make very complex parts quickly
• Disadvantages,
- Requires a powerful computer
- Some operations difficult
4.2.4 Solid - B-Rep (Boundary Representation)
• This still bears a remote resemblance to Surface Modelling.
• Major differences are that,
- inside/outside is defined for each surface
- the edges, and vertices of touching faces are defined
• Advantages,
- can store very complex geometries
- easy to propagate changes to faces, edges and vertices
- can easily generate and store complex surfaces
- many systems support this method, such as PARASOLIDS, ACIS, etc.
- can be used to mimic CSG
• Disadvantages,
- high Level information is still not present in model
- requires a powerful computer
- hard to recognize some simple features like a block
• A BRep object is pictured below,
• Each feature in a B-Rep object can be varied independently
• Geometry is kept in parallel with the object topology. One example of a data structure is seen below.
• A common data focus uses the edges of an object to define the shape (vertices and faces can also be used)
• Euler operations can be used to build an object.
• We can check to see if the solid model is valid using the basic Euler equation, or the more involved Euler-Poincare tologoical equation. These equations must be satisfied for the models to be valid.
• When developing solid modelers we can use the Euler operations to ensure that the model stays topographically valid at all times.
4.2.5 Solid - CSG
• Does not calculate lines/vertices/faces when storing part geometries
• Uses primitive shapes such as planes, blocks, spheres, cylinders, wedges, torii, etc. to model shapes
• The primitives can be rescaled to meet requirements
• Uses a basic set of operators to combine or cut with the primitives,
Union - Both primitives are joined into one (boolean OR)
Intersection - The part of the primitives which overlaps (boolean AND)
Not - The inverse of a shape
Assemble - Parts may overlap without being joined
Difference - The area of one primitive is removed from another
• Basic common primitives are,
- blocks
- cylinders
- wedges
- tetrahedrons
- spheres
- torii
- cones
• Advantages,
- Very compact representation
- Primitive shapes match human though processes
- Very fast when creating parts with standard geometrical features
• Disadvantages,
- Slow because all interpretation is done at once
- may be difficult to incorporate irregular surfaces
• Used in systems like PADL2, Romulus, Build, etc.
• CSG designs can be stored in trees
• Various types of CSG operators are possible based on closure of sets. In particular we can consider two boxes that touch, but don’t overlap.
• Halfspaces can be used for defining boundaries of an object.
4.2.6 Tessellated Models
• Space is broken down as a regular/irregular grid.
• locations in space are marked as occupied/empty/partially filled.
• this method is most common when using scanners such as CAT and MRI that collect data in voxels (these are small rectangular volumes)
4.2.7 Features
• The designer would simply define a part in terms of fundamental manufacturing features, such as chamfers, through slots, blind slots, etc.
• Very high level, but can complicate additions of unanticipated features, like a ridge in a car hood.
• Advantages,
- very intuitive and easy to use
- can simplify other aspects of CIM (eg. If a standard feature is used there will be a standard process plan to make that feature).
- emphasizes the use of standard components.
• Disadvantages,
- restrictive when dealing with nonstandard features
- interaction of features can be hard to estimate
- a complete set of all possible features would be very large
• There are two levels of features commonly used in these systems,
- micro
- macro
• A set of standard features for rotational parts might be,
• Macro Features,
- cylinder
- taper
• External Features
- rotational fillet
- thread
- square neck
- chamfer
- shoulder
- external radius
- key seat
- spline
- flat
- thread
• Internal Features
- internal taper
- internal slot]
- internal tapered radial slot
- internal round slot
- countersink
- internal spline
- woodruff keyseat
• A set of prismatic features might be,
• Macro Features,
- box
• External Features
- linear chamfer
- linear round
- linear v slot
- linear slot
- linear round slot
- linear t-slot
• Internal features
- rectangular pocket
- linear fillet