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.)

- very easy to store and alter

- well suited to line based problems

- does not require a powerful computer

- easy to perform traditional drafting

- 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.

- gives appearance of solids.

- well known, and fast software and hardware for drawing.

- 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.

- Can make very complex parts quickly

- 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

- 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

- 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

- Very compact representation

- Primitive shapes match human though processes

- Very fast when creating parts with standard geometrical features

- 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.

- 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.

- 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

- square neck

- chamfer

- shoulder

- key seat

- spline

- flat

• 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