Parametric Design

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Parametric Design

• 09 Solid Modeling

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Solid

• Adjective
• Not soft or yielding
• Not hollow
• Unadulterated or unmixed
• Of strong and secure construction
• Noun
• Something solid
• A three dimensional geometric figure of object
• A substance that resists moderate stress and
  deformation, unlike a liquid or a gas
• Encarta Dictionary

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Some history

• Early 70s
• Ian Braid Ph.D. thesis at the University of
  Cambridge in England introduced the BUILD system
• ROMULUS, Parasolid, ACIS
• the Compac system U of Berlin, Germany
• Proren U of Ruhr, Germany
• Bruns Euclid system France
• Engelis Euklid system Switzerland
• TIPS-1 Hokkaido University, Japan
• GeoMap University of Tokyo, Japan
• Shapes Draper Labs, US
• Synthavision system US
• GLIDE Chuck Eastman, Carnegie-Mellon,
  Architectural application and database
• PADL University of Rochester
• Aristides A. G. Requicha, GEOMETRIC MODELING A
  First Course

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Goals of Solid Modeling

• Problems of graphical models lack of robustness,
  incompleteness, limited applicability.
• Complete representation of solid objects that are
  adequate for answering any geometric questions
  (from robots) without help of human user.
• Two major issues integrity and complexity

Soonhung Han, Kaist
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Properties of solid modeling

• Expressive power
• Validity manufacturability,
• Unambiguity and uniqueness
• Description languages operations for
  construction
• Conciseness storage requirement
• Computational ease and applicability Computing
  power requirements

Soonhung Han, Kaist
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Solid model

• Create only complete representation of solid
  objects
• Unambiguous representations for solids
• Requirements for mathematical properties for
  solid model
• Rigidity
• Finiteness
• Solidity
• Closure under Boolean operation
• Finite describability
• Boundary determinism

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Required mathematical properties

• Rigidity
• Fixed distance and angle in Euclidean space
• Preserving distances and angles
• Finiteness
• Finite extents of physical objects
• Solidity
• No dangling faces or edges
• Closure under Boolean operation
• Boolean operations applied to solids should
  produce other solid
• Result of Boolean operation can be used as input
  to other Boolean operations
• Subtraction and addition of solid are guaranteed
  to produce solid

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Required mathematical properties

• Finite describability
• Finite amount of data should be able to describe
  point sets used to model solids
• Boundary determinism
• Boundary of a solid should define the solid
  unambiguously

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Types of solid models

• Decomposition models
• Constructive models
• Boundary models
• Non manifold models

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Decomposition model

• Represent a point set as a collection of simple
  objects from a fixed collection of primitive
  object types, combined with a single gluing
  operation

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Types of decomposition model

• Exhaustive enumeration
• All space is evenly blocked out a 3D matrix
  stores information on the material (or lack of)
  in each block
• Voxel
• Cellular decomposition
• Irregular cells block out only the object
• FEM (Finite Element Meshes) for FEA
• Space subdivision
• recursively block out all space
• Octree representation

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Exhaustive Enumeration

• Special case of decomposition where primitives
  are cubical in shape
• Uniformly-sized volume elements called voxels
• Used extensively in computer graphics and medical
  graphics
• Efficient but requires significant storage
• Accuracy limited unless voxels are extremely small

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Voxel
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Cellular decomposition

• Variety of basic cell types and a single
  combination operator glue
• Individual cells are usually created as
  parameterized instances of cell types
• Cells may be any object that topologically
  equivalent to a sphere (no holes)

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FEM
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Space subdivision

• Devised to overcome huge memory consumption of
  exhaustive enumeration
• Adaptive subdivision scheme based on the three
  dimensional grid of an exhaustive enumeration

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Quadtree
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Octree
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Constructive model

• Represent a point set as combination of primitive
  point sets.
• Each of the primitives is represented as an
  instance of primitive solid type
• Operations to combine primitives
• Union
• Intersection
• set difference
• Very efficient in terms of storing information
• Visualization required individual curves and
  surfaces to be combined and evaluated
• Very expensive computationally

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Types of constructive models

• Half-space model
• Functions f(x,y,z) divide space into 2 halves
  (flt0 or fgt0 _at_(x,y,z))
• Halves can be bounded (sphere) or unbounded
  (infinite cylinder)
• CSG(Constructive Solid Geometry) model
• User only interacts with bounded solid objects
• Computer models bounded solids as combinations of
  half spaces
• Nearly all constructive modeling uses CSG- easier
  for user

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Half-space model
Infinite Cylinder
Planes
Finite Cylinder
z lt b
2
2
2
x y lt r
z gt a
Department of Mechanical Engineering, The Ohio
State University
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CSG model

• Primitives

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CSG model

• CSG tree

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CSG model

• CSG tree

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CSG model

• CSG tree

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CSG model
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CSG model
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CSG model
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CSG model
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Boundary model

• Represent a point set in terms of its boundary.
• Boundary of solid surfaces
• Boundary of surface faces
• Boundary of face curve
• Most solid modeling packages use the boundary
  representation for storing the models
• Solid is considered to be bounded by a set of
  faces
• Faces have a compact mathematical representation
• Plane
• Toroid
• Cylinder
• Parametric surface such as a Bezier surface

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Boundary model
edge
face
vertex
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Space subdivision
Department of Mechanical Engineering, The Ohio
State University
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B-Rep graph
Topology
Geometry
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Non-manifold Models

• Heat transfer from 1 part touching another
  touching faces
• Crack propagation 1D and 2D elements within a
  solid part
• Hybrid material Sandwich panel
• 2 types of non-manifold modelers
• Include lower dimensional entities in a solid
  model data structure
• Concentrate on composite models gt cellular model

Soonhung Han, Kaist
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Non-manifold data structure
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Non-manifold model
B
A