Chapter 3: Geometric Modeling

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Chapter 3Geometric Modeling
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Agenda

• Introduction to geometric modeling
• Wireframe modeling
• Surface Modeling
• Solid Modeling
• Parametric and Variational Design
• Computer-Aided Engineering Analysis (CAE)
• CAD/CAM data exchange

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Introduction to Geometric Modeling

• The geometric information about an object
  essentially includes types of surfaces and edges
  and their dimensions and tolerances
• Geometric modeling refers to a set of techniques
  concerned mainly with developing efficient
  representations of geometric aspects of a design
• The requirements of geometric modeling trade-off
  between storage and fast access
• The completeness of the part representation
  topological and geometric data
• Easy to use by designers
• Rendering capability
• Why is knowledge of geometric modeling necessary
• Object oriented nature and limited database
• The knowledge of structure and technique of the
  software to fully understand software manual
• High level of understanding the CAD process
• Geometric modeling approaches
• 2D view drawing
• 3D models wireframe surface and solid modeling

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Wireframe Modeling

• Wireframe modeling uses points curves and so
  forth to define objects
• Characteristics of wireframe modeling
• Simple and straightforward in concept
• Contain only low-level information
• The virtual edges are not usually provided
• Ambiguous representations of real objects may be
  created
• require more user effort to input necessary
  information than that of solid models. Provide
  limited information
• Wireframe entities points and lines
• Analytic wireframe entities
• Synthetic wireframe entities

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Analytic Curves

• Analytic entities points lines arcs circles
  ellipses parabolas hyperbolas
• Example Create the wireframe model of the
  following object by utilizing a CAD/CAM system
• Solution To create the wireframe of this part
• Create the 2D profile of the part
• Sweep the profile across a space distance in the
  direction perpendicular to the profile plane 3D
  model of the base of the part
• Repeat the process to create the pocket
• Create the two holes create one hole first then
  use of some edit commands to duplicate the holes

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Synthetic Curves

• Synthetic curves curves that are constructed by
  many curve segments
• Requirements of a good representation of
  engineering objects
• Easy to control of the continuity of the curves
  to be designed
• Requires less storage to represent a curve
• Less computation time and no computational
  problems
• Easy to input by user
• Continuity the smoothness of the connection of
  two curves or surfaces at the connection points
  or edges
• C0 continuity connecting two curves
• C1 continuity the gradients at the joining point
  are the same
• C2 continuity (curvature continuity) the
  gradients and the center of curvature are the
  same
• Types of synthetic curves provided by major
  CAD/CAM systems
• Hermite cubic spline
• Bezier curves
• B-spline curves
• Rational B-splines
• Nonuniform rational B-splines

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Type of Continuities
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Hermite Cubic Spline

• Each segment is approximated by a parametric
  cubic function (interpolation techniques)
• Why
• A cubic polynomial is the minimum-order
  polynomial function that generates C0 C1 C2
  continuity curves
• A cubic polynomial is the lowest-degree
  polynomial that permit inflection within a curve
  segment and allows representation of non planar
  space curves
• Higherorder polynomials have some drawbacks
  such as oscillation about control points and are
  uneconomical in terms of storing information and
  computation
• The general form of a cubic function
• r V(t) a0 a1t a2t2 a3t3 0 t 1

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Bezier Curves

• The shape of a Bezier curve is controlled by
  control points. The Bezier curves do not pass
  through all the given data points except the
  first and the last control point ( approximation
  techniques)
• The curves pass through the first and last
  control points
• The tangents at the first and last points are in
  the directions of the first and last segments of
  the characteristic polygon.
• The Bezier curve has the convex hull property
  the entire curve lies within the characteristic
  polygon

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Bezier Curves
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Other Synthetic Curves

• The B-spline is considered a generalization of
  the Bezier curve local control is an interesting
  feature of B-spline curves
• Rational B-splines (RBSs) are generalizations of
  B-splines. Each control point has an associated
  weight to control the behavior of the curve
• The nonuniform rational B-spline (NURBS) is a
  class of RBS. Using a NURBS a designer can model
  free-form surfaces by defining a mesh of control
  points. NURBS is now used in many software
  packages.

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Surface Modeling

• Surface modeling define the objects by their
  bounding faces. Surface modeling systems contain
  definitions of surfaces edges and vertices.
• Advantages
• Contain more information wireframe information
  connection of two surfaces etc.
• Can be used to determine the curtter path
• Offer better graphic interaction
• Disadvantages
• Do not provide the topology of the entities (can
  not distinguish the interior and exterior of an
  object)
• The collection of surfaces may not define a
  physical part

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Surface Entities

• Plane Surface
• Ruled (lofted) Surface
• Surface of Revolution
• Tabulated Cylinder
• Bezier Surface and B-spline Surface

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Surface Representations

• Implicit Equation F(xyz)0
• Explicit Equation V xyzT xyf(xy)T

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Surface Representations

• Parametric equation of a surface
• V(st) xyzT X(st) Y(st) Z(st)T
• smin s smax tmin t tmax
• Parametric representation of synthetic surfaces
• Hermite bicubic surface patch
• Bezier surface patches patch
• Uniform cubic B-spline surfaces control points
• Surfaces are normally defined in patches each
  patch corresponds to a rectangular domain in s-t
  space

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Solid Modeling

• In a solid modeling system objects are defined
  directly by primitive shapes called building
  blocks.
• Representation schemes for solid modeling
• Boundary representation (BREP) for complex
  designs
• Constructive solid geometry (CSG) easy to
  create simple objects
• Sweep representation
• Primitive instancing
• Cell decomposition
• Analytical solid modeling

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Boundary Representation

• Describe the geometry of an object in terms of
  its boundaries namely vertices edges and
  (orientable) surfaces
• Basic entities for BREP face edge and vertex
• Validation of BREP model using Eulers law to
  ensure that a real object if formed or bounded
• polyhedron F E V 2
• polyhedral with passageways and holes
• F E V L 2(B G)
• where F faces E edges V vertices L
  faces inner loops B bodies G genera (
  torus through hole)
• Database the object-body-genus-face
  loop-edge-vertex

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Example of Boundary Representation
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Example

• Verify Eulers law for the two parts in the
  following figures
• Solution
• F E V 5 9 6 2
• Assume that 2 cylinders are approximated by 2
  cubic holes (12 edges for each cubic hole 4 end
  loops 2 genuses)
• F E V L 21 54 36 5 -2
• 2(B G) 2(1 - 2) -2

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Constructive Solid Geometry (CSG)

• A solid object is constructed by simple solid
  objects and Boolean operators under tree
  structure
• Advantages easy the structure is concise and
  less storage
• Disadvantages slow in displaying the objects
• Solid entities (primitives) block cylinder
  cone sphere etc.
• Half-spaces is considered the basic elements of
  primitives H V f(V)lt0 V E3
• The point set V f(V) 0 surface f(V)lt0
  solid f(V)gt0 empty.
• Common half-spaces planar cylindrical
  spherical conical .
• E.g. a block 6 planar half-spaces using AND
  operators
• Regularized set operations UNION INTERSECTION
  DIFFERENCE

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Constructive Solid Geometry (CSG)
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CSG Illustrative Example
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Sweep Representation

• A solid is defined in terms of volumes swept out
  by two-or-three-dimensional laminae as they move
  along a curve ( path)
• Translational sweep a planar 2D laminae is moved
  a distance in space in a perpendicular direction
  to the plane of the laminae
• Rotational sweep rotating the laminae about an
  axis at a given angle

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Primitive Instancing Method

• Construct an object that has the same topology as
  a potential primitive but different geometry
• E.g. a bolt can be define by BOLT primitive and
  their basic parameters ( number of sides
  length pitch diameter) specified by the designer

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Cell Decomposition Scheme

• Represent a solid object by dividing its volume
  into smaller volumes or cells.
• The cuboid cells are often chosen and all cells
  are identical
• Three types of cell empty full partial. The
  partial cells may be further decomposed into
  empty full or partial partial cell size
  resolution
• Decomposition schemes
• Simple regular grid slicing the 3D space into an
  array of equal-sized and regularly spaced cells
• Vfull Vobject Vfull Vpartial
• Octree adaptive grid a hierarchical subdivision

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Analytical Solid Modeling (ASM)

• A parametric representation of a solid by the
  tensor product formulation of parametric solid or
  hyperpatch
• The variable point of the solid
• V(stu) xyz x(stu) y(stu)
  z(stu)
• Where smin ssmax tminttmax uminuumax
• A general solid can be represented by the
  following polynomial

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Parametric and Variational Design

• Capability of a CAD/CAM system to support the
  modifications in the geometric models and
  dimensions

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Computer-Aided Engineering Analysis

• Engineering analysis is concerned with analysis
  and evaluation of engineering product designs
• Finite-element analysis (FEA) is used to analyze
  and study functional performance of an object by
  dividing it into a number of small building
  blocks (finite elements). E.g. the object
  structures stresses and deflection are predicted
  by FEA.
• Divide the object into a grid of elements
  (square cube etc.)
• The FE program has information of the elements to
  write the governing equations in the form of a
  stiffness matrix
• The unknowns for each element are the
  displacement at the node points
• The FE program assembles the stiffness matrices
  for these simple elements to form the global
  stiffness matrix for the entire model
• This stiffness matrix is solved for the unknown
  displacements given the known forces and boundary
  conditions

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Computer-Aided Engineering Analysis

• Steps in applying FEA
• Discretization of the given continuum (object)
• Selection of the solution approximation
• Development of element matrices and equations
• Assembly of the element equations
• Solution for the unknown at the nodes
• Interpretation of the result
• Static dynamic and natural frequency analysis
  determine stress deflections strains of the
  structure caused by
• Fixed load (Static analysis)
• Changing load (Dynamic analysis)
• Vibrations (Natural frequency analysis)
• Heat transfer analysis determine the temperature
  distribution
• Plastic analysis analyze the elements of the
  plastic injection molding process such as the
  plastic part runner geometry material
  properties mold gate and vent locations cooling
  system and molding temperatures and pressures to
  identify potential problems and obtain optimum
  part. Mold and process design early in the
  development process

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Computer-Aided Engineering Analysis

• Fluid flow analysis analyze various
  characteristics of fluid flow such as flow rate
  diffusion dispersion and consolidation for the
  purpose of piping system design
• Motion analysis (kinematic analysis) is the
  analysis of geometric properties ( displacement
  velocity acceleration) of a mechanism to produce
  a desired motion by 3D simulation of an object in
  motion.
• Tolerance analysis is the process of determining
  the proper assignment of tolerances
• Design optimization the analysis process for
  achieving the some specific design objectives
  (goals)

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CAD/CAM Data Exchange

• Using a neutral format file is the best solution
  to establishing communication between dissimilar
  CAD/CAM systems
• IGES ( Initial Graphics Exchange Specification)
• The basic elements of IGES are entities
  geometric entities (shape curve surface) and
  nongeometric entities ( such as relation between
  various entities)
• Each entity is assigned a number
• 1 599 and 700-5000 specific assignments
• 600-699 and 10000-99999 user-defined entities
• Two Cartesian Coordinate systems MCS and WCS
• IGES reserve numbers 100-199 to define geometric
  entities. Each entity has two main types of data
  directory data (type of entity) and parameter
  data (parameter of entities) and some other
  related data are provided
• In case of entities are not covered by IGES
  approximate conversions

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CAD/CAM Data Exchange

• IGES file structure
• Flag section standards name version and
  conversion errors
• Start section name of source and target CAD/Cam
  systems
• Global section global information of the stored
  entities
• Directory entry section entities name
• Parameter data define entities
• Terminate section a checking record

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CAD/CAM Data Exchange

• PDES (Product Data Exchange Standard or Product
  Data Exchange using STEP)
• PDES is done in terms of applications IGES
  utilizes entities as basic elements
• 3 layers
• application layer application model description
  and information are expressed
• logical layer provide a consistent and computer
  independent description
• physical layer structure and format of the
  exchange file itself to keep efficiency in file
  size
• DXF (Drawing exchange file autoCAD) interchange
  AutoCAD drawing and other programs. DXF file
  structure
• Header section general information
• Tables section definitions of named items
• Block section block definition entities
• Entities section the drawing entities including
  any block reference
• END OF FILE