# Introduction to Geometric Modeling CADCAMCAE OBJECTIVES OF GEOMETRIC

Introduction to Geometric Modeling CAD/CAM/CAE

OBJECTIVES OF GEOMETRIC MODELLING • To understand requirements for the information that is generated during the geometric modelling stage. • Study various types of geometric models possible • Develop various methodologies used for geometric construction such as sweep, surface models, solid models, etc. • Recognize the various types of surfaces and their application as used in geometric modelling • Appreciate the concept of parametric modeling which is the current mainstay of most of the 3 D modeling systems • Develop the various mathematical representations of the curves used in the geometric construction • Discuss the various CAD system requirements that need to be considered while selecting a system for a given application • Understand the concept of rapid prototyping and the various methods available for the purpose.

BENEFITS OF GEOMETRIC MODELLING • Stored in mathematical form, so modification can be easily done • by commands like move, rotate, scale

GEOMETRIC MODELLING • A prototype is needed for testing and optimizing design. • Costly • Time consuming • CAD eliminates the need of developing prototype. • Assists to evaluate the design. • “Computer compatible mathematical description” of the geometry of the object is called as geometric modelling. • CAD software allows the mathematical description of the object to be displayed as the image on the screen of computer

STEPS FOR CREATING GEOMETRIC MODEL Steps through which designer create GM by using CAD software: 1) Creation of basic geometric objects: • By using commands like points, lines, and circles 2) Transformations of the elements: • By using transformation commands of the geometric elements like scaling, translation etc. 3) Creation of the geometric model: • Commands that cause integration of the objects or elements of the geometric model to form the desired shape • Merge, shell, loft, sweep

STEPS FOR CREATING GEOMETRIC MODEL • During the process of geometric modeling the computer • converts various commands from the software into mathematical models, • stores them as the files and • finally displays them as the image. • The geometric models created by the designer can be opened at any time for reviewing, editing or analysis.

METHODS OF CREATING GEOMETRIC MODEL • Wire Frame modeling • Solid modeling • Surface modeling

Taxonomy Geometric Modeling Wire Frame Modeling Surface Modeling Voxels Solid Modeling CSG Winged Edge B-rep Halfedge

WHY TO DRAW 3 D MODELS? • 3 D models are easier to interpret. • Simulation under real-life conditions. • Less expensive than building a physical model. • 3 D models can be used to perform FEA (stress, deflection, thermal…. . ). • 3 D models can be used directly in manufacturing, Computer Numerical Control (CNC). • Can be used for presentations and marketing.

3 D MODELING There are three basic types of three-dimensional computer geometric modeling methods: • Wireframe modeling – modeling the curves of the part. • Surface modeling – Model the surfaces of the part, but without knowledge of material. • Solid modeling – full solid representation

WIREFRAME MODELING • W F models are drawn by using lines and curves. • Connected by using their point coordinates or vertices. • Simplest way of 3 D representation. • • Difficult to understand visualize the 3 D model. Still used to generate simple geometric shapes and models.

WIREFRAME MODELING • • Entities used to generate model are classified as: • Analytical curves: line, circle, hyperbola, parabola, E • Synthetic curves: Hermite, Bezier, B-spline, Rational Basic curves are combined with desired continuity like C 0, C 1, C 2 to get the required smoothness of the curve. • Entities required are Line, Ellipse, Circle, Spline, Polygon etc

WIREFRAME MODELING • Contains information about the locations of all the points (vertices) and edges in space coordinates. • Each vertex is defined by x, y, z coordinate. • Edges are defined by a pair of vertices. • Faces are defined as three or more edges. • Wireframe is a collection of edges, • there is no skin defining the area between the edges.

WIREFRAME MODELING Geometric model represented in wireframe model

WIREFRAME MODELING • A cube can be generated with twelve edges generated from eight vertices.

WIREFRAME MODELING Advantages: • Can quickly and efficiently convey information than multi-view drawings. • The only lines seen are the intersections of surfaces. • Can be used as input for CNC machines to generate simple parts. • Contain most of the information needed to create surface, solid and higher order models.

WIREFRAME MODELING Disadvantages: • • Do not represent an actual solids (no surface and volume). Cannot model complex curved surfaces. Cannot be used to calculate dynamic properties. Ambiguous views

WIREFRAME MODELING • Uniqueness problem • Ambiguity problem

WIREFRAME MODELING

WIREFRAME MODELING Of the various forms of representing the objects in geometric models, wire frames is the most basic There are three types of wire frame geometric modeling: 1) 2 D: It stands of two dimensional view and is useful for flat objects. 2) 21/2 D: • It gives views beyond the 2 D view and permits viewing of object that has no sidewall details. • Provides extruded depth to 2 D model without any detailing on its depth 3) 3 D: The three dimension representation allows complete threedimensional viewing of the model with highly complex geometry.

WIREFRAME MODELING

WIREFRAME MODELING

SURFACE MODELING A surface model represents the skin of an object, these skins have no thickness or material type. • Surface models define the surface features, as well as the edges, of objects. • A mathematical function describes the path of a curve (parametric techniques). • Surfaces are edited as single entities.

SURFACE MODELING

SURFACE MODELING • External shape of object can be obtained with no information about internal shape. • E. g. sheet metal cover, car body exteriors. • The basic sketching tools required for surface generation are : • extruded surface • revolved surface • swept surface • offset surface

SURFACE MODELING • A cube can be generated as surface model with six faces instead of 12 edges

SURFACE MODELING ADVANTAGES: • Eliminates ambiguity and non-uniqueness present in WFM • Renders the model for better visualization and presentation • Objects appear more realistic. • Provides the surface geometry for CNC machining. • Provides the geometry needed for mold and die design. • Can be used to design and analyze complex free-formed surfaces (ship hulls, airplane wings, car bodies, …). • Surface properties such as roughness, color and reflectivity can be assigned and demonstrated.

SURFACE MODELING DISADVANTAGES: • Surface models provide no information about the inside of an object. • Complicated computation, depending on the number of surfaces.

SOLID MODELS • In the solid modeling, the solid definitions include vertices (nodes), edges, surfaces, weight, and volume. • The model is a complete and unambiguous representation of a precisely enclosed and filled volume. • It may be assessed for mass property calculations, analysis, manufacturing, inspection, QC etc. • Modeling software like SW, I, PE, I etc. are available. • Most of the software provides various options to generate all three types of models. (WF, S & S)

SOLID MODELS • Analysis automation and integration is possible only with solid models has properties such as weight, moment of inertia, mass. • Solid model consist of geometric and topological data • Geometry shape, size, location of geometric elements • Topology connectivity and associativity of geometric elements

SOLID MODELS ADVANTAGES: • Has all the advantages of surface models (uniqueness, nonambiguous, realistic, surface profile) plus volumetric information. • Allows the designer to create multiple options for a design. • 2 D standard drawings, assembly drawing and exploded views are generated form the 3 D model.

SOLID MODELS ADVANTAGES: • Can easily be exported to different Finite Element Methods programs for analysis. • Can be used in newly manufacturing techniques; – CIM, CAM and DFM & DFA • Mass and volumetric properties of an object can be easily obtained; total mass, mass center, area and mass moment of inertia, volume, radius of gyration, …

SOLID MODELS DISADVANTAGES: • More intensive computation than wireframe and surface modeling. • Requires more powerful computers (faster with more memory and good graphics), not a problem any more.

WHY SOLID MODELING? • Recall weakness of wireframe and surface modeling – Ambiguous geometric description – incomplete geometric description – lack topological information – Tedious modeling process – Awkward user interface

Methods of Creating Solid Models • Boundary Representation (B-rep), mostly used in finite element programs. • Constructive Solid Geometry (CSG), CAD packages; Unigraphics, Auto. CAD – 3 D modeler. • Parametric Modeling, CAD packages; Unigraphics, Solid. Works, Inventor by Auto. Desk, Pro/Engineer, ….

BASIC PRIMITIVE SOLID:

SET THEORY Union Difference Intersection

BOOLEAN OPERATION • Union – The sum of all points in each of two defined sets. (logical “OR”) – Also referred to as Add, Combine, Join, Merge

BOOLEAN OPERATION • Difference – The points in a source set minus the points common to a second set. (logical “NOT”) – Set must share common volume – Also referred to as subtraction, remove, cut

BOOLEAN OPERATION • Intersection – Those points common to each of two defined sets (logical “AND”) – Set must share common volume – Also referred to as common

BOOLEAN OPERATION Subtract Union Intersection

· Volumetric and Mass properties of an object can be easily obtained. Corresponding mass properties are obtained if density is included.

Geometry • Geometry is the actual dimensions that defines the entity of an object • It is called as metric information • It includes • length of line • Angle between lines • Centre of circle • Radius of circle • Geometry concerns size and shape of an object

• Topology generalizes many distance related concepts, such as continuity, compactness and convergence. • In topology we can consider two wholly different shapes in geometry as the same because we can pull or push the lines or move the vertices. • It is the way in which entities are associated and connected.

Constructive Solid Geometry (CSG) Primitive Solids

Constructive Solid Geometry (CSG) • The simplest solid objects used for the representation are called primitives. • Typically they are the objects of simple shape: • cuboids, cylinders, prisms, pyramids, spheres, cones • The set of allowable primitives is limited by each software package. • Some software packages allow CSG on curved objects while other packages do not.

Constructive Solid Geometry (CSG) • Constructive solid geometry (CSG) is a technique used in solid modeling • Constructive solid geometry allows a modeller to create a complex surface or object by using Boolean operators to combine objects. • Often CSG presents a model or surface that appears visually complex, • but is actually little more than cleverly combined or decombined objects. • CSG defines a model in terms of combining basic and generated (using extrusion and sweeping operation) solid shapes.

Solid Modeling Example Using CSG Union Plan your modeling strategy before you start creating the solid model Cut

Constructive Solid Geometry (CSG) CSG objects can be represented by binary trees, where leaves represent primitives, and nodes represent operations. In this figure, the nodes are labeled for intersection, for union, and - for difference.

Constructive solid geometry (CSG) - advantage • CSG is powerful with high level command. • Easy to construct a solid model – minimum step. • CSG modeling techniques lead to a concise database less storage. – Complete history of model is retained and can be altered at any point. • Can be converted to the corresponding boundary representation.

Constructive solid geometry (CSG) - disadvantage • Only boolean operations are allowed in the modeling process with boolean operation alone, the range of shapes to be modeled is severely restricted not possible to construct unusual shape. • Requires a great deal of computation to derive the information on the boundary, faces and edges which is important for the interactive display/ manipulation of solid.

Solid Modeling Boundary Representation (B-rep) • A solid model is formed by defining the surfaces that form its boundary (edges and surfaces) • The face of a B-rep represents an oriented surface, there are two sides to the surface; solid side (inside) and void side (outside), unlike faces in a wireframe. • B-rep model is created using Euler operation • Many Finite Element Method (FEM) programs use this method. Allows the interior meshing of the volume to be more easily controlled.

• B-rep is a method to create solid models of physical objects • B-rep solid is represented as a volume contained in a set of faces. • It also contains topological information that defines the relationship between faces. • In B-rep a solid is bounded by its surface and has its exterior and interior clearly defined. • As it includes such topological information, a solid is represented as a closed space in 3 D space. • The geometry can be described by its boundaries like vertices, edges and surfaces. • Each face is bounded by edge and each edge is bounded by

A B-rep model of an object consists of faces, edges, vertices, loops, genes (handle) and body. • Vertex: A unique point (ordered triplet) in space. • Edge : A finite, non-self intersecting space curve bounded by two vertices that are not necessarily distinct. • Face : Finite, connected, non-self intersecting region of a closed, orientable surface bounded by one or more loops.

• Loop : An ordered alternating sequence of vertices and edges. A loop defines non-self intersecting piecewise closed space curve which may be a boundary of a face. • Loop is a hole in a face

• Body : An independent solid. Sometimes called a shell has a set of faces that bound single connected closed volume. A minimum body is a point (vortex) which topologically has one face one vortex and no edges. • A point is therefore called a seminal or singular body. • Genus : it is topological name for the number of handles or through holes in an object

The total information present in a B-rep model is classified into topological and geometrical data. , The topological part of the data provides the relationship among its objects such as vertices, edges and faces similar to that used in WFM Geometric information is usually equations of edges and faces.

TYPES OF MODEL IN THIS SCHEME There are two types of solid models in this scheme a) Polyhedral solid or object • Consists of planer faces connected at straight edges which in turn are connected at vertices. b) Curved solid or object: • Similar to polyhedral objects but with curved faces and edges.

Polyhedral objects are classified in to four class: a) First class: • do not have holes • each face is bounded by a single set of connected edges b) Second class • face may have loop c) Third class • objects with holes that are not through d) Fourth class • Through holes • Topologically holes are called as handle • Topological name for no. of handles in object is genus

Euler Equation Euler proved that polyhedra are topologically valid if they satisfy the following equation F – E + V – L = 2 ( B – G ) The simpler version of the above equation is F – E + V = 2

Curved and Faceted B-rep model • A closed cylindrical face has one edge and two vertices. • A spherical face has one vertex and no edges. • A boundary model of a closed cylinder has three faces (top, bottom and cylindrical face) two vertices and one edge connecting two vertices. • The other edges are for visualization called as limb. • The boundary model of a sphere consists of one face one vertex and no edges

CONSTRAINT BASED MODELLING

CONSTRAINT BASED MODELLING

CONSTRAINT BASED MODELLING

CONSTRAINT BASED MODELLING

CONSTRAINT BASED MODELLING

CONSTRAINT BASED MODELLING

FEATURE BASED MODELLING • A feature is defined as a shape and operation to build parts. • Shape is a 2 D sketch • Operation is activity that converts a sketch into 3 D shape

FEATURE BASED MODELLING STEPS TO CREATE OBJECT BY FEATURE 1. Create sketches 2. Create features 3. Use features to build parts

FEATURE BASED MODELLING EXTRUDE

FEATURE BASED MODELLING REVOLVE

FEATURE BASED MODELLING SWEEP

FEATURE BASED MODELLING LOFT

FEATURE BASED MODELLING PATTERN

FEATURE BASED MODELLING SPIRAL

FEATURE BASED MODELLING • Shaft • Pipe • Boss • Cut • Hole • Slot • Pocket • Shell • Fillet • Chamfer • Rib • Spring

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