Chapter 4: Model and Techniques for Geometric Modelling PDF

Chapter 4: MODEL AND TECHNIQUES FOR GEOMETRIC MODELLING Requirements Of Geometric Modeling 2  Functions Of Geometric Modeling Design analysis: – Evaluation of areas and volumes. – Evaluation of mass and inertia properties....

Chapter 4: MODEL AND TECHNIQUES FOR GEOMETRIC MODELLING Requirements Of Geometric Modeling 2  Functions Of Geometric Modeling Design analysis: – Evaluation of areas and volumes. – Evaluation of mass and inertia properties. – Interference checking in assemblies. – Analysis of tolerance build-up in assemblies. – Analysis of kinematics — mechanics, robotics. – Automatic mesh generation for finite element analysi s. Drafting: – Automatic planar cross sectioning. – Automatic hidden line and surface removal. – Automatic production of shaded images. – Automatic dimensioning. – Automatic creation of exploded views for technical illustrations. Fig: Exploded views of Gear Pump 3  Manufacturing: – Parts classification. – Process planning. – Numerical control data generation and verification. – Robot program generation. Production Engineering: – Bill of materials. – Material requirement. – Manufacturing resource requirement. – Scheduling. Inspection and Quality Control: – Program generation for inspection machines. – Comparison of produced part with design. 4  Geometric Models Two-dimensional (2D) Three-dimensional (3D) Three (3) principal classifications of geometric models: 1. Line model 2. Surface model 3. Solid or Volume model 5 6  4.1 Wireframe Modeling System A wireframe model is a visual presentation of a 3D or physical object used in 3D computer graphics. A 3D wireframe modeling system differs from a 2D draughting system where the locations in space are defined by X,Y and Z co-ordinates. The term wireframe comes from designers using metal wire to represent the 3D shape of solid objects. The shape of the object is defined by a collection of points (vertices) and a set of edges. The object is projected onto the computer screen by drawing lines at the location of each edge. The system uses the same geometric entities i.e. lines, circles, arcs and curves as does a 2D system. 7  The computer has no knowledge of the surface shape between the edges or of what is solid and what is not. Always visually ambiguous. Fig: A geometric model presented in wireframe model 9  Disadvantages Of Wireframe Modeling System Tend to be not realistic. Ambiguity – complex model difficult to interpret No ability to determine computationally information on mass properties (e.g volume, mass, moment etc) and line of intersect between two faces of intersecting models. No guarantee that the model definition is correct, complete or manufacture able. 10  Advantages Of Wire-frame Easy to construct. Most economical in term of time and memory requirement. Often used for previewing objects in an interactive scenario and low-end designing and manufacturing system. Possible to draw some impossible solid objects. Fig: Impossible objects that can be modeled using wireframe model  4.2 Surface Modeling System The surface model is constructed essentially from surfaces such as planes, rotated curved surfaces (ruled surfaces) and very complex surfaces. The aim with a surface modeling system is to completely define the surface form of an object in such a way that the computer can calculate accurately the XYZ co- ordinates of any point on the surface. Some surfaces cannot be defined by analytical techniques → car bodies, ship hulls, die cavity surfaces and decorative surfaces styled for aesthetic. How? → modeled through a series of control points and other boundary conditions which specify the nature and surface desired. Common type of surfaces used in CAD systems → ruled surfaces, Bezier surfaces, B-spline surfaces and NURBS. 13 Surface Models 13 Fig: Various types of surfaces used in geometric modeling 15 T Types of Curved Surfaces Single curved surfaces Double curved surfaces Translation and revolution 16  Creating Surfaces CAD surface design packages provide commands to create surfaces by various methods. However, a CAD package may not support all types of surfaces depends on niche market for the package. Resultant surfaces can be classified into three types: 1. Generic surfaces 2. Free form surfaces 3. Derived surfaces 17  1. Generic Surfaces Created by sweeping a generator curve along a vector or about an axis. Only the generator curve influences the form of a surface. Two (2) types of generic surface: 1. Tabulated cylinder – Sweeping a generator curve along a vector produces a surface tabulated cylinder. – The magnitude of a vector controls the size of the surface. 2. Surface of revolution – Revolving a generator curve about an axis produces a surface of revolution. – The angle of revolution controls the size of a surface → angle of 360 degrees is the default. 18 Tabulated cylinder 19 Surface of revolution 20  2. Free Form Surfaces Uses more than one curve to create a free form surface. The surface is modeled through a series of control points and boundary conditions. Five (5) types of free form surface: 1. Ruled surface – Created from two or more curves that form opposite boundaries of the surface. – The selection of the starting point on the curve for the surface determines the actual surface obtained. Fig: Ruled surface 19 2. Surface patch – Created from four curves that enclose an area and share coincident ends. Fig: Surface patch 3. Net surface – Created from a network of curves. – A cloud of data points mat be used to create a net surface. Fig: Net surface 22 4. Swept surface – A cross-section profile curve is swept along a guide curve to create a swept surface. Fig: Swept surface 5. Lofted surface (cross section) – Uses one or more different cross-section profiles, a spine curves and guide curves. – The cross-section profiles are always normal to the spine curve. – Guide curve limit the surface boundaries. Fig: Lofted surface 23  3. Derived Surfaces Created from one or more existing surfaces. Most common derived surfaces: 1. Offset surface – Derived from a given surface by using the offset distance (i.e. thickness). – An offset surface is like a parallel surface. – Used to develop additional surfaces and forms at a given offset distance. Fig: Offset surface 24  3. Derived Surfaces 2. Filleted surface – Most common type of derived surface. – A fillet surface is a surface swept by a ball moves in continuous contact with two surfaces. – Filleting often involves trimming original surfaces to the fillet surface boundaries and done automatically. – Design original form surface → blend with filleting operation. Fig: Filleted surface 25  BEZIER CURVES To obtain a more free form design for aesthetic surfaces that satisfy some requirements, the modeling techniques need to provide more flexibility for changing the shape. This can be achieved by the use of Bezier curves named after P. Bezier, the designer of the French car company Renault, who invented the procedure in the 1960's. Uses the vertices as control points for approximating the generated curve. The curve will pass through the first and last point with all other points acting as control points. 26  BEZIER CURVES Bézier curve and the associated control polygon 27  BEZIER CURVES Changing the position of the individual control points in space, will alter the control polygon. Very flexible process and is widely used for the design of aesthetic surfaces. More control points → higher flexibility of the curve. Modification of Bezier curve by tweaking the control points 28 BEZIER CURVES 29  B-SPLINE CURVES The Bezier curve is considered as a single curve controlled by all the control points. As a result, increase in the number of control points, the order of the polynomial representing the curve increases. To reduce this complexity, the curve is broken down into more segments with better control exercised with individual segments, while maintaining a simple continuity between the segments. Alternative → use a B-spline. Whenever a single vertex is moved, only those vertices around that will be affected while the rest remains the same. 30 B-SPLINE CURVES B-spline curve 31  CURVE REPRESENTATION FORMULATION Presentation of curve geometry can be carried out in two forms: Implicit form Parametric form The implicit form is convenient for two dimensional curves of first order and second order. Typical curves that can be covered are lines, arcs and circles. The familiar form of such usage Would be Y = mX + c ( straight lines) – this is explicit Where X and Y represent the coordinates of the Cartesian 2D system. However for higher order curves, parametric form will more suitable. In parametric form, the Curve is represented as X = x(u) Y = y(u) Z = z(u) Where X,Y,Z are the coordinates on the space curve, with corresponding functions x,y,z are the Polynomials in a parameter. 32  CIRCLE REPRESENTATION Cartesian Coordinate System A coordinate on a line can be easily determined through the Cartesian coordinates, but a point on the circle can not be identified directly in the Cartesian System. The circle’s equation is defined by its center and radius. 33  Going through Parametric Equation Through the Example of A Circle To facilitate the understanding of the students on how parametric equations are formed, the example of a circle is given. Alternative parametric equations for a 1 circle – Part 1/2 https://www.youtube.com/watch?v=vx BjntzXAu4 2 This video shows us how the alternative parametric equation 2 for the circle is derived. Its given to enhance your understanding on this matter. Alternative parametric equations for a circle - Part 2/2 https://www.youtube.com/watch?v=w3ibA6EfDAs ❖ It should be noted here that the equation of a straight line polynomial was used as a basis function to form the circle’s parametric curve equation 34  * Follow up notes from P.N Rao text book, Ch 4,pg 109-112 The general parametric representation of a curve (Fig 4.46) is given as p = p(u) Where u is the parameter with which the Cartesian coordinates will represented as X = x(u); Y = y(u); Z = z(u) The above equation also can be written in the vector form as, p = p(u) = [ x(u) y(u) z(u) ] Where u is the parameter that can have any value from -1 to +1. Thus a curve segment can be defined as the coordinates given by continuous parameter, Single valued polynomials of the form X = x(u); Y = y(u); Z = z(u) 35  Bezier Equation Formulation Bernstein Polynomial Determining the polynomial order (n) of a Bezier Curve: Polynomial is a sum of expressions of variables and terms The number of control points (P) – 1. describing a function. Ex: If the are 4 control points, then the variable term order of the curve is (4-1)=3, therefore four control points gives us the curve of the third polynomial order, n=3. The example given in the notes is to derive Expand each of the Bernstein polynomial to form the matrix a Bezier equation for the curve of the 3rd polynomial order. 36  Bezier Equation Formulation Bernstein Polynomial Determining the polynomial order (n) of a Bezier Curve: Polynomial is a sum of expressions of variables and terms The number of control points (P) – 1. describing a function. Ex: If the are 4 control points, then the variable term order of the curve is (4-1)=3, therefore four control points gives us the curve of the third polynomial order, n=3. The example given in the notes is to derive Expand each of the Bernstein polynomial to form the matrix a Bezier equation for the curve of the 3rd polynomial order. 37  Bezier Equation Formulation Bezier Curve Derivation Video: https://www.youtube.com/watch?v=2OSEy o9oS6E Bernstein Matrix Parametric Matrix Point Matrix Solving the matrix above will give us the Bezier equation in the form of: P(u) = [ x(u) y(u) z(u) ] Attached is derivation for the Bezier Curve Formulation With an Example. 38  Applications Of Surfaces To create photo realistic images by applying foreground and background colors; different light sources; assigning translucency and opacity and texture attributes. The above characteristics make surfaces a powerful industrial design tool. 31 Applications Of Surfaces 31  4.3 SOLID MODELING A solid model is a computer description of a closed, solid, 3D shape represented by a data structure within which the 3D material can be completely and unambiguously defined. Composed of combinations of primitive objects → complete, unambiguous (clear- cut), physically realizable and modifiable. A solid model is always open to further modification by Boolean combination with other shapes. Solid modeling technology is particularly suited to the automation of many manufacturing and analysis tasks. 41  Types Of Solid Modeling Representation Two (2) general types of models: 1. Constructive Solid Geometry (CSG) Model 2. Boundary Representation Model 42  Constructive Solid Geometry (Csg) Model The CSG model representation uses a tree structure (often called a CSG tree) to define a solid model part. Comprises of Boolean combinations of solid primitives → block, cylinder, cone, sphere, torus and other simple solid shapes. Boolean operators → union (add), difference (subtract) and intersection, are used to combine primitives step by step to describe a solid model part. 43 Solid Modeling Primitives 44 Boolean Operations 45 Boolean Operations 46  Boolean Operations Boolean operators and their effect on model construction Video on Boolean Operations in CATIA: https://youtu.be/H8654WzUggA 47  Boolean Operations In ray casting: Operation Surface Limit Union A, D Intersection C, B Difference (2-1) B, D Boolean operators and their effect on model construction 48  Boolean Operations CSG tree illustrating part modeling steps using primitives 49  Boolean Operations A binary tree consisting of geometrical primitives, transformations and symbols representing Boolean operators represent the CSG object. The database of a CSG solid model stores primitives and Boolean operations only. The CSG method enables one to build the complex shapes quickly but only within limitations of the set of primitives available. 50  Boundary Representation (B-rep) Model A type of solid modeling in which faces are combined to create a solid model of a part. Represents a solid model (an enclosing surface) by describing its boundaries by faces. Faces are represented by their bounding edges and vertices. The geometry of the object can be described by its boundaries, namely vertices, edges and surfaces. Each face is bounded by edges and each edge is bounded by vertices. Faces can be formed by either straight-line object or curve segments. 51  Boundary Representation (B-rep) Model Basic elements of a B-rep model 52  Commercial Solid Modelers Due to the need of engineering and manufacturing support functions such as drafting, analysis and manufacturing, a more detailed data structure is highly desirable than just the primitive and Boolean operator based CSG data structure. However, the primitive and Boolean operation functionality of CSG representation is included in most B-rep modelers. Most commercial solid modelers are B-rep types. 53  Applications Of Solid Models Visualization Mass property calculations Product Assembly modeling and interference detection Drafting and product documentation Rapid prototyping Structural analysis Mechanism analysis Tool Design & Analysis CNC & CMM Part Programming & Simulation Robotics & Simulation Virtual Prototyping and Manufacture 54 4.4 Surface Connectivity Information 55 Typical Surface Display With The Parametric Variables u and v 56  4.4 Surface Connectivity Information A surface can be considered as a continuous set of points approximating a small elemental surface (like a plane) among each point. The mathematics involved is much more complex compared to the curves representation. From the curve representation, we extend it to write the surface representation as: 57  4.5 Hybrid Modeling Hybrid modeling combines digital shape sampling and processing with the power of computer-aided design to provide an innovative solution for parametric reverse engineering of complex parts. Digital shape sampling and processing (DSSP) converts 3D scan data into digital models for design, visualization, analysis, and manufacturing. Combining the advantages of DSSP with feature based modeling provides a hybrid method of modeling. Traditional CAD works well when modeling from scratch, limitations arise when it’s used to reconstruct complex surfaces → Hybrid modeling fixes these limitations. 58  4.5 Hybrid Modeling Reconstructing complex objects in CAD takes a lot of time and there is no guarantee that the finished model will be accurate. Some cases, it is almost impossible to use a feature-based approach to reconstruct surfaces → so difficult to identify and quantify parameters that control the object’s shape. The hybrid modeling method provides full parametric control over the shape. 59  Hybrid Modeling Applications These hybrid models accurately capture and reconstruct the shape of a physical part, and are ideal for applications to: – Capture physical designs and prototypes – Reproduce legacy parts and tooling – Replicate complex and organic shapes – Prepare as-built models for CAE applications – Enable mass-customization of unique components (dental applications, hearing aids, etc.) – Preserve historical and cultural artifacts. 60 Example of Modeling an Impeller using Hybrid Modeling 1. Scanning the Impeller 2. Cleanup and Repair 3. Extract Curves and Datum's 4. Combining Free-Form Surfacing 5. Trimming and Blending 6. Generating final Result 61 Example of Modeling an Impeller 62  Hybrid Modeling Advantages Uses 3D investments Hybrid modeling takes advantage of existing modeling systems and the skills of CAD users and experts, and enhance it with 3D measurement and rapid surfacing. Helps quickly create new designs Hybrid measurement and modeling approach greatly reduces the amount of time needed to copy an existing design. In some cases, hybrid modeling takes only hours as opposed to days or weeks using conventional techniques. 63  Hybrid Modeling Advantages Produces native parametric CAD geometry Using the CAD system, various aspects of the design can be parametrically driven by numerical values or constraints such as assembly-mating conditions. Generates accurate results Free-form surfaces can be generated from point clouds containing millions of sample points. This technique reproduces fine point in surface structures that would otherwise be lost. The process is repeatable because scanned data is less operator dependent than typical hand measurements. 64