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Under Graduate Course on

Computer Aided Design & Manufacturing

Geometric ModelingShibendu Shekhar RoyRobotics & A.I Lab. (RAIL)

Department of Mechanical Engineering

( ME 502 )

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Geometric ModelingConcept

Design & Analysis Theories/ Tools

Computer GraphicsAlgorithmCAD

…To Automate the Design Process

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Geometric ModelingGeometric modeling plays a crucial role in the overall application of CAD-CAM-CAE system

Information entered through geometric modeling is utilized in number of downstream applications like:

Design Analysis

Drafting

Evaluation of centroid, area (cross-sectional & surface) & volume

Estimate the mass & inertia properties

Interference checking in assemblies

Analysis of tolerance build-up in assemblies

Kinematic/ Dynamic analysis & simulation

Finite element analysis for stress, vibration, thermal & optimization

Automatic 2D view generation

Automatic planar cross-sectioning

Automatic dimensioning

Requirements of Geometric Modeling

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Geometric Modeling

Production & Industrial Engineering

Manufacturing Part classification & Process planning

NC tool path generation & verification

Manufacturing process simulation

Robot program generation

Preparation of Bill of Material (BOM)

Material requirement planning

Manufacturing resource requirement

Scheduling

Marketing

Computer-Aided Inspection & Quality control etc…..

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Geometric ModelingHence, It is important that the geometric model generated should be as clear & comprehensive as possible so that the other modules of CAD-CAM-CAE system are able to use this information in the most optimal way.

Geometric model

Three Dimensional (3-D)

Two Dimensional (2-D)

Solid Modeling

Surface Modeling

Wireframe Modeling

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Geometric Model

Three Dimensional (3-D)

Two Dimensional (2-D)

Utility of 2-D model lies in many of the low end drafting packages which is required for preparing manufacturing drawings

Their utility is limited because of their inherent difficulty in representing complex objects.

The 3-D geometric modeling has the ability to provide all the information required for CAD-CAM-CAE applications

A 3-D geometric model should be an unambiguous representation of an object.

A 3-D model should be complete to all engineering function from documentation (drafting & shading) to engineering analysis to manufacturing.

Basic Requirement

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In this method the complete object is represented by number of lines, points, arcs & curves and their connectivity relationships

Wireframe Modeling

Disadvantages

The construction of a wireframe model is simple It does not require much computer time & memory. It can be used for simple NC tool path generation

Advantages

It can not be used for calculation of mass, inertia propertiesThe interpretation of wireframe models having many edges is very difficult

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The surface model is constructed essentially from surfaces such a s planes, rotated curved surfaces & even very complex synthetic surfaces.

Surface creation on existing CAD system usually requires wireframe entities as a start

Surface & wireframe form the core of all existing CAD system

Surface Modeling

Disadvantages

Surface model of an object is a relatively more complete & less ambiguous representation than its wireframe model This method is very much useful for specific non-analytical surfaces ( free-form surface/ sculptured surfaces) such as those used for modeling automobile & airplane bodies & turbine blades etc. From an application point of view, surface models can be utilized in Finite Element Modeling, NC tool path generation, sectioning & interference detections.

Advantages

The calculation of mass & inertia properties would be difficult

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Solid model of an object is a more complete representation than surface model, as all the information required for engineering analysis & manufacturing can be obtained with this technique.

Solid modeling produces accurate design,provides complete 3D definitionImproves the quality of designImproves visualizationhas potential for functional simulation of the system

Advantages

Solid Modeling

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Wireframe Modeling

Wireframe entities

Synthetic Curves

Analytic Entities / Curves

Point

Lines

Arcs

Ellipse

Circles

Conics Parabolas

Hyperbolas

NURBS (Non-Uniform Rational B-spline)

Bezier curves

Cubic spline

B-spline

- are described by a set of data points (i.e. control points)

- are described by analytic equationsSynthetic CurveAnalytic Curve

<Geometric Primitives>

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Synthetic Curve

Analytic Curve

are defined as those that can be described by analytic equations such as lines, circle, conics etc.

provide very compact forms to represent shapes & simplify the computation of related properties such as areas & volume.

Analytic curves are usually not sufficient to meet today’s geometric design requirements of complex mechanical parts like automobile bodies, aeroplane wings, propeller blades, bottles etc.

That require synthetic curves & surfaces (free-form surfaces)

are defined as those that can be described by a set of data points (i.e. control points) such as Splines, Bezier curve etc.

Synthetic curves provide designers with greater flexibility & control of a curve shape by changing the positions of the one or more data points or control points.

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Synthetic Curve

The need for synthetic curves in design arises on two occasions:

i) when a curve is represented by a collection of measured data points (in case of Reverse Engineering) [graphical visualization of experimental data]

ii) when an existing curve must change to meet new design requirements.

Synthetic Curve Construction Techniques :

Interpolation technique Curve passes through the data points

Curve do not passes through the data pointsApproximation technique

Mathematically, synthetic curves represent a Curve-fitting problem to construct a smooth curve

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- Produce curves that do not pass through the given data points. Instead, these points are used to control the shape of the resulting curves

- Curve resulting from this technique pass through the given data points; curve itself is called Interpolant

-Ex. : Bezier Curve- Ex. : Cubic Spline

Approximation TechniqueInterpolation Technique

Most often, approximation techniques are preferred over interpolation techniques in free-form curve design due to the added flexibility & the additional intuitive feel provided by approximate technique.

P0

P1 P2

P3

P0

P1

P2

P3

P0

P12P5

P4

P3

P2

P1 P8

P7

P6

P11

P10

P9

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Mathematical Representation of Curves

Curve can be described mathematically by

Parametric equation

Non-parametric equation

Explicit form

Implicit form

Non-parametric equation : Explicit form

If the co-ordinates ‘y’ & ‘z’ of a point on the curve are expressed as two separate functions of the third co-ordinate ‘x’ <independent variable>. This curve representation is known as Non-parametric Explicit form.

Non-parametric Explicit form of a general 3-D curve

P (x,y,z)

P

Position vector of a point ‘P’ on the curve = P = [x y z]T= [x f(x) g(x) ]T

X

Y

Z

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Non-parametric equation : Implicit form

If the co-ordinates ‘x’, ‘y’ & ‘z’ are related together by two functions, a non-parametric implicit form results

Non-parametric Implicit form of a general 3-D curvef1 (x,y,z) = 0

f2 (x,y,z) = 0

Parametric representation of curves

P (x,y,z)

P(u)

X

Y

Z

In parametric form, each point on a curve is expressed as a function of a parameter “u”. This parameter acts as a local co-ordinate for points on the curve.

P’(u)umax

umin

Parametric form of a general 3-D curve

Position vector of a point ‘P’ on the curve = P(u) = [x(u) y(u) z(u) ]T

The parametric curve is bounded by two parametric variable values umin and umax.

Where, umin≤ u ≤ umax

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Advantages :

Parametric Representation of Curves

In case of commonly used curves (such as circle, conics), these equations are polynomials rather than equations involving roots.Hence, the parametric form is not only more general but it is also well suited to computation in geometric modeling.

Parametric geometry can be easily expressed in terms of vectors & matrices which enables the use of simple computation techniques to solve complex analytic geometry problem.

To check whether a given point lies on the curve or not, reduces to finding the corresponding ‘u’ values & checking whether that value lies in the stated ‘u’ range.

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Blending is used to construct composite curve. Blending of two curves implies the joining of two curves subjected to the satisfaction of continuity condition.

Various Continuity requirements can be specified at data points to impose various degrees of smoothness of the resulting curve.

The order of continuity becomes important when a complex curve is modeled by several curve segments pieced together end-to-end.

Order of Continuity

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Order of Continuity

Zero-order Continuity

First-order Continuity

Second-order Continuity

(C0 - continuity) (C1 - continuity) (C2 - continuity)

Position Continuous curve

Slope Continuous curve

Curvature Continuous curve

C1 – Continuous is “Smoother” than C0 – Continuous at the joining pointC2 – Continuous is “Smoother” than C1 – Continuous at the joining point

Ex.: Cubic Spline Ex.: B-Spline

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Cubic Splines

Synthetic Curves

Splines are used to interpolate to given data i.e. based on Interpolation technique.

A spline is a piecewise parametric representation of the geometry of a curve with a specified order of continuity.

Cubic splines use a parametric equations of 3rd degree with the first order continuity maintained at the intersection point of the curve.

Name from the traditional drafting tool called “Splines” or “French Curves”

P0

P0.8P1

P.0.4

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Cubic Splines

Vector form3

ii

i 0P(u) u

=

= ∑C

2 30 1 2 3P(u) u u u= + + +C C C C

TP(u) = U C

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3

1uuu

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

U

Matrix form 0

1

2

3

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

CC

CCC

The parametric equation of a cubic spline segment is given by

Cubic splines use cubic polynomial.

0 u 1≤ ≤

Where, C= Coefficients Vector

u= parameterCi= Polynomial Coefficients

Cubic polynomial has four coefficients & thus requires four conditions to evaluate .

- Uses two data points at its ends & two tangent vectors at these points

- Uses four data points

Hermite Cubic SplineCubic Spline

P0

P0.8P1

P.0.4

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Bezier CurvesBased on approximation techniques.Developed by P. Bezier, Designer of French car firm Regie Renault (1962).

Used in his Software system (UNISURF) to define the outer panels of several Renault cars .

Bezier curve uses the vertices of Bezier Characteristic polygon (or control polygon) as control points for approximating the generated curve. The curve will pass through the first & last point with all other points acting as control points. The curve is also always tangent to the first & last polygon segment.

Control Polygon

Control Points

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Bezier CurvesThe degree of Bezier curve is related to the number of data points; (n+1) points define an nth degree Bezier curve.

If no. of data points is 4 [n+1=4], Then n=3 i.e degree of curve =3 (cubic)

Control Polygon

Control Points

This is widely used for the design of aesthetic surfaces. The flexibility of the curve becomes more with more control points.

Cubic Bezier Curve

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Bezier Curves

Control Polygon

Control Points

Mathematically, for (n+1) control points, the Bezier curve is defined by polynomial of degree n:

P(u) is a point on the curve, Pi is a control point

Bi,n(u) are the Berntein polynomials

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C(n,i) is the binomial coefficient

Bezier Curves

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Bezier Curves

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Major difference between the Bezier Curve & Cubic Spline Curve

i. The degree of Bezier curve is variable & is related to the number of data points; (n+1) points define an nth degree curve which permits higher-order continuity. This is not the case for cubic splines where the degree is always 3 <cubic> for a spline segment.

ii. The shape of Bezier curve is controlled by its data points only. First derivatives are not used in the curve development as in the case of Hermite cubic spline.

iii. The Bezier curve is smoother than the cubic spline because it has higher order derivative.

Bezier curve is considered as a single curve controlled by all the control points. As a result, with an increase in the number of control points, the degree of the polynomial representing the curve increases & this provides for global modification effect rather than local.

To reduce this complexity, the curve is broken down into more segments with better control excised with individual segments, while maintaining a simple continuity between the segments.

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B-Spline CurvesSingle piecewise parametric polynomial curve through any number of control points with the degree of polynomial selected by Designer.

It provides the ability to add control points without increasing the degree of the curve.

B-spline exhibit a local control of the curve shape i.e. whenever a single vertex is moved, only those vertices around that will be affected while rest remains the same.

B-Spline CurvesApproximate a given set of data points

Interpolate a given set of data pointsUseful in displaying design & engineering analysis results such as Stress/displacement distribution

Useful for modeling free-form surfaces

In contrast to Bezier curve, the theory of B-spline curve separates the degree of the resulting curve from the no. of the given control points.

Four control points can always produce a cubic Bezier curve but four control points can produce linear, quadratic or cubic B-spline curve.

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Mathematically, B-spline curve defined by (n+1) control points Piis given by

B-Spline Curves

Where ‘k’ controls the degree (k-1) of the resulting polynomial & also continuity of the curve.

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B-Spline Curves

Local control of B-spline curve

It provides the ability to add control points without increasing the degree of the curve.

B-spline exhibit a local control of the curve shape i.e. whenever a single vertex is moved, only those vertices around that will be affected while rest remains the same.

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are generalization of the curve & surface theories.

are almost exclusively used by modern CAD-CAM-CAE systems to provide a unified approach to formulate & represent curves & surfaces.

provide a convenient design tool to create smooth curves & surfaces interactively.

Rational Curve: is defined by algebraic ratio of two polynomials while a non-rational curve is defined by one polynomial.

NURBS Non-Uniform Rational B- Spline

n

i i i,ni 0

n

i i,ni 0

P w B (u)P(u) ;

w B (u)

=

=

=∑

∑Rational form of Bezier Curves

where wi is the weighting factor for each of the vertex.

0 ≤ u ≤ 1

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Disadvantages

NURBS are considered a unified representation that can be define both synthetic (like Bezier, B-spline etc.) & analytic (i.e. circle, conics etc.) curves & surfaces. Any curve or surface can be formulated using NURBS. It can represent all curves, surfaces, & solid entities, allowing unification & conversion from one CAD system to another via exchange standards (like IGES,, STEP etc.). Their related algorithm are stable & accurate. This unified representation also have the advantage of reducing the database complexity & the number of procedures required in CAD system for display & manipulation of geometric entities.

Advantages

Simple curves (like arcs, circles, conics) require more data to define as NURBS than traditional way.

NURBS

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Curve Manipulations

Blending

Segmentation

Displaying

Trimming

Evaluating points on curves

Intersection

Transformation

The effective use of analytic & synthetic curves in a design environment depends mainly on their manipulation.

Translation

Rotation

Scaling

Mirror/ Reflection

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Curve Manipulations

Blending

Segmentation

Displaying

Trimming

Evaluating points on curves

Intersection

Transformation

Translation

Rotation

Scaling

Mirror/ Reflection

provides the Designer with a means of visualizing geometric models.

is used to construct composite curves from various types of curve segments. Two curve segment should be continuous at the joint.

replacing one existing curve by one or more curve segments of the same curve type such that the shape of the composite curve is identical to that of the original curve.

is a very useful function which can truncate or extend a curve

the intersection point of two curve in space requires the solution of the parametric equations of curves.

Curve parametric eqn. is used to evaluate points on it.

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Surface Modeling

Surface entities

Synthetic Surface Entities

Analytic Surface Entities

Plane surface

Ruled surface/ Lofted surface

Surface of revolution

Tabulated cylindrical surface

Coons patch

B-Spline surface

Bi-cubic Hermite spline surface

Bezier surface or patch

Curve segment : is the fundamental building block for curve entities

Surface patch : is the fundamental building block for surfaces

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Plane Surface

Ruled Surface

is the simplest surface which requires three non-coincident points to define a plane.

The plane surface can be used to generate cross-sectional view by intersecting a surface model with it.

is a surface constructed by transitioning between two or more curves by using linear blending between each section of the surface

It interpolates linearly between two boundary curves that define the surface.

Lofted Surfaceis a surface constructed by transitioning between two or more curves by a smooth i.e. higher order blending between each section of the surface.

Used for modeling engine manifolds, turbine blades etc.

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Surface of Revolutionis an axi-symmetric surface that can model axi-symmetric objects.

is generated by a rotating a planar wireframe entity in space about the axis of symmetry a certain angle.

Tabulated Cylindrical Surfaceis a surface generated by translating a planar curve a certain distance along a specified direction (axis of cylinder).

Plane of the curve is perpendicular to the axis of the cylinder.

is used to generate surfaces that have identical curved cross-sections.

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B-spline Surface

Bi-cubic Hermite Spline Surface

Bezier Surface

It connects four corner data points & utilizes a bi-cubic equation.

is a synthetic surface that approximates given input data i.e. it doesnot pass through all given data points.

Allows only global control of the surface.

is a synthetic surface that can either approximate or interpolate given input data.

Permits local control of the surface.

Coons patches / surfaceCoons patch or surface is obtained by blending four boundary curves.

is used to create surfaces using curves that form closed boundaries.

the single patch can be extended in both the directions by adding further patches.

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Surface Manipulations

Blending

Segmentation

Displaying

Trimming

Evaluating points & curves on surfaces

Intersection

Transformation

Projection

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Solid Modeling

Solid model are known to be informationally more complete, valid & unambiguous representation of objects than its wireframe or surface model.

The completeness & unambiguity of a solid model are attributed to the fact that CAD database stores both its geometry & topology.

Solid models are complete & unambiguous but they are not unique because same object may be constructed in various ways.

Solid modeling has been acknowledged as the technological solution to automating & integrating design & manufacturing functions like…

Design / Engineering Analysis Drafting MRPCAPP, CNC tool path generation

Computer-Aided Inspection & Quality control

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Geometry

A solid model of an object consists of both the geometrical & topological data of the object.

- Geometry of an object defines the actual dimensions of its entities.

Topology

- Topology of an object defines the connectivity & associativity of the entities.- it determines the relational information between entities.

From user point of view,Geometry is visible.Topology is considered to be non-graphical relational information that is stored in solid model databases & are not visible to users.

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Geometry

Topology

The geometry that defines the object is the - the length of lines L1, L2, L3- the angle between the lines- radius ‘R’ & the centre ‘P’ of semi-circle

The topology of the object can be stated as- L1 shares a vertex (point) with L2 & C1- L2 shares a vertex with L1 & L3- L3 shares a vertex with L2 & C1- L1 & L3 donot overlap- ‘P’ lies outside the object

L1

C1L2

L3

RP

L1

C1

L2

L3

RP

L1

C1

L2

L3

RP

L1

C1

L2

L3

RP

Same Geometry but Different Topology Same Topology but Different Geometry

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Solid ModelingModern CAD systems offer two approaches to creating solid models :

Feature Approach

Primitive Approach

- allows designers to use pre-defined shapes (primitives) as building blocks to model solid object.

- Boolean operation are used to combine the primitives to create complex objects.

- is limited by the restricted shapes of primitives

- Features are more flexible as they allow the construction of more complex solid than what the primitive offer.

Feature is defined as a geometric shape (i.e. feature profile or 2-D sketch) and feature operation to build parts.

Feature Feature Profile (Sketch) Feature Operation+

The entities required for Feature profile (Sketch) are wireframe & surface entities

Feature Operation is an activity that converts the Sketch (2-D) into 3-D shape

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Constraints

Geometric Constraints Dimension Constraints

CoincidenceConcentricity

Fix

Parallelism

Tangency

Geometric constraints is defined as a geometric condition that relates two or more sketch entities.

Equidistant pointMid point

Horizontal

Perpendicular

Vertical

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Constraint-based Feature ModelingSelect a Sketch Plane

Sketching the feature profile (2D profile)

Create Base Feature

Apply Geometric & Dimensional Constraints

Apply feature operations

Combine feature to build part- is considered the best tool to create solids because of its ability to edit & change the

shape of the solid in the future by using its relations & constraints. It has flexibility to create complex shape.

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Feature Operations

Revolve

Sweep

Extrude

Loft

Hole

Pocket

Common Feature Operations & Features

Slot

Shell

Protruded Feature

Rib

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Feature Operations

Fillet

Chamfer

Pattern

Common Feature Operations & Features

Thread

Rectangular Pattern

Circular Pattern

Transformation

Translation

Rotation

Scaling

Mirror

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Extruded or Protruded feature

Common Features

Extrude command is used to create a solid object by extruding a given closed profile.

It requires a closed profile (sketch of cross-section of solid object) & an extrusion vector (length & direction).

The extrusion direction is always perpendicular to the sketch plane of the profile.

Revolved featureRevolve command is used to create an axi-symmetric objects by revolving the selected geometry about an axis.

It requires a profile (sketch of cross-section of solid object) & a revolution vector (axis & angle of revolution).

The axis of revolution is always in the sketch plane of the profile.

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Common Features

is a generalization of extrusion.

Creates a feature by moving a sketch (closed or open) along a linear or non-linear path, which is not necessarily perpendicular to the sketch plane.

Use to blend multiple cross-section (two or more) along the linear or non-linear guide curve to create a solid.

Sweep feature

Loft feature

Hole featureis equivalent to subtracting a cylinder from a solid.

Simple and Taper hole

Counter-bored hole

Counter-sunk hole

Threaded hole

Type of Hole feature

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Common Features

Removes material from solid.

Shell operation is used to create hollow or thin-walled solids by removing material out.

The input to a shell operation is the faces to be removed & a wall thickness.

Rectangular slot

T- slot

Dovetail slot

Type of Slot feature

Slot feature

Shell feature

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Common Features

Rib featureis a special type of extruded feature.

Creates an extension of geometric element by creating a feature connecting the two faces of the part.

The input to create a rib is a contour & a thickness

Pattern feature

Rectangular Pattern

Circular Pattern

Used to create multiple copy of given feature in rectangular (linear) array.

Used to create multiple copy of given feature in circular array.

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Common Features

are used to smoothen (round) the sharp edges of solid.

input to create a constant radius fillet feature is to select the edges to be filleted & fillet radius.

Fillet

Chamferuse to remove sharp edges (or corners) from parts

input to create a chamfer is angle-distance, distance-distance.

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Solid Representation Schemes

Constructive Solid Geometry (CSG)

Boundary representation (B-rep)

Sweeping

Half spaces

Analytic Solid Modeling (ASM)

Cell decomposition

Spatial enumeration

Octree encoding

Primitive Instancing

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Boundary representation (B-rep)B-rep is based on the topological notion that an object is boundedby a set of faces.

A face is a closed, orientable & bounded (by edges) surface. Each face is bounded by edges & each edge (bounded curve) is bounded by vertices.

B-rep model of an object consists of faces, edges, vertices.

Vertices (v)

Edge (E)

Face (F)

- The database of a boundary model contains both its topology & its geometry. Topology & Geometry are interrelated & cannot be separated entirely from each other

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Disadvantages

B-rep is very appropriate to construct solid models of unusual or complex shapes ( like turbine blades, auto body etc.) that are difficult to build using primitives. It is relatively simple to convert a B-rep model into a wireframemodel because the model’s boundary definition is similar to the wireframe definition.

Advantages

It requires large amounts of storage because it stores the explicit definition of model boundaries i.e. faces, edges & vertices which tend to grow fairly fast for complex model.

B - rep

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Constructive Solid Geometry (CSG)CSG model is based on the topological notion that a physical object can be divided into set of primitives (basic elements & shapes) that can be combined in a certain order following a set of rules (Boolean operation) to form the object.Primitives are considered as building block

Primitives are simple, basic shapes which can be combined by a mathematical set of Boolean operations to create the complex solid object.

Cylinder ConeBlock WedgeSphere

Primitives

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Intersection (∩)

Union (U)

Difference (-)

Boolean Operations

Used to combine or add two primitives/

objects

Intersecting two primitives give a

shape equal to their common value

Used to subtract one primitive from other & results in shape equal

to the difference in their volume

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Disadvantages

It is easy to construct out of primitives & Boolean operation. It is concise & requires minimum storage.

Advantages

Major disadvantage of CSG is in its inability to represent sculptured surfaces i.e. non-analytical (synthetic surface) cannot be modeled using CSG scheme.

CSG

- Modern CAD systems provide both approaches to increase their modeling domain

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Solid Model-based Down-stream Application

Solid Model-based Drafting

Solid Model-based Mechanism Design

Solid Model-based Assembly Design

Finite Element Modeling from Solid Models

CNC machining based on Solid Modeling …