Engineering Curves and CAGD

7/22/2019 Engineering Curves and CAGD

1/62

Engineering curves and CAGD

Rubn Dorado Vicente

Department of mechanical & mining engineering

University of Jan

III Engineering Week, Schmalkalden 2013


2/62

Contents

Computer Aided Engineering

Computer Aided Geometric Design CAGD

Mathematical model of a curve

Useful engineering curves

Bzier, B-spline and NURBS representation

Fitting: approximation and interpolation


3/62

CAGD Curve

s

Eng. curves CAD representation Fittin

g

Conclusions

Idea

Concept

& Design

Analysis

Simulation

Prototype

Tests Manufacturing

Product

Computer aided engineering

III Engineering week, Schmalkalden 2013

CAE= CAxx

,

x = process aided by computer

DesignCADbased on CAGD: computational representation of shapes

Analysis + virtual prototypingCAA

ManufacturingCAM


4/62

CAGD Curve

s


g

Conclusions

CAE advantages

Design errors lead to high final costs

CAE reduces the need for prototyping: cost & time to take market

Delayslost sales and lock-in



5/62

CAGD Curve

s


g

Conclusions

Examples of first mover advantages

Operating systems: Microsoft

1981, DOS: Unstable, unsafe, "backward compatibility

CAD software: AutoCAD

First CAD software

QWERTY keyboard

Designed to avoid a mechanical problem

Alternative combustion engine

Contamination, mechanically

complex


Source: Itedo


6/62

CAGD Curve

s


g

Conclusions

CAE: disadvantages

There are many CAD format Lost time & errors

Solution for geometry: Neutral formats like IGES, STEP

Data maintenance: Hardware + software, High costs



7/62

CAGD Curve

s


g

Conclusions

Finite element analysis


Source: engineering.com

Numerical solution

of mechanical problems

Application in structural,

thermal and dynamic studies

Idea: divide and conquer

Discretization in elements Analytical CAD models.


8/62

CAGD Curve

s


g

Conclusions

Finite Element Analysis software


Software Company Analysis

ANSYS INC Fluids, structural, thermal, dynamic.

Dassault Systemes Fluids, structural, thermal, dynamic.

Siemens Fluids, structural, thermal, dynamic.

Open source Fluids, structural, thermal

Open source Dynamic


9/62

CAGD Curve

s


g

Conclusions

Computer aided manufacturing


Help to generate ISO code from CAD models

Manufacturing automatization

Control and inspection

CAD software admits CAM plugins

Example DELCAM within SolidWorks

Tool-path generation based on CAGD

An adequate tool-path can reduce time and forces.

Source: delcam.com


10/62

CAGD Curve

s


g

Conclusions

CAM software


Software Company Analysis

CATIA Mold &

Tooling Design

Dassault Systemes Machining, mold design, plastic

injection

NX CAM 9 Siemens Machining, inspection

DELCAM Delcam Machining and inspection

Blendef Open source Milling


11/62

CAE Curve

s


g

Conclusions

Computer Aided Geometric Design

It is a discipline dealing with computational aspects ofgeometric objects G. Farin (2002)

How model engineering curves and surfaces in a computer

It became a scientific & engineering discipline in 1974

Try to reduce errors and time by the use of a pc

Main CAGD advances:

Theory of Bzier curves and B-spline techniques

Other CAGD applications:

Solid modelling, Geographic information systems, Medicalimaging, Computer gaming, Scientific visualization



12/62

CAE Curve

s


g

Conclusions

Traditional vs. computer design

CAD reduces time and cost

it allows to represent real 3D models

No material and size limitations as paper planes

Lot of different computer graphic solutions

2D-Design

Technical illustrations

Metal sheetSolid modelling



13/62

CAE Curve

s


g

Conclusions

Design software 2D


Software Company web

AutoCAD AutoDesk AutoDesk.com

Corel Designer Corel Corp. Corel.com

PTC Creo Schematic PTC Corp. Ptc.com

MicroStation

Powerdraft

Bentley Bentley.com

LibreCAD Open source Librecad.org


14/62

CAE Curve

s


g

Conclusions

Technical illustrations 2D & 3D

Software Company web

Adobe Illustrator Adobe adobe.com

Corel Draw Corel Corp. Corel.com

PTC Creo Illustrate PTC Corp. Ptc.com

InkScape Open source inkscape.org


Source: Ptc corp.


15/62

CAE Curve

s


g

Conclusions

Surface & Solid Modelling

Software Application Company web

RhinoCeros NURBS curves & surfaces Robert McNeel& Associates

rhino3D.com

SolidWorks, Catia Surfaces & Solid Modelling Dassault Systemes 3ds.com

Creo Elements Surfaces & Solid Modelling PTC Corp. Ptc.com

FreeCAD Solid Modelling Open source freecadweb.org


Source: linuxaideddesign.


16/62

CAE CAGD Eng. curves CAD representation Fittin

g

Conclusions

Curve models

Curve: A continuous map from one dimension to n-dimensionspace.

They have useful applications:

CAD Modelling uses:Define contours of orthographic projections

Define wireframe models in CAD: rotations and translations ofa curve profile generate revolution and swept surfaces.

Engineering curves: conics, Cycloid, Spirals, Helix & so on

Scientific visualization via approximation or interpolation


Source: wikimedia


17/62


g

Conclusions

Single valued curves

Definition: Graphic of a function

a vertical line cuts the curve once

Examples:



18/62


g

Conclusions

Single valued curves: Problems

This representation is not valid after a rotation

Simple and useful curves are not single valued



19/62


g

Conclusions

Parametric curves

Definition:



20/62


g

Conclusions

Parametric curves

Advantages:

Solve the aforementioned drawbacks of single valued curves

Single valued (case x=t)parametric curve

Simple modelling and visualization:

1. give values (t)points c(t)

2. Join c(t) (linear segments)

Computer graphic cards draw 100 Msegments / s

Disadvantages: It is complicated to define areas

Therefore it is difficult to distinguish inside and outside



21/62


g

Conclusions

Polynomial parametric curves


c(t)= {x(t),y(t)}, x(t), y(t)are degreenpolynomials.


22/62


g

Conclusions

Rational parametric curves


c(t)= {x(t),y(t)}, x(t), y(t)are degreenrational functions.

c(t)= p(t)

w(t),q(t)

w(t)

, p(t), q(t), w(t)are degreenpolynomials.

Advantages of Polynomial-Rational Parametric Curves

Extremely fast point evaluation:

Additions, products (and division for rational)

Operations implemented within microprocessor

GFLOPS (G Floating Point Operations per Second)

Parametric Rational Polynomial


23/62


g

Conclusions

Rational para. curves: Examples



24/62


g

Conclusions

Implicit curves

Those plane curves / f(x,y)=0 Algebraic curvesf is a degree n polynomial

Advantages: inside-outside classification. Disadvantages:

It is difficult to model free form curves.

It is also complex to extend this representation to 3D.



25/62


g

Conclusions

Definition: non algebraic curves

Main drawback: they do not admit an exact computerrepresentation

Polynomial-rational parametric curvesstandard CAD

representation

Solution:polynomial-rational fitting

Detection:intersection between a line and a transcendental

curveinfinite points

Examples:Logarithmic and Archimedes spiral, helix, catenary

Transcendental curves



26/62


g

Conclusions

Complex shapes Piecewise parametric curves

Spline curves



27/62


g

Conclusions

Spline continuity type.Mechanical applications: G0or G1tool paths (conventionalmachines), G2vehicles and High speed machining

Geometric continuity Gr



28/62


g

Conclusions

Evaluation via Curvature k(s), s=Arc length parameterization

Quality assessment


Smooth curve if k(s):

is a continuous function

has few and monotonic segments

ends at specified points

i i i l i


29/62

CAE CAGD Curve

s

CAD representation Fittin

g

Conclusions

Engineering curves

We have two main types of Engineering curves:

Curves with CAD representation:

Polynomial and rationalApplicationsDesign

Curves without CAD representation:

Transcendental curves (examples in the following slides)

CAD incorporation via interpolation or approximation

Lot of engineering applications


CAE CAGD C CAD i Fi i C l i


30/62

CAE CAGD Curve

s


g

Conclusions

Engineering curves: Trochoids

Curve described by a fixed point as a circle rolls

outside a circle (Epitrochoid)

inside a circle (Hypotrochoid)

They admit a rational representation


CAE CAGD C CAD t ti Fitti C l i


31/62

CAE CAGD Curve

s


g

Conclusions

Trochoids applications

Design of roots blower rotors.

They handles large quantities of air at a small pressure diff.

Wankel engine: few vibration & mechanical stress at high rpm

Design of gear tooth profiles


Rotary blower

Mazdas Wankel engine



32/62

CAE CAGD Curve

s


g

Conclusions

Catenary




33/62

CAE CAGD Curve

s


g

Conclusions

Clothoid spiral


Curvature is proportional to distance.

Parameterization: Fresnel integrals (Transcendental curve)

Application: road design, roller coasters

Speed = Constant

c(t)=C(t)

S(t)

,C(t)= cos(

2 2 )dz

0

t

S(t)= sin( 2 2 )dz0t

.

CAE CAGD Curve CAD representation Fittin Conclusions


34/62

CAE CAGD Curve

s


g

Conclusions

Involute or evolvent


Definition:Path follow by the end of a string which is woundtangent to a profile.

It is a transcendental curve.

Application:design of gear teeth



35/62

CAE CAGD Curve

s


g

Conclusions

Offset


Definition:parallel curve cd(t)

In general, it is a non polynomial/rational curve

cd(t)= c(t) dn(t),

n(t)normal

+ up, - down

.



36/62

CAE CAGD Curve

s


g

Conclusions

Offset: Applications


Tool path and road design

Cam design



37/62

CAE CAGD Curve

s


g

Conclusions

Offset: representation problems


Self-intersections

CAE CAGD Curve Eng curves Fittin Conclusions


38/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Standard CAD representation

We can incorporate polynomial curves intoCAD but there are different ways to describe a

polynomial.

Our first choice is the monomial basis, but is it

the best representation for CAD?

Next slides try to answer this question




39/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Monomial basis

Definition:A way to describe a polynomial using a linearcombination of monomials

Main advantages:

Simple algebraic manipulation: additions, products,

derivation

OK to approximate functions around a point (Taylor, t = 0)

Efficient computing algorithm (Horner)

Everybody learned this representation in high school


Monomialbasis: 1,t,t2 ,...,tn{ },

Polynomialrepresentation: a(t)= aiti

i=0

n

.



40/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Monomial basis

Disadvantages:

Coefficients have not geometric meaning

Poor numerical properties: small errors in aigaps

Gaps and non geometric meaning are unacceptable for CAD




41/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Bzier curves

Provide a polynomial representation: where coefficients have geometric meaningControl

points

without numerical problemsno gaps

End control points = curve end points.




42/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Examples of Bzier curves




43/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Bernstein polynomials

Definition:

Bzier: combination of control pts via Bernstein polynomials

biControl points

effect of biin the curve shape.

bipulls the curve(u=i/nMaximum )


Bi

n(u) = ni

1 u( )n i ui .

b(u) = biBi

n

(u)i=0

n

.

Bi

n(u)

Bi

n(u)

CAE CAGD Curve Eng. curves Fittin Conclusions


44/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Bzier curves: geometric properties

Affine invariance. Reposition-scale-rotation so ontransformation of control points


Other Application

Typography:

S S S



45/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

End point interpolation. The curve pass through the end controlpointsno gaps

Linear precision. If bi are alignedline

Convex hull property. B(t) lies in the convex hull of the controlpolygon. Application: Interference checking.

Operationslike integration and derivation via additions,products and divisions of control points

Good numerical properties

Bzier curves: geometric properties




46/62

CAE CAGD Curve

s

Eng. curves Fittin

g

Conclusions

Bzier curves: smoothness conditions




47/62

s

g

g

Bzier curve: Derivation




48/62

s

g

g

Bzier curves: Integration




49/62

s

g

g

Design Example: cam profile


Hypothesis. Constant angular speed

Conditions:

Constraint forcesy() is C1

Avoid jerky()is C2

= cte



50/62

s

g

g

Rational Bzier curves


Several conic curves (ellipse, circle, hyperbola) can not be

represented by means of a polynomial Bzier curve.

Solution:Rational Bzier curvea perspective transformation



51/62

s g

Rational Bzier curves


Effect of control points reposition Effect of weight modification



52/62

s g

Conic construction via Rational Bzier curves


C on ic a rc

C en tr e ( ci rc le ), f oc us : C

Angle 2Weights: w0=w2=1,w1

w1

[0,1] E ll ip se (w2= cos( ) , C i r cl e )

1 Parabola

[1,] Hyperbola



53/62

s g

Conics applications


Source: Alternative-energy-tutorials.com

Source: radartutorial.eu



54/62

s g

B-spline curves


Definition: Spline of Bzier curvessame Bzier properties

Applicationcurves with complex shape

d(u) = diN

i

n(u)

i=0

n

. Nin

(u)piecewisebasis



55/62

s g

B-spline versus Bzier


Bzier B-spline

DesignPseudo local shape

controlLocal control

Continuity

guaranteeComplex Simple (polynomial degree - 1)

Evaluation costN control points increase

computational cost

It doesnt dependon number

of control points

Implementation Straightforward Complex



56/62

s g

Example: Degree n=3 B-spline


Continuityn-1=2

Number of control pointsp=7

9 knots, Number of pieces mm=p-n=4



57/62

s g

NURBS: Non uniform rational B-spline


Definition: Spline of Rational Bzier curves with a non uniform

knot sequence.

CAE CAGD Curve Eng. curves CAD representation Conclusions


58/62

s

Fitting

Definition:Construction of a NURBS that approximates a non

polynomial-rational curve Main fitting techniques:

Approximation. NURBS passes close to a set of points

Interpolation. NURBS satisfies a set of conditions




59/62

s

Lagrange Interpolation

Lagrange interpolation: the interpolation curve passes

through an uniformly distributed set of points Drawbacks:

Runges phenomenon

New datare-compute


Original curve

Degree 4 Lagrange Interpolation

Degree 8 Lagrange Interpolation

Source: wikipedia



60/62

s

Hermite Interpolation

Definition: the interpolation curve passes through an non uniformly

distributed set of pointsdata: end points and their derivate Applications:

Approximation of boundary problems

Useful to control continuity




61/62

s

Approximation

The construction passes close to a set of data.

Complex implementation: iterative algorithm.

Basic approximation technique: Least squared method.


CAE CAGD Curve Eng. curves CAD representation Fittin


62/62

s g

Conclusions

CAGD decrease design errors & contribute toreduce the time to take the market.

CAGD is devoted to Engineering design via curvesand surfaces.

There are different types of curve representations Bzier B-Spline NURBS is the CAD standard

Advantages: local shape control, fast evaluation andstraightforward implementation of differentoperations

To incorporate transcendental curvesInterpolation or approximation (fitting)

Documents

Engineering Curves and CAGD