Engineering Curves and CAGD

III Engineering Week, Schmalkalden 2013

Engineering curves and CAGD

Rubén Dorado Vicente Department of mechanical & mining engineering University Univer sity of Jaén

Contents •

Computer Aided Engineering

•

Computer Aided Geometric Design CAGD

•

Mathematical model of a curve

•

Useful engineering curves

•

Bézier, B-spline and NURBS representation

•

Fitting: approximation and interpolati i nterpolation on

Contents •

Computer Aided Engineering

•

Computer Aided Geometric Design CAGD

•

Mathematical model of a curve

•

Useful engineering curves

•

Bézier, B-spline and NURBS representation

•

Fitting: approximation and interpolati i nterpolation on

CAGD

Curve s

Eng. curves CAD representation

Fittin g

Conclusions

Computer aided engineering Idea

Product

Concept

Analysis

Prototype

& Design

Simulation

Tests

Manufacturing

CAE =

CAx, x

x = process aided by computer Design  CAD based on CAGD: computational representa representation tion of shapes Analysis + virtual prototyping  CAA ManufacturingCAM III Engineering week, Schmalkalden 2013

CAGD

Curve s


CAE advantages

•

Design errors lead to high final costs

•

CAE reduces the need for prototyping:

Fittin g

Conclusions

cost & time to take market

Delays  lost sales and lock-in

III Engineering week, Schmalkalden 2013

CAGD

Curve s


Fittin 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


CAGD

Curve s


CAE: disadvantages

•

There are many CAD format

Fittin g

Conclusions

Lost time & errors

Solution for geometry: Neutral formats like IGES, STEP •

Data maintenance: Hardware + software, High costs III Engineering week, Schmalkalden 2013

CAGD

Curve s


Finite element analysis •

Fittin g

Conclusions

Numerical solution

of mechanical problems Application in structural, thermal and dynamic studies •

Idea: “divide and conquer” Discretization in “elements ”≠

Source: engineering.com

Analytical CAD models.


CAGD

Curve s


Fittin g

Conclusions

Finite Element Analysis software Software

Company

Analysis

ANSYS INC

Fluids, structural, thermal, dynamic.

Dassault Systemes


Siemens


Open source

Fluids, structural, thermal

Open source

Dynamic


CAGD

Curve s


Fittin 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 Source: delcam.com

•

Tool-path generation based on CAGD An adequate tool-path can reduce time and forces.


CAGD

Curve s


Fittin g

CAM software Software

Conclusions

Company

Analysis

Dassault Systemes

Machining, mold design, plastic injection

NX CAM 9

Siemens

Machining, inspection

DELCAM

Delcam

Machining and inspection

Blendef

Open source

Milling

CATIA Mold & Tooling Design


CAE

Curve s


Fittin g

Conclusions

Computer Aided Geometric Design •

“It is a discipline dealing with computational aspects of geometric 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 Bézier curves and B-spline techniques Other CAGD applications: Solid modelling, Geographic information systems, Medical imaging, Computer gaming, Scientific visualization III Engineering week, Schmalkalden 2013

CAE

Curve s


Fittin 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 sheet Solid modelling


CAE

Curve s


Design software 2D Software AutoCAD

Corel Designer

PTC Creo Schematic

MicroStation Powerdraft

LibreCAD

Fittin g

Conclusions

Company

web

AutoDesk

AutoDesk.com

Corel Corp.

Corel.com

PTC Corp.

Ptc.com

Bentley

Bentley.com

Open source

Librecad.org


CAE

Curve s


Fittin g

Conclusions

Technical illustrations 2D & 3D

Source: Ptc corp.

Software Adobe Illustrator Corel Draw PTC Creo Illustrate InkScape

Company

web

Adobe

adobe.com

Corel Corp.

Corel.com

PTC Corp.

Ptc.com

Open source

inkscape.org


CAE

Curve s


Fittin g

Conclusions

Surface & Solid Modelling

Source: linuxaideddesign.

Software RhinoCeros

Application

Company

web

NURBS curves & surfaces

Robert McNeel & Associates

rhino3D.com

Dassault Systemes

3ds.com

PTC Corp.

Ptc.com

Open source

freecadweb.org

SolidWorks, Catia Surfaces & Solid Modelling Creo Elements Surfaces & Solid Modelling FreeCAD

Solid Modelling


CAE CAGD


Curve models

Fittin g

Conclusions

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

They have useful applications: CAD Modelling uses: Define contours of orthographic projections Define wireframe models in CAD: rotations and translations of a curve profile generate revolution and swept surfaces. •

•

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

•

Scientific visualization via approximation or interpolation III Engineering week, Schmalkalden 2013

CAE CAGD


Single valued curves •

Definition: Graphic of a function a vertical line cuts the curve once

•

Examples:


Fittin g

Conclusions

CAE CAGD


Fittin g

Conclusions

Single valued curves: Problems •

This representation is not valid after a rotation

•

Simple and useful curves are not single valued


CAE CAGD


Parametric curves Definition:


Fittin g

Conclusions

CAE CAGD


Parametric curves

Fittin g

Conclusions

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 III Engineering week, Schmalkalden 2013

CAE CAGD


Fittin g

Conclusions

Polynomial parametric curves c(t ) = { x (t ), y (t )},

x (t ), y (t ) are degree n polynomials.


CAE CAGD


Fittin g

Conclusions

Rational parametric curves c(t ) = { x (t ), ), y (t )}, )}, x (t ), y (t ) are degree n rational functions. p(t ) q(t ) , , p(t ), q(t ), w (t ) are degree n polynomials. c(t ) = w (t ) w (t )

Parametric

Rational

Polynomial

Advantages Advant ages 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) III Engineering week, Schmalkalden 2013

CAE CAGD


Fittin g

Conclusions

Rational para. curves: Examples


CAE CAGD


Implicit curves •

•

•

•

Fittin g

Conclusions

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

Advantages: inside-outside classifica classification. tion. Disadvantages: It is difficult to model free form curves. It is also complex to extend this representation representation to 3D. III Engineering week, Schmalkalden 2013

CAE CAGD


Transcendental curves

•

•

Fittin g

Conclusions

Definition: non algebraic curves Main drawback: they do not admit an exact computer representation Polynomial-rational parametric curves  standard CAD representation

•

•

•

Solution: polynomial-rational fitting Detection: intersection between a line and a transcendental curve  infinite points Examples: Logarithmic and Archimedes spiral, helix, catenary III Engineering week, Schmalkalden 2013

CAE CAGD


Spline curves Complex shapes

Fittin g

Conclusions

Piecewise parametric curves


CAE CAGD


Geometric continuity G

Fittin g

Conclusions

r

Spline continuity type. Mechanical applications: G 0 or G1 tool paths (conventional machines), G2 vehicles and High speed machining


CAE CAGD


“Quality” assessment

Fittin g

Conclusions

Evaluation via Curvature k(s), s=Arc length parameterization ìis a continuous function ï Smooth curve if k ( s) : íhas few and monotonic segments ïends at specified points î


CAE CAGD

Curve s

CAD representation

Engineering curves

Fittin g

Conclusions

We have two main types of Engineering curves: •

Curves with CAD representation: Polynomial and rational Applications  Design

•

Curves without CAD representation: Transcendental curves (examples in the following slides) CAD incorporation via interpolation or approximation Lot of engineering applications III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

CAD representation

Fittin 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 III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

CAD representation

Trochoids applications

Fittin g

Conclusions

Rotary blower Mazda’s Wankel engine •

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 III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

CAD representation

Catenary


Fittin g

Conclusions

CAE CAGD

Curve s

CAD representation

Clothoid spiral •

•

Conclusions

Curvature is proportional to distance. Parameterization: Fresnel integrals (Transcendental curve) t ìC (t ) 2 cos( ) d 2 ò æ C (t ) ö ï 0 , . c(t ) ç t è S (t ) ø÷ í ï S (t ) ò sin( 2 2 ) d 0 î Application: road design, roller coasters Speed = Constant =

=

=

•

Fittin g


CAE CAGD

Curve s

CAD representation

Involute or evolvent

•

Conclusions

Definition: Path follow by the end of a string which is wound tangent to a profile. It is a transcendental curve.

•

Fittin g

Application: design of gear teeth III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

CAD representation

Offset

•

Fittin g

Conclusions

Definition: parallel curve cd(t ) ìn(t ) normal . cd (t ) = c(t ) ± d n(t ), í î+ up, - down

•

In general, it is a non polynomial/rational curve III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

CAD representation

Offset: Applications

Tool path and road design

Cam design III Engineering week, Schmalkalden 2013

Fittin g

Conclusions

CAE CAGD

Curve s

CAD representation

Fittin g

Conclusions

Offset: representation problems Self-intersections


CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Standard CAD representation •

•

We can incorporate polynomial curves into CAD 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


CAE CAGD

Curve s

Eng. curves

Fittin g

Monomial basis •

Conclusions

Definition: A way to describe a polynomial using a linear combination of monomials

{

2

Monomialbasis: 1,t ,t ,...,t

n

}, n

Polynomialrepresentation: a(t )

a

=

i •

i

t

i

.

0

=

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 III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Monomial basis

•

•

Fittin g

Conclusions

Disadvantages: –

Coefficients have not geometric meaning

–

Poor numerical properties: small errors in ai  gaps

Gaps and non geometric meaning are unacceptable for CAD III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Bézier curves •

Conclusions

Provide a polynomial representation: –

–

•

Fittin g

where coefficients have geometric meaning  Control points without numerical problems  no gaps

End control points = curve end points.


CAE CAGD

Curve s

Eng. curves

Fittin g

Examples of Bézier curves


Conclusions

CAE CAGD

Curve s

Eng. curves

Bernstein polynomials •

•

Definition:

B

n

i

æ (u) = ç è

n i

ö 1 ( ÷ ø

u

)

n i

u

i

Fittin g

Conclusions

.

Bézier: combination of control pts via Bernstein polynomials n n

b(u) =

b B (u). i

i

i =0

–

bi  Control points

–

B (u)

n

i

effect of bi in the curve shape.

bi pulls the curve (u=i/n  Maximum

n

B (u) i

)


CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Bézier curves: geometric properties Affine invariance. Reposition-scale-rotation so on  transformation of control points

Other Application Typography:

S III Engineering week, Schmalkalden 2013

S S

CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Bézier curves: geometric properties End point interpolation. The curve pass through the end control points  no gaps Linear precision. If bi are aligned  line Convex hull property. B(t) lies in the convex hull of the control polygon. Application: Interference checking. Operations like integration and derivation via additions, products and divisions of control points Good numerical properties III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Bézier curves: smoothness conditions


CAE CAGD

Curve s

Eng. curves

Bézier curve: Derivation


Fittin g

Conclusions

CAE CAGD

Curve s

Eng. curves

Bézier curves: Integration


Fittin g

Conclusions

CAE CAGD

Curve s

Eng. curves

Fittin g

Design Example: cam profile

Hypothesis. Constant angular speed

=

cte

Conditions: Constraint forces  y(θ) is C1 Avoid jerk  y(θ) is C2 III Engineering week, Schmalkalden 2013

Conclusions

CAE CAGD

Curve s

Eng. curves

Rational Bézier curves

Fittin g

Conclusions

Several conic curves (ellipse, circle, hyperbola) can not be represented by means of a polynomial Bézier curve.

Solution: Rational Bézier curve  a perspective transformation III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Rational Bézier curves

Effect of control points reposition

Fittin g

Conclusions

Effect of weight modification


CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Conic construction via Rational Bézier curves

ìCentre (circle), focus: C ï Conic arc ® íAngle 2 ïWeights: w = w = 1,w 0 2 1 î

ì[0,1] ® Ellipse (w 2 = cos( ), Circle) ï w Îí1 ® Parabola 1 ï[1, ¥] ® Hyperbola î


CAE CAGD

Curve s

Eng. curves

Conics applications

Fittin g

Conclusions

Source: radartutorial.eu

Source: Alternative-energy-tutorials.com


CAE CAGD

Curve s

Eng. curves

Fittin g

B-spline curves

Conclusions

Definition: Spline of Bézier curves  same Bézier properties Application  curves with complex shape n n

d(u) =

d N (u). i

i

N

n

i

(u) ® piecewisebasis

i =0


CAE CAGD

Curve s

Eng. curves

B-spline versus Bézier

Fittin g

Bézier Design Continuity guarantee

Conclusions

B-spline

Pseudo local shape control

Local control

Complex

Simple (polynomial degree - 1)

Evaluation cost

N control points increase It doesn’t depend on number computational cost of control points

Implementation

Complex Straightforward III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

Example: Degree n=3 B-spline

Continuity  n-1=2 Number of control points  p=7 9 knots, Number of pieces m  m=p-n=4 III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s

Eng. curves

Fittin g

Conclusions

NURBS: Non uniform rational B-spline

Definition: Spline of Rational Bézier curves with a non uniform knot sequence. III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s


Conclusions

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 III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s


Conclusions

Lagrange Interpolation Original curve Degree 4 Lagrange Interpolation Degree 8 Lagrange Interpolation

Source: wikipedia

•

•

Lagrange interpolation: the interpolation curve passes through an uniformly distributed set of points Drawbacks: –

Runge’s phenomenon

–

New data  re-compute III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s


Conclusions

Hermite Interpolation

•

•

Definition: the interpolation curve passes through an non uniformly distributed set of points  data: end points and their derivate Applications: –

Approximation of boundary problems

–

Useful to control continuity III Engineering week, Schmalkalden 2013

CAE CAGD

Curve s


Conclusions

Approximation

•

The construction passes close to a set of data.

•

Complex implementation: iterative algorithm.

•

Basic approximation technique: Least squared method. III Engineering week, Schmalkalden 2013

Engineering Curves and CAGD

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