)( )()( kk f xg )( )()()( kkk f xHs

)( )()( kkf sx )()()( kkk sσ

)()1()1( kkk ggy

)()(

)()()(

kTk

Tkkk

yσ

σσA

)()()(

)()()()()(

kkTk

kTkTkkk

yHy

HyyHB

)0(x

)()()1()1( kkkk BAHH

A Method for Determining Turbine Airfoil Geometry Parameters from a Set of Coordinates

ME 597 Project I – Spring 2006Purdue School of Engineering and Technology, IUPUIAndrew White, BPMME

Project Advisors: Dr. Hasan Akay, IUPUI Ed Turner, Rolls-Royce Corporation

Presented on April 27, 2006

Outline

• Project Terminology• Problem Description• Project Objective• Overview of Solution – Two Approaches • Previous Work• Optimization Challenges• Project Optimization Process• Objective Function Description • Quasi-Newton Optimization Methods

– DFP– NLPQL

• Results: – Objective Function Mapping– 1 Parameter Optimization– 3 Parameter Optimization

• Conclusions• Future Study

Turbine Airfoils of a Rolls-Royce Trent 1000

gas turbine engine

This project focuses on the cross-sectional shapes of Turbine Airfoils.

Throat

Sp

phi

Project Terminology

• Airfoil– Leading/trailing edge– Pressure/Suction surface

• Parameters– BETA1 – β1

– DELTABETA1 – Δβ1 – LE (a/b)

• Objective function– Baseline vs. New

• Target vs. Starting airfoil• Commercial vs. In-house code• Mapping vs. Optimizing

(Objective function)Pressure Surface

Leading Edge

Trailing Edge

Suction Surface

x y0.00E+00 0.00E+00

-1.09E-02 1.48E-02

-2.18E-02 2.96E-02

-3.28E-02 4.43E-02

-4.38E-02 5.91E-02

-5.49E-02 7.37E-02

-6.60E-02 8.83E-02

-7.72E-02 1.03E-01

-8.85E-02 1.17E-01

… …

Test Case Profiles Com pared

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x

y

XPTST, YPTST,

XPTSC, YPTSC,

Problem Description

• Determine the best set of design parameters that match given airfoil coordinates through automation by optimization methods.

• Currently performed manually with GUI

optimizer

afopt

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x

y

XPTST, YPTST,

XPTSC, YPTSC,

tds

Target/StartMatched Airfoils

Parameters

Project Objectives

• 3 main objectives for the problem:– Develop an optimizable objective function– Use a commercial optimization/design code to test

objective function behavior – Understand the theory/programming of an in-house

numerical optimization code and set it up for future optimization of the present problem in place of the commercial code

Overview of Solution – Two Approaches

Commercial Code• Optimization: NLPQL

(Non-Linear Programming by Quadratic approximation of the Lagrangian)

• GUI• Relatively “easy”• Less control, must understand

available methods• Single user per $xx,000 license

In-house code• Optimization: DFP

(Davidon-Fletcher-Powell)

• FORTRAN• Must understand code• More control over code, uses

trusted optimization method• Many users

Solve shape matching through optimization by…

• This project used the commercial code to develop an objective function while preparing the in-house code for future work

Previous Work

• Previous work on project– Shape matching with a commercial code previously

attempted with little success– Trouble shooting discontinuities in design system – Previous objective functions based on airfoil shape

• Point-to-point distance• Area• Perimeter• Center of mass

– Baseline function: Point-to-point distance– RRC in-house optimization code

Remove discontinuities in design system

Optimization Challenges• Errors in mathematical formulation of

models which no mere parameter adjustments can hope to compensate for*

• Objective function sensitivity/behavior can be difficult to predict in entire design space

• Can be difficult to tell if problems are due to objective function or mathematical model

• Optimizer algorithms generally perform more and more poorly the larger the number of varying parameters*

• Choosing the “right” optimization routine

“The best optimization routine is the one you know best.” --Papalambros and Wilde, Principles of Optimal Design

*Source: http://nsr.bioeng.washington.edu/PLN/Members/butterw/JSIMDOC1.6/Contents.stx/User_Intro.stx

Airfoil coordinates from parameters

Starting Design Parameters

Airfoil Design System (generates airfoil geometry coordinates

from 15 input design parameters)

Objective Function(Airfoil comparator function)

Objective function value, F

NLPQL/DFP optimizer(changes parameters to minimize F)

Target airfoil coordinates

Modified design parameters

Project Optimization Process

Optimization Overview

• Optimization requires:– Objective function: what to optimize– Optimization routine: how to let a computer make the

objective function as small as possible

• Next:– Objective function description– Basics of quasi-Newton optimization methods– Two quasi-Newton methods used in this project:

• DFP – Davidon-Fletcher-Powell • NLPQL – Non-Linear Programming by Quadratic

approximation of the Lagrangian

Objective Function Description

• Scalar expression that should approach zero when the two airfoils match

• Objective functions:– Baseline: Point-to-point distance– Energy Measure– New Energy Measure

regularcurve EEE

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x

y

XPTST, YPTST,

XPTSC, YPTSC,

22 )()( yxd

regularcurve EEE

Curvature & Curvature Error

-50

-40

-30

-20

-10

0

10

0 0.2 0.4 0.6 0.8 1

s

K(s

) (c

urv

atu

re)

0.00001

0.0001

0.001

0.01

0.1

1

10

100

EC

UR

V (

curv

atu

re e

rro

r)

Target Curvature (Kt)

Current Curvature (Kc)

ECURVE

Parameterized Airfoil

-50

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10

0 0.2 0.4 0.6 0.8 1

0.00001

0.0001

0.001

0.01

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1

10

100

Target Curvature (Kt)

Current Curvature (Kc)

ECURVE

TargetParameterizedDifference Measure

Target Airfoil

Objective Function Description (cont’d)

• Energy Measure objective function from computer vision shape recognition application (Cohen et al)

S PQcurve dssKsKE

2))()'((2

1

dss

sPsQE

PCregular

2))()'((

regularcurve EEE

)(/1)( sKs P

Discretization for computer code by trapezoidal quadrature formula…

where,

b

a

k

jjj xfxfhxTdxxf

11 )()(

2

1)()(

N

i

iP

iQ

iP

iQcurvature KKKK

NE

1

2112 )()(2

1

)1(

1

2

1

sQ

P s = 0

s = 1

s'Q(s') – P(s)

KQ(s')

KP(s)

Objective Function Description (cont’d)

• Modified Energy Measure objective function:– Removed radius of curvature coefficient– Integrated curvature on pressure and suction surfaces only

0

0.1

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0.3

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0.7

0.8

0.9

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2

YPTST,

YPTSC,

target, pressure side stop

"target, suction side start"

regularcurve EEE regularcurve EEE Energy Measure New Energy Measure

Removed curvature from integral on

leading edge

Quasi-Newton Optimization Methods

• What are Quasi-Newton methods?†

– Quasi-Newton methods build up curvature information (i.e. 2nd derivative) at each iteration to formulate a quadratic model problem of the form:

– The optimal solution for this problem occurs when the partial derivatives of x go to zero, i.e.,

– The optimal solution point, x*, can be written as

– Quasi-Newton methods approximate H-1 using f(x) and grad f(x) to build up curvature information with an iterative updating technique.

H = Hessian matrix c = gradient vectorb = constant scalar

†Source: http://www.mathworks.com/access/helpdesk/help/toolbox/optim/ug/f137.htmlx

Quadratic Model approximation of “Design surface”

bxcHxx TT

x2

1min

0**)( cHxxf

cHx 1*

Optimization Method I – DFP

• DFP algorithm* – Davidon-Fletcher-Powell1. Initial design, )0(x . 2. Calculate initial objective function value, )( )0(xf

3. Approximate, nIH )0( .

Then the kth iteration becomes: 4. Gradient at )(kx : )( )()( kk f xg

5. Search direction: )( )()()( kkk f xHs

6. Optimum step size k minimize )( )()( kkf sx

7. Update the design: )()()1( kk

kk sxx

8. Calculate )( )1( kf x compare to )( )(kf x to check convergence 9. Update the Hessian matrix approximation based on new design:

)()()1()1( kkkk BAHH

where )()(

)()()(

kTk

Tkkk

yσ

σσA and

)()()(

)()()()()(

kkTk

kTkTkkk

yHy

HyyHB

and )()()( kkk sσ )()1()1( kkk ggy

10. Continue iterating until convergence criteria met (|grad f | < , |g| < , etc)

*Source: References [1] through [4]

Optimization Method II – NLPQL

• NLPQL algorithm* (Nonlinear Programming by Quadratic approximation of the Lagrangian)

• Quasi-Newton, Direct, sequential quadratic programming method

• Like DFP, NLPQL – uses quadratic approximation of the function – Approximation formula for the Hessian called

BFGS (Broyden-Fletcher-Goldfarb-Shanno)

*Source: iSIGHT online documentation

Results Overview

• Objective function behavior (“mapping”)

• 1 Parameter Optimization

• 3 Parameter Optimization

Results: Objective Function MappingB

ET

A1

Energy Measure2nd term of E. M.

(no curvature) Baseline

DE

LTA

BE

TA

1LE

New Energy Measure

regularcurve EEE regularcurve EEE 22 )()( yxd regularEE

Results: 1 Parameter Optimization

BETA1 DELTABETA1 LE

Initial Value

Target

Initial Value 60 60 2.5New 39.6517 (0.2%) 43.9715 (2.3%) 1.0987 (1.9%)Baseline 38.3027 (3.2%) 44.4478 (3.4%) 1.0943 (2.3%)Target Value 39.58 43.00 1.12Limits

BETA1 DELTABETA1 LE

15-90 deg 1-90 deg 0.5-2.5

Initial Value 60 60 2.5New 38.1174 (3.7%) 56.8614 (32.2%) 1.4075 (25.7%)Baseline 39.8089 (0.6%) 58.4366 (35.9%) 1.4035 (25.3%)Initial Value 30 30 0.75New 40.8302 (3.2%) 29.6253 (31.1%) 0.8717 (22.2%)Baseline 37.1521 (6.1%) 29.5973 (31.2%) 0.8445 (24.6%)Target Value 39.58 43.00 1.12Limits

Hig

h S

tart

Lo

w S

tart

BETA1 DELTABETA1 LE

15-90 deg 0.5-2.51-90 deg

Results: 3 Parameter Optimization

New(Modified Energy Measure)

Baseline

Objective Functions Low Start High Start

Results: 3 Parameter Optimization (cont’d)

Baseline objective function

Low start

High start

Matched

Matched

Results: 3 Parameter Optimization (cont’d)

• Ran a single trial of Adaptive Simulated Annealing (ASA) algorithm on 3 Parameters

• Baseline objective function• %Error reduced by half in

DELTABETA1 and LE• Took 67 min. with 2201 iterations

(SunBlade 2000)• Compared to 3-5 min. and 110

iterations for NLPQL• Results visually the same (see

result plot at right)

Single Parameter Mapping of BETA1

ASA Optimization: Plot of BETA1

Conclusions

• Matching airfoil shapes through optimization is feasible

• Quasi-Newton methods are fast and will work if objective function behaves smoothly

• New objective function showed similar results to Baseline function with NLPQL optimizer

• Improvements can be made to New function as design system discontinuities are fixed—curvature can be re-introduced to leading edge

• 3 parameters:– BETA1 is strongest parameter and achieves smallest

%Error in final values– Visually close for both objective functions

Future Study

• Required:– Complete in-house code and run comparative study

to results of commercial code– Further trouble shooting of design system– Add curvature back into leading edge with cusps

removed from model– Increase number of parameters to optimize– Determine how close is close enough

• Possible:– Scaling parameters (BETA1) or turn individual

parameters off as they narrow in on target value– Consider other algorithms or combinations of

algorithms– Limitations on achieving various target airfoil shapes

Acknowledgements

• Dr. Hasan Akay, ME Department

• Ed Turner, Rolls-Royce mentor

• Larry Junod, Rolls-Royce mentor

• Dr. Steve Gegg, Rolls-Royce

• Dr. Asok Sen, Math Department

References• [1] Fletcher and Powell, A Rapidly Convergent Descent Method for

Minimization, The Computer Journal, 1963, July• [2] Hamming, Richard W., Introduction to Applied Numerical

Analysis, Hemisphere Publishing Corp., 1989• [3] Arora, Jasbir S., Introduction to Optimum Design, Elsevier

Academic Press, 2004• [4] Vanderplaats, Garret N., Numerical Optimization Techniques for

Engineering Design: With Applications, McGraw Hill, Inc., 1984• [5] Cohen, I., Ayache, N., Sulger, P., Tracking Points on Deformable

Objects Using Curvature Information, Proceedings from the 2nd European Conference on Computer Vision, 1992

• [6] Heath, M., Scientific Computing: An Introductory Survey, 2nd ed, McGraw Hill, 2002

• [7] www.mathworks.com• [8] http://nsr.bioeng.washington.edu/PLN/Members/butterw/JSIMDOC1.6/Contents.stx/User_Intro.stx

• [9] iSIGHT On-line documentation files