a method for determining turbine airfoil geometry parameters from a set of coordinates
DESCRIPTION
A Method for Determining Turbine Airfoil Geometry Parameters from a Set of Coordinates. ME 597 Project I – Spring 2006 Purdue School of Engineering and Technology, IUPUI Andrew White, BPMME Project Advisors: Dr. Hasan Akay, IUPUI Ed Turner, Rolls-Royce Corporation - PowerPoint PPT PresentationTRANSCRIPT
)( )()( 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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-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
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
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-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
-40
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0
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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0.5
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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