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

1. 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 Presented on April 27, 2006

2. 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

4. Project Terminology Airfoil Leading/trailing edge Pressure/Suction surface Parameters BETA1 � �1 DELTABETA1 � ?�1 LE (a/b) Objective function Baseline vs. New Target vs. Starting airfoil Commercial vs. In-house code Mapping vs. Optimizing (Objective function) Move to after problem description?Move to after problem description?

5. Problem Description Determine the best set of design parameters that match given airfoil coordinates through automation by optimization methods. Currently performed manually with GUI

6. 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

7. 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

8. 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 See weekly update 7 for Energy MeasureSee weekly update 7 for Energy Measure

9. 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

10. Project Optimization Process

11. 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

12. 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

13. Objective Function Description (cont�d) Energy Measure objective function from computer vision shape recognition application (Cohen et al)

14. Objective Function Description (cont�d) Modified Energy Measure objective function: Removed radius of curvature coefficient Integrated curvature on pressure and suction surfaces only

15. 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.

16. Optimization Method I � DFP DFP algorithm* � Davidon-Fletcher-Powell 2 min2 min

17. 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) 1 min1 min

18. Results Overview Objective function behavior (�mapping�) 1 Parameter Optimization 3 Parameter Optimization .25 min NLPQL � Sequential Quadratic Programming ASA � Simulated Annealing .25 min NLPQL � Sequential Quadratic Programming ASA � Simulated Annealing

19. Results: Objective Function Mapping 2 min2 min

20. Results: 1 Parameter Optimization 2 min2 min

21. Results: 3 Parameter Optimization 3 min3 min

22. Results: 3 Parameter Optimization (cont�d) 0.5 min0.5 min

23. 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) 1 min1 min

24. 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 1 min1 min

25. 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 1 min1 min

26. 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

27. 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