2. Optimization with surrogates Based on cycles. Each consists of sampling design points by simulations, fitting surrogates to simulations and then optimizing an objective. Zooming (This lecture) – Construct surrogate, optimize original objective, refine region and surrogate. – Typically small number of cycles with large number of simulations in each cycle. Adaptive sampling (Lecture on EGO algorithm) – Construct surrogate, add points by taking into account not only surrogate prediction but also uncertainty in prediction. – Most popular, Jones’s EGO (Efficient Global Optimization). – Easiest with one added sample at a time.

3. Design Space Refinement Design space refinement (DSR): process of narrowing down search by excluding regions because – They obviously violate the constraints – Objective function values in region are poor – Called also Reasonable Design Space. Benefits of DSR – Prevent costly simulations of unreasonable designs – Improve surrogate accuracy Techniques – Use inexpensive constraints/objective. – Common sense constraints – Crude surrogate – Design space windowing Madsen et al. (2000) Rais-Rohani and Singh (2004)

4. Radial Turbine Preliminary Aerodynamic Design Optimization Yolanda Mack University of Florida, Gainesville, FL Raphael Haftka, University of Florida, Gainesville, FL Lisa Griffin, Lauren Snellgrove, and Daniel Dorney, NASA/Marshall Space Flight Center, AL Frank Huber, Riverbend Design Services, Palm Beach Gardens, FL Wei Shyy, University of Michigan, Ann Arbor, MI 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 7-12-06

5. Radial Turbine Optimization Overview Improve efficiency and reduce weight of a compact radial turbine – Two objectives, hence need the Pareto front. – Simulations using 1D Meanline code – Polynomial response surface approximations used to facilitate optimization. Three-stage DSR 1.Determine feasible domain. 2.Identify region of interest. 3.Obtain high accuracy approximation for Pareto front identification.

6. Variable and Objectives VariableDescription MINMAX RPMRotational Speed80,000150,000 ReactPercentage of stage pressure drop across rotor 0.450.70 U/C isenIsentropic velocity ratio0.500.65 Tip FlwRatio of flow parameter to a choked flow parameter 0.300.48 Dhex %Exit hub diameter as a % of inlet diameter 0.100.40 AnsqrFracUsed to calculate annulus area (stress indicator) 0.501.0 Objectives Rotor WtRelative measure of “goodness” for overall weight EtatsTotal-to-static efficiency

7. Constraint Descriptions Constrai ntDescriptionDesired Range Tip SpdTip speed (ft/sec) (stress indicator)≤ 2500 AN^2 E08 Annulus area x speed^2 (stress indicator)≤ 850 Beta1Blade inlet flow angle0 ≤ Beta1 ≤ 40 Cx2/UtipRecirculation flow coefficient (indication of pumping upstream) ≥ 0.20 Rsex/RsinRatio of the shroud radius at the exit to the shroud radius at the inlet ≤ 0.85

8. Optimization Problem Objective Variables – Rotor weight – Total-to-static efficiency Design Variables – Rotational Speed – Degree of reaction – Exit to inlet hub diameter – Isentropic ratio of blade to flow speed – Annulus area – Choked flow ratio Constraints – Tip speed – Centrifugal stress measure – Inlet flow angle – Recirculation flow coefficient – Exit to inlet shroud radius Maximize η ts and Minimize W rotor such that

9. Phase 1: Aproximate feasible domain Design of Experiments: Face- centered CCD (77 points) – 7 cases failed – 60 violated constraints Using RSAs, dependences determined for constraints – Variables omitted for which constraints are insensitive – Constraints set to specified limits 0 < β 1 < 40 React > 0.45 Infeasible Region Range limit Feasible Region

10. Feasible Regions for Other Constraints Two constraints limit a the values of one variable each. All invalid values of a third constraint lie outside of new ranges Fourth constraint depend on three variables. Feasible Region Infeasible Region Feasible Region Infeasible Region

11. Refined DOE in feasible region New 3-level full factorial design (729 points) using reduced ranges. 498 / 729 were eliminated prior to Meanline analysis based on the two 3D constraints. 97% of remaining 231 points found feasible using Meanline code.

12. Phase 2: Windowing based on objectives Shrinking design space by limits on objectives Used two DOEs – Latin Hypercube Sampling (204 feasible points) – 5-level factorial design using 3 major variables only (119 feasible points) – Total of 323 feasible points – The refined cloud defines a Pareto front. Approximate region of interest Note: Maximum η ts ≈ 90% 1 – η ts W rotor W rotor vs. η ts W rotor 1 – η ts

13. Use different surrogates to estimate accuracy Five RSAs constructed for each objective minimizing different norms of the difference between data and surrogate (loss function). – Norm p = 1,2,…,5 – Least square loss function (p = 2) – Pareto fronts differ by as much as 20% – Further design space refinement is necessary 1 – η ts W rotor

14. Design Variable Range Reduction Design Variable Description MINMAXMINMAX Original Range Final Ranges RPMRotational Speed80,000150,000100,000150,000 ReactPercentage of stage pressure drop across rotor 0.450.680.450.57 U/C isenIsentropic velocity ratio0.50.630.560.63 Tip FlwRatio of flow parameter to a choked flow parameter 0.30.650.30.53 Dhex%Exit hub diameter as a % of inlet diameter 0.10.40.10.4 AnsqrFracUsed to calculate annulus area (stress indicator) 0.50.850.680.85

15. Phase 3: Construction of Final Pareto Front and RSA Validation For p = 1,2,…,5 Pareto fronts differ by 5% - design space is adequately refined Trade-off region provides best value in terms of maximizing efficiency and minimizing weight Pareto front validation indicates high accuracy RSAs Improvement of ~5% over baseline case at same weight 1 – η ts W rotor 1 – η ts W rotor

16. Summary Response surfaces based on output constraints successfully used to identify feasible design space Design space reduction eliminated poorly performing areas while improving RSA and Pareto front accuracy Using the Pareto front information, a best trade-off region was identified At the same weight, the RSA optimization resulted in a 5% improvement in efficiency over the baseline case