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Optimization with surrogates Zooming – Construct surrogate, optimize objective, refine region and surrogate, repeat. – Danger: Miss optima. Adaptive sampling – Construct surrogate, add points by balancing exploration and exploitation, repeat. – Most popular, Jones’s EGO – Easiest with one added sample at a time.

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Optimization cycle concept 1.Define region in design space 2.Evaluate objective and constraints at a set of points (Design of experiments) 3.Construct surrogates for expensive objective function and constraints 4.Perform optimization based on surrogates 5.Refine surrogate and go back to step 1 if convergence not achieved and another cycle is affordable

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Theoretical considerations for zooming Process may not converge to true (even local) optimum There are algorithms that are guaranteed to converge to a local optimum but they are limited (see publications by Natalia Alexandrov) It is useful to reduce size of design space (every function is quadratic in a small enough region) Choice between surrogates depends on density of sampling

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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 Benefits of DSR – Prevent costly analysis of infeasible designs – Improve surrogate model accuracy Techniques – Design space reduction – Reasonable design space – Design space windowing Madsen et al. (2000) Rohani and Singh (2004)

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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

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Radial Turbine Optimization Overview Perform optimization to improve efficiency of a compact radial turbine – Increase turbine efficiency while maintaining low turbine weight – Polynomial response surface approximations used to facilitate optimization Three-stage optimization using 1- D Meanline code 1.Determination of feasible design space 2.Identify region of interest 3.Obtain high accuracy approximation for Pareto front identification

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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

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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

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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

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Phase 1: Determine 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 – Corresponding bounds on design variables determined Constraint boundary approximations developed to help determine feasible design space

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Feasible Regions for Three Constraints RSA evaluation determines two 1-D constraints Ranges of design variables reduced to match 1-D constraint boundaries All invalid values of a third constraint lie outside of new ranges Thus, three of five constraints automatically satisfied by range reduction of two design variables Feasible Region Infeasible Region

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Feasible Regions for Two Constraints New 3-level full factorial design (729 points) 498 / 729 were eliminated prior to Meanline analysis based on new variable constraints 97% of remaining 231 points found feasible using Meanline code Feasible Region Infeasible Region 0 < β 1 < 40 React > 0.45 Infeasible Region Range limit Feasible Region

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Phase 2: Design Space Refinement Eliminate undesirable areas by shrinking design space 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 Approximate region of interest Note: Maximum η ts ≈ 90% 1 – η ts W rotor W rotor vs. η ts W rotor 1 – η ts

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Use loss function to estimate accuracy Five RSAs constructed for each objective using general loss function – Parameter p = 1,2,…,5 – Least square loss function (p = 2) – Pareto fronts differ by as much as 20% – Design space refinement is necessary 1 – η ts W rotor

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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

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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

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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

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