Optimization cycle concept

Optimization with surrogatesZoomingConstruct 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, Joness EGOEasiest with one added sample at a time.

Optimization cycle conceptDefine region in design spaceEvaluate objective and constraints at a set of points (Design of experiments)Construct surrogates for expensive objective function and constraintsPerform optimization based on surrogatesRefine surrogate and go back to step 1 if convergence not achieved and another cycle is affordable

2Theoretical considerations for zoomingProcess 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

3Design Space RefinementDesign space refinement (DSR): process of narrowing down search by excluding regions because They obviously violate the constraints Objective function values in region are poorBenefits of DSRPrevent costly analysis of infeasible designsImprove surrogate model accuracyTechniquesDesign space reductionReasonable design spaceDesign space windowing

Madsen et al. (2000)Rohani and Singh (2004)

4Radial Turbine Preliminary Aerodynamic Design OptimizationYolanda MackUniversity of Florida, Gainesville, FL

Raphael Haftka, University of Florida, Gainesville, FLLisa Griffin, Lauren Snellgrove, and Daniel Dorney, NASA/Marshall Space Flight Center, ALFrank Huber, Riverbend Design Services, Palm Beach Gardens, FLWei Shyy, University of Michigan, Ann Arbor, MI

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit7-12-06

5Radial Turbine Optimization OverviewPerform optimization to improve efficiency of a compact radial turbine Increase turbine efficiency while maintaining low turbine weightPolynomial response surface approximations used to facilitate optimizationThree-stage optimization using 1-D Meanline codeDetermination of feasible design spaceIdentify region of interestObtain high accuracy approximation for Pareto front identification

6Variable and ObjectivesVariableDescriptionMINMAXRPMRotational Speed80,000150,000ReactPercentage of stage pressure drop across rotor0.450.70U/C isenIsentropic velocity ratio0.500.65Tip FlwRatio of flow parameter to a choked flow parameter0.300.48Dhex %Exit hub diameter as a % of inlet diameter0.100.40AnsqrFracUsed to calculate annulus area (stress indicator)0.501.0ObjectivesRotor WtRelative measure of goodness for overall weightEtatsTotal-to-static efficiency

7Constraint DescriptionsConstraintDescriptionDesired RangeTip SpdTip speed (ft/sec) (stress indicator) 2500AN^2 E08Annulus area x speed^2 (stress indicator) 850Beta1Blade inlet flow angle0 Beta1 40Cx2/UtipRecirculation flow coefficient (indication of pumping upstream) 0.20Rsex/RsinRatio of the shroud radius at the exit to the shroud radius at the inlet 0.85

8Optimization ProblemObjective VariablesRotor weightTotal-to-static efficiencyDesign VariablesRotational SpeedDegree of reactionExit to inlet hub diameter Isentropic ratio of blade to flow speedAnnulus areaChoked flow ratio ConstraintsTip speedCentrifugal stress measureInlet flow angleRecirculation flow coefficientExit to inlet shroud radius

Maximize ts and Minimize Wrotor

such that

9See pages 407 413 of Hill and Peterson for full explanationPhase 1: Determine feasible domainDesign of Experiments: Face-centered CCD (77 points)7 cases failed60 violated constraintsUsing RSAs, dependences determined for constraintsVariables omitted for which constraints are insensitiveConstraints set to specified limitsCorresponding bounds on design variables determinedConstraint boundary approximations developed to help determine feasible design space

10Feasible Regions for Three ConstraintsRSA evaluation determines two 1-D constraintsRanges of design variables reduced to match 1-D constraint boundariesAll invalid values of a third constraint lie outside of new rangesThus, three of five constraints automatically satisfied by range reduction of two design variables

Feasible RegionInfeasible Region

11Feasible Regions for Two ConstraintsNew 3-level full factorial design (729 points)498 / 729 were eliminated prior to Meanline analysis based on new variable constraints97% of remaining 231 points found feasible using Meanline code

Feasible RegionInfeasible Region

0 < 1 < 40React > 0.45

Infeasible RegionRange limitFeasible Region

12Phase 2: Design Space RefinementEliminate undesirable areas by shrinking design spaceUsed two DOEsLatin 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 interestNote: Maximum ts 90%1 tsWrotor Wrotor vs. ts

Wrotor 1 ts

13Use loss function to estimate accuracyFive RSAs constructed for each objective using general loss functionParameter p = 1,2,,5Least square loss function (p = 2) Pareto fronts differ by as much as 20%Design space refinement is necessary

1 tsWrotor

14Design Variable Range ReductionDesign VariableDescriptionMINMAXMINMAXOriginal RangeFinal RangesRPMRotational Speed80,000150,000100,000150,000ReactPercentage of stage pressure drop across rotor0.450.680.450.57U/C isenIsentropic velocity ratio0.50.630.560.63Tip FlwRatio of flow parameter to a choked flow parameter0.30.650.30.53Dhex%Exit hub diameter as a % of inlet diameter0.10.40.10.4AnsqrFracUsed to calculate annulus area (stress indicator)0.50.850.680.85

15Phase 3: Construction of Final Pareto Front and RSA ValidationFor p = 1,2,,5 Pareto fronts differ by 5% - design space is adequately refinedTrade-off region provides best value in terms of maximizing efficiency and minimizing weightPareto front validation indicates high accuracy RSAsImprovement of ~5% over baseline case at same weight

1 tsWrotor

1 tsWrotor

16SummaryResponse surfaces based on output constraints successfully used to identify feasible design spaceDesign space reduction eliminated poorly performing areas while improving RSA and Pareto front accuracyUsing the Pareto front information, a best trade-off region was identifiedAt the same weight, the RSA optimization resulted in a 5% improvement in efficiency over the baseline case

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