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
17