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 ID: 675315
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Slide1
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. Slide2
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 poorCalled also Reasonable Design Space.Benefits of DSRPrevent costly simulations of unreasonable designsImprove surrogate accuracyTechniques
Use inexpensive constraints/objective.
Common sense constraints
Crude surrogateDesign space windowing
Madsen et al. (2000)
Rais-Rohani
and Singh (2004) Slide3
Radial Turbine Preliminary Aerodynamic Design Optimization
Yolanda MackUniversity of Florida, Gainesville, FLRaphael 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, MI42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit7-12-06Slide4
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 Determine
feasible domain.
Identify region of interest.Obtain high accuracy approximation for Pareto front
identification.Slide5
Variable and Objectives
Variable
Description
MIN
MAX
RPM
Rotational Speed
80,000
150,000
React
Percentage of stage pressure drop across rotor
0.45
0.70
U/C isen
Isentropic velocity ratio
0.50
0.65
Tip Flw
Ratio 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 efficiencySlide6
Constraint Descriptions
Constraint
Description
Desired Range
Tip Spd
Tip speed (ft/sec) (stress indicator)
≤ 2500
AN^2 E08
Annulus area x speed^2 (stress indicator)
≤ 850
Beta1
Blade inlet flow angle
0
≤
Beta1
≤ 40
Cx2/
Utip
Recirculation flow coefficient (indication of pumping upstream)≥ 0.20Rsex/RsinRatio of the shroud radius at the exit to the shroud radius at the inlet≤ 0.85Slide7
Optimization Problem
Objective VariablesRotor weightTotal-to-static efficiencyDesign VariablesRotational SpeedDegree of reactionExit to inlet hub diameter Isentropic ratio of blade to flow speedAnnulus areaChoked flow ratio ConstraintsTip speed
Centrifugal stress measure
Inlet flow angle
Recirculation flow coefficient
Exit to inlet shroud radius
Maximize
η
ts
and
Minimize
W
rotor
such that
Slide8
Phase 1:
Aproximate 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 limits
0 <
β
1
< 40
React
> 0.45
Infeasible Region
Range limit
Feasible RegionSlide9
Feasible Regions for
Other ConstraintsTwo constraints limit a the values of one variable each. All invalid values of a third constraint lie outside of new rangesFourth constraint depend on three variables.
Feasible Region
Infeasible Region
Feasible Region
Infeasible RegionSlide10
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.Slide11
Phase 2:
Windowing based on objectivesShrinking design space by limits on objectivesUsed two DOEsLatin Hypercube Sampling (204 feasible points)5-level factorial design using 3 major variables only (119 feasible points)Total of 323 feasible pointsThe refined cloud defines a Pareto front.
Approximate region of interest
Note: Maximum
η
ts
≈ 90%
1 –
η
ts
W
rotor
W
rotor
vs.
ηtsWrotor 1 – ηtsSlide12
Use
different surrogates to estimate accuracyFive RSAs constructed for each objective minimizing different norms of the difference between data and surrogate (loss function).Norm p = 1,2,…,5Least square loss function (p = 2) Pareto fronts differ by as much as 20%Further design space refinement is necessary
1 –
η
ts
W
rotor
Slide13
Design Variable Range Reduction
Design Variable
Description
MIN
MAX
MIN
MAX
Original Range
Final Ranges
RPM
Rotational Speed
80,000
150,000
100,000
150,000
React
Percentage of stage pressure drop across rotor
0.45
0.680.450.57U/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 diameter0.10.40.10.4
AnsqrFrac
Used to calculate annulus area (stress indicator)
0.5
0.85
0.68
0.85Slide14
Phase 3: Construction of Final Pareto Front and RSA Validation
For 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 –
η
ts
Wrotor
1 –
η
ts
W
rotor
Slide15
Summary
Response 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