Optimization with surrogates - PowerPoint Presentation

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