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.
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
– Called also Reasonable Design
Space.
• Benefits of DSR
– Prevent costly simulations of
unreasonable designs
– Improve surrogate accuracy
Madsen et al. (2000)
• Techniques
– Use inexpensive
constraints/objective.
– Common sense constraints
– Crude surrogate
– Design space windowing
Rais-Rohani and Singh (2004)
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
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
1. Determine feasible domain.
2. Identify region of interest.
3. Obtain high accuracy approximation
for Pareto front identification.
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
Maximize ηts and Minimize Wrotor
such that
80000  RPM  150000
0.45  React  0.70
0.50  U/C isen  0.65
0.30  Tip Flw  0.48
0.10  Dhex%  0.40
0.50  AnsqrFrac  1.0
• Constraints
–
–
–
–
–
Tip speed
Centrifugal stress measure
Inlet flow angle
Recirculation flow coefficient
Exit to inlet shroud radius
Tip Spd  2500 ft/sec
AN 2  850 m 4 /s 2
0  1  40
0.20  Cx2/Utip
Rsex/Rsin  0.85
Phase 1: Aproximate feasible domain
• Design of Experiments: Facecentered CCD (77 points)
– 7 cases failed
– 60 violated constraints
• Using RSAs, dependences
determined for constraints
Tip Spd  Tip Spd U/C isen 
AN 2  AN 2  AnsqrFrac 
1  1  React ,U/C isen, Tip Flw 
Cx2/Utip  Cx2/Utip  RPM ,U/C isen, AnsqrFrac 
Rsex/Rsin  Rsex/Rsin  AnsqrFrac,U/C isen, Dhex% 
– Variables omitted for which
constraints are insensitive
– Constraints set to specified
limits
0 < β1 < 40
React > 0.45
Infeasible Region
Feasible Region
Range limit
Feasible Regions for Other Constraints
Infeasible Region
Feasible Region
• Two constraints limit a the
values of one variable each.
• All invalid values of a third
constraint lie outside of new
ranges
• Fourth constraint depend on
three variables.
Feasible Region
Tip Spd  Tip Spd U/C isen 
AN 2  AN 2  AnsqrFrac 
Rsex/Rsin  Rsex/Rsin  AnsqrFrac,U/C isen, Dhex% 
Infeasible Region
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.
• Five RSAs constructed for
each objective minimizing
different norms of the
difference between data
and surrogate (loss
function).
– Norm p = 1,2,…,5
– Least square loss function
(p = 2)
– Pareto fronts differ by as
much as 20%
– Further design space
refinement is necessary
1 – ηts
Use different surrogates to estimate
accuracy
Wrotor
N
N
yˆ   0    j x j    jj x  
j 1
j 1
n
2
j
L   yi  yˆi
i 1
p
N

i j 2
x xj
ij i
Design Variable Range Reduction
Design
Variable Description
MIN MAX
Original
Range
MIN MAX
Final
Ranges
RPM
Rotational Speed
80,000
150,000
100,000
150,000
React
Percentage of stage
pressure drop across rotor
0.45
0.68
0.45
0.57
U/C isen
Isentropic velocity ratio
0.5
0.63
0.56
0.63
Tip Flw
Ratio of flow parameter to a
choked flow parameter
0.3
0.65
0.3
0.53
Dhex%
Exit hub diameter as a % of
inlet diameter
0.1
0.4
0.1
0.4
AnsqrFrac
Used to calculate annulus
area (stress indicator)
0.5
0.85
0.68
0.85
Wrotor
1 – ηts
• 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 – ηts
Phase 3: Construction of Final Pareto
Front and RSA Validation
baseline
Wrotor
trade-off region
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