Optimization cycle concept
Optimization with surrogatesBased 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, Joness EGO (Efficient Global
Optimization).Easiest with one added sample at a time.
Optimization with surrogates goes through cycles, with each
cycle consisting of performing a number of simulations, fitting a
surrogates to these simulations (possibly using also simulations
from previous cycles) and then optimizing based on the surrogates.
A termination test is undertaken, and if it is not satisfied,
another cycle is undertaken. The optimization algorithm used to
solve the optimization problem in each cycle is less critical than
in optimization without surrogates, because the objective functions
and constraints are inexpensive to evaluate from the
surrogates.
There are two broad strategies for surrogate based optimization,
zooming and adaptive sampling. In the zooming approach each cycles
involves a relatively large number of simulations, so that the
total number of cycles is usually small. In each cycle the original
optimization problem is solved except that the objective function
and/or the constraints are replaced by surrogates. When the optimum
of the cycle is obtained, the design space for future simulations
is focused on the region near that approximate optimum (the zooming
part). The zooming approach is the subject of this lecture.
The other strategy is called adaptive sampling. Its best known
algorithm is called EGO (efficient global optimization) and it is
the subject of another lecture. In adaptive sampling we add one or
small number of simulations in each cycle. The points selected for
sampling are based not only on the surrogate value but also on its
estimate of the uncertainty in its predictions. So points with high
uncertainty have a chance of being sampled even if the surrogate
prediction there is not as good as at other points with less
uncertainty.1Design 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 poorCalled also Reasonable Design Space.Benefits of
DSRPrevent costly simulations of unreasonable designsImprove
surrogate accuracyTechniquesUse inexpensive
constraints/objective.Common sense constraintsCrude surrogateDesign
space windowing
Madsen et al. (2000)Rais-Rohani and Singh (2004)
2An important step in constructing surrogates is limiting the
region where they need to be accurate hence where simulations are
performed. This avoids wasted simulations in regions where the
design is grossly infeasible, or where the objective function is
substantially non-optimal. The main benefit of a reduced design
domain is to improve the surrogate model accuracy.
Excluding portions of the design space can be done using a
partial set of constraints and/or objective functions that are
inexpensive computationally. For example, in many problems the
objective function, weight or cost , is easy to calculate, while
some constraints require costly simulations. We can then exclude
any region where the objective is more than 20% poorer than the
best feasible objective known so far. Often an initial crude
surrogate is used for that purpose.
Alternatively there are simple common-sense constraints. The top
figure shows an example (Madsen et al. 2001) where the shape of a
diffuser was designed with two design variables defining a cubic
shape function. An intuitive understanding that the polynomial
needed to be monotonic reduced substantially the design domain.
In addition, it is possible to reduce the design space by
windowing as shown in the bottom figure, moving the window as
indicated by the approximate optimum found in that window.
Raisi-Rohani and Singh, Comparison of global and local response
surface techniquesin reliability-based optimization of composite
structures , Struct. Multidisc. Optim. , 26, 333-345, 2004.Madsen,
J.I., Shyy, W., and Haftka, R.T., Response Surface Techniques for
Diffuser Shape Optimization, AIAA Journal, 38(9), pp. 1512-1518,
2000Balabanov, Giunta, Golovidov, Grossman, Mason, Watson, and
Haftka, Reasonable Design Space Approach to Response Surface
Approximation, J. Aircraft 36(1), 308-315, 1999.Radial 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
3Radial Turbine Optimization OverviewImprove efficiency and
reduce weight of a compact radial turbine Two objectives, hence
need the Pareto front.Simulations using 1D Meanline codePolynomial
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.
4The optimization sought to improve two objective functions for
a radial turbine: Efficiency and weight. Simulations were performed
by a one-dimensional code called Meanline, and they were fitted
using polynomial response surface (linear regression, see
lecture).
When you have two objective functions, you seek to find the
points where one objective cannot be improved without hurting the
other objective. The curve connecting these points is called the
Pareto front. It provides the tradeoff between the two
objectives.
The design-space refinement procedure proceeded here in three
steps. The first step was to use crude surrogate to identify the
feasible domain (where all constraints are satisfied). The second
step was to identify the region of interest that contains the
Pareto front. The third step was to zoom on that region in order to
get an accurate front.Variable 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
5Constraint 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
6Optimization 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
7See pages 407 413 of Hill and Peterson for full
explanationPhase 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 < 1 < 40React > 0.45
Infeasible RegionRange limitFeasible Region
8
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 RegionInfeasible Region
Feasible RegionInfeasible Region
9Refined DOE in feasible regionNew 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.
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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 interestNote: Maximum ts 90%1 tsWrotor
Wrotor vs. ts
Wrotor 1 ts
11After reducing the design region based on feasibility we
reduce it further by rejecting designs that have poor values of
either objective. For the efficiency we insist on at least 80% (the
best point is 90%). This excludes about a third of the design
points used for the surrogate. A limit on the rotor weight excludes
approximately another third.
We then refine the surrogate based on generating more design
points in the middle region. This time we combine a 5-level
full-factorial design in the three most influential design
variables with a latin hypercube desgin (LHSm see lecture on
space-filling designs of experiments). Out of the 125 points of the
full factorial design, 119 are feasible. 204 points come from the
LHS design.
The figure on the right shows the original 224 simulations, the
selected region to focus on, and the distribution of the 323 new
points.
Each cloud of points has a boundary on the bottom left which is
the set of points where one objective cannot be improved without
sacrificing the other objectives. This is the so called Pareto
front. 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 tsWrotor
12To estimate the accuracy of the Pareto front it was generated
from five different surrogates. All of them were quadratic
polynomials, but each minimized a different norm of the difference
between the data and the polynomial coefficients. (so called loss
function L). Least square fit corresponds to p=2, the average
absolute difference corresponds to p=1, and p=5 is very close to
minimizing the maximum difference.
The difference between the Pareto fronts obtained from the five
surrogates is substantial, showing that further refinement in the
design space is needed. This can be done by fitting the surrogates
over a narrower range of the design variables.Design 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
13The table shows the reduced region in design space. One range
was not reduced at all, while some reduced by approximately a
factor of 2, for a total reduction to about 6% of the original
volume.Phase 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
14SummaryResponse 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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