Full text of DOI:10.2514/2.5991

JOURNAL OF PROPULSION AND POWER
–
Vol. 18, No. 3, May June 2002
Shape Optimization of Supersonic Turbines Using Global
Approximation Methods
Nilay Papila¤ and Wei Shyy†
University of Florida, Gainesville, Florida 32611-6250
and
Lisa GrifŽ n‡ and Daniel J. Dorney§
NASA Marshall Flight Center, Huntsville, Alabama 35812
There is growing interest to adopt supersonic turbines for rocket propulsion. However, this technology has not
been actively investigated in the United States for the last three decades. To aid design improvement, a global
(
)
optimization framework combining the radial-basis neural network NN and the polynomial response surface
(
)
(
)
RS method is constructed for shape optimization of a two-stage supersonic turbine, involving O 10 design
variables. The design of the experiment approach is adopted to reduce the data size needed by the optimization
task. The combined NN and RS techniques are employed. A major merit of the RS approach is that it enables one
to revise the design space to perform multiple optimization cycles. This beneŽ t is realized when an optimal design
approaches the boundary of a predeŽ ned design space. Furthermore, by inspecting the in uence of each design
variable, one can also gain insight into the existence of multipledesign choices and select the optimumdesign based
on other factors such as stress and materials consideration.
Nomenclature
basis functions
total weight of the turbine
coefŽ cients of the linear combinations or weights
payload increment
We select the use of the global optimization technique because
it has several advantages : 1 Calculation of the local sensitiv-
15
)
h
W
w
1pay
´
=
=
=
=
=
=
)
ity of each design variable is not required. 2 Information col-
lected from various sources and by different tools can be utilized.
)
)
3 Multi-criterionoptimization is offered. 4 The existenceof mul-
)
tiple design points and tradeoffs can be handled. 5 Tasks may be
total to static efŽ ciency
total to total efŽ ciency
t–
t–
s
t
)
easily performed in parallel. 6 The noise intrinsic to numerical
´
and experimentaldata can often be effectivelyŽ ltered. Approxima-
tions techniques are critical to the optimization procedure. Among
alternative global approximation techniques, the RSM has gained
the most attention due to its  exibility in handling different types
Introduction
URBINE performancedirectlyaffectsenginespeciŽ c impulse,
thrust-to-weight ratio, and reliability in a rocket propulsion
T
16
of information. The RSM expresses the objective and the con-
system. In the last three decades, supersonicturbines have not been
designed for rocket propulsion in the United States. There is grow-
ing interest to reconsider this technology for space transport. De-
signing a multistage turbine is a labor-intensivetask because of the
substantial number of variables involved in the problem. Clearly, a
formal optimization methodology will be valuable to help meet the
design goals by maximizing the performance objective while ad-
dressing the structures and materials considerations. A number of
papers, using approachessuch as sensitivityevaluations,1¡5 genetic
straint by simple functions, often polynomials, which are Ž tted to
theselectedpoints.OnecanuseRBNN-basedandpolynomial-based
RSM techniquesto model the relationshipbetween designvariables
and objective/constraint functions of the overall approach. In the
present work, the RBNN is employed to supply additional data to
help construct improved polynomialrepresentationof the response
(
)
surface RS . In other words, RBNN feeds data into polynomials
before the optimization task is conducted.Such a practice has been
shown to be beneŽ cial12¡14 and can help to assess the adequacy
of the RS model when there is insufŽ cient amount of information
available.
algorithms,6¡8 or responsesurface methods RSMs ,9¡12 have been
published in this area. In this work, a global optimization method-
ology, based on the radial-basis neural networks RBNN and the
polynomial-basedRSM under development,13;14 is employed to fa-
cilitate design optimization for supersonic turbines intended for
(
)
(
)
With the aid of the global optimization technique, both prelim-
inary and detailed shape designs of a supersonic turbine are con-
sidered. The main purpose for preliminaryoptimization is to deter-
mine the optimum conŽ guration in terms of the number of stages,
sizing, revolutionsper minute, and compatibilitybetween operating
variables, as reported in Ref. 17. Based on these Ž ndings, shape
optimization is conducted for each vane and blade, based on the
(
)
reusable launch vehicle RLV applications.
Presented as Paper 2001-1065 at the AIAA 39th Aerospace Sciences
–
Meeting and Exhibit, Reno, NV, 8 11 January 2001; received 27 March
–
(
)
Navier Stokes computational uid dynamics CFD simulations.
In any optimization process, especially with global optimiza-
tion techniques, the amount and the distribution of data required
is a critical issue. To help address this question, we have investi-
2001; revision received 18 December 2001; accepted for publication 26
December 2001. This material is declared a work of the U.S. Government
and is not subject to copyright protection in the United States. Copies of
this paper may be made for personal or internal use, on condition that the
copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,
222 Rosewood Drive, Danvers, MA 01923; include the code 0748-4658/02
$10.00 in correspondence with the CCC.
(
)
gated the issues related to the design of experiments DOE . Typ-
18
(
)
ically, the face-centered composite design FCCD is employed.
This approach can be quite costly as the number of design vari-
ables increases. Additional criteria can be established to help im-
¤Graduate Student, Department of Aerospace Engineering, Mechanics
and Engineering Science. Student Member AIAA.
prove the efŽ ciency and effectiveness of the construction of the
†Professor and Chair, Department of Aerospace Engineering, Mechanics
and Engineering Science. Fellow AIAA.
19
(
)
RS by employing the concept of orthogonal arrays OA
and
d-optimality.20 These methods will be discussed in the following
sections.
‡Engineer, Space Transportation Directorate. Member AIAA.
§Engineer, Space Transportation Directorate. Associate Fellow AIAA.
509

510
PAPILA ET AL.
Fig. 1 Schematic of the RS based optimization procedure.
Approach
As shownin Fig. 1, the entireoptimizationprocesscan be divided
linearity decreases the computational cost and provides an advan-
tage when the simulations have substantial amount of numerical
noise. However, one can also designa linear NN model. The RBNN
is such an example. In terms of the ability of Ž ltering noise from
experimental data, polynomial-basedRSM has certain advantages
over NNs. However, if the number of neurons used to design the
NN is not the same as the data, then, by deŽ nition, Ž ltering is
also performed by the NN. When it comes to handling complex
functions, NN are more  exible to Ž t complex functions. This ad-
vantage is particularly noticeable if the level of numerical noise is
low.
)
)
intothreeparts:1 datageneration,2 polynomial-orNN-generation
)
phase for establishingan approximation,and 3 optimizer phase. In
the Ž rst phase, the representation of design space is decided. In
(
)
the second phase, polynomials or NN or combined models are
generated with the available training data set. Finally, in the third
phase, the optimizer uses the polynomial or NN approximations
during the search for the optimum until the Ž nal convergedsolution
is obtained.
The optimization technique follows previous works for opti-
mizing  uid machinery, such as diffuser, injector, and airfoil, as
A growing number of papers have been published to combine
(
NN- and polynomial-based RSM approximations for example,
–
–
presented in Refs. 13 15, 17, and 21 23. The NN technique and
the polynomial RSM are integrated to offer enhanced optimization
capabilities by Shyy et al.14 Optimization of a supersonic turbine
for preliminary design, using the polynomial RSM, is presentedby
Papila et al.17 Papila et al.21 investigated the effect of data size and
relativemerits between polynomialand NN-basedRSM in handling
varying data characteristics.Issues related to numerical noises and
the interaction between CFD models and RSM are addressed by
Madsen et al.22 Tucker et al.23 made a Ž rst effort to apply RSM for
injector optimization. In Refs. 13 and 14 a comprehensive update
of the conceptsand applicationsof the global optimizationmethod
is offered, including the mentioned examples.
–
)
Refs. 11 14 and 24, 25 . For example, the work done by Rai and
Madavan,11;24;25 Madavan et al.,12 and Shyy et al.14 suggests that
NN can be effectivelyused to supplementthe existing training data
to help generatea more accuratepolynomial.RBNN may lack satis-
factory Ž ltering propertiesin some cases, as stated by Papila et al.21
and Vaidyanathanet al.26 However, once trained,RBNN can gener-
ate additionaldesigndataeasilyto feedthe polynomial-basedRSM.
This approach is also employed in this work.
RBNN
NN are massively parallel computational systems compri-
sing simple nonlinear processing elements with adjustable
interconnections.27;28 The predictiveability of the network is stored
in the interunit connection strengths called weights obtained by a
processof adaptationto, or learning from, a set of training patterns.
Overview of the RS Literature
Polynomial-based RS techniques are attractive because Ž nding
the polynomial coefŽ cients is a linear regression process. The

PAPILA ET AL.
511
17
Trainingof a networkrequiresrepeatedcyclingthroughthe dataand
continuesuntil the error target is met or until the maximum number
of neurons is reached. This paper focuses on RBNN, which is a
multilayer network with hidden layers of radial-basis function and
a linear output layer. Radial-basis functions RBF are activation
functions for which the response decreases or increases monoton-
ically with distance between the input and the RBF’s center. The
distancebetween two pointsis determinedby the differenceof their
coordinatesand by a set of parameters.29
signiŽ cantly reduce the number of experimental conŽ gurations.
Finally, the d-optimaldesign minimizes the generalizedvarianceof
the estimatesthatis equivalentto maximizingthe determinantof the
moment matrix.18 The d-optimal design approach makes use of the
knowledge of the properties of the polynomial model in selecting
the design points. This criterion tends to emphasize the parameters
with the highest sensitivity.34
(
)
Optimization Procedure
A main advantage of the RBNN approachis the ability to reduce
The focus in the present work is the interplay between the num-
ber of design variables and the predictive capability and input re-
quirement of the RSM. Microsoft Excel SOLVER35 is used as an
optimization toolbox together with the polynomial RS technique
throughoutthe work. SOLVER appliesthe generalizedreducedgra-
dient optimization method to Ž nd the maximum or minimum of a
the computational cost due to the linear nature of RBNN.30 An
( )
h
RBNN is a linear model because the basis functions , in Eq. 1 ,
and any parameters,which they might contain,are Ž xed throughthe
training process:
m
X
36
function with given constraints.
( )
1
f x D
. /
h x
wj j . /
j D 1
Results and Discussion
where w are the coefŽ cients of the linear combinations or weights.
For linearRBNN models,a least-squaresprinciplethatminimizes
Preliminary Design
For preliminary design, we have considered single-, two-, and
three-stage turbines with the number of design variablesincreasing
from 6 to 11 then to 15, in accordance with the number of stages.
Thepreliminarydesigndataareobtainedbyusingthecomputercode
MEANLINE.37 The MEANLINE code performs one-dimensional
analyses that employ loss correlations gleaned from experimental
databases. It predicts performance, calculates gas conditions and
velocity triangles, generates a  owpath elevation, and estimates the
turbopump weight. It also provides an initial spanwise distribution
of rowexitanglesbased on the assumptionof constantaxialvelocity
and conservationof radial momentum.
m
the sum-squared errors can be applied for training, and then un-
known weights can be solved based on the mean square error of the
training set. However, the criterion of the mean square error of the
training set is unlikely to achieve reasonable results for predicting
the unknown input. For this purpose, test data should be used when
selectingthe model. This is the basicform of crossvalidation.29;30 In
this study, the RBNN model is designed by using the functions de-
velopedby Orr29¡31 for MATLAB®:32 Among severalchoicesgiven
by Orr,29 the RBNN method based on ridge regression30 is used be-
cause it can optimize the RBF widths to improve the data Ž tting
capability. To choose between competing models, the maximum
marginallikelihoodmethod31 is used as a model selectioncriterion.
The optimum preliminary design is obtained by using
polynomial-basedRSM coupled with a MEANLINE analysis. The
resultsindicatethatthe two-stageturbinegivesthe bestperformance
based on the combined measures between efŽ ciency and weight.
Polynomial-BasedRS Techniques
(
)
This conŽ guration is adopted for detailed vane stator and blade
The polynomial-basedRSM models the system with assumed or-
derand unknowncoefŽcients.The solutionfor the set of coefŽcients
that best Ž ts the training data is a linear least-squareproblem,18 and
so it is computationally straightforward. The linearity also allows
us to use statisticaltechniquessuch as DOE to Ž nd efŽ cient training
sets. Statisticaltechniquesare also availablefor identifyingpolyno-
mial coefŽ cients that are not well characterized by the data. These
coefŽcients are discardedto improve the predictivecapabilityof the
polynomial. Finally, these statistical tools allow us to estimate the
error of the polynomial-basedmodel at points not used for training,
that is, its prediction error.
(
)
rotor shape optimization.
Detailed Shape Design
In detailed design, the shapes of the vanes and blades are gener-
ated. To generate optimum detailed designs, CFD analysis, RBNN,
and polynomial-basedRSM are used.Ideally,the detailedoptimiza-
tion would be conducted to have both stages, in three dimensions,
optimized simultaneously. However, the sheer number of design
variablesmade this approach impractical. Therefore, Ž rst the mean
airfoil contours of each component are optimized. Airfoil contours
are generatedusinga geometrygeneratordevelopedforthistaskthat
couldread a matrix of designvariables,generateand plotthe airfoil,
and write a preliminary input Ž le for the CFD grid generator.37 Be-
cause a substantialportionof the loss in a supersonicturbinecan be
attributed to interactionsbetween the Ž rst two rows, unsteady CFD
calculationsare performedfor the Ž rststage,runningparametricson
the vane Ž rst with the baseline blade, and then performingthe blade
parametrics with an optimized vane. For detailed design, CFD cal-
culationsare performed using WILDCAT, a parallelized, unsteady,
In this study, the RSs are generated by standard least-squaresre-
gression using JMP, a statistical analysis software with a variety of
statisticalanalyses functions in an interactive manner.33 The global
Ž t and prediction accuracies of the RSs are assessed through sta-
tistical measures such as the coefŽ cient of multiple determination
2
(
)
, its adjusted value to account for the degrees of freedom in the
R
2
(
)
, and the rms-error variation.
Rad j
model
DOE
–
quasi-three-dimensional Navier Stokes code that utilizes moving
To minimize the effect of the noise on the Ž tted polynomial and
NN, and to improve the representationof the designspace, the DOE
procedure can be used to select the data to be used in construction
of the RSs or trainingof the RBNN. There are a number of different
designs of experiment techniques in the literature, as reviewed in
Ref. 34. Issues related to our approach are reviewed in Ref. 17 and
will not be repeated here. In this study, FCCD, OA, and d-optimal
design methods are used to select the effective design space. The
FCCD is widely used for Ž tting second-order RSs.18 It selects de-
signpointsat the cornersof the designspace,centerof the faces,and
C N C
grids to simulate the rotor motion.6;37;38
First Vane
There are seven design variables for detailed design of the Ž rst
(
)
vane. These are leading edge LE pressure side height/axial chord
(
)
H L
L
= , uncoveredturning UT , and Beziercurvecontrolhandles, 1,
L F L R L
L
3 , 3 , 4, and 5. The design variableranges are selected for
the designoptimization;theyare decidedin consultationwith indus-
(
try colleagues with strong expertise in the area see Acknowledg-
the centerof thedesignspace.Therefore,FCCD yields2N
2
1
ment . For this case, FCCD yields 143 possible designpoints. After
)
N
points,where isthenumberofdesignvariables.On theotherhand,
an OA is a fractional factorial matrix that assures a balanced com-
parisonof levelsof any factoror interactionoffactors.19 Becausethe
points are not necessarily at vertices, OA tools can be more robust
than the FCCD. Based on the design of experiment theory, OA can
the addition of six more levels, as shown Fig. 2, 437 design points
are obtained. The d optimality for a reduced cubic model with 40
coefŽcients is used to decrease the data size to half. However, some
of these designs have resulted in negative thickness. Consequently,
203 CFD solutions are generated for RS model construction.

512
PAPILA ET AL.
FCCD in two dimensions
Enhanced FCCD in two dimensions
Fig. 2 Design of experiment for the Ž rst vane.
)
)
b
a
)
c
(
)
)
(
)
Fig. 3 Comparisonof the optimumdesign variables DV for the Ž rst vane: a original design space cycle 1 , where the design variables are bounded
)
(
)
)
(
)
between ¡ 1 and +1, b extended design space cycle 2 , and c NN-enhanced design space cycle 3 .
L
L
Theobjectivefordetaileddesignoptimizationisthestagetotal-to-
total efŽ ciency, ´t–t. The minimum thicknessthat should be greater
or equal to 0.055 is the only design constraint of the Ž rst vane
optimization.
design variables, namely, 4 and 5. As shown in Table 2, 16 new
datapointsare added to the database.The reducedcubicpolynomial
RS is again constructedbased on the extended design space.
In cycle 3, a techniquebased on the NN-enhancedRSM is devel-
oped. The procedures can be summarized as follows:
A three-cycleoptimizationstrategyhas been adopted for the Ž rst
vane. The statistical summary of the RS models resulting from the
three cycles is shown in Table 1. In cycle 1, we have employed the
stepwise regression method to create a reduced cubic polynomial
model, with 40 coefŽ cients, to identify the optimal design points.
The design variables corresponding to the optimized vane shape
are shown in Fig. 3. Note that the optimum design approaches the
boundaryof the designspace.Naturally,it seems fruitfulto consider
expandingthe design space to further improve the design by gener-
ating additionalinformationin an extendedregion. Accordingly,in
cycle 2, the designspace is modiŽ ed by extendingthe rangesof two
)
1 RBNN is constructedfor the subsetof 177 pointsfrom the 203
data pointsfrom cycle 1 and is tested by using the rest of the 26 data
points.
2 The trained RBNN is used to predict 87 additionaldata points
50 OA data points plus 37 previously excluded FCCD data .
)
(
)
These data are added to the existing data set of 219 points used in
cycle 2.
)
3 In cycle 3, with the enhanced data, the reduced cubic poly-
nomial RS is constructed and tested by using the same 26 data
points.

PAPILA ET AL.
513
)
4 The optimum design is determined for the enhanced design
the same performance. This outcome means that other considera-
tions such as stress distribution and manufacturing difŽ culties can
be usefully incorporatedinto the Ž nal selection stage.
space in cycle 3.
The results of the optimal designs from all cycles and the CFD
results for cycles 2 and 3 are summarized in Table 3. Note that
the second and third optimizationcycles converges to the same op-
timal design. Based on the diagnosis, for example, contour plots,
it appears that the RS is quite  at with respect to certain design
variables, for example, 4 and 5, in regions close to the optimal
designs.Consequently,small deviationsin othervariablescan cause
noticeableshifts. However, there are multiple designpoints, includ-
ing the best data point from the CFD runs, which offer essentially
Figure 3 shows the comparison of the optimal design variables
(
)
¡
C
[normalized to 1; 1 range] for reduced cubic models for all
design cycles. Figure 4 shows that the designs convergeto the same
conŽ guration between cycles 2 and 3. On the other hand, as shown
in Fig. 5, there are multiple designpointsthat exhibitessentiallythe
L
L
(
)
same performance within the modeling uncertainty , albeit with
much different geometries. To help better understand these obser-
vations,we assess the sensitivityof the two geometricvariables
(
L
4
)
L
and 5 in Fig. 6, which shows that the performance of the vane is
L
very insensitive to 5. The two shapes shown in Fig. 5 are mainly
Table 1 Quality the polynomialRS models for the original
L
caused by 5, which explains why two very different vane shapes
give essentiallythe same performance.From the structural point of
(
)
(
)
(
)
cycle 1 , extended cycle 2 and NN-enhanced cycle 3
design spaces for the Ž rst vanea
L
view, the value of 5 to producea thickerproŽ le is preferable.This
Reduced cubic
Reduced cubic
Reduced cubic
example demonstrates that the global optimization technique en-
ables one to compare designsbased on insightinto the entire design
space, which allows a suitable selection to be made with additional
factors considered.
Figure 7 shows the absolute Mach contours of the two vanes
shown in Fig. 5. From Figs. 7a and 7b, observe that the Ž rst vane
shape is optimized to a contour with the airfoil throat at the trailing
edge and with no concave curvatureon the suction side as opposed
model, 203 data model, 219 data model, 306 data
(
)
(
)
(
)
cycle 3
Head
cycle 1
cycle 2
R2
0.8
0.8
1.10
203
0.8
0.8
1.10
219
0.7
0.7
1.30
306
Ra2d j
rms-error, %
Observations
(
)
or sum weights
a Mean of response is normalized by the baseline ´t–
:
t
(
)
Table 3 Optimum designs for the original cycle 1 , extended
a
(
)
(
)
cycle 2 , and NN-enhanced cycle 3 design spaces for the Ž rst vane
(
)
(
)
Table 2 Comparison of the original cycle 1 , extended cycle 2 , and
Original design Extended design NN-enhanced design
(
)
NN-enhanced cycle 3 design spaces for the Ž rst vane
Cycle 1 Cycle 2 Cycle 3
Lower Upper Lower Upper Lower Upper
(
)
(
)
(
)
space cycle 1
space cycle 2
space cycle 3
Variable
203 data
219 data
306 data
H=L
UT
L1
L3F
L3R
L4
0.791
0.2
2
0.791
0.2
2
0.791
0.2
2
Design variable
limit
limit
limit
limit
limit
limit
H=L
UT
L1
L3F
L3R
L4
0.79
1.19
0.79
1.19
0.79
1.19
0.441
2
1.25
2
0.441
2
1.35
2.5
0.441
2
1.35
2.5
¡1.20 ¡0.20 ¡1.20 ¡0.20 ¡1.20 ¡0.20
0.36
0.44
0.25
0.05
0.13
2.00
1.76
2.00
1.25
2.00
0.36
0.44
0.25
0.01
0.01
2.00
1.76
2.00
1.35
2.50
0.36
0.44
0.25
0.01
0.01
2.00
1.76
2.00
1.35
2.50
L5
´
1.042
N/A
1.044
1.031
1.043
1.031
t–t RSM
´
t–t CFD
L5
aAll design variables and ´t– are normalized by the baseline values.
t
)
)
b
a
)
(
Fig. 4 Summary of the three optimization cycles for the Ž rst vane: a optimal cross sections optimal designs of cycles 2 and 3 coincide because
)
)
(
they converge to the same conŽ guration and b optimum design variables where ¡ 1 and +1 show the minimum and maximum values of the design
)
variables in the original design space, respectively .

514
PAPILA ET AL.
( )
L L
4, 5, and 7; channel convergence ratio CCR ; and the cir-
L
to the more conventional supersonic vane. The optimized contour
results in continued expansion and accelerationon the suction side
downstreamof the throat.This additionalaccelerationappearsto re-
duce undesirablevane-to-bladeinteractionby attenuating the blade
leading-edgeshock sweeping across the aft portion of the vane suc-
tion side.39
(
)
cle factor CF . The ranges of the design variables are determined
in consultation with industry colleagues with strong expertise in
(
)
the area see Acknowledgment . Again, the stage total-to-totalefŽ -
ciency ´t–t is selected as the objectivefunction to be maximized for
the blade design. For the Ž rst blade, three different constraintsneed
be satisŽ ed:
First Blade
)
1 The minimum thicknessshould be greater or equal to trailing-
edge diameter.
There are 11 designvariablesfor detaileddesignof the Ž rst blade.
H L
These are LE pressure side height/axial chord = ; LE metal angle
)
)
.1:9 CF should be smaller than 2.7 because if
L C L F C
£
2
1
2
this parameter is more than 2.7, the airfoil develops a kink in the
suction side of the LE.
L
L F L R
¯1; UT; six different Bezier curve control handles, 1, 2 , 3 ,
)
3 The suction side radius of curvature must be greater than Ž ve
times that of the trailing-edge radius. This also helps to eliminate
unrealistic airfoil shapes such as those with kinks.
For this case, a Ž ve-level OA-based procedure has been adopted
to select 250 desgin points for the CFD code to generate solutions.
A software tool developed by Owen19 is used for constructing OA
designs. Because these points are located at either the edge or the
center of the design space, additional 23 FCDD data points at the
center of the faces are added. Finally, 38 data points along the main
diagonal of the design space is generated. In short, 311 possible
design points are selected to generate the CFD solutions. However,
only 139 out of the 250 OA selected data points, 21 out of the 23
FCCD selecteddatapoints,and 17 out of the 38 diagonal-baseddata
pointsare obtainedfrom the CFD tool. Therefore,only 177 possible
design points, out of the 311 total, have been collected. The cases
are not completed because the airfoil shapes are unrealistic, which
cause either excessively high losses, or prevent the grid generation
processfrom runningappropriately.These nonphysicalcases create
holes in the design space, which cause difŽ culties in constructing
RSs. Figure8 shows the distributionof theseholeswithinthe design
space. As one can see from Fig. 8, the difŽ culty is along the diag-
onal ends, as well as on the edges of the design space. With only
177 data points generated for 11 design variables, RBNN is used
to produce additional information, as described in the following
paragraphs.
Fig. 5 Multiple possibili-
ties for the Ž rst vane; both
designs offer comparable
efŽ ciency, but the thicker
proŽ le is preferred from
structural considerations.
Fig. 6 EfŽ ciency contours for 203 data of cycle 1, showing that L5 has little impact on the performance of Ž rst vane.

PAPILA ET AL.
515
)
)
b Optimum conŽ guration
a Baseline conŽ guration
Fig. 7 Instantaneous absolute Mach number contours for two Ž rst vane designs of essentially equal performances.
Fig. 8 Summary of the cases for both successful and unsuccessful CFD simulations for the Ž rst blade.
( )
which 17 are from the CFD runs originallyselectedas the test data
RBNN is trained based on the 160 data points from the men-
tioned177CFD-generateddata points,with the remaining17 points
reserved to tune the conŽ guration of the NNs.
The trained RBNN is used to generate additional 2204 design
points. As shown in Fig. 9, 2070 of them are selected using FCCD
datamodiŽ ed,and the remaining134of themare the casesforwhich
CFD code does not converge.
and 21 are from the NN-enhanced data. The statistical summary
for both models, with and without NN-enhanced data, is shown in
Table 4.
In the course of performing the optimization task, it is realized
that out of the 11 design variables, 5 of them, responsible for the
shape in the rear portion of the upper blade surface, can be opti-
mized Ž rst because various plausibleshapes convergeto the similar
values of them. Accordingly,one can reduce the design space from
Full quadratic and reduced cubic RS models are constructed for
the Ž rst blade.Atotalof38datapointsareusedto test bothmodelsof

516
PAPILA ET AL.
Table 4 Quality of the polynomialRS model for the Ž rst blade
Quadratic model
Reduced cubic
(
without NN-data Quadratic model
model NN-
(
)
(
)
plain CFD data ,
NN-enhanced
enhanced data ,
cycle 3
)
Summary of Ž t
cycle 1
data , cycle 2
R2
0.78
0.57
0.87
0.86
0.97
0.97
R2Ad j
rms-error, %
Observations or
sum weights
3.82
160
1.74
2343
0.81
2343
(
)
Table 5 Optimum cases for the
Ž rst blade obtained by reduced
cubic model with NN-enhanced
data for modiŽ ed variable rangesa
Variable
Blade-1
Blade-2
´
´
H=L
¯
UT
L1
L2F
L3R
L4
L5
L7
1.04
1.05
¡25
0.94
0.63
0.59
1.48
1.55
0.68
1.78
0.66
1.06
3.78
1.07
1.04
¡25
0.99
0.63
0.67
1.36
1.6
t–t RSM
t–t CFD
Blade 1
1
0.68
1.44
0.76
1.06
3.78
CCR
CF
a All design variables and ´t– are normal-
t
ized by the baseline values.
Blade 2
Fig. 10 Optimum shapes for the Ž rst blade obtained by reduced cubic
model with NN-enhanced data for modiŽ ed design variable ranges.
Conclusions
A methodology to improve the supersonic turbine performance
has been developed for preliminary and detailed design. A global
optimizationframework combiningthe RBNN and the polynomial-
based RSM is constructed.Based on the optimized preliminary de-
sign, shape optimization is performed for the Ž rst vane and blade
of an optimized two-stage supersonic turbine. The results obtained
can be summarized as follows.
Fig. 9 Schematic of the additional data generated for the Ž rst blade
in two dimensions: , is original FCCD and , is modiŽ ed FCCD data
to be added by NN.
)
1 The design of experiment approach is critical in reducing the
data size needed by the optimization task.
11 to 6 variables. Two designs are presented with the Ž ve Ž xed de-
sign variables and different six variables, as shown in Table 5 and
Fig. 10. With six design variables Ž xed, the two blade shapes yield
very comparable performance. Their shapes in the upper rear por-
tion are, as expected, virtually the same. On the other hand, their
front portionsare noticeablydifferent,with blade 1 exhibitingmore
variationsthanblade2. Again, with the insightofferedby the global
model, we have an opportunity to evaluate not only the sensitivity
of each design variable, but also additional tradeoffsbetween com-
parable designs.
)
2 A major merit of the global optimization approach is that it
enablesone to reviseadaptivelythedesignspaceto performmultiple
optimization cycles.
)
3 As evidenced by the three optimization cycles conducted for
the Ž rstvane,capabilitiesto reŽ ne adaptivelytheoptimizationscope
and procedure can be critical when an optimal design approaches
the boundary of a predeŽ ned design space. For the Ž rst blade, the
optimization process indicates that 5 out of the 11 design variables
can be chosen Ž rst, leaving the design space to reduce to have 6

PAPILA ET AL.
517
variables. Based on the reduced set of design variables, the opti-
mization task can be handled more effectively.
4 In the present study, the design variables and their ranges are
10Burman, J., Gebart, B., and Martensson, H., “Development of a Blade
Geometry DeŽ nition with Implicit Design Variables,” AIAA 2000-0671,
Jan. 2000.
)
11Rai, M. M., and Madavan, N. K., “Improving the Unsteady Aerody-
namic Performance of Transonic Turbines Using Neural Networks,” AIAA
2000-0169, Jan. 2000.
selected in consultation with industry colleagues well experienced
in the Ž eld. It is found that a number of cases create airfoil shapes
thatare unrealistic.Consequently,theyeithercauseexcessivelyhigh
losses, or prevent the grid generation process from running appro-
priately. These nonphysical cases create holes in the design space,
which cause difŽ culties in constructing RSs. The combined NN,
RS techniques,and multiple optimizationcycles help address these
fundamentalbarriers.
12Madavan, N. K., Rai, M. M., and Huber, F. W., “Neural Net-Based
Redesign of Transonic Turbines for Improved Unsteady Aerodynamic Per-
formance,” AIAA 99-2522, June 1999.
13Shyy, W., Papila, N., Vaidyanathan, R., and Tucker, P. K., “Global
Design Optimization for Aerodynamics and Rocket Propulsion Compo-
–
nents,” Progress in Aerospace Sciences, Vol. 37, No. 1, 2001, pp. 59
118.
)
5 By inspecting the in uence of each design variable, one can
14Shyy, W., Tucker, P. K., and Vaidyanathan, R., “Response Surface
and Neural Network Techniques for Rocket Engine Injector Optimization,”
AIAA Paper 99-2455, June 1999.
also gain insight into the existence of multiple design choices and
select the optimum design based on other factors such as stress and
materials considerations.
15Shyy, W., Papila, N., Tucker, P. K., Vaidyanathan, R., and GrifŽ n, L.,
“Global Design Optimization for Fluid Machinery Applications,” Proceed-
ing of the Second International Symposium on Fluid Machinery and Fluid
)
6 NN-enhancedRSM helpsto improve the accuracyof the RSM,
useful for smoothing out the designspace, and allows the optimiza-
tion task to be conducted with smaller number of CFD runs, as
illustrated in Ž rst blade example.
–
Engineering, 2000, pp. 1 10.
16Sobieszczanski-Sobieski, J., and Haftka, R. T., “Multidisciplinary
Note that the optimal shapes involve a substantialnumber of de-
signvariablesat thelimit ofthe assigneddesignspace.This situation
did not happen because of the lack of experience;the design space
was deŽ ned in direct consultation with the colleagues from rocket
propulsion industry see Acknowledgment . For new technologies,
it is not unusual that our initial judgment needs to be reŽ ned, as
is the case here. In such situations, the capability of assessing the
characteristicsof the entire designspace is criticallyimportant.The
otherside of the coin is that, to performsuchoptimizationtaskswith
conŽ dence, one needs to be able to estimate the Ž delity of the RS
model. In addition to the information presented here, the relevant
issues have been addressed in Refs. 22, 40, and 41, which may be
consulted.
Aerospace Design Optimization: Survey of Recent Developments,” Struc-
–
tural Optimization, Vol. 14, No. 1, 1997, pp. 1 23.
17Papila, N., Shyy, W., GrifŽ n, L., Huber, F., and Tran, K., “Preliminary
DesignOptimizationfora SupersonicTurbineforRocketPropulsion,”AIAA
Paper 2000-3242, July 2000.
(
)
18Myers, R. H., and Montgomery, D. C., Response Surface
Methodology—Process and Product Optimization Using Designed Experi-
–
ments, Wiley, New York, 1995, pp. 208 279.
19Owen, A., “Orthogonal Arrays for: Computer Experiments, Integra-
–
tion and Visualization,” Statistica Sinica, Vol. 2, No. 2, 1992, pp. 439
452.
20Unal, R., Lepsch,R. A., and McMillin,M. L., “ResponseSurface Model
Building and Multidisciplinary Optimization Using D-Optimal Designs,”
AIAA Paper 98-4759, Sept. 1998.
21Papila, N., Shyy, W., Fitz-Coy, N., and Haftka, R. T., “Assessment of
Neural Net and Polynomial-Based Techniques for Aerodynamic Applica-
tions,” AIAA Paper 99-3167, June 1999.
Acknowledgments
22Madsen, J. I., Shyy, W., and Haftka, R. T, “Response Surface Tech-
niques for Diffuser Shape Optimization,” AIAA Journal, Vol. 38, No. 9,
This work has been supported by NASA Marshall Space Flight
Center. We appreciate the useful discussions with F. Huber, River-
–
2000, pp. 1512 1518.
(
bend Design Services, Palm Beach Gardens, Florida private com-
23Tucker, P. K., Shyy, W., and Sloan, J. G., “An Integrated Design/
Optimization Methodology for Rocket Engine Injectors,” AIAA Paper 98-
3513, July 1998.
)
munication, February 2000 and K. Tran, The Boeing Company,
(
Rocketdyne, Canoga Park, California private communication,
24Rai, M. M., and Madavan, N. K., “Application of ArtiŽ cial Neural
Networks to the Design of Turbomachinery Airfoils,” AIAA Paper 1998-
1003, Jan. 1998.
)
February 2000 .
25Rai, M. M., and Madavan, N. K., “Aerodynamic Design Using Neural
Networks,” AIAA Paper 98-4928, Aug. 1998.
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