NASA Technical Reports Server (NTRS) 20030007917: Aerodynamic Shape Optimization Using Hybridized Differential Evolution

8 Pages·0.57 MB·English

by  NASA Technical Reports Server (NTRS)

Extended abstract of a paper submitted forreview toward presentation atthe AIAA 21st Applied Aerodynamics
Conference, Orlando, FL,June 23-26, 2003.
AERODYNAMIC SHAPE OPTIMIZATION USING HYBRIDIZED
DIFFERENTIAL EVOLUTION
Nateri K. Madavan*
NASA Ames Research Center, Moffett Field, California
ABSTRACT niques that are based on nontraditional approaches such
as evolutionary algorithms and neural networks. Various
An aerodynamic shape optimization method that uses approaces based on neural networks, 3'4 neural networks
an evolutionary algorithm known at Differential Evolu- in conjunction with response surfaces, 5'6'7'8'9 genetic
tion (DE) in conjuction with various hybridization strat- algorithms, 1°'11'12 and genetic algorithms in conjunc-
egies is described. DE is a simple and robust tion with neural networks, 13"14among others, have been
evolutionary strategy that has been proven effective in reported in the literature. These techniques offer several
determining the global optimum for several difficult advantages over traditional optimization methods. The
optimization problems. Various hybridization strategies references cited above primarily deal with airfoil shape
for DE are explored, including the use of neural net- optimization for turbomachinery applications. A com-
works as well as traditional local search methods. A
plete review of the literature and the comparative merits
Navier-Stokes solver is used to evaluate the various of all these different optimization techniques is beyond
intermediate designs and provide inputs to the hybrid the scope of this article. The reader is refered to the ref-
DE optimizer. The method is implemented on distrib- erences cited as the starting point for a more exhaustive
uted parallel computers so that new designs can be literature search.
obtained within reasonable turnaround times. Results
This paper deals with an aerodynamic shape optimi-
are presented for the inverse design of a turbine airfoil
zation method that uses an evolutionary algorithm
from a modern jet engine. (The final paper will include
known at Differential Evolution. 15DE is a simple and
at least one other aerodynamic design application). The
robust evolutionary strategy that has been proven effec-
capability of the method to search large design spaces
tive in determining the global o_timum for several diffi-
and obtain the optimal airfoils in an automatic fashion is
cult optimization problems. 16 Its application in
demonstrated.
aeronautics, however, has been rather limited. It has
INTRODUCTION been used in the predictive control of aircraft dynam-
ics. 17It has been used in conjunction with a potential
Remarkable progress has been made in recent years in flow solver in the inverse design of turbomachinery air-
the ability to design airfoil shapes that are optimal with foils; 18the same authors have also presented a hybrid-
regard to certain desired characteristics. This progress ized version 19 that combines DE with a local direct
has been achieved by combining improved methods for simplex search method to minimize the number of
the simulation of complicated flow fields with efficient objective function evaluations using the potential flow
numerical optimization techniques and by harnessing solver.
the powerful capabilities of modem computers. Both
While population-based approaches such as DE are
steady and unsteady Navier-Stokes and Euler solvers
robust and capable of locating the global optimum in
have been combined with various traditional optimiza-
difficult objective function landscapes, they often suffer
tion techniques (gradient-based methods, 1'2 response
from slow convergence once in the vicinity of the global
surfaces, etc.) to obtain optimal airfoil designs.
optimum and require many objective function evalua-
More recently, there has been considerable interest in tions. This limits their applicability in aerodynamic
the development of airfoil shape optimization tech- design optimization where the objective function evalu-
ations typically are obtained from high-fidelity simula-
Copyright©2002bytheAmerican InstituteofAeronauticsand tions (e.g., Navier-Stokes simulations) that are
Astronautics,Inc.NocopyrightisassertedintheUnitedStatesunder computationally expensive. This paper deals with
Title17,U.S.Code.TheU.S.Government hasaroyalty-freelicense hybridization strategies that use rapidly convergent, but
toexerciseallrightsunderthecopyrightclaimedhereinforGovern- less robust, local optimization methods in conjunction
mentpurposes.Allotherrightsarereservedbythecopyrightowner. with the global and robust DE algorithm to effectively
*ResearchScientist,NASAAdvancedSupercomputingDivision. reduce overall computing time requirements in aerody-
SeniorMember,AIAA. namic shape optimization
1
American Institute of Aeronautics and Astronautics
VariouhsybridizatiosntrategiefosrDEareexplored,
tation where new trial parameter vectors {Yl ..... Ym}
includingtheuseofneuranl etworkasswellastradi-
for the next generation g + 1 are generated according to
tionallocalsearcmhethodAs,Navier-Stokseoslveirs
the following mutation scheme:
usedtoevaluattehevariousintermediadteesignasnd
For l = 1,m ; i = 1,n generate
provideinputstothehybridDEoptimizeOr.nehybrid-
izationstrategtyhatisexploreudsesaneuranletwork Yl = x l(g)+F, x (g)-x (g)
thatis trainedonthecomputationfaluliddynamics
1 l 1
(CFD)simulatiodnataandactsasasurrogatfeorthe
In the above cz1, cz2, c_3 are distinct elements of
Navier-Stokseoslverintheobjectivefunctionevalua-
{1,2 ..... m} randomly selected for each l, and
tions.Otherstrategieinscorporavteariouslocalsearch
F e [0,2] is a parameter that controls the amplification
methodasndwillbedescribeindthefinalpaperT.he of the differential variation. Other variants that either
methodisimplementeodndistributepdaralleclomput-
use the difference between more than two parameter
erssothatnewdesigncsanbeobtainewdithinreason-
vectors or keep track of the best parameter vector at
ableturnarountdimes.Resultasrepresentefodrthe
each generation and use it in the mutation scheme have
inversdeesignofa
turbine airfoil from a modern jet also been developed 15and used with varying success in
engine. (The final paper will include at least one other specific applications.
aerodynamic design application). The capability of the DE is similar to other recombinative ES approaches
method to search large design spaces and obtain the in that it also uses discrete rec'ombination. The strategy
optimal airfoils in an automatic fashion is demonstrated.
adopted in differential evolution is to modify the trial
DESIGN OPTIMIZATION METHOD parameter vectors {Yl ..... Ym} to generate parameter
vectors {Zl..... Z.m} as follows:
Differential Evolution Forl = 1,m;i = 1,n generate
Differential Evolution is an evolutionary strategy
i y! with probability Pc
(ES) developed for single-objective optimization in con-
zl
tinuous search spaces. It is conceptually simple and pos- xi(g ) with probability 1- Pc
sesses good convergence properties that have been
demonstrated in a variety of applications. 15Details of where Pc is a parameter that controls the proportion
the algorithm can be found elsewhere; 15'20only its main of perturbed elements in the new population. Note that
features are summarized here. the mutation and recombination operations described
above can lead to new vectors that may fall outside the
The approach uses a population P that contains m n-
boundaries of the variables. Various repair rules can be
dimensional real-valued parameter vectors, where n is
used to ensure that these inadmissible vectors do not
the number of parameters or decision variables:
enter the population. A simple strategy, which is the one
P = {_l..... L.}
adopted here, is to delete these inadmissible vectors and
The population is usually initialized at generation form new ones until the population is filled.
g = 0 in arandom fashion: The selection scheme used in DE is deterministic but
P(0) = IX1(0) ..... Xm(0)}, g = 0 differs from methods usually employed in standard ES
The population size m is maintained constant approaches. Selection is based on local competition
throughout the optimization process. Differential evolu- only, with the modified trial parameter vector competing
tion is thus similar to a (It, _,) ES21with bt and L equal against one population member (the comparison vector)
to m. 22 The method however differs from standard ES and the survivor entering the new population g + 1 as
follows:
approaches in several respects as described below.
As with all ES-based approaches, mutation is the key For l = 1,m
ingredient of differential evolution. The basic idea is to
generate new parameter vectors for the subsequent gen-
eration by using weighted differences between two (or Ycl(g + 1) = x2Il(g) eilfse f(zl) <f(xl(g))
more) parameter vectors selected randomly from the
current population to provide appropriately scaled per- where f denotes the corresponding objective function
turbations that modify another parameter vector (or, (or fitness) value. This greedy selection criterion results
comparison vector) selected from the same population. in fast convergence; the adaptive nature of the mutation
This idea has been implemented in various forms but the operator, in general, helps safeguard against premature
form discussed and used here is the classical implemen- convergence and allows the process to extricate itself
from local optima. The generation counter is incre-
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menteadndtheprocesissrepeateudntilsomestopping tries with a common set of geometrical parameters is
criteriaaresatisfied.
essential. Variations of the airfoil geometry can be
obtained then by smoothly varying these parameters.
Hybridization Strategies
Geometrical constraints imposed for various reasons,
such as structural, aerodynamic (e.g., to eliminate flow
1. Using Neural Networks
separation), etc., should be included in this parametric
While the DE algorithm is quite efficient and effective
representation as much as possible. Additionally, the
in exploring the entire design space in search of the opti-
smallest number of parameters should be used to repre-
mal solution, the algorithm has atendency to slow down
sent the family of airfoils.
as it approaches the vicinity of the optimal solution.
The airfoil geometry parametrization method
Many subsequent function evaluations are then required
described in Rai and Madavan 7 is used here. A total of
to obtain the exact optimum. The computational effort
13 geometric parameters were used to define the airfoil
required by these function evaluations in the vicinity of
geometry in the current study. These parameters are
the optimum can be nearly eliminated using a hybrid
listed below (see Fig. 2for illustration):
approach that combines the DE algorithm with a neural
network that is trained on the CFD simulation data. 23 1.Leading edge and trailing edge airfoil metal angles
The trained neural network is then used to evaluate the (2 parameters).
objective function with negligible computational effort 2.Eccentricity of upper leading edge ellipse (1 param-
and expense instead of using the CFD solver. While eter).
hybridization is always useful, itmust be done with cau-
3. Angles defining the extent of the leading edge
tion. Here the neural network is used only in the latter
ellipses (2 parameters).
stages after the entire population has evolved to the gen-
4. Semi-minor axes values at the leading edge (2
eral vicinity of the optimal solution. Thus the neural net-
parameters).
work is used as a "local" response surface with validity
only in asmall region of the design space. This makes it 5.Angles defining the extent of the trailing edge circle
easier to train the neural network and improves its gen- (2 parameters).
eralization abilities. For the inverse shape optimization 6. Airfoil y-coordinate values at about 50% chord on
problem, the neural network is trained on the sum-of- the upper and lower surfaces (2 parameters).
squares error between the actual pressure computed by
7. Airfoil y-coordinate values at about 75% chord on
the CFD solver and the target pressure at various points
the upper surface (1 parameter).
on the airfoil. A three-layer feed-forward neural net-
8. Airfoil stagger angle (1 parameter).
work as shown in Fig. 1is used here.
This method of generating the airfoil surface provided
2. Using Local Search Methods the necessary variations in airfoil geometry required by
the optimization procedure.
The final paper will include results using hybridization
strategies based on local search methods. CFD Simulation Methodology
A Navier-Stokes solver was used to perform the flow
Constraint Handling
simulations (direct function evaluations) that serve as
For use in aerodynamic design we have incorporated
inputs to the optimization process. The solver used is a
an efficient constraint handling mechanism into the DE modified version of the ROTOR-2 computer code 24and
algorithm. It is a parameter-less approach that helps
solves the two-dimensional, Navier-Stokes equations
steer the algorithm away from infeasible regions of the
around a single airfoil in a cascade (with spanwise peri-
design space. Physical constraints (e.g: maximum air-
odic boundary conditions) for a given set of inlet and
foil thickness) are imposed, as well as aerodynamic con-
exit conditions. Multiple grids are used to discretize the
straints (e.g. wavy surfaces). Airfoil geometries that do
flow domain; an inner "0" grid that contains the airfoil
not violate the constraints but for which the CFD solver
and an outer "H" grid that conforms to the external
fails to converge are also deemed infeasible and treated
boundaries as shown in Fig. 3.
accordingly by the constraint handling mechanism.
Figure 3also shows the grid system used to discretize
Airfoil Geometry Parametrization the flow domain. Each airfoil has two grids associated
with it: an inner "O" grid that contains the airfoil and an
Geometry parameterization and prudent selection of
outer "H" grid that conforms to the external boundaries.
design variables are critical aspects of any shape optimi-
For the analyses performed here, each inner O grid has
zation procedure. Since this study focuses on airfoil
151 points in the circumferential direction and 41 points
redesign, the ability to represent various airfoil geome-
3
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inthewail-normadlirectionE.achouterHgridhas101
engine. Several flow and geometry parameters were
pointsintheaxialdirectioannd41points
in the trans- also supplied and used in the design process. The design
verse direction. For the sake of clarity, only some of the objective function was formulated as the equally-
grid points are shown in Fig. 3. weighted sum-of-squares error between the target and
The dependent variables are initialized to freestream actual pressure obtained during the optimization process
at 45 locations on the airfoil. Some of these results have
values and the equations of motion are then integrated
subject to the boundary conditions. The flow parameters been presented earlier. 23
that are specified are the pressure ratio across the turbine The initial design space was chosen to be quite large
(ratio of exit static pressure to inlet total pressure), inlet to allow a wide range of airfoil shapes to be explored. A
temperature and flow angle, flow coefficient, and unit sampling of some of the initial airfoil geometries is
Reynolds number based on inlet conditions. shown in Fig. 4. In order to hold the CFD function eval-
uations to a reasonable number, 6 (instead of 13) design
Design Objective Formulation
variables were used in the initial stages of the design.
Various design objectives can be incorporated in the The population of 50 members were then evolved using
current procedure depending on the optimization prob- the DE algorithm.
lem being solved. For the inverse turbine airfoil design The DE algorithm without any hybridization strategy
shown here, the design objective function was formu-
was considered first. Figure 5 shows the pressure distri-
lated The design objective function was formulated as
bution for the optimal airfoil that represents the best air-
the equally-weighted sum-of-squares error between the
foil obtained after 13 generations using the DE
target and actual pressure obtained during the optimiza-
optimization method with 6 design parameters. The
tion process at various locations on the airfoil.
algorithm is able to approach the target distribution
within about 650 function evaluations. Note that in the
Implementation on Distributed Parallel
early stages of evolution several of the airfoil geometries
Computers
were infeasible and hence were not evaluated by the
In order to reduce overall design time, the procedure CFD solver, After 13 generations, the number of design
has been implemented on a distributed parallel com- variables is increased to 13 and the population evolved
puter. The results in this article were obtained on the further. The optimal pressure distribution shown in
SGI Origin 3000 and the Cray SV1 distributed parallel Fig. 5 was obtained after an additional 13 generations
computers at NASA Ames Research Center. Parallel and agrees well with the target distribution. The airfoil
implementation of the method is quite straightforward geometries corresponding to the optimal designs with
and relies on the simultaneous computation of multiple, both 6 and 13design parameters are shown in Fig. 6.
independent aerodynamic simulations on separate pro-
The convergence history of the baseline DE
cessors. A script-based procedure is used that invokes a
algorithm 23is shown in Fig. 7 which plots the popula-
variable number of processors depending on processor
tion mean and variance as a function of the number of
availability and the size of the population used. The
generations. Both the mean and the variance decrease
number of processors can also be adjusted as the design
with generation number except for the slight increase
proceeds. The current setup is based on a "master-slave"
corresponding to the switch from 6to 13 design parame-
arrangement, with the master handling the tasks of set-
ters. The figure also shows that convergence of the
ting up the simulations, neural network training for the
algorithm is slower in the vicinity of the optimal solu-
hybrid method, and farming out of the aerodynamic
tion. This is typical of many other evolutionary algo-
computations to the other "slave" processors. Since the
rithms and highlights the need for a hybrid approach
aerodynamic computations are independent of each
that combines the global search and exploration capabil-
other, no communication between the processors is
ities of the DE algorithm with other algorithms that can
required until the computations are completed. The
converge rapidly to an optimum when initialized in the
slave processors then communicate their results to the
local neighborhood. As described earlier, a different
master which then performs the necessary calculations
hybrid approach to reducing overall design time was
to determine the members of the next population.
adopted here in the DE-NN method where the popula-
RESULTS tions after the first few generations (using 13 design
parameters) were used to train a neural network.
The design method was used in the inverse design of a Figure 8shows the envelope of pressure data around the
turbine airfoil with a specified pressure distribution. The target pressure distribution that was used to train the
target pressure distribution was obtained at the midspan neural network. Note, however, that the network was
of a turbine vane from a modern Pratt and Whitney jet trained directly on the sum-square-error and not on the
4
American Institute of Aeronautics and Astronautics
individuaplressurdeata.Thedataenvelopien Fig.8 ASME/SAE/ASEE Joint Propulsion Conference and
representhtsevariatioonfthepressurdeistributionfosr Exhibit, Los Angeles, CA, Jan. 20-24, 1999.
therangeofairfoilgeometrieinstheneuranl etwork
9Rai, M. M., Madavan, N. K., and Huber, E W.,
trainingsetandismeanttoconvetyhelocalnatureof
"Improving the Unsteady Aerodynamic Performance of
theneuranletworrkesponsseurfacienthevicinityofthe
Transonic Turbines Using Neural Networks," AIAA
optimaslolutionT.hepressurdeistributiofnortheopti-
Paper No. 2000-0169, 38thAerospace Sciences Meeting
malairfoildesignobtainebdytheDE-NNapproacish
and Exhibit, Reno, NV, Jan. 10-13, 2000.
showninFig.9andcomparewsellwiththetargeatnd
l°Obayashi, S., and Takanashi, S., "Genetic Optimi-
optimaDlEdesignpressudreistributions.
zation of Target Pressure Distributions for Inverse
FINALPAPER Design Methods," AIAA Journal, Vol. 34, No. 5, pp.
881-886, 1996.
Somepreliminaryresultsusingonehybridization
llDennis, B. H., Egorov, I. N., Han, Z.-X., Dulikrav-
strategfyortheDEalgorithmarepresenteTdh.efinal
ich, G. S., and Poloni, C., "Multi-Objective Optimiza-
paperwill includeresultsusingalternativsetrategies.
tion of Turbomachinery Cascades for Minimum Loss,
Additionadlemonstraticoonmputatioinllsustratintghe
Maximum Loading, and Maximum Gap-to-Chord
useofthemethoidndirec(tnotinversed)esignarealso
Ratio," AIAA Paper No. 2000-4876, Sept. 2000.
underway.
lZQuagliarella, D., and Cioppa, A. D., "Genetic Algo-
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24Rai, M. M., and Madavan, N. K., "Multi-Airfoil
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Figure 1. Schematic of the three-layer feed-forward
neural network used in the hybrid DE-NN method.
o
8.
o
o
o, I I I I I I
'T,
-0.25 O. 0 0.75 1.00 1.25
000 0.25 x./Sc
Figure 2. Schematic of a generic airfoil showing
location of control points on the airfoil surface and the
defining angles used in the parametrization of the airfoil
geometry.
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American Institute of Aeronautics and Astronautics
1.0
I I I
i [ !![!!!:-_, irfoil
I I I IIlll_
f f f f ff_f;_
1 I t Iltft:_
o
5
i i i iiiiii!!_
i ] i i i IIIEII]II'_I
Outer H-Grid
Inner O-
0.0 0.2 0.5 0.8 1.0
_C
Figure 3. Representative turbine airfoil geometry and
computational grid used in the CFD simulations.
Figure 5. Airfoil pressure loading for the optimal
design.
i [ 0.2 1 I i I I I
0.0-
,-- x.
-0.2 \ ".,
_-0.4 \ "" "" "_"_
-0.6-
1 -0.8-
Optimal 6Parameter Design (DE)
: -1.0 I I I I I I
0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0
X/C X/C
Figure 4. Sampling of initial airfoil geometries. Figure 6. Airfoil geometry for the optimal design.
7
American Institute of Aeronautics and Astronautics
I I
1I°°
U. 2 switchto13
,_I O"
.__.
0
10-_
-- Optima/Design (DE-NN) |
• OptimalDesign(DE) l
• P&WTargetDesign /
104 I I I I I , I I I /
5 10 15 20 25 3O 0.0 0.2 0.5 0.8 1.0
Generation Number #c
Figure 7. Convergence history for the baseline DE Figure 9. Airfoil pressure loading for the optimal design
algorithm (no hybridization strategy included). using the hybrid DE-NN approach.
1.0 I I
• P&WTarget Design
_-:_!_-!3! EnvelopeofNeural
',;_;_.;L_._NetwonkTrainingData
I I I I
0.0 0.2 0.4 0.6 0.8 1.0
x/c
Figure 8. Envelope of airfoil loadings used to train the
neural network for the hybrid DE-NN algorithm.
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