MDO is an important tool for designers and its use is spreading out as new implemen-tations establishes. The focus of this methodology is to bring together disciplines involvedwith the design to work all their variables concomitantly, at an optimization environmentto obtain better solutions. It is possible to use MDO in any stage of the design process,that is in the conceptual, preliminary or detailed design, as long as the numerical modelsare �tted to needs of each of this stages. This work describes the development of a MDOcode for the conceptual design of aeroelastic aircraft wings. As a tool for the designer at theconceptual stage, the numerical models must be fairly accurate and fast. The aim of thisstudy is to analyse the use of a neural network based metamodel for the utter predictionof aircraft wings in the MDO code, instead of a conventional model itself, what may a�ectsigni�cantly the computational cost of the optimization. Well trained neural networks areable to provide accurate results within short time, making them very useful for the typeof the proposed methodology. The neural metamodel is prepared using aeroelastic codebased on �nite element model coupled with linear strip aerodynamics. The MDO processis achieved using genetic algorithm. Two case studies are presented to evaluate the perfor-mance of the MDO code, revealing that the metamodel approach does improve the overalloptimization process.
Nomenclature
EIr or t Bending sti�ness value at root or tip, Nm2
GJr or t Torsion sti�ness value at root or tip, Nm2
CEI or GJ Coe�cient for the bending or torsion sti�ness distributionb Wing span, mS Wing area, m2
� Wing leading edge sweep angle, degree� Wing aspect ratioVcrit Flutter critical speed, m=sM Mass, kg
Aircraft design by means of the multidisciplinary design optimization (MDO) approach is proceededusing aerodynamic, structural, ight mechanics and systems integrated mathematical models. This practiceallows the designer to act on parameters that usually are not directly related, e.g., aerodynamic shape andstructural strength, uid-structure interaction analysis and mechanisms. In this context, it seems reasonableto impose aeroelastic constraints to a MDO methodology, as a �rst step into a new MDO environment toassist conceptual design. Usually, aeroelastic veri�cations during aircraft design occur in an advanced stage.1
By bringing aeroelastic constraints to the conceptual design phase, the ability of the designer on de�ningaircraft features, avoiding possible cost increases due to changes on future stages may be improved.
Typical MDO schemes usually face problems with high computational cost, which prevents proper search-ing on the design space and complicates the disciplines integration. An alternative to deal with the challengeof reducing computational e�ort during extensive design space exploration is the use of metamodeling con-cept.2 A metamodel is a model of a model, which assumes simpli�ed mathematical forms to approximatecomputation-intensive functions. Metamodeling techniques are able to improve the understanding of inputand output variables relationship, to provide tools for optimization and design space exploration that arefaster then the conventional computer analysis, and �nally they simplify the integration of computationalcodes.3 Hence, a metamodel for utter speed prediction seems very suitable for the proposed MDO tool.4
The aim of this work is to study an aeroelastic constraint-based MDO scheme for the design of wings usingan arti�cial neural network (ANN) metamodel. This study considers structural and geometrical variables asthe design ones, for �xed aerodynamic parameters. Critical utter speed and wing structural mass are theevaluation parameters of a genetic algorithm (GA). A neural network metamodel is used to predict the uttervelocity, thereby speeding up the overall MDO process. The complete MDO process consists of integratingthe following mathematical models: a �nite element model (FEM) for structural dynamics, an unsteadyaerodynamic model based on strip theory and linear potential ow, a PK-method for utter prediction, anda GA optimization routine. The metamodel-based MDO is achieved by substituting the FEM, aerodynamic,and utter prediction method by an arti�cial neural network (NN).
In this work results and discussion on the performance of the NN metamodel-based MDO are presented.Particular attention to the gains on computing time, when using metamodels, will be payed. Other importantissue on metamodeling is related to the database for the neural network training process, as well as the trainitself. Such issues are also discussed.
II. MDO Scheme
The MDO process is organized as follows. Design variables and constraints are set by the user. Withbending and torsion sti�ness distributions, it is possible to perform a numerical structural analysis in orderto obtain the modal features. Additional information of non-structural masses on the wing, representingexternal bodies, as tip tanks, engines, pods, weapons, and even the skin are considered as lumped massesand inertia moments. At this point, the structural mass may already be estimated. After the structuraldynamic analysis, with additional information on aerodynamics, the aeroelastic solution for utter may beperformed. The GA is responsible for the creation of the initial population and uses the results obtained bythe evaluation models to qualify and assess the evolution of the population. Figure 1 illustrates the MDOschemes, where it has been clearly illustrated that the NN metamodel performs the tasks that the aeroelasticmodel would do.
III. Aeroelastic Solution
The wing structural model has been achieved with a FEM code based on the Bernoulli-Euler beam elementrestricted to 3 degrees of freedom per node. The FEM model element considers bending as represented bythe translations in z-axis and the rotations in x-axis, while the rotations in y-axis describe beam torsion.Lumped mass and inertia moment elements may be located at the nodes to represent non-structural weightand other bodies attached to the wing. Conventional FEM discretization process is considered, where Hermitepolynomials are used as shape functions to calculate beam element deformation. Then, sti�ness and masselement matrices can be assessed and combined to produce the global FEM structural model.5 For utter
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Vortex LatticeAerodynamic Model
AeroelasticStrip TheoryTheodorsen
EigenvalueProblem:K-Method
V – g – fAeroelastic Solution
U (modes)
GeneticAlgorithmOptimizer
StructuralModelFEM
Vcrit
M
New Population
END
No
Yes
GeneticAlgorithmOptimizer
Initial Population
Vcrit
M
New Population
END
No
Yes
ArtificialNeuralNetwork
Initial Population
Figure 1. Complete and NN metamodel-based MDO schemes.
prediction, the structural model must be conveniently transformed to modal variable space, that is:
[Mgen] f��(t)g+ [Kgen] f�(t)g = 0 ; (1)
where [Mgen] and [Kgen] are the generalized mass and sti�ness matrices, respectively, and f�(t)g is the modalcoordinate vector.
The FEM code also provides the calculation of the structure total mass that, together with the dy-namic features, represent the complete set of structural information necessary to the aeroelastic model. Theaeroelastic model to the prediction of utter speed6 has used the strip theory approach to account for theunsteady aerodynamic loading. A simple swept and tapered wing model, where the fuselage and other bodiesare added as lumped masses, is adopted in this work. Figure 2 shows the aerodynamic model in terms ofstrips together with FEM modeling nodes of the structure. The distribution must be in a way that eachFEM node coincides with an aerodynamic strip, so that each element contribution to the utter predictionmethod may be computed. Flutter critical speed prediction is performed by means of the PK-method,6,7
x
y
Figure 2. Wing model schematics (aerodynamic strips and FEM nodes).
and interpolated from V -g-f curves.
IV. NN Metamodel
An arti�cial neural network is a mathematical tool inspired in the brain of animals.8 The feedforwardANN is composed of processing units called neuron, similar to the neurons of living creatures. The mostusual model of neuron is the perceptron, shown in Fig. 3. In this model the input signals are multiplied by
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the weights (also known as synaptic weights, resembling the synapse of biological neurons) that measure thenet connections importance. They are added and the result is compared to the bias (or threshold), creatingthe activation potential. On this an activation function is applied, after which the neuron output is reached.According to Fig. 3, the output may be mathematically described as:
oj = ’
�j +
nXi=1
xi � wij
!: (2)
As activation function the sigmoidal functions, such as the hyperbolic tangent,
’ (x) =2
1 + e�2�x � 1 : (3)
Figure 3. Arti�cial neuron model.
Feedforward neural network architecture represents how neurons are connected. This feature makes themmore complex, capable of learning more information. The training of an ANN is the learning process whereknowledge is given to it. In this process the synaptic weights are adjusted to accomplish the expected resultsfrom the net when given variables are inputed. There are several training algorithms known, among whichthe Levenberg-Marquardt back-propagation is one of the most e�cient for multilayer networks.
Neural networks are suitable for the concept of metamodeling, because their ability to work as surrogatesof complex systems. Here, the NN metamodel (cf. Fig. 4(a)), considers as input variables those that describethe wing bending and torsion sti�ness distributions. For bending sti�ness the value for EI at the wing root(EIr), and tip (EIt), a coe�cient (CEI) that allows the calculation of a second degree polynomial thatdescribes the sti�ness distribution semi-spanwise. Similarly, for the torsion sti�ness features the values areGJr, GJt and CGJ . Moreover, wing geometry parameters such as span (b), area (S), sweep angle (�) andtaper ratio (�) are also considered. The NN model output variables are the utter critical speed (Vcrit) andstructural mass (M).
EIrEItSEI
GJr
GJt
SGJ
b
S
Λ
λ
Vcrit
MNN
(a)
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3x 10
5
EI −
GJ
[N*m
2 ]
Semi−span [m]
(b)
Figure 4. NN metamodel input and output variables (a) and examples of wing bending and torsion sti�nessspanwise polynomials distributions (b).
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Figure 4(b) shows some of the possible bending or torsion distributions supposed as polynomials (bothare assumed to have the same range value), which may be combined in di�erent ways to create the trainingdatabase.
V. Genetic Algorithm Optimization
GAs9 are a type of evolution based search algorithm that manipulate sets of possible encoded solutions fora problem. This approach has been very popular in the �led of optimization, particularly in MDO practices.
The GA operators used here were reproduction (by roulette wheel), crossover (with random number ofbreaking points), mutation, and elitism. The GA individuals are composed by the structural variables EIr,EIt, CEI , GJr, GJt and CGJ , which was the only ones used in a previous work by the authors,4 and willalso include the geometrical variables span (b), area (S), sweep angle (�) and taper ratio (�), all encodedinto chromosomes, as shown in Fig. 5. From the sti�ness polynomials distributions, the values for each�nite element (cf. Fig. 2) may be calculated. Table 1 presents the working range of the design variables.Furthermore, the non-dimensional �tness function F is assumed,
F = 0:7Vcrit � 0:35M + 70 ; (4)
where M is the total structural mass, Vcrit is the utter critical speed and the constant term is addedto adjust the �tness range. From a multi-objective point of view this approach is called Weighted SumMethod.10{12
Table 1. Design variables ranges.
Variable Minimum Maximum Unit
EIr 164; 300 279; 050 Nm2
EIt 30; 000 93; 500 Nm2
CEI �0:45 0:45 �GJr 164; 300 279; 050 Nm2
GJt 30; 000 93; 500 Nm2
CGJ �0:45 0:45 �b 20:0 21:5 m
S 21:0 22:5 m2
� 0 3 o
� 0:7 1:0 �
EIr (8 bits) EIt (7 bits) SEI (3 bits) GJr (8 bits) GJt (7 bits) SGJ (3 bits)
The following sections present results from neural network metamodeling and GA performance and alsofrom the two proposed case studies.
A. NN Metamodel Training Database
The creation of the database to train the NN has been made using the Latin-Hypercube theory of Design-of-Experiments. It distributes the 10 variables to create the previously de�ned number of 2000 solutions ina way to properly explore the solutions domain, what is very important for creating a NN training database.
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This population with 2000 individuals is evaluated with the aeroelastic model to obtain a database withtheir 10 input and 2 output variables. The NN training process has been attained via Levenberg-Marquardtbackpropagation algorithm. The NN metamodel architecture of 30-20-2 neurons per layer has been assumedand trained, revealing good generalization capability, with solutions having errors of no more then 5 %for both output variables. A statistical evaluation was performed for 200 solutions and compared to theaeroelastic model, from which were obtained for Vcrit a mean error of 1:77 % and a standard deviation of4:96 % and for M a mean error of 0:25 % and a standard deviation of 0:30 %. Here two case studies arepresented and for each one, a di�erent NN set of weights had to be created (two distinct training processes),but the architecture is exactly the same.
B. GA performance
After statistical analyses, the GA parameters were 4% of mutation probability, 50 individuals in the pop-ulation and 10 individuals for the elitism for the �rst case study (CS1), 3% of mutation probability, 50individuals in the population and 2 individuals for the elitism for the second case study (CS2). Figures 6(a)and 6(b) show 10 runs through 300 generations with the aforementioned parameters, starting from di�erentinitial populations for CS1 and CS2. It can be seen that in all runs the optimization leads to very similarsolutions, what indicates that it is probably leading to the global optimal solution and demonstrates therobustness of the GA.
0 50 100 150 200 250 30080
90
100
110
120
130
140
150
Fitn
ess
Generation
(a)
0 50 100 150 200 250 30070
75
80
85
90
95
Fitn
ess
Generation
(b)
Figure 6. Fitness evolution of 10 GA runs, starting from di�erent populations, for CS1 (a) and CS2 (b).
C. Case Studies
The MDO scheme has been veri�ed for two wing structural con�gurations. The �rst case to be studied (CS1)is based on a high aspect ratio wing, shown in Fig. 7(a). This wing has lumped masses to represent non-structural material and bodies. All lumped masses are of 4 kg and have a moment of inertia of 3:5 kgm2,except for the central one (representing a fuselage), which is of 140 kg and has a moment of inertia of100 kgm2. The second case (CS2) follows the same features of the previous wing, but with 2 large lumpedmasses at the tips, representing tip tanks, as shown in Fig. 7(b). In this case, the extra lumped masses areof 30 kg and have a moment of inertia of 5 kgm2.
The optimization evolution for the �rst case is shown in Fig. 8(a), where the best individual �tness andthe average �tness of the population are pictured. A reasonable enhancement of the best individual �tnesscan be seen, leaving a value of less then 89, improving it to almost 142. The behavior of the average showsa great variation, which is a good indicative that the solution domain is being widely explored (anothersign of robustness). This behavior is due to crossover and mutation operators that make great changesin individuals during the optimization. The evolution of the planform variables (S, � and �) for CS1 isshown in Figure 9(a). This plot includes the variables values at intervals of 10 generations. Figure 10(a)shows the bending and torsion distributions of the best individual obtained in CS1, which has resulted ina �tness of 141:19, Vcrit of 139:59 m=s and M of 75:79 kg. Table 2 lists the values of the other variables(b, S, � and �) for this individual. To illustrate the variability of the population, Figure 11 shows the
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(a)
(b)
Figure 7. Wing modelings for the �rst case study { CS1 (a) and for the second case study { CS2 (b).
last population of NN metamodel-based MDO for CS1, presenting the EI and GJ distributions for all theindividuals, from the best (1) to the worst (50). The e�ect of elitism can also be seen in this �gure, withthe �rst ten individuals being very similar. This kind of implementation allows identical individuals to takepart in the reproduction process as if they were not the same. This feature must be analysed carefully, as itmay degrade the optimization quality. However, in this case the result is considered appropriate, given thepopulation variability as pictured in Fig. 11.
0 200 400 60060
80
100
120
140
160
Generations
Fitn
ess
Best IndividualAverage of Individuals
(a)
0 200 400 60050
60
70
80
90
100
Generations
Fitn
ess
Best IndividualAverage of Individuals
(b)
Figure 8. Evolution for CS1 (a) and CS2 (b).
Table 2. Design variables of best individuals for CS1 and CS2.
Variable CS1 CS2 Unit
b 20:0 20:0 m
S 21:5 22:5 m2
� 1:0 0:0 o
� 1:0 1:0 �
MDO evolution for CS2 is presented in Fig. 8(b) and Fig. 9(b). As in CS1, the optimization performeda considerable enhancement in the best individual and seems to satisfactorily explore the solution domain.The best individual obtained here (cf. Fig. 10(b) and Table 2) has �tness of 92:91, Vcrit of 98:94 m=s andM of 132:41 kg. Last population individuals are shown in Fig. 12.
Figures 13(a) and 13(b) show the evolution of the nondominated solutions frontiers through the genera-tions. It was considered 5 generation intervals to create the frontiers. As seen, the optimization algorithmimplemented was not capable to explore the nondominance frontiers appropriately. For CS2 some solutionsare even lost from one curve to the next.
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0 200 400 600
20
21
22
23
S [m
2 ]
Generation
Λ [o ] −
λ []
0 200 400 600
0
1
2
3SΛλ
(a)
0 200 400 600
20
21
22
23
S [m
2 ]
Generation
Λ [o ] −
λ []
0 200 400 600
0
1
2
3
SΛλ
(b)
Figure 9. Evolution of S, � and � for CS1 (a) and CS2 (b).
0 5 10 15 200
0.5
1
1.5
2x 105
Semi−span [m]
Stif
ness
[N.m
2 ]
EIGJ
(a)
0 5 10 15 200
0.5
1
1.5
2
2.5
3x 105
Semi−span [m]
Stif
ness
[N.m
2 ]
EIGJ
(b)
Figure 10. Best individuals for CS1 (a) and CS2 (b).
The accuracy and readiness that results were provided demonstrate the reason for interest in metamod-eling techniques applied in MDO. Whenever a more precise solution is desired (for example in preliminarydesign), the same scheme could be used just by changing the database for NN training. If high �delity nu-merical models provided the results for its creation with a large and smartly distributed number of solutions,certainly the generalization would be more precise, with the same cheap-to-run ability. On a PC compatiblecomputer (INTELTM PENTIUMTM Dual Core, 3.2 GHz processor and 2 Gb of RAM), the average executiontime for 600 generations was approximately 40 s for the metamodel optimization, a great enhancement whencompared to previous implementation4 using the complete model, which takes about 1 h to run the sameoptimization.
VII. Conclusions
The use of NN metamodel in a MDO scheme is investigated. The creation of a database for trainingthe NN metamodel may be a hard task once it has to reach as much as possible the entire solution domain,or there will not be enough information for the NN to learn the issue. The Latin-Hypercube technique ofdesign of experiments was included to help database achievement and presented good distribution of samplesthroughout the solutions domain. The NN metamodel has the advantage of easily integrating disciplinesin one robust code, as shown in this work by the NN capability of assimilating knowledge to provide Vcrit
and M from structural and geometrical parameters. The GA has presented satisfactory robustness andresults have shown a trend to global optimal solutions, although it may be improved for exploring betterthe nondominance frontiers. Results demonstrate the need for other algorithms to search for the paretofront, where the compromise solutions lie on. The proposed metamodel-based MDO scheme has led to goodoptimal solutions for both case studies (CS1 and CS2), obtaining coherent features for the best individuals inthe population. It is a much faster approach for the given problem when compared to other implementation
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010
2030 0 10 20 30 40 50
0
1
2
3x 10
5
Order in PopulationSemispan [m]
EI [
N.m
2 ]
010
2030 0 10 20 30 40 50
0
1
2
3x 10
5
Order in PopulationSemispan [m]
GJ
[N.m
2 ]
Figure 11. Last population of the optimization of CS1.
010
2030 0 10 20 30 40 50
0
1
2
3x 10
5
Order in PopulationSemispan [m]
EI [
N.m
2 ]
010
2030 0 10 20 30 40 50
0
1
2
3x 10
5
Order in PopulationSemispan [m]
GJ
[N.m
2 ]
Figure 12. Last population of the optimization of CS2.
based on complete model. With such a tool, aircraft designers could have adequate information about theaeroelastic behavior of a wing at the conceptual design stage, and also they could explore and learn moreabout the design variables domain.
Acknowledgements
The authors acknowledge the �nancial support of CNPq during the tenure of this research work.
References
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20 40 60 80 100 120 14060
80
100
120
140
160
180
Vcrit [m/s]
M [k
g]
0 ~ 2021 ~ 5051 ~ 100101 ~ 300301 ~ 600
130 132 134 136 138 140
80
90
100
110
120
(a)
40 50 60 70 80 90 100 110 12060
80
100
120
140
160
180
200
220
Vcrit [m/s]
M [k
g]
0 ~ 2021 ~ 5051 ~ 100101 ~ 300301 ~ 600
(b)
Figure 13. Evolution of the nondominated frontiers for CS1 (a) and CS2 (b).
11Deb, K., Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, Ltd., Chichester, 2001.12Marler, R. and Arora, J., \Survey of Multi-Objective Optimization Methods for Engineering," Struct. Multidisc. Optim.,
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