Robust and Reliability-Based Design OptimizationFramework for Wing Design
Ricardo M. Paiva,∗ Curran Crawford,† and Afzal Suleman‡
University of Victoria, Victoria, British Columbia V8P 5C2, Canada
DOI: 10.2514/1.J052161
This paper outlines an architecture for simultaneous analysis, robustness, and reliability calculations in aircraft
wing design optimization.Robust design optimization and reliability-based design optimization are unified in amixed
formulation, which streamlines the setup of optimization problems and aims at preventing foreseeable
implementation issues in uncertainty-based design while ensuring that the performance hit of robustness/reliability
assessments is kept to aminimum. To avoid the extra computation time that would be the result of a direct evaluation
approach to nondeterministic optimization, Kriging surrogate models are employed, and an alternative
implementation of the reliability subproblem is also proposed. The sigma point method is used to compute statistical
moments in the robust objective function. The computational effort of reliability analysis is further reduced through
the implementation of a coordinate change in the respective optimization subproblem to solve for the distance from
the current iterate to the most probable point of failure. Robustness and reliability-based optimization is tested on
both simple analytic problems and more complex wing design problems, across a range of statistical variation,
revealing that performance benefits can still be achieved while obeying precise probabilistic constraints.
Nomenclature
AOA = angle of attackb∕2 = semispanCD = drag coefficientCL = lift coefficientcroot = chord at wing rootctip = chord at wing tipD = drag forceF = robust objective functionf = objective functionG = robust constraint functiong = gravitational accelerationgi = constraint functionL = lift forceNRV = number of random variablesNSP = number of sigma pointsn = load factornDV = number of design variablesnd = number of deterministic constraintsng = number of constraintsnrc = number of reliability constraintsP� : : : � = probability of an eventPf = probability of failureR = cruise rangeSFC = specific fuel consumptionT = target valueV∞ = cruise speedWfuel = fuel weightWi = weights for sigma point methodWother = other components weight
T HE increasing competitiveness in the aerospace industry hasmanufacturers searching for designs that are robust in the sense
that they still perform well in off design conditions (flight conditionsor load uncertainty, for example), as well as reliable in the way thatthey present a low probability of failure. Moreover, in early stagesof the design, many parameters are yet unknown or poorlycharacterized. The classical approach to structural design employingsafety factors has frequently proved to be overly conservative, thusleaving room for improvement and achieving an edge over com-petitors. Reliability-based design optimization (RBDO) is thereforeaimed at meeting specific probabilistic targets for objectives and/orconstraint functions. Robust design optimization (RDO) on the otherhand, minimizes the sensitivity of objective and constraints to off-design parameters.RBDO and, in particular, reliability-based multidisciplinary
design optimization require the otherwise deterministic optimizationproblems to be augmented with a reliability analysis subproblem(more on this topic in Sec. II.B of this paper). Advanced reliabilityanalysis techniques have been incorporated in computer programssuch as CALREL [1], STRUREL [2], PROBAN [3], RELSYS [4],andNESSUS [5], whereas an example of the state of the art in RBDOtools is represented by ProFES [6], which can be coupled withcommercial tools for finite element analysis. Mainstream finite ele-ment software suites such as ANSYS® or NASTRAN® also includetheir own stochastic analysis tools.Bearing in mind the dichotomy between RDO and RBDO, a
number of authors have advocated combining the objective functionapproach of RDO andwith the reliability constraints of RBDO [7–9].That approach is also adopted in this work and developed into amultidisciplinary analysis and design optimization tool for aircraftwing design that can perform nondeterministic optimization using ahybrid algorithm. This architecture employs both RDO and RBDOaspects to search for robust optima while obeying reliability-typeconstraints. Named robust and reliability-based design optimization
(R2BDO), it is an effort to reconcile the search for designs that aresimultaneously less sensitive to nonstandard conditions while alsoachieving a preset measure of reliability/probability of failure. Thismethodology proves to be more versatile than RBDO, while at thesame time being more accurate than RDO in estimating failure atnondeterministic limit functions.Previous authors have evaluated a number of approaches for
computing the statistical moments for the RDO objective [8]. In thiswork, a different approach, the sigma point method (SP), isemployed. The application of anR2BDO approach has been used forsingle-discipline structural problems (e.g., [9]) but not often tomultidisciplinary problems. Jun et al. [10] did look at a wing aero-structural design problem with a similar hybrid approach but with asimplified structural box model and an artificial neural network(ANN) surrogate model. In contrast, the current work employs adetailed finite element method structural model and a Krigingsurrogate model that has been shown to be better suited to thisproblem than ANN [11]. In addition, a new implementation of theclassical RBDOsubproblem inwhich there is a change of coordinates(to hyperspherical coordinates) is presented here, which allowsthe reduction of the dimensionality of such subproblem whileeliminating its equality constraint (as discussed in Sec. II.B). Theresult is a much more computationally efficient reliability analysis.The R2BDO architecture is verified and evaluated using both
analytic and more complex and realistic wing design problems, andthe results are compared with those of a gradient-based deterministicoptimization approach.Wing design problems are defined and solvedusing a purposely developed code, capable of multidisciplinaryanalysis and optimization. Disciplines such as aerodynamics andstructural analysis can be coupled in this fully standalone application.
II. Robust and Reliable Design
To incorporate uncertainties in design optimization impliessolving a suitably modified version of the deterministic designoptimization problem. Themethodologies to achieve this are dividedinto two main groups: RDO and RBDO. The (somewhat) subtledifferences between RDO and RBDO are not mentioned oftenenough in the literature (an exception being [12]), and as such, aclarification is in order.In an RDO formulation, the goal is to optimize the response of a
system about a mean value: maximizing robust performance whileminimizing its sensitivity to random parameters, which is usuallyattained by minimizing standard deviation of the response, alongwith itsmean. In thisway, the performance in off-design conditions isalso given consideration during the optimization process.On the other hand, RBDO approaches provide a way of designing
while taking into account safety margins. In other words, theoptimization can be performedwhile having a particular risk inmind:target reliability. Besides Monte Carlo (MC), reliability may beestimated through the first- or second-order reliability methods(FORM and SORM), for instance.Although both architectures necessarily require more function
evaluations than the equivalent deterministic optimization problem,RDO can be performed on an unconstrained problem, whereasRBDO is by definition performed on constrained problems only.In this work, when necessarily dealing with the probability
distributions of multiple variables, two assumptions will be made.The first is that the uncertain variables are assumed to be normallydistributed. This assumption implies no loss of generality becausenonnormal random variables for which the probability distribution isknown can be subjected to an appropriate transformation to makethem normal (the Rosenblatt transformation, for instance [13]). Theobjective/constraints in an optimization problem are then obtainedthrough function composition. The second assumption, however, ismore restrictive, in the way that all the variables are assumedindependent; in other words, no correlation can exist between them.Though problems in which the covariance matrix is not diagonal canbe solved through an equivalent subproblem inwhich all variables areindependent (by means of diagonalization of the covariance matrixand subsequent coordinate transformation) [14] and then all the
methods developed henceforth can be applied, the impact on thesequence of operations is considerable, as may be the influence onthe final results.
A. Robust Design Optimization
A generic statement for a deterministic optimization problemcan be
minxf�x�
subject to: gi�x� ≤ 0 i � 1; : : : ; ng
xLBk ≤ xk ≤ xUBk k � 1; : : : ; nDV (1)
where x is the vector of design variables (DVs), xLB and xUB are thelower and upper bounds on the DVs, respectively.Reformulation of the same problem in an RDO perspective would
yield [15]
minμx
F�μf�x; r�; σf�x; r��
subject to: Gi�μgi�x; r�; σgi�x; r�� ≤ 0 i � 1; : : : ; ng
where μ and σ represent the mean and standard deviation of thequantities in the subscript (DVs, objective, or constraints). Thesemaybe computed as per Eqs. (3) and (4). In this formulation, the otherwisedeterministic parameters r are allowed a random distribution (in atypical design problem these may constitute material properties, forinstance), and the design variables x can now be either deterministicor random. The robust objective and constraints are now functions ofthe mean and standard deviation of objective and constraints, whichin turn depend on the probabilistic distribution of the variables. Thebounds on the DVs are now themselves established in terms of theirprobability of residing inside the preset interval (although they caneasily be reformulated as deterministic constraints back againbecause the probabilistic distribution of the variables is known):
μf�x; r� �Z �∞−∞
: : :
Z �∞−∞
f�t�px;r�t� dt (3)
σf�x; r� �Z �∞−∞
: : :
Z �∞−∞�f�t� − μf�x; r��2px;r�t� dt (4)
here f represents a function of interest andpx;r is the joint probabilitydensity function (PDF) (vector t has total dimension equal to that of xplus that of r). The analytical evaluation of the integrals in Eqs. (3)and (4) is impossible in most practical cases, and for that reason, anumerical procedure is required. Among the various techniques thatmay be used are MC methods, the Taylor-based method of moments[15], and the sigma point technique, which is the one used in thiswork and described next.The SP method is a derivative of the Taguchi method, used in
statistical tolerancing since 1978 [16]. The idea behind SP is that it iseasier to match an input distribution (SP is typically defined for anormal distribution) than to linearize (or in general, approximate) anonlinear mapping [17,18]. To compute the integrals in Eqs. (3) and(4), SP employs a procedure similar to Gaussian integration but oneinwhich the sample locations and respectiveweights are optimized tomatch the first moments of the input probability distribution.For a given vector of random variables �X1; X2; : : : ; XNRV
�,representing the uncertainty about a certain point (mean is thezero vector), a candidate set of sigma points must respect thefollowing [19]:
W0 �XNSP
i�1Wi � 1 (5)
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XNSP
i�1Wi1S
m1
1 2Sm2
1 : : : NRVSmNRVi � E�Xm1
1 Xm2
2 : : : XmNRVNRV�
∀ �m1; : : : ; mNRV�: 1 ≤ m1 � : : : �mNRV
≤ k (6)
whereWi are the weights; nSi is the nth coordinate of the ith sigmapoint (coordinates are relative to the central point, μx, i.e., the meanvalues of the design variables, in which theweight isW0); andNSP isthe number of sigma points minus the origin. The condition stated inEq. (6) should be repeated for all integer values k up to the orderrequired for the approximation (starting from one). The order of theapproximation also dictates the minimum number of points in a setthat satisfies these conditions. Minimum point sets are relatively easyto find for multivariable problems with up to three random variables.Higher-dimensionality sets usually require the usage of a nonlinearsolver. Even then, the solution may not be unique [19].For a set of independent random variables, SP can be reduced to
sampling along the coordinate axis, at a distance from the point ofinterest that is only dependent on the standard deviation in thatdirection (hence the designation of the method as sigma point). ForGaussian random vectors, a deterministic procedure is available todetermine the sigma point set for calculation of mean and standarddeviation of a function of said vector [15,17,20]:
χ0 � μx (7)
χi� � μx ��������������������������NRV � K�
p�������Σx
p�i; i � 1; : : : ; NRV (8)
χi− � μx −�������������������������NRV � K�
p�������Σx
p�i; i � 1; : : : ; NRV (9)
W0 �K
NRV � K(10)
Wi � Wi� � Wi− �1
2�NRV � K�; i � 1; : : : ; NRV (11)
where �������Σxp�i is the ith row in the square root of the covariance
matrix. In case the variables are independent, this term reduces to thestandard deviation in the ith direction: σxi .The value of the real constant K should be set so that
NRV � K � 3, the kurtosis of the standard normal distribution.Based on this set, the estimators (denoted by )̂ for the mean andstandard deviation are [21]
μ̂f�χ0� � W0f�χ0� �XNRV
i�1Wi�f�χi�� � f�χi−�� (12)
2σ̂2f�χ0� �XNRV
i�1Wi�f�χi�� − f�χi−��2
�XNRV
i�1�Wi − 2W2
i ��f�χi�� � f�χi−� − 2f�χ0��2 (13)
The derivatives of these estimates with respect to the design variableset are
∂μ̂f∂xk�χ0� � W0
∂f�χ0�∂xk
�XNRV
i�1Wi
�∂f�χi��∂xk
� ∂f�χi−�∂xk
�(14)
∂σ̂2f∂xk�χ0� �
XNRV
i�1Wi
�∂f�χi��∂xk
−∂f�χi−�∂xk
��f�χi�� − f�χi−��
�XNRV
i�1�Wi − 2W2
i ��∂f�χi��∂xk
� ∂f�χi−�∂xk
− 2f�χ0�∂xk
��f�χi��
� f�χi−� − 2f�χ0�� (15)
The set then contains 2NRV � 1 points so that each uncertaintyquantification carries an additional penalty in terms of functionevaluations. If the number of evaluations required to computesensitivities in a finite difference estimate for gradient-basedoptimization is given by NDV, a total of �2NRV � 1��NDV � 1�evaluations is required for each design point. This makes RDOproblem solving a daring prospect in terms of computationalresources, if finite differencing is used. In the case of independentrandom variables, it would certainly be beneficial if some of thesigma points could be superimposed with points used for finitedifferencing. However, in general, the distances from the centralpoint at which the evaluations for sigma points and finite differencingare made are several orders of magnitude apart.
B. Reliability-Based Design Optimization
An equivalent statement to Eq. (1) in RBDO is [22,23]
minxf�x; r�
subject to: grci �x; r� ≤ 0 i � 1; : : : ; nrc
gdj �x� ≤ 0 j � 1; : : : ; nd
xLBk ≤ xk ≤ xUBk k � 1; : : : ; nDV (16)
The constraints set is now divided into reliability constraints grci andother design constraints gdj (for which a reliability target is notestablished). Alternatively, the objective function may be defined interms of the probability of the original function exceeding/notexceeding a certain target (T), which is to be minimized [12]:
minxP�f�x; r� − T ≥ 0� or P�T − f�x; r� ≥ 0�
subject to: grci �x; r� ≤ 0 i � 1; : : : ; nrc
gdj �x� ≤ 0 j � 1; : : : ; nd
xLBk ≤ xk ≤ xUBk k � 1; : : : ; nDV (17)
The reliability constraints are of the form
grci � Pfi − Pallowi� P�gi�x; r� ≥ 0� − Pallowi
(18)
P�gi�x; r� ≥ 0� �Zgi�x;r�≥0
px;r�t� dt (19)
effectively ensuring that the probability of the originallydeterministic constraint gi being violated is at the most Pallowi
: theallowable probability of failure. Determining the probability offailure Pfi requires either sampling (e.g., Monte Carlo methods) ortechniques such as the FORM and the SORM [22]. Although FORMis widely used in reliability analysis, SORM has seen little practicaluse as it requires higher-order information on the objective functionand constraints [12,22–24]. The next section is therefore dedicated todescribing FORM in more detail.In essence, FORM consists of creating a linear approximation to
the limit state function g�r� (r now being a generalized set of randomvariables, which encompasses the uncertainties in both designvariables and parameters). According to FORM, the probability offailure is evaluated (approximately) as
Pfi � Φ�−β� (20)
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where Φ is the cumulative distribution function of the standardnormal distribution and β is the distance from themost probable point(MPP) of failure to the current iterate (also called reliability index),measured in the standard normal space u (see Fig. 1). It is determinedby solving the following optimization problem (β � �uTu�12):
minu�uTu�12
subject to: g�r�u�� � 0 (21)
The vector u, in standard normal coordinates, is obtained from rthrough a transformation (which is diagonal for the case ofstatistically independent variables):
u � T�r� (22)
In this case though, the inverse transformation T−1�u� is the one thatis most useful for the computation of the limit state function inEq. (21). Realizing that, from the result of the MPP optimizationproblem, the reliability constraint may be written in terms of thereliability index β,
grci � βreqd − βi (23)
the reliability index approach (RIA) is obtained. βreqd is the specifiedreliability index. The RIA is, however, problematic in cases in whichfailure does not occur at all for a particular set of values of the designvariables, orwhen the limit state surface is far from the origin. For thatreason, the performance measure approach (PMA) was devised [25].In PMA, the inverse problem of that one stated in Eq. (21) is solvedinstead:
minu
− g�r�u��
subject to: �uTu�12 − βreqd � 0 (24)
This is not only a more robust formulation than RIA, it alsoimmediately returns the required value of the reliability constraint(grci � g��r�u��). Another important advantage is that the MPPsubproblem may be formulated in a minimax approach, effectivelyhandling multiple constraints simultaneously, something that is notpossible with RIA.A proposed alternate strategy for solving the PMA problem stated
in Eq. (24) consists in restricting the values of the vector u to lie on ahyperspherical surface, effectively eliminating the need for theequality constraint �uTu�12 − βreqd � 0. In addition, the dimension-ality of this subproblem is reduced to NRV − 1, which should allowfor faster convergence. The problem statement is then simply
minϕ
− g�r�u�Φ��� (25)
where Φ is the set of hyperspherical coordinates Φ �fϕ1;ϕ2; : : : ;ϕNRV−1g.Hence, the solution with hyperspherical coordinates requires the
use of yet another transformation to obtain the u vector back from theϕi coordinates:
u1 � βreqd cos�ϕ1�
..
.
uNRV−1 � βreqdYNRV−2
i�1sin�ϕi� cos�ϕNRV−1�
uNRV� βreqd
YNRV−1
i�1sin�ϕi� (26)
Additional major considerations in RBDO problems includesensitivity analysis in the MPP subproblem (a derivation ofsensitivities in both RIA and PMA is presented in [12]) as well as themanner in which the latter is coupled with the main optimizationproblem. In the class of double-loop methods, the determination ofthe MPP is executed at each iteration of the main optimizationalgorithm, whereas in sequential methods, the MPP subproblem iscompletely decoupled from the deterministic part of the optimization,the two problems being solved in turn until a convergence criterion issatisfied. Unilevel methods are the other extreme because theformulation of the MPP optimization problem is embedded into theoriginal problem bymeans of its Karush Kuhn Tucker conditions [26].Other methods that have been used in reliability analysis include
the mean value, advanced mean value, and their derivatives [27].These aim at approximating the probabilistic distribution of thefunction of interest given a known uncertainty distribution in theinput. The probability of failure is then computed by evaluatingthe approximate cumulative distribution function.
C. R2BDO
Constraint treatment in RDO is not transparent because thedesigner is left with a choice of weights that will ultimately definehow far from the failure surface should the average optimum lie. Thiscan be addressed by adapting the robust constraints so that theweights are calibrated in order to mimic a probabilistic constraint.This calibration is usually made using the PDF of the inputsdistribution [20,28]. Although a good approximation for very lowinput variances or quasi-linear constraints, in general, the constraintoutput distribution may greatly differ from that of the input, thusinvalidating this type of analysis. The RBDO treatment of constraintsis better suited for this task as it precisely quantifies the violation ofthe limit state by means of a probability of failure/reliability index.At the same time, probabilistic objective function targets are
frequently problematic. If the initial guess for a target is too far offfrom what an actual design can attain, the probability becomes either
Fig. 1 MPP determination (showing the contours of the joint PDF).
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too close to zero or to one, slowly varying. In practice, this means anoptimizer would tend to stop prematurely, before finding a truecandidate to local/global minimum.Amix of robust objective/reliable constraints is therefore believed
to be better suited than the original RBDO formulation for generalpurpose optimization. Essentially, in this proposed hybridformulation, the type of objective function used in RBDO (eitherdeterministic or probabilistic, as mentioned in the preceding section)is replaced by the type used in RDO, and the RBDO constraintfunctions are retained to ensure well-posed reliability constraints areattained:
minxF�μf�x; r�; σf�x; r��
subject to: grci �x; r� ≤ 0 i � 1; : : : ; nrc
gdj �x� ≤ 0 j � 1; : : : ; nd
xLBk ≤ xk ≤ xUBk k � 1; : : : ; nDV (27)
Anexample problem is solved in the next section, comparingR2BDOwith plain RDO.
III. Numerical Example
Using the classic Rosenbrock function, the following optimizationproblem is formulated:
minxf�x� � 100�x2 − x21�2 � q�1 − x1�2
subject to: g�x� � x21 � �x2 − h�2 − 1 ≤ 0 (28)
where μh � 0.5. For q � 1, the solution to the deterministicoptimization problem is x� � 0.9306, 0.8659). As a reference, usingstarting point x0 � �0.1; 0.1�, it takes 66 function evaluations toconverge to the aforementioned solution (using a sequential quadraticprogramming algorithm).
A. Alternative PMA
To ascertain the performance impact of the proposed formulationfor the FORM subproblem [Eq. (25)], the preceding problem issolved using a deterministic objective; i.e.,
minxF�x� � f�μx�
subject to: grc ≤ 0 (29)
and uncertainty is considered for the three variables in the nonlinearconstraint (βreqd � 3).As Table 1 illustrates, the performance advantage of the alternative
approach is clear, reducing the number of function evaluationsrequired by asmuch as 80%, whereas the achieved solution is exactlythe same as if using conventional PMA. The errors (ϵ) are obtained bycomparison of the real reliability with the target. The “true” value ofthe probability of failure results from a post optimality Monte Carloprobability calculation using 2 × 107 normally distributed samples(the distribution is centered on the solution point). On another note,the effect of nondeterministic analysis on performance is nonethelessvery clear: the required number of function evaluations is increasedby up to two orders of magnitude.The limitations of FORM also become apparent for higher
uncertainties. Although the results of PMA and its alternative formstill match, the error in the reliability index at the optimum nowalmost reaches 4% for the higher input variances.
The implementation of the hybrid R2BDO formulation to thepreceding problem results in the following statement:
minxF�x� � μf � σf
subject to: grc ≤ 0 (30)
where grc ≤ 0 is computed using the alternative PMA formulation;uncertainty is considered for x1, q, and h. In both cases, the objectiveis computed using SP, which performed favorably whenbenchmarked against the method of moments and Monte Carlo insimilar problems [15,29]. The goal of this example is to compare theaccuracy of RDO and R2BDO when performing optimization forrobustness and reliability simultaneously. To that end, in theconstraint for RDO,
G�x� � μg � Kσσg ≤ 0 (31)
the standard deviation of the constraint is affected by a factorKσ � 3,which emulates a desired reliability index of 3 as well.
For the sake of readability of the results presented inTables 2 and 3,the uncertainties in q and h take the same value for each test case. InFigs. 2 and 3 the results are also graphically presented for the c.o.v. ofx1 ranging from 0.005 to 0.1.Comparison of these two tables shows that the initial assumption
that RDO does not adequately model constraint reliability proves tobe correct. Although it thrives for low input variances (requiring lessfunction evaluations), the accuracy of the reliability indexcalculation suffers in comparison to the hybridR2BDO formulationas uncertainty increases. Even though the inputs are normal andboth formulations share the same objective function, constraintnonlinearity precludes RDO from attaining the desired reliability(just as easily, the estimate could have been too conservative). Foreven higher input variances, the solution becomes independent ofthe constraint behavior, as the minimum moves to well withinthe feasible region due to the uncertainty in the objective(β ≫ βreqd). In these situations, R2BDO is dominated by RDO interms of performance, due to the seemingly unnecessary reliabilitysubproblem.
In an example based on the R2BDO problem from the precedingsection, a regression Kriging model is used to perform the requiredobjective/constraint function evaluations rather than the actualfunctions themselves. Kriging was selected given that in previouswork it demonstrated superior performance/stability compared to theother two types of models hereby considered. The same adaptivesampling technique is also adopted here (Fig. 4); starting from aninitial global sample, the surrogatemodel is then successively refinedin a trust region defined around the current optimization iterate (seeFig. 5). Latin hypercube sampling is used [11].There is a choice over how to create the Kriging model for
robustness/reliability functions. The most straightforward manner inwhich to do this would be to approximate the robust objective/reliability constraints directly, based on samples taken from theresults of the SP approach and FORM subproblem solutions,respectively. However, it was observed that, as suggested in [30], thisleads to a higher modeling error than if SP and FORM are applied tosurrogates of the original (deterministic) objective and constraints.As such, and referring back to the nomenclature used before, the
Kriging model will approximate f and g, and only then areF and grc
built. The results in Tables 4 and 5 reflect the usage of a Krigingsurrogate for various uncertainty levels in the example problem (40independent runs for each problem). The initial sample is composedof 25 evaluations and from then on between four to eight samples(depending on the level of uncertainty) are taken inside a trust regioncentered around the current iterate. The regression component ofthe Kriging model is a second-order polynomial, and a Gaussiancorrelation model is used.For cases in which the influence of the reliability constraint
manifests itself, it is clear that the use of surrogate models to simulatesuch a constraint is advantageous. Considerable savings areobserved, in terms of the number of function evaluations, and in thesesituations, the rms error (measured with respect to the referencesolutions from Sec. III.B) is also within reasonable bounds. Theerrors in the reliability index (where applicable) remain mostlyunchanged,whichmeans the FORMconstraint/robust constraints arebeing adequately captured by themodel. On the other hand, when thelevel of uncertainty in the objective drives the optimum point in-ward, toward the center of the feasible region, the Kriging modelloses its edge. The solutions returned are then dispersed around thetrue optimum (some examples for both R2BDO and RDO arepresented in Figs. 6–9) and take longer to be achieved. This happensboth for RDO and R2BDO, which means the robust objective is theproblematic function in this case. The reason for this is linkedwith thefact that, for these uncertainty values, the function experiencesextremely low gradient values in a wide area around the minimummaking convergence of the optimization on the surrogate modeldifficult.Although these false optima are very close to the true solutions, if
further refinement is needed, the framework is to revert to directevaluation upon reaching the optimum on the surrogate. This is thestrategy adoptedwhen solving thewing design problems presented inthe following section.
IV. Robust and Reliability-Based Design:Simulation and Evaluation
In this section, practical application is given to the multi-disciplinary design optimization (MDO) framework in the form ofwing design problems. The code developed for the final version ofthe framework resulted in a standalone application that is providedwith a graphical user interface. It encompasses modules suchas aerodynamics and structural analysis that can be used eitherindividually or coupled for optimization [29]. A doublet-sourcelattice three-dimensional panel code is employed for the aero-dynamics calculations, whereas FEAP [31], along with a purposelybuilt meshing code, form the structural analysis component.Although uncertainty is considered in flight condition parameters
[such as angle of attack (AOA) or altitude], no uncertainty is con-sidered in geometric dimensions or material properties. This isdeemed sensible because at the conceptual design stage, manufac-turing tolerances aremost likely unknown (and even if specified, they
Fig. 5 Sampling region in the vicinity of constraints.
Fig. 4 Sampling algorithm with constraint-based filtering.
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would be of such magnitude that their implications on the designwould not be sensed by the analyzermodules in use). Uncertainties inmaterial properties are also a topic to bemore thoroughly analyzed inlater stages of the design process as they are dependent not only on thematerial itself but also on the manufacturing process and even thesupplier.
Departures from theoretical flight conditions can have bothdirect and indirect influences on the performance of a cruisingaircraft. In a representative set of uncertain parameters chosen, directinfluences are portrayed by uncertainty in the angle of attack (lineareffect onwing loading due to lift).Other indirect sources of uncertaintysuch as differences in pressure altitude, cause changes in the properties
of air (namely viscosity, density, and, by consequence, local speed ofsound), which affect both wing loading due to lift and drag.The remaining uncertain parameter is the payload (cargo or
passengers). Typical allowances for per-passenger weights at theconceptual design stage may be violated by a significant margin in
regular operations. Although payload estimation is a routineprocedure in airliner operations (some aircraft models even possessweight sensors in their undercarriage), the fact is that this serves onlyas a means to determine fuel load and ensure weight balance (forstability purposes). The question ofwhether an aircraft remains a cost
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Fig. 8 KrigingR2BDO, c.o.v. x1 � 0.05, σh � σq � 0.1 (reference solution represented by *). The contour plot is that of the robust objective functioncomputed through SP.
Fig. 10 Probability distributions for uncertain variables in case studies I and II.
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effective platform (i.e., maintains its competitiveness) for a widerange of payload values should be addressed by the designer.The normal probability distributions used to characterize the
random parameters mentioned in the preceding sections arepresented in Fig. 10. The PDF for the angle of attack is showncentered around a single value, although in case study II, it becomes adesign variable, and as such, its standard deviation is not constant.
Essentially, two design problems will be covered: one with threegeometric design variables and two uncertain parameters and anotherone with eight design variables, two uncertain parameters, and oneuncertain design variable. Altitude, μAlt � 11; 000 m; AOA,μAOA � 2 deg; and a Mach number of 0.7 characterize the expectedflight conditions for all problems.The baseline wing geometry and structure are shown in Figs. 11
and 12. The wing has one engine mounted on it for which the totalweight is 9 kN. The thrust applied at the pylon–wing interface is set to11 kN. Other loads applied at such interface are the aerodynamicforces due to the pylon–nacelle installation.
A. Case Study I: Three Design Variables, Two Random Variables
The design variable bounds and constraints for this first set ofproblems are shown in Table 6. Altitude and AOA are the uncertainparameters.The objective is to maximize the lift to drag ratio. In this case, for
RDO and R2BDO, the robust objective is
F�x; r� � −μL∕D � σL∕D (32)
Constraints are imposed on the maximum admissible stress in eachstructural component. Kreisselmeier–Steinhauser (KS) constraintaggregation is used here to save on the number of nonlinearconstraints being imposed. The constraint aggregation parameter ρ isdetermined as proposed in [32] and is initially set to a value of 50. Therequired reliability index is set to three (βreqd � 3), whichmeans that,as before (Secs. III.B and III.C), Kσ � 3 in the robust constraint forRDO. The deterministic optimumused for comparison is obtained byperforming direct evaluation of the objective/constraint functions.The uncertainty-based design problems employ a Kriging surrogateinitially and then revert to direct evaluation to refine the result. Thevalues presented for the required number of function evaluations are,nevertheless, only indicative of performance because the problemswere only repeated to ensure the result was a minima, rather than tofully evaluate numerical performance. Statistically characterizing thecomputational cost as in Sec. III.Cwould require several runs for eachproblem, which is not feasible in this case given the cost of eachindividual analysis.The results in Table 7 show the expected general trend in which
as the uncertainty in the angle of attack increases, the performance ofthe optimal configurations diminishes (as measured by μL∕D) at thesame time its standard deviation increases. This is explained by thedecrease in the aspect ratio of thewing tomaintain the design feasiblewith respect to the reliability-based stress constraint (in short, smaller
Fig. 11 Wing with mounted engine.
Fig. 12 Baseline wing structure for optimization problems.Table 6 Design variables and constraints for
Semispan, b∕2 17 m 17.988 m 17.865 m 17.995 m 17.765 m 17.352 m 17.782 m 17.206 mRoot chord, croot 6 m 5.779 m 5.636 m 5.735 m 5.360 m 6.139 m 5.376 m 5.791 mTip chord, ctip 2 m 1.5 m 1.508 m 1.500 m 1.684 m 1.501 m 1.553 m 1.684 mF −22.807 −23.697 −23.489 −23.636 −23.113 −22.943 −22.834 −22.450μL∕D — — 23.697 23.547 23.694 23.375 23.185 23.364 22.939σL∕D — — 0 0.058 0.058 0.262 0.242 0.530 0.489# evaluations — — 260 524 758 1718 617 965 662
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wings are both more robust and more reliable). The configurationsobtained through RDO are invariably less conservative. In Fig. 13, aqualitative comparison of wing shapes obtained for RDO, R2BDO,and deterministic optimization is presented.
B. Case Study II: Nine Design Variables, Three Random Variables
In this second problem, nine DVs are used, among which is theangle of attack, which doubles as an uncertain variable, together withaltitude and payload (more details in Table 8.Besides the KS stress constraint, a trimmed flight constraint is also
imposed. This accounts for the balance of forces at the start of thecruise phase (full fuel load). Equation (33) represents this equalityconstraint:
L
n�Wother �Wpayload �Wfuel � 2Wwing�− 1 � 0 (33)
where n represents the load factor (n � 1 during cruise),Wother is theempty weight of the aircraft excluding the wing structure,Wpayload isthe total payload weight (crew/passengers/cargo), Wfuel is thefuel weight at the start of the cruise phase, and Wwing is the weightof each wing. The values used for the various weight componentsare Wother � 172; 500 N; Wpayload � 120; 000 N (average value,because it is considered an uncertain parameter); and Wfuel �110; 000 N.In this instance, the goal is to maximize the cruise range defined as
R � V∞
gSFC
CL1.2CD
logW1
W2
(34)
with V∞ being the flight speed, g the gravitational acceleration, SFCthe specific fuel consumption (engine dependent, the value of 2 ×10−5 is used), CL is the lift coefficient, CD is the total wing dragcoefficient (the factor of 1.2 accounts for other, not measured, dragsources, such as fuselage and tail).W1 andW2 are the aircraft weightat the beginning and end of the cruise phase, respectively(W2 � W1 −Wfuel).The robust objective function is then established as
F�x; r� � 10−6�−μR − σR� (35)
The results are displayed in Table 9 and Fig. 14. The deterministicoptimum is obtained through the Kriging surrogate model approach.The values in parentheses in Table 9 refer to robustness analyses of
the particular configurations at either the lower or higher uncertaintylevels. Performing this type of analysis on the baseline configurationreveals that the average range is 5723∕5737 km, for the low/highuncertainty problem. The respective standard deviations are 47 and114 km (note the output distributions are not normal, though). Andso, the optimized configurations are comparatively an improvement.A similar analysis to the deterministic optimum reveals that theexpected cruise range is 6100∕6115 km with standard deviations of52∕127 km (low/high uncertainty, respectively). Hence, the robustoptima are not necessarily better performing than the deterministicone. Notwithstanding, this indicates that the reliability constraint is in
Fig. 13 Comparison of results for case I, c.o.v.: altitude 1%, AOA 5%.
Table 8 Design variables and constraints forcase study IIa
Design variable Baseline Minimum Maximum
Angle of attack, AOA 2 deg 1 deg 3 degSemispan, b∕2 17 m 16 m 18 mSweep, ΛLE 32.25 deg 22.25 deg 42.25 degRoot chord, croot 6 m 4.5 m 9 mTip chord, ctip 2 m 1.5 m 2.5 mSpar 1 root chord fraction 0.3333 0.2933 0.3733Spar 1 tip chord fraction 0.3333 0.2933 0.3733Spar 2 root chord fraction 0.6667 0.6267 0.7067Spar 2 tip chord fraction 0.6667 0.6267 0.7067
Angle of attack, AOA 2 deg 2.03 deg 1.99 deg 1.74 degSemispan, b∕2 17 m 18.000 m 18.000 m 18.000 mSweep, ΛLE 32.25 deg 22.25 deg 22.25 deg 22.25 degRoot chord, croot 6 m 4.5 m 4.512 m 5.799 mTip chord, ctip 2 m 1.5 m 1.593 m 1.559 mSpar 1 root chord fraction 0.3333 0.2933 0.2933 0.2998Spar 1 tip chord fraction 0.3333 0.3111 0.3702 0.2933Spar 2 root chord fraction 0.6667 0.6953 0.6347 0.6394Spar 2 tip chord fraction 0.6667 0.6306 0.6345 0.6621F −5.710 −6.087 −6.020 −5.841μR (5723∕5737 km) 6087 km (6100∕6115 km) 6071 km 5961 km
(6100∕6115 km)σR (47∕114 km) 0 km (52∕127 km) 52 km 120 km# evaluations — — 178 2219 2632
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effect (which essentially manifests itself in terms of lower aspectratios and AOA in the configurations obtained withR2BDO), and thedeterministic configuration does not satisfy the required reliabilityrequirement.An issue that becomes even more relevant in these higher-
dimensionality problems is the fact that there may exist severallocal minima, and as such, a global solution cannot be guaranteed.Because of the random nature of the sampling scheme, it wouldbe possible to capture different optima if several independent runswere performed.
V. Conclusions
The examples presented in this paper illustrate the impact theadditional robustness and reliability analyses have on optimizationproblems. The solutions for these differ significantly from theirdeterministic counterparts, even for moderate values of the inputuncertainty. The capability to capture these phenomena comes,however, at the price of greatly increased computational cost.Reliability calculations, in particular, require a large number offunction evaluations, which hampers overall performance. Theinclusion of surrogate model-based optimization practices proved tobe instrumental.The comparative results of RDO and R2BDO in aircraft design
problems show the same trend that was observed in the simpleranalytical problems: RDO invariably returns results that are moreoptimistic than those predicted by R2BDO. Much like what wasobserved in the Rosenbrock examples (Sec. III), although sharing acommon origin, which converges to the deterministic optimum (zerouncertainty), with increasing uncertainty, the RDO and R2BDOsolutions start to diverge. This is due to the enforcement of thereliability constraint on the yield stress inR2BDO, which in this caseturns out to be more conservative than the measure of constraintrobustness used in RDO. Granted, in these situations, validationthrough probabilities of failure is impracticable given the compu-tational effort required by a single evaluation of the wing aero-structural analysis (plus MC would require on the order of 106
samples surrounding each solution point). Futureworkwith access tomassively parallel computer resources could include a full MCvalidation of the failure probability metrics.The modular nature of the framework allows exploration in other
fields related to aircraft design optimization. Topics that could befurther developed include parallelization of the sampling procedures,the support for complete aircraft configurations, or the implemen-tation of different MDO architectures.
Acknowledgments
This work was supported in part by the Fundação para a Ciência eTecnologia under grant SFRH/BD/27863/2006, as well as byAernnova, Aerospace, S.A. The authors also acknowledge AEMDesign for allowing access to the CFSQP source code.
References
[1] Liu, P.-L., Lin, H.-Z., and Kiureghian, A. D., “CALRELUser Manual,”NASA Langley Research Center TR-UCB/SEMM-89/18, Berkeley,CA, 1989.
[2] Rackwitz, R., “The Program Package STRUREL,” Structural Reli-
ability Methods, Wiley, Chichester, England, U.K., 1996, pp. 360–367.[3] Bjerager, P., “The Program System PROBAN,” Structural Reliability
Methods, Wiley, Chichester, England, U.K., 1996, pp. 347–360.[4] Estes,A.C., and Frangopol,D.M., “RELSYS:AComputer Program for
Structural System Reliability Analysis,” Structural Engineering and
Mechanics, Vol. 6, No. 8, 1998, pp. 901–919.[5] Thacker, B. H., Riha, D. S., Fitch, S. H., Huyse, L. J., and Pleming, J. B.,
“Probabilistic Engineering Analysis Using the NESSUS Software,”Structural Safety, Vol. 28, Nos. 1–2, 2006, pp. 83–107.doi:10.1016/j.strusafe.2004.11.003
[6] Wu, Y.-T., Shin, Y., Sues, R. H., and Cesare, M. A., “ProbabilisticFunction Evaluation System (ProFES) for Reliability-Based Design,”Structural Safety, Vol. 28, Nos. 1–2, 2006, pp. 164–195.doi:10.1016/j.strusafe.2005.03.006
[7] Yadav, O. P., Bhamare, S. S., and Rathore, A., “Reliability-BasedRobust Design Optimization: A Multi-Objective Framework UsingHybrid Quality Loss Function,” Quality and Reliability Engineering
International, Vol. 26, No. 1, 2010, pp. 27–41.doi:10.1002/(ISSN)1099-1638
[8] Lee, I., Choi, K. K., Du, L., and Gorsich, D., “Dimension ReductionMethod forReliability-BasedRobustDesignOptimization,”Computersand Structures, Vol. 86, Nos. 13–14, 2008, pp. 1550–1562.doi:10.1016/j.compstruc.2007.05.020
[9] Lagaros, N. D., Plevris, V., and Papadrakakis, M., “Reliability BasedRobust Design Optimization of Steel Structures,” International Journalfor Simulation and Multidisciplinary Optimization, Vol. 1, No. 1, 2007,pp. 19–29.doi:10.1051/ijsmdo:2007003
[10] Jun, S., Jeon, Y.-H., Rho, J., Kim, J., Kim, J., and Lee, D.-H.,“Reliability-Based and Robust Design Optimization with ArtificialNeural Network,” 6th World Congress of Structural and Multidisci-
plinary Optimization, Rio de Janeiro, Brazil, May–June 2005.[11] Paiva, R. M., Carvalho, A. R. D., Crawford, C., and Suleman, A.,
“Comparison of Surrogate Models in a Multidisciplinary OptimizationFramework for Wing Design,” AIAA Journal, Vol. 48, No. 5, 2010,pp. 995–1006.doi:10.2514/1.45790
Fig. 14 Comparison of results for case II. From left to right: baseline, deterministic optimization,R2BDO (c.o.v.: AOA 5%, altitude 2%, payload 5%).
[12] Allen, M., and Maute, K., “Reliability-Based Shape Optimization ofStructures Undergoing Fluid Structure Interaction Phenomena,”Computer Methods in Applied Mechanics and Engineering, Vol. 194,Nos. 30–33, 2004, pp. 3472–3495.doi:10.1016/j.cma.2004.12.028
[13] Rosenblatt, M., “Remarks on aMultivariate Transformation,” Annals ofMathematical Statistics, Vol. 23, No. 3, Sept. 1952, pp. 470–472.doi:10.1214/aoms/1177729394
[14] Kiureghian, A. D., “First and Second-Order Reliability Methods,”EngineeringDesignReliabilityHandbook, CRCPress, BocaRaton, FL,2004, Chap. 14.
[15] Padulo, M., Forth, S. A., and Guenov, M. D., “Robust AircraftConceptual Design Using Automatic Differentiation in MATLAB,”Lecture Notes in Computational Science and Engineering, Vol. 64,2008, pp. 271–280.doi:10.1007/978-3-540-68942-3_24
[16] D’Errico, J. R., and Zaino, N. A. Jr., “Statistical Tolerancing Using aModification of Taguchi’s Method,” Technometrics, Vol. 30, No. 4,1988, pp. 397–405.doi:10.1080/00401706.1988.10488434
[17] Bertuccelli, L. F., and How, J. P., “Robust Markov Decision ProcessesUsing Sigma Point Sampling,” American Control Conference,June 2008, pp. 5003–5008.
[18] Julier, S. J., and Uhlmann, J. K., “Unscented Filtering and NonlinearEstimation,” Proceedings of the IEEE, Vol. 92, No. 3, March 2004,pp. 401–422.doi:10.1109/JPROC.2003.823141
[19] deMenezes, L., Ajayi, A., Christopoulos, C., Sewell, P., and Borges, G.,“Efficient Computation of Stochastic Electromagnetic Problems UsingUnscented Transforms,” IET Science, Measurement and Technology,Vol. 2, No. 2, March 2008, pp. 88–95.doi:10.1049/iet-smt:20070050
[20] Steiner, G., Watzenig, D., Magele, C., and Baumgartner, U.,“Statistical Robust Design Using the Unscented Transforma-tion,” International Journal for Computation and Mathematics in
Electrical and Electronic Engineering, Vol. 24, No. 2, 2005,pp. 606–619.doi:10.1108/03321640510586213
[21] Krog, L., Tucker, A., Kemp, M., and Boyd, R., “Topology Optimiza-tion of Aircraft Wing Ribs,” The Altair Technology Conference,2004.
[22] Agarwal, H., Renaud, J. E., Lee, J. C., and Watson, L. T., “A UnilevelMethod for Reliability Based Design Optimization,” 45th AIAA/ASME/
ASCE/AHS/ASC Structures, Structural Dynamics and Materials
Conference, AIAA Paper 2004-2029, 2004.[23] Frangopol, D. M., and Maute, K., “Life-Cycle Reliability-Based
Optimization of Civil and Aerospace Structures,” Computers and
Structures, Vol. 81, No. 7, 2003, pp. 397–410.doi:10.1016/S0045-7949(03)00020-8
[24] Frangopol, D. M., and Maute, K., “Reliability-Based Optimizationof Civil and Aerospace Structural Systems,” Engineering Design
Reliability Handbook, CRC Press, Boca Raton, FL, 2004, Chap. 24.[25] Tu, J., Choi, K. K., and Park, Y. H., “ANew Study on Reliability-Based
Design Optimization,” Journal of Mechanical Design, Vol. 121, No. 4,1999, pp. 557–564.doi:10.1115/1.2829499
[26] Yu, X., and Du, X., “Reliability-Based Multidisciplinary Optimizationfor Aircraft Wing Design,” Structure and Infrastructure Engineering,Vol. 2, Nos. 3–4, 2006, pp. 277–289.doi:10.1080/15732470600590333
[27] Cruse, T., Wu, Y.-T., Dias, B., and Rajagopal, K., “ProbabilisticStructural Analysis Methods and Applications,” Computers and
Structures, Vol. 30, Nos. 1–2, 1988, pp. 163–170.doi:10.1016/0045-7949(88)90224-6
[28] Koch, P. N., Yang, R.-J., and Gu, L., “Design for Six Sigma ThroughRobust Optimization,” Journal of Structural and Multidisciplinary
Optimization, Vol. 26, Nos. 3–4, 2004, pp. 235–248.doi:10.1007/s00158-003-0337-0
[29] Paiva, R., “A Robust and Reliability-Based Optimization Frameworkfor Conceptual Aircraft Wing Design,” Ph.D. Thesis, Dept. ofMechanical Engineering, Univ. of Victoria, B.C., Canada, Dec. 2010.
[30] Giunta, A. A., Eldred, M. S., and Castro, J. P., “UncertaintyQuantification Using Response Surface Approximations,” Proceedingsof the 9th ASCE Joint Specialty Conference on Probabilistic Mechanics
and Structural Reliability, Albuquerque, NM, July 2004.[31] Taylor, R. L., “FEAP—A Finite Element Analysis Program,”
Manual–Ver. 8.2, Univ. of California at Berkeley, Dept. of Civil andEnvironmental Engineering, March 2008.
[32] Martins, J. R. R. A., and Poon, N. M. K., “On Structural OptimizationUsing Constraint Aggregation,” 6th World Congress on Structural and
Multidisciplinary Optimization, Rio de Janeiro, Brazil, June 2005.
