Aircraft wing box optimization considering uncertainty in surrogate models

Struct Multidisc Optim (2010) 42:745–753DOI 10.1007/s00158-010-0532-8

INDUSTRIAL APPLICATION

Aircraft wing box optimization considering uncertaintyin surrogate models

Daniel Neufeld · Kamran Behdinan · Joon Chung

Received: 23 December 2009 / Revised: 24 March 2010 / Accepted: 10 June 2010 / Published online: 4 July 2010c© Springer-Verlag 2010

Abstract Aerospace design often involves computation-ally expensive physics based analysis methods such asComputational Fluid Dynamics (CFD) or the Finite Ele-ment Method (FEM). Since conceptual design optimizationcan require many function evaluations, simplified analysismethods are typically used. Designs optimized with sim-plified analysis methods may be found to violate designgoals when subjected to the high fidelity approaches later inthe design process. This paper presents how the uncertaintyintroduced by an approximation model in the conceptualdesign of the wing box of a generic light jet can be assessedand managed by applying Reliability Based Design Opti-mization (RBDO) in order to ensure that a feasible solutionis obtained. Additionally, the performance of several alter-native RBDO approaches are benchmarked using the wingbox conceptual design problem.

Keywords RBDO · MDO · Aircraft conceptual design

1 Introduction

Computational design optimization has enabled designersto explore the solution space of engineering problems withgreat accuracy and efficiency when compared the manual,iterative approaches traditionally used in conceptual design.Design optimization methods such as Genetic Algorithms(GAs) and Sequential Quadratic Programming (SQP) arecapable of a level of precision that enables them to pre-cisely locate optima for a given set of design equations.

D. Neufeld (B) · K. Behdinan · J. ChungRyerson University, Toronto, Ontario, Canadae-mail: [email protected]

A consequence of this level of precision is that optimumdesigns nearly always lie exactly on one or more of the con-straint boundaries (Youn et al. 2004). Any deviation in thedesigner’s assumptions such as the strength of a material,the manufacturing precision, or any approximate analy-sis methods can lead to the failure of optimized designswhen they are subjected to more rigorous analysis laterin the design process. RBDO is a method for enforcing adesired probability that a design will be viable provided thatthe sources of uncertainty can be modeled by probabilitydistribution functions.

Practical engineering problems are often characterizedby computationally costly simulation or physics based anal-ysis methods such as CFD or FEM. The design optimizationof such systems is computationally expensive since manyfunction evaluations may be required to obtain a convergedsolution. Additionally, non-smooth objective or constraintfunctions may hamper solution convergence for gradientbased optimization methods. Surrogate modeling is a pro-cedure for reducing the computational cost in optimizationby representing the high fidelity analysis methods mathe-matically. The high fidelity methods are sampled across apredetermined set of design variables. These are usuallyselected such that the design space is evenly covered. Thesurrogate models utilize the sample data to mathematicallyrepresent the design space by a best-fit equation or sets ofequations. The equations can be solved with very little com-putational expense and optimized solutions can be quicklyobtained. For these reasons, surrogate models are widelyimplemented in design optimization (Jin et al. 2001). How-ever, surrogate models only approximate the high fidelityanalysis, and therefore introduce uncertainty into the opti-mization. The optimum designs obtained using surrogatemodels may be found to be infeasible when the designis subjected to high fidelity analysis methods. RBDO can

746 D. Neufeld et al.

be used to manage the uncertainties introduced by surro-gate models in order to increase the confidence a designermay place in an optimized solution obtained using surrogatemodels.

RBDO is widely viewed as a better method for handlinguncertainty in engineering design than simply applyingsafety factors (Smith and Mahadevan 2003). RBDO sup-plants the need for safety factors by tracking the influenceof error through the design optimization process by assess-ing each active constraint to determine the likelihood offailure. RBDO replaces uncertain design variables or con-stants with a probabilistic quantity. The design constraintsare redefined as probabilistic constraints, enforcing themaximum allowable likelihood of failure rather than deter-ministic inequalities. An acceptable likelihood of failure isdefined by the designer. RBDO methods drive optimizeddesigns deeper into feasible design space by altering onlythe variables or constants that directly affect the likelihoodof failure of active constraints. Variables or constants thathave little or no influence on active constraints are notaltered. Consequentially, RBDO potentially yields betterdesigns than deterministic optimization with safety factors.RBDO has been applied to a wide variety of engineeringproblems that encounter uncertainties in material proper-ties, manufacturing tolerances, contributing equations, andothers. Youn et al. (2004) studied vehicle crash-worthinessunder an uncertain impact location on a vehicle frame con-structed with structural members having uncertain dimen-sions due to the variability in manufacturing. Deb et al.(2007) solved the same crash-worthiness problem usingevolutionary algorithms in order to enable handling multipleobjective functions including a reliability index objective.Smith and Mahadevan (2003) implemented RBDO to solvea spacecraft conceptual sizing problem considering modeluncertainty incurred due to the use of response surfaceapproximations to the system analysis equations.

2 Review of RBDO strategies

There are many approaches currently available for assessingthe reliability of probabilistic constraints in design opti-mization. Many simulation based methods exist such asMonte-Carlo sampling, Latin hypercube sampling. Thesemethods work by varying the probabilistic quantities andcounting the number of constraint failures to estimate theprobability, requiring a very large number of function eval-uations. Such methods usually rely on response surfaceapproximation models rather than system equations to esti-mate failure probabilities (Ramu et al. 2006) to reducethe computational expense. Simulation methods are stillwidely used for problems that are highly non-linear withrespect to the uncertain variables and parameters (Allen and

Maute 2004). Allen et al. implemented this approach tosolve a reliability based optimization of aeroelastic struc-tures (Allen and Maute 2004). Alternatives to simulation-based approaches include several analytical methods. Themost widely implemented approaches to reliability assess-ment are derived from the First Order Reliability Method(FORM), which operates by translating all uncertain designvariables and parameters into normal distribution spaceand calculating the distance between the nearest problemconstraint boundary to the current design point. The corre-sponding point along the constraint boundary is referred toas the Most Probable Point (M P P). The distance betweenthe design point and the M P P is defined as the Reliabil-ity Index, β, which corresponds to the number of standarddeviations from the mean value a design point lies froma constraint boundary in normal space. In FORM, theconstraint boundary is approximated as a straight tangentline, which is usually a reasonable approximation if theconstraints are not highly non-linear with respect to theuncertain variables or constants. There are several numericalapproaches for calculating the location of the M P P and thecorresponding β. The two most common methods includethe Reliability Index Approach (RIA) and the PerformanceMeasure Approach (PMA).

2.1 Reliability index approach

RIA is a direct method for calculating β (Yu et al. 1997).The uncertain variables are normalized by applying a trans-formation to all of the uncertain variables and parameters(uncertain quantities that are not changed by the optimizer).

G =0

βjr

G <0

U–Space

u1

u2

0

Infeasibleregion,

MPP

U*j

j

Fig. 1 RIA approach (Deb et al. 2007)

Aircraft wing box optimization considering uncertainty in surrogate models 747

For example, the transformation equation for a normal dis-tribution is the U = x − μ

σwhere U is the design and

uncertain variable vector in normal space and μand σare thevariable or parameter mean values and standard deviationsrespectively. The reliability index, β,is then calculated bysolving the optimization problem shown in (1) and Fig. 1,where G is a constraint function evaluated at normalizedvariable vector U .

Minimize‖U‖Subject to Gi (U ) = 0

(1)

Solving (1) calculates the co-ordinates of the M P P in nor-mal space. The distance from the M P P to the design pointis the reliability index, βi , and is calculated by (2).

βi ≈ ‖UGi =0‖ (2)

In an optimization scheme, the calculated βi values for eachconstraint are constrained by the optimizer to reach a tar-get reliability index. RIA can have convergence problemsdue to the equality constraint on the potentially non-linearproblem constraint functions (Tu et al. 1999). Despite thisdrawback, the RIA method has the advantage that the reli-ability index for each constraint can be calculated directly,unlike the PMA method, where a desired reliability indexmust be implicitly enforced for each constraint. This is par-ticularly useful for multi-objective optimizations, where it isdesired to study designs having a range of reliability indices.Deb et al. (2007) demonstrated this approach on an analyt-ical problem and a vehicle crash-worthiness problem usingevolutionary algorithms.

2.2 Performance measure approach

The PMA method, introduced by Tu et al. (1999), solvesthe inverse of the RIA optimization problem. As shown inFig. 2, the constraint level, Gi (U ) is minimized subjectto a constraint that forces the distance in normal space tothe M P P to become equal to the desired reliability indexβ, resulting in the depth into feasible design space thatmust be enforced. This is an improvement over the RIAmethod because non-linear constraint functions become aminimization problem, not an equality constraint as in theRIA method, resulting in an approach that is generallymore reliable and efficient for most applications (Younand Choi 2004). For aircraft conceptual design, manynon-linear constraint functions are enforced. These includedesired performance characteristics, stability, and others.The RIA method requires enforcing each constraint in turnto an equality condition. This leads to poor convergencecharacteristics in gradient-based optimizers, making the

G =c

βjr

G <0

G =0

U*

MPP

U–Space

u1

u2

0

Infeasibleregion, j

j

j

Fig. 2 PMA approach (Deb et al. 2007)

PMA method a more suitable approach for such problemssince each constraint function is minimized in the RBDOproblem objective function, not constrained to an equalitycondition.

Minimize Gi (U )

Subject to ‖U‖ = βt(3)

2.3 RBDO-MDO integration strategies

There are currently three basic approaches to integrate theFORM calculation in an optimization framework: doubleloop methods, sequential methods, and single loop meth-ods. Yang et al. describes and compares the performanceof these approaches in Yang and Gu (2004). The threeapproaches are briefly described as follows. Double loopmethods nest PMA or RIA reliability assessments in everyconstraint evaluation in an optimization. If RIA is used,each constraint value is replaced by the reliability indexachieved by each constraint at the current design point.For PMA, the constraints are evaluated at a shifted posi-tion corresponding to the target reliability index. Sequentialmethods include the Safety Factor Approach (SFA) (Wu andWang 1998) and the Sequential Optimization and Reliabil-ity Assessment (SORA) (Du and Chen 2004) and operateby first preforming a full deterministic optimization, thenconducting a reliability assessment of each constraint usingPMA. A modified deterministic optimization problem isformulated by shifting each constraint to be feasible atthe desired reliability index. The modified deterministic


problem is solved in sequence with the reliability assess-ment until a consistent solution is reached. Performancecomparisons have shown that sequential methods oftenrequire significantly fewer function evaluations when com-pared to the double loop approach (Yang et al. 2004; Yangand Gu 2004; Wu et al. 2001; Du and Chen 2004). Boththe single loop and double loop approaches implement exactsolutions to the FORM method by solving either the PMAor RIA methods. Single loop methods implement gradient-based approximations to estimate the FORM solution anddo not require nested reliability assessment loops. The Sin-gle Loop Single Vector (SLSV) approach was introducedby Chen et al. (1997) and subsequently enhanced by Lianget al. (2008). SLSV based methods solve a gradient basedapproximation to the PMA method which is updated atevery iteration of a gradient based optimizer.

Implementations of double loop, sequential, and sin-gle loop methods were developed and integrated under twodifferent Multi-disciplinary Design Optimization (MDO)approaches: the Multi-Discipline Feasible (MDF) methodand the Collaborative Optimization (CO) method. The MDFmethod is one of the most basic MDO architectures. Itis driven by a single global optimizer and handles inter-discipline coupling by a Multi-Discipline Analysis (MDA)loop—an iterative process that ensures that inter-disciplinecoupling variables are consistent for every optimizer iter-ation. The CO method, introduced by Braun and Kroo(1995), decouples the multi-discipline optimization prob-lem into a multi-level optimization scheme whereby localoptimizers handle each discipline independently under aglobal optimizer which provides target values, minimizesthe objective function, and ensures compatibility betweenthe disciplines at the solution. CO is generally regardedas an inefficient method for small-scale problems, but haspotential advantages for problems that have large num-bers of local variables (Braun et al. 1996; Lin 2004).For RBDO based optimization schemes, CO has an addi-tional advantage: each discipline can implement differentRBDO or deterministic optimization strategies, enablingthe use of methods that work best for each set of localequations.

3 Wing box optimization with an uncertaincontributing analysis

A wing box conceptual design problem was solved, illus-trating how uncertain discipline analysis methods influencethe optimum solution in a practical engineering problem. Aparametrized finite-element model of a generic light busi-ness jet wing box was developed and is shown in Fig. 3.Following common practice in aircraft conceptual design,a target aircraft mass and a wing subgroup mass budgetwas assumed. The target gross aircraft mass was assumedto be 5,200 kg with a wing-stored fuel capacity of 1,200kg—similar in size and performance to the Citation CJ4or the Beechcraft Premier II. The wing weight budget wasassumed to be 440 kg. The maximum von Mises stress wasconstrained to be below 360 MPa—the yield strength of alu-minum 7075 divided by the regulation specified margin of1.5. The optimization objective was to maximize the winglift-to-drag ratio at a cruise speed of 400 kts and an altitudeof 35,000 ft. The body contribution to the lift-to-drag ratiowas neglected. The FEM analysis was replaced by a Krigingsurrogate model. The model was randomly sampled to esti-mate the error probability distribution associated with themaximum stress constraint.

3.1 Problem description

The problem was formulated as a multi-discipline optimiza-tion with two contributing analysis methods: a vortex-latticeaerodynamics solver and a structures solver consisting of aKriging approximation model. This model was generatedusing a database of finite-element solutions sampled evenlyacross the design space. This surrogate model was consid-ered as an uncertain contributing analysis by assessing themodel error between FEM runs and the surrogate model.

The FEM model implements 29 member attributes repre-senting the thicknesses of the primary structural members—19 ribs, the front and rear spar, six stringers, and the upperand lower skin. The dimensionality was reduced by link-ing the attributes to common design variables as shownin Table 1. Figure 4 shows the member assignments of

Fig. 3 Wing box FEM model


Table 1 Member attribute list

Member Variable Limit (mm)

number Lower Upper

1 Rib 1 1 2 5

2 Rib 2

3 Rib 3

4 Rib 4

5 Rib 5

6 Rib 6

7 Rib 7

8 Rib 8

9 Rib 9

10 Rib 10

11 Rib 11

12 Rib 12

13 Rib 13

14 Rib 14

15 Rib 15

16 Rib 16

17 Rib 17

18 Rib 18

19 Rib 19

20 Front spar 2 10 30

21 Rear spar 3 10 30

22 Upper stringer 1 4 2 10

23 Upper stringer 2

24 Upper stringer 3

25 Lower stringer 1 5 2 10

26 Lower stringer 2

27 Lower stringer 3

28 Upper skin 6 15 30

29 Lower skin 7 15 30

the structural design variables on the cross section of thewing box. Two variables were introduced to alter the overallwing geometry: span and wing reference area, for a total of

Fig. 4 Structural discipline variables

nine system design variables. The sweep angle, taper ratio,and airfoil shape were held constant. Considering all 29attributes additional wing shape variables would potentiallyyield better designs. However, the accuracy of the surrogatemodel with 31 dimensions was found to be extremely low.

3.2 Surrogate model

Kriging models are widely used as surrogates for computa-tionally expensive analysis methods in design optimization(Venter et al. 1998; Roux et al. 1998). Kriging modelsare formulated such that they pass through every givendesign point unlike response surfaces, and are thereforeless prone to model bias. A Kriging model was calculatedfrom a database of finite element analysis solutions sam-pled across 200 evenly distributed design points. Increasingthe database beyond 200 did not appreciably improve theaccuracy of the approximation model. The model replacedthe FEM analysis in the optimization, saving computationalexpense while introducing a source of uncertainty. Theuncertainty was quantified by adding a probabilistic errorterm, defined in (4), to the optimization problem. The errorterm, εσ , represents the ratio of the model predicted max-imum von-Mises stress, σp, to the stress calculated fromFEM analysis, σF E M , using Ansys, a well known com-mercial finite element solver (DeSalvo and Swanson 1979).

εσ = σp/σF E M (4)

3.3 Model error

A database containing the variables and properties of 200random, uniformly distributed design variable vectors was

0.9 0.95 1 1.05 1.10

1

2

3

4

5

6

7

8

Data

Den

sity

Model Error Histogram

Normal Distribution

Fig. 5 Kriging model error distribution: μ = 1.0039,σ = 0.0736


Table 2 Error database sample

A1 A2 A3 A4 A5 A6 A7 b S Mass L/D σp σFEM εσ

(mm) (mm) (mm) (mm) (mm) (mm) (mm) (m) (m2) (kg) (MPa) (MPa)

3.29 20.9 20.7 4.72 7.00 20.9 21.3 10.0 8.45 220 33.4 409 393 1.040

2.45 24.3 21.7 5.17 6.82 23.1 24.1 10.6 6.73 193 37.9 708 622 1.134

4.10 17.7 17.1 5.80 9.03 24.4 25.1 10.3 8.47 242 34.9 464 456 1.018

4.21 21.5 20.2 5.22 8.50 16.4 15.8 10.8 10.6 233 35.0 443 473 0.936

2.71 22.4 19.5 5.93 6.34 25.2 27.3 8.59 7.18 220 27.6 317 309 1.024

3.69 20.1 21.1 4.49 8.17 20.7 15.0 11.1 5.93 136 41.4 1,213 1,391 0.872

2.68 20.9 25.5 4.40 5.70 14.7 27.3 11.6 8.47 224 40.1 883 853 1.036

2.36 23.8 18.5 4.61 5.11 22.2 17.5 12.6 10.4 258 41.5 633 639 0.990

2.59 18.1 16.3 4.93 5.02 21.0 18.5 9.21 8.04 191 30.1 422 407 1.036

2.87 22.2 25.0 4.25 5.18 15.2 17.6 8.91 8.90 202 28.1 365 404 0.904

generated. The database contains the geometry, mass, andthe stress obtained from the Kriging model and the FEManalysis. The ratios between the FEM-based and the pre-dicted stress were obtained. The error ratio distribution wasfound to approximate a normal distribution with a meanvalue, μ, of 1 and a standard deviation, σ , of 0.0736.The estimated distribution shape is shown in Fig. 5. Alsoshown is the histogram of the 200 data points for com-parison. Increasing the sample size beyond 200 did notsignificantly change either the distribution shape, the mean,or the standard deviation. A sample of the database isshown in Table 2 to indicate the performance of the Krigingmodel.

3.4 Solution strategy

The wing box was optimized using both MDF and CO archi-tectures using the double loop, sequential, and single loopmethods. The problem formulation under the MDF basedmethods is shown in (5) where L/D is the wing lift-to-dragratio, b and S denote the wing span and wing reference arearespectively. Structural thickness values are denoted by t .The constraints are given in Table 3.

The objective function is to maximize wing L/D suchthat the mass, M , and the approach speed Va are less thanthe limits and the probability that the stress, σ , is less thanthe limit stress greater than the target probability. Note that

Table 3 Constraints

Constraint Symbol Value

Maximum stress σ 360 Mpa

Mass M 440 kg

Approach speed Va 120 kts

every function and constraint evaluation is an MDA loop toensure compatibility between the disciplines.

maxL

D= f (b, S, t1...7) (5)

P (σ ≤ σmax ) ≥ Pgoal

M ≤ Mgoal

Va ≤ Va,goal

The solution strategy for the CO architecture is shown inFig. 6. The problem is divided into aerodynamics and struc-tures sub-problems. The aerodynamics sub-problem is afunction of only the wing span and wing area design vari-ables since the thickness values of the structural membersare not required for the aerodynamics discipline to evalu-ate. The structures discipline depends on both the memberthicknesses and the overall wing shape, and is thereforea function of all nine design variables. The system level

Fig. 6 Collaborative optimization with RBDO


0 1 2 3 4 5 628

30

32

34

Reliability Index

Win

g L/

D R

atio

0 1 2 3 4 5 62

2.5

3

3.5

4x 10

8M

axim

um v

on-M

ises

Str

ess

(Pa)

Predicted

ActualMaximum Allowed

Fig. 7 Wing box RBDO results

problem was defined as a function of the two commonvariables—wing span and area. Axillary constraints Aaero

and Astruct force consistency between the disciplines. Anysingle-discipline deterministic or RBDO approach can beused for the local optimizations.

3.5 Results

The optimization was solved for reliability indices of 0to 6 for using each RBDO-MDO approach. This corre-sponds to failure probabilities of 50% to 10−7%. Figure 7

shows the influence of increasing target reliability indiceson the location of the optimum. The predicted values ofthe maximum stress at the optimum point for each targetreliability index lie on the stress constraint boundary whenβ is zero. When β increases, the wing shape and materialthicknesses are altered, as shown in Table 4, such that thepredicted stress moves farther into feasible design space.The slenderness and span of the wing are the attributes mostsignificantly affected by increasing the reliability index. Thepredicted optimum points were evaluated using FEM anal-ysis to check for possible discrepancies between the stress

Table 4 Design variable values—MDF/sequential method

β Pf ail A1 A2 A3 A4 A5 A6 A7 b S L/D

(%) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (m) (m2)

0 50 2.30 18.1 15.2 4.71 6.02 18.4 14.9 10.3 10.6 31.9

1 16 2.30 17.9 15.2 4.68 5.92 18.5 15.0 10.0 10.6 31.1

2 2.3 2.30 17.9 15.5 4.66 5.91 18.5 15.2 9.83 10.5 30.9

3 0.13 2.30 17.9 15.7 4.57 5.81 18.5 15.5 9.64 10.4 30.3

4 3.2 × 10−3 2.30 17.8 15.8 4.56 5.80 18.6 15.5 9.48 10.4 29.7

5 2.9 × 10−5 2.30 17.8 16.1 4.47 5.71 18.8 16.0 9.30 10.2 29.1

6 9.9 × 10−8 2.30 17.8 16.5 4.45 5.70 18.8 16.2 9.16 10.1 28.5


(a) Wing Planform (b) Wing Box Cross Section

0 1 2 3 4 5

-1

-0.5

0

0.5

1

1.5

2

2.5

wing half-span (m)

win

g ch

ord

(m)

β = 0

β = 2

β = 4

β = 6

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

wing box root chord (m)

win

g bo

x ro

ot th

ickn

ess

(m)

β = 0

β = 2

β = 4

β = 6

Fig. 8 a, b Wing geometry

predictions and the actual stress. We can observe that thetrue stress values are larger than the predicted stress, indi-cating that the structure designed using the low-fidelityapproach fails when subjected to better analysis. We cansee that by using RBDO with reliability indices larger thanβ = 1.2 (88%), the design is feasible. Reliability indicesabove this yield conservative designs that fall well withinthe feasible region when subjected to the high fidelity anal-ysis. Since the standard deviation of the error was verylow, solutions up to β = 6 could be obtained. In gen-eral, however, if the approximate models are less accurate(if σ is large), the changes to the design may be verydrastic.

Figure 8a shows the influence of the reliability indexon the shape of the wing. The wing shape becomesmore conservative as the reliability index increases. Thedeterministic optimum wing shape is a longer and moreslender design than the reliable solutions. Aerodynam-ics dictates that increasing wing slenderness reduces theinduced drag. However, slender structures require morematerial (increased structural member thicknesses) to main-tain strength. Increases to the reliability level increases themargin by which the designs must exceed the specifiedstress constraint while remaining under the mass bud-get. Figure 8b shows how the wing box outer dimen-sions are increased with the reliability level. The reli-able wing designs trade aerodynamic efficiency for struc-tural efficiency by reducing the slenderness of the wingwhile remaining at or below the prescribed weight bud-get. Table 4 indicates that increases to the reliabilityindex has a slight influence on the thicknesses of struc-tural members while the wing planform shape is stronglyinfluenced.

3.6 Algorithm performance comparison

Each RBDO-MDO architecture was run with a reliabilityindex of 6 to compare the relative performance of eachapproach. We can observe from Table 5, the single loopmethods have significant performance advantages, but pre-dict an optimum L/D that is a slight outlier from themethods that use exact solutions to FORM. The CO basedapproaches require many structural function evaluations butrelatively few aerodynamics evaluations due to the reduceddimensionality of the two-variable aerodynamics sub prob-lem. As a consequence, despite having large numbers ofstructural function evaluations, the solution times for theCO methods were found to be competitive since evaluat-ing the structures approximation model is extremely fastcompared to calls to the aerodynamics solver. Since theaerodynamics sub-problem is a two-variable optimization inthe CO framework, fewer calls to the aerodynamics solverare needed.

Table 5 RBDO-MDO performance comparison

Method Function calls Objective

Aero Struct L/D

MDF/sequential 96 96 28.50

MDF/double loop 105 105 28.49

MDF/single loop 37 37 28.22

CO/sequential 42 2,362 28.61

CO/double loop 26 3,120 28.65

CO/single loop 23 1,477 28.23


4 Conclusion

The wing box of a generic light jet was optimized usingRBDO while considering the model uncertainty introducedwhen using surrogate models. The error distributions wereevaluated by uniformly sampling the Kriging models andcomparing the predicted maximum stress with FEM solu-tions. The solution indicates that under traditional optimiza-tion, the structure optimized with an approximation modelwould fail the stress constraint when subjected to highfidelity analysis. However, it was shown that RBDO with asufficiently low failure probability protects the low fidelitysolution from producing infeasible designs. Additionally,the performance of different RBDO-MDO architectureswere evaluated. The single loop approaches were the mostefficient under both MDF and CO architectures. However,the solutions deviated slightly from those from methods thatemploy exact solutions to FORM, indicating a slight lossin accuracy. The CO based method with sequential RBDOsignificantly reduced the number of aerodynamic disciplineevaluations by holding local structural variables to the struc-tural sub problem. Since the model evaluations are veryfast, the CO/sequential method exhibited solution times thatwere competitive with the other approaches despite the largenumber of structural discipline evaluations.

In many practical problems, it may not be possible tocross-check the obtained solutions with accurate analysis.Problems where the approximation models are derived fromphysical testing or observations rather than high fidelityanalysis are examples of such problems. An allowable fail-ure probability must be defined by the designer in suchcases. When selecting this limit, it should be noted thatoptimization methods always search for the best possibleobjective function performance given the provided analy-sis methods. The optimizer will therefore move into designspace that has advantage to the objective function whetherthe design is truly better than the adjacent designs or thedesign falls within a region where the analysis methodsover-predict the performance of the design. The optimizersees no distinction. This renders it more likely for a solutionto lie in a region where performance characteristics are over-predicted. This is consistent with Fig. 7, where the optimumpredicted stress is always lower than the observed stress.Deterministic optima are therefore consistently optimistic,and promise performance characteristics that are not likelyto be achieved when the design is subjected to better anal-ysis methods. Consequentially, it is important to implementprocedures such as RBDO to manage uncertain analysismethods in order to find the best possible designs that lie nocloser to the constraint boundaries than an allowable limit.A reliability index target of β = 3, which corresponds toa failure probability of 0.13%, is widely used in RBDOliterature.

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