Multidisciplinary wing design optimization considering global sensitivity and uncertainty of approximation models

Journal of Mechanical Science and Technology 28 (6) (2014) 2231~2242

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-014-0127-1

Multidisciplinary wing design optimization considering global sensitivity and

uncertainty of approximation models† Hyeong-Uk Park1, Joon Chung1,*, Kamran Behdinan2 and Jae-Woo Lee3

1Department of Aerospace Eng., Ryerson University, 350 Victoria Street, Toronto, Ontario, M5B 2K3, Canada 2Department of Mechanical & Industrial Eng., University of Toronto, 5 King's College Road, Toronto, Ontario, M5S 3G8, Canada

3Aerospace Information Eng., Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul, 143-701, Korea

(Manuscript Received April 11, 2013; Revised November 8, 2013; Accepted November 8, 2013)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract In recent years, high-fidelity analysis tools, such as computational fluid dynamics and finite element method, have been widely used in

multidisciplinary design optimization (MDO) to enhance the accuracy of design results. However, complex MDO problems have many design variables and require long computation times. Global sensitivity analysis (GSA) is proposed to assuage the complexity of design problems by reducing dimensionality where variables that have low impact on the objective function are neglected. This avoids wasting computational effort and time on low-priority variables. Additionally, uncertainty introduced by the fidelity of the analysis tools is con-sidered in design optimization to increase the reliability of design results. Reliability-based design optimization (RBDO) and possibility-based design optimization (PBDO) methods are proposed to handle uncertainty in design optimization. In this paper, the extended Fou-rier amplitude sensitivity test was used for GSA, whereas a collaborative optimization-based framework with RBDO and PBDO was used to consider uncertainty introduced by approximation models. The proposed method was applied to an aero-structural design optimi-zation of an aircraft wing to demonstrate the feasibility and efficiency of the developed method. The objective function was to maximize the lift-to-drag ratio. The proposed process reduced calculation efforts by reducing the number of design variables and achieved the target probability of failure when it considered uncertainty. Moreover, this work evaluated previous research in RBDO with MDO for the wing design by comparing it with the PBDO result.

Keywords: Multidisciplinary design optimization; Global sensitivity analysis; Extended Fourier amplitude sensitivity test; Wing design; Uncertainty; Reli-

ability-based design optimization; Possibility-based design optimization ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Numerous design variables are associated with the multid-isciplinary design optimization (MDO) of aircraft designs. In recent years, high-fidelity tools and parallel computing have been implemented with MDO to improve the accuracy of design results with enhanced computing power. Depending on the level of maturity of the design, the number of parameters required to describe an aircraft can vary from tens to thou-sands. The complex problem decomposition and large number of design variables in MDO problems require long computa-tion time. If design variables do not significantly affect the optimization result, the computational effort and time con-sumption for processing these variables may be wasted. The number of variables can be reduced, provided that the de-signer can determine which variables are important for a given design optimization task and which have little effect. Many

sensitivity analysis methods have been applied to engineering problems to identify the important parameters in design opti-mization problems [1-4]. Global sensitivity analysis (GSA) offers a comprehensive approach to model analysis and evalu-ates the effect of a factor while all other factors also vary [1, 5]. The method of global sensitivity indices was first suggested by Sobol’ (1990) [6] and then further developed by Saltelli and Sobol’ (1995) [7]. Homma and Saltelli (1996) [8] devel-oped variance-based GSA techniques that provided informa-tion on the importance of different subsets of input variables on the output variance.

Recent studies in design optimization focus on methods for handling uncertainty to increase the reliability of optimized solutions [9]. Reliability-based design optimization (RBDO) enforces probabilistic constraints in optimization problems, which propose to consider uncertainty. Moreover, it is the prevailing approach in stochastic design optimization, where information regarding any uncertain parameter is sufficient to generate accurate input of statistical distribution functions. When a sufficient amount of uncertain data cannot be ob-

*Corresponding author. Tel.: +1 416 979 5000, Ext. 7213, Fax.: +1 416 979 5056 E-mail address: [email protected]

† Recommended by Associate Editor Jeong Sam Han © KSME & Springer 2014

2232 H.-U. Park et al. / Journal of Mechanical Science and Technology 28 (6) (2014) 2231~2242

tained, the probabilistic method cannot be used for reliability analysis and design optimization. Possibility-based design optimization (PBDO) is proposed to overcome this disadvan-tage of RBDO. PBDO uses the fuzzy function for modeling uncertain parameters and is useful for cases that have insuffi-cient data for producing probability density functions [10]. Many researchers have studied reliability-based MDO meth-ods. Various MDO techniques are implemented with RBDO [11-15]. Multidisciplinary feasible and individual discipline feasible methods have been implemented, but these ap-proaches require large-scale disciplinary analysis at the system level to locate most probable points [16, 17]. Concurrent sub-system optimization (CSSO) and bi-level integrated system synthesis (BLISS) methods are used with RBDO, but the for-mulation of CSSO and BLISS is generally too complicated for implementation. Collaborative optimization (CO) was imple-mented in this research to acquire efficiency in design formu-lation with uncertainty consideration, where RBDO and PBDO were integrated. The system-level objective function was unchanged from the deterministic optimization. On the other hand, the constraints were updated from RBDO and PBDO results to incorporate uncertainty associated with vari-ous parameters. Because the compatibility between disciplines was enforced by the objective function of each local optimiza-tion, auxiliary constraints do not appear in local optimization problem statements. Therefore, modification of reliability analysis with the coupling variables and compatibility con-straints was unnecessary. As a result, the formulation of CO with uncertainty concluded to be an easier method than other MDO techniques.

The required computer codes and analysis tools for engi-neering design are becoming more complex. With the imple-mentation of MDO, massive engineering data exchange and multidiscipline-related system analysis are indispensable; hence, inefficiency in time and cost may exist without a care-ful design strategy [18, 19]. In most cases, the responses ob-tained from the analysis of either a single discipline or multi-disciplines through the system approach have numerical noises, irregularities, and discontinuities. These factors make it difficult to obtain gradient information and increase compu-tational loads [20]. Studies on approximation techniques to resolve these issues are an important approach in system de-sign using MDO. Response surface method (RSM) is a statis-tical method utilizing design of experiment theory [21, 22]. RSM constructs a multidimensional surface (response surface) by creating an experimental model with obtained data to pre-dict the response of non-experimented regions. The approxi-mation methods reduce the computational cost in the optimi-zation by mathematically representing high-fidelity analysis methods [23].

In this study, a design process with GSA, RBDO, and PBDO with MDO was proposed. This process reduced calcu-lation time and improved the probability of optimization re-sults when it considered uncertainty. Important design pa-rameters were identified using GSA. The design variables

with low sensitivity with respect to the objective function were fixed as initial values to reduce calculation time and cost for the design optimization. In addition, RBDO and PBDO methods were implemented with CO to consider the uncer-tainty introduced by the approximation models. Previous re-search used RBDO to consider the uncertainty of approxima-tion models on the wing box optimization problem [24]. However, uncertainty from low-fidelity analysis methods or approximation models can also be handled more conserva-tively by using PBDO. The wing box optimization problem in Ref. [24] was re-evaluated and compared with the results by using PBDO.

2. Design process and methods

Random sample cases for the wing box structure were ana-lyzed with ANSYS, a high-fidelity finite element analysis (FEA) tool. In this research, 148 cases were derived using the Latin hypercube method and analyzed to secure accuracy on the surrogate model, which was generated in previous re-search by using 200 sample cases [24]. However, 52 cases that violated constraints were removed in this research, and re-maining cases were kept to generate RSM for nine design variables [25]. The approximation model of the wing box structure was derived using these FEA data. An extended Fou-rier amplitude sensitivity test (e-FAST) was implemented to derive sensitivity indices. This work identified important de-sign variables of structural discipline for the objective function. The uncertain parameter of the structural analysis was gener-ated by comparing results between the approximation model and FEA. The dimensionality in the optimization problem was reduced by fixing variables with low sensitivity indices to their initial value.

CO was used for the wing box design problem [26]. The panel method was used as an aerodynamic analysis module, and the approximation model was used as the structural analy-sis module with the uncertain parameter. For this research, general CO, RBDO with CO, and PBDO with CO were for-mulated to compare the results. The proposed design process is shown in Fig. 1, and details are described in this section.

2.1 GSA

This research implemented e-FAST method to determine global sensitivity indices [6]. e-FAST is based on the original FAST method, which is more efficient than the Monte Carlo

Fig. 1. Proposed design process.

H.-U. Park et al. / Journal of Mechanical Science and Technology 28 (6) (2014) 2231~2242 2233

simulation (MCS) approach when estimating the value, vari-ance, and contribution of individual inputs to the sensitivity of function output [27-29].

e-FAST computes the contribution level of each input fac-tor to the variance of output. The sampling strategy imple-mented in e-FAST establishes a sinusoidal function of a par-ticular frequency for each input parameter. The choice of sinu-soidal function depends on the distribution of desired parame-ter values. The frequencies for each parameter have to satisfy the criteria so frequencies can be differentiated within Fourier analysis. The sinusoidal function tends to repeatedly process the same samples because of its symmetry properties of trigo-nometric functions. To remove this inefficiency, the re-sampling scheme is applied [7]. The primary advantage of e-FAST over the original FAST is the ability to calculate both the first-order sensitivity and the total-order sensitivity of each input parameter. e-FAST is robust at small sample sizes and computationally efficient [7]. In this paper, an e-FAST mod-ule was developed using visual FORTRAN language. Its per-formance was evaluated by investigating an 18-bar truss opti-mization problem (Section 3).

2.2 RSM

RSM can approximate the global optimum through building a response surface, which corresponds to the change of design variables. Generally, the second-order polynomial function in Eq. (1) is used to represent the response surface [22].

1

20

1 1 1 2

k k k k

predict i i ii ii ij i ji i i j

y b b x b x b x x-

= = = =

= + + +å å åå (1)

where x1, x2, ¼ , xk are the design variables that affect the response, bo and bi (i = 1, 2, ¼ , k) are the coefficients of the regression function, and ypredict is the predicted value of the regression function. The reliability of the response surface can

be inferred using the experimental points. It can be estimated by the adjusted R-square (R2

adj) value [25].

)1/()/(12

--

-=nSS

mnSSRy

Eadj (2)

where SSE and SSy are the error sum of squares and total sum of squares, respectively. The variable n is the number of ex-perimental points, and m is the number of response function coefficients. When the response surface model accurately predicts the observed response values, the typical values for R2

adj are between 0.9 and 1.0.

2.3 CO

CO is introduced as the decomposed and decentralized bi-level optimization method. A block diagram of CO is shown in Fig. 2. System-level optimization is responsible for provid-ing target values for global design variables (z) and system responses (y). Local disciplinary-level optimization assures that the discrepancies between disciplines vanish by enforcing compatibility constraints. It is modeled to minimize interdisci-plinary discrepancies while satisfying specific local con-straints. CO formulation can be stated at the system level as follows [26, 30]:

( )min. ,SL SLf z y

( )( )* * * *, , , , , 0i SL i SL i i j isubject to J z z y y x y z =

1,2, ,i n j i= ¹L (3)

where J represents the compatibility constraints, one for each discipline; z*, y*, and x* are the optimal disciplinary optimiza-tion level results; and subscript SL is the system level. The ith disciplinary-level optimization problem is formulated as

Fig. 2. CO architecture [26, 30].


( ) ( )2 2min. i SL i SL iJ z z y y= - + -å å

( )( ), , , , 0i i i i i j isubject to g x z y x y z £ (4)

where gi is the ith disciplinary constraint.

2.4 Design optimization with uncertainty

In this research, RBDO and PBDO methods were imple-mented to consider uncertainty from RSM [31, 32]. RBDO is suitable for uncertain parameters that have sufficient amount of information to generate the probability density function (PDF). It uses random variables for uncertainty, such as error from the environment and manufacturing tolerance [33]. A RBDO result can be improved when it has more accurate data for uncertain parameters. On the other hand, PBDO is used when uncertain parameters have insufficient information. A PBDO result is not improved when it has more information about uncertain parameters but keeps conservative results.

D. J. Neufeld et al. used RBDO to consider uncertainty from an approximation model [24]. They generated PDF from the error between FEA results and Kriging model results [24]. RBDO was used for the uncertainty of the structural analysis model. In this paper, PBDO as well as RBDO methods were applied with selected design variables. Uncertainty of the sur-rogate model can be used for RBDO or PBDO from the per-spective of a designer. Errors from the surrogate model can be handled as inherited errors of the analysis tool. In this research, RBDO and PBDO methods were implemented, and the results were compared. In addition, the previous RBDO result from D.J. Neufeld et al. [24] was evaluated with the PBDO result because RBDO is independent from the information of uncer-tain parameters.

RBDO. The basic idea behind RBDO is to employ numerical opti-

mization algorithms to obtain optimal designs that ensure reliability [31, 34, 35]. When optimization is performed with-out considering uncertainty, system failure can be induced by constraints that are active within the deterministic solution. The reliable solution lies farther inside the feasible design region than the deterministic optimization result, while satisfy-ing the targeted reliability level. In the RBDO method, uncer-tainty is modeled using probability theory. The probability distributions of random variables are obtained using statistical models. The probability of failure corresponding to a failure mode can be obtained or posed as a constraint in the optimiza-tion problem to obtain safer designs [33, 36]. The RBDO model can generally be defined as follows [31]:

( )min. f d

( )( ) ( )0 0,i tsubject to P G b£ -F - £X 1,2, ,i np= L

,L U ndvd d d R£ £ Îd nrvX RÎ (5)

where d, dL, and dU are the design variable vector, the lower boundary of design variables, and the upper boundary of de-sign variables, respectively. Gi(X) is the ith constraint function, and X is the random vector. P(•) is the probability measure, and np is the number of possible constraints. The variables ndv and nrv are the number of design vectors and random vectors, respectively. Φ(•) is the standard normal cumulative distribution function, and βt is the probability distribution and the prescribed reliability target.

PBDO. For uncertainty with insufficient information to generate

PDF for RBDO, the possibility-based method has been intro-duced in design optimization. When information is insuffi-cient for input data, the possibility-based method gives more conservative possible design results than the more probable design [32, 37, 38]. The general formula of PBDO is shown in Eq. (6) [32].

( )min. f d

( )( )0 0 ,i tsubject to G aP £ > £X 1,2, ,i np= L

L Ud d£ £d (6)

where П(•) is the possibility measure, X is the random vector, and αt is the target possibility of a failure. The variables d, dL, and dU are the design variable vector, the lower boundary of the design variables, and the upper boundary of the design variables, respectively.

In this paper, non-interactive fuzzy variables Xi were as-sumed to have their membership function Пxi(xi) satisfying the following properties: 1) unity, 2) strong convexity, and 3) boundedness [38, 39]. These three properties make it possible for non-interactive input fuzzy variables Xi, i = 1,…, nf to be uniquely transformed into fuzzy variables Vi with non-interactive isosceles-triangular membership functions, as de-scribed by the following equations.

( )1, 1 0

11 , 0 0i

i iv i i

i i

v vV v

v v+ - £ £ì

P = = -í - £ £î

1, 1,2, , .iv i nf£ = L (7)

The transformation can be written as

( )( )

1

1i

i

X L i i ii

X R i i i

X X dV

X X d

ìP - £ï= í-P >ïî

(8)

where ПxiL(Xi) and ПxiR(Xi) are the left and right sides, respec-


tively, of the membership function of the input fuzzy variable Xi. In addition, di is the maximal grade of this membership function.

CO with RBDO / CO with PBDO. In this research, two different modules, CO with RBDO and

CO with PBDO, were performed, and their results were com-pared. The system-level optimization of CO (Eq. (3)) was changed for RBDO as follows:

( )min. , ,SL SLf z y p

( )( )* * * *, , , , , , 0i SL i SL i i j isubject to J z z y y x y z p =

1,2, ,j n j i= ¹L (9)

where p' represents the uncertain parameters. The ith disci-plinary-level optimization problem changes to

( ) ( )2 2min. i SL i SL iJ z z y y= - - -å å

( )( )( ), , , , , 0i i i i i j i tsubject to P g x z y x y z p P£ ³ (10)

where P is the set of probabilities of the feasibility for each problem constraint and Pt represents the target probability of the feasibility.

CO with PBDO formulation of the system level changes as

( )min. , ,SL SLf z y X

( )( )* * * *, , , , , , 0i SL i SL i i j isubject to J z z y y x y z X =

1,2, ,j n j i= ¹L (11)

where X represents the fuzzy parameters. The formulation of the ith disciplinary-level optimization changes to

( ) ( )2 2min. i SL i SL iJ z z y y= - + -å å

( )( )( ), , , , , 0i i i i i j i tsubject to g x z y x y z X aP £ £ (12)

where Π(●) is the possibility measure and αt represents the target possibility of the failure.

Its performance was evaluated by utilizing a numerical op-timization example (Section 3).

3. Validation of numerical implementations

3.1 The 18-bar truss optimization problem

e-FAST was applied to the 18-bar truss example below from the deterministic problem proposed in E. Salajegheh and G.N. Vanderplaats [40] to evaluate the implemented GSA module. An initial set of the design variables were selected to define the truss shape and element thicknesses. The GSA module was implemented to selectively sort out variables that

hold more importance over those that have less significance on reducing dimensionality in the optimization problem. The optimization result based on the complete set of the design variables was compared with the result with reduced dimen-sionality. The objective function was to minimize the weight of the truss structure. Maximum tensile and compressive stresses in every member were constrained to be below the ultimate and buckling stress limits. The initial truss structure is shown in Fig. 3, where the x and y axis units are in inches.

The design variables included four element area variables (x1-x4) and eight variables (x5-x12) that defined the coordinates of the lower nodes [18, 40]. The definitions of the variables are shown in Table 1. A represents the element areas, x de-notes the design variable vector, and X and Y are the coordi-nates of the lower truss nodes. The tensile stress and buckling stress were considered to be the constraints. A uniform normal force was applied to nodes 1, 2, 4, 6, and 8. Table 2 shows the load conditions.

Table 1. Design variables and their range [18, 40].

Variables Definition Initial value

Lower boundary

Upper boundary

x1 A1, A4, A8, A12, A16 10.0 0.1 30.0

x2 A2, A6, A10, A14, A18 21.65 0.1 30.0

x3 A3, A7, A11, A15 12.50 0.1 30.0

x4 A5, A9, A13, A17 7.07 0.1 30.0

x5 X3 1000.0 800.0 1200.0

x6 Y3 0.0 0.0 220.0

x7 X5 750.0 510.0 800.0

x8 Y5 0.0 0.0 220.0

x9 X7 500.0 350.0 510.0

x10 Y7 0.0 0.0 220.0

x11 X9 250.0 50.0 350.0

x12 Y9 0.0 0.0 220.0 Table 2. Loading condition for 18-bar truss problem.

Node Fx (lbs) Fy (lbs) Fz(lbs)

1 0 -20,000 0

2 0 -20,000 0

4 0 -20,000 0

6 0 -20,000 0

8 0 -20,000 0

Fig. 3. Initial 18-bar truss [18, 40].


Buckling stress (σb) was defined by the Euler buckling equation (Eq. (13)). The elastic modulus (E) was assumed to be 1.0 E+4 Kpsi. The buckling coefficient (K) was assumed to be 4.0, and the allowable stress was assumed to be 20 Kpsi. L denoted the element length, and the material density was as-sumed to be 0.1 lb/in3 [18, 40].

2 .i i ib

i

K E AL

s -= (13)

GSA was performed to determine the sensitivity indices of

each design variable with respect to the objective function. Ta-ble 3 shows the results and the sensitivity rank of each variable.

The first-order sensitivity index represents the fraction of the model output variance explained by the input variation of

the given parameter. On the other hand, the total sensitivity index shows the factor’s main effect and all the interaction terms of that factor. The total sensitivity index is defined as the sum of all the sensitivity indices involving that factor. The X coordinates of nodes 5, 7, and 9 (x7, x9, x11) were less sensi-tive than any other nodal positions.

A genetic algorithm optimizer was used with a population size of 40 in this study [41]. Case 1 used all design variables as in Refs. [18, 40]. On the other hand, other cases used a differ-ent number of design variables from the GSA result (Table 4). Two design variables (x9, x11) were fixed in Case 2; four design variables (x7, x9, x11, x12) were fixed in Case 3; six design vari-ables (x6, x7, x9, x10, x11, x12) were fixed in Case 4; and only four design variables (x1, x2, x3, x4) were used in Case 5.

These results show Cases 2 and 3 within 4% of Case 1. Moreover, far fewer evaluations were required for Case 2 and Case 3, as shown in Table 4. On the other hand, Case 4 and Case 5 show almost 20% of Case 1 even though these have fewer evaluations. The case with eight design variables, Case 3, had huge improvement on the iteration number and small error in the objective function. The reduced number of design variables from the e-FAST result improved the efficiency of calculation with tolerable error.

3.2 Numerical example for RBDO with CO / PBDO with CO

In this research, the performance measure approach (PMA) was used for the reliability assessment strategy. PMA is well established and accepted for RBDO and PBDO methods [31, 32]. The performance of CO implementations with RBDO and PBDO was evaluated using the numerical example. In addition, this example showed the characteristics of RBDO and PBDO methods.

A multidisciplinary analytical example from Ahn et al., Eq.

Table 3. Sensitivity indices for 18-bar truss problem.

Design variables

1st order sensitivity index

Total sensitivity index Rank

x1 0.5832 0.7174 1

x2 0.2855 0.3512 2

x3 0.0446 0.0549 4

x4 0.0609 0.0749 3

x5 0.0065 0.0075 5

x6 0.0060 0.0073 7

x7 0.0008 0.0009 10

x8 0.0061 0.0074 6

x9 0.0002 0.0003 12

x10 0.0033 0.0041 8

x11 0.0006 0.0008 11

x12 0.0024 0.0029 9

Table 4. Comparison of design result.

Design variables Initial Case 1(12) Case 2(10) Case 3(8) Case 4(6) Case 5 (4)

x1 10.00 11.20 13.84 11.06 11.02 11.06

x2 21.65 16.63 16.76 16.60 16.60 16.60

x3 12.50 2.40 3.64 2.36 5.55 5.55

x4 7.07 7.16 4.94 7.84 7.87 7.84

x5 1000.0 831.39 750.04 878.25 835.57 -

x6 0.0 162.67 170.35 185.69 - -

x7 750.0 580.17 660.23 - - -

x8 0.0 106.01 116.56 174.54 44.76 -

x9 500.0 287.34 - - - -

x10 0.0 18.35 101.64 100.14 - -

x11 250.0 284.53 - - - -

x12 0.0 17.56 35.52 - - -

Weight (lb) 6,430.7 4,267.83 4,341.49 4,436.44 5,091.49 5,173.49

Error - - 1.73% 3.95% 19.30% 21.22%

Number of evaluation - 13,421 12,813 7,625 6,915 5,380

Improvement - - 4.53% 43.19% 48.48% 59.91%


(14), was used to validate the accuracy of RBDO and PBDO when the number of sample cases for modeling uncertain pa-rameters varied on the MDO problem [29]. This equation consisted of three subsystems with two state variables. The range of uncertain variables x1 and x2 is shown in Table 5 [12].

( )3 2 11 1

2

min. 6 exp yf x yy

æ ö= - - + - ç ÷

è ø

2 21 1 2

yy x= +

11 2 1

2exp 2.2yg y xy

æ ö= - + +ç ÷è ø

( )1 22 1 2

1

3x xy x x

y= + +

( ) ( )2 32 2 1 2 21 4 .g y y x x= - - + - - (14)

These variables were assumed to have a coefficient of

variation (COV) of 0.04 with mean values assigned by the optimizer. COV was defined as the ratio of the standard devia-tion (σ) to the mean value (μ) of a random variable. The CO formulation for this problem is shown in Fig. 4.

The RBDO with CO and PBDO with CO methods were solved for seven cases at a reliability level of 3σ. The starting vector was x0 = [1.5, 1.5], and the convergence tolerance was 10-5. This problem had two uncertain variables, so it required at least two million random numbers to estimate the probability of a failure when using MCS. The result is shown in Table 6.

The deterministic optimization result was 115.76 and had 0.40% probability when the design variables had a given dis-tribution. Each case was randomly selected from the range of

each uncertain variable. The RBDO result converged to the MCS result when the number of sample cases increased. On the other hand, the PBDO case showed similar results when it had 50 points of uncertain parameters. When the uncertain parameters had 10 cases, the RBDO result presented a differ-ence from the MCS result, but the PBDO result represented similar values when it had more cases. Thus, the RBDO result more strongly depended on the information of uncertain pa-rameters than that of PBDO. The RBDO result improved when enough information on uncertain parameters was sup-plied. On the other hand, the increased number of cases did not improve the PBDO result. Thus, the RBDO result of pre-vious research Ref. [24] can be evaluated by comparing it with the PBDO result in this research.

4. Multidisciplinary wing design using GSA and

RBDO/PBDO

The proposed method was applied to the conceptual design optimization of a light jet aircraft wing design. The wing de-sign problem evaluated the accuracy and efficiency of the proposed method. The optimization results were compared when the number of design variables was changed from the GSA result. The surrogate model was developed from sample cases of FEA. RSM was used for estimating the maximum stress and weight of the wing box [42]. A parameterized finite element (FE) model of a generic light business jet wing box was developed with ANSYS (Fig. 5).

The wing box model was automatically constructed by in-cluding the leading and trailing edge spars, the upper and lower skins, as well as the stringers and ribs by using a MATLAB function to generate the ANSYS mesh. The

Table 5. Range of variables.

Uncertain variable Lower boundary Upper boundary

x1 0 5

x2 0 5

Fig. 4. CO formulation.

Table 6. Optimization results.

Number of cases RBDO PBDO

10 193.288 122.052

50 119.959 121.676

100 119.884 121.495

200 119.813 121.492

500 119.976 121.497

1000 121.048 121.491

MCS 120.082 121.493

Fig. 5. Wing box FE model [24].


MATLAB function applied random cases of the parameter values to generate meshes on ANSYS. It developed meshes and analyzed the von Mises stress and weight for each case of the MATLAB function. In this research, the RSM of the wing weight and the Von Mises stress values were derived from the FEA results of 148 sample cases [24]. Fig. 6 shows the proce-dure for RSM development from the FEA results.

The FE model consisted of 29 member attributes represent-ing the thicknesses of the primary structural members—19 ribs, the front and rear spar, 6 stringers, the upper and lower skin, the span, as well as the wing reference area. Linking attributes to nine design variables (Fig. 7 and Table 7) reduced dimensionality. The sweep angle, taper ratio, and airfoil were fixed as constants.

4.1 Design formulation

The aircraft concept in this research was similar in size and performance to light jet aircraft, such as Cessna Mustang and

D-Jet [43, 44]. The performance targets of the conceptual wing design were selected to match values to typical light jet aircraft. In this research, higher L/D was considered while minimizing the weight of the wing. Major design parameters for the wing design are shown in Table 8. The maximum von Mises stress was constrained to below 360 MPa, correspond-ing to the yield strength of aluminum 7075 with a safety mar-gin of 1.5 as required by airworthiness standards. Optimiza-tion aimed to maximize the wing lift-to-drag ratio at a cruise speed of 400 kts with an altitude of 35000 ft while body con-tribution to the lift-to-drag ratio was neglected. FEA was re-placed by RSM. An uncertain error term was defined from differences between stress calculated using FEA and stress estimated using the approximation model.

The MDO problem was formulated with two disciplines: an aerodynamic solver using the panel method and a structural solver using RSM. RSM was generated from a database of sampled FE solutions in the design space. A total of 148 ran-dom cases of the wing box structural analysis were calculated on the FE model, and these results were used to develop the RSM model. The developed model had R2

adj = 0.98, which showed the accuracy of the approximation model.

Table 9 shows the sensitivity indices for the wing box weight, which is the objective function of the structural mod-ule. In this research, the design variables that had more than 0.004 of the total-order sensitivity index were selected. Seven design variables were selected, and two design variables (x3, x4) were fixed as the initial values. MDO formulation was generated using seven design variables. Fig. 8 shows the ar-

Table 7. Design variables for wing box and its range [24].

Member Design variable Unit Initial

value Lower limit

Upper limit

Rib 1-19 x1 in 0.1379 0.0787 0.1969

Front spar x2 in 0.7874 0.3937 1.1811

Rear spar x3 in 0.7874 0.3937 1.1811

Upper stringer 1-3 x4 in 0.2362 0.0787 0.3937

Lower stringer 1-3 x5 in 0.2362 0.0787 0.3937

Upper skin x6 in 0.8859 0.5906 1.1811

Lower skin x7 in 0.8859 0.5906 1.1811

Span length (b) x8 ft 31.0 26.0 36.0

Reference wing area (S) x9 ft2 113.25 96.50 130.0

Fig. 6. Response surface model development for wing box.

Fig. 7. Wing box shape and design variables [24].

Table 8. Design parameters for wing design.

Design parameter Given value

Gross weight of aircraft 11,464.04 lb

Wing stored fuel capacity 2,645.55 lb

Wing weight budget 970.03 lb

Max. von Mises stress 360 MPa

Cruise speed 400 kts

Cruising altitude 35,000 ft

Table 9. Sensitivity indices of design variables.

Design variables

1st order sensitivity Index

Total sensitivity index Rank

x1 0.0030 0.0059 5

x2 0.0025 0.0049 6

x3 0.0013 0.0025 8

x4 0.0012 0.0023 9

x5 0.0045 0.0089 4

x6 0.0023 0.0045 7

x7 0.0134 0.0266 3

x8 0.7087 0.9151 1

x9 0.2631 0.4569 2


chitecture of this CO formulation. The system objective function was to maximize L/D. The

variables Aaero and Astruct represent the auxiliary constraints of the aerodynamics and structural discipline, respectively; t is the structural thickness; M is the mass of the wing; and Va is the approaching speed. Tables 10 and 11 show the constraints and formulations, respectively, of each method for the structural discipline. Minimization of the weight and stress in the struc-tural analysis decreased the span of the wing geometry and reduced aerodynamic efficiency. To consider aerodynamic efficiency, the approach speed was used for the constraint of the aerodynamic discipline. Other aircraft characteristics, such as range and endurance, need the parameters from performance and thrust disciplines. However, these disciplines were not considered in this paper. On the other hand, an approach speed can be derived from the given parameters in this MDO prob-lem and used to evaluate the system objective function.

Uncertain parameters were generated from the error be-tween the FEA and RSM results. The uncertainty of the RSM of the structural discipline was assumed to be of normal distri-bution with a mean of 0.0010 and a standard deviation of 0.0932 for RBDO. In addition, the membership function was

derived for PBDO. The target reliability level of RBDO and PBDO had a 99.87% probability, whereas the deterministic optimization result had a 70.52% probability. The probability values of RBDO and PBDO were within a reasonable range in the general engineering field. The comparison of each result is shown in Table 12. Case 1 shows the deterministic optimiza-tion result, and Case 2 shows the result based on selected de-sign variables from GSA. Moreover, Cases 3 and 4 show the RBDO and PBDO results, respectively, with the selected de-sign variables.

4.2 Discussion

The fixed design variables were the thickness of the rear spar and the upper stringer. They have a minimal effect on the wing weight while satisfying the stress constraints. Case 2 reduced the number of design variables, showed an error of 3.45%, and reduced the number of iterations of Case 1 by 33.55%. Cases 3 and 4 had more iterations than Case 2 be-cause additional reliability and possibility analysis was con-sidered. The results of RBDO and PBDO specified smaller wingspan and area to satisfy the target probability of failure from the structural analysis when considering uncertainty.

As described in the previous section, the result of RBDO depended on the number of sample cases for the uncertainty simulation. In this research, RBDO and PBDO methods were implemented to evaluate the uncertainty simulation of the previous research of D.J. Neufeld et al., which used RBDO with MDO [24]. RBDO and PBDO results for the wing box design optimization problem showed similar results. Thus, the number of cases for uncertain parameters was sufficient for RBDO because the PBDO result was not dependent on the number of cases for uncertain parameters. Moreover, reduc-

Table 10. Constraints for wing box conceptual design.

Constraints Symbol Value

Maximum stress Σ ≤ 360 MPa

Mass M ≤ 970.03 lb

Approach speed Va ≤ 120 kts

Table 11. Formulations for structural discipline.

Deterministic optimization RBDO PBDO

Objective min. Astuct(b,S,t1…7) min. Astuct(b,S,t1…7) min. Astuct(b,S,t1…7)

Constraint σ ≤ σmax M ≤ Mgoal

P(σ ≤ σmax) ≥ Pt M ≤ Mgoal

Π(σ ≤ σmax) ≤ αt M ≤ Mgoal

Fig. 8. CO architecture.

Table 12. Comparison of design result.

Design vari-ables

Case 1 (Deterministic optimization)

Case 2 (Selected

design variables)

Case 3 (RBDO)

Case 4 (PBDO)

x1 0.0906 0.0906 0.0906 0.0906

x2 0.7126 0.7047 0.7047 0.7047

x3 0.5984 - - -

x4 0.1850 - - -

x5 0.2362 0.2362 0.2283 0.1811

x6 0.7244 0.8858 0.7283 0.8583

x7 0.5866 0.8858 0.6102 0.9213

x8 33.7927 33.7927 31.6273 31.4305

x9 114.0975 114.0975 111.9447 111.9447

L/D 31.9 30.8 30.3 30.1

Error - 3.45% 5.02% 5.64%

Number of iterations 158 104 112 111

Improvement - 33.55% 29.05% 29.65%


tion of design variables from GSA reduced calculation time with small error. Table 12 shows that Cases 2, 3, and 4 had a small error in objective function and had a 30% reduced itera-tion number compared with the deterministic optimization result. If the number of design variables is greater than that in this problem, such difference can be huge even though a GSA module step is added.

5. Conclusion

In this research, the reduction of dimensionality and consid-erations of uncertainty on MDO were studied to enhance the design process. GSA identifies the most important design variables and which design variables can be omitted to ac-complish the design objective. The number of design variables was reduced to decrease calculation time for optimization. The 18-bar truss optimization evaluated the accuracy of the im-plemented e-FAST module. It showed small error when the number of design variables was reduced. For the wing box conceptual design case, e-FAST reduced the number of design variables. In addition, the implemented RBDO with CO and PBDO with CO methods improved the reliability of results when uncertainty of the approximation method was consid-ered. The error between FEA and RSM results was used for uncertain parameters and applied to the structural discipline. RBDO and PBDO methods cannot show global optimum results but showed small errors when considering uncertainty. Nevertheless, these methods prevented the violation of con-straints when uncertainty existed. The result of the wing box conceptual design showed fewer iterations with a reduced number of design variables while it accomplished the targeted probability.

The GSA result can be used not only for RSM but also for FEA. The number of design variables had a large effect on calculation time for high-fidelity analysis tools, such as FEM and CFD. Thus, the proposed method is useful on the concep-tual design when it uses high-fidelity analysis tools by reduc-ing the number of design variables as fixing less effective variables. Moreover, the implementation of RBDO and PBDO when it considered uncertainty from the approximation model improved the probability of design results. In actual engineer-ing problems, the number of cases can be insufficient from experiment or high-fidelity analysis tools to derive the ap-proximation model. The developed methods, RBDO with CO and PBDO with CO, are useful to consider the error of ap-proximation models or low-fidelity analysis tools. In such cases, the proposed design process is useful at the conceptual design level when using high-fidelity analysis tools to increase the accuracy of the result.

Acknowledgments

We gratefully acknowledge the support given by the Na-tional Sciences and Engineering Research Council of Canada grant (RGPIN227747-2012) and the framework of the interna-

tional cooperation program by the National Research Founda-tion of Korea (2013K2A1A2054453) to this study.

Nomenclature------------------------------------------------------------------------

X : Fuzzy parameters ( )

iX L iXP : Left side of the membership function of the input fuzzy variable

( )iX R iXP : Right side of the membership function of the input

fuzzy variable ( )

iX ixP : Membership function of fuzzy variables A : Element areas Aaero : Auxiliary constraints of the aerodynamics discipline Astruct : Auxiliary constraints of the structural discipline b : Wing span bi : Coefficients of the regression function d : Design variable vector di : Maximal grade of membership function dL : Lower boundary of design variables dU : Upper boundary of design variables E : Elastic modulus gi : ith disciplinary constraints Gi(X) : ith constraints functions J : Compatibility constraints K : Buckling coefficient L : Element length L/D : Lift-to-drag ratio M : Mass of the wing m : Number of the response function coefficients n : Number of experimental points ndv : Number of design vectors np : Number of possible constraints nrv : Number of random vectors P : Set of the probabilities of the feasibility for each

problem constraint P(•) : Probability measure Pt : Target probability of the feasibility R2

adj : Adjusted R-square S : Wing area SL : System level SSE : Error sum of squares SSy : Total sum of squares t : Structural thickness Va : Approaching speed Vi : Fuzzy variables X : Random vector Xi : Non-interactive fuzzy variables xk : Design variables that affect the response y : System responses ypredict : Predicted value of the regression function z : Global design variables αt : Target possibility of a failure βt : Reliability target μ : Mean value of a random variable σ : Standard deviation of a random variable


σb : Buckling stress Φ(•) : Standard normal cumulative distribution function П (•) : Possibility measure

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Hyeong-UK Park has been a Ph.D. candidate of Ryerson University since August 2008 and received his MSc de-gree in 2005 at Konkuk University. His research interests include reliability-based design optimization, possibility-based design optimization, multidisci-plinary design optimization, and deriva-

tive design.

Joon Chung is an associated professor of the Department of Aerospace Engi-neering, Ryerson University. His re-search interests include the multidisci-plinary design optimization of aircraft and its systems and aircraft flight data analysis and simulation.

Kamran Behdinan is a professor of the Department of Mechanical and Indus-trial Engineering, University of Toronto. He is the Natural Science and Engineer-ing Research Council of Canada chair-man on Multidisciplinary Engineering Design. His research interests include the design and development of light-

weight structures for aerospace, automotive, and nuclear ap-plications and the multidisciplinary design optimization of aerospace and automotive systems.

Jae-Woo Lee is a faculty member of the Department of Aerospace Information Engineering at Konkuk University in Seoul, South Korea. He is a member of the National Defense Acquisition Com-mittee of Korea and an advisor to the Minister of Land and Transportation. He is the Director of the Aerospace Design

Integration Research Center at Konkuk University and has been employed as a senior researcher with the Agency for Defense Development, South Korea. His interests include aerodynamic design and optimization, multidisciplinary aero-space vehicle design and optimization, and MOD aerospace systems engineering and mission analysis.

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Multidisciplinary wing design optimization considering global sensitivity and uncertainty of approximation models