Dimension reduction method for reliability-based robust ...user.engineering.uiowa.edu/~kkchoi/RBDO_A.12.pdfdesign optimization (RDO) is a method to improve the product quality by minimizing

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www.elsevier.com/locate/compstruc

Computers and Structures xxx (2007) xxx–xxx

Dimension reduction method for reliability-basedrobust design optimization

Ikjin Lee a, K.K. Choi a,*, Liu Du a, David Gorsich b

a Department of Mechanical and Industrial Engineering, College of Engineering, The University of Iowa, Iowa City, IA 52241, United Statesb US Army RDECOM/TARDEC AMSRD-TAR-N, MS 157, 6501 East 11 Mile Road, Warren, MI 48397-5000, United States

Received 5 March 2007; accepted 30 April 2007

Abstract

In reliability-based robust design optimization (RBRDO) formulation, the product quality loss function is minimized subject to prob-abilistic constraints. Since the quality loss function is expressed in terms of the first two statistical moments, mean and variance, threemethods have been recently proposed to accurately and efficiently estimate the moments: the univariate dimension reduction method(DRM), performance moment integration (PMI) method, and percentile difference method (PDM). In this paper, a reliability-basedrobust design optimization method is developed using DRM and compared to PMI and PDM for accuracy and efficiency. The numericalresults show that DRM is effective when the number of random variables is small, whereas PMI is more effective when the number ofrandom variables is relatively large.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Reliability-based robust design optimization (RBRDO); Dimension reduction method (DRM); Performance moment integration (PMI);Percentile difference method (PDM); Sensitivity analysis; Statistical moment calculation

1. Introduction

In recent years, several approaches to integrate robustdesign [1,2] and reliability-based design [3–5] have beenproposed [6–8]. The reliability-based design optimization(RBDO) is a method to achieve the confidence in productreliability at a given probabilistic level, while the robustdesign optimization (RDO) is a method to improve theproduct quality by minimizing variability of the outputperformance function. Since both design methods makeuse of uncertainties in design variables and other parame-ters, it is very natural for the two different methodologiesto be integrated to develop a reliability-based robust designoptimization (RBRDO) method.

0045-7949/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2007.05.020

* Corresponding author.E-mail addresses: [email protected] (I. Lee), kkchoi@engi-

neering.uiowa.edu (K.K. Choi), [email protected] (L. Du),[email protected] (D. Gorsich).

Please cite this article in press as: Lee I et al., Dimension reduction mj.compstruc.2007.05.020

The product quality in robust design can be described byuse of the first two statistical moments of a performancefunction: mean and variance [9]. Thus, it is necessary todevelop methods that estimate the first two statisticalmoments of the performance function and their sensitivitiesaccurately and efficiently. The statistical moments can beanalytically expressed using a multi-dimensional integral.However, it is practically impossible to calculate the statis-tical moments of the performance function using the multi-dimensional integral. Hence, there have been variousnumerical attempts to estimate the moments more effi-ciently: experimental design [10], first order Taylor seriesexpansion [1,2,11], Monte Carlo simulation (MCS) [12],importance sampling method [13], and Latin hyper cubesampling method [14].

Monte Carlo simulation could be accurate for themoment estimation, however it requires a very large num-ber of function evaluations. Therefore, in many large-scaleengineering applications, it is not practical to use MonteCarlo simulation. The experimental design also needs a

ethod for reliability-based ..., Comput Struct (2007), doi:10.1016/

mailto:[email protected]





2 I. Lee et al. / Computers and Structures xxx (2007) xxx–xxx

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large amount of computation when the number of designvariables is large. The first order Taylor series expansionhas been widely used to estimate the first and second statis-tical moments in robust design. However, the first orderTaylor series expansion results in a large error especiallywhen the input random variables have large variations.This is because the first order Taylor series expansion doesnot use all information of the probability density functions(PDF) of input random variables.

To overcome the shortcomings explained above, threemethods have been recently proposed: the univariatedimension reduction method (DRM) [15,16], performancemoment integration (PMI) [8], and percentile differencemethod (PDM) [6,7]. In this paper, RBRDO using the uni-variate DRM is proposed and the calculation of the statis-tical moments and their sensitivities using PMI is derived.In addition, the results of RBRDO using the univariateDRM are compared with those designs obtained usingPMI and PDM. Both DRM and PMI are directly estimat-ing the statistical moments. On the other hand, in PDM,the robustness is achieved through a design objective inwhich the variation of the design performance is approxi-mately evaluated through the percentile performance differ-ence between the right and left tails of the performancedistribution [6]. Thus, three methods can be compared interms of how accurately these methods can find an opti-mum design to minimize the variance of the performancefunction. Hence, in this paper, three methods are evaluatedby comparing the variances at the optimum designs. PMIand DRM are also compared in terms of how accuratelyand efficiently estimate the statistical moments of the per-formance function. For the comparisons, several examplesincluding one-dimensional and two-dimensional perfor-mance functions, and a large-scale engineering problemare used. These comparisons illustrate that the univariateDRM is the most accurate and efficient method when thenumber of design variables is small and PMI is a betteroption when the number of design variables is relativelylarge.

For the inverse reliability analysis of RBDO, theenriched performance measure approach (PMA+) [5] andits numerical method, the enhanced hybrid mean value(HMV+) [4] are utilized.

Fig. 1. Comparison of conventional and robust design optimum [1].

2. Fundamental concept of robust design

2.1. Reliability-based robust design

In general, a conventional (deterministic) design optimi-zation problem can be formulated to

minimize hðdÞsubject to GiðdÞ 6 0; i ¼ 1; . . . ; nc

dL6 d 6 dU ; d 2 Rndv

ð1Þ

where h(d) is the cost function, Gi is the ith constraint, andd is the design variable vector; and nc and ndv are the num-


ber of constraints and design variables, respectively. Theoptimum design of the conventional optimization problemis the deterministic optimum that could be sensitive to thevariation of input design variables and other parameters.

Due to the variation of design variables and other param-eters, the performance function h(d) also has variation.Thus, in robust design, the robustness of a design objectivecan be achieved by simultaneously ‘‘optimizing the meanperformance lH and minimizing the performance variancer2

H ’’ [6]. In other words, the goal of robust design is to findthe most insensitive design to the variation of the designvariables and other parameters. Since robust design is fun-damentally considering the variations of the design variablesand other parameters, it is very natural to integrate robustdesign and reliability-based design in one formulation. Thisdesign optimization is called reliability-based robust designoptimization (RBRDO) and can be formulated to

minimize f ðlH ; r2H Þ

subject to P ðGiðX; dÞ > 0Þ 6 Uð�btiÞ; i ¼ 1; . . . ; nc

dL6 d 6 dU ; d 2 Rndv and X 2 Rnrv

ð2Þ

where f ðlH ; r2H Þ is the cost function, d = l(X) is the design

vector, X is the random vector, and Gi is the ith probabilis-tic constraint. Quantities nc, ndv, nrv and bti are the numberof probabilistic constraints, design variables, random vari-ables, and the ith target reliability index, respectively. De-tailed explanation about RBDO can be found in [3–5]. Inthis paper, the enriched performance measure approach(PMA+) [5] is introduced to perform inverse reliabilityanalysis of the constraints.

Fig. 1 compares a conventional design optimization witha RDO for a one-dimensional performance function. Withthe same variability of a design variable, the robust opti-mum shows less variation of the performance functionh(d) than the conventional design optimum.

2.2. Three types of cost function

Since the cost function in Eq. (2) depends on lH and r2H

for robust optimum design in RBRDO, it is a bi-objective


I. Lee et al. / Computers and Structures xxx (2007) xxx–xxx 3

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optimization problem. The optimum of the bi-objectiveoptimization depends on the weight on each term in thecost function. However, since the main goal of this paperis not focused on determination of the weights, interestedreaders can refer to [17] for more details.

The cost function f ðlH ; r2HÞ in Eq. (2) can be formulated

in various ways based on engineering application types[8,9]. The following are three important cost function typesfor reliability based robust design.

(1) Nominal-the-best type

f ðlH ; r2H Þ ¼ w1

lH � ht

lH0� ht0

� �2

þ w2

rH

rH0

� �2

; ð3Þ

where ht and ht0are the target nominal value and the

initial target nominal value of the performance func-tion h(X) respectively, and w1 and w2 are weights tobe determined by the designer. To reduce the dimen-sionality problem of two objectives, each term is nor-malized by the initial value lH0

and rH0.

(2) Smaller-the-better type

f ðlH ; r2H Þ ¼ w1 � sgnðlHÞ �

lH

lH0

� �2

þ w2

rH

rH0

� �2

: ð4Þ

(3) Larger-the-better type

f ðlH ; r2H Þ ¼ w1 � sgnðlHÞ �

lH0

lH

� �2

þ w2

rH

rH0

� �2

: ð5Þ

3. Three methods for reliability based robust design

As shown in Eqs. (3)–(5), the main concern of RBRDOis how accurately and efficiently the statistical momentsand their sensitivities of the performance function h(X)can be estimated. Analytically, the kth statistical momentof the performance function can be obtained using the fol-lowing integration:

EðfhðXÞgkÞ ¼Z 1

�1� � �Z 1

�1fhðXÞgkfXðxÞdx; ð6Þ

where fX(x) is a joint probability density function (PDF) ofthe random parameter X. As stated before, it is practicallyimpossible to calculate the statistical moments of the perfor-mance function using Eq. (6) especially when the dimensionof the problem is relatively large. For numerical evaluationof Eq. (6), three methods have been recently proposed.These methods are briefly introduced in the following sec-tions and compared. More importantly, sensitivity analysisof statistical moments is derived and evaluated for accuracy.It should be noted that these three methods assume that in-put variables are statistically independent of each other.

3.1. Dimension reduction method

The dimension reduction method [15,16,18] is a newlydeveloped technique to calculate statistical moments of


the output performance function. There are severalDRMs depending on the level of dimension reduction:(1) univariate dimension reduction, which is an additivedecomposition of N-dimensional performance function intoone-dimensional functions; (2) bivariate dimension reduc-tion, which is an additive decomposition of N-dimensionalperformance function into at most two-dimensional func-tions; (3) multivariate dimension reduction, which is anadditive decomposition of N-dimensional performancefunction into at most S-dimensional functions, whereS 6 N. In this paper, the univariate DRM is used forcomputation of statistical moments and their sensitivi-ties. Computational efficiency of DRM is discussed inSection 3.1.2.

3.1.1. Basic concept of univariate dimension reduction

method

In the univariate DRM, any N-dimensional perfor-mance function h(X) can be additively decomposed intoone-dimensional functions as

hðXÞ ffi hðXÞ

�XN

i¼1

hðl1; . . . ; li�1; xi; liþ1; . . . ; lN Þ

� ðN � 1Þhðl1; . . . ; lN Þ ð7Þ

where li is the mean value of a random variable Xi and N isthe number of design variables. For example, ifh(X) = h(x1,x2), i.e., N = 2, then the univariate additivedecomposition of h(X) is

hðXÞ ffi hðXÞ � hðx1; l2Þ þ hðl1; x2Þ � hðl1; l2Þ: ð8Þ

Using the univariate DRM, one N-dimensional integra-tion in Eq. (6) becomes N one-dimensional integrations,which will reduce the number of function evaluations sig-nificantly when the number of design variables is large.This reduction of the number of function evaluations isexplained in Section 3.1.2. The one-dimensional numericalintegration can be calculated using the moment-basedintegration rule (MBIR) [19], which is similar to Gaussianquadrature [20]. According to MBIR, the kth statisti-cal moment of a one-dimensional function can be obtainedas

EðfhðXÞgkÞ ¼Xn

i¼1

wihkðxiÞ; ð9Þ

where wi are weights, xi are quadrature points (realizationsof a random variable X) and n is the number of weights andquadrature points. If PDF of the design variables is given,then these weights wi and quadrature points xi can be ob-tained using MBIR. For the standard normal input ran-dom variable with three quadrature points, the weightsand quadrature points are shown in Table 1 [19].

Using Eqs. (7) and (9), the mean value and variance ofthe performance function h(X) can be obtained as


Table 1Weights and quadrature points for standard normal

Quadrature points Weights

x1 x2 x3 w1 w2 w3

�ffiffiffi3p

0ffiffiffi3p

1

6

4

6

1

6


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lH �E½hðXÞ�

ffiEXN

i¼1

hðl1; . . . ;li�1;X i;liþ1; . . . ;lN Þ�ðN �1Þhðl1; . . . ;lN Þ( )

ð10Þ

ffiXn

j¼1

XN

i¼1

wji hðl1; . . . ;li�1;x

ji ;liþ1; . . . ;lN Þ�ðN �1Þhðl1; . . . ;lN Þ

r2H �E½ðhðXÞ�lH Þ

2� ¼E½h2ðXÞ��l2H

ffiEXN

i¼1

h2ðl1; . . . ;X i; . . . ;lN Þ�ðN�1Þh2ðl1; . . . ;lN Þ( )

�l2H

ð11Þ

ffiXn

j¼1

XN

i¼1

wji h

2ðl1; . . . ;xji ; . . . ;lN Þ�ðN �1Þh2ðl1; . . . ;lN Þ�l2

H

The estimation of statistical moments using the univar-iate DRM involves two approximations. As shown inEqs. (10) and (11), the univariate DRM approximates theperformance function h(X) using the sum of one-dimen-sional functions. If hðXÞ ¼

PNi¼1hiðxiÞ where hi(xi) is any

function of xi only, then the approximation is exact. How-ever, if there are off-diagonal or mixed terms, then there issome error that results from approximating off-diagonalterms using sum of one-dimensional functions. To reducethis error, the bivariate DRM or multivariate DRM canbe used. The second approximation involves the numericalintegration using the weights and quadrature points. Basedon Gaussian quadrature theory [20], n quadrature pointsand weights give a degree of precision of 2n � 1. Hence,if three quadrature points and weights for each variableare used, the numerical integration error for a quadraticperformance function will disappear. If the performancefunction is highly nonlinear, then three quadrature pointsmay not be sufficient to estimate the moments of the per-formance function. In this case, the error can be reducedif the number of quadrature points is increased.

3.1.2. Computational efficiency

Even though the accuracy is the most important con-cern, it is also important to efficiently estimate statisticalmoments of the performance function for large-scale prob-lems. In general, when the output moments are estimatedusing the univariate DRM and MBIR, the number of func-tion evaluations required is

FE ¼ n� N þ 1; ð12Þ


where n is the number of quadrature points and N is thenumber of design variables. If the distributions of all inputdesign variables are symmetric, e.g., normal distribution oruniform distribution, and the number of design variables isodd, then the required number of function evaluations isreduces to

FE ¼ ðn� 1Þ � N þ 1: ð13Þ

Therefore, when the number of design variables is large,the reduction becomes significant compared to the numberof function evaluation in directly integrating Eq. (6), whichis nN. However, although the reduction becomes significantwhen N is large, the number of function evaluations is stillincreasing proportionally to the number of design variablesas shown in Eq. (13).

If bivariate DRM is used to estimate the first and secondoutput moments, then the number of function evaluationswill increase exponentially to

FE ¼ NðN � 1Þ2

� n2 þ n� N þ 1: ð14Þ

For example, if the number of design variables is 5 andthe number of quadrature points is 3, then the number offunction evaluations by the univariate DRM is 16 fromEq. (12) and the number of function evaluations by bivar-iate DRM is 106 from Eq. (14). Both of the numbers areless than 35 = 243, which is the required number of func-tion evaluations for the numerical integration of Eq. (6)by including the mixed variable terms. However, the num-ber of function evaluations by the univariate DRM is sig-nificantly less than the number of function evaluationsfor bivariate DRM. For this reason, the univariate DRMis used to estimate statistical moments in this paper.

3.1.3. Sensitivity of statistical moments

To obtain a robust design, not only the values of the firstand second statistical moments but also the sensitivities ofthese moments are needed. Using Eq. (6) and Rosenblatttransformation [21] from the design space (x-space) tothe standard Gaussian space (u-space), which can bedescribed as FX(x) = U(u), sensitivities of the mean andvariance of the performance function with respect to thedesign variable li can be derived as

olH ðlÞoli

¼ o

oli

Z 1

�1� � �Z 1

�1hðxÞfXðxÞdx

� �

¼ o

oli

Z 1

�1� � �Z 1

�1hðxðu; lÞÞ/U ðuÞdu

� �

ðRosenblatt transformationÞ

¼Z 1

�1� � �Z 1

�1

ohðxðu; lÞÞoli

/UðuÞdu ð15Þ



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¼Z 1

�1� � �Z 1

�1

ohðxðu;lÞÞoxi

oxiðui;liÞoli

/U ðuÞdu

ðIndependencyÞor2

H ðlÞoli

¼ o

oli

Z 1

�1� � �Z 1

�1h2ðxÞfXðxÞdx

� �� ol2

H

oli

¼ o

oli

Z 1

�1� � �Z 1

�1h2ðxðu;lÞÞ/U ðuÞdu

� �� ol2

H

oli

ðRosenblatt transformationÞ

¼Z 1

�1� � �Z 1

�1

oh2ðxðu;lÞÞoli

/UðuÞdu� ol2H

olið16Þ

¼Z 1

�1� � �Z 1

�1

oh2ðxðu;lÞÞoxi

oxiðui;liÞoli

/U ðuÞdu� ol2H

oli

ðIndependencyÞ

where u is the standard normal variable. The input vari-ables are assumed to be independent for the derivationsof Eqs. (15) and (16).

To calculate oxiðui;liÞoli

in Eqs. (15) and (16), Rosenblatttransformation shown in Table 2 is used. For example, ifthe input variable is normally distributed, then Table 2shows that xi can be expressed as xi = li + riui. Since ri

is fixed and ui is independent of an input mean li,oxiðui ;liÞ

oli¼ 1 is obtained. For Gumbel and uniform distribu-

tion, the same result oxiðui;liÞoli¼ 1 is obtained from Rosenbl-

att transformation. For the Lognormal and Weibulldistribution, oxiðui;liÞ

olican be approximated to be 1.

By using the inverse transformation from u-space tox-space, the assumption oxiðui ;liÞ

oliffi 1, and Eqs. (10) and

(11), (15) and (16) can be further approximated by

olH ðlÞolk

ffiXn

j¼1

XN

i¼1

wji �

ohðxÞoxk

��x¼ðl1;...;x

ji ;...;lN Þ

� ðN � 1ÞohðxÞoxk

��x¼l

;

ð17Þ

or2H ðlÞolk

ffiXn

j¼1

XN

i¼1

wji �

oh2ðxÞoxk

��x¼ðl1;...;x

ji ;...;lN Þ

� ðN � 1Þoh2ðxÞoxk

��x¼l

� ol2H

olk: ð18Þ

Table 2Probability distribution and its transformation between x and u-space

Parameters

Normal l = mean; r = standard deviation

Log-normal �r2 ¼ ln 1þ rl

� 2� �

; �l ¼ lnðlÞ � 0:5�r2

Weibull l ¼ vC 1þ 1k

�; r2 ¼ v2 C 1þ 2

k

�� C2 1þ 1

k

�� Gumbel l ¼ mþ 0:577

a ; r ¼ pffiffi6p

a

Uniform l ¼ aþb2 ; r ¼ b�affiffiffiffi

12p

a UðUÞ ¼ 1ffiffiffiffi2ppRU�1 e�

u2

2 du.


Since the univariate DRM does not use sensitivities ofthe performance function evaluated at the quadraturepoints to estimate the moments, additional function evalu-ations are needed for the sensitivity analysis using Eqs. (17)and (18).

3.2. Performance moment integration (PMI)

3.2.1. Derivation of performance moment integration

The multi-dimensional integral in Eq. (6) for statisticalmoments can be rewritten using Rosenblatt transformationas

EðhkðXÞÞ ¼Z 1

�1� � �Z 1

�1hkðxÞfXðx; lÞdx

¼Z 1

�1� � �Z 1

�1hkðxðu; lÞÞ/U ðuÞdu ð19Þ

which can also be written in terms of the output distribu-tion as

EðhkðXÞÞ ¼Z 1

�1� � �Z 1

�1hkðxðu; lÞÞ/U ðuÞdu

¼Z 1

�1hkfH ðh; lÞdh; ð20Þ

where fH(h) is PDF of a performance function h(X). Sincethe cumulative distribution function (CDF) of the perfor-mance function can be expressed in terms of the standardnormal CDF using the following transformationFH(h) = U(t), Eq. (20) becomes

EðhkðXÞÞ ¼Z 1

�1hkfH ðh; lÞdh ¼

Z 1

�1hkðt; lÞ/ðtÞdt ð21Þ

where the parametric variable t is the distance from the ori-gin in u-space to the most probable point (MPP) as shownin Fig. 2.

Hence, the multi-dimensional integral can be rewrittenby a one-dimensional integral. Similar to the univariateDRM, the performance moment integration (PMI) makesuse of three quadrature points and weights to approximatethe one-dimensional integration in Eq. (21). A differencebetween the two methods is that quadrature points of theunivariate DRM lie on the xi-axis, whereas quadraturepoints of PMI lie on the MPP locus [3,22]. Therefore, thenumber of quadrature points in the univariate DRM

PDF Transformation

f ðxÞ ¼ 1ffiffiffiffi2pp

re�0:5 x�l

r½ �2

X = l + rU

f ðxÞ ¼ 1ffiffiffiffiffiffi2pxp

�re�0:5 ln x��l

�r½ �2 X ¼ expð�lþ �rUÞ

f ðxÞ ¼ km ðxm Þ

k�1e�ðxvÞ

kX ¼ v½� lnðUð�UÞaÞ�

1k

f ðxÞ ¼ ae�aðx�mÞ�e�aðx�mÞX ¼ m� 1

a ln½� lnðUðUÞÞ�

f ðxÞ ¼ 1b�a ; a 6 x 6 b X = a + (b � a)U(U)


Fig. 2. Approximation of CDF Using MPP Locus [3].


ARTICLE IN PRESS

increases as the number of design variables increases asshown in Eq. (12), whereas the number of quadraturepoints in PMI does not change since the integration is per-formed in the output space.

Since t follows the standard normal distribution, theweights and quadrature points in Table 1 can be used todiscretize Eq. (21) as

EðhkðXÞÞ ¼Z 1

�1hkðt; lÞ/ðtÞdt

ffi 1

6� hkðt; lÞ

��t¼�

ffiffi3p þ 4

6� hkðt; lÞ

��t¼0

þ 1

6� hkðt; lÞ

��t¼ffiffi3p : ð22Þ

By changing the order of calculation, Eq. (22) becomes

EðhkðXÞÞ ¼Z 1

�1hkðt; lÞ/ðtÞdt

ffi 1

6� fhðt; lÞgk��

t¼�ffiffi3p þ 4


t¼0

þ 1


t¼ffiffi3p

¼ 1

6� hkð�

ffiffiffi3p

; lÞ þ 4

6� hkð0; lÞ þ 1

6� hkð

ffiffiffi3p

; lÞ:

ð23Þ

Using the first order reliability method (FORM) [23,24]and MPP locus illustrated in Fig. 2, each term in Eq. (23)can be approximated as two function values at two MPPsand a function value at the design point. The function val-ues at MPPs can be obtained using the inverse reliabilityanalysis PMA to

maximize hðUÞ;subject to kUk ¼

ffiffiffi3p ð24Þ

The optimum result of Eq. (24) is denoted as hmaxbt¼

ffiffi3p ,

which can be used to approximate hðffiffiffi3p

; lÞ in Eq. (23).The term hð�

ffiffiffi3p

; lÞ in Eq. (23) can be approximated bythe optimum result obtained by minimizing h(U) in Eq.


(24) and denoted as hminbt¼

ffiffi3p . The term h(0;l) in Eq. (23)

can be approximated by h(lX), which is the performancefunction value at the design point. Hence, using these func-tion values and Eq. (23), the statistical moments of a per-formance function can be calculated as

EðhkðXÞÞ ffi 1

6� hmin

bt¼ffiffi3p

� kþ 4

6� hkðlXÞ þ

1

6� hmax

bt¼ffiffi3p

� k: ð25Þ

Consequently, the mean value and variance can be esti-mated by

lH ffi1

6hmin

bt¼ffiffi3p þ 4

6hðlXÞ þ

1

6hmax

bt¼ffiffi3p ;

r2H ffi

1

6ðhmin

bt¼ffiffi3p Þ2 þ 4

6h2ðlXÞ þ

1

6ðhmax

bt¼ffiffi3p Þ2 � l2

H :

ð26Þ

Thus, PMI is very efficient when the number of designvariables is relatively large.

3.2.2. Sensitivity of statistical moments

Similar to the sensitivity calculation in DRM, from Eqs.(21), (23) and (25), the sensitivities of the mean and vari-ance of the performance function with respect to a thedesign variable li can be derived as

olH

oli

¼ o

oli

Z 1

�1hðt; lÞ/ðtÞdt ¼

Z 1

�1

ohðt; lÞoli

/ðtÞdt

ffi 1

6

ohminbt¼

ffiffi3p

oliþ 4

6

ohðlXÞoli

þ 1

6

ohmaxbt¼

ffiffi3p

oli

ffi 1

6

ohminbt¼

ffiffi3p

oxMPPi� oxMPP

i

oliþ 4

6

ohðlXÞoli

þ 1

6

ohmaxbt¼

ffiffi3p

oxMPPi� oxMPP

i

oli

ð27Þ

¼ 1

6

ohðxÞoxi

��x¼xmin

MPP

� oxMPPi

oliþ 4

6

ohðxÞoxi

��x¼lX

þ 1

6

ohðxÞoxi

��x¼xmax

MPP

� oxMPPi

oli



ARTICLE IN PRESS

or2H

oli¼ o

oli

Z 1

�1h2ðt;lÞ/ðtÞdt�ol2

H

oli¼Z 1

�1

oh2ðt;lÞoli

/ðtÞdt�ol2H

oli

ffi 1

6

oðhminbt¼

ffiffi3p Þ2

oliþ4

6

oh2ðlXÞoli

þ1

6

o hmaxbt¼

ffiffi3p

� 2

oli�ol2

H

oli

ffi 1

6

o hminbt¼

ffiffi3p

� 2

oxMPPi

�oxMPPi

oliþ4

6

oh2ðlXÞoli

þ1

6

o hmaxbt¼

ffiffi3p

� 2

oxMPPi

�oxMPPi

oli�ol2

H

olið28Þ

¼ 1

6

oh2ðxÞoxi

��x¼xmin

MPP

�oxMPPi

oliþ4

6

oh2ðxÞoxi

��x¼lX

þ1

6

oh2ðxÞoxi

��x¼xmax

MPP

�oxMPPi

oli�ol2

H

oli

Since no explicit equation is available for xMPPi , it is not

possible to analytically obtainoxMPP

ioli

. However, the termoxMPP

ioli

can be approximated as following. Using Rosenblatt trans-

formation in Table 2, it is clear that xMPPi is a function of

uMPPi and li written as xMPP

i ¼ T�1ðuMPPi ; liÞ where

T : x! u is the transformation. UsingouMPP

ioliffi 0,

oxMPPioli

can

be approximated byoxMPP

ioliffi 1. The verification of the

approximationoxMPP

ioliffi 1 for various distributions using

the following example hðXÞ ¼ 1� 80X 2

1þ8X 2þ5

with the target

reliability bt = 3 and finite difference method with 1% per-turbation is given in Table 3.

By using the approximationoxMPP

ioliffi 1, which is a similar

to oxiðui ;liÞoliffi 1 in DRM, sensitivities of the mean and vari-

ance of the performance function with respect to li canbe obtained as

olH

oliffi 1

6

ohðxÞoxi

��x¼xmin

MPP

þ4

6

ohðxÞoxi

��x¼lX

þ1

6

ohðxÞoxi

��x¼xmax

MPP

;

or2H

oliffi 1

6

oh2ðxÞoxi

��x¼xmin

MPP

þ4

6

oh2ðxÞoxi

��x¼lX

þ1

6

oh2ðxÞoxi

��x¼xmax

MPP

�ol2H

oli

ð29Þ

Table 3Verification of assumption

oxMPP1

ol1ffi 1 using hðXÞ ¼ 1� 80

X 21þ8X 2þ5

Distribution xMPP xMPPa xMPP1 � xMPP

1

l1 � l1

N(5,0.3) (5.7368,5.5168) (5.7890,5.5136) 1.044LN(5,0.3) (5.7923,5.5207) (5.8443,5.5172) 1.040Weibull(5,0.3) (5.5289,5.4501) (5.5799,5.4482) 1.020Gumbel(5,0.3) (6.2823,5.2749) (6.3362,5.2689) 1.078U(5,0.3) (5.5044,5.4972) (5.5546,5.4989) 1.004

a xMPP means xMPP obtained from 1% perturbation of l1.


Since the sensitivities of the performance function on theright hand side of Eq. (29) are used during the inverse reli-ability analysis described in Eq. (24), no additional func-tion evaluations are required to calculate sensitivitiesusing Eq. (29).

3.3. Percentile difference method (PDM)

Like PMI, PDM uses the results of the inverse reliabilityanalysis [6,7]. PMI utilizes the function values at two MPPs(hmax

bt¼ffiffi3p and hmin

bt¼ffiffi3p Þ obtained from the inverse reliability

analysis and the function value at the mean lX to approx-imate the multidimensional integration in Eq. (6), whereasPDM ‘‘uses the difference between the function values attwo MPPs to represent the variation of the performancefunction’’ [6]. Hence, the RBRDO formulation usingPDM is to

minimize f ðhðlXÞ; hp1� hp2

Þsubject to PðGiðX; dÞ > 0Þ 6 Uð�btiÞ; i ¼ 1; . . . ; nc

dL6 d 6 dU ; d 2 Rndv and X 2 Rnrv

ð30Þ

where p1 is a right-tail percentile, p2 is a left-tail percentileand, in general, p1 + p2 = 1. When p1 = 0.95 and p2 = 0.05[6,7], hp1

and hp2in Eq. (30) are calculated from the inverse

reliability analysis with a target reliability index 1.645(bt = U�1(p1) = 1.645), that is, hp1

¼ hmaxbt¼1:645 and hp2

¼hmin

bt¼1:645.

As shown in Fig. 3, the idea of PDM is simple and couldbe viewed as meaningful, but it has rather serious short-comings. If the performance function is not monotonic, itmay not be possible to use hp1

� hp2as a measurement of

robustness. In a non-monotonic performance functioncase, two MPPs obtained from the inverse reliabilityanalysis may not approximate the left-tail and right-tailpercentile accurately because the inverse reliability analy-sis searches MPPs on the surface of the hyper-sphere in

Fig. 3. Basic concept of robust design using percentile difference method[6].



ARTICLE IN PRESS

u-space. For example, if h(X) = X2 and X � N(0,1) and thetarget reliability bt is 1.645, then two MPPs become 1.645and �1.645. Thus, two percentile performances hp1

andhp2

are identical. In contrast to PDM, PMI and the univar-iate DRM show the correct moment estimation of the per-formance function h(X) = X2. Thus, PDM-based RBRDOmay identify a wrong global minimum when there are sev-eral local minima, as demonstrated in Section 4.2. Moresignificantly, there is no one percentile that can be usedin PDM to identify all local optima correctly as shown inSection 4.2.

Sensitivity of the cost function with respect to a designvariable li can be calculated using a similar procedure asPMI

ohðlXÞoli

¼ ohðxÞoxi

��x¼lX

ohp1

oli� ohp2

oliffi ohp1

oxMPPi� oxMPP

i

oli� ohp2

oxMPPi� oxMPP

i

oli

ffi ohðxÞoxi

��x¼xMPP

p1

� ohðxÞoxi

��x¼xMPP

p2

:

ð31Þ

3.4. Comparison

Two criteria to identify which method is effective forrobust design optimization (RDO) are computational effi-ciency and accuracy of the moment estimation. In termsof computational efficiency, both PMI and PDM willrequire the same number of function evaluations if thesame inverse reliability analysis method is used. In general,if the number of design variables is large, Eq. (12) showsthat DRM requires more function evaluations than PMIand PDM. However, an advantage of using the univariateDRM is that the univariate DRM does not require sensitiv-ity information (i.e., no MPP search) in estimating themoments. Hence, the univariate DRM can reduce the num-ber of function evaluations during line searches.

The objective of the univariate DRM and PMI is toapproximate the multi-dimensional integration in Eq. (6).That is, both methods attempt to transform the multi-dimensional integration into a readily computable numeri-cal integration. However, PDM does not use any numericalintegration, instead it uses the difference of percentile per-formances. Thus, PDM may yield wrong results when theperformance function is non-monotonic. Both PMI and

Table 4Comparison of the first and second moments of Eq. (32)

Mean (lH)

PMI DRM NI

h1 �5.6500 �5.5000 �5Error (%) 2.73 0No. of F.E. 7 + 7b 2 · 2 + 1

a NI means numerical integration.b 7 + 7 means 7 function evaluations and 7 sensitivity calculations.


PDM may have a difficulty to find MPPs when the perfor-mance function is non-monotonic and highly nonlinear. Onthe other hand, DRM may accurately estimate themoments of the performance function regardless of the per-formance function type.

In terms of accuracy of the moment estimation, the uni-variate DRM yields better results in most cases than PMI.If the performance function is highly nonlinear, then theunivariate DRM with three quadrature points may notaccurately estimate the second moment. In this case, theerror can be reduced if more quadrature points are usedin the univariate DRM. However, PMI with more quadra-ture points than 3 may not necessarily yield more accurateresults. This is because function values at quadraturepoints, which are obtained using FORM and MPP search,are approximations. More details of comparison withnumerical examples are given in the following section.

4. Numerical examples

In this section, four cases of comparisons are carried outusing numerical examples. In Section 4.1, the univariateDRM and PMI are compared in terms of accuracy and effi-ciency in estimation of the moments and their sensitivitiesof a performance function. PDM is excluded in Section4.1 since it cannot estimate the moments of the perfor-mance function. In Section 4.2, DRM, PMI, and PDMare compared using a one-dimensional fourth order poly-nomial for identification of correct robust optimum design.In this one-dimensional problem, PMI and the univariateDRM with three quadrature points can be considered tobe the same method. In Section 4.3, comparison of threemethods is carried out using a two-dimensional fourthorder polynomial for design optimization. In Section 4.4,a side impact crashworthiness example is used for the com-parison of DRM and PMI in terms of the number of thefunction evaluations in a large-scale engineering problem.

4.1. Comparison of PMI and DRM for computation of

moments and sensitivities

For the first example, the performance function is

h1ðXÞ ¼ 1� X 21X 2

20; ð32Þ

where Xi � N(5, 1) for i = 1, 2. As shown in Table 4, bothDRM and PMI provide good estimation of the mean value

Variance ðr2H Þ

a PMI DRM NI

.5000 8.4623 7.9355 8.31751.74 4.587 + 7 2 · 2 + 1


Table 5Sensitivity of mean value using PMI and DRM for Eq. (32)

PMI DRM Analytic

olH ðdÞod1

olH ðdÞod2

olH ðdÞod1

olH ðdÞod2

olH ðdÞod1

olH ðdÞod2

Sensitivity �2.5475 �1.2823 �2.5000 �1.3000 �2.5000 �1.3000Error (%) 1.90 1.36 0.00 0.00Additional no. of F.E. 0 2 · 2 + 1


ARTICLE IN PRESS

and standard deviation in comparison with the exactnumerical integration results. The reason DRM has a lar-ger error in estimation of standard deviation is becausethe performance function in Eq. (32) has an off-diagonalterm only. As mentioned in Section 3.1.1, if the perfor-mance function has off diagonal terms only, then the uni-variate additive decompositions of the moments in Eqs.(10) and (11) may contain significant errors.

For this example, PMI yields reasonable estimation ofthe moments because the design variables are normally dis-tributed, which means that the inverse reliability analysisdoes not require non-linear transformations from x-spaceto u-space, and the performance function is monotonic atthe given design. In the same token, the sensitivities inTables 5 and 6 have similar errors as Table 4.

The total number of function evaluations for PMI toevaluate the mean and standard deviation is 7 + 7 as shownin Table 4, where the first 7 is the number of function eval-uation for MPP search and the second 7 is the number ofsensitivity calculation for MPP search. The number of

Table 6Sensitivity of variance using PMI and DRM for Eq. (32)

PMI DRM Analytic

or2H ðdÞod1

or2H ðdÞod2

or2H ðdÞod1

or2H ðdÞod2

or2H ðdÞod1

or2H ðdÞod2

Sensitivity 3.9211 2.5894 3.7500 2.5500 3.9000 2.5500Error (%) 0.54 1.54 3.85 0.00

Table 7Comparison of the first and second moments of Eq. (33)

Mean (lH)

PMI DRM NI

h2 �1.0594 �1.1167 �1Error (%) 5.13 0No. of F.E. 5 + 5 3 · 2 + 1

Table 8Sensitivity of mean value using PMI and DRM for Eq. (33)

PMI

olH ðdÞod1

olH ðdÞod2

Sensitivity �0.1209 �0.5254Error (%) 9.30 1.49Additional no. of F.E. 0


function evaluations for DRM is 5. Since the design vari-ables are normally distributed and the number of quadra-ture points is odd, Eq. (13) is used for the total numberof function evaluations.

PMI does not require additional function evaluationsfor the sensitivity analysis of moments because PMI usesthe sensitivity information in MPP search. However,DRM does require additional function evaluations for sen-sitivity analysis, thus the total number of function evalua-tions needs to be doubled in DRM as shown in Table 5.

Since the first example contains an off-diagonal termonly and the design variables are normally distributed,the second example is modeled as

h2ðXÞ ¼ 1� ðX 1 þ X 2 � 5Þ2

30� ðX 1 � X 2 � 12Þ2

120; ð33Þ

where Xi � Gumbel(5,1) for i = 1, 2. The performancefunction in Eq. (33) contains both off-diagonal terms anddiagonal terms, and the degree of the polynomial perfor-mance function is 2. Therefore, it can be expected that

Variance ðr2H Þ

PMI DRM NI

.1167 0.3357 0.3774 0.383312.42 1.545 + 5 3 · 2 + 1

DRM Analytic

olH ðdÞod1

olH ðdÞod2

olH ðdÞod1

olH ðdÞod2

�0.1333 �0.5333 �0.1333 �0.53330.00 0.003 · 2 + 1

Table 9Sensitivity of variance using PMI and DRM for Eq. (33)

PMI DRM Analytic

or2H ðdÞod1

or2H ðdÞod2

or2H ðdÞod1

or2H ðdÞod2

or2H ðdÞod1

or2H ðdÞod2

Sensitivity 0.0724 0.1033 0.0883 0.1149 0.0884 0.1162Error (%) 18.10 11.10 0.11 1.12



ARTICLE IN PRESS

DRM may yield better results for this example. Asexpected, Tables 7–9 illustrate that DRM is accurate inestimation of the moments and their sensitivities.

On the other hand, PMI yields somewhat larger errorsin estimation of the moments and their sensitivities. Thisis because the design variables follow Gumbel distribution.In such a case, the inverse reliability analysis requires non-linear transformations from x-space to u-space, whichmakes the performance function become highly non-linearand the FORM error become larger.

Since the Gumbel distribution is not symmetric, Eq. (12)is used for the total number of function evaluations forDRM.

4.2. Comparison of PMI, DRM and PDM for identification

of robust optimum design

In this section, three methods are compared in detail forproper identification of robust optimum design, using aone-dimensional example. RDO can be formulated to

minimize r2H

subject to 0 6 X 6 5ð34Þ

where h3(X) = (X � 4)3 + (X � 3)4 + 10 and X � N(l, 0.4).Again note that in one-dimensional problem, PMI andthe univariate DRM with three quadrature points can beconsidered to be the same method. Fig. 4a illustrates theshape of the performance function and Fig. 4b illustrates

Fig. 4. Shape and variance o

Table 10Position and value of optimum using three methods for Eq. (34)

PMI and DRM

3 pts 5 pts

Left min. xmin 1.463 1.483r2

H or hp1� hp2

3.397 4.361Right min. xmin 3.405 3.359

r2H or hp1

� hp21.075 1.220


the variances obtained from DRM and PMI and percentiledifferences from PDM.

As mentioned before, in this example, PMI and DRMwith three quadrature points can be considered to be thesame method since the design variable is normally distrib-uted and there is no FORM error in a one-dimensionalfunction. As shown in Fig. 4b, PMI and DRM with threequadrature points can approximate the variance of the per-formance function very well. On the other hand, PDM withvarious percentiles cannot estimate the moments. More sig-nificantly, the location of the optimum point changesdepending on the percentiles used. In fact, there is no onepercentile that can be used to accurately identify the loca-tion of both local minima simultaneously in Fig. 4b.Table 10 shows that the best percentile should be locatedbetween 2r and 3r for the left local minimum and the bestpercentile should be located between 1.645r and 2r for theright local minimum. In Fig. 4b, ‘Measure for variance’ isused instead of variance. It is because PDM cannot esti-mate the variance of the performance function and usespercentile differences as the measure for the variance.

Another problem of using PDM for a highly non-linearperformance function such as Eq. (34) is that PDM mightnot be able to identify which local minimum is the globalminimum when there is more than one local minimum.As shown in Table 10, the results of PDM with three differ-ent percentiles indicate that the value of the cost function atthe left minimum in Fig. 4b is less than the value at theright minimum, which is wrong.

f performance function.

PDM NI

1r 1.645r 2r 3r

1.236 1.315 1.376 1.622 1.4850.000 0.000 0.000 0.000 4.4033.464 3.397 3.341 3.037 3.3581.375 3.239 4.759 10.645 1.234


Table 11Properties of random variables of Eq. (35)

Random variable Std. dev. Distr. type dL d dU

x1 0.400 Normal 1.000 8.000 10.000x2 0.400 Normal 1.000 8.000 10.000

Table 12Optimum design and cost comparison for Eq. (35)

d1 d2 r2H or hp1

� hp2Analytic variance

DRM (3pts) 3.423 5.004 1.138 1.326DRM (5pts) 3.378 4.997 1.316 1.289PMI 3.407 5.010 1.075 1.308PDM (1.645r) 3.396 5.003 3.239 1.299

Fig. 5. Accuracy of PMI and DRM with five quadrature points.

Fig. 6. Contour of performance function h(X).

Table 13Optimum design and cost of Eq. (35) with various percentiles

Percentile d1 d2 Percentile difference Analytic variance

0.5r 3.491 4.998 0.548 1.4651.0r 3.465 4.994 1.375 1.4001.5r 3.413 4.996 2.724 1.3151.645r 3.396 5.003 3.239 1.2992.0r 3.341 4.995 4.759 1.2862.5r 3.227 5.005 7.495 1.4263.0r 3.046 5.099 10.652 2.024

Fig. 7. Vehicle side impact problem.


ARTICLE IN PRESS

Table 10 also shows that PMI and DRM with threequadrature points yields some errors in finding the locationof the optimum and estimating the value of the optimum.This is because the performance function is a polynomialof degree 4, thus three quadrature points may not be suffi-cient. In this case, DRM and PMI with five quadraturepoints are a good option to achieve accuracy. The accuracyof DRM and PMI with five quadrature points is illustratedin Table 10 and Fig. 5.

4.3. Comparison of PMI, DRM, and PDM for RBRDO

For the purpose of comparison among the three meth-ods, the cost function of the smaller-the-better type in Sec-tion 2.2 is used and weights are given as w1 = 0 and w2 = 1.Then, RBRDO can be formulated to [7]:

minimize r2H

subject to P ð�x1 � x2 þ 6:45 > 0Þ 6 Uð�btÞP ð1 6 xi 6 10ÞP UðbtÞ; i ¼ 1; 2

ð35Þ

where h(x) = (x1 � 4)3 + (x1 � 3)4 + (x2 � 5)2 + 10, Xi �N(li, 0.4) for i = 1, 2 and bt = 3.

Fig. 6 illustrates the contour of the performance func-tion h(X) in the formulation (35) and Table 11 shows theproperties of the random variables.

As shown in Table 12, DRM with 5 quadrature pointsshows the best result in terms of locating the minimum var-iance and the estimation of the variance has the smallesterror. DRM with 3 quadrature points and PMI show errorin estimation of the variance since the performance func-tion is fourth order polynomial as explained in Section3.1.1. PDM with 1.645r shows better result than DRMwith 3 points and PMI. However, as shown in Table 13,the optimum design varies depending on the percentilesused. For this problem, a percentile close to 2.0r showsthe smallest variance of the performance function, whichdoes not mean that the percentile (2.0r) is the best for allproblems as shown in the previous example.


4.4. RBRDO for side impact crashworthiness

The RBRDO model of crashworthiness for vehicle sideimpact shown in Fig. 7 is formulated to



ARTICLE IN PRESS

minimize w1

MM0

þw2

rH

rH0

� �2

subject to Pðabdomen load> 1:0 kNÞ6Uð�btÞPðupper=mid=lowerVC> 0:32 m=sÞ6Uð�btÞPðupper=mid=lower rib deflection> 32 mmÞ6Uð�btÞ

ð36ÞPðpubic symphysis force; F > 4:0 kNÞ6Uð�btÞPðvelocity of B-pillar at mid-point> 9:9 mm=msÞ6Uð�btÞPðvelocity of front door at B-pillar> 15:7 mm=msÞ6Uð�btÞdL6 d6 dU ; d2R9 and X2R11;bt ¼ 2

where M is the mass of the vehicle door and the perfor-mance function h(x)is the lower rib deflection. The detailedequations for the mass of vehicle door and constraints canbe found in [25].

There are eleven random parameters and nine parame-ters out of the eleven random parameters are design param-

Table 14Properties of design and random parameters of Eq. (36)

Random variable Std. dev. Distr. type dL

1. B-pillar inner (mm) 0.100 Normal 0.5002. B-pillar reinforce (mm) 0.100 Normal 0.4503. Floor side inner (mm) 0.100 Normal 0.5004. Cross member (mm) 0.100 Normal 0.5005. Door beam (mm) 0.100 Normal 0.8756. Door belt line (mm) 0.100 Normal 0.4007. Roof rail (mm) 0.100 Normal 0.4008. Mat. B-pillar inner (GPa) 0.006 Normal 0.1929. Mat. floor side inner (GPa) 0.006 Normal 0.19210. Barrier height (mm) 10.000 Normal 10th a11. Barrier hitting (mm) 10.000 Normal

Table 15RBRDO results using DRM for side impact problem

Initial design Optim

Mass Var. Analytic variance Mass

30.83 2.361 2.374 28.17

Table 16RBRDO results using PMI for side impact problem

Initial design Optim

Mass Var. Analytic variance Mass

30.83 2.380 2.374 27.77

Table 17Optimum design comparison for side impact problem

d1 d2 d3 d4

Initial 1.000 1.000 1.000 1.000DRMa 0.949 1.350 0.529 1.500PMIa 0.876 1.350 0.527 1.500DRMb 1.359 1.350 0.529 1.500PMIb 1.360 1.350 0.529 1.500

a For both DRM and PMI, w1 = w2 = 0.5 is used.b For both DRM and PMI, w1 = 0.0, w2 = 1.0 is used.


eters. The design parameters are the thickness (d1 � d7) andmaterial properties of critical parts (d8, d9) as shown inTable 14.

Tables 15 and 16 show RBRDO results using DRMwith 3 points and PMI where equal weights(w1 = w2 = 0.5) are used. Both methods show significantreduction in the robust objective and very good accuracyin estimation of the variance. However, a total number offunction evaluation (54 + 54) in PMI to estimate the vari-ance is much less than (209 + 95) in DRM with 3 points.When equal weights (w1 = w2 = 0.5) are used, two opti-mum designs seem to be a little bit different, but this differ-ence is due to the error of variance estimation andcharacteristic of a bi-objective optimization. If the objec-tive is changed to minimize the variance only, that is,w1 = 0.0 and w2 = 1.0, then two optimum results arealmost identical as shown in Table 17.

d dU

1.000 1.5001.000 1.3501.000 1.5001.000 1.5002.000 2.6251.000 1.2001.000 1.2000.300 0.3450.300 0.345

nd 11th random variables are not regarded as design variables

um design

Var. Analytic variance No. of F.E.

1.458 1.471 209 + 95

um design

Var. Analytic variance No. of F.E.

1.488 1.492 54 + 54

d5 d6 d7 d8 d9

2.000 1.000 1.000 0.300 0.3000.975 1.200 0.400 0.280 0.1920.961 1.200 0.400 0.305 0.1922.000 1.200 1.000 0.192 0.3002.000 1.200 1.000 0.192 0.300



ARTICLE IN PRESS

5. Discussion and conclusion

Three methods (PMI, PDM, and univariate DRM) arecompared in terms of efficiency and accuracy for computa-tion of the statistical moments and their sensitivities. Tocompare the accuracy in estimation of the statisticalmoments of the performance function, two polynomial per-formance functions with two design variables are employed.In this comparison, PDM is excluded since PDM cannotestimate the moments of the performance function. Thecomparison shows that DRM can accurately estimate thestatistical moments of the performance function for thedesign variables with both non-normal and normal distri-butions. On the other hand, PMI can accurately estimatethe statistical moments of the performance function forthe design variables with normal distributions. For non-normally distributed design variables, PMI shows someerrors since non-linear transformations make the perfor-mance function become highly non-linear. For RBRDO,a highly nonlinear performance function was used for com-parison purposes. Both the one-dimensional and the two-dimensional examples show that, in most cases, PMI andDRM can identify the optimum design and estimate thecost function accurately, whereas the optimum design ofPDM varies depending on the percentile used, and PDMhas identified a wrong global minimum. To achieve betteraccuracy, DRM with five quadrature points can be used.

PMI and PDM yield the same efficiency if the sameinverse reliability analysis is used to find MPPs. Non-line-arity of the performance function affects the total numberof function evaluations most significantly in RBRDO usingPMI and PDM. In estimation of the statistical momentsusing DRM, the number of design variables affects thetotal number of function evaluations most significantly.Hence, if the number of design variables is large, it is rec-ommended to use PMI, compared to DRM, for RBRDO.

Acknowledgement

Research is supported by the Automotive Research Cen-ter that is sponsored by the US Army TARDEC.

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Dimension reduction method for reliability-based robust ...user.engineering.uiowa.edu/~kkchoi/RBDO_A.12.pdfdesign optimization (RDO) is a method to improve the product quality by minimizing