Sampling-Based Stochastic Sensitivity Analysis Using Score Functions for RBDO Problems with Correlated Random Variables

Transcription

1 Proceedngs of the ASME 00 Internatonal Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference IDETC/CIE 00 August 5 8, 00, Montreal, Canada DETC Samplng-Based Stochastc Senstvty Analyss Usng Score unctons for RBDO Problems wth Correlated Random Varables In Lee, yung. Cho, Yooeong Noh, Lang Zhao Department of Mechancal & Industral Engneerng, College of Engneerng The Unversty of Iowa, Iowa Cty, IA 54, USA lee@engneerng.uowa.edu, cho@engneerng.uowa.edu, noh@engneerng.uowa.edu, lazhao@engneerng.uowa.edu Davd Gorsch US Army RDECOM/TARDEC Warren, MI , USA davd.gorsch@us.army.ml ABSTRACT Ths study presents a methodology for computng stochastc senstvtes wth respect to the desgn varables, whch are the mean values of the nput correlated random varables. Assumng that an accurate surrogate model s avalable, the proposed method calculates the component relablty, system relablty, or statstcal moments and ther senstvtes by applyng Monte Carlo smulaton (MCS to the accurate surrogate model. Snce the surrogate model s used, the computatonal cost for the stochastc senstvty analyss s neglgble. The copula s used to model the ont dstrbuton of the correlated nput random varables, and the score functon s used to derve the stochastc senstvtes of relablty or statstcal moments for the correlated random varables. An mportant mert of the proposed method s that t does not requre the gradents of performance functons, whch are nown to be erroneous when obtaned from the surrogate model, or the transformaton from -space to U-space for relablty analyss. Snce no transformaton s requred and the relablty or statstcal moment s calculated n -space, there s no approxmaton or restrcton n calculatng the senstvtes of the relablty or statstcal moment. Numercal results ndcate that the proposed method can estmate the senstvtes of the relablty or statstcal moments very accurately, even when the nput random varables are correlated. EYWORDS Stochastc Senstvty Analyses, Score unctons, Monte Carlo Smulaton, Copula, Surrogate Model, RBDO. INTRODUCTION Relablty-based desgn optmzaton (RBDO and relablty-based robust desgn optmzaton (RBRDO have been wdely appled to varous engneerng applcatons such as stampng [,], vehcle desgn wth durablty [3,4], and nose, vbraton, and harshness (NVH analyss [5,6], where accurate senstvtes of performance functons are avalable. If accurate senstvtes are avalable n a complex physcal system, then the most probable pont (MPP based relablty analyss, whch ncludes the rst Order Relablty Method (ORM [7-0], the Second Order Relablty Method (SORM [,], and the MPP-based Dmenson Reducton Method (DRM [3-5], can be used for approxmately assessng the relablty of the system, whch s used as a probablstc constrant n both RBDO and RBRDO. urthermore, the frst and second statstcal moments of the performance functon, whch are used n the obectve functon of RBRDO, are approxmated usng the frst-order Taylor seres expanson [6-7] or the mean-based DRM [8-0]. However, for engneerng applcatons where accurate senstvtes of performance functons are not avalable, such as drvetran, crashworthness, mcro or nano mechancs, or flud-structure nteracton, the MPP-based relablty analyss, whch uses the senstvtes of performance functons to fnd the MPP, cannot be drectly used. Instead, surrogate models

2 have been wdely used to carry out desgn optmzaton for the engneerng applcatons where senstvtes are unavalable [-3]. Once an accurate surrogate model s avalable for the desgn optmzaton, the drect Monte Carlo smulaton (MCS [4] can be appled to the accurate surrogate model to estmate the relablty or statstcal moments of the system wth neglgble computatonal burden. The relablty or statstcal moment wll be called the probablstc response n ths paper. To use the probablstc response obtaned usng the MCS for the desgn optmzaton, senstvtes of the probablstc response are stll requred, whch can be obtaned usng the fnte dfference method (DM or usng the senstvtes of the surrogate model. However, snce the probablstc response s obtaned from the MCS, the DM may requre a sgnfcant number of samples to obtan accurate senstvtes due to the statstcal nose of the MCS. On the other hand, even f the surrogate model s very accurate for the response value, the senstvtes obtaned from the surrogate model are nown to be naccurate, and accordngly t s not a good dea to use them for the desgn optmzaton. The man obectve of ths paper s to propose a new samplng-based RBDO for correlated random nputs [5-7], whch does not requre obtanng the senstvtes of the performance functons and ther senstvtes from the surrogate model. Instead, stochastc senstvty analyss usng the score functon, whch was proposed for the ndependent random varables [8] or correlated Gaussan random varables [9], s used to derve the senstvtes of the probablstc response wth correlated random varables. The proposed senstvty analyss does not requre the transformaton from the orgnal desgn space to the standard normal space, whch means that there s no approxmaton or restrcton n calculatng the senstvtes of the probablstc response. Snce the generaton of the accurate surrogate model s beyond the scope of ths paper, t wll not be explaned here. or ths, the dynamc rgng (D-rgng method developed n Ref. 3 s used n ths paper. Numercal examples demonstrate that the senstvtes developed n the paper agree very well wth the senstvtes obtaned from the DM wth 50 mllon samples for the MCS. The paper s organzed nto four man parts. The frst part, Secton, explans how to mathematcally express the component, the system probablty of falure, and the statstcal moments and ther senstvtes. The second part, Secton 3, shows how to derve the score functon for statstcally ndependent or correlated random varables. The thrd part, Secton 4, shows the formulaton of the samplngbased RBDO and RBRDO. The last part, Secton 5, demonstrates wth numercal examples the accuracy of the proposed senstvtes compared wth the DM results. nally, the proposed method s ntegrated wth the D-rgng method to carry out RBDO and compared wth four other RBDO methods to valdate and pont out the mert of the proposed method.. RELIABILITY, STATISTICAL MOMENTS AND SENSITIVITY. Relablty and Statstcal Moments A relablty analyss, for both the component and the system level, nvolves calculaton of the probablty of falure, denoted by P, whch s defned usng a mult-dmensonal ntegral P( ψ P[ ] I ( f ( ; d EI ( N x x ψ x ( where ψ s a vector of dstrbuton parameters, whch usually ncludes the mean (µ and/or standard devaton (σ of the random nput,, T N, P represents a probablty measure, s the falure set, (; x ψ s a ont probablty densty functon (PD of, and the expectaton operator. The falure set s defned as x: G ( x 0 for component relablty analyss of the f E represents and th constrant functon G (x, and : NC x G ( 0 x NC x: G ( 0 x for seres system and parallel system relablty analyss of NC performance functons, respectvely [9,30]. I ( x n Eq. ( s called an ndcator functon and defned as I, x ( x ( 0, otherwse μ,, N, s used as a desgn vector, the vector of dstrbuton parameters ψ s smply replaced wth µ for the dervaton of the senstvty. In a fashon smlar to Eq. (, the q th statstcal moment of a performance functon H(x s defned as In ths paper, snce the mean of, T q q m ( μ E[ H ( ] H ( x f ( x; μ dx, q N and thus, Eqs. ( and (3 can be wrtten n a generalzed form as h( μ E[ g( ] g( x f ( x; μ dx N whch s called a probablstc response [9]. In Eq. (4, h( μ and g(x represent P ( μ and I ( x, respectvely, for relablty analyss, and h( μ and g(x represent m ( μ and q H ( x, respectvely, for statstcal moment analyss.. Stochastc Senstvty Analyss The senstvty of the probablstc response h( μ wth respect to s consdered n ths secton. or the dervaton of the senstvty, the followng four assumptons, whch are nown as the regularty condtons, are requred [8,9].. The ont PD f (; x μ s contnuous.. The mean,,, N, where M s an open nterval on. q (3 (4

3 f ( ; 3. The partal dervatve x μ exsts and s fnte for all x and. In addton, h( μ s a dfferentable functon of µ. 4. There exsts a Lebesgue ntegrable domnatng functon r(x such that f ( ; g( x x μ r( x (5 for all µ. Wth the four assumptons satsfed, tang the partal dervatve of Eq. (4 wth respect to yelds h( μ g( f ( ; d N x x μ x and the dfferental and ntegral operators can be nterchanged due to Assumpton 4 and the Lebesgue domnated convergence theorem [9,3], gvng h( μ f ( x; μ g( d N x x ln f ( x; μ g( x f ( ; d N x μ x ln f ( ; x μ Eg( x snce g(x s not a functon of. The partal dervatve of the log functon of the ont PD n Eq. (7 wth respect to s nown as the frst-order score functon for and s denoted as (6 (7 ( ln f ( ; ( x; μ x μ. (8 s Therefore, t s requred to now the frst-order score functon to derve the senstvty of the probablstc response, whch s ether the relablty or the statstcal moments. The dervaton of the frst-order score functon for ndependent and correlated random varables wll be shown n the subsequent secton. 3. SENSITIVITY ANALYSIS BY SCORE UNCTION 3. or Independent Random Varables Consder a random nput,, T N whose components are statstcally ndependent random varables. Then, the ont PD of s expressed as multplcaton of ts margnal PDs as N f ( x ; μ f ( x ; (9 where f ( x; s the margnal PD correspondng to the th random varable. Therefore, for statstcally ndependent random varables, the frst-order score functon for s expressed as ln ( ; ( ln f ( ; f x ( x; μ x μ. (0 s Snce the margnal PD and cumulatve dstrbuton functon (CD are avalable analytcally as lsted n Table, where ( and ( are the standard normal CD and PD, respectvely, gven by u u ( u ( d exp d, ( the dervaton of Eq. (0 s straghtforward for normal, lognormal, and Gumbel dstrbutons. Normal Lognormal Gumbel Webull Unform Table. Margnal PD, CD, and Its Parameters PD, f ( x CD, ( x Parameters x 0.5[ ] e x e x e ln x 0.5[ ] ( x ( x e x ( ln x ( x e e x ( x v ( e v e, a x b b a, ln[ ( ], ln( , 6 v(, v [ ( ( ] x a a b b a, b a However, for the case of Webull dstrbuton, the dervaton s not straghtforward snce two dstrbuton parameters, and ν, are coupled as shown n Table. The score functon for Webull dstrbuton s wrtten as ( x x s ( x; μ ln ( ln ( x ln ( x x ( ln ( whch requres the calculaton of and snce and ν are functons of. As shown n Table, both and requre the evaluaton of the gamma functon defned as

4 z t z t e dt 0 ( and ts dervatve. By tang the partal dervatve on two parameters of Webull dstrbuton and usng the dervatve of the gamma functon, we can obtan and as t t e lntdt 0 t t e lntdt 0 (3 and nsertng Eq. (3 nto Eq. ( yelds the frst-order score functon for Webull dstrbuton. Table summarzes the frst-order score functons for four margnal PDs. and for lognormal dstrbuton n Table can be easly derved usng the defnton of and shown n Table, and and for Webull dstrbuton n Table are shown n Eq. (3. Table. rst-order Score uncton for for Independent Random Varables Margnal rst-order Score uncton, s ( ( ; Dstrbuton x μ x Normal ln Log-normal x (ln x ( x Gumbel e x ( ln Webull x x ( ln nally, the unform dstrbuton, as shown n Table, s not contnuous on, whch maes the ont PD n Eq. (9 dscontnuous. Ths dscontnuty of the unform dstrbuton volates the frst assumpton explaned n Secton.. Thus, the score functon cannot be used f a random varable follows unform dstrbuton. Assumng follows the unform dstrbuton and all the components of are statstcally ndependent random varables, then the senstvty of the probablstc response wth respect to shown n Eq. (6 can be wrtten as h( μ b( ( ( ; ( ; N ( a g x f x f x μ dxdx (4 where and μ are vectors of the random varables and ts means, respectvely, wthout the th component. Snce two ntegral lmts, a and b, are functons of the dfferental varable µ, the dfferental and ntegral operators cannot be nterchanged drectly. Instead, the Lebnz ntegral rule [33] gves a formula for dfferentaton of a defnte ntegral whose lmts are functons of the dfferental varable, whch s gven by b( z b( z f b f( x, z dx dx f( b( z, z za( z a( z z z a f( a( z, z z Applyng the Lebnz ntegral rule to Eq. (4 yelds h( μ g( x, b g( x, a ( ; N f d x μ x ba g(, b g(, a E b a (5 (6 whch s the senstvty of the probablstc response wth respect to µ when follows unform dstrbuton. 3. or Correlated Input Random Varables Consder a bvarate correlated random nput, T. Then, the ont PD of s expressed as [6,7] Cuv (, ; f ( x ; μ f ( x; ( ; f x uv (7 C ( u, v; f ( x; f ( x ;, uv where C s a copula functon, u ( x ; and v ( x ; are CDs for and, respectvely, and θ s the correlaton coeffcent. Table 3 lsts commonly used copula functons [6,7]. Copula Type Table 3. Commonly Used Copula unctons Copula uncton, Cu, v Clayton u v / AMH uv u v ln / uv uv u v u v sw s w exp dsdw uv u v ran e e e GM Gaussan Independent

5 The partal dervatve of the copula functon wth respect to the margnal CDs u and v s called the copula densty functon and denoted as Cuv (, ; cuv (, C, uv ( uv, ; uv (8 and dsplayed n Table 4. As shown n Table 4, the ont PD for ndependent random varables explaned n Eq. (9 s a specal case of Eq. (7 where the ndependent copula functon s used. Table 4. Copula Densty unctons Copula Type Copula Densty unctons, cuv (, ; Clayton ( ( ( uv u v AMH ran u v u v uv 3 uv e ( uv e GM Gaussan Independent ( u ( v ( uv e e e e u v ( u ( v exp ( u ( v ( u ( v exp ( Accordngly, usng Eq. (7, the frst-order score functons n Eq. (8 for a correlated bvarate nput are expressed as ln ( ; ( ln f ( ; ln cuv (, ; f x ( x; μ x μ. (9 s The dervaton of the frst term of the rght-hand sde of Eq. (9 s straghtforward usng Table 4 and lsted n Table 5, and the second term of the rght-hand sde of Eq. (9 s dentcal to Eq. (0, so t can be obtaned from Table. One can see from Table 5 that Eq. (9 s dentcal to Eq. (0 f the ndependent copula s used. In Table 5, the partal dervatve u of the margnal CD wth respect to µ, s easly obtaned from Table and s shown n Table 6. Table 5. Log-dervatve of Copula Densty uncton Copula ln cuv (, ; Type ( ( u u Clayton u u v v( v 3 ( v u AMH uvuvuv uv ran ( u ( uv e e u ( u ( v ( uv e e e e GM (v u uv Gaussan ( u ( v ( u u ( ( u ( ( u( Independent 0 Table 6. Partal Dervatves of Margnal Dstrbuton wth respect to µ Margnal u Partal Dervatves of Margnal Dstrbuton, Dstrbuton Normal Log-normal Gumbel Webull Unform x x ln 0.5[ ] exp d e ln (ln d e ( x ( x e x ( x x e ln b a In a smlar way to that explaned n Secton 3., consder a bvarate correlated random nput, T where s assumed to follow the unform dstrbuton. Then, the senstvty of the probablstc response wth respect to for the bvarate correlated random nput can be wrtten as h( μ b( a( g( x cuv (, ; f ( x; f ( x; dxdx Applyng the Lebnz ntegral rule to Eq. (0 yelds (0 h( μ b( g(, x c( u, v; E[ dx a( ba g(, bc (, b g (, ac (, a ] b a ( and Eq. ( s dentcal wth Eq. (6 f the ndependent copula s used snce cuv (,;. 4. or Both Independent and Correlated Input Random Varables Consder a random nput,, T N where M pars of bvarate correlated random varables exst. Then, the ont PD of s expressed as

6 f ( x ; μ M c ( u N, v ; f ( x ; ( from Eqs. (9 and (7. Tang the partal dervatve on both sdes of Eq. ( yelds the frst-order score functons for a general random nput as ( ln f ( ; s ( x; μ x μ ln c (, ; ln f ( x ; u v. (3 Thus, f s a correlated random varable, Eq. (3 s dentcal to Eq. (9, and f s statstcally ndependent, Eq. (3 s dentcal to Eq. (0. Smlarly, f follows the unform dstrbuton, Eq. ( can be used to calculate the senstvty of the probablstc response wth a general random nput. 5. ORMULATION O SAMPLING-BASED RBDO The mathematcal formulaton of a general RBDO problem s expressed as mnmze Cost( d PG P NC (4 nrv d dd, dr and R subect to [ Tar ( 0],,, L U ndv T where d { d } μ( s the desgn vector, whch s the mean value of the N-dmensonal random vector T Tar ={,,, N } ; P s the target probablty of falure for the th constrant; and NC, ndv, and nrv are the number of probablstc constrants, desgn varables, and random varables, respectvely. The mathematcal formulaton of a general RBRDO s gven by mnmze f ( H, H subect to [ Tar ( 0],,, L U ndv where and H PG P NC (5 nrv d dd, dr and R H are the mean value and varance of the performance functon H(, respectvely. To carry out RBDO and RBRDO usng Eqs. (4 and (5, respectvely, t s requred to now the functon value and ts senstvtes at a gven desgn. However, n most engneerng applcatons, t s very dffcult, f not mpossble, to obtan accurate senstvtes. or engneerng applcatons where accurate senstvtes are not avalable, surrogate models have been wdely used to carry out desgn optmzaton. Once an accurate surrogate model s avalable for the desgn optmzatons, the MCS can be appled to estmate the relablty or statstcal moments of the system wth neglgble computatonal burden. Denote the surrogate models for constrant functon G ( and performance functon H( as Gˆ ( and H ˆ (, respectvely. Then, by applyng the MCS to the surrogate model Gˆ (, the probablstc constrants n Eqs. (4 and (5 can be approxmated as P P G I x P (6 ( Tar [ ( 0] ˆ ( where s the MCS sample sze, x ( s the th realzaton of, and the falure set ˆ for the surrogate model s defned as ˆ : ˆ x G( x 0. Senstvty of the probablstc constrant n Eqs. (4 and (5 s obtaned usng the score functon explaned n Secton 3 as P x x μ (7 ( ( ( I ˆ ( s ( ; ( ( where s ( x ; μ s obtaned usng Eq. (3. In a smlar manner, the statstcal moments n Eq. (5 can be approxmated as ( x and ( x, (8 ˆ ( ˆ ( ˆ H H Hˆ H Hˆ respectvely, and ther senstvtes are and Hˆ ˆ ( x ( x ; μ (9 Hˆ ( ( ( H s x x μ (30 ˆ ( ( ( ˆ ( ( ; H H s Hˆ, respectvely. As shown n Eqs. (7, (9, and (30, the senstvty calculaton usng the score functon and MCS does not requre the senstvty of the surrogate models, whch s nown to be naccurate even f the surrogate model accurately approxmates the functon values. urthermore, the computaton of the senstvty usng the score functon does not nclude any approxmaton except the statstcal nose due to the MCS, whch can be avoded usng a suffcently large MCS sample sze. In addton, ths senstvty analyss does not requre the transformaton from the orgnal desgn space to the standard normal space, whch usually maes the performance functon become hghly nonlnear, especally when the random nput follows non-gaussan margnal dstrbuton and s correlated. Therefore, the senstvty analyss usng the score functon and MCS wll be very accurate and computatonally effcent for engneerng applcatons wth correlated random nput once accurate surrogate models are avalable; ths wll be verfed through numercal examples n the subsequent secton. 6. NUMERICAL EAMPLES

7 Numercal studes are carred out n ths secton to verfy the stochastc senstvtes derved n Secton 3 usng the score functon. or the benchmar senstvty to test the accuracy of the proposed method, the DM usng the MCS wth 50 mllon samples s used. The stochastc senstvtes of the component probablty of falure and statstcal moments are compared wth the DM results n Secton 5. and Secton 5., respectvely. Secton 5.3 llustrates how the proposed senstvty combned wth an accurate surrogate model can be used to solve an RBDO problem. 6. Senstvtes of Component Probablty of alure Consder a -D hghly nonlnear polynomal functon, whch was studed n Ref. 5, G 3 4 ( ( Y 6 ( Y ( Y 6 Z (3 Y where Z As shown n Table 7, and follow Webull and normal dstrbutons, respectvely. The Webull dstrbuton wth the scale parameter of 4 and shape parameter of 5 has a mean of and a standard devaton of Two random varables and are assumed to be correlated wth each copula shown n Table 3. Except for the ndependent copula case where the correlaton coeffcent s always zero, the correlaton of coeffcent of τ=0.3 s used. or the senstvty calculaton of the component probablty of falure usng the score functon, the sample sze for the MCS s mllon, whereas the senstvty calculaton usng the DM employs 50 mllon samples for the MCS. Ths s because the DM uses the dfference of two probabltes of falure obtaned from the MCS, whch could nclude relatvely large statstcal nose f a small number of samples are used. Table 7. Propertes of Random Varables Random Varables Dstrbuton Dstrbuton Parameters Webull ν=4 =5 Normal µ=.5 σ=0.3 Tables 8 and 9 show the comparson of senstvtes wth respect to the mean of the non-normal random varable and the normal random varable, respectvely. Both tables llustrate that the senstvtes of the probablty of falure obtaned usng the score functon n Eq. (7 and MCS wth mllon samples agree very well wth the senstvtes obtaned usng the DM and MCS wth 50 mllon samples. As mentoned prevously, ths good agreement s very obvous snce the senstvty dervaton explaned n Secton 3 does not nclude any approxmaton except the statstcal nose due to the MCS. Hence, regardless of the margnal dstrbuton and copula types of random varables, the stochastc senstvty usng the score functon s very accurate, and t s also computatonally effcent snce the MCS uses a relatvely small sample sze compared wth the senstvty analyss usng the DM, and no fnte dfference perturbaton sze s nvolved. Table 8. Senstvty of Probablty of alure w.r.t Senstvty Copula Score uncton Clayton AMH ran GM Gaussan Independent DM wth perturbaton sze 0.% Table 9. Senstvty of Probablty of alure w.r.t Senstvty Copula Score uncton Clayton AMH ran GM Gaussan Independent DM wth perturbaton sze 0.% To verfy whether the proposed stochastc senstvty analyss wors for hgh-dmensonal problems, consder a 4-D polynomal functon G ( (3 The propertes of the random varables n Eq. (3 are shown n Table 0. or ths problem, two random varables and are assumed to be correlated wth each copula shown n Table 3, and two random varables 3 and 4 are assumed to be statstcally ndependent. The relatvely large correlaton coeffcent (τ=0.7 for and s used except for the ndependent copula. However, snce the AMH copula cannot deal wth a correlaton coeffcent larger than /3 [7], the AMH copula s excluded n ths example. Table 0. Propertes of Random Varables Random Standard Dstrbuton Mean Varables Devaton Normal Normal Normal Normal Table compares two senstvtes of the probablty of falure wth respect to. As n the prevous example, two senstvtes agree very well wth each other. Accordngly, t can be sad that the proposed stochastc senstvty analyss for the probablty of falure s very accurate, regardless of the type of performance functons and copulas.

8 Table. Senstvty of Probablty of alure w.r.t Senstvty Copula Score uncton Clayton ran GM Gaussan Independent DM wth perturbaton sze 0.% 6. Senstvtes of Statstcal Moments Consder the same -D hghly nonlnear polynomal functon shown n Eq. (3 for the senstvty comparson of statstcal moments. and follow Webull and normal dstrbutons, respectvely, wth the same dstrbuton parameters as shown n Table 7. The senstvtes of the frst two statstcal moments, mean and varance, of the performance functon n Eq. (3 are obtaned usng Eqs. (9 and (30, respectvely. Agan, mllon MCS samples are used for the senstvty calculaton usng Eqs. (9 and (30, that s, =,000,000. or the senstvty of the statstcal moment usng the DM, the statstcal moment s obtaned frst usng the MCS wth 50 mllon samples. Then, by perturbng the mean of by 0.%, the perturbed statstcal moment s obtaned, and the senstvty s obtaned usng the dfference between two statstcal moments. So, for a problem wth two random varables as n Eq. (3, the senstvty analyss usng the DM requres three MCSs wth 50 mllon samples. The computatonal tme for the senstvty analyss usng the DM s 750 tmes that of the proposed senstvty analyss, and ths dfference wll ncrease as the number of random varables ncreases. Table compares two senstvtes of statstcal moments wth respect to. Regardless of the copula type used, senstvtes of two statstcal moments obtaned usng the score functon and DM agree very well wth each other. ( d d 0 ( d d 0 mnmze C( d 30 0 Tar subect to PG ( ( d ( 0 P.75%, ~ 3 L U d dd, dr and R where three constrants are gven by G ( G ( ( Y 6 ( Y ( Y 6 Z 80 G3 ( 8 5 where Z (33 (34 Y , whch are drawn n gure. The propertes of two random varables are shown n Table 3, and they are assumed to be correlated wth the Clayton copula (τ=0.5. As shown n Eq. (33, the target probablty of falure ( P Tar s.75% for all constrants. Table 3. Propertes of Random Varables Random Dstrbuton d L d O d U Standard Varables Devaton Normal Normal Table. Senstvty of Statstcal Moments wth respect to Copula Score uncton Mean DM wth perturbaton sze 0.% Score uncton Varance DM wth perturbaton sze 0.% Clayton ran GM Gaussan Independent Samplng-Based RBDO usng Proposed Stochastc Senstvty To see how the proposed senstvty analyss wors for an RBDO problem, consder a -D mathematcal RBDO problem, whch s formulated to gure. Shape of Constrant and Cost unctons To apply the proposed senstvty analyss to an RBDO problem, accurate surrogate models need to be utlzed. or that purpose, the D-rgng method wth sequental samplng proposed by Zhao et al. [3] s used because, n terms of accuracy, the D-rgng method outperforms exstng methods such as the polynomal response surface method, the radal bass functon (RB method, the support vector regresson (SVR method, and the unversal rgng method. Assumng the constrant functons n Eq. (34 are not gven analytcally,

9 whch s usually descrbed as the blac-box constrant, the surrogate models for the constrant functons are frst generated usng the D-rgng method. Then, usng the generated surrogate models, the probabltes of falure for three constrants and ther senstvtes are obtaned usng the MCS wth mllon samples. Table 4 compares the numercal results of fve dfferent RBDO methods. The frst three results are obtaned from the so-called MPP-based RBDO, whch requres the senstvty of the constrant functons for the MPP search and desgn optmzaton. Ths MPP-based RBDO ncludes the ORM, the DRM wth three quadrature ponts, whch s denoted as DRM3 n Table 4, and the DRM wth fve quadrature ponts, whch s denoted as DRM5. The last two results are obtaned from the samplng-based RBDO, whch uses the MCS for the estmaton of the probablty of falure and ts senstvty. The samplng-based RBDO usng the D-rgng method s the proposed method, and for the comparson of the accuracy of the proposed method, the result of the samplng-based RBDO usng the true functons gven n Eq. (34 s also shown n the table. Tar Table 4. Comparson of Varous RBDOs ( P.75% MPP- Based RBDO Samplng -Based RBDO Methods Cost Optmum Desgn P,% MCS P,% uncton Call ORM , DRM , DRM , D-rgng , True uncton , N.A. rom the table, t can be seen that the probablty of falure of the second constrant (.835% at the optmum desgn obtaned usng the ORM s not close to the target probablty of falure (.75%. Ths s because the second constrant s hghly nonlnear as shown n gure and the ORM cannot accurately estmate the probablty of falure of hghly nonlnear functons. To mprove the accuracy of the probablty of falure at the optmum desgn, the MPP-based DRM wth three or fve quadrature ponts can be used; Table 4 shows that the MPP-based DRM ndeed mproves the accuracy of the probablty of falure at the optmum desgn. However, to obtan a more accurate optmum desgn, more quadrature ponts are requred, such as the DRM7, etc. The ORM uses 5 functon evaluatons and 5 senstvty calculatons, whereas the MPP-based DRM wth fve quadrature ponts uses 46 functon evaluatons and 0 senstvty calculatons to obtan the optmum desgn, and the number of functon evaluatons for the MPP-based DRM wll be ncreased as the number of quadrature ponts ncreases. On the other hand, the samplng-based RBDO, whch uses the D-rgng method and the proposed stochastc senstvty analyss, shows very accurate optmum desgn; yet t requres only 57 samples for the accurate optmum desgn. Wthout the senstvty of the performance functons, the samplng-based RBDO can obtan a very accurate optmum desgn and the optmum desgn s very close to the optmum desgn obtaned usng the true functons. Ths means that the D-rgng method generates very accurate surrogate models for the true functons. rom ths example, t can be sad that once accurate surrogate models are avalable, the samplng-based RBDO usng the proposed senstvty analyss yelds very accurate optmum desgns wth good effcency. More detaled dscusson on the samplng-based RBDO, whch ncludes hgher dmensonal engneerng applcatons such as MA Abrams tan roadarm [5], needs to be carred out and s ongong. 7. CONCLUSIONS The stochastc senstvty analyss of the probablstc constrants and statstcal moments wth respect to mean values of correlated random varables usng the score functon and MCS s carred out n ths study. Snce t does not requre the senstvty of performance functons or even the senstvty of surrogate models, the proposed senstvty analyss yelds very accurate senstvty estmaton regardless of the margnal and copula types of the random nput once accurate surrogate models are avalable. urthermore, the proposed method uses only one MCS at a gven desgn to obtan the probablty of falure and ts senstvty or statstcal moments and ther senstvtes smultaneously, whereas the DM uses N+ MCS to obtan the senstvtes of the probablstc response, where N s the number of random varables. Thus, the proposed senstvty analyss usng the score functon s far more effcent than the DM. In addton, the proposed senstvty analyss s more accurate than the DM when the same MCS sample sze s used. Hence, the proposed stochastc senstvty analyss combned wth accurate surrogate models, whch are obtaned n ths paper usng the D-rgng method, s recommended for RBDO of engneerng applcatons where accurate senstvtes of performance functons are not avalable. Numercal examples show the accuracy of the senstvty results and demonstrate that samplng-based RBDO usng the proposed senstvty analyss and accurate surrogate models by the D-rgng method yelds a very accurate optmum desgn wth good effcency. 8. ACNOWLEDGEMENT Research s ontly supported by the Automotve Research Center, whch s sponsored by the U.S. Army TARDEC and ARO Proect W9N These supports are greatly apprecated. 9. REERENCES. Y,., Cho,.., m, N.H., and Botn, M.E., Desgn Senstvty Analyss and Optmzaton for Mnmzng Sprngbac of Sheet-ormed Part, Int. J. of Numercal Methods n Engneerng, Vol. 7, pp , Youn, B.D., Cho,.., and Y,., Performance Moment Integraton (PMI Method for Qualty Assessment n Relablty-Based Robust Optmzaton, Mechancs Based Desgn of Structures and Machnes, Vol. 33, No., pp. 85-3, Youn, B.D., Cho,.., Yang, R-J, Gu, L., Relablty- Based Desgn Optmzaton for Crashworthness of Vehcle Sde Impact, Structural and Multdscplnary Optmzaton, Vol. 6, No. 3-4, pp. 7-83, Youn, B.D., Cho,.., and Tang, J., Structural Durablty Desgn Optmzaton and Its Relablty

10 Assessment, Internatonal Journal of Product Development, Vol., Nos. 3/4, pp , Dong, J., Cho,.., Wang, A., Zhang, W., and Vlahopoulos, N., Parametrc Desgn Senstvty Analyss of Hgh requency Structural-Acoustc Problems Usng Energy nte Element Method, Int. J. of Numercal Methods n Engneerng, Vol. 6, pp. 83-, Dong, J., Cho,.., Vlahopoulos, N., Wang, A., and Zhang, W., Desgn Senstvty Analyss and Optmzaton of Hgh requency Radaton Problems Usng Energy nte Element and Energy Boundary Element Methods, AIAA Journal, Vol. 45, No. 6, pp , Haldar, A., and Mahadevan, S., Probablty, Relablty and Statstcal Methods n Engneerng Desgn, John Wley & Sons, New Yor, NY, Madsen, H. O., ren, S., and Lnd, N. C., Methods of Structural Safety, Prentce-Hall Inc., Englewood Clffs, NJ, Hasofer, A. M. and Lnd, N. C., An Exact and Invarant rst Order Relablty ormat, ASCE Journal of the Engneerng Mechancs Dvson, Vol. 00, No., pp. -, Dtlevsen, O., and Madsen, H.O., Structural Relablty Methods, John Wley & Sons Ltd., Chchester, Hohenbchler, M., and Racwtz, R., Improvement of Second-Order Relablty Estmates by Importance Samplng, ASCE Journal of Engneerng Mechancs, Vol. 4, No., pp , Bretung,., Asymptotc Approxmatons for Multnormal Integrals, ASCE Journal of Engneerng Mechancs, Vol. 0, No. 3, pp , Rahman, S., and We, D., A Unvarate Approxmaton at Most Probable Pont for Hger-Order Relablty Analyss, Internatonal Journal of Solds and Structures, Vol. 43, pp , We, D., A Unvarate Decomposton Method for Hgher-Order Relablty Analyss and Desgn Optmzaton, Ph. D. Thess, The Unversty of Iowa, Lee, I., Cho,.., Du, L., and Gorsch, D., Inverse Analyss Method Usng MPP-Based Dmenson Reducton for Relablty-Based Desgn Optmzaton of Nonlnear and Mult-Dmensonal Systems, Computer Methods n Appled Mechancs and Engneerng, Vol. 98, No., pp. 4-7, als, M., Hacer,., and Lews,., A Comprehensve Robust Desgn Approach for Decson Trade-Offs n Complex Systems Desgn, ASME Journal of Mechancal Desgn, Vol. 3, No., pp. -0, Su, J., and Renaud, J. E., Automatc Dfferentaton n Robust Optmzaton, AIAA Journal, Vol. 35, No. 6, pp , Rahman, S., and u, H., A Unvarate Dmenson- Reducton Method for Mult-Dmensonal Integraton n Stochastc Mechancs, Probablstc Engneerng Mechancs, Vol. 9, pp , u, H., and Rahman, S., A Generalzed Dmenson- Reducton Method for Mult-dmensonal Integraton n Stochastc Mechancs, Internatonal Journal for Numercal Methods n Engneerng, Vol. 6, No., pp , Lee, I., Cho,.., Du, L., and Gorsch, D., Dmenson Reducton Method for Relablty-Based Robust Desgn Optmzaton, Specal Issue of Computer & Structures: Structural and Multdscplnary Optmzaton, Vol. 86, pp , Youn, B.D. and Cho,.., A New Response Surface Methodology for Relablty-Based Desgn Optmzaton, Computers and Structures, Vol. 8, Nos. -3, pp. 4-56, Zhang, T., Cho,.., Rahman, S., Cho,., Perry, B., and Shal, M., and Heta, D., A Response Surface and Pattern Search Based Hybrd Optmzaton Method and Applcaton to Mcroelectroncs, Structural and Multdscplnary Optmzaton, Vol. 3, No. 4, pp , m, C. and Cho,.., Relablty-Based Desgn Optmzaton Usng Response Surface Method wth Predcton Interval Estmaton, ASME Journal of Mechancal Desgn, Vol. 30, No., pp. -, Rubnsten, R. Y., Smulaton and Monte Carlo Method, John Wley & Sons, New Yor, Noh, Y., Cho,.., and Du, L., Relablty Based Desgn Optmzaton of Problems wth Correlated Input Varables Usng Copulas, Structural and Multdscplnary Optmzaton, Vol. 38, No., pp. -6, Noh, Y., Cho,.., and Lee, I., Reducton of Transformaton Orderng Effect n RBDO Usng MPP- Based Dmenson Reducton Method, AIAA Journal, Vol. 47, No. 4, pp , Noh, Y., Cho,.., and Lee, I., Identfcaton of Margnal and Jont CDs Usng the Bayesan Method for RBDO, Structural and Multdscplnary Optmzaton, Vol. 40, No., pp. 35-5, Rubnsten, R. Y., Shapro A., Dscrete Event Systems Senstvty Analyss and Stochastc Optmzaton by the Score uncton Method, John Wley & Sons, New Yor, Rahman, S., Stochastc Senstvty Analyss by Dmensonal Decomposton and Score unctons, Probablstc Engneerng Mechancs, Vol. 4, pp , McDonald, M., and Mahadevan, S., Desgn Optmzaton wth System-Level Relablty Constrants, Journal of Mechancal Desgn, Vol. 30, No., 403, Zhao, L., Cho,.., Lee, I., A Metamodel Method usng Dynamc rgng and Sequental Samplng, submtted, January, Browder, A., Mathematcal Analyss: An Introducton, Sprnger-Verlag, New Yor, Hab, O., Introducton to Calculus and Classcal Analyss, Sprnger-Verlag, New Yor, 997.