MPP-Based Dimension Reduction Method for RBDO Problems with Correlated Input Variables

Transcription

1 MPP-Based Dmeso Reducto Method for RBDO Problems wth Correlated Iput Varables Yoojeog Noh, K.K. Cho, ad Ikj Lee 3 Departmet of Mechacal & Idustral Egeerg College of Egeerg, he Uversty of Iowa Iowa Cty, IA 54, U.S.A. I relablty-based desg optmzato (RBDO problems wth correlated put varables, a jot cumulatve dstrbuto fucto (CDF eeds to be obtaed to trasform, usg the Roseblatt trasformato, the correlated put varables to depedet stadard Gaussa varables for the relablty aalyss. However, a true jot CDF requres fte umber of data to be obtaed, so ths paper, a copula s used to model the jot CDF usg margal CDFs ad correlato parameters obtaed from samples, whch are avalable practcal applcatos. Usg the jot CDF modeled by the copula, the trasformato ca be carred out based o the frst order relablty method (FORM, whch has bee commoly used relablty aalyss. However, the FORM may yeld dfferet relablty aalyss results wth some errors for dfferet trasformato orderg of put varables due to the oleartes of dfferetly trasformed costrat fuctos. For ths, the most probable pot (MPP based dmeso reducto method (DRM, whch more accurately ad effcetly calculates the probablty of falure tha the FORM ad the secod order relablty method (SORM, respectvely, s proposed to use to reduce the effect of trasformato orderg the verse relablty aalyss, ad thus RBDO. o study the effect of trasformato orderg o RBDO results, several umercal examples are tested usg two dfferet relablty methods, the FORM ad DRM. dv Nomeclature Number of radom varables Number of desg varables,, Vector of desg varables, d d d dv,, Vector of radom varables, X X X,, Realzato of vector, x x x d [ ] X [ ] x X [ ] x x DRM FORM-based MPP DRM-based MPP U [,, Vector of depedet stadard Gaussa varables, U U U ] u Realzato of vector U, u [,, u u ] [,, Y Vector of correlated stadard Gaussa varables, Y Y Y ] y Realzato of vector Y, [,, y y y ] ρ j Pearso s correlato coeffcet betwee X ad X j P ρ Covarace matrx of x, { j} Graduate Research Assstat, e-mal: oh@egeerg.uowa.edu. Roy J. Carver Professor, Correspodg Author, e-mal: kkcho@egeerg.uowa.edu. 3 Graduate Research Assstat, e-mal: lee@egeerg.uowa.edu. Amerca Isttute of Aeroautcs ad Astroautcs 09407

2 P Covarace matrx of y, { ρ j} Φ( Φ P X ( X ( X ( Margal Gaussa CDF P Multvarate Gaussa CDF wth P F Margal CDF of X F f Jot CDF of X,, X X Jot PDF of X,, X X,, F Codtoal CDF of X X θ Matrx of correlato parameters of X,, X C ( θ τ Copula wth θ Kedall s tau I. Itroducto may RBDO problems, put radom varables, such as the materal propertes ad fatgue parameters, are Icorrelated. -3 For the RBDO problem wth the correlated put varables, the jot CDF of the put varables should be avalable to trasform the correlated put varables to the depedet stadard Gaussa varables, by usg the Roseblatt trasformato, 4 to carry out the verse relablty aalyss. However, dustral applcatos, ofte oly the margal CDFs ad lmted pared sampled data are avalable usg expermetal testg, ad the put jot CDF s very dffcult to obta. I ths paper, a copula, whch lks the jot CDF ad margal CDFs, s used to model the jot CDF. Sce the copula oly requres margal CDFs ad correlato parameters, whch are ofte avalable dustral applcatos, to model the jot CDF, the jot CDF ca be readly obtaed. hus, t s valuable to use the copula for modelg the jot CDFs practcal applcatos wth correlated put varables. Oce the jot CDF s obtaed usg the copula, the Roseblatt trasformato ca be utlzed to trasform the orgal radom varables to the depedet stadard Gaussa varables for the verse relablty aalyss. For the verse relablty aalyss, the FORM s most ofte used. O the other had, depedg o the types of the jot put CDF, f a dfferet order of Roseblatt trasformato s used, eve the costrat fucto that was mldly olear wth respect to the orgal radom varables could become hghly olear terms of the depedet stadard Gaussa varables. Frst, obvously, f the put varables are depedet (.e., the jot CDF s a smple multplcato of the margal CDFs, there s o effect of trasformato orderg. Secod, f the put varables have the jot CDF modeled by a Gaussa copula, the effect of trasformato orderg stll does ot exst because the Gaussa copula makes the Roseblatt trasformato become lear, whch s depedet of ordergs. However, f the put varables have a o-gaussa jot CDF modeled by a o-gaussa copula, whch ofte occurs dustral applcatos, 5, sce the Roseblatt trasformato becomes hghly olear, the dfferet orderg ca sgfcatly affect the olearty of the trasformed costrats. I ths case, f the FORM s used, the verse relablty aalyss results could be very dfferet for the dfferet orderg sce the FORM uses a lear approxmato of the costrat to estmate the probablty of falure. hs effect of trasformato orderg RBDO wll be uacceptable ad make the user sgfcatly cocered. o reduce the depedecy of the verse relablty aalyss result ad thus the RBDO result o the orderg of the Roseblatt trasformato, t s proposed to use the MPP-based DRM, 6 for the verse relablty aalyss ths paper. Wth the accuracy of the verse relablty aalyss usg the DRM eve for hghly olear costrat fuctos, t s show that the RBDO results are becomg less depedet o the Roseblatt trasformato orderg of the put varables. II. Modelg of Jot CDF usg Copula As metoed earler, f the put varables are correlated, t s ofte too dffcult to obta the true jot CDF practcal dustral applcatos wth oly lmted expermetal data. I ths paper, a copula s used to model the jot CDF usg margal CDFs ad correlato measures that are calculated from the expermetal data. he defto of copula ad the correlato measures assocated wth copulas are explaed ths secto. Amerca Isttute of Aeroautcs ad Astroautcs 09407

3 A. Defto of Copula Copula s orgated from a Lat word for lk or te that coects two dfferet thgs. I statstcs, the defto of copula s stated by Roser (999: Copulas are fuctos that jo or couple multvarate dstrbuto fuctos to ther oe-dmesoal margal dstrbuto fuctos. Alteratvely, copulas are multvarate dstrbuto fuctos whose oe-dmesoal margs are uform o the terval [0, ]. (,..., Accordg to Sklar s theorem, f the radom varables have a jot dstrbuto FX X x x wth margal F X ( x,..., FX x, the there exsts a -dmesoal copula C such that dstrbutos, (,...,,..., F x x C F x F x θ X,..., X X X ( where θ s the matrx of the correlato parameters of x,..., x. If margal dstrbutos are all cotuous, the C s uque. Coversely, f C s a -dmesoal copula ad FX ( x,..., FX x are the margal dstrbutos, the F (,..., X X x x s the jot dstrbuto. 7 By takg the dervatve of Eq. (, the jot probablty desty fucto (PDF f,, (,, X X x x s obtaed as where c u (,, u ( C u,, u u u ( f X X X Π X,,,, x,, x c X F x θ F x f x wth ( u F ( x, ad f X X x s the margal PDF for,,. A copula oly requres margal CDFs ad correlato parameters to model a jot CDF, so the jot CDF ca be readly obtaed from lmted data. I addto, sce the copula decouples margal CDFs from the jot CDF, the jot CDF modeled by the copula ca be expressed terms of ay type of margal CDF. hat s, havg margal Gaussa CDFs does ot mea that the jot CDF s Gaussa. hus, t s desrable to be able to model the jot CDF of correlated put varables wth mxed types of margal CDFs, whch ca ofte occur dustral applcatos. 5 o model the jot CDF usg the copula, the correlato parameters eed to be obtaed from expermetal data as see Eqs. ( ad (. Sce varous types of copulas have ther ow correlato parameters, t s desrable to have a commo correlato measure to obta the correlato parameters from the expermetal data. B. Correlato Measures o measure the correlato betwee two radom varables, Pearso s rho ad Kedall s tau ca be used. Pearso s rho was frst dscovered by Bravas 846, 8 ad was developed by Pearso 896, 9. Pearso s rho dcates the degree of lear relatoshp betwee two radom varables as ρ XY (, Cov X Y σ σ X Y (3 where σ X ad σ Y are stadard devatos of X a d Y, respectvely, ad Co v X, Y s the covarace betwee X ad Y. Sce Pearso s rho oly dcates the lear relatoshp betwee two radom varables, t s vald oly whe the jot CDF s Gaussa. Pearso s rho also ca be used as correlato measure the jot CDF modeled by Gaussa copula, because the Gaussa copula s orgated from a jot Gaussa CDF. If the margal CDFs are Gaussa, the the jot CDF modeled by the Gaussa copula s the jot Gaussa CDF. he Gaussa copula allows geeratg a jot Gaussa CDF wth o- margal Gaussa CDFs as CΦ ( u,, u P ΦP ( Φ ( u,, Φ ( u P, u I (4 3 Amerca Isttute of Aeroautcs ad Astroautcs 09407

4 s the margal CDF of X for,,, P s the covarace matrx cosstg of correlato where u F ( x X coeffcets, Pearso s rho, betwee correlated put varables. Φ ( represets the margal stadard Gaussa CDF, exp x Φ x, ad ΦP ( s the jot Gaussa CDF defed as π Φ ( x P / exp ( π P x- μ for [,, ] μ P x - x x x wth a mea vector μ [ μ,, μ ]. However, Pearso s rho caot be a good measure for a olear relatoshp betwee two radom varables, whch ofte occurs practcal egeerg applcatos. If the gve data follows a jot o-gaussa CDF modeled by a o- Gaussa copula, aother correlato measure s ecessary. Ulke Pearso s rho, Kedall s tau does ot requre the assumpto that the relatoshp betwee two radom varables s lear. Sce the Kedall s tau measures the correspodece of rakgs betwee correlated radom varable, t s called a rak correlato coeffcet. he Kedall s tau was frst troduced by Kedall 938, 0 ad s defed as ( [ ] I dc( u v τ 4 C u, vθ, (5 where I I I I 0, ad Eq. (5 s the populato verso of Kedall s tau. he sample verso of Kedall s tau s c d s t ( c d/ c+ d (6 where c represets the umber of cocordat pars, d s the umber of dscordat pars, ad s s the umber of samples. Usg the estmated Kedall s tau, the correlato parameter of the copula, θ, ca be calculated because Kedall s tau ca be expressed as a fucto of the correlato parameter as show Eq. (5. he explct fuctos of Eq. (5 for some copulas are preseted Ref.. Cosder a o-gaussa copula, whch uses a rak correlato coeffcet such as Kedall s tau as the correlato measures. Ulke the Gaussa copula, the Archmedea copula s costructed a completely dfferet way. A mportat compoet of costructg Archmedea copula s a geerator fucto ϕ θ wth a correlato parameter θ. If ϕ θ s a cotuous ad strctly decreasg fucto from [ 0, ] to [ 0, such that ϕ θ ( 0 ad ϕ 0 ad the verse ϕ θ s completely mootoc o [ 0,, the the Archmedea copula ca be defed as, 7 θ (,, θ ϕθ ϕθ + + ϕθ ( u C u u u 4 Amerca Isttute of Aeroautcs ad Astroautcs (7 for. Each Archmedea copula has a correspodg uque geerator fucto ϕθ, whch provdes a multvarate copula as show Eq. (7. Oce the geerator fucto s provded, the Kedall s tau ca be obtaed as θ θ ( t ϕ t τ + 4 dt (8 0 ϕ Usg Eq. (8, the correlato parameter θ ca be expressed terms of Kedall s tau. he Archmedea copula ca be used for a multvarate CDF. But t s hard to expad to a -dmesoal copula because, as show Eq. (7, t has oe geerator fucto, ad thus has the same correlato parameter eve f varables are correlated wth dfferet correlato coeffcets. Hece, most copula applcatos cosder bvarate data. For multvarate data, the data are aalyzed par by par usg a bvarate copula. hs paper also cosders a bvarate copula. More detaled formato o Kedall s tau s preseted Ref. 0.

5 Icludg the Gaussa copula ad Archmedea copula, there exst varous kds of copulas; thus, selectg a approprate copula s ecessary to correctly model a jot CDF based o the gve expermetal data. As metoed earler, to model a jot CDF usg a copula, the margal CDFs ad correlato parameters eed to be obtaed. he margal CDFs are ofte kow to follow specfc CDF types; for example, some materal propertes such as fatgue parameters are kow to follow logormal CDFs. O the other had, selectg a approprate copula that best descrbes the gve expermetal data s ot a smple problem. Sce addressg two ssues (effect of trasformato orderg ad detfcato of the rght copula together s complcated ad requres legthy dscusso, ths paper, oly the effect of trasformato orderg s addressed, ad the jot CDFs modeled by copulas are assumed to be exact. he detfcato of the rght copula s addressed Refs. ad detal. III. Effect of rasformato Orderg RBDO Based o the detfed jot CDF, the put varables eed to be trasformed to depedet stadard Gaussa varables for the verse relablty aalyss RBDO usg the Roseblatt trasformato. Whe the put varables are depedet or the jot CDF s modeled by a Gaussa copula, the orderg of put varables does ot affect the trasformato. However, whe the jot CDF s modeled by a o-gaussa copula, dfferet ordergs of put varables cause dfferet trasformatos for the verse relablty aalyss, whch leads to dfferet RBDO results. hs ssue wll be addressed ths secto. A. Roseblatt rasformato for RBDO he RBDO problem ca be formulated to m. cost( d ar s.t. PG ( ( X > 0 P,,, c F dv d μ( X, dl d d U, d R ad X R (9 where X s the vector of radom varables; d s the vector of desg varables; G ( X represets the costrat fuctos; P s the gve target probablty of falure for the th costrat; ad c, dv, ad are the umber of ar F probablstc costrats, umber of desg varables, ad umber of radom varables, respectvely. he probablty of falure s estmated by a mult-dmesoal tegral of the jot PDF of the put varables over the falure rego as P ( G X > 0 f ( x dx,,, c (0 G ( X > 0 where x s the realzato of the radom vector X. However, sce t s dffcult to compute the mult-dmesoal tegral aalytcally, approxmato methods such as the FORM or the SORM are used. he FORM ofte provdes adequate accuracy ad s much easer to use tha the SORM, ad hece t s commoly used RBDO. Sce the FORM ad SORM requre the trasformato of the correlated radom put varables to the stadard Gaussa varables, the Roseblatt trasformato s used. Usg a performace measure approach (PMA+ 3, the th costrat ca be rewrtte, from Eq. (9, as X PG P G ar [ ( X > 0] F 0 0 x ( where G ( x s the th costrat fucto evaluated at the most probable pot (MPP x X-space. Usg the FORM, Eq. ( ca be rewrtte as ( β G x PG [ ( X > 0] Φ 0 0 t ( where PF ar ( t Φ β ad β t s the target relablty dex. 5 Amerca Isttute of Aeroautcs ad Astroautcs 09407

6 o satsfy the feasblty of the costrat, the MPP eeds to be estmated for each costrat by solvg the followg optmzato problem: max. g ( u s.t. u β t (3 where g ( u s the th costrat fucto that s trasformed from the orgal space (X-space to the stadard Gaussa space (U-space,.e., g( u G ( x( u G( x. he optmum of Eq. (3 s called the MPP, whch s deoted by u U-space or x X-space. If the costrat fucto at the MPP, g (u, s less tha or equal to zero, the the th costrat Eq. (9 s satsfed for the gve target relablty. hus, Eq. (9 ca be rewrtte as m. cost( d G x c s.t. 0,,, L U dv d d d d X, R ad R (4 As show Eq. (3, the correlated put varables eed to be trasformed to the depedet stadard Gaussa varables usg the Roseblatt trasformato, whch s defed as the successve codtog: ( u FX ( x ( u FX ( x x ( u FX ( x x, x,, x ( uj FX ( x,,,, j j x x x xj ( u FX ( x x, x,, x (5 If the orderg of the varable x s chaged to the varable x j, Eq. (5 ca be rewrtte as ( u FX ( x ( u FX ( x x ( u FX ( x,,, j j x x x ( uj FX ( x x, x, x, xj, x+,, xj ( u FX ( x x, x,, x (6 hus, f the umber of varables s, there are! ways of trasformg the orgal varables to depedet stadard Gaussa varables. Eve though there are may dfferet ways to trasform the orgal varables to depedet stadard Gaussa varables, sce the Roseblatt trasformato s exact, f the verse relablty aalyss the depedet stadard Gaussa space s exact, the we should obta the same results. However, f 6 Amerca Isttute of Aeroautcs ad Astroautcs 09407

7 the FORM s used for the verse relablty aalyss, certa orders of trasformato mght yeld more errors tha other orders of trasformato accordg to the put jot CDF type. B. Effect of rasformato Orderg for Varous Iput Jot CDF ypes he jot CDFs of put radom varables ca be categorzed as follows: depedet jot CDF, jot CDF modeled by a Gaussa copula, ad jot CDF modeled by a o-gaussa copula. For varous put jot CDF types, to study the effect of trasformato orderg the verse relablty aalyss, t s ecessary to vestgate whether the same MPP s obtaed for dfferet trasformato ordergs. However, sce the MPPs deped o costrat fuctos, t s ot coveet to compare the MPPs for all costrat fuctos. Istead of comparg the MPPs, comparg target hyperspheres u β t trasformed from U- to X-space for dfferet trasformato ordergs s more approprate because obtag the same target hyperspheres meas obtag the same MPPs X space, whch lead to same RBDO results. Frst, whe the put varables are depedet, the trasformed target hypersphere from U to X-space ca be obtaed usg Eq. (5, whch meas the Roseblatt trasformato wth a gve orderg as ( ( Φ FX x + + Φ FX x + + Φ FX x j j + + Φ FX x βt u u u (7 Whe the order of the varable target hypersphere s obtaed as x s terchaged to the varable ( ( F x j for the secod orderg, the trasformed ( Φ FX x + + Φ FX x j j + + Φ X x + + Φ FX x βt u u u (8 whch results the same trasformed target hypersphere wth Eq. (7. hus, there s o effect of trasformato orderg whe the put varables are depedet. Secod, cosder whe the put varables are correlated wth a jot CDF modeled by the Gaussa copula, whch s defed Eq. (4. I the jot CDF modeled by the Gaussa copula, each varable Φ FX ( x for,, s the stadard Gaussa varable wth the covarace matrx P. Let y Φ FX ( x. Sce the j reduced correlato coeffcet ρ betwee Y ad Y j s dfferet from the correlato coeffcet ρ j betwee ad X j j, t eeds to be calculated from ρ. he reduced correlato coeffcet ρ s obtaed from the correlato coeffcet ρj usg the followg equato: where Ξ ( X / X μ σ X ( y, y ; ρ ρ E Ξ Ξ ξξ φ dydy (9 j j j j j s the ormalzed radom varable of X ad j j ξ s the realzato of the mplct Eq. (9 requres a teratve process that s very tedous to solve, Eq. (9 s approxmated by Ξ X. However, sce ρ R ρ j j j (0 where Rj a+ bv + cv + dρj + eρj + fρjv + gvj + hvj + kρjvj + lvv j, V ad varato ( V σ / μ V j are the coeffcets of for each varable, ad the coeffcets deped o the types of put varables. Whe the margal CDFs are Gaussa so that the jot CDF becomes Gaussa, the reduced correlato coeffcet s the same as the orgal correlato coeffcet, whch meas R j. For varous types of put varables, the correspodg coeffcets are gve Refs. 4 ad 5. he maxmum error of the estmated correlato coeffcet obtaed from Eq. (0 s ormally much less tha %, ad eve f the expoetal margal CDF or egatve correlato s 7 Amerca Isttute of Aeroautcs ad Astroautcs 09407

8 volved, the maxmum error the correlato coeffcet s at most up to %. 5 herefore, the approxmato provdes adequate accuracy wth less computatoal effort. For the jot CDF modeled by a Gaussa copula, the Roseblatt trasformato s lear as u L y ( where y represets correlated stadard Gaussa varables ad L - s the verse of a lower tragular matrx L obtaed from the Cholesky decomposto of P. hat s, P LL ad each etry of the matrx L s obtaed as l j l k, j k j ρj ll k jk / ljj, > j k ( Usg Eq. (, the trasformed target hypersphere ca be expressed as uu y L L y y P y (3 If the order s chaged,.e., the order of the th ad j th varables are terchaged (<j the Roseblatt trasformato, the trasformed target hypersphere s chaged to uu y P y y L L y (4 where,,,,,, y y yj y y ad L s obtaed from the Cholesky decomposto of P, whch s the reduced covarace matrx of y. hat s, P ad P are P ρ ρ ρ j ρ ρ ρj ρ ρj ρ sym. ρ j ad ρ ρ j ρ ρ ρj ρ ρ ρj ρ j P (5 sym. ρ o show that the terchaged orderg provdes the same trasformed target hyperspheres ( u u u u, we eed to use y stead of y Eqs. (3 ad (4. For ths, aother matrx L eeds to be troduced stead of L. he matrx L ca be obtaed by terchagg the th row ad colum wth the j th row ad colum of L, respectvely, as followg. - - uu y P y y L L y y L Ly y P y (6 8 Amerca Isttute of Aeroautcs ad Astroautcs 09407

9 where L l 0 0 l l 0 0 l l l 0 0 lj lj lj ljj 0 l l lp lq l ad L l 0 0 l l 0 0 lj lj lj ljj 0. l l l 0 0 l l lp lq l P o show that P or LL LL, cosder two arbtrary correlato coeffcets of P. For ay arbtrary k th colum, the etry at the th row of P, P ad the oe the j th row of P, ( P are ρ jk ad ρ k, respectvely as show Eq. (5. Sce ρ jk ad ( k k ρ are the etres at the j th row ad k th colum of P ad at the th row ad k th colum of P, respectvely, all etres of P are same as those of as followg: P jk jk P ρ ( P th ro w of L kth col. of L j th row of. L L kth col of ( P k jk ( P ρk j th row of k th col. of th row of k th col. of P L ( P k jk L L L k jk (7 (8 hs meas the trasformed target hyperspheres are the same eve for dfferet trasformato ordergs of put varables. herefore, the Roseblatt trasformato s depedet of orderg for the jot CDF modeled by a Gaussa copula. Fally, cosder whe the put varables have a jot CDF modeled by a o-gaussa copula. For example, let two radom varables have a jot CDF modeled by the Clayto copula, whch s oe of the Archmedea X, X ~ N 0,. he Clayto copula s defed as copulas, wth the margal Gaussa CDFs ( θ / θ θ θ C u, v u + v, for θ > 0 (9 where the geerator s ϕ ( t ( t θ θ, uφ ( x, vφ ( x, ad θ s the correlato parameter of the θ Clayto copula. I the Clayto copula, usg the Kedall s tau τ obtaed from samples, θ ca be expressed as τ θ (30 τ Usg the Clayto copula, the Roseblatt trasformato ca be carred out two dfferet ways as ( u F ( x Φ X x θ θ θ / θ X x ( u F ( x x Φ ( x ( x Φ +Φ (3 ad Φ u F x Φ x X θ θ θ / θ X ( u F ( x x ( x ( x ( x Φ Φ +Φ (3 9 Amerca Isttute of Aeroautcs ad Astroautcs 09407

10 Usg Eqs. (3 ad (3, the target hypersphere ca be expressed terms of x ad x uu u + u x + Φ Φ x Φ x +Φ x / θ θ θ θ βt (33 ad uu u + u x + Φ Φ x Φ x +Φ x / θ θ θ θ βt (34 hese dfferet trasformato ordergs wll provde dfferet RBDO results. Further, for the o-gaussa jot CDF modeled by a o-gaussa copula, the Roseblatt trasformato becomes hghly olear, whch caot be hadled accurately by the FORM. A more accurate method tha the FORM for the estmato of the probablty of falure the relablty aalyss s ecessary to reduce the effect of the trasformato orderg o the RBDO results. IV. Method to Resolve Effect of rasformato Orderg Usg MPP-based DRM he MPP-based relablty aalyss such as the FORM, 6-8 ad the SORM, 9,0 has bee a very commoly used for relablty assessmet. However, whe the costrat fucto s olear or mult-dmesoal, the relablty aalyss usg the FORM could be erroeous because the FORM caot hadle the complexty of olear or mult-dmesoal fuctos. Relablty aalyss usg the SORM may be accurate, but the secod-order dervatves requred for the SORM are very dffcult ad expesve to obta dustral applcatos. O the other had, the MPP-based DRM acheves both the effcecy of the FORM ad the accuracy of the SORM. 6 he DRM s developed to accurately ad effcetly approxmate a mult-dmesoal tegral. here are several DRMs depedg o the level of dmeso reducto: uvarate dmeso reducto, bvarate dmeso reducto, ad multvarate dmeso reducto. he uvarate, bvarate, ad multvarate dmeso reducto dcate a addtve decomposto of -dmesoal performace fucto to oe, two, ad s-dmesoal fuctos ( s, respectvely. I ths paper, the uvarate DRM s used for calculatg probablty of falure due to ts smplcty ad effcecy. he uvarate DRM s carred out by decomposg a -dmesoal costrat fucto G(X to the sum of oe-dmesoal fuctos at the MPP as, 6,, G( X Gˆ ( X G( x,, x, X, x,, x ( G( x + (35 where x { x, x,, x } s the FORM-based MPP obtaed from Eq. (3 ad s the umber of radom varables. I the verse relablty aalyss, sce the probablty of falure caot be drectly calculated U-space, a costrat shft a rotated stadard Gaussa space (V-space eeds to be defed as G G G s ( v ( v ( v (36 where ( v ( x( v probablty of falure usg the MPP-based DRM s calculated as, 6 v {0,, 0, β} s MPP V-space ad G G. he, usg the shfted costrat fucto, the P DRM F s G ( v Φ ( β + φ( v dv b Φ ( β (37 s s g( u where G ( v G (0,, 0, v,0,, β s a fucto of v oly ad b. u 0 Amerca Isttute of Aeroautcs ad Astroautcs 09407

11 Equato (37 ca be approxmated as usg the momet-based tegrato rule, 3, smlar to Gaussa quadrature, 4 P DRM F N s j wjφ β + j b ( β Φ G ( v (38 j where v represets the j th quadrature pot for v, w j deote weghts, ad N s the umber of quadrature pots. he quadrature pots ad weghts for the stadard Gaussa radom varables v are show able. Whe the quadrature pot s ( N ad weght s, Eq. (38 becomes G ( v w ( β ( β Φ( β Φ ( β Φ ( β s Φ + DRM b Φ F P (39 s s where w ad v 0 by able ad G ( v G (0 0. Equato (39 s the same as the probablty of falure calculated by the FORM. herefore, t ca be sad that the probablty of falure calculated by the FORM s a specal case of the oe calculated by the DRM wth oe quadrature pot ad weght. able. Gaussa Quadrature Pots ad Weghts, 6. N Quadrature Pots Weghts ± ± ± a Cocave Fucto b Covex Fucto Fgure. DRM-based MPP for Cocave ad Covex Fuctos. 6 Amerca Isttute of Aeroautcs ad Astroautcs 09407

12 Usg the estmated DRM P F correspodg relablty dex β DRM ca be defed as obtaed from the MPP-based DRM for the shfted costrat fucto G s (x, the β DRM Φ DRM ( P F (40 whch s ot the same as the target relablty dex β Φ t ( P because the olearty of the costrat ar F DRM fucto s reflected the calculato of P F. Hece, usg β DRM, a ew updated relablty dex β up ca be defed as β β +Δ β β + ( β β up cur cur t DRM (4 where β cur s the curret relablty dex. he recursve form of the Eq. (4 s +Δ + ( ( k+ ( k ( k β β β β βt βdrm (4 (0 where β βt at the tal step. Usg ths updated relablty dex, the updated MPP ca be foud by usg a teratve MPP search or usg a approxmato. If a teratve MPP search wth the updated relablty dex s used, the updated MPP s called the true DRM-based MPP ad s deoted by x DRM, whch meas the updated MPP s the optmum soluto of Eq. (3 usg β up stead of β t ; however, the procedure wll be computatoally expesve. Accordgly, to mprove the effcecy of the optmzato, the updated MPP ca be approxmated as, 5 (k+ (k+ a β a β uk+ uk or v k k+ v ( k k (43 β β a assumg that the updated MPP v k+ s located alog the same radal drecto v N as the curret MPP v k V-space, as show Fg.. he updated MPP obtaed from Eq. (43 s called the DRM-based MPP ad s used to check whether or ot the optmum desg satsfes the costrat. he locato of the DRM-based MPP for a cocave ad covex fucto s show Fgs..a ad.b, respectvely. Smlar to the FORM, usg the DRM-based verse relablty aalyss, the RBDO formulato Eq. (9 ca be rewrtte as mmze cost( d subject to G ( x 0,,, c DRM d d d, d R ad X R L U dv (44 x DRM x where s the MPP obtaed from the DRM whle s the oe obtaed from the FORM. V. Numercal Examples o observe how the orderg of trasformato affects the RBDO results for the jot o-gaussa CDFs modeled by o-gaussa copulas, two- ad four-dmesoal mathematcal problems are tested. A. wo-dmesoal Problem Assume two put varables are correlated by a Clayto copula wth Kedall s tau τ 0.5. he RBDO formulato s defed as Amerca Isttute of Aeroautcs ad Astroautcs 09407

13 m. cost( d d + d ( ( X ( βt s.t. P G 0 Φ,,,3 0 d, d 0, β.0 t ( G ( X X 0.900X X X + / 0 X + + G X X.8 /30 X X /0 G ( X 80/ 8X + X (45 where the margal CDFs are Gaussa ( X, X ~ N(5.0,0.3. Deote the tal orderg as orderg ad the terchaged orderg ( x x as orderg. a Orderg b Orderg Fgure. rasformed Costrats ad arget Hypersphere U-space Usg Clayto Copula. As show Fg., the target hyperspheres U-space are the same, but the costrat fuctos X-space are dfferetly trasformed U-space accordg to the dfferet trasformato ordergs. Sce the trasformato of the o-gaussa copula s hghly olear, some trasformed costrat fuctos become hghly olear U- space. For orderg (Fg..a, the frst costrat fucto s mldly olear ear the MPP (, but the secod costrat fucto s hghly olear ear the correspodg MPP ( u g, whch yelds a large FORM error. O the other had, for orderg (Fg..b, two costrat fuctos are mldly olear ear the MPPs ( u g ad u g, so that the FORM estmates the probablty of falure more accurately tha that of the secod costrat fucto wth the frst orderg. he FORM results for dfferet ordergs are show able. I able, Case FORM- ad Case FORM- dcate the FORM wth ordergs ad, respectvely. As expected, whe the FORM s used for the orderg (FORM-, the probablty of falure for the secod costrat ( P f s poorly estmated (much less tha target probablty of.75%. As a result, the optmum desg pots obtaed usg the FORM wth dfferet ordergs are deed dfferet as show the thrd ad fourth colums of able. If the MPP-based DRM wth three quadrature pots, deoted as DRM3- ad DRM3- for ordergs ad, respectvely, s used, the dfferece betwee optmum desg results s reduced from 0.06 to ad the DRM provdes a more accurate estmato of the probabltes of falure (closer to.75% for both ordergs. If the umber of quadrature pots s fve (DRM5- ad DRM5-, the the optmum desg pots are much closer to u g 3 Amerca Isttute of Aeroautcs ad Astroautcs 09407

14 each other for both ordergs ad the probablty of falure calculato also becomes more accurate. hus, the MPPbased DRM deed reduces the effect of trasformato orderg o the RBDO results. able. RBDO Results Obtaed from Clayto Copula. opt Case Cost Optmum desg pots d d opt G G G 3 f P (% P f (% FORM , FORM , DRM , DRM , DRM , DRM , For the same problem Eq. (45, assume that two put varables are ow correlated wth a Frak copula, whch belogs to the Archmedea copula, wth Kedall s tau τ 0.5. he Frak copula s gve as C( u, vθ l + θ θu θv ( e ( e e θ (46 he correlato parameter θ ca be calculated from Kedall s tau by solvg the followg equato, 4 t θ τ dt θ θ 0 t (47 e As observed the prevous example, whe the jot CDF modeled by a o-gaussa copula s used, the MPPs obtaed from dfferetly trasformed costrat fuctos provde dfferet RBDO results accordg to the dfferet orderg of put varables. (a Orderg (b Orderg Fgure 4. rasformed arget Hyperspheres U-space a Orderg b Orderg Fgure 3. rasformed Costrats ad arget Hypersphere U-space Usg Frak Copula. 4 Amerca Isttute of Aeroautcs ad Astroautcs 09407

15 Due to the trasformato of the jot CDF modeled by a o-gaussa copula, the costrat fuctos X-space are dfferetly trasformed to those U-space for dfferet ordergs, ad some trasformed costrat fuctos are hghly olear. For orderg, the frst costrat fucto s hghly olear ear the MPP ( whle for orderg, the secod costrat fucto s hghly olear ear the MPP ( ug (Fg. 3.a, (Fg. 3.b. herefore, for orderg, the FORM error s large at the frst costrat (FORM-, whle for the secod orderg, t s large at the secod costrat (FORM-. As show able 3, probabltes of falure Pf for orderg ad P f for orderg are close to the target probablty (.75%, whereas P f for orderg ad P f for orderg are ot. Whe the MPP-based DRM wth three quadrature pots s used, the probablty of falure becomes closer to the target probablty for both ordergs (DRM3- ad DRM3-. he DRM wth fve quadrature pots provdes the most accurate calculato of the probablty of falure (DRM5- ad DRM5-3. he optmum desg pots obtaed from the DRM are deed smlar to each other compared wth those obtaed from the FORM for dfferet ordergs. hus, the DRM s ecessary to reduce the effect of trasformato orderg ad to provde accurate RBDO results. If the umber of correlated varables s larger tha two, the effect of trasformato orderg ad accurate estmato of probablty of falure mght be more sgfcat, ad thus usg the FORM the relablty aalyss mght be more urelable. I the ext secto, ths ssue wll be further addressed through a fourdmesoal problem. ug able 3. RBDO Results Obtaed from Frak Copula. opt opt Case Cost Optmum desg pots d d G G G P 3 f (% P f (% FORM , FORM , DRM , DRM , DRM , DRM B. Four-dmesoal Problem hs example s the four-dmesoal modfed Rose-Suzuk problem, 6 ad the RBDO s formulated to d d 5 + d d 5 + d3 d3 4 + d4 d m. cost( d 45 ( ( X ( βt s.t. P G 0 Φ,,,3 0 d, d, d, d 0, β.0 G G G 3 ( X ( X ( X 3 4 t ( 9 + ( + ( + ( X X X X X3 X3 X4 X4 68 X X X X X3 X3 X4 X 5 X( 9 X + X( X + X3( 0 X3 + X4 95 ( + ( 0 + ( 0 + ( 4 (48 Assume that the frst ad secod varables are correlated wth the Gumbel copula ad the thrd ad fourth varables are correlated wth the A copula, whch the Gumbel ad A copula belog to the Archmedea copula. he Gumbel copula s defed as θ / θ { } θ (, exp ( l + ( l C u v θ u v (49 5 Amerca Isttute of Aeroautcs ad Astroautcs 09407

16 where uφ ( x ad v ( x Φ wth X, X ~ N(5.0, 0.3. Kedall s tau τ 0.5 s assumed for both copulas ad the correlato parameter s obtaed as θ ( τ. he A copula s defed as θ C( u, vθ + ( u + ( v θ / θ (50 Lkewse, Φ, u x 3 v Φ x 4 wth X3, X4 ~ N(5.0,0.3 ad θ. 3 τ Sce the umber of put varables s four ad two pars of varables are correlated, four dfferet ordergs are possble the trasformato. I able 4, FORM- dcates FORM wth the tal orderg, whch meas the orderg s ot chaged. FORM- s the case wth the terchaged orderg of x ad x, ad FORM-3 s the oe wth terchaged orderg of x 3 ad x 4. FORM-4 s the case where the ordergs of all varables are terchaged, whch meas x ad x are terchaged ad x 3 ad x 4 are terchaged. As see able 4, for all ordergs, the probabltes of falure Pf ad Pf 3 are poorly estmated whe the FORM s used. Eve though the calculato of probablty of falure for the fourth orderg s the most accurate, Pf ad P f 3 are stll much larger tha the target probablty,.75%. Compared wth the two-dmesoal example, the FORM error for the four-dmesoal case s more sgfcat. Whe the MPP-based DRM wth three quadrature pots s used (DRM3-,, 3, ad 4, the dfferece betwee the probabltes of falure becomes smaller tha whe the FORM s used. Whe fve quadrature pots are used (DRM5-,, 3, ad 4, the MPP-based DRM estmates the probabltes of the falure more accurately tha the case wth three quadrature pots. hus, the MPP-based DRM s ecessary to reduce the orderg effect o RBDO results. able 4. RBDO Results of Rose-Suzuk Problem Obtaed from Gumbel ad A Copula. Case Cost Optmum desg pots G G G P 3 f (% P f 3 (% FORM , 5.63, 6.68, FORM , 5.59, 6.68, FORM , 5.56, 6.639, FORM , 5.584, 6.646, DRM , 5.586, 6.574, DRM , 5.56, 6.58, DRM , 5.564, 6.59, DRM , 5.545, 6.590, DRM , 5.584, 6.573, DRM , 5.558, 6.58, DRM , 5.554, 6.593, DRM , 5.544, 6.59, Method-: Orderg (orgal orderg Method-: Orderg ( X X Method-3: Orderg 3 ( X3 X 4 Method-4: Orderg 4 ( X X ad X 3 X 4 VI. Cocluso I RBDO problems, the jot CDF eeds to be used the Roseblatt trasformato for the verse relablty aalyss. However, sce the jot CDFs are dffcult to obta, copulas are proposed to model the jot CDFs ths paper. Icorporatg the copula cocept, the jot CDF ca be categorzed as depedet jot CDF, jot CDF 6 Amerca Isttute of Aeroautcs ad Astroautcs 09407

17 modeled by Gaussa copula, ad jot CDF modeled by o-gaussa copula. Whe the put varables are depedet or correlated wth Gaussa copula, the verse aalyss results are the same for dfferet trasformato ordergs of the put varables. However, whe the correlated put varables wth jot CDFs are modeled by a o-gaussa copula, dfferet trasformato ordergs could lead to hghly olear costrat fuctos. hus, t becomes a sgfcat challege to accurately carry out the verse relablty aalyss usg the FORM. hus, the MPP-based DRM, whch ca hadle the olear costrats, s proposed to be used ths paper for the RBDO of problems wth correlated put varables wth jot CDFs modeled by o-gaussa copulas. Numercal examples show that whe the MPP-based DRM s used, the dfferece betwee the RBDO results usg dfferet trasformato ordergs s reduced as well as the accurate estmato of probablty of falure s acheved. Ackowledgmets hs research s supported by the Automotve Research Ceter, whch s sposored by the U.S. Army ARDEC. Refereces Soce, D.F., Semar otes: Probablstc Aspects of Fatgue, 003, URL: [cted May 8 008]. As, C., Probablstc Lfe Predcto Ist as Easy as It Looks, Joural of ASM Iteratoal, Vol., No., 004, pp Efstratos, N., Ghocel, D.M., ad Sghal, S., Egeerg desg relablty hadbook, CRC press, New York, Roseblatt, M., Remarks o a Multvarate rasformato, A. Math Stat., Vol. 3, No. 3, 95, pp Pham, H. (ed., Sprger Hadbook of Egeerg Statstcs, Sprger, Lodo, Lee, I., Cho, K.K., Du, L., ad Gorsch, D., Dmeso Reducto Method-Based MPP for Relablty-Based Desg Optmzato of Hghly Nolear Mult-Dmesoal Systems, Specal Issue of Computer Methods Appled Mechacs ad Egeerg: Computatoal Methods Optmzato Cosderg Ucertates, to appear, Nelse, R.B., A Itroducto to Copulas, Sprger, New York, Bravas, A., Aalyse Mathematque sur les Probabltes des Erreurs de Stuato du Pot, Memores par dvers Sava, Vol. 9, 846, pp Pearso, K., Mathematcal Cotrbutos to the heory of Evoluto. III. Regresso, Heredty ad Pamxa, Phlos. ras. Royal Soc. Lodo Ser. A, Vol. 87, 896, pp Kedall, M., A New Measure of Rak Correlato, Bometrka, Vol. 30, 938, pp Huard, D., Év, G., ad Favre, A.C., Bayesa Copula Selecto, Computatoal Statstcs & Data Aalyss, Vol. 5, No., 006, pp Noh, Y., Cho, K.K., ad Du, L., Selecto of Copula to Geerate Iput Jot CDF for RBDO, 34th ASME Desg Automato Coferece, August 3-6, 008, New York Cty, New York. 3 You, B.D., Cho, K.K., ad Du, L., Erched Performace Measure Approach for Relablty-Based Desg Optmzato, AIAA Joural, Vol. 43, No. 4, 005, pp Madse, H.O., Krek S., ad Ld N.C., Methods of Structural Safety. Pretce-Hall, Eglewood Clffs, New Jersey, Dtlevse, O., ad Madse, H.O., Structural Relablty Methods. Wley, New York, Hasofer, A.M., ad Ld, N. C., A Exact ad Ivarat Frst Order Relablty Format, ASCE Joural of the Egeerg Mechacs Dvso, Vol. 00, No., 974, pp u, J., ad Cho, K.K., A New Study o Relablty-Based Desg Optmzato, Joural of Mechacal Desg, Vol., No. 4, 999, pp u, J., Cho, K. K., ad Park, Y.H., Desg Potetal Method for Relablty-Based System Parameter Desg Usg Adaptve Probablstc Costrat Evaluato, AIAA Joural, Vol. 39, No. 4, 00, pp Hohebchler, M., ad Rackwtz, R., Improvemet of Secod-Order Relablty Estmates by Importace Samplg, ASCE Joural of Egeerg Mechacs, Vol. 4, No., 988, pp Bretug, K., Asymptotc Approxmatos for Multormal Itegrals, ASCE Joural of Egeerg Mechacs, Vol. 0, No. 3, 984, pp Rahma, S., ad We, D., A Uvarate Approxmato at Most Probable Pot for Hger-Order Relablty Aalyss, Iteratoal Joural of Solds ad Structures, Vol. 43, 006, pp We, D., A Uvarate Decomposto Method for Hgher-Order Relablty Aalyss ad Desg Optmzato, Ph.D. Dssertato, Mechacal & Idustral Dept., he Uversty of Iowa, IA, Amerca Isttute of Aeroautcs ad Astroautcs 09407

18 3 Xu, H., ad Rahma, S., A Momet-Based Stochastc Method for Respose Momet ad Relablty Aalyss, Proceedgs of d MI Coferece o Computatoal Flud ad Sold Mechacs, Cambrdge, MA, July 7-0, Atkso, K.E., A Itroducto to Numercal Aalyss, Joh Wley & Sos, New York, 989, Chaps 5. 5 Ba-abbad, M.A., Nkolads, E., ad Kapaa, R. K., New Approach For System Relablty-Based Desg Optmzato, AIAA Joural, Vol. 44, No. 5, 006, pp Schttkowsk, K., More est Examples for Nolear Programmg Codes, Sprger-Verlag, New York, Amerca Isttute of Aeroautcs ad Astroautcs 09407