7--55 Relablty Based Desg Optmzato wth Correlated Iput Varables Copyrght 7 SAE Iteratoal Kyug K. Cho, Yoojeog Noh, ad Lu Du Departmet of Mechacal & Idustral Egeerg & Ceter for Computer Aded Desg, College of Egeerg, The Uversty of Iowa, Iowa Cty, IA 54, USA ABSTRACT Relablty-based desg optmzato (RBDO, whch cludes desg optmzato desg space ad verse relablty aalyss stadard ormal space, has bee recetly developed uder the assumpto that all put varables are depedet because t s dffcult to costruct a jot probablty dstrbuto fucto (PDF of put varables wth lmted data such as the margal PDF ad covarace matrx. However, sce real applcatos, t s commo that some of the put varables are correlated, the RBDO results mght cota a sgfcat error f the correlato betwee put varables for RBDO s ot cosdered. I ths paper, Roseblatt ad Nataf trasformatos, whch are the most represetatve trasformato methods ad have bee wdely used the relablty aalyss, have bee studed ad compared terms of applcablty to RBDO wth correlated put varables. It s detfed that Nataf trasformato s oe of copulas ad more applcable tha Roseblatt trasformato. Usg umercal examples, t s also show that the correlato of put varables sgfcatly affects the RBDO results.. INTRODUCTION The RBDO process requres two level optmzatos: the desg optmzato the desg space ad the verse relablty aalyss the stadard ormal space. Thus, the trasformato betwee the put radom varables ad the stadard ormal radom varables s ecessary for the verse relablty aalyss RBDO. Roseblatt ad Nataf trasformatos are the most represetatve trasformato methods ad wdely used for the relablty aalyss. Eve though Roseblatt trasformato s very well used for applcatos RBDO, Nataf trasformato has ot bee used as much. Roseblatt trasformato [] s a mathematcally exact method ad requres complete formato of the put varables such as jot CDF or codtoal CDF [4, 5]. O the other had, Nataf trasformato [, 3] s a approxmate method that oly requres the covarace matrx ad margal CDF [4, 5]. If the put varables of the RBDO problem are depedet, ether method ca be used because two methods have the same trasformato formulato. However, f ot, they may yeld dfferet RBDO results depedg o the put formato. Addtoally, sce the orderg of the put varables should ot affect the RBDO result, t s mportat to select a method that s ot affected by the orderg of the put varables durg trasformato. I may RBDO problems, the put radom varables such as the materal propertes are correlated. To solve the RBDO problems wth the correlated put varables, a jot PDF or cumulatve dstrbuto fucto (CDF of put varables should be avalable. However, ofte dustral applcatos, oly lmted formato such as the margal dstrbutos ad covarace are practcally obtaed, whereas the put jot probablty dstrbuto fuctos (PDF are very dffcult to obta [4, 5]. Thus, the lterature, most RBDO studes have assumed all put radom varables are depedet ad mostly Roseblatt trasformato method has bee cosdered RBDO. I ths paper, t s foud that the RBDO results ca be sgfcatly affected by the correlato of put varables. Thus, dfferet trasformato methods are vestgated for possble applcato to the RBDO of problems wth correlated put varables. I addto, t s foud that Roseblatt trasformato s mpractcal for problems wth correlated put varables due to dffculty of costructg a jot PDF from the margal dstrbutos ad covarace. Moreover, the relablty aalyss results deped o the orderg of put varables durg the trasformato. Nataf trasformato, whch belogs to the copula famly, ca costruct the jot PDF from the lmted formato such as margal dstrbutos ad covarace ad t s depedet of orderg of the put varables. Thus, t s applcable to practcal dustral problems wth correlated put varables, where oly the margal PDFs ad covarace are avalable. Further, whe put varables are mxed wth dfferet types of dstrbutos, a jot PDF or CDF s rarely kow a mathematcal formulato, but t ca be approxmated usg Nataf trasformato.
Sce Nataf trasformato s orgated from Gaussa copula, t has a dsadvatage that t may ot be applcable to o-gaussa dstrbutos except the ormal ad logormal dstrbuto. However, because the ormal ad logormal dstrbutos cover majorty of practcal dustral applcatos, Nataf trasformato s applcable to a broad class of RBDO problems. Therefore, ths study, Nataf trasformato s used to develop a RBDO method for desg problems wth correlated radom put varables. Numercal examples are used to demostrate the proposed method ad t s show that the correlated radom put varables do sgfcatly affect the RBDO result.. RELIBILITY BASED DESIGN OPTIMIZATION (RBDO Ths s a example of a Ma Headg secto. Ths secto wll clude sub-sectos. The RBDO problem s formulated to m. cost( d ( βt s.t. PG ( ( Φ,,, NC p NDV d μ(, dl d du, d R ad R where deotes the vector of radom varables, d s the desg varables, G ( represets the probablstc p costrats of radom varables, βt s the target relablty dex, NC, NDV, ad NRV are the umber of probablstc costrats, umber of desg varables, ad umber of radom varables, respectvely. The falure of the probablstc costrat s expressed by the CDF F of G as [6] Gp p ( ( ( p FG Φ( βt P G p where the falure of the probablstc costrat s estmated by a multple tegral of the jot PDF of put varables over the falure rego as F ( f ( x dx dx,,, NC (3 G G p ( where s the radom varable ad x s the realzato of the radom varable. However, sce t s dffcult to compute these multple tegral exactly, approxmato methods such as the Frst Order Relablty Method (FORM or Secod Order Relablty Method (SORM are used. Sce FORM ofte provdes adequate accuracy ad s much easer to use tha SORM, t s commoly used RBDO. Sce FORM requres trasformato of the orgal radom varables to the stadard ormal varables, Roseblatt trasformato or Nataf trasformato s ofte used. NRV ( ( For the verse relablty aalyss, the probablstc costrat equato ( ca be rewrtte, through the verse trasformato, as G ( ( β p F Φ G t To obta the probablstc performace measureg p, a verse relablty aalyss ca be carred out by maxmzg the performace fucto G stadard ormal space subject to the L orm of the stadard varables equal to the target relablty dex as max. G( U s.t. U β Usg the ehaced hybrd mea value (HMV+ method, the value of the probablstc costrat o the most probable pot (MPP ca be estmated [6]. 3. TRANSFORMATION METHODS Two trasformato methods are studed ths paper for applcato to RBDO for problems wth correlated put varables: Roseblatt ad Nataf trasformatos. Nataf trasformato s detfed as oe of Gaussa copula. 3.. ROSENBLATT TRANSFORMATION Roseblatt trasformato s a well-kow trasformato method that maps the orgal varables oto the stadard ormal varables. It s defed by the followg successve codtog: u F x ( ( Φ Φ u F x x u F x x x x t (,,, Φ where s umber of put varables, F ( x,,..., x x x dcate the CDF of codtoal o x, x,, x, ad Φ ( s the verse CDF of the stadard ormal varables. Based o equato (6, whe the jot PDF or codtoal CDF s are kow, the probablty of falure ca be exactly estmated usg Roseblatt trasformato. I the case of depedet varables, the probablty of falure ca be obtaed from the jot PDF, whch s the multplcato of margal PDF s of each varable. I addto, aalytcally, the result of Roseblatt trasformato s ot affected by the orderg as ca be see the followg equato; (4 (5 (6
( P f x,..., x dx... dx f... g( > F,..., g( > g( > gu ( > ( ( ( f x f x x f x x,... x dx... dx ( u,..., u ( x,..., x fu (,... U u, u,..., u... dx dx ϕ... ϕ du... du U U However, eve though Roseblatt trasformato has several advatages, t may ot be wdely applcable to practcal egeerg problems due to followg reasos. Frst, the jot PDF or codtoal CDF s should be avalable for all varables to estmate the probablty of falure, whch s ofte too expesve or dffcult to obta real applcatos. Rather, the margal CDF ad covarace are commoly avalable. Also, whe the dstrbuto types of put varables are mxed,.e., some of the varables are logormal ad others are expoetal or Gumbel, t s ot possble to express the jot PDF a mathematcal formulato. Thus, Roseblatt trasformato ca be used oly for lmted cases where all put varables have ormal dstrbuto ad a jot PDF or codtoal CDF s are provded. Fally, the result of Roseblatt trasformato s supposed to be theoretcally depedet of the orderg, but the estmato of probablty of falure mght be dfferet whe FORM s used, due to the approxmato error of FORM. Ths behavor s dscussed later usg a umercal example. 3. COPULA The copula s orgated from a Lat word for lk or te that coects two dfferet thgs. I statstcs, the copula s a trasformato that uses the margal dstrbutos ad covarace to costruct a jot PDF. Accordg to Sklar s theorem [7], f the radom varables have margal dstrbutos, the there exsts a - dmesoal copula C such that (,..., (,..., ( ( F x x C F x F x,..., If margal dstrbutos are all cotuous, the C s uque. Coversely, f C s a -dmesoal copula ad are the margal dstrbuto fuctos, the the jot dstrbuto s a -dmesoal fucto wth margal dstrbutos F (,..., x ( F x [7]. 3.. Gaussa Copula (Nataf Trasformato Usg Sklar s theorem, Gaussa copula, whch s also called Nataf trasformato, ca be obtaed as ( (,..., Φ Φ (,..., Φ ( F,... x x F x F x (7 (8 (9 where F trasformato from (,..., s the margal CDF of x. Cosder the to Y as ( y Φ F x,,..., The, sce the jot PDF s defed by f x x,..., ( F F y y x x y y x x ( ( ( y y f x F where ad φ ( x φ y,p, Gaussa y y copula ca be obtaed by dfferetatg equato (9 as (,..., f x x ( y,..., y φ ( y,p' ( x,..., x ( ( φ ( y, P ' ( y φ( y f x f x φ ( where the margal CDF s F,,, are kow ' ad P' { ρj } are the reduced covarace matrx to be estmated. Thus, Y s a -dmesoal stadard ormal varable wth a jot probablty desty fucto φ ( y,p'. Therefore, the jot PDF ca be calculated usg the margal PDF s of the put varables ad the reduced covarace matrx. The reduced correlato coeffcet betwee two varables ca be estmated by a teratve process usg the double tegral as ' (, ; j j j j j j ρ E Z Z z z φ y y ρ dy dy ( (3 where Z μ / σ. However, sce the teratve process s very tedous ad ukows are wth the double tegral, equato (3 s approxmated by ' ρ R ρ j j j (4 to obta the reduced coeffcet. I equato (4, R j s approxmated by a polyomal as R a+ bv + cv + dρ + eρ + fρ V where V ad j ( V σ / μ j j j j + gv + hv + kρ V + lvv j j j j j (5 V are the coeffcet of varato for each varable ad the coeffcets deped o the types of put varables. For dfferet types of put varables, the correspodg coeffcets are gve Refs. 3, 4 ad 5. The maxmum error of the estmated
correlato coeffcet obtaed from equato (4 s ormally much below %, ad eve f the expoetal dstrbuto or egatve correlato s volved, the maxmum error the correlato coeffcet s at most up to % [4, 5]. Therefore, the approxmato provdes adequate accuracy wth less computatoal effort. ( ( ( ( x F Φ a u x F Φ a u + a u ( ( x F a u a u a u Φ + + + ( As stated Secto, the verse relablty aalyss s carred out usg the trasformed stadard ucorrelated ormal varables U. Sce the relatoshp betwee the orgal correlated varables ad the correlated stadard ormal varables Y s gve equato (, the ext step s to trasform the correlated stadard ormal varables Y to the ucorrelated stadard ormal varables U usg a lear trasformato. Cosder the followg lear equato. YA+BU (6 where Y ~ N(,I has the reduced correlato matrx ΣY P' ad U~ N(,I has the covarace matrx Σ I. The mea of Y ca be calculated as U E[ Y ] E[ A+BU ] A+BEU [ ] A (7 I the same way, the covarace matrx of Y ca be calculated as [ ] Var [ ] T T P' Σ Var A+BU BU BΣ B BB (8 Y Sce the covarace matrx of Y s postve defte, P' ca be decomposed to the lower ad upper tragular matrx usg Cholesky factorzato. Therefore, the matrx B ca be cosdered as a lower tragular matrx. Equato (8 the ca be rewrtte as a a a a a a a a T P' Σ Y BB a a a a ρ' ρ' ρ ρ ' ' ρ' ρ' (9 Usg equatos ( ad (6, the relatoshp betwee orgal varables ad correlated stadard varables U ca be obtaed as U where the etres a j the lower tragular matrx are expressed as the reduced correlato coeffcets. Sce the orgal varables ca be expressed by depedet stadard ormal varables, the relablty aalyss ca be carred out by substtutg equato ( to the performace fucto G p equato (. As prevously stated, sce Nataf trasformato uses just margal CDF s of the put varables, t s ot affected by the orderg of varables ulke Roseblatt trasformato. Further, f oly the covarace matrx ad margal dstrbuto are avalable, Nataf trasformato s the oly way to costruct the jot PDF of the put varables. 3... Applcablty of Nataf Trasformato Oe of the advatages of Nataf trasformato s to be able to costruct a approxmate jot PDF for varous types of the correlated put varables. As a result, may dustral problems wth varous types of the correlated put varables ca be solved usg RBDO. It s oted that Nataf trasformato ca approxmate a jot PDF of ormal ad logormal varables more accurately tha the jot PDF of other o-ormal varables because Nataf trasformato s orgated from Gaussa copula. Cosder the mxed ormal ~ N( μ, σ ad logormal LN( λ, ξ put radom varables. Suppose the jot PDF of ormal ad logormal varables s gve as f ( x, x πσξ η x l x λ μ x η ξ σ x μ exp η σ ( σ where ξ l + l( + κ, μ ρ κ λ l μ ξ, ad η ρ '. The correlato ξ coeffcet ρ ' betwee the correlated stadard ormal varables y ad y s obtaed from the correlato coeffcet ρ betwee x ad x by
ρ κ ( + κ ρ ' l ( Usg Nataf trasformato, the correspodg jot PDF ca be approxmated as ( ( ( y φ( y f x f x f ( x, x φ y, y, φ ( ρ ' ' y ρyy + y exp ' ' πσ ξ ρ ( ρ x ' ' ( ρ + ( ρ y y y exp ' ' πσ ξ ρ ( ρ x y η y exp ' πσ ξ ρ x η x μx where y, y f ( σ x x ad f ( x ormal varable gve as f f ( x ( x l x λ ξ y (3. I equato (3, are the margal PDFs of the x ad logormal varable x, whch are x μ exp σ π σ l x λ exp ξ π ξ x (4 Note equato (3 s the exactly same as the equato (. Thus, Nataf trasformato ca costruct the exact jot PDF for ormal ad logormal varables twodmesoal case. Usg Nataf trasformato, the jot PDF of two logormal varables ca be approxmated as ( ( ( y φ( y f x f x ' f ( x φ ( ρ, x y, y, φ (6 ' y ρ + yy y exp ' ' πξ ξ ρ ( ρ xx The reduced correlato coeffcet betwee logormal varables s obtaed as ' ρ ( + ρ κ κ l ( + κ ( + κ l l (7 Sce equatos (5 ad (6 have the same formulato except the correlato coeffcets, Nataf trasformato ca accurately approxmate a jot CDF of logormal varables f the dfferece betwee the reduced correlato coeffcet ad the orgal correlato coeffcet s small. Fgure shows the relatve error betwee the orgal correlato coeffcet ad the reduced correlato coeffcet. If two logormal put varables are postvely correlated, the dfferece betwee the reduced correlato coeffcet ad the orgal correlato coeffcet s small. Whe two put varables are depedet, sce the orgal correlato coeffcet s zero, the reduced correlato coeffcet s also zero. That s, the relatve error s ot defed at ρ Fgure. However, for egatve correlato coeffcets, the relatve error betwee the orgal correlato coeffcet ad the reduced correlato coeffcet s sgfcat. Thus, f two varables are postvely correlated or depedet, the jot CDF ca be accurately estmated usg Nataf trasformato, but for the egatvely correlated put varables the jot CDF may ot be accurate. Next cosder two logormal varables, L N( λ, ξ ad LN( λ, ξ. Suppose the jot PDF s gve as f ( x, x πξ ξ ρ y ρ y y + y exp ( ρ x x (5 l x λ l x λ where y, y, ξ ( + κ l, ξ ξ ξ ( + κ l, λ l μ ξ, ad λ l μ ξ.
Relatve error, (ρ'-ρ/ρ x (% 9 8 7 6 5 4 3 Relatve error CDF's (% 9 8 7 6 5 4 3 ρ-.9 ρ-.5 ρ-. ρ. ρ.5 ρ.9 - -.8 -.6 -.4 -...4.6.8 Correlato coeffcet (ρ 3 4 5 6 Relablty dex (β Fgure. Relatve Error betwee Orgal Correlato Coeffcet ad Reduced Correlato Coeffcet To vestgate how the relatve error betwee the reduced correlato coeffcet ad the orgal correlato coeffcet affects accuracy of the estmated jot CDF, Fgure shows the relatve error betwee the exact jot CDF ad approxmate jot CDF s obtaed from Nataf trasformato for dfferet correlato coeffcets. The relatve error s calculated by (, (, F ( x, x F x x F x x Relatve error(% F ( x, x F ( x, x (8 where ad are the approxmate ad exact jot CDFs of the logormal varables, respectvely. As ca be see Fgure, the more egatve correlato logormal varables s, the more sgfcat the relatve errors CDF s are. For postve correlato coeffcets, eve f logormal put varables are hghly correlated, the relatve error CDF s s less tha 5% as show Fgure. The maxmum error (5% occurs ear ρ.5. The reaso that, for hgh postve correlato coeffcet, eve though the dfferece betwee the reduced correlato coeffcet ad the orgal correlato coeffcet s small, the correlato sgfcatly affects estmato of the jot CDF. O the other had, for low postve correlato coeffcet values, eve f the values of the orgal correlato coeffcet ad the reduced correlato coeffcet are rather dfferet, the effect of the correlato s eglgble estmatg the jot CDF. Thus, the md-rage correlato ( ρ.5 causes a maxmum error whe the jot CDF s estmated. Fgure. Relatve Error betwee Exact Jot CDF ad Approxmate Jot CDF Obtaed from Nataf Trasformato. Therefore, Nataf trasformato s very effectve for the problem wth postve correlated ad depedet logormal put varables, but ot applcable to the oes wth egatve correlated logormal put varables. I addto, f the target relablty dex β t RBDO s requred to be less tha three, Nataf trasformato s stll applcable to some egatve correlato coeffcet. Ths ssue s dscussed Secto 4. 3.3. NUMERICAL COMPARISON OF TWO TRANSFORMATIONS I ths secto, a mathematcal example wth correlated two put varables wth expoetal PDF s used to demostrate how Roseblatt ad Nataf trasformatos affect the relablty aalyss results. Ths example s troduced by Hohebchler ad Rackwtz [8] ad t was dscussed by Madse [9]. The jot PDF of the expoetal put varables s defed as ( x x x x ( x x x x + + exp, f ( x, x x, x (9, otherwse The margal CDF s of ad obtaed as The codtoal CDF s of x are ( exp( ( exp( F x x F x x (3 o x ad o
( ( + exp ( + ( ( + exp ( + F x x x x x x F x x x x x x (3 The lmt-state fucto s gve as ( g x, x 8 3x x (3 Usg Roseblatt trasformato, the orgal varables ca be trasformed to the depedet stadard ormal varables two dfferet ways as u ( x Φ exp ( ( u x x x x Φ + exp + (33 ad ( x ( ( u Φ exp Φ u + + x exp x xx (34 Smlarly, usg Nataf trasformato, the depedet stadard ormal varables ca be obtaed terms of the orgal varables two dfferet ways as - x u Φ e - x - x Φ e ρ ' Φ e u ρ ' (35 - Φ x u e - Φ x - ρ ' Φ x e e (36 u ρ ' As ca be see equato (33, (34, (35 ad (36, Roseblatt trasformato uses the margal CDF s ad codtoal CDF s of the put varables whle Nataf trasformato uses oly margal CDF s of put varables for trasformato from to U. Whe Nataf ad Roseblatt trasformatos are used, the lear lmt state fucto equato (3 becomes olear as show Fgure 3. I Fgure 3, the sold le (orderg ad the dashed le (orderg dcate the lmt state fuctos obtaed from Roseblatt trasformato for the orderg ad of the put varables. The dotted le (orderg ad the dash-dot le (orderg dcate the lmt state fuctos obtaed from Nataf trasformato for two dfferet ordergs of the put varables. Fgure 3. Lmt Sate Fuctos for Dfferet Trasformatos ad Ordergs Whe Roseblatt ad Nataf trasformatos are used relablty aalyss, for the orderg, almost the same MPP pots are obtaed U space, but for the orderg, dfferet MPP pots are obtaed due to the olearty of the lmt state fucto. Accordgly, the results such as MPP pots ad the probablty of falure become dfferet for dfferet trasformatos ad the dfferet ordergs of the put varables as ca be see the Table. Table. Relablty Aalyss Results for Dfferet Trasformatos ad Ordergs Idep. Roseblatt trasformato Nataf trasformato Ord. Ord. Ord. Ord. MPP * 4.945 6. 5.97 6. 5.953 MPP * MPP * MPP *.583..4..67 U.46.78 -.3.795 -.487 U.795.64.399.67.368 β.586.78.649.796.796 P 4.863.73 4.37 E-.59.59 E- f E-3 E-3 3 E-3 3 Error (% 65.39 8.5 37.3.9.95 MCS.94E-3 MPP MPP * MPP space, U MPP U space, ad MCS Mote Carlo Smulato As see Table, whe Roseblatt trasformato s used, the probablty of falure from the orderg s the most accurate, but the result from the orderg s the most accurate. O the other had, whe Nataf trasformato s appled, the results are very smlar for dfferet ordergs eve though they have FORM errors. I fact, to solve the accuracy of the FORM, more precse approach such as SORM based approach ca be used. However, t s ot developed yet ad beg
vestgated for our ext research. I addto, the Roseblatt trasformato result s obtaed usg the gve codtoal CDF s that are ot usually avalable real applcato whereas Nataf trasformato result s obtaed usg the margal CDF s ad covarace matrx that are ormally avalable. Thus, Nataf trasformato s more practcal to use tha Roseblatt trasformato. Table also compares the case that two put varables are depedet wth the oe that two put varables are correlated. As show the table, the assumpto that two correlated varables are depedet could lead to wrog results such that the MPP pots, relablty dex, ad probablty of falure have sgfcat errors whe compared wth the Mote Carlo smulato result. O the other had, f the correlato put varables s cosdered the relablty aalyss, the errors the relablty aalyss results are rapdly reduced. Thus, t s very mportat to cosder the correlato carryg out relablty aalyss ad RBDO. The above results show that Nataf trasformato s better tha Roseblatt trasformato, but t stll has some approxmato errors estmatg the jot PDF for the expoetal varables. Ths s dscussed ext secto. 4. LIMITATION OF NATAF TRANSFORMATION Nataf trasformato s applcable to ormal ad logormal varables, but t s formatve to study whether t s also applcable to other dstrbutos, sce desg varables real dustral problems could clude other types of dstrbutos. For stace, for two expoetal varables, whch could be used to aalyze the relablty of a electroc system, the orgal jot PDF s gve as [] {( θx( θx θ} ( θ + + f ( x + +, x exp x x x x, x, x, otherwse (37 The correspodg expoetal jot CDF s ( x x e e + exp x + x + θ xx, F( x, x x, x (38, otherwse Usg Nataf trasformato, a approxmate jot PDF ca be obtaed as f ( x f ( x (, φ(,, ρ ' φ( y φ( y f x x y y e φ x x e ( y φ( y φ ( y, y, ρ' x x (, (, F x x f x x dx dx (39 The approxmate expoetal jot CDF ca be estmated by tegratg the approxmate jot PDF equato (38 as (4 x x where y Φ e, ad y Φ e. The correlato coeffcet for the expoetal varables s tz ( ( / / θ ρj + e E θ θ (4 where E z e t dt. If the parameter θ vares from. to., the correlato coeffcet ρ rages from. to.4366. I ths example, t s assumed that ρ.4366. F(x,x Φ(-β.5.4.3.. Approxmate CDF Exact CDF 3 4 5 6 Relablty dex (β Fgure 4. Exact ad Approxmate Expoetal CDFs The approxmate expoetal jot CDF obtaed usg Nataf trasformato s compared wth the exact expoetal jo CDF at dfferet relablty dex levels, where the relablty dex s obtaed from β ( F( x, x Φ (4 where the CDFs are evaluated at the same values of x ad x Fgure 4. As ca be see Fgure 4, the overall CDF s seem to be almost detcal but the relatve error s sgfcat the terval from β. to 6. as show Fgure 5. As
the relablty dex creases, the relatve error of the approxmate jot CDF creases rapdly as show Fgure 5. to be.4366 as Fgure 4. As ca be see Fgure 7, the dfferece betwee the approxmate jot CDF ad exact jot CDF seems smaller tha the result show Fgure 4. The tred ca be also observed the relatve error as see Fgure 8. As metoed before, f the target relablty dex RBDO s less tha three, the relatve error seems to be acceptable..5 Exact CDF Approxmate CDF.4 F(x,x Φ(-β.3.. Fgure 5. Relatve Error of Expoetal Jot CDF Obtaed Sug Nataf Trasformato at Dfferet Relablty Idex Levels Further, at a certa target relablty dex, e.g., β 3., the relatve error s sgfcat for all values of the correlato coeffcet (Fgure 6. Thus, the Nataf trasformato may ot be approprate for the expoetal varables due to the large relatve error the CDF. 3 4 5 6 Relablty dex (β Fgure 7. Exact ad Approxmate Logormal CDF s 6 5 Relatve error (% 4 3 3 4 5 6 Relablty dex (β Fgure 8. Relatve Error Logormal Dstrbuto Versus Relablty Idex Fgure 6. Relatve Error Versus Correlato Coeffcet for β 3. To carry out a smlar study for the jot CDF of correlated logormal put varables, cosder equato (5 Secto 3... Fgure 7 shows comparso of the exact jot CDF ad the approxmate jot CDF of logormal varables obtaed from Nataf trasformato at dfferet relablty dex levels. For the purpose of comparso wth the prevous case of the jot expoetal CDF, the correlato coeffcet s selected Lke Fgure 6, the relatve error of the approxmate jot CDF of logormal varables also ca be calculated for a rage of the correlato coeffcet at certa target relablty dex, e.g., β 3.. Fgure 9 shows that the relatve error s sgfcat for the low values of egatve correlato coeffcet, but t s small for the correlato coeffcet betwee.3 ad. Thus, Nataf trasformato s applcable for the problem wth postvely correlated logormal varables ad for some values of egatve correlato coeffcets.
Relatve error ( % 9 8 7 6 5 4 3 to. at. tervals. I Fgure, the crcle dcates the orm of the stadard radom varables U, where the radus s the target relablty dex whe the put varables are depedet. However, whe put varables are correlated, the crcle becomes a ellpse, whch has ether postve or egatve agle accordg to the sg of the correlato coeffcet. For selected correlato coeffcets that rage from to, relabltybased optmum desgs are obtaed as show by the sg Fgure. - -.8 -.6 -.4 -...4.6.8 Correlato coeffcet (ρ Fgure 9. Relatve Error CDF Versus Correlato Coeffcets Istead of usg Nataf trasformato, t mght be possble to obta a expoetal copula for expoetal varables, lke Nataf trasformato beg a Gaussa copula. I fact, may researches o the expoetal copula have bee performed, but the curretly developed expoetal copula does ot seem to be applcable to RBDO because t s ot cotuous ad does ot have a wde rage of correlato coeffcets [-6]. Thus, t s ecessary to develop a ew copula that s applcable for correlated oormal varables. It wll be our ext research topc ad wll be vestgated. However, the ormal ad logormal dstrbutos cover broader applcato areas such as materal propertes (stregth, lfetme, chemcal process, fatgue, crack propagato, ad loads [7-8]. Thus, terms of applcablty to real dustral applcatos, Nataf trasformato s valuable. 5. NUMERICAL EAMPLES 5. MATHEMATICAL EAMPLE Cosder a mathematcal problem wth put radom varables, ~ N(5.,.3,,. The RBDO formulato s defed to ( ( ( d m. cost ( d d 3.3 + 3. ( ( s.t. P Gp Φ βt,,,3 d, d, βt 3. G ( / p ( ( G ( + 5 / 3 / p ( G ( 8/ + 8 + 5 p3 (43 Usg the verse relablty aalyss (PMA+ RBDO, the relablty-based optmal desg ca be obtaed for the dfferet correlato coeffcet that rages from -. Fgure. Optmal Desg Pots ad Cost for Idepedet ad Correlated Cases. Table shows the optmum desg, optmum cost ad actve costrats for the dfferet correlato coeffcets. For the correlato coeffcet rage from -.6 to.4, the optmal desg pots are o a le as show Fgure ad have two actve costrats as show Table. As see the table, the optmum desgs ad the correspodg optmum costs sgfcatly deped o the correlato coeffcets. Table. Optmal Desg Pots for the Mathematcal Problem Correlato Actve Cost coef. Cost. -. 3.95 3.448. G p -.8 3.85 3.47.79 G p -.6 3.96 3.367.47 G, G p p -.4 3.74 3.34.8 G, G p p -. 3.354 3.35. G, G p p. 3.43 3.85.98 G, G p p. 3.59 3.54.7 G, G p p.4 3.65 3.3.53 G, G p p.6 3.694 3..3 G p.8 3.743 3.6.64 G p
. 3.79 3.94.37 p G ( + δ P K 8P D d N Δ 4 dg 3 (44 5. COIL SPRING PROBLEM I ext example, a egeerg problem s preseted to show how the correlato put varables affects RBDO results. Col sprgs are wdely used practcal applcatos. The desg objectve of the col sprg s to mmze the mass to carry a gve axal load such that desg satsfes the mmum deflecto ad allowable shear stress requremet, ad surge wave frequecy s above the lower lmt [9]. The secod costrat s that the shear stress the wre should ot be larger tha τ a, whch s formulated as 8 kp( D + d 8 P( D + d 4( D + d d.65d τ + τ 3 3 a πd πd 4D D+ d (45 where k s Wahl stress cocetrato factor. The thrd costrat requres that the surge wave frequecy of the sprg should be hgher tha ω as ω d G ω π + ρ ND ( d (46 Usg the data ad ormalzed costrats for the col sprg problem, the RBDO formulato s defed to Fgure. Col Sprg I ths example, fve desg parameters, whch are the mea er dameter of col sprg ( D, wre dameter ( d, umber of actve cols ( N, shear modulus ( G, ad mass desty of materal ( ρ, are selected. Other data are gve as: weght desty of 3 sprg materal, γ.85 lb /., shear modulus, 7 G (.5 lb/., allowable shear stress, τ a 8, lb /., umber of actve cols, Q, appled load, P lb, mmum sprg deflecto, Δ.5., ad lower lmt of surge wave frequecy. The desg ad radom varables such as umber of actve cols (, Col er dameter (, wre dameter ( 3, mass desty of materal ( 4, ad Shear modulus ( 5 have ormal dstrbutos ad have propertes show Table 3. Table 3. Propertes of Desg ad Radom Varables for the Col Sprg Radom Stadard dl d du var. dev.. 7. 3....8..5 3...5.3 4. E-4.85 E-4 7.38E-4.E-4 5.E+7.E+7.5E+7 3.E+7 As stated before, to carry out a gve axal load wthout materal falure the col sprg, three costras must be satsfed. The frst costra s that the deflecto δ uder the load P should be at least Δ as ( ( ( ( d ( ( ( β m. mass 5 d + Q π ( d + d d d s.t. PG ( Φ,,,3 G G G p 3 8 Px ( + x x. x x Δ 3 p 4 3 5 3 3 4 ( 4x + 3x 8 Px ( + x3 3.65x3. + + πx3τa 4 x ( x + x3 x3 x5. π x ( x + x w x p 3 p3 3 t 4 (47 I the maufacturg process, t s assumed that the col er dameter ad the wre dameter are correlated, ad thus the correlato coeffcet betwee those two varables s cosdered for RBDO. Table 4. RBDO Results for Dfferet Correlato Coeffcets ρ 3...4.6.7.7E+.99E+.8E+.798E+.784E+.37E+.34E+.37E+.7E+.E+ 3.E+.E+.E+.6E+.4E+ 4.85E-4.85E-4.85E-4.85E-4.85E-4 5.5E+8.5E+8.5E+8.5E+8.5E+8 G..... p G -.55 -.7 -.85 -.63 -.5 p G -.389 -.6 -.993 -.43 -.8 p 3 cost 4.863 4.386 3.85 3.4.95 Table 4 shows the optmal desg pots, costrats, ad cost for the dfferet correlato coeffcet. As see
the table, the optmum desgs ad costs sgfcatly deped o the correlato coeffcets. To mmze the mass of the sprg, the mass desty goes to lower boud ad the shear modulus does ot chage because the thrd costrat s always actve ad the shear modulus does ot affect the cost. From these two examples, t s clear that the correlato should be cosdered RBDO of real applcatos. 6. CONCLUSION I ths paper, a RBDO method that deals wth the correlato of put varables s proposed. For ths, two represetatve trasformato methods, Roseblatt trasformato ad Nataf trasformato, are vestgated for applcablty to RBDO problems wth correlated put varables. Roseblatt trasformato s a mathematcally exact trasformato method, but t has lmted applcatos sce the trasformato of orgal radom varables to stadard ormal varables ca be carred out oly whe a jot CDF or codtoal CDF s are avalable. Thus, t s ot applcable to real applcato whe the covarace matrx ad margal dstrbuto are oly avalable. A alteratve Nataf trasformato, whch s a Gaussa copula, s foud to be practcally applcable. Nataf trasformato ca costruct a exact jot PDF whe put varables are ormal or whe the ormal ad logormal varables are combed. Further, Nataf trasformato ca accurately costruct a jot CDF whe logormal varables have postve or eve some egatve correlatos. Addtoally, RBDO results are depedet of ordergs of the put varables whe Nataf trasformato s used, whereas t s ot the case for Roseblatt trasformato. I ths paper, usg Nataf trasformato, RBDO s carred out to solve umercal examples wth correlated put varables to demostrate that the correlato the put varables sgfcatly fluece the optmum results of RBDO. 7. ACKNOWLEDGMENTS Ths research s supported by the Automotve Research Ceter that s sposored by the U.S. Army TARDEC. 8. REFERENCES. Roseblatt, M., Remarks o a Multvarate Trasformato, The Aals of Mathematcal Statstcs, Vol. 3, 95, pp. 47-47.. Nataf, A., Détermato des dstrbutos de probabltés dot les marges sot doées, Comptes Redus Hebdomadares des Séaces de l Académe des Sceces, Vol. 55, 96, pp. 4-43. 3. Lu, P-L. ad Der Kuregha, A., Multvarate dstrbuto models wth prescrbed margals ad covaraces, Probablstc Egeerg Mechacs, Vol., No., 986, pp. 5-4. Melchers, R.E., Structural Relablty Aalyss ad Predcto, d edto, J. Wley & Sos, New York, 999. 5. Dtlevse, O. ad Madse, H.O., Structural Relablty Methods, J. Wley & Sos, New York, 996. 6. You, B.D., Cho, K.K., ad Du, L., Adaptve Probablty Aalyss Usg A Ehaced Hybrd Mea Value Method, Structural ad Multdscplary Optmzato, Vol. 9, No., 5, pp. 34-48. 7. Roser B.N., A Itroducto to Copulas, Sprger, New York, 999. 8. Hohebchler, M. ad Rackwtz, R., Noormal Depedet Vectors Structural Relablty, Joural of Egeerg Mechacs Dvso, ASCE, 7, 98, pp. 7-38. 9. Madse, H.O., Krek, S, ad Ld N.C., Methods of Structural Safety, Eglewood Clffs, NJ: Pretce- Hall, 986.. Kotz, S., Barlakrsha, N., ad Johso, N.L., Cotuous Multvarate Dstrbuto, Vol., d edto,.. Gembel, E.J., Two systems of Bvarate Extremal Dstrbutos, 35 th Sessos of the Iteratoal Statstcal Isttute, No. 69, Beograd, 965.. Marshall, A.W. ad Olk. I., A Multvarate Expoetal Dstrbuto, Joural of the Amerca Statstcal Assocato, Vol. 6, pp. 967, 3-44. 3. Freud, J.E., A Bvarate Exteso of the Expoetal Dstrbuto, Joural of the Amerca Statstcal Assocato, Vol. 56, 96, pp. 97-977. 4. Basu, A.P. ad Su, K., Multvarate Expoetal Dstrbutos wth Costat Falure Rates, Joural of Multvarate Aalyss, Vol. 6, 997, pp. 59-69. 5. Block, H.W. ad Basu, A.P., A Cotuous Multvarate Expoetal Exteso, Joural of the Amerca Statstcal Assocato, Vol. 69, 974, pp. 3-37. 6. Raftery, A.E., A Cotuous Multvarate Expoetal Dstrbuto, Commucatos Statstcs-Theory ad Methods, Vol. 3, 984, pp. 947-965. 7. Tobas, P.A. ad Trdade, D.C., Appled Relablty, d edto, CRC Press, 995. 8. Hah, G.J. ad Shapro, S.S., Statstcal models Egeerg, Joh Wley & Sos, 994. 9. Arora, J.S., Itroducto to Optmum Desg, d edto, Elsever academc press, 4