Integration of Reliability- And Possibility-Based Design Optimizations Using Performance Measure Approach

Transcription

1 SAE Keynoe Paper: Cho, K.K., Youn, B.D., and Du, L., Inegraon of Relably- and Possbly-Based Desgn Opmzaons Usng Performance Measure Approach, 005 SAE World Congress, Aprl 11-14, 005, Dero, MI Inegraon of Relably- And Possbly-Based Desgn Opmzaons Usng Performance Measure Approach Kyung K. Cho, and Lu Du Cener for Compuer-Aded Desgn, Deparmen of Mechancal & Indusral Engneerng, College of Engneerng, he Unversy of Iowa, Iowa Cy, IA 54, USA Copyrgh 004 SAE Inernaonal Byeng D. Youn Deparmen of Mechancal Engneerng, Unversy of Dero Mercy, Dero, MI , USA ABSRAC Snce deermnsc opmum desgns obaned whou consderng uncerany lead o unrelable desgns, s val o develop desgn mehods ha ae accoun of he npu uncerany. When he npu daa conan suffcen nformaon o characerze sascal dsrbuon, he desgn opmzaon ha ncorporaes he probably mehod s called a relably-based desgn opmzaon (RBDO). I nvolves evaluaon of probablsc oupu performance measures. he enrched performance measure approach (PMA+) has been developed for effcen and robus desgn opmzaon process. hs s negraed wh he enhanced hybrd mean value (HMV+) mehod for effecve evaluaon of non-monoone and/or hghly nonlnear probablsc consrans. When suffcen nformaon of npu daa canno be obaned due o resrcons of budges, facles, human, me, ec., he npu sascal dsrbuon s no belevable. In hs case, he probably mehod canno be used for relably analyss and desgn opmzaon. o deal wh he suaon ha npu unceranes have nsuffcen nformaon, a possbly (or fuzzy se) mehod should be used for srucural analyss. A possbly-based desgn opmzaon (PBDO) mehod s proposed along wh a new numercal mehod, called maxmal possbly search (MPS), for fuzzy (or possbly) analyss and employng he performance measure approach (PMA) ha mproves numercal effcency and sably n PBDO. he proposed RBDO and PBDO mehods are appled o wo examples o show her compuaonal feaures. Also, RBDO and PBDO resuls are compared for mplcaons of hese mehods n desgn opmzaon. 1. INRODUCION Due o he exensve effors of engneerng dscplnes over las hree decades, desgn gudelnes and/or sandards have been modfed o ncorporae he concep of uncerany no an early desgn sage. In response o hese new desgn requremens, varous mehods have been developed o rea unceranes n engneerng analyss and, more recenly, o carry ou desgn opmzaon wh relably. here are wo dfferen ypes of unceranes: aleaory and epsemc unceranes [1]. Aleaory uncerany s classfed as objecve and rreducble uncerany, whereas epsemc uncerany s a subjecve and reducble uncerany ha sems from lac of nowledge on daa. In general, a large amoun of daa s employed o consruc aleaory uncerany, such ha an uncerany s accuraely quanfed wh a dsrbuon ype and s parameers. Ofen, s very dffcul o collec suffcen daa for uncerany quanfcaon due o he resrcons of resources (budges, facles, human, me, ec.). hus, s desrable ha dfferen desgn mehodology s ulzed for dfferen ypes of uncerany. For aleaory uncerany, a relably-based desgn opmzaon (RBDO) [-5] has been used o consder engneerng unceranes n a desgn process. In RBDO, a prmary concern has been how o mae compuaonally affordable, whle mananng numercal accuracy and sably. Compuaonal effcency and sably have been mproved by usng he enrched performance measure approach (PMA+), where four mprovemens are made over he orgnal PMA: as a way o launch RBDO a a deermnsc opmum desgn, as an effcen probablsc feasbly chec, as an enhanced hybrd-mean value (HMV+) mehod, and as a fas relably analyss under he condon of desgn closeness [5]. In areas where s no possble o produce accurae sascal nformaon, he probablsc mehods are no

2 approprae for srucural analyss and desgn opmzaon subjec o physcal unceranes, snce mproper modelng of uncerany could cause greaer degree of sascal uncerany han hose of physcal uncerany [6]. o handle epsemc uncerany when modelng physcal uncerany wh nsuffcen nformaon, possbly-based (or fuzzy se) mehods have recenly been nroduced n srucural analyss and desgn [7]. In ad of he operaonal framewor of possbly heory [8-13], a fuzzy analyss enals he followng four aracve feaures. 1) he fuzzy analyss preserves he nrnsc random naure of physcal varables hrough her membershp funcons. ) Exended fuzzy operaons are smpler han hose requred o use probably, especally when a number of varables are nvolved. 3) I has been poned ou ha, when lle nformaon s avalable for npu daa, he possbly-based mehod s beer snce provdes a more conservave desgn han he probablsc desgn ha s conssen wh he lmed avalable nformaon [13,14]. hs s a desrable mer, snce a conservave opmum desgn s preferred when accurae sascal nformaon s no avalable. 4) Possbly analyss provdes a sysem-level possbly unle relably analyss [14]. Usng PMA, a new formulaon of PBDO has been formulaed o mprove numercal effcency, sably, and accuracy [7]. o resolve dsadvanages of he verex mehod and he mullevel-cu mehod, a new maxmal possbly search (MPS) mehod [7] has been proposed for a possbly (or fuzzy) analyss, such ha evaluaes possbly consrans effcenly and accuraely for nonlnear srucural applcaons. hs paper presens an negraed desgn plaform of boh RBDO and PBDO ha employ PMA for more effecve desgn formulaons. wo examples are used o show compuaonal feaures of PMA+ for RBDO wh aleaory npu unceranes and MPS for PBDO wh epsemc npu unceranes. In addon, RBDO and PBDO resuls are compared for mplcaons of hese mehods n desgn opmzaon.. RELIABILIY-BASED DESIGN OPIMIZAION.1 RBDO MODEL FOR PERFORMANCE MEASURE APPROACH For general engneerng applcaons, he RBDO model [1-5] can be formulaed as mn Cos( d) s.. PG ( ( dx ( )) > 0) Φ( β ) 0, = 1,, np L d d d n where d = [ ] = µ ( X) R s he desgn vecor, nr d U X = [ X ] R s he random vecor, and n, nr and np are he number of desgn varables, random varables, and probablsc consrans, respecvely. he desgn consrans are descrbed by he probably P( ) of he falure even G ( dx ( )) 0. he sascal descrpon of he consran volaon s characerzed by he cumulave dsrbuon funcon F () as G where PG ( ( X) 0) = 1 F (0) Φ( β ) G F (0) = f ( x ) dx dx, = 1,, np G G ( X) 0 X In Eq. (3), f X( x ) s he jon probably densy funcon of all random varables. Is evaluaon requres a relably analyss where mulple negraons are nvolved, as shown n Eq. (3). Some approxmae probably negraon mehods have been developed o provde effcen soluons, such as he frs-order relably mehod (FORM) [15,16], or he asympoc second-order relably mehod (SORM) [17,18] wh a roaonally nvaran measure as he relably. FORM ofen provdes adequae accuracy and s wdely used for desgn applcaons. hose relably mehods requre a ransformaon [19,0] from he orgnal random parameer X o he sandard normal random parameer U. he performance funcon G ( X) n X-space can hen be mapped ono G(X) = G( (X)) G(U) n U-space. he probablsc consran n Eq. () can be expressed as a performance measure hrough he nverse ransformaon of F () as [3-5] G p G 1 G ( dx ( )) = F (1 Φ( β )) 1 G 1 = F ( Φ( β )) 0 where G p s he h probablsc consran. In Eq. (4), he probablsc consran n Eq. (1) can be replaced wh he performance measure, whch s referred o as he performance measure approach (PMA) [3-5]. hus, he RBDO model usng PMA can be redefned as n (1) () (3) (4)

3 mnmze Cos( d) subjec o G ( dx ( )) 0, = 1,,, p L U d d d np (5) hese wo search pons. For he nerpolaon, a paramerc coordnae s nroduced as ( 1) ( ) U= s () u + u and U = β, s, 0 (8). RELIABILIY ANALYSIS MODEL OF PMA [3-5] Relably analyss n PMA can be formulaed as he nverse of relably analyss n he relably ndex approach. he frs-order probablsc performance measure s obaned from a nonlnear G p,form opmzaon problem n U-space, defned as maxmze G( U) subjec o U β = where he opmum pon on he arge relably surface s denfed as he mos probable pon (MPP) u β β wh β = he prescrbed relably = u β β ucer (KK) necessary condon of Eq. (6) s β= β = β β= β G( ) G( β β u u u = ) = (6). he Karush-Kuhn- Any general opmzaon algorhm can be employed o solve he opmzaon problem n Eq. (6). However, an enhanced hybrd mean value (HMV+) frs-order mehod s well sued for PMA due o s sably and effcency [5,1]..3 ENHANCED HYBRID MEAN VALUE (HMV+) MEHOD Even hough he hybrd mean value (HMV) mehod [3] performs well for mldly nonlnear monoone performance funcons, could fal o converge for hghly nonlnear and non-monoone oupu performance funcons. o mprove numercal sably and effcency, an enhanced HMV (HMV+) mehod has been proposed by revsng he prevous HMV algorhm [5,1]. In hs HMV+ algorhm, f he value of he performance funcon s decreased a he nex search pon, he performance funcon s approxmaed along he arc (wh consan probably level β ι ) beween he curren pon and nex search pon o fnd a new search pon where he approxmaed performance funcon aans s maxmum value. Deal numercal procedure and numercal resuls can be found n refs. 5 and 1, where has shown ha he HMV+ mehod mproves numercal effcency and sably subsanally n relably analyss for hghly nonlnear performance funcon. he performance and s sensvy values a wo search ( ) ( 1) pons, uhmv+ and uhmv+, are used o nerpolae he performance funcon along he arc regon beween (7) where ( 1) ( ) ( 1) ( ) 4 ( ) (1 ) β β + u u + u u + s () = ( 1) ( ) ( 1) ( ) 4 ( ) (1 ) β β u u u u + s () = he sensvy of he performance funcon wh respec o he paramerc coordnae can be obaned usng a chan rule as dg G U G ds ( 1) ( ) = = u + u d U U d In Eq. (10), he sensves of performance funcon are ( 1) ( ) evaluaed a wo search pons u and u. hen, ( 1) ( ) g( u ), g( u ), ( 1) ( ) dg( u ) dg( u, and ) d d he enrched PMA (PMA+) has been proposed o enhance numercal effcency whle mananng sably n he RBDO process [5]. PMA+ s an exenson of PMA by negrang hree ey deas: as a way o launch RBDO a a deermnsc opmum desgn, as a probablsc feasbly chec, and as a fas relably analyss under he condon of desgn closeness. he overall desgn 3 (9) (10) are used o nerpolae he performance funcon usng a cubc polynomal as G () = a + a+ a + a ( + 1) u he nex search pon s obaned where he approxmaed performance funcon G s maxmum, as ( + 1) ( 1) ( ) u = s ( ) u + u a = G ( ) s maxmum, for β > 0 where s mnmum, for β < 0 (11) Noe ha s() may no be unque for a gven value of, f > 1 (s, >0; s, 4 β 4 ( 1) ( ) β ( u u ) ), as shown n Eq. (9). hs could be rue when he angle composed of hree ( 1) (0) ( ) pons u, u, and u s more han 90, whch can ( 1) ( ) be expressed mahemacally as u u < 0. One of hese wo s values, whch yelds greaer performance funcon value wll be seleced..4 ENRICHED PERFORMANCE MEASURE APPROACH [PMA+]

4 procedure n PMA+ for RBDO s frs o oban he deermnsc opmum desgn effcenly, and hen carry ou relably-based desgn opmzaon. he feasbly of probablsc consrans n RBDO can be denfed by usng he MV frs-order mehod ha provdes an allowable degree of accuracy for he purpose of consran volaon. Once he feasbly saus of probablsc consrans s denfed by he MV frs-order mehod, a refned relably analyss s performed usng he enhanced hybrd mean value (HMV+) frs-order mehod o evaluae ε-acve and volae consrans. he MV frs-order mehod based feasbly chec for probablsc consrans subsanally mproves he numercal effcency of he RBDO process. Durng RBDO eraons, suffcen nformaon s generaed whle evaluang he cos and probablsc consrans, and updang he desgn. Some of hs nformaon could be reused o evaluae probablsc consrans effcenly a he nex desgn eraon usng he condon of desgn closeness. In oher words, under he condon ha wo consecuve RBDO desgn eraons are close enough, he relably analyss can be effcenly carred ou by sarng from he MPP obaned a he prevous eraon, nsead of a he mean value pon of he curren desgn eraon. hs fas relably analyss mehod s negraed wh he HMV+ mehod o evaluae probablsc consrans effcenly. 3. POSSIBILIY-BASED DESIGN OPIMIZAION 3.1 PBDO MODEL FOR PERFORMANCE MEASURE APPROACH For general engneerng applcaons, he PBDO model can be formulaed as [7] mnmze Cos( d) subjec o Π ( G ( dx ( )) > 0) α, = 1,, np nr L d d d n U (1) where d = [ d ] R s he desgn vecor, X = [ X ] R s he fuzzy random vecor where he fuzzy random varable Π X x X has he membershp funcon ( ) and he maxmal grade [13] max{ Π ( x )} = d, X α s a arge possbly of falure, and n, nr, and np are he number of desgn varables, fuzzy random varables, and possbly consrans, respecvely. Fuzzy random parameers consdered n hs paper are assumed o be non-neracve. he possbly consrans are descrbed by a possbly Π () α for he falure even G ( dx ( )) > 0. he PMA approach, whch s developed for RBDO as shown n Secon, s applcable o he PBDO model o formulae as [7] mnmze Cos( d) subjec o GΠ ( dx ( )) 0, = 1,,, np L U d d d 3. GENERAION OF MEMBERSHIP FUNCION (13) he generaon of he npu membershp funcons of he fuzzy varables usng he avalable lmed se of daa s a very mporan sep of he possbly analyss and PBDO. Several mehods have been proposed dependng on he number and he nd of he daa avalable. hs paper nroduces wo procedures: 1) generang he pseudoprobably densy funcon of he fuzzy varable from he avalable daa; ) generang he membershp funcon of he fuzzy varable from he pseudo-probably densy funcon Generaon of Pseudo-Probably Densy Funcon If he only avalable nformaon for he npu fuzzy varable s he judgmen of expers (subjecve) wh he mos lely value and he nerval correspondng o he ceran confdence levels, he pseudo-probably densy funcon can be generaed usng he framewor of Program-Evaluaon and Revew echnque (PER) analyss []. If he fuzzy varable has a random naure bu he avalable daa are no suffcen o assgn he probably of elemenary evens, he pseudo-probably densy funcon can be esmaed usng ernelsmoohng mehod [3]. Noce ha, snce he npu daa s no suffcen, he pseudo-probably densy funcon canno be used drecly n he probably-based mehod. 3.. Generaon of Membershp Funcon o generae he membershp funcon of he fuzzy varable from he pseudo-probably densy funcon, wo mehods are used n hs paper. he probably-possbly conssen prncple and he leas conservave prncple are used o generae he membershp funcon from he pseudo-probably densy funcon. he probably-possbly conssen prncple says ha, he probably of one even canno be less han he possbly of hs even [14]. he membershp funcon sasfyng he probably-possbly conssen prncple s no unque. he leas conservave one s chosen such ha he membershp funcon s no oo much conservave. If he pseudo-cumulave dsrbuon funcon of a fuzzy varable s FX ( x), hen he membershp funcon of he fuzzy varable sasfyng he probably-possbly 4

5 conssen prncple and he leas conservave prncple s unque: FX( x) x { x : FX( x) 0.5} Π X ( x) = FX( x) x { x : FX( x) > 0.5} R (14) If he membershp funcon generang by he above way s no conservave enough, Savoa s way s a good alernave one [8]: where and Π ( x) = F ( x ) + f ( x)( x x ) + F ( x ) (15) fx X X L X R L X ( x) s he pseudo-probably densy funcon x L and R f ( x ) = f ( x ) = f ( x). X L X R X x s choosng such ha 3.3 POSSIBILIY ANALYSIS MODEL OF PMA In hs paper, s assumed ha he non-neracve npu fuzzy varables X have s membershp funcon ( ) sasfyng hree properes [8,14,4]: (1) uny, Π X x () srong convexy, and (3) boundedness. hese hree properes enable non-neracve npu fuzzy varables X o be unquely ransformed o an sosceles rangular membershp funcon as u + 1, 1 u 0 Π U ( u ) = = 1 u, u 1 1 u, 0 u 1 he ransformaon can be wren as Π X d, L( X) 1 X U = 1 Π X, R( X) X > d (16) (17) where Π X, L( x) and Π X, R( x) are he lef sde and rgh sde of he membershp funcon of he npu fuzzy varable X, respecvely, and d s he maxmal grade of he membershp funcon. hus, evaluaon of he possbly consran requres an nverse fuzzy analyss usng PMA, whch s formulaed as maxmze G ( U) subjec o U 1 α (18) where he opmum pon on he arge possbly doman U 1 α s denfed as he mos possble pon (MPP) u α wh he prescrbed possbly of falure α. he dfference beween relably analyss [3,5] and fuzzy analyss [8-14,4-6] s ha, he mos probable pon (MPP) n relably analyss s based on FORM, whch means he relaed probably s frs order approxmaon, whereas he mos possble pon (MPP) n possbly analyss s exac, along wh he relaed possbly. Anoher dfference s he search doman enclosed by he arge confdence level. he relably analyss has nr-dmensonal hyper-sphere as s search doman, U β, whereas he possbly analyss has nr-dmensonal hyper-cube as s search doman, U 1 α, ha maes he numercal compuaon smpler, compared o he relably analyss. Snce he MPP search n possbly analyss s dfferen from he one n relably analyss, a new numercal mehod has been proposed n he nex secon o solve he problem (16). 3.4 GENERAION OF PSEUDO-PROBABILIY DENSIY FUNCION For a non-monoonc response whn he range of npu fuzzy parameers, he fuzzy analyss could be naccurae usng he verex mehod [8] or compuaonally expensve usng he mullevel-cu mehod or oher compuaonal schemes such as a global opmzaon mehod [6]. hus, n hs paper, he maxmal possbly search (MPS) mehod has been proposed for fuzzy analyss o ensure numercal effcency and accuracy n PBDO. hs mehod frs aemps o fnd an MPP usng he maxmal possbly search, snce n majory of cases he MPP s lely o be on he verex of he arge possbly doman U 1 α. If he maxmal possbly search does no yeld a soluon a a verex, hen he maxmal possbly search s negraed wh an nerpolaon n search of he MPP on he edge or n he neror doman of he hyper-cube Maxmal Possbly Search he maxmal possbly search algorhm s as followng: Sep 1. Se he eraon couner = 0 wh he convergence parameer ε = 10. Se he poner j = 1. (0) (0) Le u = 0. Calculae he performance G (u ) and he 3 sensvy (u ). Le d = G( u ). G (0) ( ) ( ) Sep. Compue he nex pon as ( + 1) ( ) u = π sgn( d ) where π = 1 α and sgn( X ) = (sgn( X ),sgn( X ),,sgn( Xnr )) f nr 1 X = [ X ] R. Le = + 1. Sep 3. Calculae he performance sensvy G (u ( ) ) ( ) ( ) ( 1) and he (u ). Le d = G( u ) + βd where 5

6 ( ) ( 1) β = ( G( u ) / G( u ) ). If ( ) sgn( ( u )) = sgn( u ), s he maxmum pon and ( ) ( j) sop. If G( u ) G( u ), le j = and go o Sep. Oherwse, go o Sep 4. Sep 4. Go o Sep 5, wh ( j ) u, j (u ) and j ( u ) Maxmal Possbly Search wh an Inerpolaon he proposed maxmal possbly search wh an nerpolaon algorhm s as followng: () l ( ) Sep 5. Le l = 0 and d = G( u j ). Go o Sep 6. Sep 6. Calculae he new pon of he doman usng he sar pon () l drecon d. Le = + 1. ( + 1) u Sep 7. Calculae he performance sensvy (u ). If on he boundary ( j ) u and he search G (u ( ) ) and he G ( ) sgn( ( u )) = sgn( u), for u = π or u = π x G ( ) ( u ) < ε, for π < u < π x possbly search fnds MPP effcenly, he MPS mehod wll be effcen whle robus for nonlnear and nonmonoonc responses. 4. INEGRAION OF RBDO AND PBDO 4.1 WHY PBDO FOR EPISEMIC UNCERAINY? I s apparen ha RBDO mus be performed when he npu sascal nformaon s suffcen. However, whou modelng physcal npu unceranes properly due o resrcon of resources, a probablsc approach may lead o an unsafe desgn, as shown n Fg. 1. Hsograms of wo unceranes are bul wh nsuffcen raw daa (50 samples from normal dsrbuon: X1 ~ N(.0,1.0 ), X ~ N(4.0,1.0 ) ), and hey are presumably modeled as a normal dsrbuon hrough a fng echnque. An occupan safey consran wh a lnear model s defned as G 500. Nney-wo percen relably wh mprecse sascal daa urns ou o be seveny-seven relably wh precse sascal nformaon. Less conservave (or aggressve) esmaon of falure rae due o lac of daa could lead o a dangerous desgn. Sysem nonlneary and smaller amoun of daa wll ncrease he error assocaed wh a relably level. X 1 ~ N(1.75,0.99 ) X ~ N(3.64,0.93 ) hen s he maxmum pon and sop. Oherwse, go o Sep 8. Sep 8. Usng j G ( ) ( u ), ( u ), j o consruc he hrd order polynomal sragh lne beween ( j ) (u ) and PG () G u ( ) ( ) on he u, and u where s he parameer for he lne. Calculae he maxmum pon ( + 1) for hs polynomal. Le u be he pon on he lne correspondng o. Le = + 1. ( ) Occupan Safey Performance G(X) = 10X 1 +10X +390 Sep 9. Calculae he performance G (u ( ) ) and he sensvy (u ). Chec he convergen crera usng he equaon n Sep 7. If convergen, sop. Oherwse, le he new conjugae drecon be ( l+ 1) ( ) ( l) d = G( u ) + βd where ( ) ( ) β = ( G( u ) / G( u ) ). Le j =, l = l + 1, and go o Sep 6. CDF PDF he proposed MPS mehod s suffcen for he monoonc responses. he proposed MPS mehod wh an nerpolaon wll be used only when he maxmal possbly search fals. Snce n mos cases he maxmal Fgure 1. Erroneous Desgn Employng Lac of Informaon 6

7 4. SRUCURE OF INEGRAED RBDO AND PBDO RBDO and PBDO can be negraed n one desgn plaform, as shown n Fg.. Aleaory uncerany mus be deal wh a probably approach, whereas epsemc uncerany can enal a possbly (or fuzzy) approach. Boh RBDO and PBDO employ PMA o mprove numercal effcency, sably, and accuracy. he nal desgn s d = [5.0,5.0]. Random properes are X1 ~ N( µ 1,0.3) and X ~ N( µ,0.3) for RBDO, and fuzzy varables are modeled usng Eq. (14) for PBDO, as shown n Fg. 3. (0) Daa Level Analyss Desgn Opmum Aleaory Uncerany Relably Analyss: HMV+ RBDO Inegraed Framewor of RBDO & PBDO Epsemc Uncerany Possbly Analyss: MPS PMA PBDO Opmum Desgn w/ Confdence Fgure. Srucure of Inegraed RBDO and PBDO 5. NUMERICAL EXAMPLES OF RBDO AND PBDO 5.1 NONLINEAR MAHEMAICAL EXAMPLE [5] Consder he followng mahemacal model for boh RBDO and PBDO wh desgn varables d = [ d1, d] = [ µ 1, µ ]. he RBDO problem s defned as mnmze ( d + + = 1 d 10) ( d d 1 10) Cos( d) subjec o PG ( ( X) > 0) Φ( β ), = 1,,3 or Π ( G ( Y) > 0) α, = 1,,3 0 d 10 & 0 d 10 1 where β =.0 (σ desgn), α = 0.03, and nonlnear performances are defned as 1 X X G1( X) = 1 0 (19) 3 4 G ( X) = 1 + ( Y 6) + ( Y 6) 0.6( Y 6) + Z G3 ( X) = 1 80 (0) ( X 1 + 8X+ 5) Y = X X Z = 0.46 X X Fgure 3. Membershp Funcon of Random Inpu ables 1 and dsplay he numercal resuls of RBDO and PBDO, respecvely. In hese ables, NFE represen he number of funcon evaluaons, wh he same number of sensvy analyses. he RBDO provded σ desgn, whch s componen level relably. Unle RBDO, has been found ha he deermnsc opmzaon does no mprove he PBDO effcency, due o he dfference n HMV+ and MPS. Even hough he second consran s non-monoonc and hghly nonlnear, boh HMV+ mehod for RBDO and MPS mehod for PBDO behave very well n erms of numercal sably. Noe ha he same npu random properes are used for opmum desgns of RBDO and PBDO. However, when a relably analyss s carred ou for he PBDO resul usng he Mone Carlo smulaon mehod he PBDO resul, he sysem level relably s 99.4 % (.5-sgma). In oher words, PBDO provdes a much conservave desgn n erms of cos and confdence level, compared o RBDO. However, hey share he smlar feasbly behavor of he probablsc consrans. Opmzaon hsores of RBDO and PBDO are shown n Fgs. 4 and 5, respecvely. able 1. RBDO Resuls of Mahemacal Problem Ier. Cos d 1 d G 1 G G 3 NFE NA De Op Ac. Ac. Inac. 50 able. PBDO Resuls of Mahemacal Problem Ier. Cos d 1 d G 1 G G 3 NFE

8 Op Ac. Ac. Inac. 44 aleaory npu unceranes and he PBDO wh epsemc npu unceranes. Normal dsrbuon and 5 % coeffcen of varaon s used for random varables, and fuzzy varables are modeled usng Eq. (14) for PBDO, as shown n Fg. 3. Fgure 6. Sde Impac Crashworhness Fgure 4. RBDO Hsory for Mahemacal Example he vehcle sde mpac s employed o show compuaonal feaures of PMA+ for RBDO wh aleaory npu unceranes and MPS for PBDO wh epsemc npu unceranes. en performance measures are used o deermne he occupan safey. he desgn objecve s o enhance sde mpac crash performance wh a σ relably, whle mnmzng he vehcle wegh. Oher han he σ relably requremen, oher feaures of he RBDO problem of crashworhness for sde mpac s defned he same way as n Refs. 7, and 9. able 3 lss he nal, deermnsc opmum, relably-based opmum, and possbly-based opmum desgns. he RBDO provded σ desgn (97.7% relably), whch s componen level relably. When a relably analyss s carred ou for he PBDO resul usng he Mone Carlo smulaon mehod, he sysem level relably s 98 %. Agan, PBDO provdes a much conservave desgn n erms of cos and confdence level, compared o RBDO. he objecve hsores for RBDO and PBDO are shown n Fg. 7. Agan, PBDO yelds a conservave desgn, n erms of he cos and confdence level, compared o RBDO. Fgure 5. PBDO Hsory for Mahemacal Example 5. RBDO EXAMPLE FOR VEHICLE SIDE IMPAC [7] he large-scale desgn applcaon shown n Fg. 6 s used o nvesgae compuaonal feaures of he RBDO wh aleaory npu unceranes and he PBDO wh epsemc npu unceranes. he desgn objecve s o enhance overall performance of sde-crash wh a arge confdence level whle mnmzng he vehcle wegh. Deal dscussons of sde-crash modes are summarzed n Ref. 7. he opmal Lan hypercube mehod used a oal of 33 samples o generae a sample of desgn pons for consrucng he sepwse regresson response surface [8]. In hs sudy, approxmaons of he responses obaned from response surface mehod are regarded as exac responses of vehcle sde mpac o nvesgae compuaonal feaures of he RBDO wh able 3. Desgns for Sde Impac Crashworhness b Inal De. Op. Rel. Op. Pos. Op

9 Fgure 7. RBDO and PBDO Cos Hsory for Sde Impac 6. DISCUSSIONS AND CONCLUSIONS hs paper presened an negraed desgn formulaon for RBDO and PBDO usng he performance measure approach (PMA). When he npu daa conan suffcen nformaon o characerze sascal dsrbuon, RBDO wh he enrched performance measure approach (PMA+) was employed for a desgn opmzaon. On he oher hand, when suffcen nformaon of npu daa s no avalable due o resrcons of resources, he npu sascal dsrbuon s no accurae. In hs suaon, he probably mehod could provde an unrelable desgn due o mprecse sascal nformaon. For naccurae sascal nformaon, he possbly-based desgn opmzaon (PBDO) has been nroduced by negrang a possbly (or fuzzy se) mehod o he desgn plaform. For PBDO, he maxmal possbly search (MPS) mehod was developed, and showed s effcency and sably for hghly nonlnear and nonmonoonc performance responses. PBDO provdes more conservave desgn han RBDO, n erms of confdence level and cos. Usng PMA, aleaory and epsemc unceranes can be deal wh by RBDO and PBDO n he negraed desgn plaform. Fuure research wll address a challenge of how o handle suaons ha an engneerng sysem conans boh aleaory and epsemc unceranes smulaneously. 7. ACKNOWLEDGEMENS Research s parally suppored by he Auomove Research Cener sponsored by he U.S. Army ARDEC. 8. REFERENCES 1. Helon, J. C., Uncerany and sensvy analyss n he presence of sochasc and subjecve uncerany, Journal of Sascal Compuaon and Smulaon, Vol. 57, 1997, pp Yu, X., Cho, K.K., and Chang, K.H., A Mxed Desgn Approach for Probablsc Srucural Durably, Journal of Srucural Opmzaon, Vol. 14, No. -3, pp , Youn, B.D., Cho, K.K., and Par, Y.H., Hybrd Analyss Mehod for Relably-Based Desgn Opmzaon, Journal of Mechancal Desgn, ASME, Vol. 15, No., pp. 1-3, 003 and Proceedngs of 001 ASME Desgn Engneerng echncal Conferences: 7 h Desgn Auomaon Conference, Psburgh, PA, Lee,.W., and Kwa, B.M., A Relably-Based Opmal Desgn Usng Advanced Frs Order Second Momen Mehod, Mech. Sruc. & Mach., Vol. 15, No. 4, pp , Youn, B.D., Cho, K.K., and Du, Lu, Enrched Performance Measure Approach (PMA+) for Relably-Based Desgn Opmzaon Approaches, AIAA Journal, In Press, Ben-Ham, Y., and Elshaoff, I., Convex mehods of uncerany n appled mechancs, Elsever, Amserdam, Cho, K.K., Du, L., and Youn, B.D., A New Fuzzy Analyss Mehod for Possbly-Based Desgn Opmzaon, 10 h AIAA/ISSMO Symposum on MAO, AIAA , Albany, NY, Augus Savoa, M., Srucural relably analyss hrough fuzzy number approach, wh applcaon o sably, Compuers & Srucures, Vol. 80, 00, pp Dubos, D., Prade, H., Possbly heory: an approach o compuerzed processng of uncerany, Plenum Press, New Yor, NY, Rao, S., Opmum desgn of srucures n a fuzzy envronmen sysem, AIAA Journal, Vol. 5, 1987, pp onon, F., and Bernardn, A., A random se approach o he opmzaon of unceran srucures, Compuers & Srucures, Vol. 68, 1998, pp Ferrar, P., and Savoa, M., Fuzzy number heory o oban conservave resuls wh respec o probably, Compuer Mehods n Appled Mechancs and Engneerng, Vol. 160, 1998, pp Rao, S. S., Descrpon and opmum desgn of fuzzy mechancal sysems, ASME, Journal of Mechansm, ransmssons, and Auomaon n Desgn, Vol. 109, 1987, pp Nolads, E., Cudney, H. H., Chen, S., Hafa, R.., and Rosca, R., Comparson of probablsc and possbly heory-based mehods for desgn agans caasrophc falure under uncerany, Proceedngs of Madsen, H.O., Kren, S., and Lnd, N.C., Mehods of Srucural Safey, Prence-Hall, Englewood Clffs, NJ, Palle,.C. and Mchael J. B., Srucural Relably heory and Is Applcaons, Sprnger-Verlag, Berln, Hedelberg, Breung, K., Asympoc Approxmaons for Mulnormal Inegrals, Journal of Engneerng Mechancs, ASCE, Vol. 110, No. 3, pp , ved, L., Dsrbuon of Quadrac Forms n Normal Space-Applcaon o Srucural Relably, Journal of Engneerng Mechancs, ASCE, Vol. 116, No. 6, pp ,

10 19. Racwz, R. and Fessler, B., Srucural Relably Under Combned Random Load Sequences, Compuers & Srucures, Vol. 9, pp , Hohenbchler, M. and Racwz, R., Nonnormal Dependen Vecors n Srucural Relably, Journal of he Engneerng Mechancs, ASCE, Vol. 107, No. 6, pp , Youn, B.D., Cho, K.K., and Du, L., Adapve Probably Analyss Usng An Enhanced Hybrd Mean Value (HMV+) Mehod, Journal of Srucural and Muldscplnary Opmzaon, Vol. 8, No. 3, Valappan S, Pham D., Consrucng he membershp funcon of a fuzzy se wh objecve and subjecve nformaon, Mcrocompuers n Cvl Engneerng, Vol. 8, 1993, pp Wand M. P., Jones M.C., Kernel Smoohng, Chapman and Hall, London, Zadeh, L. A., Fuzzy Ses, Informaon and Conrol, Vol. 8, 1965, pp Shh, C. J., Ch, C. C., and Hsao, J. H., Alernave α-level-cus mehods for opmum srucural desgn wh fuzzy resources, Compuers & Srucures, Vol. 81, 003, pp Möller, B., Graf, W., Beer, M. Fuzzy srucural analyss usng α -level opmzaon, Compuaonal Mechancs, Vol. 6, 000, pp Gu, L., Yang, R-J., ho, C.H., Maows, M., Faruque, O., and L, Y., Opmzaon and Robusness for Crashworhness of Sde Impac, In. J. Vehcle Desgn, Vol. 6, No. 4, Myers, R.H. and Mongomery, D.C., Response Surface Mehodology, John Wley & Sons, New Yor, NY, Youn, Byeng D., Cho, K. K., Gu, L., and Yang, Ren- Jye, "Relably-Based Desgn Opmzaon for Crashworhness of Sde Impac," Journal of Srucural and Muldscplnary Opmzaon, Vol. 6, No. 3-4, pp. 7-83, 004. v Mos possble pon (MPP) for possbly analyss, s() Paramerc coordnaes for MPP search ransformaon marx from X space o U space θ Conjugae drecon facor for maxmal possbly search 9. DEFINIIONS, ACRONYMS, ABBREVIAIONS X Random or fuzzy varable; X = [X 1, X,, X n ] x Realzaon of X; x = [x 1, x,, x n ] U Independen and sandard normal random varable u Realzaon of U; u = [u 1, u,, u n ] V Non-neracve fuzzy random varable wh an sosceles rangular membershp funcon v Realzaon of V; v = [v 1, v,, v n ] β Relably ndex α Possbly ndex G Performance funcon; consdered fal f G > 0 PA ( ) Probably of even A Φ () Sandard normal dsrbuon Π( A) Possbly of even A u Mos probable pon (MPP) for relably analyss 10