The First Order Saddlepoint Approximation for. Reliability Analysis

Transcription

1 AIAA Joral, Vol. 4, No. 6, 004, pp The Frst Order Saddlepot Approxmato for Relablty Aalyss Xaopg D Uversty of Mssor Rolla, Rolla, MO Ags Sdjato + Ford Motor Compay, Dearbor, MI Abstract I the approxmato methods of relablty aalyss, o-ormal radom varables are trasformed to stadard ormal radom varables. Ths trasformato teds to crease the olearty of a lmt-state fcto ad hece reslts less accrate relablty approxmato. The Frst Order Saddlepot Approxmato for relablty aalyss s proposed to mprove the accracy of relablty aalyss. By approxmatg a lmt-state fcto at the Most Lkelhood Pot the orgal radom space ad employg the accrate saddlepot approxmato, the proposed method redces the chace of creasg olearty of the lmt-state fcto. Ths approach geerates more accrate relablty approxmato tha the Frst Order Relablty Method wthot creasg the comptatoal effort. The effectveess of the proposed method s demostrated wth two examples ad s compared wth the Frst ad Secod Order Relablty Methods. Assstat Professor, Departmet of Mechacal ad Aerospace Egeerg, 870 Mer Crcle, dx@mr.ed, Member AIAA. + Maager, Sbsystem Egeerg C, V-Ege Egeerg, Powertra Operatos, 500 Oakwood Blvd, asdja@ford.com

2 Nomeclatre E = expectato F = cmlatve dstrbto fcto f = probablty desty fcto g = lmt-state fcto H = Hessa matrx k = ma crvatre of the lmt-state fcto at K = cmlat geeratg fcto t = mber of tractable radom varables ~t = mber of tractable radom varables P = probablty p f = probablty of falre R = relablty t = saddlepot x = realzato of radom varable X X = vector of radom varables X = radom varable x = Most Probable Pot or Most Lkelhood Pot x-space Y = system respose = realzato of radom varable U U = vector of stadard ormal radom varables U = stadard ormal radom varable = Most Probable Pot or Most Lkelhood Pot -space β = relablty dex Φ = cmlatve dstrbto fcto of stadard ormal dstrbto Φ = verse cmlatve dstrbto fcto of stadard ormal dstrbto φ = probablty desty fcto of stadard ormal dstrbto = gradet I. Itrodcto Nmercal smlatos are rotely sed to captre the physcal pheomea detal to predct egeerg system behavors ad to redce the mber of physcal testg. Sce the performace ad relablty of egeerg systems are drectly affected by the certates of model parameters ad model strctres, t s ecessary to cosder certates wth the comptatoal smlatos the desg process order to esre hgh relablty. Typcal applcatos clde relablty-based desg -4 ad tegrated desg for relablty ad

3 robstess 5-8. De to hgher relablty reqremets of a egeerg system, the accracy of the calclato of relablty or the probablty of falre becomes very crtcal. The tradtoal Mote Carlo Smlato 9 s geerally accrate f a sffcet mber of smlatos are sed. However, for hgh relablty, a excessvely large mber of smlatos are ofte eeded. Ths hgh comptatoal demad s ofte prohbtve for complex egeerg smlatos sch as Fte Elemet Aalyss ad Comptatoal Fld Dyamcs. To overcome the shortcomg of the expesve comptatoal cost, approxmato methods have bee developed 0-8 sch as the Frst Order Relablty Method (FORM) ad the Secod Order Relablty Method (SORM) to redce the mber of fcto evalatos (smlato rs). Compared to Mote Carlo Smlato, both FORM ad SORM are mch more effcet, especally whe the relablty s extremely hgh. Geerally, SORM s more accrate tha FORM bt eeds more comptatos tha FORM. I spte of ts sefless, FORM s ofte ot accrate eogh may cases. Ths aroses a trade-off cosderato betwee the effcecy ad accracy ad leads to the eed for a more accrate relablty aalyss method wthot large comptatoal demad. To meet ths eed, we propose a ew approxmato method for relablty aalyss Frst Order Saddlepot Approxmato (FOSPA). FOSPA s geerally more accrate tha FORM, ad some cases more accrate tha SORM, whle matag the same order of magtde of comptatoal effort as FORM. I the ext secto, we wll trodce the theoretcal ad mathematcal backgrod of ths paper, cldg FORM, SORM, ad the Saddlepot Approxmato. Thereafter, we wll preset the proposed FOSPA detal ad examples to demostrate ts effectveess. The dscsso ad coclso wll be gve more depth at the ed of ths paper.

4 II. Methods for Probablty Evalato Essetally, the evalato of relablty or the probablty of falre by FORM ad SORM s to estmate a probablty, or the Cmlatve Dstrbto Fcto (CDF) of a radom varable whch s a fcto (.e., a lmt-state fcto) of other radom varables (basc varables) provded that the dstrbtos of the later varables are gve. Saddlepot Approxmato 9 was orgally developed for related prpose,.e., to approxmate CDF of statstcs of a radom varable (e.g., mea of radom varable). I the followg dscsso, we wll brefly revew FORM, SORM, ad the Saddlepot Approxmato. Thereafter, we wll dscss the eed to exted Saddlepot Approxmato to relablty aalyss. A. SORM ad FORM The relablty s defed as The probablty of falre s gve by { } R= P g( X ) 0 () { } p = R= P g( X ) < 0 () f If the jot Probablty Desty Fcto (PDF) of X s f x, the probablty of falre s evalated wth the tegral pf = Pg { ( X) < 0} = fx( x) dx (3) g( x) < 0 The lmt-state fcto g( X ) s sally a olear fcto of X ; therefore, the tegrato bodary s olear. Sce the mber of radom varables practcal applcatos s sally hgh, mltdmesoal tegrato s volved. De to these complextes, there s rarely a closed-form solto to Eq. (3); t s also ofte dffclt to evalate the probablty wth 3

5 mercal tegrato methods. Whe the comptato cost of the lmt-state fcto s relatvely cheap, Mote Carlo tegrato s ofte appled to the problem. However, whe Mote Carlo smlato s ot comptatoally affordable, approxmato methods sch as Frst Order Relablty Method (FORM) 0 ad Secod Order Relablty Method (SORM) have become the methods of choce practcal applcatos. These approxmato methods volve the followg steps:. Trasformato of radom varables form ther orgal radom space to a stadard ormal space. Optmzato process to fd the Most Probable Pot (MPP) the desg pot wth the hghest cotrbto to the tegral calclato Eq. (3) 3. Lear ( FORM) or qadratc approxmato ( SORM) of the lmt-state fcto the stadard ormal space at the MPP 4. Calclato of probablty sg ormal dstrbto tal approxmato I the frst step, the orgal radom varables = { X X X } trasformed to a set of radom varables = { U U U } X,,, ( x-space) are U,,, ( -space) whose elemets follow a stadard ormal dstrbto. The trasformato s gve by 0 : = F F x, (4) { ( ) } x The probablty tegrato s the rewrtte as P{ g( X) < 0} = f( ) d (5) g( ) < 0 It s oted that after the trasformato, the tegrato Eq. (5) -space s detcal to the tegrato Eq. (3) x-space wthot ay loss of accracy, ad the cotors of the tegrad f ( ) become cocetrc hyper spheres. The motvato for sg the trasformato 4

6 formlato Eq. (5) stead of Eq. (3) to calclate probablty of falre wll become clear the followg dscsso. I order to make the tegrato calclato Eq.(5) easer, addto to makg the tegrad more reglar (cocetrc hyper crcle cotors), the tegral bodary g( ) s also approxmated learly wth the frst order Taylor expaso as g ( ) g( ) + ( )( ) U U (6) or wth the secod order Taylor expaso as where T g( U) g( ) + ( )( U ) + ( U ) H( )( U ) (7) s the expaso pot. Eq. (6) s sed FORM ad Eq. (7) s sed SORM. To redce the loss of accracy to a mmm degree, t s atral to expad the fcto g( U ) at a pot that has the hghest cotrbto to the probablty tegrato. Therefore, the Most Probable Pot (MPP) s cosdered as the expaso pot. The MPP s the pot o the srface of g ( U ) = 0 for whch PDF of U s at ts maxmm. Maxmzg the jot PDF of U o the srface of g ( U ) = 0, otg that the ( ) s a cocetrc hyper sphere, we have the followg formlato for locatg the MPP, f m sbject to g( ) = 0 (8) where stads for the orm (legth) of a vector. Geometrcally, the MPP s the shortest dstace pot from srface g ( ) = 0 to the org -space ad the mmm dstace (6), the probablty of falre s approxmated by FORM as β = s called "relablty dex". From Eqs. (5) ad 5

7 p = Pg { ( X ) < 0} =Φ ( β ) (9) f From Eqs. (5) ad (7), the Secod Order Relablty Method (SORM) gves the followg approxmato, = ( ) p = Pg { ( X ) < 0} =Φ ( β) + βκ (0) f Geerally, sce the approxmato of lmt-state SORM (see Eq. (7)) s better tha that FORM, the accracy of SORM s hgher tha that of FORM (see Eq. (6)). / B. Saddlepot Approxmato Daels 9 trodced the Saddlepot Approxmato techqe for approxmatg dstrbto of statstcs (e.g., mea) by tegrato of ts desty estmate. Sce Daels work, especally after 980, research ad applcatos ths area have vastly creased ~30. Istead of drectly approxmatg the probablty tegrato Eq., Saddlepot Approxmato ses a Forer verso formla ( a tegral form) to approxmate a Probablty Desty Fcto (PDF). Let Y be a radom varable dstrbted accordg to the desty fcto f(y). The Momet Geeratg Fcto of Y s defed as, + ( ) ad the Cmlat Geeratg Fcto (CGF) of Y s defed as,, ξ y M ( ξ ) = e f y dy () K ( ξ) log{ M( ξ) } = () To restore f(y) from K( ξ ), we ca apply the verse Forer formla 6

8 + x y f ( y) = M( x) e dx p + = exp{ K( x) xy} dx p (3) Usg expoetal power seres expasos to evalate the tegral Eq. (3) ad Hermte polyomals approxmato, Daels 9 arrved to the so-called saddlepot approxmato to f(y) as, f ( y) = exp K t ty p K" ( t) { ( ) } (4) where K"(t) s the secod dervatve of the CGF wth respect to t, where t s the saddlepot correspodg to the solto to the followg eqato K'( t) = y. (5) The cetral dea of dervg Eq. (4) s to choose the tegral path passg throgh the saddlepot of the tegrad, where the tegrad s approxmated. Sce the saddlepot s a extreme pot, the fcto of tegrad falls away rapdly as we move from ths pot. Ths, the flece of eghborg pots o the tegral Eq. (3) s dmshed 9. Iterested reader shold coslt Gots ad Casella 9 for a good explaato of saddlepot approxmato. For the comprehesve methodology, oe ca refer to Ref. 30. Althogh the theory of Saddlepot Approxmato s qte complex, ts se, especally the CDF approxmato verso, s farly straghtforward. The approxmato of CDF of Y by the saddlepot approxmato derved by Lgaa ad Rce 3 s, FY = PY { y} =F ( w) + φ( w) w v (6) or alteratvely by Bardorff-Nelse 4, 7

9 v FY = PY { y} =F w+ log w w (7) where { [ ]} / w = sg() t ty K() t, (8) ad { "( )} / v= t K t (9) whch sg (t) = +, -, or 0, depedg o where t s postve, egatve, or zero. Daels 9 dscsses the exstece ad propertes of the real roots to Eq. (5), po whch the saddlepot approxmato depeds, ad coclded that the saddlepot approxmato ca be sed wheever t les wth the restrcted rage assmed by K'( t ) where Eq. (5) has a qe real root. From Eqs. (6) ad (7) we see that the CDF of Y s approxmated sg stadard ormal dstrbto as show by the se of CDF ad PDF of the stadard ormal dstrbto Eqs. (6) ad (7). Wood, et al. 4 derved a geeral saddlepot formla where the ormal-base dstrbto s replaced by a geeral-base dstrbto. As dcated by may prevos researches (for example, Ref. 9), the saddlepot approxmato yelds extremely good accracy for CDF, especally for the tal area of a dstrbto, whle t reqres oly the process of fdg oe saddlepot wthot ay tegrato. I terms of accracy ad effcecy, there s a great potetal to exted ths techqe to relablty aalyss ad evetally to probablstc egeerg desg. Sce the Saddlepot Approxmato method volves the CGF ad ts dervatves, the major reqremet for applcatos of the techqe s the tractablty (.e., the exstece of a CGF) of the dstrbto of radom varable Y. For a egeerg applcato, Y s a system 8

10 performace (.e., lmt-state fcto) whch s depedet o basc radom varables X,.e, Y = g( X ). The key to apply the saddlepot approxmato to a geeral performace Y s to fd the CGF of Y provded that dstrbtos of X are gve. I ths artcle, a geeral Frst Order Saddlepot Approxmato (FOSPA) method s developed wth the capablty of evalatg the CDF of a lmt-state fcto accrately for ay cotos dstrbtos of basc varables. III. The Frst Order Saddlepot Approxmato Relablty Method The calclato error of probablty of falre of FORM comes from the lear approxmato (Eq. (6)) to the lmt-state state fcto -space. The error of SORM comes from two sorces, oe s the qadratc approxmato (Eq. (7)) to the lmt-state fcto - space ad the other s the approxmato of probablty tegrato for the approxmated lmtstate fcto the qadratc form. For detaled dscsso o the error of FORM ad SORM, please refer to Ref. 3. Eve thogh FORM gves a accrate solto to the probablty tegrato for the approxmated lmt-state fcto (a lear fcto), t s geerally less accrate tha SORM becase of the lear approxmato. The fact that SORM s geerally more accrate tha FORM mples that the accracy of the lmt-state fcto approxmato s very mportat to esre hghly accrate relablty estmato. Thogh the o-ormal to ormal trasformato makes t possble ad easy to calclate the probablty of falre aalytcally (wthot smlatos), the trasformato geerally creases the olearty of a lmt-state fcto becase the trasformato Eq. (4) s olear. For example, f a lmt-state s a lear fcto of o-ormal radom varables, after the trasformato sg Eq. (4), t wll become a olear fcto of stadard ormal radom varables. If the approxmato to the lmt-state fcto at the MPP -space caot captre 9

11 the olearty well, the accracy of the probablty approxmato wll become acceptable. To redce the accracy loss to the mmm extet, we eed to avod or redce the chace of creasg the olearty de to the trasformato of radom varables. I other word, we may cosder approxmatg a lmt-state fcto the orgal x-space or avod ecessary trasformato as mch as possble. To address the aforemetoed cocers, we propose the Frst Order Saddlepot Approxmato Method (FOSPA) to mprove the accracy of relablty aalyss whle matag the same effcecy as FORM. I FOSPA, the lmt-state fcto s learzed the orgal radom space at the so-called Most Lkelhood Pot (MLP) f all the radom varables are tractable, the the Saddlepot Approxmato ca be drectly appled. If some of the radom varables do ot have CGF, they are trasformed to other radom varables that have CGF before the learzato. I the followg, we wll dscss the FOSPA three cases: ) all the radom varables are tractable, ) some of the radom varables are tractable, ad 3) oe of the radom varables s tractable. Strctly speakg, by tractable we mea that a radom varable has a closed-form of CGF; otherwse, we call the radom varable tractable. At the ed of ths secto, we wll preset a geeral procedre ad comptatoal aspect of FOSPA mplemetato. A. Case All the Radom Varables Are Tractable The lmt-state fcto g( X ) s frst learzed at some pot x, amely, the terval bodary of Eq. (3) s approxmated by a hyper plae at the expaso pot x. Smlar to the cocept of the MPP, x s chose sch that the jot PDF of X s at ts maxmm vale o the bodary of the lmt-state g ( X ) = 0; ths pot s called the Most Lkelhood Pot (MLP). I 0

12 other words, the MLP s the pot o the bodary g ( X ) = 0, whch has the hghest cotrbto to the probablty of falre p = f ( x) dx. f g( x) < 0 x The followg model s sed to detfy the MLP x, max f( x) x = sbject to g( x) = 0 (0) The lear form of g( X ) at x s g( X) ( x )( X x ) () The the CGF of g( X ) s gve by Kt () = K() t () = where K() t s the CGF of ( )( ) x X x. The frst ad secod dervatves of Kt () are ' '() K () t = K t = (3) ad '' "() () = K t = K t (4) respectvely. eqato Accordg to Eq. (5), the saddlepot t s detfed by the solto to the followg ' '() = () = 0 = K t y K t y (5)

13 Oce the saddlepot t s detfed, the probablty Pg { ( X ) y} ca be calclated from Eq. (6) wth the followg eqatos ad / w = sg() t { [ ty K() t ]} = sg() t ty K () t =, (6) / / '' { "()} v= t K t = t K () t (7) = The CGFs of some commo dstrbtos are lsted Table. For more detals, please refer to Ref. 3. Table CGF of Some Dstrbtos Dstrbto PDF CGF ( x µ ) σ Normal f( x) = e Kt () = µ t+ σ t πσ t Expoetal f( x) = βe β x Kt ( ) = l β Uform f( x) ( ) l bt at Kt = e e l b a l t b a x µ x µ Type I Extreme σ σ f( x) = e exp e Vale (Gmbel) Kt ( ) = µ t+ log Γ( σt) σ x / χ f( x) = x e Kt ( ) = l Γ / ( t) ( / ) Gamma = ( ) ( ) ( ) β ( α) α α βx f( x) x e = Kt ( ) = α ( ) ( ) Γ { l β β t} / B. Case Some of the Radom Varables Are Tractable Some radom varables may ot have a closed-form (.e., tractable) CGF, for example, Webll dstrbto ad logormal dstrbto. There are two ways to approach tractable CGF: ) Approxmate the CGF sg polyomal expasos 33 or ) trasform the radom

14 varable to aother radom varable wth tractable CGF. The later approach s adopted ths paper for the prpose of smplcty. Oe possble trasformato s smlar to the oe sed FORM ad SORM as show Eq. (4) whch s the trasformato from a radom varable wth tractable CGF to a stadard ormal varable. I geeral, ay dstrbto wth tractable CGF ca be sed for the trasformato. Let the set of varables whch have tractable CGF be X t = { set of varables wthot tractable CGF be X ~t = { t X ; =,,, t } ad the ~t X j ; j =,,, ~t }. After the o-ormal ormal trasformato, X ~t s trasformed to a set of stadard ormal varables U = { U ; j =,,, ~t }. The, the formlato for searchg the MLP { t, } x becomes j t ~ t t max ( ) ( ) t f x f j x, = j= t sbject to g( x, ) = 0 (8) t After learzato, the lmt-state fcto at the MLP {, } = t x, j= j t x, x s gve by ~ ( ) (, ) = t t t g t t g g X q X U t ( X x ) + ( U j j) (9) x Becase the lmt-state Eq. (9) s a lear combato of tractable radom varables, the saddlepot approxmato method Eqs. () - (7) ca be appled cojcto wth Eq. (9) to evalate the probablty of falre. C. Case 3 Noe of the Radom Varable Is Tractable Whe all radom varables are tractable, they mst be trasformed to selected tractable radom varables sch as stadard ormal varables. If all the radom varables are 3

15 trasformed to stadard ormal varables, after the trasformato, the model of searchg the MLP becomes max φ( ) = sbject to g( ) = 0 (30) whch s eqvalet to the model Eq. (8) for the MPP search. Therefore, the solto to the model Eq. (30) s exactly the MPP defed the model (8). At the MLP, the learzato of the lmt-state fcto s gve by g g X U (3) ( ) = ( ) = Appedx shows that the calclated probablty of falre from Saddlepot Approxmato based o Eq. (3) s the same reslt as that of FORM. I other words, FORM s detcal to FOSPA whe all radom varables are trasformed to stadard ormal varables. Therefore, FORM s a specal case of FOSPA. D. The Geeral Procedre ad Comptato Implemetato of FOSPA The procedre of FOSPA s smmarzed as follows. a. Determe whether a radom varable has tractable or tractable CGF ad form two sets of radom varables, oe set wth tractable CGF, X t, ad the other set wthot tractable CGF, X ~t. Trasform the later set to stadard ormal varables U. t b. Solve the model Eq. (8) to detfy the MLP {, } x. c. Learze the lmt-state fcto at the MLP as show Eq. (9). d. Formlate the saddlepot eqato ad solve t to obta the saddlepot t. e. Use Eqs. (5)-(9) to fd the probablty of falre. 4

16 It s oted that f all the radom varables are tractable, X ~t wll be a empty set ad the problem belogs to Case ad f oe of the radom varables s tractable, X t wll be a empty set ad the problem belogs to Case 3 where the same reslt as FORM wll be obtaed. To make the mercal comptato process of FOSPA more stable, several practcal measres may be cosdered ad some of them are brefly dscssed here. The varables Eqs. (0) ad (8) for MLP search are ormalzed by the meas ad stadard devatos of the radom varables. Ths ormalzato makes the desg varables the same scales. Note that ths ormalzato s a lear trasformato ad wll ot affect the olearty of the lmt-state fcto bt wll help the covergece of the teratve process of fdg the MLP. To avod the objectve fctos of MLP search Eqs (0) ad (8) becomg too small, oe may choose to se the atral logarthm of the objectve fctos. To avod sglartes Eqs. (8) ad (9), oe may se the reverse sg of the lmt-sate fcto whe a sqare root of a egatve vale occrs. Cosderg that there s a strog eed to mmze the mber of lmt-state evalatos so that the techqe s practcal for comptatoally expesve egeerg smlato models (e.g., Fte Elemet Aalyss ad Comptatoal Fld Dyamcs), we compare the effcecy of the methods by cotg the mber of fcto evalatos of lmt-state fcto. Sce FOSPA ses smlar optmzato formlato to fd the MLP as FORM for the MPP, ad ses less olear costrat fctos, the comptatoal effort (measred by the mber of fcto evalatos) of FOSPA s less tha or at most the same as that of FORM. 5

17 IV. Nmercal Examples I ths secto, two examples are sed to demostrate the effectveess of the proposed method. The frst example s assocated wth a lear lmt-state fcto ad the other wth a olear lmt-state fcto. We wll compare the accracy ad effcecy amog FOSPA, FORM, ad SORM. If o theoretcal solto exsts, we wll se the reslt of Mote Carlo smlato wth a relatvely large sample sze as a referece. I the followg examples, the frst order ad secod order dervatves are evalated mercally wth fte dfferece method. Becase of ths fte dfferece calclato, SORM, whch reqres secod order dervatve formato, has a heret effcecy terms of the mber of lmt-state evalatos. A. Example : Lear Lmt-State Fcto follows A lear lmt-state fcto s gve by a sm of depedet radom varables as g( X ) = ( + a ) X (3) = where a s a costat ad Case : all radom varables are tractable X are depedet radom varables. Let each of the radom varables follows a stadard expoetal dstrbto wth CDF F( x ) = exp( x ) (33) For ths specfc example, the theoretcal solto ca be fod. The probablty of falre pf = Pg { ( X ) < 0} s lsted Table ad depcted Fg. for =. 6

18 Table Probablty p = Pg { ( X) < 0} for = f a FORM SORM FOSPA Exact Isert Fg. here Fg. Probablty of falre whe = Fg. shows the probablty of falre for dfferet vales of a. The probablty of falre chages the rage roghly betwee 0.4 ad 0 as a vares. The crves of FOSPA ad the exact solto almost overlap each other over the whole rage of the probablty. Ths dcates that FOSPA s evely good over the rage of probablty of falre. SORM s more accrate tha FORM, bt whe the probablty of falre s hgh (for example 0.4), SORM s ot accrate as show Fg.. The accracy of solto from SORM creases as the probablty of falre becomes lower. Ths pheomeo coforms to the fact that SORM s oly accrate at the tal of a dstrbto de to ts asymptotc approxmato to the probablty tegrato. I ths example wth lear lmt-state fcto ad tractable CGF radom varables, the reslts show that FOSPA s the most accrate method. Fg. shows that whe a = 3.5, the orgal lear lmt-state fcto becomes hghly olear after the trasformato to stadard ormal dstrbtos reqred by both FORM ad SORM. The lear approxmato of FORM s far away from the trasformed olear 7

19 lmt-state fcto -space ad eve the qadratc approxmato SORM caot very well captre the olearty of the trasformed lmt-state fcto. Therefore, both FORM ad SORM are ot as accrate as FOSPA ths example. Sce FOSPA ses the orgal lear lmt-state fcto wthot the crease of olearty ad Saddlepot Approxmato reslts a hgh accracy approxmato. That s, the overall accracy of FOSPA s speror to FORM ad SORM. Isert Fg. here Fg. Lmt-state fcto x ad spaces The reslt for hgher dmeso wth =0 s lsted Table 3 ad depcted Fg. 3. The reslt stll shows that the FOSPA s mch more accrate tha FORM ad SORM. The related detaled eqatos sed ths example are gve Appedx. Table 3 Probablty p = Pg { ( X) < 0} for =0 f a FORM SORM FOSPA Exact e-6.96e-4.9e-4.9e e-7.05e e e e e-5.47e-5.47e-5 Isert Fg. 3 here Fg. 3 Probablty of falre whe =0 8

20 Case : Some radom varables are ot tractable I the followg case, we choose X 3 to follow a Webll dstrbto whch does ot have a closed-form CGF. The dstrbto formato s show Table 4. Table 4 Iformato of radom varables Varable Parameter Parameter Dstrbto X. Expoetal a X. Expoetal X 3.5 Webll b a For a expoetal dstrbto, Parameter s the mea. b For a Webll dstrbto, Parameters ad are parameters a ad b, respectvely, the PDF of a Webll dstrbto b b ax f( x) abx e =. Sce X 3 s ot tractable, t s trasformed to a stadard ormal varable before the Saddlepot Approxmato s appled. Mote Carlo Smlato (MCS) s employed ad ts reslt s sed as a referece for comparso of the accracy of other methods. The mber of smlatos the Mote Carlo s 0 6. The calclated probablty of falre s show Table 5. It s oted that FOSPA s the most accrate method ad SORM s more accrate tha FORM. Wth FORM ad SORM, the trasformato of {,, } {,, } 3 X X X to a stadard ormal varable 3 U U U makes the orgal lear lmt-state fcto become olear terms of {,, } U U U. O the other had, FOSPA oly volves the trasformato of X 3 to a stadard 3 ormal varable U 3. That s, the orgal lmt-state s oly olear terms of U 3 ad the remag terms of X ad X are kept lear. As a reslt of the mmm crease of olearty of the lmt state, FOSPA s more accrate tha FORM ad SORM. The mbers of fcto evalatos sed by FOSPA, FORM, ad SORM (cldg fte dfferece calclato ad teratos to fd MLP/MPP) are 5, 37, ad 57, respectvely. I ths case, the mmm crease 9

21 of olearty also helps FOSPA to be the most effcet method for fdg the MLP whle SORM s the least effcet method for ths specfc case. Table 5 Probablty of falre for case a FORM SORM FOSPA MCS Pg { ( X ) < 0} N a a Nmber of fcto evalatos Case 3: All the radom varables are ot tractable I the followg case, all radom varables follow Webll dstrbtos as show Table 6. Sce a Webll dstrbto does ot have tractable CGF, the trasformato from { X, X, X } to a stadard ormal varable {,, } 3 U U U s reqred. 3 Table 6 Iformato of radom varables Varable Parameter Parameter Dstrbto X.5 Webll X.5 Webll X 3.5 Webll As expected, FOSPA has the same reslt as FORM as show Table 7. The MLP from FOSPA ad the MPP from the FORM are detcal,.e., MLP MLP MLP MPP MPP MPP { x, x, x3 } { x, x, x3 } {.887,.887,.887 } = =. I ths case SORM s the most accrate method becase the secod order approxmato SORM provdes a better approxmato to the lmt-state fcto -space. Table 7 Probablty Pg { ( X ) < 0} for case 3 a FORM SORM FOSPA MCS Pg { ( X ) < 0} N

22 B. Example : Nolear Lmt-State Fcto Cosder the lmt-state fcto of a shaft a speed redcer defed as 3 F L g( ) S T π D 6 X = + (34) 3 where S s the materal stregth, D s the dameter of the shaft, F s the exteral force, T s the exteral torqe, ad L s the legth of the shaft. The lmt-state fcto represets the dfferece betwee the stregth ad the maxmm stress. The varable formato s gve Table 8. Table 8 Dstrbtos of Radom Varable Varables Parameter Parameter Dstrbto Dameter D 39 mm 0. mm Normal a Spa L 400 mm 0. mm Normal Exteral force F 500 N 350 N Gmbel b Torqe T 50 Nm 35 Nm Normal Stregth S 70 MPa 80 MPa Uform c a For ormal dstrbto, Parameters ad are mea ad stadard devato respectvely. b For Gmbel dstrbto, Parameters ad are mea ad stadard devato respectvely. c For a form dstrbto, Parameters ad are lower ad pper bods respectvely Ths problem belogs to Case where all the radom varables are tractable. The reslts of probablty of falre as compared wth MCS (0 6 smlatos) are show Table 9. Refereced to MCS, FOSPA geerate the most accrate solto wth the least comptatoal demad. Table 9 Probablty Pg { ( X ) < 0} FORM SORM FOSPA MCS Pg { ( X ) < 0} N The above reslt dcates that FOSPA provdes accrate CDF estmate at the rght tal of the dstrbto of the lmt-state fcto. To llstrate the accracy of FOSPA over the whole

23 dstrbto rage, the CDF of the lmt-state fcto at the left tal ad ear the meda are also calclated ad gve Tables 0 ad, respectvely. From Tables 9 ad 0, t s oted that FOSPA s also speror to FORM ad SORM at both tals terms of accracy ad effcecy. Table shows that FOSPA also prodces reasoably accrate CDF estmate arod the meda of the dstrbto whle both FORM ad SORM have very large errors. Ths example demostrates that FOSPA s evely accrate over the whole dstrbto ad therefore beefcal for geeratg a complete dstrbto of a performace (lmt-state fcto). Table 0 Probablty at the tals of dstrbto FORM SORM FOSPA MCS 7 Pg { ( X ) < } N Table Probablty ear the meda FORM SORM FOSPA MCS 7 Pg { ( X ) <.48 0 } N V. Dscsso I ths secto, we smmarze the proposed FOSPA method wth detaled dscsso o ts accracy ad effcecy comparso to FORM ad SORM. Based o the dscsso, recommedatos for selectg the relablty aalyss methods der varos crcmstaces wll be provded the ext secto. Saddlepot approxmato s a accrate method for estmatg CDF of a radom varable f ts CGF s kow. The cetral dea of the proposed FOSPA s to approxmate the CGF of a geeral lmt-state fcto throgh learzato of lmt-sate fcto. The learzato s

24 codcted at the Most Lkelhood Pot (MLP) the pot where the jot PDF of the radom varables s at ts maxmm vale for a gve lmt-state vale. If a radom varable does ot have a closed form CGF (tractable), t s trasformed to aother radom varable wth a tractable CGF before the learzato. I ths paper, a tractable radom varable s trasformed to a stadard ormal varable. It s worthwhle otg that other types of radom varables wth tractable CGF ca also be sed for the trasformato. Oce the lmt-state fcto s the form of a lear combato of tractable varables, the CGF of the lmt-state fcto s easly obtaed. The saddlepot s the solto to the eqato of the frst dervatve of the CGF eqal to the lmt-state vale. Thereafter, the saddlepot approxmato solto s sed to approxmate the probablty of falre or the relablty. I cotrast to FORM that codcts learzato of the trasformed stadard ormal space (whch mposes olear trasformatos), FOSPA learzes the lmt-state fcto the orgal space of tractable radom varables. As a coseqece to mmzg radom varable trasformato, FOSPA redces the chace of creasg the olearty of the lmt-state fcto. Therefore, the learzato of the lmt-state fcto FOSPA gves a more accrate approxmato tha that of FORM. Geerally, FOSPA s more accrate tha FORM except the followg cases where they are eqvalet ) all radom varables have tractable CGF ad they are trasformed to stadard ormal varables; ) all tractable radom varables are ormally dstrbted ad all the tractable radom varables are trasformed to stadard ormal varables; ad 3) all radom varables are ormally dstrbted. I the aforemetoed three cases, the MLP from FOSPA s detcal to the MPP from FORM ad, therefore, both methods have the same accracy. I ths sese, FORM s a specal case of FOSPA. 3

25 It s geerally recogzed that SORM s more accrate tha FORM althogh there are few coterexamples; however, there s o sch drect coclso abot the comparso betwee FOSPA ad SORM terms of ther accracy. Oe method s more accrate tha the other depedg o the problem der cosderato. Geerally speakg, whe the lmt-state fcto s less olear terms of orgal radom varables or the o-ormal to ormal trasformato creases olearty of the lmt-state fcto sgfcatly, FOSPA may have a hgher accracy tha SORM. The search of the MLP eeds a teratve process where the lmt-state fcto s evalated repeatedly. Sce searchg a MLP s a smlar task as searchg a MPP, t s expected that FOSPA has at most the same order of magtde of comptatoal demad as that of FORM. I may cases, searchg the MLP s more effcet tha searchg the MPP sce the costrat fcto the optmzato model of the MLP s more lear tha that of the MPP. It shold be oted that the search of the saddlepot does ot cosme ay lmt-state fcto evalatos. Becase SORM eeds the secod order dervatve of a lmt-sate fcto, t s geerally mch less effcet whe the dervatve s evalated mercally. Cosderg the same comptatoal effort ad hgher accracy of FOSPA compared to FORM, oe may choose FOSPA for a relablty aalyss. Whe hgher accracy s eeded, oe shold also cosder the fact that depedg o the learty of the lmt-state ad radom dstrbto, SORM s ot always better tha FOSPA terms of accracy. The comptatoal effcecy, accracy, ad mplemetato smplcty of the proposed method make t attractve for real world relablty aalyss. Oe of the athors has extesvely appled the proposed method to varos comptatoally tesve smlato models sed atomotve ege desg (for example, Hoffma et al. 34 ). As dcated the Example, FOSPA ca also be sed 4

26 to geerate accrate CDF assocated wth a rage of lmt-state vales. Ths s accomplshed by emeratg the lmt-state vales, perform learzato at all MLP's assocated wth the lmt state vales, ad the calclate the probablty sg Eq. (6) or (7). Usg ths approach, FOSPA ca accrately calclate the CDF at both tals as well as arod the meda (or mea) of a dstrbto. To frther mprove the accracy, the Secod Order Saddlepot Approxmato ca be cosdered ad the key to the ew developmet s how to detfy the CGF of secod order approxmato of a lmt-state fcto. VI. Coclso I smmary, the proposed Frst Order Saddlepot Approxmato method for relablty aalyss s a attractve alteratve to the exstg relablty aalyss methods FORM ad SORM. Oe may cosder the followg facts whe selectg the relablty methods: FORM s a specal case of FOSPA ad the later s more accrate tha the former wth less or at most the same comptatoal effort. If the lmt-state fcto the orgal space s less olear tha that of stadard ormal trasformed space, FOSPA may be more accrate tha SORM. SORM s less effcet (.e., t reqres more fcto evalatos) tha FOSPA ad FORM. Appedx : FORM s a Specal Case of FOSPA If oe of the radom varables s tractable, FORM prodces the same reslt as FOSPA whe stadard ormal trasformato s employed. After a lmt-state fcto g( X ) s approxmated by a lear fcto Eq. (3), the CGF of q( U ) s gve by 5

27 6 () g g K t t t = = = + (A) ad ts dervatve s ' () g g K t t = = = + (A) The saddlepot s obtaed from ' () 0 K t = g t g = = = (A3) The CGF at the saddlepot becomes () g Kt g = = = (A4) ad ts secod order dervatve wth respect to the saddlepot s gve by '' () g K t = = (A5) Sbstttg (A4) ad (A5) to Eqs. (8) ad (9) yelds / g w g β = = = = (A6) ad

28 g g / = g = / = v = = = β g g = = (A7) respectvely. Combg Eqs. (A6), (A7), ad Eq. (6) reslts whch s the same reslt of FORM. { } P g < 0 =Φ ( β ) (A8) ) FOSPA Appedx : Case of Example The CGF of the lmt-state fcto of Example s gve by ad ts dervatves are Kt ( ) = l( t) ( + a t ) (A9) K ' () t = ( + a ) t (A0) ad '' ( + a ) K () t = (A) respectvely. Solvg K ' () t = 0prodces the saddlepot a t = > 0 + a (A) Combg Eqs. (A9) throgh (A), we obta 7

29 / w = sg() t { ( ty K() t } = l( ) + a + a / (A3) { '' ()} / v = t K t = a (A4) ad the probablty of falre v / a pf =Φ w+ l( ) =Φ { c+ a } + l( ) / / w w { c+ a } { c+ a } (A5) where c = l + a (A6) ) FORM The MPP { } =, =,, s gve by a exp + = Φ (A7) ad the relablty dex s calclated by 3) SORM The, the probablty of falre s p f β = (A8) ( β ) ( ) The probablty of falre by SORM s gve by =Φ =Φ (A9) Φ ( ) f( ) Φ ( ) p f =Φ ( ) + (A0) Ackowledgmet 8

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