A New Inverse Reliability Analysis Method Using MPP-Based Dimension Reduction Method (DRM)

Transcription

1 roceedng of the ASME 007 Internatonal Degn Engneerng Techncal Conference & Computer and Informaton n Engneerng Conference IDETC/CIE 007 September 4-7, 007, La Vega, eada, USA DETC A ew Inere Relablty Analy Method Ung M-Baed Dmenon Reducton Method () Ikjn Lee, Kyung K. Cho, Lu Du Department of Mechancal & Indutral Engneerng, College of Engneerng The Unerty of Iowa, Iowa Cty, IA 54, USA lee@engneerng.uowa.edu, kkcho@engneerng.uowa.edu, ludu@engneerng.uowa.edu Dad Gorch AMSTA-TR- (MS 63) U.S. Army TARDEC Warren, MI , USA gorchd@tacom.army.ml ABSTRACT There are two commonly ued relablty analy method of analytcal method: lnear approxmaton - rt Order Relablty Method (ORM), and quadratc approxmaton - Second Order Relablty Method (SORM), of the performance functon. The relablty analy ung ORM could be acceptable for mldly nonlnear performance functon, wherea the relablty analy ung SORM uually neceary for hghly nonlnear performance functon of mult-arable. Een though the relablty analy ung SORM may be accurate, t not derable to ue SORM for probablty of falure calculaton nce SORM requre the econd-order entte. Moreoer, the SORM-baed nere relablty analy ery dffcult to deelop. Th paper propoe a method that can be ued for multdmenonal hghly nonlnear ytem to yeld ery accurate probablty of falure calculaton wthout requrng the econd order entte. or th purpoe, the unarate dmenon reducton method () ued. A three-tep computatonal proce propoed to carry out the nere relablty analy: contrant hft, relablty ndex () update, and the mot probable pont (M) approxmaton method. Ung the three tep, a new -baed M obtaned, whch compute the probablty of falure of the performance functon more accurately than ORM and more effcently than SORM. KEYWORDS Inere Relablty Analy; Dmenon Reducton Method (); rt Order Relablty Method (ORM); Second Order Relablty Method (SORM); Sytem Leel Relablty.. ITRODUCTIO In recent year, there hae been arou attempt to deelop enhanced relablty analy method to accurately compute the probablty of falure of a performance functon. The mot popular relablty analy method are () analytcal method and () mulaton or amplng method. The analytcal method nclude the M-baed method and D approxmaton method. The M-baed method nclude the rt Order Relablty Method (ORM) [,] and the Second Order Relablty Method (SORM) [3,4]. ORM or SORM compute the probablty of falure by approxmatng the performance functon G ( X) ung the frt or econd order Taylor ere expanon at the mot probable pont (M). Snce the ORM or SORM-baed method requre M earch, the entty analy uually ued for the method. When the entte are not aalable, the repone urface method can be ued [5,6]. The probablty denty functon (D) approxmaton method [7-9] ealuate D of the performance functon by aumng a general dtrbuton type and then, ung the approxmated D, the method ealuate the probablty of falure of the performance functon. The mulaton or amplng method, uch a Monte Carlo mulaton () [0], mportance amplng method [], and Latn hypercube amplng method [], can be readly ued for the probablty of falure calculaton nce thee method do not requre any analytc deraton.

2 Among thee method, the M-baed method tll ery popular. Howeer, the relablty analy ung ORM could be ery well erroneou f the mult-dmenonal performance functon are hghly nonlnear. Th becaue ORM approxmate the performance functon ung a lnear functon, whch cannot reflect complcty of nonlnear and mult-dmenonal functon. Although the relablty analy ung SORM may be accurate, t not derable to ue nce SORM requre the econd-order entte, whch are dffcult and ery expene to obtan n practcal engneerng problem. The accuracy of the repone urface method tll challengng, epecally for the hghly nonlnear problem that requre hgh relablty, een though the method could be effcent. The mulaton or amplng-baed method could be accurate, howeer they requre a ery large number of functon ealuaton. The D approxmaton can ge accurate reult only f the probablty dtrbuton can be repreented by the aumed dtrbuton functon, and moreoer the method hould be combned wth the repone urface method for the degn optmzaton [9], whch may hae accuracy problem. The dmenon reducton method () [3,4] ha been recently propoed to repreent a mult-dmenonal functon ung the um of lower dmenonal functon. Becaue of t wde applcablty, ha been appled to robut degn [5,6], relablty-baed degn optmzaton (RBDO) [7], and D approxmaton [8,9]. or the robut degn, mean-baed ued to calculate the tattcal moment of the performance functon. Th moment calculaton ung mean-baed alo ued to approxmate D of the performance functon for relablty analy [9]. The relablty analy ung D approxmaton method and mean-baed requre accurate moment calculaton, D approxmaton, and repone urface generaton, whch could be lmtaton of the method. Mbaed propoed for the relablty analy and RBDO, whch more accurate than mean-baed for the relablty analy [7]. Howeer, M-baed ha not been ued for the nere relablty analy, whch eental for the performance meaure approach (MA) of RBDO [8]. In th paper, the probablty of falure calculaton ung unarate propoed to deelop an enhanced nere analy method that accurate for mult-dmenonal hghly nonlnear problem. Snce the nere relablty analy ung M-baed tll requre M earch, t can be categorzed a an M-baed method. A three-tep computatonal proce propoed to accurately and effcently carry out the nere relablty analy ung : contrant hft, relablty ndex () update, and M update. or the effcent nere relablty analy, the enhanced hybrd mean alue (HMV+) [9] ued n th paper. The probablty of falure etmated by compared wth the probablty of falure etmated by ORM and SORM. alo ued for the benchmark purpoe. Thee comparon llutrate that the -baed relablty analy can decrbe the probablty of falure of the performance functon more accurately than the ORM-baed relablty analy and more effcently than the SORM-baed relablty analy.. ROBABILITY O AILURE USIG. M-Baed Dmenon Reducton Method The [3,4,7,0] a newly deeloped technque to accurately and effcently approxmate a mult-dmenonal ntegral. There are eeral method dependng on the leel of dmenon reducton: () unarate dmenon reducton, whch an addte decompoton of - dmenonal performance functon nto one-dmenonal functon; () barate dmenon reducton, whch an addte decompoton of -dmenonal performance functon nto at mot two-dmenonal functon; (3) multarate dmenon reducton, whch an addte decompoton of -dmenonal performance functon nto at mot S-dmenonal functon, where S. In th paper, the unarate ued for computaton of probablty of falure becaue of t mplcty and effcency. In the unarate, any -dmenonal performance functon G( X) can be addtely decompoed nto onedmenonal functon at the M of the random ector X a G( X) G ( X) Gx (,, x, X, x,, x ) ( ) G( x ) = + T where x ={ x, x,, x } the ORM-baed M of the performance functon G ( X) and the number of random arable. or example, f G( X) = G ( X, X) wth =, then the unarate addte decompoton of G ( X) X G( X) G( ) G( X, x ) + G( x, X ) G( x, x ) A hown n Eq. (), the unarate approxmate the performance functon G ( X) ung the um of onedmenonal functon. Conequently, f there are off-dagonal or mxed term n the performance functon G ( X), then there hould be ome error that reult from approxmatng offdagonal term ung um of one-dmenonal functon. To reduce th error, the barate or multarate can be ued [3,4].. Rotated Standard ormal V-Space Conder a performance functon G ( X) that depend on T X={ X, X,, X } and whoe M denoted by T x ={ x, x,, x }. Snce the relablty analy performed n the tandard normal U -pace obtaned ung Roenblatt tranformaton [], the M n U -pace T denoted by u ={ u, u,, u } and defned a the cloet pont on the lmt tate functon G ( U) = 0 to the orgn n U-pace (mean n X-pace ). The dtance from the M to the orgn commonly called the Haofer-Lnd relablty ndex [,] and denoted by. () ()

3 To obtan the rotated tandard normal V-pace U-pace, contruct an orthonormal matrx from R whoe th column u α,.e., R = [ R α ], where T R atfe ( α ) R = 0 and the number of random arable [7,0]. Ung an orthonormal tranformaton u = R, can repreent the rotated tandard T normal V-pace wth the M = {0,, 0, }. The orthonormal matrx R can be found, for example, by Gram- Schmdt orthogonalzaton. Howeer, the orthogonal matrx R not unquely determned. gure how U-pace and V-pace for =..3. robablty of alure Ung ORM and SORM To calculate the probablty of falure of the performance functon G( x) ung ORM and SORM, t neceary to fnd M on the lmt tate functon n the tandard normal U-pace a hown n g.. The M can be found by olng the followng optmzaton problem to mnmze u ubject to G( u) = 0 After fndng the M, the Haofer-Lnd relablty ndex can be obtaned by meaurng the dtance between the M and the orgn. Ung the relablty ndex, ORM can approxmate the probablty of falure ung a lnear approxmaton of the performance functon a (4) ORM Φ( ) (5) u where Φ ( u) = exp ξ dξ π the cumulate dtrbuton functon (CD) of the tandard Gauan random arable. The M obtaned by olng Eq. (4) alo ued for the probablty of falure calculaton ung SORM. Ung a quadratc approxmaton of the performance functon n U-pace and the rotatonal tranformaton from U-pace to V-pace explaned n Secton., the probablty of falure can be obtaned ung SORM a [3,4,0] SORM Φ Φ φ ( ) ( ) I A (6) ( ) gure. Standard ormal U-Space and Rotated Standard ormal V-Space [7].3 ORM and SORM A relablty analy ental calculaton of probablty of falure, denoted by, whch defned ung a multdmenonal ntegral [] [ G( X) 0] f ( x) dx > = X G ( X) > 0 T where X={ X, X,, X } an -dmenonal random ector, G( X) the performance functon uch that G ( X) > 0 defned a falure, and fx ( x) a jont D of X. Snce ealuaton of Eq. (3) ery dffcult or ometme mpoble to obtan n real engneerng applcaton, many method hae been uggeted to approxmate Eq. (3): repreentately, ORM and SORM. In th ecton, ORM and SORM to approxmate Eq. (3) are brefly reewed and a new method to approxmate Eq. (3) ung explaned and compared wth ORM and SORM n Secton 4.4. (3) A A T where A = = R HR, H the Hean A G A U matrx ealuated at the M, R the rotaton matrx uch that u = R, and φ() the D of a tandard Gauan random arable..3. Error n ORM-baed Relablty Analy Although ORM ha been wdely ued for the relablty analy and nere relablty analy due to t mplcty and effcency, ORM could be erroneou f the multdmenonal performance functon hghly nonlnear a hown n the followng example. Conder G( X) = a X + = X where X ~ (0,) for =,. The performance functon ha an M at x = {0,, 0, } T and the probablty of falure by ORM ORM Φ ( ) regardle of a and. If =, then the probablty of falure by ORM become (7)

4 ORM Φ( ) =.750%. Th probablty of falure can be compared wth the reult obtaned ung for dfferent a and, repectely, a hown n Table and. Table. robablty of alure by When = a = 0. a = 0.5 a =.0 a =.0.595%.89% 0.945% % Table. robablty of alure by When a = 0. = = 3 = 5 = 0 MC S.595%.309% 0.546% % rom Table and, t can be een that the probablty of falure obtaned ung ORM ha gnfcant error when a performance functon hghly nonlnear (.e. larger a ) and epecally for a hgh dmenonal problem (.e. larger ). Thee error can be mproed by SORM nce SORM ue a quadratc approxmaton of the performance functon. Howeer, Hean matrx requred to calculate the probablty of falure n Eq. (6) ung SORM, whch ery dffcult or ometme mpoble to accurately etmate n real engneerng applcaton. or th reaon, SORM ha been lmtedly appled n engneerng applcaton..4. robablty of alure Calculaton Ung.4.. Inere Relablty Analy The relablty analy preented n Secton.3. called Relablty Index Approach (RIA) nce t fnd the relablty ndex ung Eq. (4). The adantage of RIA that the probablty of falure for the performance functon can be calculated at a gen degn, for example, ung Eq. (5) and (6). Howeer, t well known that the nere relablty analy n erformance Meaure Approach (MA) [8] much more effcent than relablty analy n RIA. MA doe not calculate the probablty of falure drectly. Intead, MA judge whether or not a gen degn atfe the probabltc contrant wth a gen target relablty ndex t by olng the followng optmzaton problem maxmze G( u) ubject to u = Snce Eq. (8) the nere problem of Eq. (4), th called the nere relablty analy. The optmum pont of Eq. (8) alo called M and denoted by u. If the contrant functon alue at the M, G( u ), le than zero, then the probabltc contrant atfed wth the gen target relablty t and target probablty of falure Φ( t ) by ORM. The nere relablty analy ung SORM much more dffcult and ha not been deeloped yet. Moreoer, t requre the econd order entty..4. robablty of alure Calculaton Ung and Inere Relablty Analy t (8) In the rotated tandard normal V-pace, the probablty of falure n Eq. (3) can be rewrtten a [ G ( V ) 0] f ( ) d > = V G G G G( V) > 0 Snce the nere relablty analy doe not calculate the probablty of falure, a contrant hft concept ntroduced for the probablty of falure calculaton uch that ( ) ( ) ( ) where G ( ) a hfted performance functon and T = {0,, 0, } the ORM-baed M n V -pace wth a gen relablty ndex. By applyng the M-baed explaned n Secton. to G ( ), G ( ) can be approxmated at the M a G G = (9) (0) G ( ) G ( ) ( ) ( ) ( ) () where G( ) G(,,,, +,, ). By the defnton of G ( ) n Eq. (0), G ( ) zero, thu, we obtan ( ) G G ( ) G ( ) = G ( ) + G ( ) = = () Due to the rotatonal tranformaton of the coordnate a hown n g., the th unarate component G( ) can be lnearly approxmated [0]. Th lnear aumpton of G( ) along -ax alo ued for the probablty of falure calculaton n SORM [3, 4]. Ung the lnear aumpton, Eq. (9) can be wrtten a 0 [ G ( V ) > 0] [ b + bv + G ( V ) > 0] (3) Snce functon alue and gradent at the M are ued durng the nere relablty analy, b 0 + bv can be rewrtten ung the frt order Taylor ere expanon at the M a = G ( ) G ( ) b b G ( ) ( ) where 0 + = + = G( ) = ( ) = = = = = = (4) G ( ) G( ) G ( ) 0, b, and =. Inertng Eq. (4) nto Eq. (3) yeld

5 = [ b ( V ) + G ( V ) > 0] = bv [ > b G ( V)] = (5) Snce the gradent b at M alway pote due to maxmzaton n Eq. (8), Eq. (5) can be rewrtten, by ddng both de by b and ung the ymmetry of the tandard normal dtrbuton nce V ~ (0,), a V G V )] [ > ( b = = V [ < + b G V = = E[ Φ( + G ( V))] b = ( )] (6) where E the expectaton operator. Snce Eq. (6) a dmenonal ntegraton, Eq. (6) can be further mplfed by applyng to the ntegrand of Eq. (6) a where b G ( ) Φ ( + ) φ( d b ) = Φ ( ) T T G( u) ( u ) G( u) ( α ). (7) = = u u Detaled u=u deraton of Eq. (7) can be found n Ref. 7. Ung the moment-baed ntegraton rule (MBIR) [3], whch mlar to Gauan quadrature [4], Eq. (7) further approxmated a where G ( ) ( ) n j wjφ + = j = b Φ ( ) j are quadrature pont, u=u (8) are weght, and n the number of quadrature pont and weght. Snce are tandard normal random arable, quadrature pont and weght n Table 3 can be ued to calculate Eq. (8) [4]. Table 3. Gauan Quadrature ont and Weght n Quadrature ont Weght ± ± ± w j or a pecal cae of Eq. (8), aume n =, whch mean one quadrature pont and weght, then Eq. (8) can be wrtten a G ( ) w Φ ( + ) ( ) b Φ = = = =Φ( ) Φ ( ) Φ ( ) (9) where w = and = 0 by Table 3 and G ( ) = G (0) = 0. Equaton (9) the ame a the probablty of falure by ORM. Therefore, we can ay that the probablty of falure calculaton by ORM a pecal cae of the probablty of falure calculaton by when one quadrature pont and weght ued. The number of addtonal functon ealuaton needed to ealuate Eq. (8) bede the M earch gen by ( ) ( n ). Hence, the total number of functon ealuaton neceary for Eq. (8) # of E for M earch + ( ) ( n ) (0) Snce the probablty of falure calculaton ung requre ntegraton n Eq. (7), accuracy of the probablty of falure etmaton can be ealy acheed by ncreang the number of quadrature pont and weght n Eq. (8). In th cae, the probablty of falure by requre only functon j alue at the quadrature pont, whch are G ( ) n Eq. (8). Conequently, the accuracy of the reult can be mproed by ncreang the number of quadrature pont f neceary, whch doe not requre any entty. Th comparon wll be dcued n detal ung numercal example n Secton IVERSE RELIABILITY AALYSIS USIG The objecte of the -baed nere relablty analy to fnd a new -baed M, denoted by x, ung the ORM-baed M denoted by. A tated n x ORM Secton.4., the nere relablty analy doe not calculate the probablty of falure drectly, ntead, t judge whether or not a gen degn atfe the probabltc contrant by checkng a performance functon alue at M. Howeer, the probabltc contrant may not be atfed een though the contrant alue at the ORM-baed M G( x ) le ORM than zero. Th becaue the probablty of falure calculated by ORM may hae gnfcant error epecally for the multdmenonal and hghly nonlnear performance functon. In th ecton, a new method propoed to fnd a -baed M ung the ORM-baed nere relablty analy, whch ued for the next degn teraton of RBDO and thu yeld an accurate optmum degn een for hghly nonlnear ytem. 3. -Baed M ndng a new -baed M cont of three tep: contrant hft, relablty ndex update, and M update. A. Contrant Shft

6 A explaned n Secton, the contrant hft concept ntroduced uch that G G G G G G ( x) ( x) ( x ) or ( ) ( ) ( ) to make the hfted performance functon ( x) () at M become zero, that, G ( x ) = G ( ) = 0. Th performance functon hft carred out to calculate the probablty of falure ung the nere relablty analy at the current degn nce the nere relablty analy cannot etmate the probablty of falure drectly. Ung the contrant hft n Eq. () and probablty of falure calculaton n Eq. (8), the probablty of falure of the hfted performance functon accurately computed and compared wth the target probablty of falure, denoted by. The dfference between the computed probablty of falure and ued to update the relablty ndex. B. Relablty Index Update After computng the probablty of falure ung for the hfted performance functon G ( x), the correpondng relablty ndex obtaned ung = Φ ( ). It lkely that not the ame a T the target relablty ndex Φ ( ar t = ) and a new updated relablty ndex up obtaned ung the dfference between two relablty ndce and lnear hft of the relablty ndex a G probablty of falure by. After terately dong th procedure untl conerged, a -baed M can be obtaned where the ame a. Howeer, th terate procedure wll be computatonally ery expene f a new M earch carred out eery tme an updated relablty ndex obtaned. To achee effcency, the updated M correpondng to the probablty of falure by obtaned wthout carryng out a new M earch a [5] u u up up T up cur or up cur {0,,0, up} cur = cur along the ame radal drecton wth the current M u cur n U-pace a llutrated n g.. After fndng the approxmately updated M, the ame terate procedure explaned aboe can be performed untl conerge to. The M obtaned through the terate procedure called the -baed M, whch wll be ued to ealuate whether the degn atfe the probabltc contrant or not. Two method to fnd the -bae M are compared n term of effcency n Secton 4. through numercal example. Snce both method nclude approxmaton, conergence hould be checked a well. up (3) That, t aumed that the updated M u located () ( ) up cur t where the ame a cur the relablty ndex at the current tep, whch t at the ntal tep. or example, conder a performance functon n g. and et target probablty of falure a =Φ( ) =Φ( ). In th paper, a performance t functon defned a concae near M f ORM-baed relablty analy oeretmate the probablty of falure and conex near M f ORM-baed relablty analy underetmate the probablty of falure. Snce the performance functon n g. concae near the M, the probablty of falure calculated by wll be maller than the target probablty of falure, whch mean > t. Hence, ung Eq. (), a maller relablty ndex up wll be obtaned becaue > t. Th mean that a maller relablty ndex than ORM-baed relablty ndex hould be ued to atfy the target probablty of falure for the concae performance functon and ce era for the conex performance functon. C. M Update Method Ung th updated relablty ndex, we can carry out a new nere relablty analy to fnd a better M whch atfe the gen target probablty of falure. After fndng a new M, contrant hft agan ued to compute the gure. Approxmaton of Updated M 3. Sytem Leel Relablty Analy When there are more than two performance functon, ytem leel probablty of falure defned a m y = G( X) > 0 (4) = where m the number of performance functon. Howeer, nce the rght hand de of Eq. (4) not eay to compute

7 numercally, the ytem leel probablty of falure approxmated by the um of the probablty of falure (ung Dtleen frt-order upper bound) [5] a wthout requrng the econd-order entty calculaton and a effcently a ORM wthout lo of accuracy - the error of the ORM reult about 5% a hown n Table 4. y m (5) = where the probablty of falure for th performance functon. The ytem leel probablty of falure approxmated by Eq. (5) mean that the nterecton part of each performance functon gnored. In th paper, the ytem leel probablty of falure approxmated by the um of the probablty of falure by a m m y = = (6) and compared wth the ORM-baed ytem leel probablty of falure. 4. UMERICAL EXAMLES The -baed probablty of falure computaton erfed by comparng wth ORM and SORM-baed probablty of falure computaton n Secton 4.. The probablty of falure obtaned ung ued a a benchmark tet. Secton 4. how the terate procedure to obtan the -baed M that atfe the gen target probablty of falure ung the three-tep proce. In Secton 4.3, the ytem leel probablty of falure ealuated and compared ung and ORM. (a) 4. Comparon of ORM, SORM and for robablty of alure Calculaton or the frt example, a hghly nonlnear fourth order polynomal functon G ( ) ( Y 6) ( Y 6) 0.6 ( Y 6) 3 4 X = Z (7) where Y X = Z X and X ~ (4,0.3) and X ~ (3,0.3), ued for the probablty of falure computaton. The relablty ndex of =.645 ued for the ORM-baed nere relablty analy. gure 3 (a) how the hfted and orgnal performance functon and g. 3 (b) how the approxmated functon by ORM, SORM, and at the M n V-pace. In Table 4, wth three and fe quadrature pont are ued to ealuate Eq. (8). rom Table 4, t can be een that wth fe quadrature pont can the mot accurate method for th example. In fact, th reult een more accurate than the SORM reult, compared wth the reult, whch can be condered a exact. In term of effcency, ORM how the bet effcency, whch alway true nce SORM and requre the ORM-baed M. Howeer, the addtonal number of functon ealuaton for bede the M earch doe not requre entty analy. Hence, can etmate the probablty of falure a accurately a SORM (b) gure 3. erformance uncton n X-pace and V-pace Table 4. robablty of alure by Varou Method ORM SORM 3 pt 5 pt, % E 7 7+Hean mean the number of functon and entty analy for M earch and 4 functon ealuaton for do not requre entty analy or the econd example, a four dmenonal quadratc functon

8 G ( X) = X X X X X + X + X + X (8) where X ~ (5, 0.4) for =,,3,4, ued for the falure rate computaton. The relablty ndex of =.645 ued for the ORM-baed nere relablty analy. A decrbed n Eq. (0), the total number of functon ealuaton for the -baed relablty analy wll ncreae a the number of random arable ncreae. Snce the performance functon n Eq. (8) ha four random arable, ( 4 ) (3 ) = 6 functon ealuaton are requred for wth three quadrature pont; and (4 ) (5 ) = functon ealuaton are requred for wth fe quadrature pont a hown n Table 5. Agan, thee functon ealuaton do not requre entty analy. Een though the number of functon ealuaton ncreae for, tll how the bet accuracy compared wth the reult regardle of the number of nput random arable. Table 5. robablty of alure By Varou Method ORM SORM 3 pt 5 pt, % E +Hean +6 + mean the number of functon and entty analy for M earch 6 and functon ealuaton for do not requre entty analy 4. Inere Relablty Analy Ung The two-dmenonal performance functon n Eq. (7) agan ued for the conergence tet n th ecton. or the gen target probablty of falure =.75%, the correpondng relablty ndex obtaned from = Φ (0.075) =, whch the ntal relablty ndex n Table 6 and 7. The relablty ndex ued for the ORMbaed nere relablty analy to fnd M. After fndng the ORM-baed M, the probablty of falure calculated ung and compared wth the target probablty of falure. Snce the etmated probablty of falure by, whch.409% n Table 6, maller than the target probablty of falure, the next relablty ndex hould be maller than the ntal relablty ndex. Ung Eq. (), the updated relablty ndex obtaned a = ( ) up cur t = ( Φ (0.0409) ) =.8058 (9) where cur = t = nce t the ntal teraton and = Φ ( ) = Φ (0.0409). The updated relablty ndex hown n Table 6 and 7. Ung the M update explaned n Secton 3., the frt -baed M canddate obtaned a ( ,.7936) n Table 6. By terately performng th procedure, fnally -baed M can be obtaned a ( 4.509,.798) where the probablty of falure etmaton by,.75%, almot the ame wth the target probablty of falure.750%. In th example, the updated relablty ndex decreang nce the performance functon concae near M pont, whch mean that ORM-baed relablty analy oeretmate the probablty of falure. Table 7 how terate way of fndng -baed M ung new M earch method, whch mean that new M earch carred out after obtanng the updated relablty ndex. Snce th requre M earch at eery teraton, t become expene to fnd the -baed M a hown n Table 6 and 7. The total number of functon ealuaton needed for Table 6 5, whch nclude functon and entty analy, plu, whch requre only functon ealuaton. Wherea, the total number of functon ealuaton needed for Table 7 5, whch nclude functon and entty analy, plu 6, whch requre only functon ealuaton. Both method conerge ery fat wthn three teraton. In addton, two M obtaned ung M approxmaton and new M earch method are cloe to each other a hown n Table 6 and 7, whch mean that the M approxmaton method can be effectely ued to fnd the -baed M wthout requrng further M earch. Th reducton of the number of functon ealuaton play a gnfcant role when t appled to RBDO. Table 6. Iterate Way of ndng -Baed M Ung M Approxmaton Iter. x,, Approx % % E (4.5547,.774) (4.5009,.7936) (4.5030,.797) (4.509,.798) mean the number of functon and entty analy for M earch mean the number of functon ealuaton only for Table 7. Iterate Way of ndng -Baed M Ung ew M Search Iter. x,, ORM % % E (4.5547,.774) (4.508,.89) (4.50,.899) mean the number of functon and entty analy for M earch mean the number of functon ealuaton only for When the performance functon n Eq. (8) ued and the target probablty of falure gen by =.75%, the terate way of fndng -baed M ung M approxmaton and new M earch method are dentcal

9 nce M obtaned ung new M earch method le on the radal drecton from the orgn to M n U-pace. Table 8 llutrate that the ntal relablty ndex, whch obtaned from ORM-baed relablty analy, ncreae gnfcantly from.0000 to.459 and a a reult, the ealuated probablty of falure by reduced from to.748, whch almot the ame wth the target probablty of falure.750. Th fact mean that f the ORM-baed relablty analy ued for a degn optmzaton problem, t could fnd a wrong optmum degn nce t probablty of falure etmaton contan gnfcant error. In th example, the updated relablty ndex ncreang nce the performance functon conex near M pont, whch mean that ORM-baed relablty analy underetmate the probablty of falure. the ytem leel probablty of falure come from the probablty of falure etmaton for each performance functon. A hown n Table 0, the major error of the ORMbaed ytem leel probablty of falure due to t wrong etmaton of the probablty of falure for the econd performance functon. Snce the number of ample n the nterecton area 705 out of mllon, the probablty that both performance functon fal %, whch gnorable compared to the ytem leel probablty of falure of 3.740% obtaned ung. Table 8. Iterate Way of ndng -Baed M for 4-D Quadratc uncton Iter. xorm and x,, Approx % % (5.400,4.600,4.600,4.600) (5.488,4.5,4.5,4.5) (5.483,4.57,4.57,4.57) (5.483,4.57,4.57,4.57) Comparon of ORM and for Sytem Leel Relablty Snce the ytem leel probablty of falure decrbed n Eq. (5) contan an aumpton that the nterecton part of the falure regon can be gnored, two cae ung two dmenonal performance functon are adapted n th ecton to erfy whether the aumpton reaonable or not. The frt cae when the target relablty ndex equal to two and the econd cae when the target relablty ndex equal to three. Two performance functon are X X G ( X) = G ( ) ( Y 6) ( Y 6) 0.6 ( Y 6) 3 4 X = + + Z Y X and X ~ (.0,0.3). where =, Z X (30) X ~ (5.0,0.3) 4.3. or Relablty Index of =.0 Snce the relablty ndex two, the probablty of falure etmate by ORM for each performance functon ORM =Φ( ) =.75% a hown n Table 9. Ung Eq. (5), the ORM-baed ytem leel probablty of falure the um of two probablte of falure, whch 4.550% a hown n Table 0. Smlarly, the ytem leel probablte of falure by wth three and fe quadrature pont are 4.5% and 3.905%, repectely. Compared wth reult 3.740%, wth fe quadrature pont how the bet accuracy. Thu, n th example, the mot error contrbuted to gure 4. Sytem Leel robablty of alure Ung for =.0 Table 9. Component Leel robablty of alure for =.0 ORM (3pt) (5pt), G ( X) % G ( ) Table 0. Sytem Leel robablty of alure for =.0 ORM (3pt) (5pt) y, % or Relablty Index of = In the cae that the relablty ndex large, the effect of the nterecton part become een more gnorable. A hown n g. 5, the probablty both performance functon fal 0.00%. Conequently, t can be concluded for th example that the ytem leel probablty of falure can be approxmated a the um of each probablty of falure epecally when the relablty ndex relately large. Snce the probablty of falure by ORM are erroneou for each performance functon, the ORM-baed ytem probablty of falure ha the larget error among the three method.

10 the nterecton part of the falure regon gnored. or the example hown n Secton 4.3, t clear that the error of the probablty of falure etmaton for each performance functon much larger than the error due to gnorng the nterecton part of the falure regon, epecally when the relablty ndex large. or the next tep, RBDO ung the -baed nere relablty analy currently beng carred out. 6. ACKOWLEDGEMET Reearch upported by the Automote Reearch Center that ponored by the U.S. Army TARDEC. gure 5. Sytem Leel robablty of alure Ung for = 3.0 Table. Component Leel robablty of alure for = 3.0, % ORM (3pt) (5pt) G ( ) G ( ) Table. Sytem Leel robablty of alure for = 3.0 ORM (3pt) (5pt), % y 5. DISCUSSIOS AD COCLUSIO Three method to ealuate the probablty of falure ung ORM, SORM, and are compared n term of effcency and accuracy. In term of effcency, the probablty of falure calculaton by ORM the bet nce the probablty of falure calculaton by SORM and ue the M of the ORM-baed nere relablty analy. Howeer, a hown through the example n th paper, the probablty of falure calculaton by ORM could be ery erroneou n partcular when the mult-dmenonal performance functon hghly nonlnear. Een though SORM can ealuate the probablty of falure more accurately than ORM, SORM ha lmted applcaton nce t requre the econd-order entte. On the other hand, the probablty of falure calculaton by a accurate a SORM, and ometme een better than SORM, wthout requrng the econd-order entte. Moreoer, th accurate probablty of falure etmaton ued to fnd the -baed M, whch can dentfy falure regon of the performance functon better than ORMbaed M. A three-tep computatonal proce propoed to fnd out the -baed M ung the nere relablty analy: contrant hft, the relablty ndex update, and the M update. Conergence tet performed to ee whether the -baed M obtaned ung the three tep conerge or not. In addton, the ytem leel probablte of falure by and ORM are compared ung reult. or th, 7. REERECES. Haldar, A., and Mahadean, S., robablty, Relablty and Stattcal Method n Engneerng Degn, John Wley & Son, ew York, Y, Haofer, A. M. and Lnd,. C., An Exact and Inarant rt Order Relablty ormat, ASCE Journal of the Engneerng Mechanc Don, Vol. 00, o., pp. -, Hohenbchler, M. and Rackwtz, R., Improement of Second-Order Relablty Etmate by Importance Samplng, ASCE Journal of Engneerng Mechanc, Vol. 4, o., pp , Bretung, K., Aymptotc Approxmaton for Multnormal Integral, ASCE Journal of Engneerng Mechanc, Vol. 0, o. 3, pp , Jn, R., Du, X. and Chen, W., The Ue of Metamodelng Technque for Degn under Uncertanty, Structural and Multdcplnary Optmzaton, Vol. 5, o., pp , Youn, B. D. and Cho, K. K., A ew Repone Surface Methodology for Relablty-Baed Degn Optmzaton, Computer and Structure, Vol. 8, pp. 4-56, Roenblueth, E., ont Etmate for robablty Moment, Appled Mathematc Modelng, Vol. 5, pp Du, X., and Huang, B., Uncertanty Analy by Dmenon Reducton Integraton and Saddlepont Approxmaton, Journal of Mechancal Degn, Vol. 8, pp. 6-33, Youn, B. D., X, Z., Well, L. J., and Lamb, D. A., Stochatc Repone Surface Ung The Enhanced Dmenon Reducton (edr) Method or Relablty- Baed Robut Degn Optmzaton, III European conference on Computatonal Mechanc, Lbon, ortugal, Ln, C. Y., Huang, W. H., Jeng, M. C., and Doong, J. L., Study of an Aembly Tolerance Allocaton Model Baed on Monte Carlo Smulaton, Journal of Materal roceng Technology, Vol. 70, pp. 9-6, Rubnten, R. Y., Smulaton and Monte Carlo Method, John Wley & Son, ew York, 98.. Walker, J. R., ractcal Applcaton of Varance Reducton Technque n robabltc Aement, the Second Internatonal Conference on Radoacte Wate Management. Wnnpeg, Mant, Canada, pp. 57-5, 986.

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