Probabilistic Sensitivity Analysis for Novel Second-Order Reliability Method (SORM) Using Generalized Chi-Squared Distribution

Transcription

1 th World Congress on Structural and Multdscplnary Optmzaton May 9 -, 3, Orlando, lorda, USA Probablstc Senstvty Analyss for ovel Second-Order Relablty Method (SORM) Usng Generalzed Ch-Squared Dstrbuton Davd Yoo, Ijn Lee, and Hyunyoo Choo 3 Unversty of Connectcut, Storrs, Connectcut, USA, davd.yoo@engr.uconn.edu Unversty of Connectcut, Storrs, Connectcut, USA, lee@engr.uconn.edu 3 Unversty of Iowa, Iowa Cty, Iowa, USA, hyunyoo-cho@uowa.edu. Abstract Relablty-based desgn optmzatons (RBDO) requre evaluaton of senstvtes of probablstc constrants. o develop RBDO utlzng the recently proposed novel second-order relablty method (SORM) that mproves the conventonal SORM n terms of accuracy, the senstvtes of the probablstc constrants at the most probable pont (MPP) are requred. hus, ths study presents the senstvty analyss of the novel SORM at MPP for more accurate RBDO. Durng the analytc dervaton n ths study, t s assumed that the Hessan matrx does not change due to change of dstrbuton parameter. he calculaton of the senstvty based on the analytc dervaton requres evaluaton of probablty densty functon (PD) of a lnear combnaton of non-central ch-square varables, whch s obtaned by utlzng general ch-squared dstrbuton. In terms of accuracy, the proposed probablstc senstvty analyss s compared wth the fnte dfference method (DM) usng the Monte Carlo smulaton (MCS) through numercal examples. he numercal examples demonstrate that the analytc senstvty of the novel SORM agrees very well wth the senstvty obtaned by DM usng MCS when a performance functon s quadratc n U-space and nput varables are normally dstrbuted. It s further tested that senstvty of a hgher order performance functon n terms of how the proposed assumpton the Hessan s constant affects the accuracy of the senstvty.. Keywords: Senstvty Analyss, Most Probable Pont (MPP), ovel Second-Order Relablty Method (SORM), General Ch-squared Dstrbuton 3. Introducton Relablty analyss and relablty-based desgn optmzaton (RBDO) have been successfully appled to dverse engneerng applcatons such as structural engneerng analyss [-7]. he man goal of the relablty analyss s to evaluate probablty of falure of a performance functon that s used as a probablstc constrant of RBDO wth both effcency and accuracy. Methods of obtanng the probablty of falure are commonly categorzed nto samplng-based and most probable pont (MPP)-based methods. MPP-based frst-order relablty method (ORM) [8-] very effcently calculates the probablty of falure, however, accuracy s sacrfced for hghly non-lnear performance functons and hgh-dmensonal nput varables. MPP-based second-order relablty method (SORM) [-5] mproves ORM n terms of accuracy, however t s computatonally expensve compared to ORM snce t requres the computaton of the Hessan matrx. MPP-based dmenson reducton method (DRM) [6-8] can be also used for approxmately assessng the relablty of a system whch s used as a probablstc constrant n RBDO. SORM that mproves accuracy of ORM stll contans three types of errors: () error due to approxmatng a general nonlnear lmt state functon by a quadratc functon at the MPP n standard normal U-space, () error due to approxmatng the quadratc functon n U-space by a parabolc surface, and (3) error due to calculaton of the probablty of falure after mang the prevous two approxmatons. On the other hand, the recently proposed novel SORM contans only type () error. he general dea of the novel SORM s to compute the probablty of falure usng non-central or general ch-squared dstrbuton [9]. Smlarly to other MPP-based methods, the novel SORM frst nvolves MPP search after transformng all random varables n orgnal X-space to standard normal U-space through the Rosenblatt transformaton []. After a quadratc approxmaton at MPP n U-space, the novel SORM does not use further approxmaton of a quadratc functon. Instead, the novel SORM converts a quadratc falure functon of standard normal varables to a lnear combnaton of non-central ch-square varables by orthogonal transformaton. Snce every random varable n U-space s a standard normal varable, probablty of falure of a quadratc functon n U-space can be evaluated usng a lnear combnaton of non-central ch-square varables. o carry out RBDO utlzng relablty analyss method, senstvtes of probablstc constrants wth respect to desgn varables are requred, and many wors have been devoted to derve the senstvty of the probablstc constrant [-7]. hus, ths study presents the senstvty analyss of the novel SORM for more accurate RBDO. Snce the novel SORM performs relablty analyss at MPP, senstvtes of probablstc constrants at MPP wth

2 respect to dstrbuton parameters are derved durng the senstvty analyss n ths study. o calculate the senstvty based on the senstvty analyss, t s necessary to evaluate probablty densty functon (PD) of a lnear combnaton of non-central ch-square varables, whch s obtaned utlzng the general ch-squared dstrbuton n ths study. hs paper s organzed as follow. Secton revews the relablty analyss usng the novel SORM. Secton 5 presents the senstvty analyss usng the novel SORM. Secton 6 provdes the numercal examples where analytc senstvtes are compared wth the fnte dfference method (DM) usng Monte Carlo smulaton (MCS) n terms of accuracy. Secton 7 summarzes ths study and dscusses about the future research.. Revew of Relablty Analyss Usng ovel SORM Relablty analyss usng the novel SORM s revewed n ths secton to help readers understand senstvty analyss of the novel SORM that wll be explaned n Secton 5 [9].. uadratc Approxmaton of Performance uncton A relablty analyss nvolves calculaton of probablty of falure, denoted by mult-dmensonal ntegral [8,7] P, whch s defned usng a where P s a probablty measure; random varables and x are the realzatons of P [ ( ) ] ( ) P G X fx x dx () G( X) X ={ X, X,, X } s an -dmensonal random vector where X ; X are G X s the performance functon such that G( X ) s defned as falure; and fx () s a jont probablty densty functon (PD) of X. In conventonal SORM, for the calculaton of the probablty of falure n Eq. (), G( X ) n Eq. () s transformed to U-space through the Rosenblatt transformaton, then t s approxmated by ts second-order aylor seres expanson at MPP (u * ), whch can be wrtten as * * * G( X) g( U) g ( U) g ( U u ) ( U u ) H( U u ) * * * * g u u Hu g u HU U HU () usng the gradent vector and the Hessan matrx (H) evaluated at MPP. u * n Eq. () s obtaned by solvng the followng optmzaton problem to mnmze u. (3) subject to g. A relablty ndex, denoted by, s defned as dstance from the orgn to u * and s gven by u / * * * u u u. () Usng the falure defnton G( X ), u * can be wrtten as * g u g α (5) where α s the normalzed gradent vector at u *. hen, dvdng Eq. () by g and usng Eq. (5) yeld U H H H α α α α U U U. g g g (6) g g ( )

3 . Orthogonal ransformaton of uadratc uncton Consder an orthogonal transformaton u = y where s the matrx of the egenvectors of H and y R s an -dmensonal vector of standard normal random varables, y, whch are statstcally ndependent to each other snce u s statstcally ndependent [9]. Usng the orthogonal transformaton, Eq. (6) can be transformed to [] ˆ ( Y) g g ˆ a a Y Y AY (7) where three quanttes a, a, and  are gven by a H α α (8) g R, H a α I R, (9) g and ˆ A dag,,, R, () g respectvely where s the th egenvalue of H. Consequently, the probablty of falure n Eq. () can be calculated usng Eqs. (7)-() as g ( ) ˆ U g( Y) P ˆ P P P g( ) g g Y P g a g a Y Y. () where a and Y are th component of a and standard normal varables, respectvely. Snce u and y are the ndependent standard normal varables, any orthogonal transformaton does not change the probablty of falure [9]. hus, Eq. () can be rewrtten as where the th non-centralty parameter s gven by P P Y g a () ga (3) Snce Y s the standard normal varable, Y Y s a non-central ch-square varable and becomes a general ch-square varable whose PD and CD are avalable as shown n Ref. [8]. Hence, the probablty of falure n Eq. () can be smplfed as P P g a g a () 3

4 where s the CD of..3 General Ch-Squared Dstrbuton he dstrbuton of n Eq. () can be obtaned by usng a general ch-squared dstrbuton. Let for,,, for,,, and for,,. hen, the lnear combnaton n Eq. () can be expressed as (5) Z U V where are dstnct egenvalues, Z ~ ( ), and,. (6) U Z V Z he PDs of U and V n Eqs. (5) and (6) are obtaned by Ruben n Ref. [9]. Addtonally, the exact PD and CD of the general ch-square varable usng Whttaer s functon [3] are obtaned by Provost and Rudu [8]. he PD and CD of are not presented n ths paper due to ther complextes. 5. Senstvty Analyss Usng ovel SORM he senstvty of the probablty of falure wth respect to a dstrbuton parameter θ can be obtaned by tang the dervatve of Eq. () as denotng dp, dq d q d f q,. (7) d d d d ga n Eq. () as q. In Eq. (7), f general ch-squared dstrbuton explaned n Secton.3. Here, t should be noted that s the PD of, whch s obtaned by usng n Eq. () s not only functon of q but also functon of, whch s vector of the non-centralty parameter n Eq. (3). o calculate d Eq. (7), needs to be obtaned based on the defnton of q, and thrd-order dervatve of performance d functon or dervatve of the H s thus requred to be obtaned. Consder the generalzed egenvalue problem of the Hessan matrx H such that H = where and are the dagonal matrx of the egenvalues and the matrx of the egenvectors of H, respectvely, and A represents the d matrx that s symmetrc and postve defnte. hen, accordng to the egenvalue perturbaton theory, d s obtaned usng the property of the matrx A as [3] where d H j H j Hj H A j j (8) d H H j j j j j H j s the element of H at the th row and the j th column; s the th egenvector correspondng to ; s the th component of ; s perturbaton sze, whch s chosen so that t s always much smaller than ; and s the Kronecer delta. Snce the dervatve of the Hessan matrx H s not avalable n SORM, t s assumed n j H j d ths study that n Eq. (8) s. Consequently, n Eq. (8) becomes. d Usng the assumpton, the senstvty of the probablty of falure n Eq. (7) can be wrtten as

5 dp, d d g da d q g da a d g fq, a g d d d d d d d d d g da d q, g da a d g f q, a g. d d d d d d (9) Equaton (9) can be then rewrtten based on the defnton of stated n Eq. (3) as q, da a dp da d g d g da d g d g f q, g δ δ a a g. d d d d d d d d () As shown n Eq. (), d a d g da,, and d d d need to be derved for the senstvty analyss of the probablty of falure. rom Eq. (9) and the assumpton made n the prevous paragraph, d a d n Eq. () becomes da d g d d d d H α H α Hα Hα I Hα I d g d d g g () d g d g d where d α can be derved based on the defnton of α stated n Eq. (5) as d dα d g dg g d g dg α d g g d d g g d d g d g d. () Utlzng the chan rule, d g d n Eq. () can be wrtten as * dg g du H (3) d d where d * u can be obtaned by tang dervatve of * u n Eq. (5) wth respect to θ as d * du d dα α. () d d d d g d n Eq. () can be obtaned as d g dg α. (5) d d hen, usng Eqs. (3), () and (5), Eq. () can be rewrtten as 5

6 dα I αα g d dα. d g H α d d (6) hus, t s obtaned I αα H I αα dα g d I Hα. (7) d g g d he lmt-state functon n the orgnal X-space s expressed as [] G g u; x. (8) Due to the fact that dstrbuton parameter θ has no nfluence on the lmt state functon expressed n Eq. (8), dervatve of the left hand sde of Eq. (8) wth respect to θ s zero, whch mples dg u; g du g. d d (9) hus, t s derved usng Eq. (9) as g g du g. d (3) By nsertng Eq. (3) nto Eq. (7), t s obtaned I αα H I αα dα du d I g. d g g Hα d d (3) he relablty ndex β can be expressed based on Eq. (5) as * αu. (3) hen, by tang dervatve of Eq. (3) wth respect to θ yelds * * d dα * du du u α α (33) d d d d dα owng to the fact that and α are mutually orthogonal and d drectly by dfferentaton of αα or by * u α. he orthogonalty can be verfed / α u * g gg u * * g g 3 g d d g d d u d d d g d g d d g g g g. g d g d (3) he lmt-state functon at MPP n U-space s gven by g u * ; and the dfferentaton of t gves 6

7 * dg g du g. d d (35) hen, dvdng both sdes of Eq. (35) by g gves * * * g du g g du g du g α. (36) g d g g d g d Usng Eqs. (9) and (36), Eq. (33) can be rewrtten as d g du. d g α d (37) uu * nally, da d n Eq. () can be obtaned usng Eq. (8) as d α Hα d g g α Hα da d d α Hα d d d d g d g d d dα α Hα d g d g d g d g d α Hα α H. (38) hen, usng Eqs. () and (5), Eq. (38) can be expressed as α α Hα α H α I αα. (39) d d g d g d da d d α Hα d he senstvty of the probablty of falure wth respect to θ s therefore gven by where dp, da d g d g da d q g da a d g f q, g δ δ a a g d d d d d d d d d, da dg da da q g a d g f q g δ α δ a a g d d d d d d () da d d d g H α Hα Hαα I d g () d g d g d d g du d d g α H α () d d d d and d g d, d d, d α da, and d d are obtaned from Eqs. (5), (37), (3) and (39), respectvely. d u n Eqs. (37) d and () s obtaned by the Rosenblatt transformaton, whose th component s obtaned as 7

8 du d X x ;. (3) d d or normally dstrbuted ndependent random varables and when dstrbuton parameter s j th mean of random du varable, n Eq. (3) becomes d where j s the Kronecer delta. o calculate Eq. (), d d q, du j () d j j s also requred to be calculated, whch s obtaned usng DM as ' q, q, q, d d ' (5) ' where s perturbed value of. o calculate general ch-squared dstrbuton n Secton.3. ', '. Durng the calculaton of ', d d q, n Eq. (5),, q s frst calculated usng q s then calculated after settng the approprate value for q, another MPP search, whch s teratve algorthm and thus can sgnfcantly affect effcency of the calculaton, s not nvolved and all other parameters except, whch are d q, used to calculate q,, reman constant. hus, effcency s not lost durng the calculaton of n d Eq. (5). Addtonally, accuracy s not lost ether by settng very small perturbaton sze for q, and ', q are exactly calculated. ' due to the fact that 6. umercal Examples he frst two numercal studes are carred out n ths secton to verfy the proposed senstvty analyss for both low-dmensonal and hgh-dmensonal cases. he last numercal study s carred out to test that the senstvty of hgher-order performance functon n terms of how the proposed assumpton that the Hessan does not change wth respect to dstrbuton parameter affects the accuracy of the senstvty calculaton. 6. Senstvty Usng ovel SORM for wo-dmensonal Inputs In ths numercal example, the senstvty of the probablty of falure wth respect to the desgn pont s obtaned usng the analytc dervaton n Secton 5 for the two-dmensonal performance functon. he means of the random varables are used as desgn pont. Consder the followng D performance functon gven n X-space as G( X ) X X X X.5X X 3, (6) 8

9 6 G = G(X) = X - where ~, X gure. Performance uncton X and they are statstcally ndependent to each other, whch s also shown n g.. he performance functon n Eq. (6) n U-space becomes g( U ) U U U U.5 U U 3 (7) usng the Rosenblatt transformaton. Eq. (7) s further transformed to χ -space as [9] gˆ L Z Z Z (8) 3 where Z 3 ~ 9 and Z ~. he probablty of falure then can be calculated usng Eq. (8) and numercal ntegraton as [9] P f ( z) dz f ( z ) dz f ( z ) dz 88.6 Z z fz ( z ) dz.65% gˆ ( ).6 L Z Z z Z Z 5 (9) able. umercal Values of Parameters Parameter umercal Value f q,.66 dq / d / 3,, /.996, / d.77,.77 d q d d he senstvty of the probablty of falure n Eq. (9) wth respect to the means of the random varables s then calculated usng Eq. (7), the general ch-squared dstrbuton n Secton.3, and the analytc dervaton n Secton 5. he parameters, whch are necessary to calculate dp, are obtaned based on the dervaton n Secton 5 and d they are shown able. Usng the result,.66. dp dp d s calculated as.66. Lewse, d s calculated as able. Comparson of Senstvty Calculaton 9

10 Proposed Senstvty DM usng MCS ( M) dp / d dp / d he calculated senstvty s then compared wth the senstvty obtaned by DM usng MCS, whch s shown n able. Accordng to able, the senstvty calculated based on the senstvty analyss n Secton 5 s very close to the senstvty obtaned by DM usng MCS, whch s owng to the fact that the performance functon s perfectly quadratc. 6. Senstvty Usng ovel SORM for Hgh-Dmensonal Inputs gure. Schematc Dagram of Cantlever ube In ths numercal example, the senstvty of the probablty of falure wth respect to the desgn pont s obtaned usng the analytc dervaton n Secton 5 for the hgh-dmensonal performance functon. Consder the cantlever tube shown n g. subjected to external forces,, and P, and torson [3]. he hgh-dmensonal performance functon s defned as the dfference between the yeld strength of 9MPa and the maxmum stress max, whch s gven as where G max 9MPa X (5) max s the maxmum von Mses stress on the top surface of the tube at the orgn, whch s gven by (5) max x 3 zx where the normal stress x can be obtaned as P sn sn L cos L cos d x d d t d d t 6 (5) and the shear stress xz can be obtaned as xz d d d t 6, (53) able 3. Propertes of Random Varables Varables Mean Standard Devaton Dstrbuton ype X () t mm. mm ormal X ( ) d mm. mm ormal X3( L ) mm 3 mm ormal X ( L ) 6 mm 3 / mm ormal

11 X5( ) 3..3 ormal X6( ) 3..3 ormal X ( ) 7 P.. ormal X ( ) 8 9. m 9. m ormal respectvely. he propertes of the random varables used n Eqs. (53)-(56) are gven n able 3 and the means of random varables are used as desgn pont. Also, they are all statstcally ndependent to each other. wo angles are assumed to be fxed at 5 and. able. Comparson of Senstvty Calculaton DM Proposed Senstvty usng MCS ( M) Error (%) dp / d dp / d dp / d dp / d dp / d dp / d dp / d dp / d Usng the above nformaton and the analytc dervaton n Secton 5, the senstvty s calculated, and the obtaned senstvty s then compared wth the senstvty obtaned by DM usng MCS. Based on the mostly small percent errors shown n able, t can be concluded that the proposed senstvty analyss accurately calculates senstvty. or a few of the random varables, relatvely large errors compared to the prevous example are generated, whch s owng to the fact that the performance functon s not quadratc n ths example. 6.3 Senstvty Usng ovel SORM for Hgher-Order Performance uncton In ths numercal example, the senstvtes of the probablty of falures wth respect to the three desgn ponts are obtaned usng the analytc dervaton n Secton 5 for the hgher-order performance functon. he hgher-order or hghly non-lnear performance functon n X-space s gven as G X Y Y YZ (5) where Y X Z he propertes of the random varables at three desgn ponts are also X able 5. Propertes of Random Varables at hree Desgn Ponts X X Dstrbuton ype Dstrbuton ype Desgn Pont ormal 3.8 ormal.8 Desgn Pont ormal.8 ormal 5.8 Desgn Pont 3 ormal 7.8 ormal.8

12 5 Desgn Pont Most Probable Pont uadratc Approxmaton G = 5 Desgn Pont Most Probable Pont uadratc Approxmaton 5 Desgn Pont Most Probable Pont uadratc Approxmaton U U G = U G = U U U (a) (b) (c) gure 3. uadratc Approxmatons n U-space at (a) Desgn Pont, (b) Desgn Pont, and (c) Desgn Pont 3 shown n able 5 and the means of the random varables are the desgn ponts. hey are also all statstcally ndependent to each other. In able 5, standard devatons for both random varables at every desgn pont are ntentonally set to be large to generate notceable amount of error n ths example. As prevously stated, the hghly non-lnear performance functon gven n Eq. (57) s not quadratc, and quadratc approxmatons performed at three desgn ponts n U-space are shown n g. 3. Desgn Pont Desgn Pont Desgn Pont 3 able 6. Comparson of Senstvty Calculaton at hree Desgn Ponts dp / d dp / d 6. dp / d.38 dp / d.69 dp / d.653 dp / d ovel SORM DM usng MCS ( M) Error (%) Usng the above nformaton and the analytc dervaton n Secton 5, the senstvty s calculated at each desgn pont, and t s then compared wth the senstvty obtaned by DM usng MCS, whch s shown n able 6. As shown n able 6, the generated errors are not large and they are wthn range from to %. 7. Conclusons In ths study, the senstvty of the probablty of falure wth respect to dstrbuton parameter usng the novel SORM has been derved through the senstvty analyss. Durng the dervaton of the senstvty, senstvty of egenvalue of the Hessan matrx s frst requred. Based on the egenvalue perturbaton theory, senstvty of egenvalue wth respect to dstrbuton parameter s obtaned. However, snce t s nherent n SORM that the thrd-order dervatve of performance functon s not avalable, t s assumed that senstvty of egenvalue wth respect to dstrbuton parameter s zero. Wth the assumpton, the senstvty of the probablty of falure wth respect to dstrbuton parameter n addton to all the necessary parameters are derved through senstvty analyss. he calculaton of the derved senstvty ncludes calculaton of the senstvty of CD of lnear combnaton of non-central ch-square varables wth respect to non-centralty parameter by DM, whch, however, does not requre any teratve algorthm and repeatng calculaton of the parameters except the non-centralty parameter after perturbaton. hus, effcency s not lost. Accuracy s not lost ether snce CD of lnear combnaton of non-central ch-square varables before and after perturbaton are exactly calculated n novel SORM. he calculaton of the derved senstvty also requres probablty densty functon (PD) of a lnear combnaton of non-central ch-square varables, whch s obtaned by utlzng general ch-squared dstrbuton. In numercal examples, the derved senstvty s appled to calculate senstvty n both low- and hgh- dmensonal examples. or the low-dmensonal example that s perfectly quadratc and the hgh-dmensonal example that s not quadratc, the obtaned senstvtes based on the proposed senstvty analyss are very close to those obtaned by DM usng MCS. o further test the assumpton how the Hessan matrx does not change affects the accuracy of the calculaton of the senstvty, the last numercal example s carred out wth hgher-order or hghly non-lnear

13 performance functon. he generated errors are not large and they are wthn acceptable ranges wth the largest one below %. In concluson, the senstvty calculated by the proposed senstvty analyss s effcent and accurate, and the assumpton that the egenvalue of the Hessan matrx does not change wth respect to dstrbuton parameter does not sgnfcantly affect the accuracy. In future research, RBDO utlzng the senstvty analyss n ths study wll be performed. Also, the problem detected whle dong research for ths study that probablty of falure and ts senstvty are not obtanable when one or some of the egenvalues for the Hessan matrx s/are zero are necessarly to be resolved. 8. References [] Engesser, M., Buhmann, A., rane, A.R., and Korvn, J.G., Effcent Relablty-based Desgn Optmzaton for Mcro-Electromechancal systems, IEE Sensors Journal, Vol., o. 8, pp ,. [] Grujcc, M., Araere, G., Bell, W.C., Marv, H., Yalavarthy, H.V., Pandurangan, B., Haque, I., and adel, G.M., Relablty-Based Desgn Optmzaton for Durablty of Ground Vehcle Suspenson System Components, Journal of Materals Engneerng and Performance, Vol. 9, o. 3, pp. 3-33,. [3] Young, Y.L., Baer, J.W., and Motley, M.R., Relablty-Based Desgn Optmzaton of Adaptve Marne Structures, Composte Structures, Vol. 9, o., pp. -53,. [] Acar, E. and Solan, K., System Relablty-Based Vehcle Desgn for Crashworthness and Effects of Varous Uncertanty Reducton Measures, Structural and Multdscplnary Optmzaton, Vol. 39, o. 3, pp. 3-35, 9. [5] Huang, M.., Chan, C.M., and Lou, W.J., Optmal Performance-Based Desgn of Wnd Senstve all Buldngs Consderng Uncertantes, Computers & Structures, Vol , pp. 7-6,. [6] Youn, B.D., Cho, K.K., Yang, R.J., and Gu, L. Relablty-Based Desgn Optmzaton for Crashworthness of Vehcle Sde Impact, Structural and Multdscplnary Optmzaton, Vol. 6, o. 3-, pp. 7-83, [7] Dong, J., Cho, K.K., Vlahopoulos,., Wang, A., and Zhang, W., Desgn Senstvty Analyss and Optmzaton of Hgh requency Radaton Problems Usng Energy nte Element and Energy Boundary Element Methods, AIAA Journal, Vol. 5, o. 6, pp , 7. [8] Haldar, A. and Mahadevan, S., Probablty, Relablty and Statstcal Methods n Engneerng Desgn, John Wley & Sons, ew Yor, Y,. [9] Hasofer, A.M. and Lnd,.C., An Exact and Invarant rst Order Relablty ormat, ASCE Journal of the Engneerng Mechancs Dvson, Vol., o., pp. -, 97. [] u, J. and Cho, K.K., A ew Study on Relablty-Based Desgn Optmzaton, Journal of Mechancal Desgn, Vol., o., pp , 999. [] u, J., Cho, K.K., and Par, Y.H., Desgn Potental Method for Relablty-Based System Parameter Desgn Usng Adaptve Probablstc Constrant Evaluaton, AIAA Journal, Vol. 39, o., pp ,. [] Bretung, K., Asymptotc Approxmatons for Mult-normal Integrals, ASCE Journal of Engneerng Mechancs, Vol., o. 3, pp , 98. [3] Hohenbchler, M. and Racwtz, R., Improvement of Second-Order Relablty Estmates by Importance Samplng, ASCE Journal of Engneerng Mechancs, Vol., o., pp , 988. [] Adhar, S., Relablty Analyss Usng Parabolc alure Surface Approxmaton, ASCE Journal of Engneerng Mechancs, Vol. 3, o., pp. 7-7,. [5] Zhang, J. and Du, X., A Second-Order Relablty Method Wth rst-order Effcency, Journal of Mechancal Desgn, Vol. 3, o., paper #:6,. [6] Rahman, S., and We, D., A Unvarate Approxmaton at Most Probable Pont for Hgher-Order Relablty Analyss, Internatonal Journal of Solds and Structures, Vol. 3, pp , 6. [7] Lee, I., Cho, K.K., Du, L., and Gorsch, D., Inverse Analyss Method Usng MPP-Based Dmenson Reducton for Relablty-Based Desgn Optmzaton of onlnear and Mult-Dmensonal Systems, Computer Methods n Appled Mechancs and Engneerng, Vol. 98, o., pp. -7, 8. [8] Xong,., Greene, S., Chen, W., Xong, Y., and Yang, S., A ew Sparse Grd Based Method for Uncertanty Propagaton, Structural and Multdscplnary Optmzaton, Vol., o. 3, 9, pp ,. [9] Lee, I., oh, Y., and Yoo, D., A ovel Second-Order Relablty Method (SORM) Usng oncentral or Generalzed Ch-Squared Dstrbutons, Journal of Mechancal Desgn, Vol. 3, o., pp. 9-~,. [] Rosenblatt, M., Remars on A Multvarate ransformaton, he Annals of Mathematcal Statstcs, Vol. 3, pp. 7-7, 95. [] Dtlevsen, O. and Madsen, H.O., Structural Relablty Method, John Wley & Sons, Chchester, UK, 996. [] Lee, I., Cho, K.K., and Gorsch, D., Senstvty Analyses of ORM-based and DRM-based Performance Measure Approach (PMA) for Relablty-based Desgn Optmzaton (RBDO), Internatonal Journal for 3

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