Reliability-Based Design Optimization Under Stationary Stochastic Process Loads

Transcription

1 Engneerng Optmzaton 015: 1-17 DOI: /030515X Relablty-Based Desgn Optmzaton Under Statonary Stochastc rocess Loads hen Hu and Xaopng Du 1 Department o Mechancal and Aerospace Engneerng Mssour Unversty o Scence and Technology Abstract Tme-dependent relablty-based desgn ensures the satsacton o relablty requrements or a gven perod o tme, but wth a hgh computatonal cost. Ths work mproves the computatonal ecency by extendng the sequental optmzaton and relablty analyss (SORA) method to tme-dependent problems wth both statonary stochastc process loads and random varables. The challenge o the extenson s the dentcaton o the most probable ponts (M) assocated wth tme-dependent relablty targets. Snce the drect relatonshp between the M and relablty target does not exst, ths work denes the concept o equvalent M, whch s dented by the extreme value analyss and the nverse Saddlepont approxmaton. Wth the equvalent M, the tme-dependent relablty-based desgn optmzaton s decomposed nto two decoupled loops: determnstc desgn optmzaton and relablty analyss, and both are perormed sequentally. Two numercal examples are used to show the ecency o the proposed method. Keywords: Tme-dependent relablty; optmzaton; stochastc process; West 13th Street, Toomey Hall 7, Rolla, MO 65409, U.S.A., Tel: , e-mal: dux@mst.edu 1

2 1. Introducton Tme-dependent relablty Rt () provdes the probablty that a system or component works properly ater t has been put nto operaton or a perod o tme [0, t ]. Let X = [ X1, X,, X n ] be a vector o random varables, x = [ x1, x,, x n ] be a realzaton o X, and Y () t = [ Y(), t Y (), t, Y ()] t be a vector o stochastc processes, and then Rt () s dened by 1 m Rt ( ) = r{ G= g( XYt, ( )) < 0, t [0, t]} (1) where g( XYτ, ( )) s a lmt-state uncton, and G s the response. G < 0 ndcates a workng state. The tme-dependent probablty o alure s computed by p () t = 1 Rt (), or p ( t) = r{ g( XYt, ( )) 0, t [0, t]} () For specal problems wth only random varables X, the relablty becomes tme ndependent or constant. A typcal tme-ndependent RBDO model s gven by mn ( d) ( d, μ ) X R s.t. r{ g ( dx, ) 0} [ p ], = 1,,, n gdj ( d) 0, j = 1,,, nd p (3) In the above model, ( d ) s the objectve uncton, and d s a vector o determnstc desgn varables. X= [ XR, X] s a vector o random varables wth X R beng random desgn varables and X beng random parameters. The mean values o X R, or μ X R, are usually treated as desgn varables. g () s a constrant uncton or whch relablty s concerned, and [ p ] s the permtted probablty o alure. g () s a determnstc constrant uncton. Dj

3 Solvng the above RBDO model s tme consumng because the relablty analyss or r{ g ( dx>, ) 0} s embedded wthn the optmzaton. Many methods have been proposed to mprove the computatonal ecency. The commonly used methods are the sequental sngleloop methods, ncludng the ecent relablty and senstvty analyss method (Wu and Wang 1996) and the Sequental Optmzaton and Relablty Analyss (SORA) method (Du and Chen 004). These methods decouple the RBDO process nto a determnstc optmzaton process and a relablty analyss process. The decoupled processes make RBDO more ecent. When tme-dependent uncertantes are nvolved (Wang and Wang 01), the RBDO model or a perod o tme [0, t ] becomes mn ( d) ( d, μ ) X R s.t. r{ g ( dxy,, ( t)) 0, t [0, t]} [ p ], = 1,,, n gdj ( d) 0, j = 1,,, nd p (4) Tme-dependent RBDO s much more dcult than ts tme-ndependent counterpart due to two reasons. Frst, t s dcult to obtan accurate relablty analyss results. Developng accurate tme-dependent relablty methods s stll an ongong research topc (Mourelatos et al. 015, Wang et al. 015, Jang et al. 014). A bre revew about tme-dependent relablty methods s avalable n (Hu et al. 01). Second, RBDO s much more tme consumng because o the hgher computatonal cost o tme-dependent relablty analyss (L, Mourelatos, and Sngh 01, Sngh, Mourelatos, and L 010, Wang and Wang 01). Methodologes or tme-dependent RBDO have been proposed and used n many applcatons. For nstance, Kuschel and Rackwtz (Kuschel and Rackwtz 000) developed a structure desgn optmzaton model by usng the outcrossng rate method or tme-dependent relablty analyss. Mourelatos et al. (L, Mourelatos, and Sngh 01) ntroduced the tme-dependent relablty 3

4 analyss nto the lecycle cost optmzaton. Wang and Wang (Wang and Wang 01) proposed a sequental desgn optmzaton method based on a nested extreme response method. A RBDO model was also developed n (Rathod et al. 01) or problems wth relablty degradaton over tme. The accuracy and ecency o the exstng tme-dependent RBDO methodologes can be urther mproved. For example, most o the current methods mbed the relablty constrants n the optmzaton ramework (L, Mourelatos, and Sngh 01, Rathod et al. 01), and ths may requre a large number o uncton evaluatons. SORA s a easble way to mprove the ecency by decouplng the relablty analyss rom optmzaton by employng the Frst-Order Relablty Method (FORM). The drect applcaton o SORA to tme-dependent problems, however, may not be accurate. In ths work, we extend SORA to tme-dependent RBDO based on the concept o equvalent Most robable ont (M). The man contrbutons o ths work nclude the ollowng: (1) the extenson o SORA so that tme-dependent RBDO wth statonary stochastc processes can be solved ecently and accurately, () a new concept o equvalent M, whch allows or decouplng determnstc optmzaton rom tme-dependent relablty analyss, and (3) an ecent approach to searchng or the equvalent M. Ths paper s organzed as ollows. The orgnal SORA s revewed n Secton. The new tme-dependent SORA s dscussed n Secton 3, ollowed by the numercal procedure n Secton 4. Two numercal examples are gven n Secton 5, and conclusons are made n Secton 6.. Revew o SORA The orgnal SORA s or the tme-ndependent RBDO dened n Eq. (3) (Du and Chen 004). SORA perorms RBDO wth sequental cycles o determnstc optmzaton and 4

5 relablty analyss. Ater an optmal desgn pont s ound rom the determnstc optmzaton loop, FORM s perormed at ths pont n the relablty analyss loop. The result o the relablty analyss s then used to reormulate a new determnstc optmzaton model or the next cycle so that the relablty wll be mproved. The process contnues cycle by cycle tll convergence (Du and Chen 004). The determnstc optmzaton n the k-th cycle s ormulated as mn ( d) [ d, μ ] X R s.t. g ( d, T( u )) 0, = 1,,, n gdj ( d) 0, j = 1,,, nd ( k 1) M, p (5) where ( k 1) u M, s the M or the -th probablstc constrant rom the relablty analyss n the (k- 1)-th cycle. T () s the transormaton operator gven by x= T ( u), where u s a vector o the realzaton o standard normal varables U. The result o the optmzaton s the optmal pont ( k) ( k) [, X ] R d μ. Then relablty analyss or the nverse M search s perormed at ( k) ( k) [, X ] R d μ or each o the probablstc constrant unctons. The M u s obtaned rom the ollowng model: ( k ) M, max g ( d, T( u )) ( k ) ux, s.t. ux, = β X, (6) n whch stands or the norm o a vector, and β s called a relablty ndex and s gven by n whch β = Φ 1 ([ p ]) (7) Φ 1 () s the nverse cumulatve densty uncton (CDF) o a standard normal varable. The M u corresponds drectly to the permtted probablty o alure [ p ] as shown ( k ) M, n Eq. (7). I the constrant uncton at ( ) u k M, s less than 0, p wll be less than [ p ]. Thereore, 5

6 g ( d, T( u )) 0 n the determnstc optmzaton leads to the satsacton o the relablty ( k ) M requrement. Ater the k-th cycle, no convergence s reached, the (k+1)-th cycle s perormed. Studes (Cho and Lee 010, Rouh and Ras-Rohan 013) show that SORA perorms well or RBDO problems where FORM s appled. It mght also be applcable or tme-dependent RBDO problems. However, the applcaton s not straghtorward snce there s no drect correspondence o the M to the permtted tme-dependent probablty o alure. Major modcatons o SORA are needed or tme-dependent RBDO. In ths work, we ocus on modyng SORA or tme-dependent RBDO nvolvng only statonary stochastc processes and random varables. 3. Tme-Dependent SORA (t-sora) In ths secton, we rst ntroduce the man dea o tme-dependent SORA (t-sora) and then ts detals. 3.1 Overvew o t-sora In the lmt-state uncton g ( XYτ, ( )), the components o Y( τ ) are ndependent statonary stochastc processes. Snce the dstrbuton o Y( τ ) on [0, t ] does not change, so does the M o g ( XYτ, ( )). Fg. 1 shows the lowchart o t-sora. As t-sora nherts the eatures o the orgnal SORA, the steps o the two methods are smlar. The entre optmzaton s perormed cycle by cycle tll convergence. Each cycle conssts o decoupled determnstc optmzaton and tme-dependent relablty analyss. However, the major derence or challenge s that the M correspondng to the permtted probablty o alure [ p ] cannot be drectly used n the determnstc optmzaton. The reason s explaned as ollows. 6

7 One cycle Intal desgn Determnstc optmzaton d, μ X Tme-dependent relablty analyss Equvalent M u Formulate a new optmzaton model N Converge? Y Optmal desgn Fg. 1. Flowchart o t-sora Wth Y( τ ), all the random varables at τ become = [ XYτ, ( )]. The M * u z s ound by max g ( d, T( u )) u 1 s.t. u = Φ ([ p ]) (8) where u = [ u, u ] s a vector o the realzatons o standard Gaussan random varables X Y U = [ UX, UY] wth U X and rom X and Y( τ ), respectvely. Then usng the orgnal SORA, we have U Y beng the transormed standard Gaussan random varables * r{ g( dxy,, ( ) g( d, T( u z ))} p τ ) = [ ] (9) However, the probablty * r{ g( XY, ( t)) g ( d, T( u z )), t [0, t]} over [0, t ] s always greater than or equal to the tme nstantaneous probablty g dxyτ) g d, T u z ) (L, * r{ (,, ( ) ( ( ) } Mourelatos, and Sngh 01, Sngh and Mourelatos 010a), and thereore * r{ g( XY, ( t)) g( d, T( u z )), t [0, t]} [ p ] (10) As a result, the constrant * g (, ( )) 0 d T u z n the determnstc optmzaton can satsy r{ g ( dxyτ,, ( )) 0} [ p ] at only tme nstant τ, and t may not satsy the tme-dependent relablty requrement r{ g( XYt, ( )) 0, t [0, t]} [ p ] over the entre perod o tme [0, t ]. 7

8 u To address the above challenge, we propose a concept o equvalent M and denote t by. It s the M at whch the lmt-state uncton satses r{ g( XY, ( t)) g ( d, T( u )), t [0, t]} = [ p ] (11) u can be obtaned by addng the above condton to the nverse M search model. The new model s gven by max g ( T( u )) [ u, β ] s.t. u = β r{ g( XY, ( t )) g( T( u )), t [0, t]} = [ p ] (1) The relablty ndex β s also treated as a desgn varable n the M search snce t cannot be predetermned. Then the task o tme-dependent relablty analyss s to search or the equvalent Ms wth Eq. (1). Dong so, however, s too computatonally expensve. In ths work, we develop an ecent algorthm or Eq. (1). Next, we dscuss how to use the equvalent M n the determnstc optmzaton and then how to search or the equvalent M. 3. Determnstc optmzaton Usng the equvalent M, we ormulate the determnstc desgn optmzaton or the k-th cycle as mn ( d) ( d, μ ) X R s.t. g ( d, T( u ) 0, = 1,,, n gdj ( d) 0, j = 1,,, nd ( k 1), p (13) n whch s the equvalent M or the -th relablty constrant. The optmzaton model s ( k 1) u, smlar to that o the orgnal SORA. The only derence s that the Ms are replaced by the equvalent Ms. As dscussed above, the use o the equvalent Ms n constrants 8

9 g ( d, T( u ) 0, = 1,,, n ensures the satsacton o tme-dependent relablty ( k 1), p requrements. 3.3 Tme-dependent relablty analyss The task heren s to search or the equvalent Ms or g ( XY, ( τ )) = g ( ), where = [ XYτ, ( )], gven desgn varables [ d, μ ]. As ndcated n Eq. (1), there are two research X R ssues. The rst s how to calculate the tme-dependent probablty r{ g( XY, ( t)) g ( d, T( u )), t [0, t]}, and the second s how to solve Eq. (1) ecently Calculaton o r{ g( XY, ( t)) g ( d, T( u )), t [0, t]} The task s to calculate the tme-dependent probablty r{ g( XYt, ( )) c, t [0, t]}, where c= g (, ( )) d T u on the condton that c s known. Tme-dependent relablty methods have been extensvely studed (Hu and Du 01, L and Mourelatos 009, Sngh, Mourelatos, and Nkolads 011a, Sngh and Mourelatos 010b, Sngh, Mourelatos, and Nkolads 011b). Amongst them, the most commonly used one s the upcrossng rate method wth the Rce s ormula (Rce 1944). Ths method s accurate when the probablty o alure s low, but may be naccurate when the probablty o alure s hgh. Many mprovements have been made or the Rce s ormula, such as consderng the correlaton between upcrossngs (Madsen and Krenk 1984), makng emprcal correctons to the Rce s ormula (Vanmarcke 1975), and employng mportant samplng (L, Mourelatos, and Sngh 01). A bre revew o the upcrossng rate method s gven n Appendx A. In ths work, we use the rst order samplng approach (FOSA), whch completely removes the osson assumpton used n the Rce s ormula. The accuracy and ecency are sgncantly mproved. Detals o FOSA s avalable n Re. (Hu and Du 015). 9

10 3.3. Equvalent M search Exstng M search algorthms cannot be used or Eq. (1) due to the constrant r{ g( XY, ( t)) g( d, T( u )), t [0, t]} = [ p]. Drectly solvng the model n Eq. (1) wll be tme consumng because t nvolves a double-loop procedure. Usng the same strategy as SORA, we propose to perorm the M search wth a sequental procedure. The dea s to separate the M search rom the probablty calculaton, one s or searchng or u, and the other s or searchng or β. At rst, the nverse M search s perormed gven β, wth the ollowng model: max g ( T( u )) u s.t. u = β (14) It s the regular M search and produces the M u gven β. Then the next analyss s perormed to update β gven u. The purpose s to nd a new β so that r{ g( XY, ( t)) g( d, T( u )), t [0, t]} = [ p]. How to update β wll be dscussed n Sec The two analyses are perormed repeatedly tll convergence as shown n Fg.. Ater convergence, the M search produces the equvalent M u. Intal β M search max g ( T( u )) u s.t. u = β u Relablty ndex updatng Fnd β such that r{ g( XY, ( t )) g ( d, T( u )), t [0, t]} = [ p ] N β Converge? Y u Fg.. Tme-dependent M search 10

11 3.3.3 Relablty ndex updatng We now dscuss how to update β so that p ( t) = r{ g ( T( U )) 0, t [0, t]} = [ p ]. Snce FORM s used, we approxmate g ( T( U )) at u. As shown n Appendx A, ater approxmaton, the tme-dependent probablty s T p () = r{ g ( T( U )) 0 t [0, t] } r{ L( t ) = a U u, t [0,] t } (15) t T where L( τ ) = a U s a lnear combnaton o U, and a s a constant vector evaluated at u and s gven n Eq. (A). I p () t > [p ], we should obtan a new relablty ndex β so that p () t = [ p ]. Let the global maxmum o L( τ ) over [0, t ] be W ; namely p () t can then be calculated by W = max{ L( t), t [0, t]} (16) p t () t = r{ W u } (17) I p () t > [p ], we should reduce the old relablty ndex u and obtan an updated relablty ndex β such that or r{ L( t ) β, t [0, t]} = [p ] (18) r{w β} = [ ] (19) p It s obvous that β < u. The problem now becomes to nd the percentle value o W gven a probablty level [ p ]. It s a dcult task because there may not be a close-orm soluton to the dstrbuton o the extreme value W. Wang (Wang and Wang 01) proposed a 11

12 krgng model method to approxmate the extreme value dstrbuton, but the method s lmted to lmt-state unctons n the orm o g( X, t) wthout any nput stochastc processes. Heren we use a samplng method. T Recall that L( τ ) = a U s a statonary Gaussan process wth known coecents a. We can then use smulatons to obtan ts sample trajectores, and or each trajectory, we nd the maxmum value. Then the samples o W wll be avalable or the estmaton o the CDF o W. The CDF wll then produce β as ndcated n Eq. (19). The samples can be ecently generated usng the Orthogonal Seres Expanson (OSE) (hang and Ellngwood 1994), whch s gven n Appendx B. Once the samples o W are avalable, the percentle value o W n Eq. (19) s approxmated. Snce [ p ] s small, β s n the ar rght tal o the dstrbuton o W. To obtan an accurate result, we use the saddlepont approxmaton (SA) method (HU and Du 013). The detals are provded n Appendx C. Snce the samplng approach s based on L( τ ) = a U lmt-state uncton g () wll not be called Numercal procedure o the tme-dependent relablty analyss T, the orgnal We now summarze the strategy o the tme-dependent relablty analyss and ts procedure. For a general lmt-state uncton g ( XY, ( τ )) = g ( ), when t s approxmated at an M, a number o teratons are needed to solve the ollowng model: max g ( T( u )) [ u, β ] s.t. u = β r{ W β} = [ p ] (0) 1

13 It s derved rom the orgnal model n Eq. (1) when FORM s employed. The model s solved wth the procedure shown n Fg. where the M search and relablty ndex updatng are perormed separately and sequentally. The man steps are as ollows: Step 1: Intalzaton: set the ntal relablty ndex β. Snce the tme-dependent probablty o alure s usually larger than the tme-ndependent ones, based on our experence, we recommend usng the ollowng ntal value: β = Φ ( ) (1) 1 1. [ p ] Step : M search: Search or the M usng Eq. (14). The results are the M u and vector a (gven n Eq. (A)). Step 3: Update the relablty ndex: (1) Construct L( τ ) by L( τ ) = a U T. () Generate samples or L( τ ) over [0, t ]. (3) Obtan the samples o the extreme value o W. (4) Use SA to compute the relablty ndex β. Step 4: Check convergence: I the derence between the current β value and prevous β s larger than a predened tolerance, repeat Steps through 4; otherwse, set the equvalent M u = u and stop. The convergence tolerance can be set to 0.01 or or other small numbers. A more detaled lowchart s gven n Fg. 3. The above procedure s or a general lmt-state uncton. It should be executed or all the lmt-state unctons n the overall RBDO model. 4. Summary o the Numercal rocedure We now summarze the procedure o the entre RBDO and show t n Fg. 4. Step 1: Intalze parameters. (1) Dene the ntal desgn varables. () Set k = 1. Step : erorm determnstc optmzaton. I k = 1, solve determnstc optmzaton at mean values o random varables and man unctons o stochastc processes. I k > 1, ormulate the 13

14 determnstc optmzaton model usng the equvalent Ms ( k u 1),, where = 1,,, np, obtaned rom the ( k 1)-th cycle; then solve the optmzaton model gven n Eq. (13). The ( k) ( k) optmal soluton s [ d,μ ]. X [ d, μ X] Step 1: Intalzaton 1 β = 1. Φ ( p ]) [ Step : M search max g ( T( u )) u s.t. u = β u, a N Step 3: Update relablty ndex Approxmate L( τ ) usng Eq. (B1) Obtan extreme value samples Compute relablty ndex usng Eq. (C3) β Step 4: Convergence? u Y = u Fg. 3. Detaled lowchart or tme-dependent relablty analyss ( k) ( k) Step 3: erorm tme-dependent relablty analyss at [ d,μ ] ollowng the procedure n X Fg. 3. The soluton s the equvalent Ms ( ) u k,, where 1,,, np =. Step 4: Check convergence. I the lmt-state unctons satsy g ( d, T( u )) ε () ( k) ( k), where ε s a small postve number, then the optmal soluton s ound and stop. Otherwse, update the cycle counter by k = k+ 1, and repeat Steps through 4. 14

15 0 d, 0 u and 0 β Step 1 Intalze parameters k = 1 Step Determnstc Optmzaton mn ( d) [ d,μ X ] ( k 1) s.t. g ( d, T( u, ) 0, = 1,,, n gdj ( d) 0, j = 1,,, nd p ( k) ( k) [ d,μ X ] Step 3 Tmedependent relablty analyss u ( k ), N k = k+ 1 Step 4 ( k) ( k) g ( d, T( u )) ε? Fg. 4. Numercal procedure o t-sora, Y Optmal desgn [ d, μ X] Smlar to the orgnal SORA, the t-sora usually converges wthn a ew cycles, and the typcal number o cycles s between three and ve. In addton to the decouplng between optmzaton and relablty analyss, the proposed approach to the equvalent M search converges quckly, and ths also makes t-sora ast. 5. Numercal Examples 5.1 A two-bar rame under a stochastc orce A two-bar rame s subjected to a stochastc orce Ft () as shown n Fg. 5. The dstances O 1 O 3 and O 1 O are random parameters and are denoted by l 1 and l, respectvely. Falures occur when the maxmum stresses o the two bars are larger than ther materal yeld strengths S 1 and S. The dameters D 1 and D o the two bars are random desgn varables. 15

16 O A-A D 1 B-B D A l A B O 1 B l 1 Fg. 5. A two-bar rame under stochastc orce The lmt-state unctons are gven by O 3 F(t) g( dxyτ,, ( )) = 4 F( τ) l + l ( lπds) 1 (3) g( dxyτ,, ( )) = 4 F( τ) l ( lπds) 1 (4) 1 where X= [ XR, X], X R = [ D1, D], X = [ l1, l, S1, S], d = [ µ, µ ], and Y () t = [ Ft ()]. D1 D The normaton known s gven n Table 1, where STD and G stand or a standard devaton and a Gaussan process, respectvely. The auto-correlaton coecent unctons o F( τ ) s n whch ζ = 0.1 year s the correlaton length. { ( ) } F ρ ( τ, τ ) = exp ( τ τ ) ζ (5) 1 1 The objectve s to mnmze the weght o the two bars, and the RBDO model or a servce perod o [0,10] years s ormulated as 16

17 mn ( d) = pmlm 4 4 (, ) 1 D + p m 1 l + m 1 lm D d μ X s.t. r{ g ( dxy,, ( t)) 0, t [0, t]} [ p ], = 1, 0.07 m md 0.5 m, =1, (6) where [ p 1] = 0.01, [ p ] = 0.001, and t = 10 years. Table 1. Random varables and stochastc process Varable Mean STD Dstrbuton Autocorrelaton 3 D µ 1 D1 3 D µ D S a 10 S a 10 l m l 0.3m Ft () N 1 10 m Normal N/A 1 10 m Normal N/A 1.7 a Lognormal N/A 1.7 a Lognormal N/A m Normal N/A m Normal N/A 5 10 N G Eq. (5) To evaluate the accuracy and ecency o t-sora, we used three methods to solve the problem wth the same startng pont. The three methods are the t-sora wth the Orthogonal Seres Expanson (OSE) method presented n Appendx B, the double-loop method usng the same tme-dependent relablty analyss method as t-sora, and the double-loop method wth the Rce s ormula or tme-dependent relablty analyss presented n Appendx A. Next we call the latter two methods the Double (OSE) and Double (Rce). The parameters o OSE used by t-sora and Double (OSE) are gven below. The number o tme nstants that dvde [0, t ] equally: Q = 500 The number o samples generated at each tme nstant: The number o terms used n the OSE model: M = N = 10 Table shows the convergence hstory o t-sora. The optmal soluton was obtaned wthn three cycles. Ater the rst cycle, the two lmt-state unctons were much larger than zero, and

18 ths s the ndcaton that the relablty requrements were not met. Ater the thrd cycle, the two lmt-state unctons were close to zero. Then the tme-dependent probabltes o alure were almost at ther target values. Table Convergence hstory o t-sora k (m 3 ) ( µ D, µ ) 1 D (m) β 1 β g 1 ( d, T( u,1)) g ( d, T( u,)) (0.0831, ) (0.1051, ) (0.1051, 0.098) Table 3 shows the nal results rom the three methods. We use the number o uncton calls (NOFC) to measure the ecency. t-sora and Double (OSE) produced almost dentcal results. t-sora s much more ecent than the Double (OSE) and Double (Rce) methods. The ourth and th columns o Table 3 present the probabltes o alure ater the optmzaton. Snce t- SORA does not compute the probabltes o alure drectly, ther values are not avalable. The probabltes o alure o the Double (OSE) and Double (Rce) methods are computed by the OSE-based samplng method (Appendx B) and Rce s ormula (Appendx A), respectvely. The results show that the relablty constrants were satsed by the three optmzaton methods. Table 3 Optmal results Method (m 3 ) ( µ D, µ ) 1 D (m) p () 1 t p () t NOFC t-sora (0.1051, 0.098) N/A N/A 715 Double (OSE) (0.1051, 0.098) Double (Rce) (0.1066, ) To very the accuracy, we also perormed Monte Carlo Smulaton at the optmal desgn ponts n Table 3 rom the three methods. In MCS, [0, t ] was dscretzed nto 00 tme nstants, and 10 6 samples were generated at each tme nstants. Table 4 gves the percentage errors, and 18

19 Table 5 gves the 95% condence ntervals o the MCS solutons. The percentage error s computed by ε = p t p t p t (7) MCS MCS ( ) ( ) / ( ) 100% For t-sora and Double (OSE), p () t s calculated by the OSE-based samplng method, and or Double (Rce), t s calculated by the method based on Rce s method. p MCS () t s the probablty o alure obtaned rom MCS. Table 4 Accuracy comparson () MCS p 1 t p 1 () t Error (%) () MCS p t p () t Error (%) t-sora Double (OSE) Double (Rce) Table 5 95% condence ntervals o MCS solutons t-sora Double (OSE) Double (Rce) MCS p [0.009, ] [0.009, ] [0.0044, ] 1 p [8.77, 9.97] 10-4 [8.77, 9.97] 10-4 [4.96, 5.88] 10-4 MCS The results ndcate that the accuracy o t-sora s good. For the t-sora and Double (OSE) methods, at the optmal desgn ponts, the actual tme-dependent probabltes o alure are very close to the permtted ones. The probabltes o alure rom the Double (Rce) method are much lower than the permtted ones. The reason s that the Rce s ormula overestmates the probablty o alure, whch resulted n an over-desgn or ths problem. 5. A smply supported beam under stochastc loadngs A smply supported beam shown n Fg. 6 s subjected to two stochastc loadngs, whch are the stochastc orce F(t), and the unormly dstrbuted loadng qt (). The heght a and wdth b o 19

20 the cross secton are random desgn varables. A alure o the beam occurs when the stress exceeds the ultmate strength o the materal S. The weght o the beam s expected to be mnmzed under the constrant that the tme-dependent probablty o alure o the beam over 30 years s less than The lmt-state uncton o the beam s gven by ( st ) g( dxyt,, ( )) = 4 F( t) l / s + q( t) l /8 + ρ abl /8 ( ab S) 1 (8) where X= [ X, X ], X = [ ab, ], X = [ S], d = [ µ, µ ], and Y( τ) = [ F( τ), q( τ)], S s the R R ultmate strength, ρ st s the densty, and l s the length o the beam. Table 6 gves the random varables, parameters, and stochastc processes. The autocorrelaton coecent unctons o F( τ ) and q( τ ) are and a { ( ) } (, ) exp ( ) / F 1 1 b ρ τ τ = τ τ ζ (9) ρ( τ, τ) = cos( πτ ( τ)) (30) q 1 1 respectvely, where ζ = 0.8 year s the correlaton length o F( τ ). l l/ A F(t) q(t) A-A b a A The RBDO model s gven by Fg. 6. A beam under stochastc loadngs 0

21 mn ( d) = mm a b [ d, μ ] X s.t. r{ g( dxy,, ( t)) 0, t [0, t]} [ p ] mb 4 ma; 0.04 m ma 0.15 m 0.15 m mb 0.5 m (31) where [ p ] = 0.05 and t = 30 years. The RBDO model was solved by the t-sora, Double (OSE), and Double (Rce) methods wth the same ntal desgn pont. Table 7 gves the convergence hstory o t-sora. The results show that t-sora converged wth three cycles. Table 6 Varables, parameters, and stochastc processes Varable Mean STD Dstrbuton Autocorrelaton a b 8 S.4 10 a -3 µ a 0-3 µ b m Lognormal N/A 5 1 m Lognormal N/A a Lognormal N/A F( τ ) 6000 N 600 N G Eq. (9) q( τ ) 900 N/m 90 N/m G Eq. (30) l 15 m N/A Determnstc N/A ρ 78.5 kn/m 3 N/A Determnstc N/A st Table 7 Convergence hstory o t-sora k (m ) ( µ a, µ b) (m) β g( d, T( u,1)) (0.0403, ) (0.0460, ) (0.0461, 0.184) Table 8 presents the nal results rom the three methods. The results show that t-sora s much more ecent than the other two methods. 1

22 Table 8 Optmal results Method (m ) ( µ a, µ b) (m) p () t NOFC t-sora (0.0461, 0.184) N/A 156 Double (OSE) (0.0463, ) Double (Rce) (0.0478, ) Table 9 gves the probabltes o alure rom MCS at the optmal desgn ponts rom the three aorementoned methods. Table 10 presents the 95% condence ntervals o the MCS solutons. The tme nterval [0, 30] year was dvded nto 600 tme nstants, and 10 6 samples were generated at each tme nstant or MCS. The t-sora and Double (OSE) methods are more accurate than the Rce s ormula, whch overestmated the probablty o alure. The optmal desgn obtaned rom the Double (Rce) method s thereore conservatve. Table 9 Accuracy comparson p () t p MCS () t Error (%) t-sora Double (OSE) Double (Rce) Table 10 95% condence ntervals o MCS solutons t-sora Double (OSE) Double (Rce) p t [ ] [ ] [ ] MCS () Although the two examples nvolve explct lmt-state unctons, they are actually treated as black-box unctons snce the dervatves o the lmt-state unctons are evaluated numercally.

23 6. Concluson Tme-dependent Sequental Optmzaton and Relablty Analyss (t-sora) method s developed or problems wth both random varables and stochastc processes. To address the lmtaton that there s no drect connecton between tme-dependent relablty and the Most robable ont (M), t-sora uses the equvalent M, whch drectly corresponds to the requred tme-dependent relablty. Ths ensures that the overall optmzaton be solved sequentally n cycles o determnstc optmzaton and relablty analyss. The results show that t-sora can eectvely solve desgn optmzaton wth tme-dependent relablty constrants. The proposed method s based on FORM. Its accuracy s then aected by the lnearzaton made by FORM. However, the proposed method may not be lmted to FORM. I the lmt-state uncton n the transormed normal varable space s hghly nonlnear, more accurate relablty analyss methods, such as the Second Order Relablty Method (SORM), can also be used. t-sora s or problems wth only statonary stochastc processes. The same strategy may be extended to problems wth non-statonary processes. The mplementaton, however, wll be sgncantly derent and wll be more challengng. It needs a urther nvestgaton. Acknowledgement Ths materal s based upon the work supported by the Natonal Scence Foundaton through grant CMMI and the Intellgent Systems Center at the Mssour Unversty o Scence and Technology. Appendx A. Relablty analyss wth the Rce s ormula and FORM For a lmt-state uncton g ( dxy,, ( t)) = g ( d, ( t)), where () t = [ XY, ()] t, ts M s obtaned rom 3

24 mn U s.t. g ( T[ U( t)]) 0 (A1) where U () t = [ U, U ()] t s the standard normal varables assocated wth X and Y () t. X Y Ater the M u() t = [ u, u ()] t s ound, the lmt-state uncton X Y g ( dxy,, ( t)) = g ( d, ( t)) s lnearzed at the M, the tme-dependent probablty o alure gven n Eq. (5) s then approxmated by (Hu and Du 01, Hagen and Tvedt 1991): T p ( t) = r{ g ( dxy,, ( t)) 0, t [0, t]} r{ L( t) = a U ( t) > βt ( ), t [0, t]} (A) n whch βτ ( ) = u( τ) and a = u( τ )/ β. The Rce s ormula gves the upcrossng rate by (Rce 1944) ( ) + v () t = ω() t φβ ( ()) t Ψ β() t ω() t (A3) where φ() s the probablty densty uncton (DF) o a standard normal varable, and and β() t = β() t t (A4) Ψ ( x) = φ( x) xφ ( x) (A5) T ω t = + C 1 T () aa a (, tt) a (A6) n whch a = a( t) t (A7) and 4

25 ρ1( t1, t) t1 t 0 0 C 1( t1, t) = t1= t= t ρm( t1, t) t1 t t1= t= t (A8) n whch ρ l ( t1, t), l = 1,, m, are the coecents o the autocorrelaton o stochastc process U () t, and m s the number o stochastc processes. Snce the stochastc processes Y () t are Y l assumed to be statonary, a = 0, and β = 0. p s computed by (Hu and Du 01) t + { ( t t 0 )} p (t) = 1 R(0) exp v ( ) d (A9) where R( 0) = Φ ( β ) s the tme nstantaneous relablty at the ntal tme nstant. Appendx B. Orthogonal Seres Expanson (OSE) As shown n Eq. (A), the tme-dependent probablty o alure p () t s approxmated by T p ( t) = r{ G( t) = g ( dxy,, ( t)) 0, t [0, t]} r{ L( t) = a U ( t) β, t [0, t]}, where G( τ ) s a non-gaussan stochastc process and L( τ ) s a standard Gaussan stochastc process. I the maxmum value o L( τ ) over [0, t ], W, s avalable, accordng to Eq. (17), p ( t) = r{ W β}. The dstrbuton o W can be obtaned rom the samples o L( τ ), and the samples may be generated rom the OSE method. OSE approxmates L( τ ) as ollows (hang and Ellngwood 1994): M M λξ h j j = 0 j= 0 L( τ) ( ( τ)) (B1) 5

26 n whch λ s the -th egenvalue o covarance matrx Σ, j s the projecton o the -th egenvector o covarance matrx Σ on the j-th Legendre polynomal, and hj () t s the j-th Legendre polynomal, ξ, where = 1,,, M, are M ndependent standard normal varables, and Σ s a matrx wth element Σ j gven by (hang and Ellngwood 1994) t t Σ = ρ h ( t ) h ( t ) dt dt (B) j 0 0 t1 t 1 j 1 where ρ T = ac( τ, τ ) a (B3) ττ 1 1 and C( τ1, τ ) s a dagonal matrx wth the dagonal element beng the covarance o U ( ) τ 1 and T U ( τ ). Once the approxmated response L( τ ) s avalable, N samples can be generated at Q dscretzng nstants over [0, t ]. The samples are gven n matrx L N Qas below. L N Q lt ( 1, 1) lt (, 1) lt ( Q, 1) lt ( 1, ) lt (, ) lt ( Q, ) = lt ( 1, N) lt (, N) lt ( Q, N) N Q (B4) N samples o the extreme value W can then avalable through the ollowng equatons: w = max{ lt (, j), lt (, j),, lt (, j)}, where j= 1,,, N (B5) j 1 Q Appendx C. Saddlepont Approxmaton (SA) At rst, the cumulants o W are computed rom the samples by (Fsher 198) 6

27 κ1 = m1 N κ = ( Nm m1) ( N( N 1)) 3 κ 3 = (m1 3 nm1m + N m3) ( N( N 1)( N )) 6m + 1nm m 3 N( N 1) m 4 N( N + 1) mm + N ( N + 1) m κ4 = + N( N 1)( N )( N 3) N( N 1)( N )( N 3) (C1) where κ s the -th cumulant o W, samples W and are gven by m ( s = 1,,3,4) are the sums o the s-th power o the s m s N = w (C) = 1 s n whch w s the -th sample o W gven n Eq. (B5). In ths work, the rst our moments are used. Hgher order may also be used. Once κ j, j = 1,,3,4, are avalable, the relablty ndex β s updated by β= κ+ κη1! + κη! + κη 3! (C3) where η s the saddlepont, whch satses the ollowng equatons: ( ) 1 [ p ] = Φ ( z) + ( z) 1 z 1 v (C4) ' { } 1/ L L z = sgn( η) ηk ( η) K ( η) (C5) 1/ v= η K " L ( η) (C6) L 4 K ( η) = κη! (C7) = 1 and 4 " j L η κ κη j j= 3 K ( ) = + ( j )! (C8) where sgn( η ) = +1, 1, or 0, dependng on whether η s postve, negatve, or zero. 7

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