Probabilistic Engineering Mechanics. Stochastic sensitivity analysis by dimensional decomposition and score functions

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1 Probablstc Engneerng Mechancs 24 (2009) Contents lsts avalable at ScenceDrect Probablstc Engneerng Mechancs journal homepage: Stochastc senstvty analyss by dmensonal decomposton score functons Sharf Rahman College of Engneerng, The Unversty of Iowa, Iowa Cty, IA 52242, Unted States a r t c l e n f o a b s t r a c t Artcle hstory: Receved August 2007 Receved n revsed form 22 June 2008 Accepted 24 July 2008 Avalable onlne 3 July 2008 Keywords: Gradents Relablty Relablty-based desgn Robust desgn Optmzaton Monte Carlo smulaton Ths artcle presents a new class of computatonal methods, known as dmensonal decomposton methods, for calculatng stochastc senstvtes of mechancal systems wth respect to probablty dstrbuton parameters. These methods nvolve a herarchcal decomposton of a multvarate response functon n terms of varables wth ncreasng dmensons score functons assocated wth probablty dstrbuton of a rom nput. The proposed decomposton facltates unvarate bvarate approxmatons of stochastc senstvty measures, lower-dmensonal numercal ntegratons or Lagrange nterpolatons, Monte Carlo smulaton. Both the probablstc response ts senstvtes can be estmated from a sngle stochastc analyss, wthout requrng performance functon gradents. Numercal results ndcate that the decomposton methods developed provde accurate computatonally effcent estmates of senstvtes of statstcal moments or relablty, ncludng stochastc desgn of mechancal systems. Future effort ncludes extendng these decomposton methods to account for the performance functon parameters n senstvty analyss Elsever Ltd. All rghts reserved.. Introducton Senstvty analyss provdes an mportant nsght about complex model behavor,2 so that one can make nformed decsons on mnmzng the varablty of a system 3, or optmzng a system s performance wth an acceptable rsk 4. For estmatng the dervatve or senstvty of a general probablstc response, there are three prncpal classes of methods or analyses. The fnte-dfference method 5 nvolves repeated stochastc analyses for nomnal perturbed values of system parameters, then nvokng forward, central, or other dfferentaton schemes to approxmate ther partal dervatves. Ths method s cumbersome often expensve, f not prohbtve, because evaluatng probablstc response for each system parameter, whch consttutes a complete stochastc analyss, s already a computatonally demng task. The two remanng methods, the nfntesmal perturbaton analyss 6,7 the score functon method 8, have been mostly vewed as competng methods, where both performance senstvtes can be obtaned from a sngle stochastc smulaton. However, there are addtonal requrements of regularty condtons, n partcular smoothness of the performance functon or the probablty measure 9. For the nfntesmal perturbaton analyss, the probablty measure s fxed, the dervatve of a Correspondng address: Department of Mechancal & Industral Engneerng, The Unversty of Iowa, Iowa Cty, IA 52242, Unted States. Tel.: ; fax: E-mal address: rahman@engneerng.uowa.edu. The nouns dervatve senstvty are used synonymously n ths paper. performance functon s taken, assumng that the dfferental ntegral operators are nterchangeable. The score functon method, whch nvolves probablty measure that contnuously vares wth respect to a desgn parameter, also requres a somewhat smlar nterchange of dfferentaton ntegraton, but n many practcal examples, nterchange n the score functon method holds n a much wder range than that n nfntesmal perturbaton analyss. Nonetheless, both methods, when vald, are typcally employed n conjuncton wth the drect Monte Carlo smulaton, a premse well-suted to stochastc optmzaton of dscrete event systems. Unfortunately, n mechancal desgn optmzaton, where stochastc response senstvty analyses are requred at each desgn teraton, even a sngle Monte Carlo smulaton s mpractcal, as each determnstc tral of the smulaton may requre expensve fnte-element or other numercal calculatons. Ths s the prncpal reason why nether the nfntesmal perturbaton analyss nor the score functon method have found ther way n to the desgn optmzaton of mechancal systems. The drect dfferentaton method, commonly used n determnstc senstvty analyss 0, provdes an attractve alternatve to the fnte-dfference method for calculatng stochastc senstvtes. In conjuncton wth the frst-order relablty method, Lu Der Kureghan ther smlar work has sgnfcantly contrbuted to the development of such methods for obtanng relablty senstvtes. The drect dfferentaton method, also capable of generatng both relablty ts senstvtes from a sngle stochastc analyss, s partcularly effectve n solvng fnteelement-based relablty problems, when () the most probable pont can be effcently located (2) a lnear approxmaton of the performance functon at that pont s adequate. Therefore, the /$ see front matter 2008 Elsever Ltd. All rghts reserved. do:0.06/j.probengmech

2 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) drect dfferentaton method nherts hgh effcency of the frstorder relablty method, but also ts lmtatons. In contrast, the three senstvty methods descrbed n the precedng are ndependent of underlyng stochastc analyss. Ths artcle presents a new class of computatonal methods, known as dmensonal decomposton methods, for calculatng stochastc senstvtes of mechancal systems wth respect to probablty dstrbuton parameters. The dea of dmensonal decomposton of a multvarate functon, orgnally developed by the author s group for statstcal moment 2,3 relablty 4 analyses, has been extended to stochastc senstvty analyss, whch s the focus of the current paper. Secton 2 descrbes a unfed probablstc response senstvty, derves score functons assocated wth a number of probablty dstrbutons. Secton 3 presents the dmensonal decomposton method for calculatng the probablstc senstvtes, usng ether the numercal ntegraton or the smulaton method score functons. The computatonal effort requred by the decomposton method s also dscussed. Four numercal examples llustrate the accuracy, computatonal effcency, usefulness of the senstvty method n Secton 4. Secton 5 states the lmtatons of the proposed method. Fnally, conclusons are drawn n Secton Probablstc response senstvty Let (Ω, F ) be a measurable space, where Ω s a sample space F s a σ -feld on Ω. Defned over (Ω, F ), consder a famly {P θ : F 0, } of probablty measures, where θ = {θ,..., θ M } T R M s an M-dmensonal vector of determnstc parameters R M s an M-dmensonal, real, vector space. In other words, a sample pont ω Ω obeys the probablty law P θ (F) for any event F F θ R M, so that the probablty trple (Ω, F, P θ ) depends on θ. Let {X = {X,..., X N } T : (Ω, F ) (R N, B N )} wth B N as the Borel σ -feld on R N denote a famly of R N -valued nput rom vector, whch descrbes statstcal uncertantes n loads, materal propertes, geometry of a mechancal system. The probablty law of X s completely defned by a famly of jont probablty densty functons {f X (x; θ), x R N, θ R M } that are assocated wth probablty measures {P θ, θ R M }. Let y(x), a realvalued, measurable transformaton on (Ω, F ), defne a relevant performance functon of a mechancal system. It s assumed that y : (R N, B N ) (R, B) s not an explct functon of θ, although y mplctly depends on θ va the probablty law of X. The objectve of stochastc senstvty analyss s to obtan the partal dervatves of a probablstc characterstc of y(x) wth respect to a parameter, =,..., M, gven a reasonably arbtrary probablty law of X. 2.. Statstcal moments relablty Denote by L q (Ω, F, P θ ) a collecton of real-valued rom varables ncludng y(x), whch s defned on (Ω, F, P θ ) such that E y q (X) <, where q s an nteger E θ represents the expectaton operator wth respect to the probablty measure {P θ, θ R M }. If y(x) s n L q (Ω, F, P θ ), then ts qth moment, defned by the multfold ntegral m q (θ) := E θ y q (X) := y q (x)f X (x; θ)dx; q =, 2,..., () R N exsts s fnte. A smlar ntegral appears n tme-nvarant relablty analyss, whch entals calculatng the falure probablty P F (θ) := P θ X Ω F = I ΩF (x)f X (x; θ)dx := E θ IΩF (X), (2) R N where Ω F := {x : y(x) < 0} s the falure set for component relablty analyss; Ω F := {x : K k= y(k) (x) < 0} Ω F := {x : K k= y(k) (x) < 0} are the falure sets for seres-system parallel-system relablty analyses, respectvely, wth y (k) (x) representng the kth out of K performance functons, {, x ΩF I ΩF (x) := ; x R 0, x Ω \ Ω N (3) F s an ndcator functon. Therefore, expressons of both ntegrals or expectatons n Eqs. () (2) can be consoldated nto a generc probablstc response h(θ) = E θ g(x) := g(x)f X (x; θ)dx, (4) R N where h(θ) g(x) are ether m q (θ) y q (x), respectvely, for statstcal moment analyss or P F (θ) I ΩF (x), respectvely, for relablty analyss Senstvty analyss by score functons Consder a dstrbuton parameter, =,..., M, suppose that the dervatve of a generc probablstc response h(θ), whch s ether the statstcal moment of a mechancal response or the relablty of a mechancal system, wth respect to s sought. For such senstvty analyss, the followng assumptons are requred 8.. The probablty densty functon f X (x; θ) s contnuous. Dscrete dstrbutons havng jumps at a set of ponts, or a mxture of contnuous dscrete dstrbutons, can be treated smlarly, but wll not be dscussed here. 2. The parameter Θ R, =,..., M, where Θ s an open nterval on R. 3. The partal dervatve f X (x; θ)/ exsts s fnte for all x Θ R. In addton, h(θ) s a dfferentable functon of θ R M. 4. There exsts a Lebesgue ntegrable domnatng functon r(x) such that g(x) f X(x; θ) θ r(x) (5) for all θ R M. The assumptons 4 are known as the regularty condtons. Takng the partal dervatve of both sdes of Eq. (4) wth respect to gves h(θ) = R N g(x)f X (x; θ)dx. (6) By nvokng assumpton 4 the Lebesgue domnated convergence theorem 5, the dfferental ntegral operators can be nterchanged, yeldng h(θ) = g(x) f X(x; θ) dx R N = g(x) ln f X(x; θ) f X (x; θ)dx R N = E θ g(x) ln f X(X; θ) provded f X (x; θ) 0. Defne ; =,..., M, (7) (x; θ) := ln f X(x; θ), (8) whch s known as the frst-order score functon for the parameter 8. Therefore, the frst-order senstvty of h(θ) can be expressed by h(θ) = E θ g(x) θ (X; θ) ; =,..., M. (9)

3 280 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Table Log-dervatves for Gaussan lognormal dstrbutons Dstrbuton f X (x ; µ, σ ) Gaussan 2πσ exp x + 2 Lognormal a 2πx σ exp ( ) 2 x µ 2 σ ; ( ln x µ σ ) 2 ; ln f X (x ;µ,σ ) µ ln f X (x ;µ,σ ) σ ( ) x µ σ σ σ ( σ µ + ln x µ µ σ 2 σ ) σ µ + (ln x µ ) σ µ σ ( x µ σ ) 2 ( ) σ σ σ + ln x µ µ σ 2 σ σ σ + (ln x µ ) σ σ 0 < x + a σ 2 = ( ) ln + σ 2 /µ 2 µ = ln µ σ 2 /2. The partal dervatves of µ σ wth respect to µ or σ can be easly obtaned, so they are not reported here. The extenson to hgher-order partal dervatves of h(θ) s straghtforward. See Rubnsten Shapro 8, who poneered the score functon method. In general, the senstvty s not avalable analytcally, snce the response h(θ) s not ether. Nonetheless, the probablstc response h(θ) (Eq. (4)) ts senstvtes h(θ)/ (Eq. (9)) have been formulated as expectatons of stochastc quanttes wth respect to the same densty functon, facltatng ther concurrent evaluatons n a sngle stochastc smulaton or analyss. The man contrbuton of ths work s to smultaneously evaluate both the stochastc response senstvtes by an alternatve route, known as the dmensonal decomposton method, to the tradtonal Monte Carlo smulaton. The score functon method requres dfferentatng only the probablty densty functon f X (x; θ). The resultng score functons can be easly determned, n many cases, analytcally. In contrast, the nfntesmal perturbaton analyss n ts orgnal form requres dervatves or perturbaton of the performance functon, whch s always expensve n stochastc-mechancs applcatons. Furthermore, f the performance functon s not dfferentable, the regularty condton that permts nterchangeablty of dfferental ntegral operators s volated the nfntesmal perturbaton analyss wll not work. In the score functon method, g(x) can be dscontnuous for example, the ndcator functon I ΩF (x) that comes from relablty analyss but the method stll allows evaluaton of the senstvty f the densty functon s dfferentable. In realty, the densty functon s often smoother than the performance functon, therefore the regularty condtons are mlder n the score functon method than n the nfntesmal perturbaton analyss Score functons The score functons (x; θ); =,..., M depend only on the probablty dstrbuton of rom nput X = {X,..., X N } T. When the dstrbuton of X s ether ndependent or both ndependent dentcal, the expressons of the score functons smplfy slghtly. Snce a major applcaton of senstvty analyss s desgn optmzaton, where the second-moment propertes of rom nput play the role of desgn parameters, attenton s confned to the score functons assocated wth the mean stard devatons of nput Independent dstrbutons Consder a rom nput X, where the components X,..., X N are ndependent rom varables. Let X follow the probablty densty functon f X (x ; µ, σ ) for =,..., N, wth mean µ stard devaton σ. The jont densty of X s f X (x; θ) = N = f X (x ; µ, σ ), θ = {µ, σ..., µ N, σ N } T, M = 2N. Therefore, from Eq. (8), the frst-order score functons for µ σ become µ (x; θ) = ln f X (x ; µ, σ ) µ ; =,..., N (0) σ (x; θ) = ln f X (x ; µ, σ ) ; σ =,..., N, () respectvely Independent dentcal dstrbutons Furthermore, consder X, where the components X,..., X N are not only ndependent, but also follow the common probablty densty functon f (x ; µ, σ ) wth the common mean µ common stard devaton σ, so that θ = {µ, σ } T M = 2. Such dstrbutons frequently arse n manufacturng processes, where a sequence of events occur ndependently, but share the same dstrbuton parameters. In that case, the jont densty of X s f X (x; θ) = N = f (x ; µ, σ ), yeldng µ (x; θ) = N = ln f (x ; µ, σ ) µ N σ (x; θ) = ln f (x ; µ, σ ) σ = (2) (3) as the frst-order score functons. In ether case, the log-dervatves of a margnal probablty densty functon are requred n determnng the score functons. Table presents explct expressons of log-dervatves for Gaussan lognormal dstrbutons Dependent dstrbutons When X s dependent, the dervaton of score functons s generally tedous, but not dffcult. For example, when X s Gaussan wth ts mean µ := E θ X = {µ,..., µ N } T, covarance matrx Σ := E θ (X µ)(x µ) T = ρ j σ σ j, densty f X (x; θ) = (2π) N/2 Σ /2 exp (x µ) T Σ (x µ)/2, where µ σ are the mean stard devaton, respectvely, of X ρ j s the correlaton coeffcent between X X j, the frst-order score functons are µ (x; θ) = {0,...,,..., 0}Σ (x µ) (4) σ (x; θ) = Σ (x µ)t 2 σ (x µ) 2 ln Σ σ. (5) If the varables share the same mean µ, the same stard devaton σ, the same correlaton coeffcent ρ between any two dstnct varables, then µ = µ, Σ = σ 2 ( δ j )( ρ), leadng to smplfed score functons µ (x; θ) = T Σ (x µ) (6) σ (x; θ) = (x µ) T Σ (x µ) N, (7) σ where = {,..., } T δ j s the Kronecker delta. The score functons for dependent non-gaussan dstrbutons were not consdered n ths work.

4 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Dmensonal decomposton method Consder a contnuous, dfferentable, real-valued, multvarate performance functon y(x) := y(x,..., x N ), whch descrbes g(x), a functon that represents ether y q (x) for statstcal moment analyss or I ΩF (x) for relablty analyss. Let c = {c,..., c N } T be a reference pont of X y(c,..., c, x, c +,..., c S k, x S k, c S k +,..., c N ) represent an (S k)th dmensonal component functon of y(x), where S N k = 0,..., S. Then, an S-dmensonal decomposton of y(x) s 3,4 S ( ) ŷ S (x) := ( ) k N S + k k k=0 N,..., S k =; < < S k y(c,..., c, x, c +,..., c S k, x S k, c S k +,..., c N ). (8) Usng a multvarate functon theorem 3, t can be shown that ŷ S (x) n Eq. (8) conssts of all terms of the Taylor seres of y(x) that have less than or equal to S varables. The exped form of Eq. (8), when compared wth the Taylor expanson of y(x), ndcates that the resdual error n the S-varate approxmaton ncludes terms of dmensons S + hgher. All hgher-order S- lower-varate terms of y(x) are ncluded n Eq. (8), whch should therefore generally provde a hgher-order approxmaton of a multvarate functon than equatons derved from frst- or secondorder Taylor expansons. When S = 2, Eq. (8) degenerates to the unvarate bvarate approxmatons, respectvely. 3.. Decomposton wth numercal ntegraton When g(x) represents y q (x), t can also be decomposed by a convergent sequence of lower-varate approxmatons, as n Eq. (8). Therefore, for an ndependent rom vector X, an S-varate approxmaton of the qth moment m q (θ) ts senstvty m q (θ)/ can be evaluated usng stard numercal quadratures, leadng to m q (θ) S ( ) = ( ) k N S + k k k=0 N,..., S k =; < < S k j S k = j = y q (c,..., c, x (j ), c +,..., c S k, w (j ) w (j S k) S k x (j S k) S k, c S k +,..., c N ) (9) m q (θ) S ( ) = ( ) k N S + k k k=0 N,..., S k =; < < S k j S k = j = y q (c,..., c, x (j ), c +,..., c S k, x (j S k) S k, c S k +,..., c N ) (x (j t()) for =,..., M, where x (j m) m the m th varable, w (j m) m w (j ) w (j S k) S k t() ; θ) (20) s the j m th ntegraton pont of s the assocated weght that ncludes the probablty densty, m =,..., S k, t() N s an nteger ndex such that s the dstrbuton parameter of the X t() th varable, n s the number of ntegraton ponts. The score functon (x t() ; θ) s assocated wth the parameter, whch can be ether the mean µ or the stard devaton σ of the rom varable X t(). Snce X t() X t(j) for any t() t(j) are ndependent, (x t() ; θ) s a unvarate functon, as already demonstrated by Eq. (0) or (). If X comprses both ndependent dentcal rom varables, the score functon (x; θ), accordng to Eq. (2) or (3), s a lnear combnaton of unvarate functons, hence, effectvely remans unvarate. In the latter case, (x t() ; θ) n Eq. (20) should be replaced by (x; θ), whch s just another unvarate functon. Therefore, the proposed equatons, Eqs. (9) (20), for ndependent /or dentcal dstrbutons ental evaluatng at most S-dmensonal ntegrals, whch s substantally smpler more effcent than performng one N-dmensonal ntegraton, partcularly when S N. Hence, the computatonal effort n conductng moment ts senstvty analyses s sgnfcantly reduced usng the dmensonal decomposton. When S =, Eqs. (9) (20) degenerate to the unvarate approxmaton nvolvng only one-dmensonal ntegraton. When S = 2, Eqs. (9) (20) become the bvarate approxmaton entalng at most two-dmensonal ntegraton. When X ncludes dependent rom varables, the score functon (x; θ) s a multvarate functon n general. Therefore, the formulaton of the S-varate approxmaton of the senstvty for dependent varables must nclude the S-varate decomposton of the product y q (x) (x; θ). Nonetheless, t s possble to generalze Eq. (20) for senstvty analyss of a general stochastc system wth dependent rom varables, but wth an addtonal layer of approxmaton from decomposton of the score functons Decomposton wth smulaton The S-varate decomposton assocated numercal ntegraton developed should not be appled to g(x) when t represents I ΩF (x), the ndcator functon from relablty analyss. Ths s because I ΩF (x) s a dscontnuous functon, whch takes on dscrete values of 0 n the sample space Ω. In that case, consder the n-pont Lagrange nterpolaton of y(c,..., c, x, c +,..., c S k, x S k, c S k +,..., c N ) for S N k = 0,..., S, yeldng ŷ S (X) S ( ) = ( ) k N S + k k k=0 N,..., S k =; < < S k j S k = φ j (X ) φ js k (X S k ) j = y(c,..., c, x (j ), c +,..., c S k, x (j S k) S k, c S k +,..., c N ), (2) where x (j S k) S k s the j S k th sample of X S k, φ js k (X S k ) s the rom Lagrange shape functon, y(c,..., c, x (j ) c S k, x (j S k) S k, c +,...,, c S k +,..., c N ) s the assocated determnstc coeffcent. Eq. (2) provdes a convergent sequence of lower-varate approxmatons of y(x) f the Lagrange nterpolatons of component functons are convergent. Snce ŷ S (x) represents an explct approxmaton of y(x), any probablstc characterstc of y(x), ncludng ts statstcal moments probablty densty functon, can be easly evaluated by performng Monte Carlo smulaton on Eq. (2). Recall that ŷ S (x) or {ŷ (k) S (x), k =, K} are S-varate approxmatons of performance functons for component or system relablty analyss. Based on these approxmatons, let

5 282 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) ˆΩ F,S := {x : ŷ S (x) < 0}, ˆΩ F,S := {x : K k= ŷ(k) S (x) < 0}, ˆΩ F,S := {x : K k= ŷ(k) S (x) < 0} defne approxmate falure sets n relablty analyses of the component, seres system, parallel system, respectvely. Therefore, the Monte Carlo estmates of the falure probablty P F (θ) ts senstvty P F (θ)/, employng S-varate approxmatons of the falure sets, are P F (θ) L = E θ I ˆΩ F,S (X) = lm I L L ˆΩ F,S (x (l) ) (22) P F (θ) = Eθ I θ ˆΩ F,S (X) (X; θ) = lm L L L l= l= I ˆΩ F,S (x (l) ) (x (l) ; θ), (23) respectvely, where L s the sample sze, x (l) s the lth realzaton of X, { I ˆΩ F,S (x (l), x ) = (l) ˆΩ F,S 0, x (l) (24) Ω \ ˆΩ F,S s an approxmate ndcator functon correspondng to S-varate decompostons of performance functons. Both ndependent dependent rom varables can be accounted for, provded that ther realzaton can be generated. Settng S = or 2 n Eqs. (22) (23), the unvarate or bvarate approxmatons of the falure probablty ts senstvty can be nvoked. A Monte Carlo smulaton on an S-varate approxmaton ŷ S (X) also leads to ts qth moment m q (θ) = E θ ŷq S (X) = lm L ts senstvty m q (θ) = Eθ ŷ q S θ (X)s() (X; θ) = lm L L L l= L L ŷ q S (x(l) ) (25) l= ŷ q S (x(l) ) (x (l) ; θ), (26) therefore provdes an alternatve means to numercal ntegraton. The proposed methods nvolvng unvarate (S = ) bvarate (S = 2) approxmatons, where an n-pont numercal ntegraton yelds senstvty of moments or where an n-pont Lagrange nterpolaton assocated Monte Carlo smulaton produce senstvty of moments or relablty, are defned as the unvarate bvarate decomposton methods n ths paper Computatonal effort The unvarate bvarate decomposton methods for calculatng senstvty of moment or relablty of a mechancal system requre evaluaton of determnstc coeffcents: y(c), y(c,..., c, x (j), c +,..., c N ), y(c,..., c, x (j ), c +,..., c 2, x (j 2) 2, c 2 +,..., c N ) for,, 2 =,..., n j, j, j 2 =,..., n. Hence, the computatonal effort requred by the decomposton method can be vewed as numercally calculatng a mechancal response y(x) at several determnstc nputs defned by ether ntegraton ponts (for moments) or user-selected sample ponts (for relablty). Therefore, the total cost for the unvarate decomposton entals a maxmum of nn + functon evaluatons. For the bvarate decomposton, a maxmum of N(N )n 2 /2 + nn + functon evaluatons are requred. If the ntegraton or sample ponts nclude a common pont n each coordnate x (see the forthcomng example secton), the numbers of functon evaluatons reduce slghtly. 4. Numercal examples Four numercal examples nvolvng mathematcal functons sold-mechancs/structural problems are presented to llustrate the proposed decomposton methods, for obtanng frst-order senstvty of the moment or relablty. Whenever possble, a forward fnte-dfference method, wth one percent perturbaton the drect Monte Carlo smulaton, was employed to evaluate the accuracy computatonal effcency of the decomposton methods. The sample szes for the drect Monte Carlo smulaton the embedded Monte Carlo smulaton of the decomposton method vary from 0 5 to 0 8, dependng on the examples, but they are dentcal for a specfc problem. The score functons assocated wth the probablty dstrbutons employed n these examples are provded n Secton 2.3. The dmensonal decomposton was formulated n the orgnal space (x space) of the rom nput. The reference pont c assocated wth the decomposton methods was fxed at the mean nput. A fve-pont Gauss Hermte ntegraton three- to fve-pont Lagrange nterpolaton schemes were selected. For an n-pont Lagrange nterpolaton, sample ponts (c,..., c, x (j), c +,..., c N ) (c,..., c, x (j ), c +,...,, c 2 +,..., c N ) for unvarate bvarate decompostons, respectvely, were unformly dstrbuted, where x (l) k = c k c 2, x (j 2) 2 (n )/2, c k (n 3)/2,..., c k,..., c k +(n 3)/2, c k +(n )/2, k =,, 2, l = j, j, j 2. For response or senstvty analyss, (n )N + N(N )(n ) 2 /2 + (n )N + functon evaluatons are nvolved n the unvarate bvarate methods, respectvely. 4.. Example : Moments The frst example nvolves senstvty analyss of moments of a quadratc performance functon y(x) = X 2 + X X X X 2 + 2X 2 X 3 + 4X 3 X, (27) where X N(µ, σ 2 ); =, 2, 3 are three ndependent dentcally dstrbuted Gaussan rom varables wth the common mean µ common stard devaton σ θ = {µ, σ } T. All fnte-order moments of y(x) can be obtaned exactly, of whch, the frst three moments are: m (µ, σ ) := E θ y(x) = 3σ 2 +0µ 2, m 2 (µ, σ ) := E θ y2 (X) = 27σ 4 +00µ 4 +98σ 2 µ 2, m 3 (µ, σ ) := E θ y3 (X) = 405σ µ 2 σ µ 4 σ µ 6. Therefore, senstvtes of these moments wth respect to µ or σ can be obtaned exactly to evaluate the accuracy of the decomposton methods. When solvng ths problem by the decomposton methods, both numercal ntegraton (Eqs. (9) (20)) smulaton (Eqs. (25) (26)) approaches were employed. A fve-pont Gauss Hermte ntegraton a threepont Lagrange nterpolaton schemes were selected. The sample sze L = 0 6. Table 2 lsts the frst three moments m q := E θ y q (X) ther frst-order senstvtes m q / µ m q / σ for q =, 2, 3, when () µ = ; σ = 0.3 (2) µ = ; σ = 0.6, representng moderate large nput uncertantes, respectvely. The tabulated results came from the decomposton methods wth both approaches the exact soluton. The agreement between the results of the decomposton methods usng numercal ntegraton or smulaton the exact soluton s excellent. The results mprove when swtchng from the unvarate to the bvarate method, as expected. Regardless of the method selected, the magntudes of moments senstvtes for σ = 0.6, compared wth those for σ = 0.3, ncrease, as they should for a larger stard devaton; but more mportantly, a comparson

6 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Table 2 Moments of a quadratc functon senstvtes Decomposton wth ntegraton () Moderate nput uncertanty (µ =, σ = 0.3) Decomposton wth smulaton Unvarate Bvarate Unvarate Bvarate Exact m m m m / µ m 2 / µ m 3 / µ m / σ m 2 / σ m 3 / σ (2) Large nput uncertanty (µ =, σ = 0.6) m m m m / µ m 2 / µ m 3 / µ m / σ m 2 / σ m 3 / σ wth the exact soluton demonstrates that the decomposton methods, n partcular the bvarate method, can ndeed account for large nput uncertantes, a desrable property of any approxmate soluton Example 2: Relablty Consder two relablty problems: () a component relablty problem wth the performance functon y(x) = , 3 00 (28) X = where X N(µ, Σ) s a 00-dmensonal, Gaussan rom vector wth mean vector µ = µ{,..., } T covarance matrx Σ = σ 2 ( δ j )( ρ), θ = {µ, σ } T ; (2) a system relablty problem wth three performance functons: y () (X) = X 2 + 2X 3 + X 4.5, y (2) (X) = X + X 2 + X 4 + X 5 2.4, y (3) (X) = X + 2X 3 + 2X 4 + X , (29) where X s an ndependent, lognormal rom varable wth mean µ stard devaton σ, θ = {µ, σ..., µ 5, σ 5 } T. The performance functons n Eq. (29) represent three rgdplastc falure mechansms of a well-studed portal frame 6. The objectve of ths example s to evaluate the accuracy of the decomposton method n calculatng the component falure probablty P F, := P θ y(x) < 0 the seres-system falure probablty P F,2 := P θ X Ω F ; Ω F := {x : 3 k= y(k) (x) < 0}, ther respectve senstvtes. The statstcal propertes of nput are: () µ = 0, σ =, two dstnct cases of ρ = 0 ρ = 0.5 for the component relablty problem; (2) µ = σ = 0.25; =,..., 5 for the system relablty problem. Snce y for the component relablty problem s a multvarate functon, the unvarate or bvarate decomposton for a fnte value of n, regardless how large, provdes only an approxmaton. Nonetheless, usng only n = 3 L = 0 6, the unvarate bvarate estmates of P F,, P F, / µ, P F, / σ are lsted n Table 3 for both () uncorrelated (ρ = 0) (2) correlated (ρ = 0.5) nput. The exact solutons are Table 3 Component falure probablty senstvtes for µ = 0, σ = () Uncorrelated nput (ρ = 0) Unvarate Bvarate Exact P F, P F, / µ P F, / σ (2) Correlated nput (ρ = 0.5) P F, P F, / µ P F, / σ P F, = Φ( β), P F, / µ = φ( β) N/σ ( + (N )ρ, P F, / σ = φ( β) 3 ) Nµ /σ 2 + (N )ρ, where β = ( 3 ) Nµ /σ ( + (N )ρ, φ(u) = / ) 2π ( ) exp u 2 /2, Φ(u) = u φ(ξ)dξ. Compared wth the exact results, also lsted n Table 3, both versons of the decomposton method are satsfactory for solvng relablty problems nvolvng both ndependent dependent Gaussan varables. However, the bvarate approxmaton provdes a hghly accurate soluton for ths hgh-dmensonal, nonlnear problem. Table 4 lsts the system falure probablty P F,2, whch s calculated by the proposed unvarate decomposton method nvolvng at most (n = 3, N = 5) evaluatons of each performance functon the drect Monte Carlo smulaton nvolvng 0 8 samples. Both methods provde an dentcal result: P F,2 = Ths s possble, snce each performance functon n Eq. (29) s a unvarate functon, s exactly represented by a three-pont Lagrange nterpolaton, leadng to the unvarate approxmaton ˆΩ F, of the falure set that s the same as the exact falure set Ω F. Therefore, there s no need to pursue the bvarate approxmaton. It s worth notng that a value of n as large as nne was requred to produce a satsfactory unvarate approxmaton of the falure set n the Gaussan space (u space), where the transformed performance functons become hghly nonlnear 4. Therefore, a hgher-order nterpolaton can be avoded by decomposton n the orgnal space n ths problem. The advantage of the unvarate decomposton method n the x space extends to senstvty analyss, as essentally the

7 284 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Fg.. A through-wall-cracked cylnder under four-pont bendng; (a) geometry loads; (b) cracked cross-secton; (c) fnte-element dscretzaton. Table 4 System falure probablty senstvtes for µ =, σ = 0.25 Unvarate Drect MCS/FD a P F, P F,2 / µ P F,2 / µ P F,2 / µ P F,2 / µ P F,2 / µ P F,2 / σ P F,2 / σ P F,2 / σ P F,2 / σ P F,2 / σ a MCS/FD = Monte Carlo smulaton/fnte-dfference. same effort delvers the senstvtes P F,2 / µ P F,2 / σ for =,..., 5, whch are also presented n Table 4. Alternatve senstvty estmates from the fnte-dfference method nvolvng 0 8 samples for each drect smulaton run were also developed, can be found n the last column of Table 4. The agreement between the results of the unvarate method the fntedfference method s very good. It s worth notng that the fntedfference method typcally gves based senstvty estmates, where slght fluctuatons n the results are expected due to a fnte varance of the estmator Example 3: Nonlnear fracture relablty Consder a crcumferental, through-wall-cracked (TWC), nonlnearly elastc cylnder, whch s subjected to a four-pont bendng, as shown n Fg. (a). The cylnder has a md-thckness radus R = 50.8 mm, a wall thckness t = 5.08 mm, a symmetrcally centered through-wall crack wth the normalzed crack angle θ/π = /8. The outer span L o =.5 m the nner span L = 0.6 m. The cross-sectonal geometry at the cracked secton s shown n Fg. (b). The cylnder s composed of an ASTM Type 304 stanless steel, whch follows the Ramberg Osgood consttutve equaton 7 ɛ j = + ν E S j + 2ν σ kk δ j + 3 3E 2E α ( σe σ 0 ) m S j, (30) where σ j ɛ j are stress stran components, respectvely, E s the Young s modulus, ν s the Posson s rato, σ 0 s a reference stress, α s a dmensonless materal coeffcent, m s a stran hardenng exponent, δ j s the Kronecker delta, S j := σ j σ kk δ j /3 s the devatorc stress, σ e := 3S j S j /2 s the von Mses equvalent stress. Table 5 lsts the means, stard devatons, probablty dstrbutons of tensle parameters (E, α, m), four-pont bendng load (F), fracture toughness (J Ic ). All rom varables are statstcally ndependent, θ comprses all ten secondmoment statstcs of the rom nput. Also, σ 0 = MPa

8 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Table 5 Statstcal propertes of rom nput for a through-wall-cracked cylnder Rom varable Mean Stard devaton Probablty dstrbuton Elastc modulus (E), GPa Gaussan Ramberg Osgood coeffcent (α) Lognormal Ramberg Osgood exponent (m) Lognormal Four-pont bendng load (F), kn Gaussan Intaton toughness (J Ic ), kj/m Lognormal 4.4. Example 4: Relablty-based desgn optmzaton Fg. 2. A ten-bar truss; a boxed or unboxed number ndcates a member or node number. ν = 0.3. A fnte-element mesh of the quarter-cylnder model, consstng of 236 elements 805 nodes, s shown n Fg. (c). Twenty-noded, soparametrc, sold elements from the ABAQUS lbrary 8 were used wth focused sngular elements at the crack tp. Ths type of TWC cylnder s frequently analyzed for fracture evaluaton of pressure boundary ntegrty n the nuclear ndustry. For nonlnear fracture, the J-ntegral s a useful crack-drvng force that unquely characterzes the asymptotc crack-tp stress stran felds 7. Therefore, a fracture crteron, where the J exceeds the fracture toughness of the materal, can be used to calculate the probablty of fracture ntaton: P F := P θ J(X) > J Ic (X), n whch the thckness-averaged J depends only on the frst four rom varables of X defned n Table 5. The bvarate decomposton method, usng n = 5, L = 0 7, only 3 ABAQUS-aded fnte-element analyses, predcts a fracture-ntaton probablty of a value reasonably close to 5 0 4, obtaned from a J-estmaton-based Monte Carlo smulaton 9,20. The J-estmaton-based analyss entals response surface approxmaton from a small, but fxed number of determnstc fnte-element analyses, leadng to an explct expresson of the J-ntegral wth respect to the rom nput, then conductng Monte Carlo smulaton on the explct functonal form. Due to expensve, nonlnear, fnte-element analyss, the drect Monte Carlo smulaton was not feasble to verfy the low probablty n ths example. The bvarate method also predcts the senstvtes of the fracture-ntaton probablty wth respect to the means of E, α, m, F, J Ic, whch after beng scaled by respectve stard devatons, are , ,. 0 4, , , respectvely. Wth respect to the stard devatons, the scaled senstvtes are , , , , , respectvely. A comparson of scaled senstvtes wth respect to means reveals that the load fracture toughness of the materal are the most mportant nput varables. Based on the scaled senstvtes wth respect to stard devatons, the uncertantes n the load fracture toughness propertes are more sgnfcant than those n others. The fnal example demonstrates how senstvty analyss can be exploted for the optmal desgn of mechancal systems. A lnearelastc, ten-bar truss, shown n Fg. 2, s smply supported at nodes 4, s subjected to two concentrated loads of 0 5 lb at nodes 2 3. The truss materal s made of an alumnum alloy wth the Young s modulus E = 0 7 ps. The rom nput s X = {X,..., X 0 } T R 0, where X denotes the cross-sectonal area of the th bar. The rom varables are ndependent lognormally dstrbuted wth means µ, =,..., 0, each of whch has a ten percent coeffcent of varaton. From a lnear-elastc fnte-element analyss, the maxmum vertcal dsplacement v 3 (X) occurs at node 3, where a permssble dsplacement s lmted to 4 n. The truss was desgned by mnmzng ts mean volume gven that the dsplacement-based falure probablty s no greater than Φ( 2.5) = The relablty-based desgn optmzaton s formulated to mn h 0 (θ) = 360E θ X + X 2 + X 3 + X 4 + X 5 + X 8 subject to + 2 (X 6 + X 7 + X 9 + X 0 ) h (θ) := P θ 4 v 3 (X) < 0 Φ( 2.5) n. 2 0 n. 2 ; =,..., 0, (3) where the desgn vector s θ = {µ,..., µ 0 } T R 0. Snce the state functon s lnear n X, h 0 (θ) = 360µ + µ 2 + µ 3 + µ 4 + µ 5 + µ (µ 6 + µ 7 + µ 9 + µ 0 ). The ntal desgn pont s θ 0 = {3,..., 3} T n. 2, whch corresponds to the ntal volume h 0 (θ 0 ) = 2, 589 n. 3. For the decomposton methods, n = 5 L = 0 5. A modfed algorthm of feasble drectons 2 was employed to solve ths optmzaton problem, where the falure probablty ts senstvtes wth respect to the mean cross-sectonal areas were suppled by decomposton methods at each desgn teraton. Fg. 3(a) presents the optmzaton hstory, whch reveals convergence n nne teratons by both unvarate bvarate methods. The optmal volumes h 0 (θ ) acheved by the unvarate bvarate methods are 9707 n n. 3, respectvely. The optmal solutons, lsted n Table 6, suggest that redstrbuton of cross-sectonal areas can lead to not only lower volumes of the truss, but also lower falure probabltes. Fg. 3(b) tracks the evoluton of the falure probablty by each decomposton method durng the optmzaton process. 5. Lmtaton The score functons employed n the decomposton method developed here are applcable to senstvty analyss of a mechancal system, where the performance functon g does not depend on the parameter vector θ, θ solely descrbes the probablty densty f X (x; θ). Many senstvty analyses of mechancal systems, ncludng all numercal examples presented n ths paper, fall under ths class of problems, whch s the focus of the current work. However, there are also exceptons that requre delvng nto broader classes of problems. For nstance, consder a probablstc response h(θ) = E θ g(x; θ) := R N g(x; θ)f X (x; θ)dx under a general settng, where g f X both depend on θ R M the senstvty

9 286 S. Rahman / Probablstc Engneerng Mechancs 24 (2009) Conclusons outlook Fg. 3. Optmzaton hstores; (a) objectve functon; (b) probablty of falure. Table 6 Optmzaton results of a ten-bar truss Intal desgn Fnal desgn Unvarate Bvarate µ, n µ 2, n µ 3, n. 2 3 µ 4, n. 2 3 µ 5, n µ 6, n µ 7, n µ 8, n. 2 3 µ 9, n. 2 3 µ 0, n P F 0.096/0.098 a Volume, n. 3 2, a by unvarate; by bvarate. h(θ)/ s sought. Under regularty condtons smlar to those presented n Secton 2.2 followng the same argument, the correspondng score functons can be derved, but now they wll nvolve the partal dervatve g(x; θ)/ that must exst be fnte. In addton, the multvarate decomposton presented n Secton 3 needs to be developed for not only the performance functon, but also the more complcated score functons. Such developments are not trval, extensons of the decomposton method to solve ths rather generalzed class of problems are left as future efforts. A new class of computatonal methods, referred to as dmensonal decomposton methods, was developed for calculatng stochastc senstvtes of mechancal systems wth respect to probablty dstrbuton parameters. The methods are based on a herarchcal decomposton of a multvarate response functon n terms of varables wth ncreasng dmensons score functons assocated wth the probablty dstrbuton of a rom nput. The decomposton permt unvarate bvarate approxmatons of stochastc response senstvty, (2) lower-dmensonal numercal ntegratons for senstvty of statstcal moments, (3) lower-varate Lagrange nterpolatons Monte Carlo smulaton for senstvty of relablty or moments. Both the probablstc response ts senstvtes can be estmated from a sngle stochastc analyss, wthout requrng performance functon gradents. These methods can help solve both component system relablty problems. The effort n obtanng probablstc senstvtes can be vewed as calculatng the response at a selected determnstc nput, defned by ether ntegraton ponts or sample ponts. Therefore, the methods can be easly adapted for solvng stochastc problems nvolvng thrd-party, commercal fnte-element codes. Unvarate bvarate decomposton methods were employed to solve four numercal problems, where the performance functons are lnear or nonlnear, nclude Gaussan /or non- Gaussan rom varables, are descrbed by smple mathematcal functons or mechancal responses from fnte-element analyss. The results ndcate that the decomposton methods developed, n partcular the bvarate verson, provde very accurate estmates of senstvtes of statstcal moments or relablty. The computatonal effort by the unvarate method vares lnearly wth respect to the number of rom varables or the number of ntegraton or nterpolaton ponts, therefore the unvarate method s economc. In contrast, the bvarate method, whch generally outperforms the unvarate method, dems a quadratc cost scalng, makng t also more expensve than the unvarate method. Nonetheless, both decomposton methods are far less expensve than the fnte-dfference method or the exstng score functon method entalng drect Monte Carlo smulaton. The last example hghlghts the usefulness of the decomposton methods n generatng senstvtes that lead to relablty-based desgn optmzaton of mechancal systems. Compared wth the exstng drect dfferentaton method, whch can calculate senstvtes wth respect to both dstrbuton performance functon parameters, the decomposton methods n ther current form are lmted to senstvty analyss wth respect to the dstrbuton parameters only. Therefore, future effort n extendng these decomposton methods to account for the performance functon parameters should be undertaken. Acknowledgment The author would lke to acknowledge fnancal support from the US Natonal Scence Foundaton under Grant No. DMI References Melchers RE, Ahammed M. A fast approxmate method for parameter senstvty estmaton n monte carlo structural relablty. Computers & Structures 2004;82(): Au SK. Relablty-based desgn senstvty by effcent smulaton. Computers & Structures 2005;83(4): Du X, Chen W. Effcent uncertanty analyss methods for multdscplnary robust desgn. AIAA Journal 2002;40(3): Enevoldsen I, Sorensen JD. Relablty-based optmzaton n structural engneerng. Structural Safety 994;5: L Ecuyer P, Perron G. On the convergence rates of IPA FDC dervatve estmators. Operatons Research 994;42(4): Ho YC, Cao XR. Dscrete event dynamc systems perturbaton analyss. Norwell (MA): Kluwer; 99.

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