Equivalent Standard Deviation to Convert High-reliability Model to Low-reliability Model for Efficiency of Samplingbased

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1 roceedngs of the ASME 0 Internatonal Desgn Engneerng echncal Conferences & Computers and Informaton n Engneerng Conference IDEC/CIE 0 August 8 3, 0, Washngton, D.C., USA DEC Equvalent Standard Devaton to Convert Hgh-relablty Model to Low-relablty Model for Effcency of Samplngbased RBDO Ikn Lee, Kyung K. Cho* Department of Mechancal & Industral Engneerng College of Engneerng he Unversty of Iowa, Iowa Cty, IA 54, USA lee@engneerng.uowa.edu, kkcho@engneerng.uowa.edu * Correspondng author Davd Gorsch US Army RDECOM/ARDEC Warren, MI , USA davd.gorsch@us.army.ml ABSRAC hs study presents a methodology to convert an RBDO problem requrng very hgh relablty to an RBDO problem requrng relatvely low relablty by ncreasng nput standard devatons for effcent computaton n samplng-based RBDO. rst, for lnear performance functons wth ndependent normal random nputs, an exact probablty of falure s derved n terms of the rato of the nput standard devaton, whch s denoted by δ. hen, the probablty of falure estmaton s generalzed for any random nput and performance functons. or the generalzaton of the probablty of falure estmaton, two coeffcents need to be determned by equatng the probablty of falure and ts senstvty wth respect to the standard devaton at the current desgn pont. he senstvty of the probablty of falure wth respect to the standard devaton s obtaned usng the frstorder score functon for the standard devaton. o apply the proposed method to an RBDO problem, a concept of an equvalent standard devaton, whch s an ncreased standard devaton correspondng to the low relablty model, s also ntroduced. umercal results ndcate that the proposed method can estmate the probablty of falure accurately as a functon of the nput standard devaton compared to the Monte Carlo smulaton results. As antcpated, the samplngbased RBDO usng the surrogate models and the equvalent standard devaton helps fnd the optmum desgn very effcently whle yeldng relatvely accurate optmum desgn whch s close to the one obtaned usng the orgnal standard devaton. KEYWORDS Very Small robablty of alure, Samplng-based RBDO, Monte Carlo Smulaton, Score uncton, Copula, Surrogate Model.. IRODUCIO Surrogate models or meta-models have been wdely used for relablty-based desgn optmzaton (RBDO of varous engneerng applcatons when accurate senstvtes of performance functons are not avalable [-6]. When surrogate models are used for RBDO, samplng-based relablty analyss methods to evaluate probablstc constrants of RBDO are often adapted due to ther computatonal smplcty. he most straghtforward approach among samplng technques s the drect Monte Carlo smulaton (MCS [7]. Usng the MCS, the probablty of falure can be estmated by countng the number of samples wthn the falure regon and dvdng t by the total number of samples. he man concern for the MCS s the computatonal cost because t s well known that the total number of samples requred to obtan a reasonably accurate estmate s proportonal to the nverse of the probablty of falure [8], whch mples that a very large number of samples wll be requred for the MCS f the target probablty of falure s very small, for example, 4~6σ desgn.

2 o enhance the computatonal effcency of the MCS, Latn Hypercube samplng (LHS and ts modfcaton [9-3] can be used for the relablty analyss. he LHS s known to be more effcent than the MCS snce ts stratfcaton propertes allow for the estmaton of the probablty of falure wth a relatvely small sample sze [], and t s also shown that the LHS can save more than 50% of the computatonal cost of the drect MCS [3]. o further enhance the computatonal effcency of the samplng scheme, varatons of the MCS, ncludng the mportance samplng [3-5], subset smulaton [6], and drectonal samplng [7,8], have been proposed. All these mprovements of the drect MCS attempt to reduce the number of samples by ether allocatng samples more effectvely on the samplng doman or movng the samplng doman near the lmt state functon where performance functons have zero values. However, even f the number of samples s reduced by usng the effcent samplng schemes, a large number of samples are stll requred for a very small target probablty of falure. urthermore, f the samplng doman moves to the vcnty of the lmt state functon as n the mportance samplng, as many surrogate models as the number of RBDO constrants have to be generated at a gven desgn when the local wndow concept s used for the samplng-based RBDO to mprove ts accuracy. In addton, t requres addtonal computatons to move the samplng doman to the vcnty of the lmt state functon, whch makes the samplng-based RBDO further neffcent. o avod ths, a global wndow concept can be used to generate the surrogate models; however, t causes accuracy problems, especally for hgh-dmensonal problems. he man obectve of ths paper s to propose a methodology to convert an RBDO problem wth a very small target probablty of falure to an RBDO problem wth a relatvely hgh probablty of falure by ncreasng the nput standard devatons to reduce the computatonal cost of the drect MCS for the samplng-based RBDO. or ths, the exact relatonshp between the probablty of falure and nput standard devatons s derved for lnear performance functons wth ndependent normal random nputs, and then the relatonshp s generalzed for any random nput and performance functons. o derve the general relatonshp between the probablty of falure and nput standard devaton, the frst-order score functon for the nput standard devatons s ntroduced [9-]. After fndng the relatonshp, a concept of an equvalent standard devaton, whch s an ncreased standard devaton correspondng to the model wth hgh probablty of falure, s also proposed to be used for samplng-based RBDO problems. Snce the proposed method s appled to the samplng-based RBDO, accurate surrogate models are naturally used for the relablty analyss nstead of computatonally expensve computer smulatons. or the generaton of accurate surrogate models, the Dynamc Krgng method [] can be utlzed. Even f t s developed to be used wth surrogate models, the proposed method s applcable to the samplng-based RBDO usng actual computer smulatons f the model s not too computatonally demandng snce the proposed method can reduce the number of samples sgnfcantly, especally when used n conuncton wth the effcent samplng schemes. he paper s organzed as follows. Secton brefly revews the formulaton of the samplng-based RBDO and the stochastc senstvty analyss snce t helps understand the proposed method. Secton 3 shows how to derve the relatonshp between the probablty of falure and nput standard devatons for general random nputs and performance functons. Secton 4 explans the concept of the equvalent standard devaton for the samplng-based RBDO. Secton 5 llustrates wth numercal examples the effcency and accuracy of the proposed method compared wth results obtaned usng the orgnal random nput. nally, ths paper s concluded n Secton 6.. SAMLIG-BASED RBDO In ths secton, we brefly revew the concept of the samplng-based RBDO, whch wll help us understand the proposed method n Secton 3. Secton. explans the formulaton of the samplng-based RBDO and the evaluaton of probablstc constrants. Secton. revews the stochastc senstvty analyss usng the score functon for mean values. nally, Secton.3 shows accuracy of samplng technques and ustfcaton of the proposed method, whch s explaned n detal n Secton 3.. ormulaton he mathematcal formulaton of a general RBDO problem s expressed as mnmze Cost( d G C ( subect to [ ( 0],,, L U ndv d dd, dr and R where d { d } μ( s the desgn vector, whch s the mean value of the -dmensonal random vector ={,,, } ; represents the probablty measure; s the target probablty of falure for the th constrant; and C, ndv, and nrv are the number of probablstc constrants, desgn varables, and random varables, respectvely. A relablty analyss nvolves calculaton of the probablty of falure, denoted by and shown n Eq. ( as G [ ( 0], whch s defned usng a mult-dmensonal ntegral ( ψ [ ] I ( f ( ; d EI ( x x ψ x ( where ψ s a vector of dstrbuton parameters, whch usually ncludes the mean (µ and/or standard devaton (σ of the random nput ; s the falure set; f ( x; ψ s a ont nput probablty densty functon (D of ; and E represents the expectaton operator. he falure set s defned x: G ( x 0 for component relablty analyss of as the th constrant functon G (x, and : C x G ( 0 x and : C x G ( 0 x for seres system and parallel system relablty analyss of C performance functons, nrv

3 respectvely [0,3]. I ( x n Eq. ( s called an ndcator functon and defned as I, x ( x (3 0, otherwse o carry out RBDO n Eq. ( usng gradent-based optmzaton methods, t s requred to know the functon value and ts senstvtes of the probablstc constrants at a gven desgn pont. However, n most engneerng applcatons, t s dffcult to obtan accurate senstvtes. or engneerng applcatons where accurate senstvtes are not avalable, surrogate models have been wdely used to carry out desgn optmzaton. Once an accurate surrogate model s avalable for the desgn optmzatons, samplng technques such as the drect MCS or more effcent LHS can be appled to estmate the probablty of falure wth an affordable computatonal cost. Denote the surrogate model for a constrant functon G ( as G ˆ (. hen, by applyng the MCS or LHS to the surrogate model, the probablstc constrants n Eq. ( can be approxmated as he partal dervatve of the log functon of the ont D n Eq. (6 wth respect to s known as the frst-order score functon for and s denoted as ( ln f ( ; ( x; μ x μ. (7 s he frst-order score functons for specfc margnal and ont dstrbuton types are lsted n Ref. [] n detal. In a smlar manner to Eq. (4 for the probablty of falure calculaton, ts senstvty s obtaned usng the frst-order score functon n Eq. (7 and applyng samplng technques to Eq. (6 as K ( k ( ( k I ˆ ( s ( ; K k x x μ. (8.3 Accuracy of Smulaton for Relablty Analyss he percentage error of the MCS to compute the probablty of falure by Eq. (4 can be measured usng the 95% confdence nterval of the estmated probablty of falure and gven by [4] G I x (4 K ( k [ ( 0] ˆ ( K k MCS ( K 00%. (9 ( k where K s the sample sze, x s the k th realzaton of, and the falure set ˆ for the surrogate model s defned as ˆ x: Gˆ ( x 0.. Stochastc Senstvty Analyss In addton to the probablty of falure shown n Eq. (, ts senstvty wth respect to a desgn varable s requred to carry out RBDO n Eq. (. akng the partal dervatve of Eq. ( wth respect to yelds ( μ I ( f ( ; d x x μ x. (5 In Eq. (5, dstrbuton parameter ψ become μ because μ s the desgn vector and σ s assumed to be ndependent of μ. Snce the dfferental and ntegral operators can be nterchanged f the ntegrand n Eq. (5 s bounded due to the Lebesgue domnated convergence theorem [9,0], Eq. (5 can be rewrtten as ( μ f ( x; μ I ( d x x ln f ( x; μ I ( f ( ; d x x μ x. (6 ln f ( ; x μ EI ( x o estmate the percentage error of Eq. (8, Eq. (8 can be rewrtten as K f ( kf ( ( kf I ˆ ( x s ( x ; μ K k f K f ( ( k f K f s ( x ; μ s f sf K k f K (0 where K f s the number of faled samples and s f s the mean value of the score functon values for the faled samples. Hence, the percentage error for the MCS to compute Eq. (8 can be measured by MCS s f. o see the effect of the percentage error n Eq. (9 on the target probablty of falure, consder an example where the MCS sample sze K s 500,000 and.75%. hen, MCS s.85% of the target probablty of falure, whch means that there exsts 95% probablty that the probablty of falure estmated usng the MCS wll be between.33% and.37% (.e.,.85% nterval of.75% wth 500,000 samples. If the target probablty of falure s very small, for example, %, whch s called a 4σ desgn, then, 369,04,089 MCS samples are requred to satsfy the same percentage error of.85%. Or, f the MCS sample sze K s 500,000, then the percentage error becomes 50.6%, whch s too large to be used for RBDO. Even f the surrogate model s not computatonally demandng and many model evaluatons can be performed, too many samples can cause computer memory problems and make the samplng-based

4 RBDO extremely slow, especally when mplct surrogate models such as the Krgng model are used. o obtan more accurate results, the LHS can be utlzed, whch s known to be more accurate for the probablty of falure calculaton than the MCS when the same number of samples s used [9-3]. However, even wth the LHS, the sample sze should be very large to be accurate for very small probablty of falure problems. Hence, n ths paper, the MCS s used for the samplng scheme snce t does not change the man pont of the paper. As mentoned above, for a very small target probablty of falure such as 4~6σ desgn, the probablty of falure and ts senstvtes computed usng Eqs. (4 and (8, respectvely, could be naccurate unless a suffcent number of samples are used, whch may not be possble due to hgh computatonal cost. hus, the samplng-based RBDO for a hgh-relablty model could yeld a wrong optmum desgn due to naccurate estmaton of the probablty of falure. o overcome ths, t s necessary to convert the hgh-relablty model to a lowerrelablty model by ncreasng nput standard devatons and fndng an equvalent probablty of falure correspondng to the ncreased standard devatons, whch s the man purpose of the paper and wll be explaned n the subsequent secton. 3. EQUIVALE SADARD DEVIAIO OR RELIABILIY AALYSIS 3. or Independent ormal Random Varables and Lnear Lmt State uncton Consder a lnear performance functon as G( a a a a ( 0 0 where have a normal dstrbuton as ~ (,. Usng the Rosenblatt transformaton [5] from the -space to the U- space, can be expressed as U, ( and by nsertng Eq. ( nto Eq. (, the performance functon can be rewrtten n the U-space as [6] where g( U a ( U a au a a aub a { } and a (3 b a a a μa G( μ. A -D example for lnear performance functons n the - space and the U-space s llustrated n g. (a and (b, respectvely. (a -space (b U-space gure. Lnear erformance uncton he relablty ndex β s defned as the mnmum dstance from the orgn n the U-space to the lmt state functon [7], whch s defned as g( U 0, as shown n g. (b and expressed as b 0. (4 a Snce falure s defned f G( 0 and b0 G( μ, the relablty ndex β s postve when a functon value at the desgn pont s negatve, whch mples the desgn pont s located n the feasble regon. Accordngly, the relablty ndex becomes negatve f the desgn pont s located n the nfeasble doman. or a lnear lmt state functon n the U-space, the probablty of falure s analytcally gven usng the relablty ndex by ( b 0 a, (5 and snce a { a }, the probablty of falure n Eq. (5 can be expressed n terms of nput standard devatons as b0 ( σ ( a. (6 he nput standard devaton ( of the th random varable o o can be expressed as usng the rato ( where s the current standard devaton for the th random varable. hen, the probablty of falure at a gven desgn d μ( for a lnear lmt state functon n the U-space s expressed as a functon of the rato δ as b0 ( δ o ( a. (7 Hence, for a lnear lmt state functon n the U-space, the probablty of falure s exactly computed accordng to the

5 rato of the nput standard devaton (δ. Inversely, we can fnd an exact δ value correspondng to the target probablty of falure at the current desgn, whch wll be explaned n detal n Secton or General Random Inputs and erformance unctons Even f t s exact for lnear performance functons wth ndependent normal random nputs, Eq. (7 cannot be drectly used for general cases where random nputs could be nonnormal and/or correlated, or performance functons are nonlnear. herefore, to generalze Eq. (7 so that t can be used for any random nputs, ether correlated or ndependent, and nonlnear performance functons, Eq. (7 s modfed as ( δ c0 o ( c (8 where c 0 and c need to be determned usng the current probablty of falure where δ = c c d cur 0 o ( c 0 0 and ts senstvty wth respect to δ c d d ( c 3 o (9 δ=. (0 cur 3 o d0d0 ( c d hus, summng up Eq. (0 from = to = yelds cur 3 o d0d0 ( 0 c d cur d0d0 d0 δ= ( from Eq. (9. Once the left-hand sde of Eq. ( s known, Eq. ( can be easly solved for d 0 usng numercal root fndng methods such as fzero n Matlab, whch does not requre extra functon calls of the orgnal performance functons and thus s computatonally trval. After obtanng d 0, c can be obtaned usng Eq. (0. hen, the probablty of falure n Eq. (8 can be expressed usng d 0 and c as ( δ. ( cur o d0 ( c he senstvty of the probablty of falure wth respect to δ n Eq. ( s gven usng the chan rule as, (3 o o δ= σ=σ and the senstvty of the probablty of falure wth respect to s expressed n a smlar manner to Eq. (6 as ( σ f ( x; σ I ( d x x o σ=σ. (4 ln f ( ; x σ EI ( x he partal dervatve of the log functon of the ont D wth respect to s known as the frst-order score functon for and s denoted as [9,0] ( ln f ( ; ( x; σ x σ. (5 s o further derve the frst-order score functon for, frst consder statstcally ndependent dmensonal random nput. hen, the ont D of s expressed as multplcaton of ts margnal Ds as f ( x ; σ f ( x ; (6 where f ( x ; s the margnal D correspondng to the th random varable. herefore, for statstcally ndependent random varables, the frst-order score functon for s expressed as ln ( ; ( ln f ( ; f x ( x; σ x σ. (7 s Snce the margnal D s analytcally avalable as lsted n Ref., the dervaton of Eq. (7 for any dstrbuton type s very straghtforward. he frst-order score functons for for the normal, lognormal, and Gumbel dstrbutons are lsted n able where dstrbuton parameters for each D are also explaned n Ref.. able. rst-order Score uncton for for Independent Random Varables Margnal rst-order Score uncton, s ( (; D x σ x ormal Lognormal ln x (ln x

6 ( Gumbel x x e or a bvarate correlated random nput, the ont D of s expressed as [7-9], f ( x ; σ c( u, v; f ( x ; f ( x ; (8 where c s called a copula functon and defned as Cuv (, ; cuv (, ; C, uv ( uv, ; uv (9 and u ( x ; and ( ; v x are CDs for and, respectvely; θ s the correlaton coeffcent between and ; and C s a copula functon [8,30]. Commonly used copula functons and ther densty functons are lsted n Refs. and 30. Accordngly, usng Eq. (8, the frst-order score functon n Eq. (5 for the correlated bvarate nput s expressed as ln ( ; ( ln (, ; ( x ; μ cuv f x. (30 s he dervaton of the frst term of the rght-hand sde of Eq. (30 s also straghtforward from analytc forms of copula densty functons and lsted n able for the Clayton, rank, Gaussan, and ndependent copula where ( and ( are the standard normal CD and D, respectvely, gven by u u ( u ( d exp d, (3 and the second term of the rght-hand sde of Eq. (30 s dentcal to Eq. (7, so t can be obtaned from able. One can see from able that the log-dervatve of copula densty functon wth respect to s dentcal to the log-dervatve of copula densty functon wth respect to shown n able 5 u of Ref. except the term, whch s a partal dervatve of a margnal CD wth respect to and lsted n able 3. One can also see from able that Eq. (30 s dentcal to Eq. (7 f the ndependent copula s used, whch means that the ndependent random nput s a specal case of the correlated random nput where the ndependent copula s used. able. Log-dervatve of Copula Densty uncton Copula ype ln cuv (, ; Clayton ( ( u u u u v rank e e e e Gaussan ( u ( v ( u u ( ( u ( ( u( Independent 0 ( u ( uv e e u ( u ( v ( uv able 3. artal Dervatve of Margnal Dstrbuton Margnal u D x x ormal Lognormal ln x (ln x u ( x Gumbel x e 4. EQUIVALE SADARD DEVIAIO OR SAMLIG-BASED RBDO o apply Eq. ( to the samplng- based RBDO, the new target probablty of falure whch s denoted as needs to new be set up. hen, the obectve of ths secton s to fnd the equvalent standard devaton whch s defned as the ncreased standard devaton to satsfy the new target probablty of falure at the optmum desgn. However, Eq. ( cannot be drectly appled to the samplng-based RBDO snce the current probablty of falure changes durng the desgn teraton. urthermore, f the samplng scheme s utlzed to estmate the current probablty of falure and fnd the equvalent standard devaton at a gven desgn, t s a sgnfcant computatonal effort. hus, t s necessary to develop a way to fnd the equvalent standard devaton wthout usng the samplng. 4. Shft of robablty of alure Equaton ( can be used at the optmum desgn obtaned usng the orgnal random nput and the orgnal target probablty of falure denoted by as o ( δ (3 o o d0 ( c snce the current probablty of falure at the optmum desgn should be the same as the target probablty of falure. hen, the equvalent standard devaton can be found by settng ( δ new. However, ths approach apparently cannot be appled to the samplng-based RBDO snce the obectve of the proposed method s to effcently fnd the same optmum desgn usng the new target probablty of falure and the equvalent standard devaton. Hence, Eq. (3 needs to be

7 modfed to fnd the equvalent standard devaton from the begnnng of the desgn teratons. If the new target probablty of falure and equvalent standard devaton are used for the samplng-based RBDO, the probablty of falure at the optmum desgn wll be new and snce new, we need a decreasng functon of δ. Hence, o usng the nverse form of Eq. (, the probablty of falure at the optmum can be expressed as ( new 0 δ (33 d o ( c and by lettng ( δ n Eq. (33 to be o, δ s obtaned as o ( c d0. (34 new o 4. Algorthm he samplng-based RBDO launches at the determnstc optmum desgn snce t s usually closer to the RBDO optmum desgn than the ntal desgn and accordngly the computatonal effort can be reduced. At the determnstc optmum desgn, the rato δ s frst set up as and the samplng s carred out for the relablty analyss. Usng the relablty analyss result and Eq. (34, δ s updated and the updated standard devaton s used as the nput at the next desgn. or an RBDO problem wth multple constrants, even f δ s fxed as one parameter for multple random varables, t could be dfferent for dfferent actve constrants. In ths case, the maxmum δ s selected to assure a relable and safe optmum desgn. hs could be another error source of the proposed method n addton to the probablty of falure approxmaton shown n Eq. (8. hs error wll be studed n detal n Secton 5.3 usng a numercal example. By usng the samplng-based RBDO wth the equvalent standard devaton, the new target probablty of falure whch s much larger than the orgnal target probablty of falure can be utlzed resultng n the reducton of the number of samples used. gure shows the overall flowchart of the samplng-based RBDO wth the equvalent standard devaton. he soluton of Eq. (34 cannot be unquely obtaned snce there are unknowns but only one equaton. he easest way of solvng Eq. (34 s to assume all δ are the same as δ. However, there could be some cases where we cannot ncrease nput standard devatons, for example, when nput random varables cannot have negatve values. In such cases, we can set up the rato for those random varables as one and assume that the rest of the ratos are the same. After solvng Eq. (34, the equvalent standard devaton for the th random varable s gven by. (35 E o However, as mentoned before, the current probablty of falure at a desgn s not always new durng the desgn optmzaton. Hence, the performance functon needs to be shfted by α as s G ( G( (36 such that the probablty of falure at a current desgn s always. α can be easly obtaned from functon values at MCS new samples. Usng the shfted probablty of falure, Eq. (34 can be used from the begnnng and the coeffcent d 0 s obtaned by takng the partal dervatve of Eq. (33 wth respect to δ and usng Eqs. (3 and (4 where the falure set s defned as x: G( x 0. (37 As the desgn approaches the optmum desgn, the current probablty of falure approaches the new target probablty of falure and thus α converges to zero. gure. lowchart of Samplng-based RBDO wth Equvalent Standard Devaton 5. UMERICAL EAMLES umercal studes are carred out n ths secton to verfy the probablty of falure estmaton n terms of δ proposed n Secton 3 and the equvalent standard devaton for the samplng-based RBDO proposed n Secton 4. Sectons 5. and 5. show comparson studes between the MCS and proposed method for the probablty of falure estmaton usng -D and 9-D mathematcal examples, respectvely. or the -D example, both ndependent and correlated random nputs are consdered. Secton 5.3 llustrates how the proposed

8 equvalent standard devaton can be appled to solve a hghrelablty RBDO model. or all tests, to concentrate on the proposed method and elmnate errors from surrogate models, true analytc functons are used nstead of surrogate models, and the ratos (δ n Eqs. ( and (34 are assumed to be the same as δ for the smplcty of calculaton. 5. robablty of alure Estmaton Usng -D Mathematcal Example o verfy how accurately the proposed probablty of falure n terms of the nput standard devaton can approxmate the true one obtaned by the MCS wth mllon samples, consder a -D hghly nonlnear performance functon [3] shown n g. 3 and expressed as requred to obtan the same accuracy usng the orgnal standard devaton. able 4. Comparson of robablty of alure Estmated , % MCS Error, % where 3 4 G( ( Y 6 ( Y ( Y 6 Z (38 Y Z , and have (4.5,0.3 and statstcally ndependent. (,0.3, respectvely, and they are gure 4. Comparson of MCS and Estmated robablty of alure for Independent Case gure 3. Shape of Hghly onlnear erformance uncton able 4 compares the probablty of falure obtaned usng the MCS and the probablty of falure estmated by the proposed method shown n Eq. (, respectvely, whch s shown n g. 4, too. rom the table and fgure, we can see that the proposed probablty of falure estmaton works very well for a hghly nonlnear performance functon. able 4 also shows that f the nput standard devaton ncreases from 0.3 to 0.54, that s.8, the probablty of falure ncreases more than 50 tmes. Inversely, f we want the current probablty of falure to ncrease by 50 tmes, then we can fnd the correspondng standard devaton, whch s used to fnd the equvalent standard devaton. If the ncreased probablty of falure and equvalent standard devaton are used for the samplng-based RBDO, the total number of MCS samples wll reduce to less than % of the number of MCS samples o test the probablty of falure estmaton for correlated random nput, suppose that and have (4.5,0.45 and (,0.45, and they are correlated wth the Clayton copula (τ=0.5. he same performance functon n Eq. (38 s stll used for the test. able 5 and g. 5 show the result of the comparson test. In ths case, due to the correlaton effect, error between the MCS and the estmated probablty of falure becomes relatvely larger than the ndependent case. However, t s stll accurate enough up to.4, whch means the probablty of falure can be ncreased by about 0 tmes and 5% of the number of MCS samples wll be requred for the same accuracy for the samplng-based RBDO. able 5. Comparson of robablty of alure Estmated , % MCS Error, %

9 MCS Error, % gure 5. Comparson of MCS and Estmated robablty of alure for Correlated Case 5. robablty of alure Estmaton Usng 9-D Mathematcal Example o verfy whether the proposed method works for hghdmensonal problems, consder a 9-D polynomal functon, whch s known as the extended Rosenbrock functon [3] and modfed for the purpose of the probablty of falure calculaton, 8 G( ( 00( 36, 5 0 for,,9 (39 where the propertes of nne random varables are shown n able 6. or ths problem, all random varables are assumed to be statstcally ndependent. able 6. ropertes of Random Varables Random Standard Dstrbuton Mean Varables Devaton ~ 9 ormal.0 0. able 7 and g. 6 show the result of the comparson test. or hgh-dmensonal problems, the probablty of falure s very senstve to the change of the nput standard devatons as shown n able 7. Hence, n ths test, the ncrement of standard devaton s tested only up to.4. able 7 shows that by ncreasng the nput standard devaton by.4 tmes the probablty of falure ncreases almost 00 tmes wth % error. hs mples that f the ncreased standard devaton s used for the samplng-based RBDO, the total number of MCS samples reduces to less than % of the number of MCS samples requred to obtan the same accuracy usng the orgnal standard devaton. hs sgnfcant reducton of the number of MCS samples used wll be shown n the next secton. able 7. Comparson of robablty of alure ,% Estmated gure 6. Comparson of MCS and Estmated robablty of alure for Hgh Dmensonal Case 5.3 Samplng-Based RBDO wth Equvalent Standard Devaton o see how the proposed equvalent standard devaton can reduce the computatonal effort of the samplng-based RBDO, consder a -D mathematcal RBDO problem, whch s formulated to ( d d 0 ( d d 0 mnmze C( d 30 0 G d subect to ( ( ( 0, ~ 3 L U d dd, dr and R where three constrants are gven by G ( G ( ( Y 6 ( Y ( Y 6 Z 80 G3 ( 8 5 (40 (4 Y where Z , and are drawn n g. 7. he propertes of two random varables are shown n able 8, and they are assumed to be ndependent. In Eq. (40, the orgnal target probablty of falure ( o s set up as % for all three constrants, whch s a 4σ desgn. able 8. ropertes of Random Varables Random Dstrbuton d L d O d U Standard Varables Devaton ormal

10 ormal gure 7. Shape of Constrant and Cost unctons Usng Eq. (9, the number of the MCS samples to accurately estmate the target probablty of falure s 50 mllon assumng MCS 5%, whch wll make the samplngbased RBDO slow and even slower when combned wth mplct surrogate models such as the Krgng model. o carry out the samplng-based RBDO, the determnstc optmum s frst found at d dopt =(5.956, where the samplng-based RBDO wth the orgnal random nput and target probablty of falure s launched and the RBDO optmum s found at d ropt =(4.684,.847. o fnd the RBDO optmum, 5 mllon MCS samples are used when constrants are not actve and 50 mllon MCS samples are used when actve. Detaled nformaton on how to carry out the samplng-based RBDO s shown n Ref.. or the test of the equvalent standard devaton, 5 dfferent new target probabltes of falure whch are obtaned by multplyng the orgnal target probablty of falure by 0,0,30,40, and 50, respectvely, are consdered as lsted n able 9. he thrd column of able 9 shows the number of the MCS samples requred to obtan the probablty of falure wth the same accuracy ( MCS 5% as the orgnal random nput. rom the thrd column of able 9, t can be easly shown how drastcally the number of the MCS samples s reduced from 50 mllon. Wth 5 dfferent new target probabltes of falure, 5 dfferent samplng-based RBDOs are carred out wth the MCS samples shown n able 9. able 9. ew get robablty of alure Case new,% o. of MCS Samples Case M Case M Case M Case M Case M able 0 compares the samplng-based RBDO results of each case wth the one obtaned from the orgnal nput and target probablty of falure. At each optmum desgn, the MCS wth 50 mllon samples s carred out to check the accuracy of the samplng-based RBDO wth the equvalent standard devaton. able 0 shows that the maxmum error n terms of the probablty of falure estmaton becomes larger as the new target probablty of falure become larger. hs s manly because the dfference between two δs for two actve constrants s larger as the new target probablty of falure become larger, and only one δ s used for the desgn optmzaton. Another error source s the hgh nonlnearty of the second constrant functon as shown n g. 7. Even wth relatvely large error of the probablty of falure calculaton, the optmum desgns are very close to the orgnal result. able 0. Comparson of ve RBDO Cases Case Optmum MCS Max Desgn,%,% Error,% Case 4.606, Case 4.65, Case , Case , Case , Orgnal 4.684, he proposed samplng-based RBDO wth the equvalent standard devaton s orgnally developed to be combned wth surrogate models to enhance the computatonal effcency. However, t can be applcable to the samplngbased RBDO usng actual computer smulatons f the model s not computatonally demandng snce the proposed method can reduce the number of samples sgnfcantly, especally when used n conuncton wth more effcent samplng schemes than the MCS. In that case, error from surrogate models s elmnated, whch makes the proposed samplngbased RBDO more accurate. 6. COCLUSIOS o enhance the computatonal effcency of the samplngbased RBDO, a methodology to convert an RBDO problem wth very small probablty of falure to an RBDO problem wth relatvely large probablty of falure by ncreasng nput standard devatons s proposed, whch can be used n conuncton wth surrogate models and mproved samplng schemes. he frst-order score functon for the nput standard devaton s used to derve the probablty of falure n terms of the nput standard devaton for both ndependent and correlated random nputs. he derved probablty of falure s then used to update the target probablty of falure and fnd the equvalent standard devaton to be used for the samplngbased RBDO. umercal examples show the accuracy of the proposed probablty of falure n terms of nput standard devatons and demonstrate that the samplng-based RBDO wth the equvalent standard devaton yelds a smlar optmum desgn obtaned usng the orgnal random nput wth sgnfcantly enhanced computatonal efforts. o further mprove the accuracy of the proposed method, a new equvalent standard devaton concept s beng nvestgated

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