Response Surface Method Using Sequential Sampling for Reliability-Based Design Optimization

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1 Proceedngs of the ASME 9 Internatonal Desgn Engneerng echncal Conferences & Computers and Informaton n Engneerng Conference IDEC/CIE 9 August September, 9, San Dego, Calforna, USA DEC Response Surface Method Usng Sequental Samplng for Relablty-Based Desgn Optmzaton Lang Zhao, K.K. Cho and Ikjn Lee Department of Mechancal & Industral Engneerng College of Engneerng he Unversty of Iowa, Iowa Cty, IA 54, USA lazhao@engneerng.uowa.edu kkcho@engneerng.uowa.edu lee@engneerng.uowa.edu Correspondng author Lu Du Department of Cvl and Envronmental Engneerng he Unversty of Calforna, Los Angeles, CA 995, USA dul@seas.ucla.edu ABSRAC radtonal RBDO requres the senstvty for both the most probable pont (MPP) search n nverse relablty analyss and desgn optmzaton. However, the senstvty s often unavalable or dffcult to compute n complex multphyscs or multdscplnary engneerng applcatons. Hence, the response surface method (RSM) s often used to calculate both functon evaluatons and senstvty effectvely. Researchers have been developng the RSM for decades, and yet are stll searchng for an approach wth an effcent samplng method for fast convergence whle meetng the accuracy crtera. hs paper proposes a new adaptve sequental samplng method to be ntegrated wth the Krgng method for RBDO. By usng the bandwdth of the predcton nterval from the Krgng method, a new samplng strategy and a new local response surface accuracy crtera are proposed. In ths sequental samplng method, the response surface s ntated usng very few samples. An addtonal samplng pont wll then be determned by fndng the pont that has the largest absolute rato between the bandwdth of the predcton nterval and the predcted response wthn a neghborng area of current pont of nterest. he nserton of addtonal samplng wll contnue untl the accuracy crteron of the response surface n the neghborhood of the current pont of nterest s acheved. Case studes show ths proposed adaptve sequental samplng technque yelds better result n terms of convergence speed compared wth other samplng methods, such as the Latn hypercube samplng and the grd samplng, when the same sample sze s used. Both a hghly nonlnear mathematcal example and a vehcle durablty engneerng example show that the proposed RSM yelds accurate RBDO results that are comparable to the senstvtybased RBDO results, as well as sgnfcant savngs n computatonal tme for functon evaluaton and senstvty computaton. KEYWORDS Sequental Samplng Method, Response Surface Method (RSM), Krgng Method, Gaussan Stochastc Process, Predcton Interval, Relablty-Based Desgn Optmzaton (RBDO). INRODUCION Relablty-based desgn optmzaton (RBDO) has been wdely used recently for desgn of engneerng applcatons. o acheve the optmal desgn wth target relablty of performance functon, the optmzaton algorthm and relablty analyss are mplemented. In recent years, there have been varous attempts to develop enhanced relablty analyss methods to accurately compute the probablty of falure of a performance functon. he most common relablty analyss methods are () analytcal methods and () smulaton or samplng methods. he analytcal methods have two dfferent branches. One s the MPP-based method, whch ncludes the frst-order relablty method (FORM) [-4], the second-order relablty method (SORM) [5,6], and the newly developed MPP-based dmenson reducton method (DRM) [7,8,9]. he other one s the probablty densty functon (PDF) approxmaton method [-]. Among the MPP-based methods, FORM and SORM frst use frst or second order aylor seres expanson to approxmate the performance

2 functon G( X) at the MPP, and then calculate the probablty of falure usng the approxmated performance functon. he MPP-based DRM approxmates the mult-dmensonal performance functon by usng the sum of lower dmensonal functons and then calculates the probablty of falure. he PDF approxmaton method evaluates the PDF of the performance functon by assumng a general output dstrbuton type and then, usng the approxmated PDF, evaluates the probablty of falure of the performance functons. he smulaton or samplng method, such as Monte Carlo smulaton (MCS) [] s used for the probablty of falure calculaton snce these methods do not requre any analytcal formulaton. here s a tradeoff between these two dfferent methods. he analytcal method requres fewer numbers of functon evaluatons, but t requres the senstvty of both cost and performance functon [4]. hs could be a problem when dealng wth dverse engneerng applcatons. A large number of engneerng desgn problems do not have desgn senstvty avalable or t s extremely hard to obtan; ths means the senstvty-based method would not be applcable for use. On the other hand, the smulaton or samplng method requres too many functon evaluatons, and the computatonal tme could not be affordable n large-scale engneerng applcatons. Hence, the response surface method becomes desrable for such cases. In exstng response surface methods, most researchers focused on global response surface to approxmate the true performance functons [4,5]. he global response surface method requres a large number of functon evaluatons to acheve a hgh fdelty response surface and would not be applcable for complex mult-physcs or multdscplnary engneerng applcatons. In such cases, the adaptve local response surface method [6,7] s more desrable snce t adaptvely samples around the current pont of nterest, whch saves qute an amount of functon evaluaton. Another challenge s the samplng strategy for response surface. he Latn hypercube samplng method [8,9] has been appled n response-surface-based RBDO. It tres to occupy the entre desgn doman most evenly and gan as much nformaton about the true performance functons as they can. Another samplng technque, mportance samplng [,], samples around the lmt state area and predcts the response accurately around the lmt state. However, for general response surface generaton purpose, t s stll not clear that where to sample n order to maxmally mprove the fdelty of the response surface or how many samples are enough. In ths paper, we propose a new response-surface-based RBDO method that combnes the above two methods. We use FORM for relablty analyss and a proposed sequental samplng adaptve local response surface for functon evaluaton and senstvty calculaton. In ths way, we keep the effcency and accuracy of senstvty-based relablty analyss for probablty of falure and offer a way to calculate the functon evaluaton and senstvty that s more effcent and accurate compared to the exstng global response surface methods. Numercal examples demonstrate that the proposed sequental samplng-based response surface method for RBDO can acheve smlar accuracy and smlar or better results n terms of computatonal expense when compared wth tradtonal senstvty-based RBDO. At the same tme, t offers a greater capablty for dealng wth dverse and practcal engneerng applcatons where senstvty-based RBDO s not feasble.. SEQUENIAL SAMPLING ADAPIVE RESPONSE SURFACE MEHOD. Krgng Method he Krgng method has ganed a lot of nterests for generatng the response surface n recent years. In the Krgng method, the outcomes are consdered as a realzaton of a stochastc process and the predcted values are derved later by applyng stochastc process theory. Consder n sample ponts: X= [ x, x,..., x ] wth x R m n, and n responses Y = [ y( x), y( x),..., y( x n )] wth y( x ) R. In the Krgng method, the response at the samples are consdered as a summaton of two parts as Y = Fβ + e () he frst part of the rght hand sde of Eq. (), Fβ, s consdered as the mean structure of the response, where F=[ fk( x ), =,..., n, k =,..., K] s an ( n K) desgn matrx, and fk ( x) represents user-defned bass functons, whch are usually n a smple polynomal form, such as, xx,,.... In Eq. (), β = [ β, β,..., β K ] are the regresson coeffcents from the generalzed least squares regresson method. In ths paper, to use as small number of samples as possble, the bass functons are set to fk () x = andx. he second part of Eq. (), e= [( e x),( e x),...,( e xn )], s a realzaton of the stochastc process e() x that s assumed to have zero mean and covarance structure E[( e x)( e xj)] = σ R(, θ x, xj), where σ s the process varance, and θ s the process parameter whch has to be estmated from sample data. R( θ, x, x j ) s the correlaton functon of the stochastc process. For a multdmensonal problem, t becomes the multplcaton of the correlaton functons for each dmenson as m d d d d R( θ, x, x j) = Rd( θ, x, xj ), where x s the d th d = d component of x, θ s the d th component of θ. For example, f the process s assumed to be a statonary Gaussan process, d d d d d d then Rd( θ, x, xj ) = exp( θ x x j ). Another assumpton of the model n Eq. () s, Cov[ Fx ( ), e ( x )] = ; that s, the resduals are uncorrelated wth the bass functons. Under the general decomposton of Eq. (), the objectve s to predct the nose-free unbased response at a new pont of nterest x. In the Krgng method, ths predcton of response s wrtten as a lnear predctor as ŷ (x ) = w Y ()

3 where w denotes the ( = [ w( x), w( x),..., w n ( x )] n ) weght vector for predcton at x. Usng Eq. (), the unbased predcton condton Ey [ ˆ( )] = Ey [ ( x )] s expressed as x E [ yˆ (x ) y(x ) ] = E[ wy y(x ) ] = Ew [ ( Fβ + e) ( fβ + e( x))] = Ew [ e ( x ) + ( F w f ) β = E[( F w f ) β] e ] ( ) where f = [ f( x), f x,..., f K ( x)]. herefore, the unbased condton s ensured by mposng the constrant Fw = f on the predcton weghts for each pont of nterest []. Under ths constrant, w s obtaned by solvng = F λ f whch represents the Lagrangan frst-order necessary condtons of mnmzng the mean squared error (MSE) of predcton [], where λ represents the Lagrangan multplers, R s the correlaton matrx ( R) = R( θ, x, x ),, j =,..., n, where β = ( F R F) F R Y, γ = R ( Y Fβ ), and the predcton error varance [] s expressed as ) () R F w r, (4) j j and r, )] = [ R( θ, x, x ),..., R( θ, xn x s the correlaton vector between the predcton locaton x and all n samples, =,..., n. he soluton of Eq. (4) s gven by x λ = ( F R F) ( F R r f w R r Fλ = ( ) Hence the predcton s expressed as y ˆ( x ) = w Y = ( r Fλ) R Y r R Y F R r f F R F F R Y = ( ) ( ) = f β + r γ σ ˆ p ( x) = Var[ y( x) y( x)] = σ (+ wrw wr ) (7) Also, the senstvty of the predcted response s gven by where y y = = J f + r x xn yˆ '( x ),..., ( x ) β J ( x ) γ (8) ( J ( )) f ( x ) f x j = and δ x j ( J ( x )) r j j (5) (6) R( θ, x, x ) = are x the Jacobans of f and r, respectvely []. Accordng to Eqs. (6) and (8), the functon value and senstvty of response are obtaned as long as the correlaton matrx R(, θ x, x ) s avalable. In ths paper, e() x j s assumed to be a Gaussan process, hence d d d d d d Rd( θ, x, xj ) = exp( θ x xj ). he optmum value of process parameter θ s defned as the maxmum lkelhood estmator (MLE), whch s the maxmzer of ( n lnσ + ln R(θ ) ), and t s equvalent wth the formulaton as θs the mnmzer of ψ ( θ) = R( θ) n σ (9) where the process varance σ s estmated by ( ) σ = Y Fβ ( Y Fβ ) [4]. Under the assumpton of n Gaussan process, the α-level predcton nterval of response s wrtten as yˆ( x ) Z σ ( x ) y( x ) yˆ( x ) + Z σ ( x ) ( + α)/ p ( + α)/ p () herefore, the bandwdth of the predcton nterval at pont of nterest s x d( x ) = Z α σ p ( x ) () ( + )/. Sequental Samplng Method for Krgng Method One of the bggest challenges n RSM s where to take a new sample such that the mprovement of the response surface can be maxmzed. Researchers have developed numerous methods, such as the Latn hypercube samplng, mportance samplng, and the maxmum entropy method, to determne where to sample. However, none of these methods s samplng accordng to the nonlnearty of the true model whch s the crucal factor for response surface. Another challenge s that t s very dffcult to determne how many samples are necessary for the response surface whle the on-demand accuracy of the response surface s acheved. Hence t s necessary to propose a new samplng method that can use as few samples as possble and stll acheve the on-demand accuracy... Sequental Samplng Response Surface Method hs paper proposes a sequental samplng-based response surface method (SS-RSM) usng the bandwdth of the predcton nterval determned by Eq. () to solve the samplng challenge mentoned above. Frst, the nserton crteron IC s defned as d ( x) IC( x) = max, =,..., P () yˆ ( x) where P s the number of performance functons. From the Krgng method shown above, t s easy to see that the predcton nterval gven by Eq. () s an nherent and desrable measure for predcton error measurement. In addton, dvdng t by the predcted response can normalze the error measure and make t consstent for all cases. he phlosophy of the nserton crteron s that t s tryng to fnd the pont that t has the largest dscrepancy between the true response and the predcted response. However, ths

4 dscrepancy cannot be obtaned unless the true response s known. herefore, alternatvely, by defnng IC as Eq. (), we are tryng to fnd the pont wth the largest dscrepancy between the predcton nterval and the predcted response, n other words, the weakest pont n the doman where we have the least confdence on the predcton accuracy. As wll be shown later n ths paper, we can see that ths alternatve dscrepancy measurement s ndeed consstent wth the dscrepancy between the true response and the predcted response and the stepwse nserton procedure decreases the predcton error n a fast speed. o assess the accuracy of the response surface, the error defnton s gven by the average of relatve error wthn the doman as N yˆ( ) y( ) Err = x x () N y( x ) = surface (the sold pnk lne) has stll not converged to the true response (the sold blue lne) shown as Fg.. Response Samples Response Surface rue Response Largest Bandwdth Pont X (a) Intal Stage: Samples where N s the number of checkng ponts over the doman selected by the user; t usually takes a large number, such as. he sequental samplng strategy s conducted as followng: Step Intal samples are placed by grd samplng nto the desgn doman, and the responses are evaluated at these samples. Step Apply the Krgng method based on current exstng samples. Step Check the crteron IC( x ) wthn the doman. Step 4 Fnd the pont x such that IC( x ) takes the largest value and nsert one more sample at x. Step 5 Check the error defned by Eq. (); f error s larger than tolerance, repeat Step to Step 4. Otherwse stop samplng. Response Response.5 Samples.5 Response Surface rue Response Largest Bandwdth Pont X.5.5 (b) After More Samples Inserted Samples Response Surface rue Response.5 Largest Bandwdth Pont X.5 (c) After More Samples Inserted he proposed sequental samplng method can acheve the accuracy faster than other samplng methods, such as grd samplng and Latn hypercube samplng. A -D profle example s gven to demonstrate the fast convergence of ths sequental samplng-based Krgng RSM compared wth movng least squares RSM. he true functon s Response.5.5 Samples Response Surface rue Response Largest Bandwdth Pont 5 4 Y( x) =.5x.5x.5x +.5x +.x+. (4) where x [, ]. he sequental samplng method s ntated wth three evenly dstrbuted samples shown as Fg. (a). As shown n the fgures, the sold red lne, the sold green lne, and the two dashed black lnes are the true response gven by Eq. (4), the response surface predcted by Eq. (6), and the 95% predcton nterval by Eq. (), respectvely. he black stars are the ntal grd samples and the red star s the dentfed current largest bandwdth pont wthn the doman. From Fg. (a)-(d), we can see that each tme the nserton crteron correctly dentfy the pont where t has a large dscrepancy between the true response and the predcton response. From Fg. (d) we can see that after fve more samples are sequentally nserted, the response surface becomes almost the same as the true response. Meanwhle, f the movng least squares method [7] s appled to ths example, even after evenly dstrbuted samples have been used, the response X (d) After 5 More Samples Inserted Fgure. Response Surface Usng Sequental Samplng Y 4 CI_PI (Runs_Quad-95%) X Ytrue Yrsm CIL CIU PIL PIU Exper Fgure. Movng-Least-Squares- Based RSM for -D Example [7]

5 .. Comparson Study between Sequental Samplng and Other Samplng Methods An error convergence comparson study s carred out between sequental samplng, grd samplng and Latn hypercube samplng for a complex -D example. he -D functon s Y( x) = + (.96x +.46x 6) + (.96x x 6).6 (.96x+.46x 6) (5) (.46x +.96 x ) where x [5,9], x [,5.5] Among the three samplng methods, the grd samplng method samples each desgn parameter at N ponts evenly along ts axs. Hence, the number of entre samples used s N d where d s the number of desgn parameters. In tradtonal Latn hypercube samplng, N ponts n a d-dmensonal Latn hypercube are to be selected where each of the d coordnate dmensons s dscretzed to the values through N. he ponts are to be chosen n such a way that no two ponts have any coordnate value n common. A so-called mproved Latn hypercube samplng (IHS) method was proposed by Beachkofsk [5]; t attempts to spread out the samples over the entre doman as evenly as possble. In ths paper, IHS s used for comparson purposes. Snce each realzaton of Latn hypercube samplng s unque and dfferent from others, we took trals for IHS, and the maxmum, mnmum, medan and mean of the error are recorded. he error, defned as Eq. (), s calculated at evenly dstrbuted testng ponts over the entre doman after usng the Krgng method for each samplng method at dfferent sample szes. able. Error Comparson between hree Samplng Methods # of Grd IHS ( trals) Sequental Samples Samplng Mn Max Mean Medan Samplng he comparson n able shows that the error by usng sequental samplng converges more quckly to zero than by usng the grd samplng. When compared wth IHS, even though n the best case (Mn) IHS s more accurate, sequental samplng converges more quckly n the probablty sense (Mean and Medan) and yelds more accurate results as the samples sze ncreases. Another wdely used -D mathematcal example for testng metamodelng s used to demonstrate the fast convergence of the sequental samplng. hs -D functon s the so-called Bran example [6] and expressed as 5. 5 Y( x) = ( x x x 6) ( )cos( x) 4π + π + 8π + (6) 5 x, x 5 he comparson study s the same as the prevous -D example, carred out between grd samplng, Latn hypercube samplng and the sequental samplng. able shows the smlar result compared wth able. he sequental samplng method converges faster than Latn hypercube samplng n probablty sense. able. Error Comparson between hree Samplng Methods # of Grd IHS ( trals) Sequental Samples Samplng Mn Max Mean Medan Samplng In addton, consderng for multdmensonal desgn cases, IHS s computatonally far more expensve and not even affordable to apply [5]. Another dsadvantage of IHS s that the user does not know how many samples are enough for response surface untl all samples are evaluated. If the response surface does not acheve the accuracy requrement based on current samples, the user has to ncrease sample sze and calculate the Latn hypercube samplng method agan. If that s the case, those prevously evaluated samples may not be used at all whch leads to a very large waste of samplng cost. On the other hand, n sequental samplng, the user ncreases the sample sze step by step and can stop when the accuracy of response surface s acheved. All of prevous samples contrbute to the response surface; hence no sample s wasted.. SEQUENIAL SAMPLING RESPONSE SURFACE MEHOD FOR RBDO. Center Fxed Support for Local Response Surface n Relablty Analyss Durng one desgn teraton of RBDO the performance functon value and senstvty wth respect to desgn varables are requred to fnd the MPP, whch can be obtaned by the Krgng method. Snce the MPP search s conducted on a hyper-sphere n U-space where the current desgn s the orgn, the response surface doman should be a hyper-sphere that covers the desgn doman wth the radus somewhat larger than the target relablty ndex β t. Choosng ths local support nstead of the global support s because t s easer to make the response surface generated n the local support more accurate

6 than that the global support. Hence n ths paper, as shown n Fg., the local support for the relablty analyss s defned as a hyper-sphere whose radus s cβ t, where c s a scalng factor whch s.~.5 dependng the nonlnearty of the performance functons. Fgure 4. Demonstraton of Ill-condtoned Matrx Problem Because of Samplng Method n U-space Fgure. Local Support Sze of Response Surface for Relablty Analyss n U-space It s possble to generate a new local response surface wth new samples for each MPP canddate. hat s, new samples around the current MPP canddate are evaluated and a new local response surface s generated each tme. However ths approach would cause two problems. he frst problem s that t wastes samples. In fact, for relablty analyss, snce the MPP wll always be located wthn a relatvely small area around the orgn, one center-fxed local response surface s enough. he second problem s, f a new response surface s generated every tme wth new samples for a new MPP canddate, the to use prevously evaluated samples for current response surface generaton may cause the estmated covarance R matrx to be ll-condtoned and consequently fal to predct the response surface. hs ll-condtoned matrx orgnates from Krgng algorthm. In Eq. (6) we can see that the predcton of response depends on R. However, R may become ll-condtoned f the samples are too aggregated wthn a narrow space []. Fgure 4 shows how the llcondtoned matrx happens f we take samples and generate a new local response surface for each MPP canddate search. As shown n Fg. 4, after two teratons of the MPP search, ten samples were generated along the axs; then, the current MPP canddate III wll evaluate addtonal fve samples. Wthn the local support of MPP canddate III, there are eght samples that aggregate n two parallel lnes. In ths stuaton, the estmated R would become ll-condtoned []. hs stuaton would occur more often n hgh-dmensonal desgn problems. Because of these two major reasons, t s better to have the local support fxed as a hyper-sphere wth the center of orgn n U-space for relablty analyss.. Adjustment of Inserton Crteron and Error Defnton for SS-RSM RBDO As shown n Secton, the sequental samplng method s to dentfy the pont that has the largest IC among the entre desgn doman. However n the RBDO procedure, we are focusng more on the area neghborng the pont of nterest. Hence an adjustment s done to search the weakest pont around the pont of nterest, whch s shown as follows: Step Intal samples are placed by grd samplng nto the desgn doman, and the responses are evaluated at these samples. Step Apply the Krgng method based on current exstng samples. Step Check the crteron IC( x ) wthn the doman D, whch s defned as D = { x y( x) yˆ ( x ) Z α σ p ( x )} ( )/ where x s the pont of nterest. Step 4 Fnd the pont x such that IC( x ) takes the largest value and nsert one more sample at x. Step 5 Check the error; f error s larger than tolerance, repeat Step to Step 4. Otherwse, stop samplng. In Secton.., the error of the response surface s gven by Eq. () where the denomnator s the true functon value of the response. However n real problems, the functon value s not avalable at the pont of nterest. Hence the error defnton of the response surface has to be adjusted. It s noted that the error has to reflect the dscrepancy of the predcted response and the true response at the pont of nterest, whle we only have the predcton nterval bandwdth as the measure of accuracy of response. Hence the error here s defned as the average of the normalzed predcton standard devaton as N σ p ( ) Err = x N yˆ( x ) = (7) where N s the number of checkng ponts wthn a small neghborhood of the pont of nterest, { x x x < d}, and d s the threshold. From Eq. (7) we can see that the error of the current pont of nterest s defned as the average error around t; ths can be graphcally understood n Fg. 5. Recall that the purpose of the response surface s to predct the functon value and senstvty accurately at the current pont of nterest; hence there s no need to check the entre support doman, only the neghborhood around the pont of nterest has to be assured to be accurate. Moreover, checkng the entre support doman would make the accuracy crtera harder to

7 acheve and therefore would lead to more sample nserton whch s an unnecessary waste for functon evaluaton. On the other hand, f we check only the error at the pont of nterest, there s a chance that the error could be zero from Eqs. (7) and (7) f the pont of nterest s also a sample that has been evaluated. Snce the error s used as the crtera to check that whether or not the response surface s accurate enough, zero error at the pont of nterest cannot provde any confdence n the accuracy for predcted senstvty of the performance functon. herefore, a small neghborhood around the pont of nterest s most desrable for error checkng. In ths paper, the neghborng area s defned as % of the current support sze. Fgure 6. Relaton between Support Sze and Desgn Movement when the desgn movement s small, the support sze should be ncreased to nclude more prevously evaluated samples to save computatonal tme. A comparson study shows the effcency of ths adaptve support sze n savng functon evaluaton. A determnstc optmzaton problem s defned as Fgure 5. Neghborng Area of the Pont of Interest for Error Checkng. Adaptve Support Sze Effcency Strategy n Determnstc Desgn Optmzaton In the RBDO algorthm, PMA+ [7] requres to carry out determnstc optmzaton frst and then start the RBDO process. In response surface-based determnstc optmzaton, there s a tradeoff between the support sze and the accuracy of the response surface. As s well known already, a smaller support can provde a more accurate response surface than a larger support does. At the same tme, usng more of prevously evaluated samples s also desrable to save computatonal tme, and a larger support can always nclude more of prevously evaluated samples than a smaller support can. Hence t s mportant to decde a sutable support sze to generate the response surface. In ths paper, we propose an adaptve support sze method that can solve ths problem for the determnstc optmzaton process. Startng from the ntal desgn pont, the frst prorty s to fnd the correct drecton for the teraton. Hence a smaller support that can gve more accurate senstvty s desrable. As the desgn moves to the next poston, the support sze would be decded by the dstances from the prevous desgns as R = mn( R d / d, R ) (8) k k k k max where R k s the k th support sze, R max s the maxmum threshold, and d s the dstance between the k- th k desgn and the k th desgn as shown n Fg. 6. Equaton 8 mples that when the desgn movement s large, whch means two desgns are qute dfferent, the support sze should be decreased to yeld a more accurate response surface to determne the next drecton. On the other hand, mnmze C( d) subject to G ( Xd ( )) <, =,, nc where ( d+ d ) ( d d + ) C( d) = X X G ( X) = G X X X.46X 6).6 (.96X.46X ( X) = + ( ) + ( ) (.46X+.96 X) 8 G ( X) = ( 8 5) d = [5,5], [,], [,] X + X + ntal X X (9) Frst, the Krgng method usng the fxed support sze s appled to solve the determnstc optmzaton problem. Second the adaptve support sze effcency strategy s appled to solve the same problem. Fnally these two results are compared wth the result solved by usng the senstvty-based determnstc optmzaton algorthm. As shown n able, the number of functon evaluatons (NFE) conssts of the functon evaluaton (FE) and the senstvty calculaton (SC). he tme for functon evaluaton s the tme used to evaluate the functon value at a gven sample, and the tme for senstvty calculaton s the tme used to calculate the frst dervatve of the functon respect to desgn varables and random parameters. he Krgng method by usng the adaptve support sze effcency strategy not only saves number of functon evaluatons, but also yelds a more accurate optmum desgn. able. Comparson between Fxed & Adaptve Support Sze Optmum Desgn Case Cost NFE X X Fxed Support FE Adaptve Support FE Senstvty-Based FE+9 SC

8 .4 Overall Process of Sequental Samplng-Based Response Surface Method for RBDO Summarzng the three parts above, we can now apply the sequental samplng-based RSM for the RBDO problem. he SS-RSM, shown as Fg. 7, s appled to carry out the functon evaluaton and senstvty calculaton. For each pont of nterest, the support sze of current response surface s decded frst, and then the prevously evaluated samples wthn the local support are found. he sequental samplng strategy s carred out to nsert more sample ponts untl the response surface acheves the accuracy crteron. In the end the functon value and senstvty value are gven by the Krgng method. hs SS-RSM s encapsulated as a black-box n the overall RBDO procedure and s called whenever functon value and senstvty value are requred, shown n Fg. 8. Fgure 7. Flowchart of SS-RSM oolbox Fgure 8. Flowchart of SS-RSM-based RBDO 4. NUMERICAL EXAMPLES he accuracy and effcency of the sequental samplngbased response surface for RBDO s verfed by comparng t wth the senstvty-based RBDO. A -D hghly nonlnear mathematcal problem and a -D multdmensonal engneerng vehcle problem are carred out by both SS-RSMbased RBDO and senstvty-based RBDO for comparson purposes. 4. -D Hghly Nonlnear Mathematcal Example A two-dmensonal mathematcal RBDO problem s formulated to mnmze C( d) Xd d d d, d R andx R where ( d+ d ) ( d d + ) C( d) = X X G ( X) = G ( X) (.96X.46X 6) ( ) G ( X) = X + 8X + 5 L U d = [, ] and d ntal = [,], d = [5, 5] X ~ N( d,.5) for =, ar subject to PG ( ( ( )) > ) PF, =,, L U ndv = X + X 6(.96X 4.46X 6) (.46X.96 X) () where the target probablty of falure for each constrant s ar P F =.75%, =~. Snce the gven target probablty of falure s.75%, the relablty ndex of ar β = Φ ( P F ) =. s selected. In able 4, t shows that both the senstvty-based RBDO and the SS-RSM RBDO converge to the optmal desgn after two teratons. Numbers shown n column Iteraton means the current desgn teraton and the current lne search number. For example, D.O. means the determnstc optmum, and, means that t s the frst desgn wth the second lne search n RBDO. Accordng to able 4, we can see that the SS-RSM for RBDO can acheve the same RBDO optmum desgn as senstvty-based RBDO. Moreover, n ths case, the number of functon evaluatons used n response surface method s smaller than the number used n senstvty-based RBDO because when the desgn s gettng close to the optmum pont more of the prevously evaluated samples are used to save computatonal tme. Hence the SS-RSM-based RBDO s more effcent than senstvty-based RBDO n ths case. able 4. Comparson between Senstvty-Based RBDO and SS-RSM RBDO Iteraton Cost x x NFE Senstvty-Based RBDO Intal

9 D.O FE+9 SC, FE+4 SC, FE+4 SC, FE+ SC, FE+4 SC, FE+7 SC Opt FE+7 SC SS-RSM-Based RBDO Intal D.O , , , , , , Opt racked Vehcle Roadarm Problem he roadarm of a tracked vehcle s used to demonstrate the applcablty of the SS-RSM-based RBDO. he roadarm s modeled usng 57 eght-node soparametrc fnte elements (SOLID45) and four beam elements (BEAM44) of Ansys [8], as shown n Fg. 9, and s made of S44 steel wth Young s modulus E=. 7 ps and Posson s rato ν=.. he durablty analyss of the roadarm s carred out usng the Durablty and Relablty Analyss Workspace (DRAW) [9, ], to obtan the fatgue lfe contour as shown n Fg.. he fatgue lves at the crtcal nodes shown n Fg. are chosen as the desgn constrants of RBDO. Fgure 9. Fnte Element Model of Roadarm Fgure. Shape Desgn Varables of Roadarm In Fg., the shape desgn varables consst of four cross-sectonal shapes of the roadarm where the wdths (x - drecton) of the cross-sectonal shapes are defned as desgn varables, d, d, d 5, and d 7, at ntersectons,,, and 4, respectvely, and the heghts (x -drecton) of the crosssectonal shapes are defned as desgn varables, d, d 4, d 6, and d 8. Eght shape desgn random varables and four random parameters for the fatgue materal propertes are lsted n able 4. able 5. Propertes of Random Varables of Roadarm Lower Intal Upper Bound Desgn Bound COV L d d U d Random Varables Dstrbuton ype d % Normal d % Normal d % Normal d % Normal d % Normal d % Normal d % Normal d % Normal Fatgue Materal Propertes Non-desgn Uncertantes Mean COV Dstrbuton ype Fatgue Strength Coeffcent, σ 77 % Log-normal Fatgue Strength Exponent, b -.7 % Normal Fatgue Ductlty Coeffcent, ε f.4 % Log-normal Fatgue Ductlty Exponent, c -.6 % Normal he RBDO problem for the roadarm s formulated to Fgure. Fatgue Lfe Contour and Crtcal Nodes of Roadarm mnmze C( d) ar subject to PG ( ( d) ) PF,,, L U d d d where C( d) : Weght of Roadarm () L( d) G ( d) =, =,, L t > = L( d) : Crack Intaton Fatgue Lfe, L t P : Crack Intaton arget Fatgue Lfe: 5 years =Φ ( β ) =Φ ( ), =,, ar F t

10 For the determnstc desgn optmzaton process, 8-D response surface s used snce the other 4 parameters are not changng n ths process. able 6 shows the fnal RBDO optmum results as well as the computatonal tme. he computatonal tme conssts of two parts: functon evaluaton and senstvty calculaton. For senstvty-based RBDO, the senstvty calculaton per desgn varable takes about % of the tme that the functon evaluaton does n fnte element analyss. Hence, the overall computatonal tme s calculated as Comp.me=No.of FE +. No.of FE No.of Varables () able 6 shows that the SS-RSM used more computatonal tme to acheve the determnstc optmum compared wth the senstvty-based one. However, startng from the determnstc optmum pont untl the end of the RBDO optmum desgn, the SS-RSM saves more n computatonal tme than the senstvty-based one does. he reason s that at each desgn pont the relablty analyss requres a large number of functon evaluatons as well as senstvty calculatons for the MPP search. If the response surface has been generated, the functon evaluaton and senstvty calculaton would be free, whereas n senstvty-based RBDO, t has to evaluate functon value and senstvty by usng fnte element analyss and senstvty analyss, whch requre sgnfcant computatonal tme. able 6. Comparson between SS-RSM-Based and Senstvty- Based RBDO Determnstc Desgn Intal RBDO Optmzaton Parameters Desgn SEN SS-RSM SEN SS-RSM d d d d d d d d Cost Actve Constrants NFE Comp. me,,5,8,,,5,8,,,5,8,,,5,8, FE +SC FE 7FE+ 7SC 988FE CONCLUSION A sequental samplng-based response surface method s proposed to effcently generate the accurate response surface for RBDO. he new sample pont s dentfed by the largest predcton nterval bandwdth wthn the doman where the predcted response locates wthn 95% predcton nterval from the current pont of nterest. o ntegrate ths samplng method for RBDO and avod an ll-condtoned covarance matrx n the Krgng method, a center-fxed local support s used for the nverse relablty analyss. An adaptve support sze method s also proposed for response surface to rapdly acheve the determnstc optmum before the RBDO process. he proposed method s compared wth the senstvtybased RBDO method n terms of accuracy and effcency. Numercal examples show that sequental samplng-based RSM can acheve the same accurate optmum desgn as the senstvty-based RBDO does. Furthermore, the sequental samplng-based RSM can save more n terms of computatonal tme, whch s a crucal ssue n real engneerng desgn applcatons. he most mportant pont s that the proposed SS-RSM-based RBDO s applcable for broader engneerng applcatons because t does not requre the senstvty analyss of performance functons. 6. ACKNOWLEDGEMEN Research s supported by the Automotve Research Center, whch s sponsored by the U.S. Army ARDEC. 7. REFERENCES. Haldar, A., and Mahadevan, S., Probablty, Relablty and Statstcal Methods n Engneerng Desgn, John Wley & Sons, New York,.. Hasofer, A.M., and Lnd, N.C., An Exact and Invarant Frst Order Relablty Format, ASCE Journal of the Engneerng Mechancs Dvson, Vol., No., pp.-, u, J., and Cho, K.K., A New Study on Relablty- Based Desgn Optmzaton, Journal of Mechancal Desgn, Vol., No.4, pp , u, J., Cho, K.K., and Park, Y.H., Desgn Potental Method for Relablty-Based System Parameter Desgn Usng Adaptve Probablstc Constrant Evaluaton, AIAA Journal, Vol.9, No.4, pp ,. 5. Hohenbchler, M., and Rackwtz, R., Improvement of Second-Order Relablty Estmates by Importance Samplng, ASCE Journal of Engneerng Mechancs, Vol.4, No., pp.95-99, Bretung, K., Asymptotc Approxmatons for Multnormal Integrals, ASCE Journal of Engneerng Mechancs, Vol., No., pp.57-66, Rahman, S., and We, D., A Unvarate Approxmaton at Most Probable Pont for Hgher-Order Relablty Analyss, Internatonal Journal of Solds and Structures, Vol.4, No.9, pp.8-89, We, D., A Unvarate Decomposton Method for Hgher-Order Relablty Analyss and Desgn Optmzaton, Ph. D. hess, the Unversty of Iowa, Lee, I., Cho, K.K., Du, L., and Gorsch, D., Inverse Analyss Method Usng MPP-Based Dmenson Reducton for Relablty-Based Desgn Optmzaton of Nonlnear and Mult-Dmensonal Systems, Computer Methods n Appled Mechancs and Engneerng, Vol.98, No., pp.4-7, 8.. Rosenblueth, E., Pont Estmates for Probablty Moments, Proceedngs of he Natonal Academy of Scences of he Unted States of Amerca, Vol.7, No., pp.8-84, Huang, B.Q., and Du, X.P., Uncertanty Analyss by Dmenson Reducton Integraton and Saddlepont Approxmatons, Journal of Mechancal Desgn, Vol. 8, No., pp.6-, 6.. Youn, B.D., X, Z.M., and Wang, P.F., Egenvector Dmenson-Reducton (EDR) Method for Senstvty-Free

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