Doubly weighted moving least squares and its application to structural reliability analysis

Transcription

1 Doubly weghted movng least squares and ts applcaton to structural relablty analyss Jan L 1 Shangha Jao ong Unversty, Shangha 0040, Chna Nam H. Km Unversty o Florda, Ganesvlle, FL 3611, USA and Ha Wang 3 Shangha Jao ong Unversty, Shangha 0040, Chna In ths paper, we proposed a two-stage hybrd relablty analyss ramewor based on the surrogate model, whch combnes the rst-order relablty method and Monte Carlo smulaton wth a doubly-weghted movng least squares (DWMLS) method. he rst stage conssts o constructng a surrogate model based on DWMLS. he weght system o DWMLS consders not only the normal weght actor o movng least squares, but also the dstance rom the most probable alure pont (MPFP), whch accounts or relablty problems. An adaptve expermental desgn scheme s proposed, durng whch the MPFP s progressvely updated. he approxmate values and senstvty normaton o DWMLS are chosen to determne the number and locaton o the expermental desgn ponts n the next teraton, untl a convergence crteron s satsed. In the second stage, MCS on the surrogate model s then used to calculate the probablty o alure. he proposed method s appled to our benchmar examples to valdate ts accuracy and ecency. Results show that the proposed surrogate model wth DWMLS can estmate the alure probablty accurately, whle requrng ewer orgnal model smulatons. Nomenclature n = number o nput random varables N = number o expermental ponts x = vector o nput random varables = relablty ndex HL = relablty ndex by Hasoer-Lnd algorthm = mean = standard devaton g( X ) = lmt state uncton gˆ( X ) = approxmate lmt state uncton/ response surace uncton X add = new added expermental ponts n any teraton MLS = movng least square DWMLS = doubly weghted movng least square SVR = support vector regresson ANN = artcal neural networs MPFP = most probable alure pont 1 Graduate Student, School o Aeronautc and Astronautc, ljannpu@gmal.com Assocate Proessor, Department o Mechancal and Aerospace Engneerng, nm@ul.edu, 3 Proessor, School o Aeronautc and Astronautc, wangha601@sjtu.edu.cn 1

2 MCS = Monte Carlo Smulaton FORM = rst order relablty method SORM = second order relablty method H-L = Hasoer-Lnd algorthm RSM = response surace method LHD = Latn hypercube desgn FEA = nte element analyss CFD = computatonal lud dynamcs COV = coecent o varaton I. Introducton has been well recognzed that uncertantes n engneerng systems (e.g., appled loads, materal propertes and Igeometrc tolerances) can result n catastrophc alure and should be managed approprately. he tradtonal actor-o-saety approach to compensate or uncertantes oten leads to ether un-conservatve or too-conservatve desgns. Relablty analyss taes nto account these uncertantes n evaluatng system s saety, whch has become an mportant part o recent engneerng desgn. here has been a growng nterest n the use o relablty methods or structural desgn and saety assessment 1-4. However, wth the development o advanced numercal smulaton methods, whch commonly tae several hours to perorm a sngle evaluaton, classcal relablty methods can easly become mpractcal. Monte Carlo Smulaton (MCS) and ts varants demand tremendous computatonal resources that prohbt ts practcalty. On the other hand, approxmaton methods, such as rst-order relablty method (FORM) and second-order relablty method (SORM), have ssues wth relatvely poor perormance n accuracy. hereore, t seems reasonable to use a response surace or surrogate model to approxmate the perormance uncton and apply ether MCS to calculate relablty. However, relablty analyss s derent rom approxmaton problems, rom whch the surrogate model orgnated. In conventonal approxmaton problems, the general crteron o a surrogate model s to mnmze the error between the true uncton and the surrogate model n the entre doman o nterest. In relablty analyss, however, t s mportant to denty a lmt state, whch s the boundary between sae and aled regons, especally near the most probable alure pont (MPFP). In ths paper, a doubly weghted movng least squares (DWMLS) method s proposed, whch taes nto account the characterstc eature o relablty analyss. he proposed DWMLS method conssts o two weghtng schemes the rst weght consders the dstance between the samplng pont and the predcton pont, whle the second consders the dstance between the samplng ponts and the MPFP. A hybrd, two-stage relablty analyss ramewor s proposed, whch taes ull advantage o the FORM/SORM, MCS, adaptve experment desgn and DWMLS to acheve both accuracy and ecency. he remander o ths paper s organzed as ollows. In Secton II, a bre lterature revew o varous surrogate models n relablty analyss s presented. In Secton III, a general prncple o the movng least squares method s stated, ater whch the newly added weghtng system o DWMLS s detaled. In Secton IV, a new adaptve expermental desgn procedure s llustrated n detal. A complete lowchart o the proposed hybrd relablty analyss ramewor s then presented n Secton V. Four numercal examples are presented n Secton VI, whch hghlght the capabltes o the proposed method and demonstrate ts accuracy and ecency, ollowed by a summary and conclusons n Secton VII. II. Revew on surrogate models n relablty analyss he ncreased complexty o smulatons on real systems stmulates the development o surrogate models that approxmate the behavor o complex systems, mprove ther valdaton process, and ad optmzaton o the system 7. Surrogate models are developed n order to analyze expermental data and to buld emprcal models based on observatons. hese models were rst ntroduced n desgn optmzaton and appled to relablty analyss and desgn because o ther merts n ecency. Wong 5 rst proposed a complete, quadratc orm polynomal and appled t to relablty analyss. In hs wor, the number o polynomals and the requred samplng ponts ncrease rapdly wth the number o random varables. In order to reduce the number o samplng ponts, Bucher and Bourgund 3 proposed a two-teraton quadratc polynomal wthout cross-terms. Rajashehar and Ellnwood 6, and Lu and Moses 4 mproved ths approach by updatng the surrogate model parameters untl a convergence crteron was satsed. Km and Na 8 proposed a sequental approach to the surrogate model where the gradent projecton method s used to ensure that the samplng ponts are located near the alure surace. Zheng and Das 9 proposed an mproved surrogate model and appled t to

3 relablty analyss o a stened plate structure. Guan and Melchers 10 evaluated the eect o surrogate model parameter varaton on relablty. Kaymaz and McMahon 11 suggested a new surrogate model, n whch a weghted regresson method was appled nstead o normal regresson. Inspred by Kaymaz and McMahon s method, Nguyen and Seller 1 mproved the weghted regresson method where the ttng ponts were weghted accordng to ther dstance rom the true alure surace and the estmated desgn pont. However, when the lmt state uncton s hghly nonlnear, polynomal-based surrogate models can perorm poorly because they try to approxmate the perormance uncton globally. hus, advanced surrogate modelng methods were ntroduced to replace tradtonal global polynomal-based models. Cho and Grandh 13, and Km and Wang 14 ntroduced polynomal chaos expanson or relablty analyss and desgn. Papadraas et al. 15, and Hurtado and Alvarez 16 used Neural Networs to relablty analyss n conjuncton wth Monte Carlo Smulaton. Gomes and Awruch 17 compared Neural Networs wth FORM, MCS, and the Importance Samplng technque. Kaymaz 18, and Panda and Manohar 19 proposed the Krgng method to relablty analyss and compared t wth the most common surrogate models. Echard and Gayton 0 presented a Krgng enhanced MCS method n relablty analyss and the test problems have demonstrated ts ecency and accuracy. Most 1 presented an ecent adaptve response surace approach or relablty analyss, where support vector machnes were used to classy the alure and sae doman. Guo and Ba proposed a least squares support vector machne or regresson nto relablty analyss and the results demonstrate excellent accuracy and smaller computatonal cost than the relablty method based on support vector machnes. Movng least squares (MLS) s a local weghted least squares method, orgnally ntroduced by Lancaster and Salausas 3 or smoothng and nterpolatng data. Ater that, t s wdely used to obtan approxmatons n meshree methods and structural optmzaton because o ts localzed approxmaton property 4. Unle other surrogate models, the MLS has no expensve nner parameter optmzaton durng the adaptve modelng process. hereore, Krshnamurthy 5 compared the MLS method wth other local methods, such as Krgng, and ound t to be more accurate and computatonally eectve or the examples consdered. Bucher 6 carred out research amed at comparng the perormance o these response suraces and ts applcaton n relablty analyss. In Kang and Koh s research 7, MLS has made t possble to derve the approxmaton uncton closer to the lmt state uncton and exhbted mproved perormance n terms o sgncant reducton o the number o structural analyses and senstvty accuracy o the relablty ndex to the random varables. Youn and Cho 41, and Song and Lee 8 have appled the movng least squares method to relablty based desgn optmzaton and MLS perormed well n uncertanty desgn. III. Doubly weghted movng least squares (DWMLS) he basc dea o surrogate modelng n relablty analyss s to replace the perormance uncton wth an approxmate uncton, whose value can be computed easly. Compared wth the global polynomal regresson method, the movng least squares (MLS) method s a local regresson and a relatvely new surrogate modelng technque. he MLS method can be ound extensvely n the element-ree Galern method and computer graphcs. In ths Secton, the basc prncple o the MLS method s presented rst, ollowed by an ntroducton to a new weghtng scheme that wors better or relablty analyss. A. Basc prncple o movng least squares We consder a perormance uncton g(x) n an n-dmensonal space o random varables, n whch the vector o random varables s dened as x [ x 1, x,..., x n ]. In the MLS method, the perormance uncton s approxmated by where gˆ( ) ( ) ( ) x p x a x (1) px ( ) [ p1( x), p( x),..., p m ( x)] s a vector o m polynomal bass unctons, and x a1 x a x a m x s a vector o correspondng coecents. It s noted that n global regresson methods, a( ) [ ( ), ( ),..., ( )] the coecents are constant, whle they are unctons o x n MLS. In ths paper, the bass uncton px ( ) s dened usng polynomals up to the second order wthout cross terms, as px ( ) [1, x, x,..., x, x, x,..., x ] () 1 n 1 n 3

4 where the dmenson o px ( ) s m = n + 1. However, t s possble that the bass uncton can nclude cross-terms, as well as hgher order terms. he unnown coecents n (1) can be calculated by mnmzng the error between the perormance uncton and ts approxmaton at dscrete ponts. In order to do that, N sample ponts are rst selected rom the nput space; these samples are denoted by xi, I 1,, N. hen, the perormance unctons, g( x I ), are calculated at these sample ponts. hs process may nvolve numercal smulatons, such as nte element analyss or computatonal lud dynamcs. At a gven predcton pont x, the MLS technque determnes the unnown coecents by mnmzng the error between actual and approxmated values o the perormance uncton wth weghts, as N x xxi xi p xi a x (3) I 1 R( ) w( )[ g( ) ( ) ( )] he above ormula can also be wrtten n a matrx orm as: R( x) [ Pa( x) g] W[ Pa( x) g ] (4) where g, P and W are dened as: g [ g( x1), g( x),..., g( x N )] (5) p ( x1) p ( x ) P p ( xn ) N m w( x x1) w( )... 0 ( ) x x Wx w( x xn ) N N In (7), the weght w( x x ) taes the ollowng Gaussan orm: I (6) (7) w ( xx / D ) ( e I I e ) xxi DI 0 otherwse ( ) I (1 e ) x x (8) where the parameter s used to control the weght uncton curve, DI denes the doman o the nluence o pont x I, x xi s the Eucld dstance between samplng pont x I and predcton pont x, and r xxi / DI s a normalzed dstance. In Fg.1, the shape o Gaussan weght uncton w(r) wth derent values o s demonstrated. 4

5 Fgure.1. Shape o the Gaussan weght uncton wth derent control parameter he mnmum o the square error R( x) can be acheved by vanshng the partal dervatves wth respect to unnown coecents, as R( x) 0, 1,, m (9) a he condtons n Eq. (9) yeld the ollowng system o lnear equatons: where A(x) and b(x) are dened by: Axax ( ) ( ) bx ( ) (10) Ax ( ) PWxP ( ) (11) bx ( ) PWxg ( ) (1) Once the unnown coecents, a(x), are calculated by solvng Eq. (10), Eq. (1) s used to approxmate the perormance uncton. As can be seen rom the weght scheme n Eq. (8), the weght w I, whch s assocated wth samplng pont x I, decreases as x moves away rom x I. he contrbuton o those ponts whose dstance rom x s greater than DI vanshes, and thus, there s no need to nclude them n the regresson process. hereore, the dmenson o the matrces n the MLS process s much smaller than the total number o sample ponts N. It s mportant to note that at a gven predcton pont x, there must be enough sample ponts x I such that the coecent matrx A(x) should be non-sngular. Also, t should be noted that the coecents, a(x), must be calculated at every predcton pont. B. Doubly weghted movng least squares In MLS, t s ratonal to mpose a heavy weght to the ponts that are close to the predcton pont and a lght weght or more dstant ponts n order to better approxmate the perormance uncton. In relablty analyss, however, the most mportant regon s around the MPFP, because t contrbutes most to the probablty o alure. hus, we ntroduce an addtonal weghtng scheme nto MLS where the dstances o samplng ponts to the predcton pont x and to the MPFP are consdered smultaneously. As dscussed n the prevous secton, the rst weghtng scheme s based on the dstance o samplng ponts xi to the predcton pont x: 5

6 w ( x, x ) w( x x ) (13) I I I he second weghtng scheme taes nto account the dstance between the samplng ponts and the current MPFP. It ams to penalze ponts located ar rom the current MPFP. he second weght actor s expressed as: w II (, I ) exp( di ) x x (14) where d I s the dstance between the I-th samplng pont and the current MPFP. It should be noted that snce there s no normaton on the poston o MPFP n the rst stage, the conventonal MLS s appled to the rst surrogate model. hen, through the advantage o the above-mentoned two weghtng schemes, we nd the ollowng expresson sutable to obtan the weght or each samplng pont: w( xx, ) w ( xx, ) w ( x ) (15) I I I II I In ths paper, the weght matrx n (4) s replaced wth ths double weght matrx and the new surrogate model consders both the dstance rom the predcton pont and the MPFP. IV. Adaptve Desgn o Experments (DOE) A. Dscusson on DOE n relablty analyss he qualty o probablty estmates usng a surrogate model depends on not only the surrogate model tsel, but also the locaton o the ponts chosen to buld the surrogate model (desgn o experments). In the context o relablty analyss, there exst two nds o DOE strategy: sngle desgn and adaptve desgn. For the ormer, Wong 5 and Faravell 9 employed sngle actoral expermental desgn contanng n ponts to t a quadratc uncton and to estmate the alure probablty. Cheng 30 proposed a surrogate model based on the artcal neural networ and usng unorm expermental desgn n predctng alure probablty. However, n the case o a blac-box computer model, sngle expermental desgn cannot guarantee the accuracy o approxmaton, especally n doman around the lmt state surace. Hence, t would be better to start wth an ntal set o samples and gradually add more samples based on the normaton provded by prevous samples. Such a process s commonly reerred to as the adaptve desgn o experments and has receved more attentons than the ormer strategy. In relablty analyss, Bucher and Bourgund 3 rst appled a surrogate model n whch expermental ponts are chosen around the mean values o random varables, whch orm a matrx called the desgn matrx. Quadratc polynomals wthout cross terms are then used to t these expermental ponts. Among varous samplng methods, a common approach s to evaluate g( X ) at a n 1 combnaton o and, where and are the mean and standard devaton o random varables, X, and s a actor that denes the samplng range. he number o unnown coecents n the perormance uncton s n 1, gven as 0 n n 1 1 yˆ( x) a a x a x A FORM algorthm was appled to estmate the MPFP based on the above quadratc polynomal uncton, and the next desgn matrx was constructed on the new center pont determned by the ollowng expresson: μ xd xm μg( μ) g( μ) g( x ) where x m and x D ndcate the new center pont and the nterm MPFP obtaned n the prevous stage, respectvely. hs second desgn matrx s used to orm another quadratc polynomal just as the rst matrx and the nal alure probablty was estmated by t. hus, ths procedure requres a 4n 3 evaluaton o g( X ). Rajashehar and Ellngwood 6 questoned a sngle cycle o updatng s adequate, and they proposed to mprove t by usng more teratons untl a convergence crteron s satsed. he above surrogate model n structural relablty analyss has some dsadvantages: (1) wth the ncrease o random varables, the total g( X) evaluaton number ncreases excessvely ast, partcularly whle the convergence process s slow; () the result obtaned by the above procedures has been shown to be senstve to the 6 D (16) (17)

7 parameter and may not always gve an acceptable approxmaton to the true alure probablty (t s possble that the sequental center ponts durng teraton may oscllate n the doman around the true desgn pont and not converge); and (3) at derent stages o surrogate modelng, only a part o avalable normaton on all prevous g( X ) s drectly used. hus, t s consdered that the accuracy o above teratve algorthm depends manly on the characterstcs o the nonlnear perormance uncton, thereby lmtng ts applcaton. Smpson 31 concluded that a recommended expermental desgn should have a space-llng property. In the current study, an optmal unorm Latn hypercube desgn s utlzed as the ntal expermental desgn. In the ollowng adaptve experment desgn process, the dstance rom the center pont to the new sample pont s determned not by the constant parameter, but by the locaton o lmt state estmated usng all the prevous normaton o g( X ). B. Adaptve expermental desgn 1. Intal expermental desgn--latn hypercube desgn (LHD) he statstcal method o the Latn hypercube desgn (LHD) was developed to generate a set o samples rom a multdmensonal dstrbuton. he technque was rst descrbed by McKay et al. 3, and urther elaborated by Iman and Conover 33. LHD s popular n desgn and analyss o computer experments. he locaton o LHD ponts s determned through a random procedure and a complete theory can be ound n Forrester's wor 34. he goal o a good LHD s to mae the selected samplng ponts as unorm as possble to cover the entre desgn space. In ths paper, the p -crteron s selected as a unormty measure and the translatonal propagaton algorthm proposed by Vana 35 s used to obtan the optmal unorm LHD. In ths algorthm, samplng ponts are determned by mnmzng the ollowng crteron: p where d s the dstance between two sample ponts, j 1/ p N1 N p dj (18) 1 j1 x and x j, 1/ t n t dj d( x, x j ) x x (19) j 1 and p 50 and t 1were recommended by Jn 36. he space-llng property o optmal unorm LHD ensures that no two samplng ponts are too close to each other; unormly dstrbuted expermental ponts would enhance the approxmaton capacty o surrogate models. In ths paper, LHD s used as an ntal set o expermental desgns.. Adaptve expermental desgn process Based on the developed DWMLS model, the FORM algorthm can determne an nterm MPFP, whch s presumably located close to the true MPFP. he teratve process adds new expermental ponts to mprove the accuracy o the surrogate model near the MPFP. In ths paper, an addtonal samplng pont s added n the locaton o the lmt state surace n each varable drecton, startng rom the current MPFP. Let us assume that x s the MPFP at th DWMLS model. hen, the perormance uncton g( X ) s evaluated at x, and addtonal samplng ponts wll be added. Frstly, t s necessary to chec the magntude o g( x ), whch helps judge whether x s close enough to the true MPFP. A rato actor C r s ntroduced to measure the closeness o x to lmt state surace, as C r g x g( μ) ( ) (0) where μ s the mean o random varables. I C r s smaller than the threshold 0 C r (always set as 0.05), then consdered to be close enough to the lmt state surace, and the ollowng sngle pont s added to the exstng samplng ponts: x s 7

8 xadd μg( μ) g μ x ( μ) g( x ) (1) where x add s the newly added expermental desgn n 1th teraton. On the other hand, when C r s larger than the threshold, t means that x s relatvely ar away rom the lmt state surace. In such a case, samplng ponts are added, startng rom x, close to the lmt state surace n the drecton o each varable (see Fg. 4). By dong ths, the accuracy o the surrogate model s mproved locally near the current MPFP and the lmt state surace. Unle other surrogate models, the MLS has no expensve nner parameter optmzaton durng the adaptve modelng process, so a temporary DWMLS model g ( ) N 1 x can be establshed qucly based on exstng N 1 ponts. he dstance between x and the lmt state surace n the -th varable drecton,, s calculated usng the rst-order aylor seres expanson, as g( x ) g ( x) N 1 x x () hereore, the ollowng n+1 samples are added at the 1th expermental desgn: x 1 x [,0,,0] x [0,,,0] x [0,0,, ] n (3) he rst-order approxmaton n () can have a large error when the perormance uncton s hghly nonlnear and the partal dervatve becomes too small. hereore, n order to prevent rom beng too large, a threshold c t s proposed, where s the standard devaton o -th random varable and t s a constant between.0 and 3.0. Hence the adjusted samplng ponts become: x x [0,...,,...,0] c add c c x [0,...,sgn( ),...,0] (4) Ater n+1 addtonal expermental desgn ponts are determned, the perormance unctons are evaluated at these locatons. Ater updatng the MWMLS wth more ponts, FORM s used to determne a new x. he adaptve expermental desgn process repeats untl the lmt state surace can be approxmated accurately, partcularly n the regon near the MPFP. he ollowng two convergence crtera are used: where the two tolerances, and MPFP, are xed at (5) x x 1 MPFP V. Procedure o the proposed method he proposed two-stage hybrd relablty analyss method can be dvded nto (1) DWMLS enhanced FORM teratons and () MCS to calculate the alure probablty. In the rst stage, the DWMLS surrogate model s constructed and updated by adaptvely addng samplng ponts near the lmt state surace. In the second stage, MCS s perormed on the nal DWMLS to calculate the probablty o alure. Fg. summarzes the step-by-step procedures o the proposed algorthm. (1). Determne the ntal expermental desgns, X ntal : 8 1

9 he ntal number o samples s determned by N1 3n where n s the number o random varables. he ntal optmal LHD s generated n standard normal space U o random varables, so t s necessary to dene the samplng range n U space. he samplng range on each dmenson s selected by, where s xed at 4.0. hen, an optmal LHD U ntal s generated through the SURROGAES oolbox 37. he ndependent expermental ponts n U ntal are transerred nto the physcal space o mutually correlated nonnormal random varables by Nata's rule, and the ntal expermental desgn X ntal s determned. Snce the mean pont x s not ncluded n X ntal, x s added to X ntal ; thus, the number o ntal samples s N1 1. (). Compute the value o the perormance uncton at each pont o X ntal : g g( x ), {1,,..., N 1} (6) 1 (3). Calculate weght actors assgned to each pont accordng to Eq. (8), and t the rst conventonal MLS. Apply FORM based on the rst surrogate model and determne the rst relablty ndex 1 and MPFP x. (4). Calculate the values o the perormance uncton at the current MPFP and chec the closeness rato actor C r by Eq. (0). I step (7). C r s smaller than the threshold value, obtan the new addng pont 1 X add usng Eq. (1) and sp to (5). Add the new MPFP observaton evaluated rom the above step nto exstng samplng ponts and constructng a temporary DWMLS to estmate the partal dervatves to each random varables on MPFP. (6). Calculate the length by Eq. () on each drecton, compare wth (4), and add to X add. and then determne x c add based on Eq. (7). Calculate the values o the perormance uncton at gven X add and add these ponts nto the exstng samplng ponts. (8). Calculate the weght actors assgned to each pont accordng to Eq. (15) and buld a DWMLS model. (9). Apply the FORM algorthm to the DWMLS model and determne the relablty ndex HL and MPFP. (10). Repeat step (4) ~ step (9) untl the convergence crtera n Eq. (5) s satsed. (11). Perorm MCS usng DWMLS to calculate the probablty o alure. 9

10 Fgure. Flow chart or the proposed doubly weghted movng least square method or relablty analyss VI. Numercal examples In ths secton, our examples nvolvng explct and mplct unctons rom structural applcatons are presented to llustrate the ecency and accuracy o the proposed DWMLS method. Snce t s assumed that numercal smulaton s ar more expensve than developng surrogate models, the number o numercal smulatons Ns s used as a measure o ecency. he proposed DWMLS method s a hybrd relablty analyss approach where rst, adaptve expermental desgn combned wth FORM algorthm s manly amed at postonng the MPFP, and then, subsequent Monte Carlo smulaton s used to estmate the alure probablty p. hereore, the accuracy o the 10

11 proposed method would not only depend on MCS or p, but also the nterm FORM. In numercal examples, comparsons have been made wth the other relablty analyss methods presented n lterature to evaluate the perormance o the proposed method. A. A nonlnear lmt state uncton In the rst numercal example, the ollowng two-dmensonal perormance uncton s used: g( x ) exp[0.4( x ) 6.] exp[0.3x 5] 00 (7) 1 where x 1 and x are ndependent, standard normal random varables. hs example s wdely used n relablty analyss methods 11,1,7,38. Fgure. 3 ~ Fg. 5 llustrate the process o the DWMLS algorthm. Fgure. 3 shows the ntal sx LHD samples along wth the true lmt state surace. Fgure. 4 shows the ntally estmated MPFP1 along wth two addtonal samplng ponts. Fgure. 5 shows two more teratons wth MPFP and MPFP3 n a partally enlarged drawng. It s clear that the estmated MPFP converges to the true MPFP. he teraton hstory o the MPFP search usng the proposed method s shown n able 1. he perormance uncton s evaluated twelve tmes durng the three teraton cycles beore arrvng at the nal convergence. able compares the results rom the proposed method wth that o varous methods rom the lterature. It should be noted that the exact p s estmated by the Monte Carlo method wth one mllon samples and the exact soluton o HL and MPFP s estmated by the Hasoer-Lnd algorthm. he comparson o P ams at accuracy o the proposed method, whle the comparson o HL and MPFP ndcate the perormance o DWMLS n postonng and convergng to the true MPFP. As t can be seen, the results o the relablty analyss are close to each other or all methods under consderaton. hs s manly because the nonlnearty o the problem n the vcnty o the desgn pont s relatvely low. he proposed method predcts very accurate estmates o the relablty ndex and desgn pont when compared wth the FORM method. However, a slght derence n the alure probablty predcton s observed. he coecent o varaton (COV) o the estmated alure probablty P by drect MCS wth sample sze o N s 11 s 1 P (8) NP When one mllon samples are used, the COV o P s he same number o samples s used to calculate the alure probablty n DWMLS. he derence n P between MCS and the proposed DWMLS s wthn the range o COV. Compared wth other approaches, the hybrd DWMLS method has demonstrated not only the accuracy n postonng the MPFP and estmatng relablty ndex HL, but also the ecency as only twelve uncton evaluatons are needed n achevng t. able 1: Iteraton hstory o DWMLS method -Example #1 Iteraton MPFP Ns HL 1.9 [-.116, 0.699] [-.5, 0.968] [-.547, 0.94] able : Summary o results-example #1 Method MPFP p HL Monte Carlo smulaton.685 Unavalable 3.68E-3 (1.7%) 1E6 FORM (H-L algorthm).710 [-.540,0.945] 3.37E-3 7 RSM n Re [-.541,0.94] 3.36E-3 1 RSM n Re [-.567,0.860] 3.39E-3 1 RSM n Re [-.558,0.80] 3.6E-3 8 RSM n Re [-.538,0.951] 3.36E-3 1 DWMLS + MCS.710 [-.547,0.94] 3.61E-3 1 he results wth bold ont are regarded as exact results to compare wth. Ns s

12 Fgure 3. Intal LHD and mean value pont or example 1 (Step 1) Fgure 4. Illustraton o addng n new expermental ponts based on prevous MPFP or example 1(Step 6) 1

13 Fgure 5. Illustraton o addng one new expermental pont based on prevous MPFP or example 1(Step 4) B. Example : Dynamc response o a nonlnear oscllator In order to nvestgate the perormance o the proposed method n complex problems wth more random varables and greater non-lnearty, the second example deals wth a nonlnear undamped sngle degree o reedom system as presented n Fg. 6. hs example s also used n several other studes 6,0,39. he nonlnear oscllator wth random system parameters subjected to a rectangular pulse load wth random duraton and ampltude s presented, and the perormance uncton s dened by: g c, c, m, r, t, F 3r z max r F wt sn( ) mw0 where w0 ( c1 c)/ m. he random parameters o sx basc varables are lsted n able 3. (9) Fgure 6. Non-lnear oscllator - system denton and appled load able 3: Probablstc dstrbuton o random varables - Example Random varable Dstrbuton ype Mean Value Standard Devaton c 1 Normal c Normal m Normal r Normal t 1 Normal F 1 Normal

14 able 4 shows the relablty analyss results rom derent methods n the lterature along wth the proposed DWMLS method. he reported results come rom drectonal samplng (DS) and mportance samplng (IS) methods combned wth typcal surrogate models (e.g., polynomals, splnes, and neural networ), and an actve learnng method combnng Krgng and Monte Carlo smulaton (AK-MCS). It can be seen that the DWMLS method can get compettve accuracy wth the least amount o smulatons, Ns = 43. It s noted that the results reported by Schueremans and Gemert 39 and by Echard and Gayton 0 are based on nterpretng the last column o able 3 as a standard devaton, not as a COV. able 4: Summary o relablty results-example # Method Ns p Monte Carlo Method 1E6 3.89E-(1.9%) 1.76 FORM E- 1.7 Drectonal Samplng(DS) E DS+Polynomal 6 3.4E DS+Splne E DS+Neural Networ 86.8E Importance Samplng(IS) E IS+Polynomal 109.5E IS+Splne 67.7E IS+Neural Networ E MCS 7E4.83E-(.%) 1.91 AK-MCS+U 58.83E AK-MCS+EFF 45.85E DWMLS+MCS E the results come rom Re. 39 the results come rom Re. 0 C. Example 3: A cantlever beam he above two examples demonstrated the perormance o the proposed method n estmatng the alure probablty and locatng the MPFP. he proposed method can also be appled to multple perormance unctons;.e., calculatng the system probablty o alure. In such a case, adaptve expermental desgn s appled to each perormance uncton. he thrd example consders system relablty wth two lmt state suraces. A cantlever beam, as shown n Fg. 7, s subjected to a tp load o 00.0 N. wo alure crtera are consdered: () the dsplacement at the tp o the beam should be less than 0.005m, as expressed n Eq. (30), and () maxmum stress n the beam should be less than 33 MPa, as expressed n Eq. (31). HL Fgure 7. Smply supported beam 3 4PL Dsplacement lmt state : g1( X) (30) 3 Ebh 6 1PL Stress lmt state : g ( X) (31) bh In the above equatons,,, Lbhndcate the length, wdth and heght o the beam, whose probablty dstrbuton propertes are shown n able 5. he modulus o elastcty o the beam was taen to be 70.0 GPa. he system probablty o alure s dened by: 14

15 p 1 P[ g ( X) 0 g ( X ) 0] (3) 1 able 5: Probablstc characterstcs o the basc random varables o the cantlever beam Random Varable Mean(m) Standard Devaton Dstrbuton type L Normal b Normal h Normal As the closed orm expressons o perormance unctons are avalable, t s possble to estmate system relablty wth Monte Carlo smulatons. able 6 lsted the results o MCS wth a mllon samples and the result estmated by the proposed DWMLS method. It should be noted that 98 samples n the DWMLS method nclude all the ponts used to locate two lmt state suraces and correspondng two MPFPs untl convergence. In order to nspect the accuracy and ecency o the DWMLS method on ndng each MPFP, the FORM algorthm s used or each lmt state uncton to estmate an ndvdual relablty ndex or comparson n able 7. It can be concluded that the hghly nonlnear lmt state unctons, partcularly the rst, can be postoned accurately wthout a large number o samples. able 6: Comparson o alure probablty or the cantlever beam system Method p Ns Monte Carlo smulaton 8.33E-3(1.09%) 1E6 DWMLS + MCS 8.98E-3 98 able 7: Comparson o relablty ndex or the cantlever beam system g1 Method Ns HL Ns HL FORM DWMLS + MCS g D. Example 4: en-bar truss structure As a practcal example usng nte element analyss, a ten-bar truss structure (as shown n Fg. 8) s consdered. he ten-bar truss structure s a classcal structural analyss problem and wdely studed 7,40. he structure s smply supported at nodes 1 and 4, and s subjected to two concentrated loads P = 10 5 lb at nodes and 3. he truss members, whch have random cross-sectonal areas A, 1,,...,10, are made o an alumnum alloy wth Young's modulus E=10 7 ps. he nput random varables X { A1, A,..., A10} ollow normal dstrbuton and have a mean.5 n and standard devaton 0.5 n. he maxmum vertcal dsplacement ( X ), whch occurs at node 3, s lmted to 0 18 n. hereore, the perormance uncton s dened as: g v ( X) 18.0 v ( X ) (33) 0 max max he structural analyss was done n MSC/NASRAN, a commercal nte element analyss program. he estmaton results are reported n able 8. 15

16 Fgure 8. A ten-bar truss structure able 8: Falure probablty o ten bar truss Method Ns p Monte Carlo method 10E (0.5%) FORM( HL algorthm) SORM(Bretung) DWMLS + MCS Apparently, the perormance o the DWMLS method perorms well n ths hgh dmensonal relablty analyss problem. he alure probablty o the DWMLS method s comparable wth other methods and demands the least number o smulatons. VII. Conclusons In ths paper, a doubly weghted movng least squares and a two-stage hybrd relablty analyss scheme are proposed to mprove the surrogate model or relablty analyss. he proposed method provdes a larger weght to the pont near the MPFP and adds more samples closer to the lmt state surace. From the rst two benchmar examples, t was shown that, n comparson to classcal MCS and FORM wth varous surrogate models, DWMLS mproves the convergence speed, and locates the lmt state uncton more accurately wth a less number o samplng ponts. In the cantlever beam example, the proposed two-stage relablty analyss scheme was able to calculate system relablty by dentyng multple MPFPs. However, t should be noted that the proposed method s not ntended as a replacement o exstng surrogate models n relablty analyss, but as a possble complement and mprovement to these methods. Furthermore, more studes are needed to extend the proposed method to relablty based desgn optmzaton o complex systems. Acnowledgements he study reported n ths paper was unded by Chna Scholarshp Councl under support number he authors also than Pro. Raphael. Hata and Dr Felpe A. C. Vana or ther valuable comments. Reerences 1 Dtlevsen, O., Madsen, H., Structural relablty methods. Wley, 1996, Hoboen. Hurtado, J. E., Structural relablty: Statstcal learnng perspectves. Sprnger, 004, Hedelberg. 3 Bucher, C. G., Bourgund, U., "A ast and ecent response surace approach or structural relablty problems," Structural Saety, Vol. 7, 1991, pp Rachwtz, R., "Relablty analyss-a revew and some perspectves," Structural Saety, Vol. 3, 001, pp Wong, F. S., "Slope relablty and response surace method,". Journal o Geotechncal Engneerng, Vol. 111, 1985, pp Rajashhar, M., Ellngwood, B., "A new loo at the response approach or relablty analyss". Structural Saety, Vol. 1, 1993, pp Quepo, N., Hata, R.., Shyy, W., Goel,., Vadyanathan, R., ucer, P., "Surrogate-based analyss and optmzaton". Progress n Aerospace Scences, Vol. 41, 005, pp HL

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