A Linear Response Surface based on SVM for Structural Reliability Analysis

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1 APCOM & ISCM -4 th December, 03, Sngapore A Lnear Response Surace based on SVM or Structural Relablty Analyss U. Albrand, C.Y. Ma, and C.G. Koh Department o Cvl and Envronmental Engneerng, Natonal Unversty o Sngapore, 7576 Sngapore Correspondng author: umberto.albrand@nus.edu.sg Key Words: Structural Relablty Analyss, FORM, desgn pont, Response Surace, Support Vector Method, Monte Carlo Smulaton Abstract The Structural Relablty theory allows the ratonal treatment o the uncertantes and gves the methods or the evaluaton o the saety o structures n presence o uncertan parameters. The man challenge s the computatonal cost, snce the alure probablty wth respect to an assgned lmt state s gven as the soluton o a very complcated multdmensonal ntegral. The most robust procedure s the Monte Carlo Smulaton (MCS), but especally n ts crude orm s very demandng. For ths reason, wde popularty has been ganed by the Frst Order Relablty Method (FORM) by ts smplcty and computatonal ecency. However, or strongly nonlnear systems the FORM approxmaton s not very close to the exact one. To ths am, n ths paper we ntroduce a novel Lnear approxmaton o the lmt state, based on the Support Vector Method (SVM), and whch allows to mprove the FORM soluton, startng rom the knowledge o the desgn pont. Introducton Recently t has been largely recognzed that a realstc analyss o the structural systems should take nto account all the unavodable uncertantes appearng n the problem at hand. In ths context a powerul tool s represented rom the structural relablty theory (Madsen et al. 986, Dtlevsen & Madsen 999, Melchers, 999) whch gves a ratonal treatment o the uncertantes and whch allows the assessment o the evaluaton o the saety o structures n presence o uncertan parameters. The alure probablty P wth respect to an assgned lmt state s dened as P X ( x) dx () g x0 where x s an n vector collectng the basc random varables, y g( x ) s the Lmt State Functon (LSF), g( x ) 0 s the Lmt State Surace (LSS) separatng the alure set g( x ) 0 rom the sae set g( x ) 0, ( x ) s the jont probablty densty uncton o the random varables X x, x,, x. The evaluaton o the alure probablty n P s known n closed orm only or a very restrcted number o cases, n the most general case t s necessary to solve numercally a multdmensonal ntegral, whch s computatonally demandng. The most robust procedure or the evaluaton o the alure probablty s represented by the Monte Carlo Smulaton (MCS), whch however, especally n ts crude orm, requres an excessve computatonal eort or the evaluaton o the very small alure probabltes. For ths reason, wde popularty has been ganed by the Frst Order Relablty Method (FORM) by ts smplcty and computatonal ecency, moreover extensve numercal expermentaton has

2 shown that t gves good approxmatons o the alure probablty or most practcal problems. However, t s known that the FORM approxmaton s not adequate or lmt state suraces whch depart sgncantly rom lnearty around the desgn pont. In ths paper we overcome ths shortcomng wth a partcular type o Response Surace Methodology (RSM) based on the Support Vector Method (SVM) and the theory o the statstcal learnng (Vapnk 995, Burges 998). The basc dea o the RSM s the buldng o a surrogate model o the target lmt state uncton, dened n a smple and explct mathematcal orm; once the Response Surace (RS) s bult, t s possble to substtute the RS wth the target LSF, and then t s no longer necessary to run demandng nte element analyses; startng rom ths denton, FORM tsel s a partcular knd o RS, whch approxmates the LSS wth the hyperplane passng through the desgn pont u and normal to the desgn pont drecton, the latter beng the ray jonng the orgn o the standard normal space wth the desgn pont. The RS models can be bult to nd the desgn pont wth reduced computatonal cost (Bucher & Burgound 990, Albrand & Der Kureghan 0); recently, many alternatve response surace methodologes have been proposed, whose am s the mprovement o the FORM approxmaton (Bucher & Most, 008; Albrand & Rccard 005, Albrand & Rccard 008, Albrand, Impollona & Rccard 00). To the latter cathegory belong the RS approaches based on the SVM (Hurtado 004; Albrand & Rccard 0). Usng the SVM the relablty problem s treated as a classcaton approach (Hurtado & Alvarez 003), snce we are not nterested to the exact value o the LSF, but only to ts sgn. Thereore the samples are labelled wth the value (sae sample) and (unsae sample) and ths requste s less strong than approxmatng the exact value o the LSF. In the exstng approaches based on SVM, the mprovement o the FORM soluton s obtaned by choosng non-lnear models or approxmatng the lmt state; conversely, n ths paper, we adopt a smple lnear model, where the constrants o correct classcaton are relaxed, acceptng thereore that some ponts may be msclassed (Albrand 0) The startng model s bult choosng a set o sample ponts along the desgn pont drecton, the latter beng the drecton o probablstc nterest. In ths way, startng rom the knowledge o the desgn pont, t s possble to approxmate the lmt state wth an hyperplane close to FORM but secant to the lmt state, gvng rse to an alternatve Lnear response surace based on SVM (LSVM), and gvng a better approxmaton than FORM. Structural Relablty Analyss as a classcaton approach Usually the multdmensonal ntegral () s very dcult to be evaluated, and then some approxmate technques are used. As a rst step, a probablstc coordnate transormaton s done toward the standard normal space and the alure probablty s gven as P U ( u) du () g u0 In () the ntegrand uncton s the multvarate normal standard probablty densty uncton (pd), whle the ntegraton doman s the regon alure g( u ) 0. Accordng to (), the alure probablty can be obtaned usng the Monte Carlo Smulaton (MCS), consderng a set o N samples u, u,, un and evaluatng the rato between the number N o samples belongng to the alure regon g( u ) 0 and the total number o smulated samples N :

3 N N P, MCS I g( j ) 0 N N u (3) j where I[ ] s an ndcator uncton, equal to g( u ) 0 and zero otherwse. The qualty o the obtaned approxmaton s determned evaluatng the coecent o varaton o P, MCS ; t s however well known that the MCS, especally n ts crude orm, requres an excessve computatonal eort. Alternatvely, a good estmate o () s obtaned by approxmatng the target Lmt State Functon (LSF) y g( u ), usually complcated and mplct, wth an approxmate model y g ( u), called Response Surace (RS). Once the RS s bult, t s no longer necessary to run demandng Fnte Element Analyses, but we can use the surrogate model. In the ollowng we wll consder only problems where the lmt state surace has nether peak or valleys, nor multple desgn ponts. From (3) t s seen that we are not nterested to the value y g( u ) o the LSF, but to ts sgn z sgn[ g( u )], so that the ponts u belongng to the sae regon have the value z, and the ponts u belongng to the alure regon have the value z. It s easy to see that the buldng o a surrogate model y g ( u) such that t satses only the sgn constrants z sgn[ g( u )], s equvalent to the buldng o a RS whch models drectly the LSS. Then, accordng to the RS Methodology, the estmated alure probablty can be evaluated usng (3), by substtutng the ndcator uncton I[ g( u ) 0] wth I[ g ( u ) 0]. j j A uncton whch separates the ponts belongng to the sae set rom the ones belongng to the alure set, s named classer, snce attrbutes a class ( sae or alure ) to each pont. The LSS g u s the target classer, whle a RS whch approxmates the LSS, s able to classy 0 correctly only a lmted number o ponts. Clearly, because the RS works well, t s necessary that t classes correctly the ponts at least n the regon o probablstc nterest. j A Lnear Response Surace based on SVM Let be known a set o m samplng ponts u, u,, um, whle y, y,, ym and z, z,, zm be the correspondng values o the LSF y g( u ) and sgns z sgn[ g( u )], respectvely. Suppose that the target LSS s lnear, g( u) a u c, then the samplng ponts u are lnearly separable. Consder now the approxmated LSS gˆ( u) w u b, where w determnes the orentaton o the plane, whle the scalar b determnes the oset o the plane rom the orgn. Clearly, when the number o support ponts converges toward nnty m, then the lnear classcaton uncton becomes concdng wth the target LSS,.e. w a, b c. Conversely, or a lmted number o support ponts, there are nnte possble planes that classy the ponts correctly. Intutvely, a hyperplane that passes too close to the samplng ponts wll be less lkely to generalze well or the unseen data, whle t seems reasonable to expect that a hyperplane that s arthest rom all ponts wll have better generalzaton capabltes. Gven a set o m samplng ponts, the margn s dened as the mnmum dstance between ponts belongng to derent classes. Thereore, the optmal separatng hyperplane s the one maxmzng the margn. Recall rom the elementary geometry that the dstance o a pont u rom the hyperplane gˆ( u ) 0 reads as w u b w ; notcng that gˆ( u) w u b 0 s nvarant under a postve rescalng, we choose the soluton or whch the uncton yˆ gˆ( u ) becomes one or the ponts 3

4 closest to the boundary,.e. w u b. The couple o hyperplanes gˆ ( u) w u b, u g( u ) 0 and gˆ ( u) w u b, u g( u ) 0, are called canoncal hyperplanes (or support hyperplanes). The dstance rom the closest ponts to the boundary gˆ( u ) 0 s w, and the margn becomes M w, as shown n Fgure (a). s u a u b w w u b w b w gˆ( u) 0 0 gˆ( u) 0 u u Fgure. (a) Lnear SVM, (b) Non-lnearly separable SVM Maxmzng the margn s equvalent to mnmze w, gvng rse to the ollowng Quadratc Programmng (QP) problem mn w w, b s. t. z w u b,,,, m (4) where the nequalty constrants are equvalent to w u b, u g( u ) 0, and w u b, u g( u ) 0. It s here noted that (4) s a standard convex optmzaton problem, so the unqueness o the soluton s guaranteed and moreover there are many robust algorthms that can eectvely solve t. Among the m samplng ponts, the support vectors are the m SV ponts lyng on the support hyperplanes w u b ; n Fgure (a) the support vectors are represented rom the lled markers. It s seen that only the support vectors contrbute to denng the optmal hyperplane, thus, the complete samplng set could be replaced by only the m SV support vectors, and the separatng hyperplane would be the same. Suppose now that the LSS s non-lnear, so that t s not possble to denty an hyperplane whch correctly classes all the samplng ponts. To ths am, we relax the constrants o (4) by ntroducng the slack varables 0, gvng rse to w u b, u g( u ) 0 and w u b, u g( u ) 0. The varables gve a measure o the departure rom the condton o correct classcaton. In partcular, when 0 the pont s well classed but alls nsde the margn, whle when the pont s not well classed. Fnally, 0 the pont s 4

5 correctly classed, see Fgure (b). Under ths hypothess, the optmal separatng hyperplane has maxmum margn wth mnmum classcaton error. The optmzaton problem (4) becomes m mn w w, b, ξ s. t. z wu b,,,, m 0,,, m (5) Outlne o the procedure It wll be used the ollowng teratve procedure: Probablstc transormaton toward the n-dmensonal standard normal space, as t s usually done n relablty analyss; Evaluaton o the desgn pont wth the FORM approxmaton P u and o the correspondng relablty ndex, FORM, see Fgure. u 3 Choce o a set o samplng ponts u k, k,, along the desgn pont drecton, together u u. Classcaton o alure and sae ponts belongng to the drecton through zk sgn[ g( u k )] ; 4 Buldng o a rst LSVM model through (5), together wth ts margns; 5 Choce o a new pont u k nsde the margn and ts classcaton; 6 Buldng o the SVM model through (5) and evaluaton o the estmated alure probablty. Snce the LSVM s a lnear response surace we have P, LSVM LSVM, LSVM beng the dstance o the LSVM rom the orgn o the standard normal space. 7 I the convergence on P s acheved stop, else go to step 5. u LSVM u FORM g u 0 LSVM Fgure. FORM and LSVM u 5

6 Example Statc Analyss o a -do rame As a rst example, we appled the proposed approach to a shear type rame wth two-stores, subjected to two determnstc concentrated loads, F F F wth random stnesses k k( x) and k k( x), beng F, k k and the luctuatons x, x modelled as two ndependent Gaussan random varables wth zero mean and standard devaton 0.0 (Fgure 3a). Be X ( x, x ) the horzontal dsplacement o the second storey, and the consdered lmt state s g( x, x) X ( x, x ) X,lm, wth X,lm.5. The alure probablty s 3 P obtaned usng MCS, wth an estmated coecent o varaton equal to P %, and requred,050,848 analyses. The correspondng generalzed relablty ndex s P.348. G a) F b) 5 4 LSVM k k F 3 u k k u FORM g u u Fgure 3. Example : (a) -do shear-type rame, (b) LSS, FORM and LSVM As shown n Fgure 3(b), the LSS s nonlnear, and The FORM soluton gves a alure probablty 3 P, FORM.593 0, wth a relatve error e.97%. The LSVM acheves the convergence FORM ater 38 samples, obtanng.344 and correspondng alure probablty P LSVM, (relatve error elsvm 0.46% ). In Fgure 4(a) we represent the obtaned alure probabltes n terms o the number o samples, whle n Fgure 4(b) the correspondng relatve errors are shown P 30 P,FORM P,LSVM 0 Falure Probablty 0-4 Error (%) e FORM e LSVM samples samples Fgure 4. Example : (a) Falure Probablty, (b) Relatve error, n terms o the number o samples 6

7 Example Nonlnear random waves Let S XX ( ) a wave spectrum parttoned nto N components o equal nterval. The secondorder wave elevaton s represented as (Longeut-Hggns 963, Moarezadeh & Melchers 006) N N N (6) u u u n n nm n m n n m where u, u,, u N are normal standard random varables, whle and are the rst and second order terms o the nonlnear sea elevaton,, 0, n n n N nm mn n, m n m, nn, m N max n, m n m, nn, m n, mn 0 g g n, m,,, N (7) beng, wth n G n G the spectral densty uncton o the sea elevaton. As a case study, we consdered the JONSWAP spectrum wth a shape actor o unty 5 4 G a exp[.5( p ) ], where p 0.4 rad/sec s the peak requency, a s a scalng actor that s selected so that the area under the spectrum s 5 m (Low 03). The spectrum s dvded nto N 40 harmonc components rom 0. to. rad/sec. It s requred to evaluate the probablty that the wave elevaton s greater than 5 m, that s P =Prob[ 5] ; the Lmt State Functon s G 5 u u, whle the number o basc random varables s n N 80. It s here noted that ( u ) s a quadratc uncton o the normal standard random varables, rom whch t ollows the nonlnearty o the Lmt State Surace. 4 The exact soluton obtaned wth a MCS s wth a coecent o varaton 5% and t requred 3,863,373 samples. The FORM approxmaton s P P P FORM,.5 0 wth a relatve error eform.%, whle the LSVM, ater 355 samples, gves 4 P, LSVM.0 0 ( elsvm.8% ). In Fgure 5 we represent the obtaned alure probabltes and the correspondng relatve errors n terms o the number o samples P 0 e FORM P,SVM 5 e SVM 0 Falure Probablty 0-4 Error (%) samples samples Fgure 5. Example : (a) Falure Probablty, (b) Relatve error, n terms o the number o samples 7

8 Conclusons FORM s a powerul tool or Structural Relablty Analyss, and n most cases o practcal nterest t gves a good approxmaton o the alure probablty. However, or lmt state suraces whch depart sgncantly rom lnearty around the desgn pont, the FORM soluton may be nadequate. In ths paper we presented a Lnear response surace based on SVM, called LSVM, whch startng rom the desgn pont drecton, allows an mprovement o FORM, requrng a reduced number o samplng ponts. Two smple numercal examples showed the accuracy and eectveness o the presented procedure. Acknowledgements The authors would lke to acknowledge the research grants (R ) unded by the Agency or Scence, Technology and Research o Sngapore. Reerences Albrand U. An alternatve lnear response surace mprovng FORM soluton or stochastc dynamc analyss, Relablty and Optmzaton o Structural Systems, IFIP WG 7.5 Workng conerence, 4-7 June 0, Yerevan, Armena Albrand U. & Der Kureghan A. (0), A Gradent-Free Method or Determnng the Desgn Pont n Nonlnear Stochastc Dynamc Analyss, Probablstc Engneerng Mechancs, 8, -0. Albrand U., Impollona N. & Rccard G. (00). Probablstc Egenvalue Bucklng Analyss solved through the Rato o Polynomal Response Surace, Computer Methods n Appled Mechancs and Engneerng, 99(9-), Albrand U. & Rccard G. (005), Bounds o the probablty o collapse o rgd-plastc structures by means o Stochastc Lmt Analyss, 9th Internatonal Conerence on Structural Saety and Relablty, Rome, Italy, 9-3 June 005 Albrand U. & Rccard G. (008), The use o stochastc stresses n the statc approach o probablstc lmt analyss. Internatonal Journal or Numercal Methods n Engneerng, 73(6), Albrand U. & Rccard G. (0). Structural Relablty Analyss solved through a SVM-based Response Surace, ICASP 0, The th Internatonal Conerence on applcatons o Statstcs and Probablty n Cvl Engneerng, -4 August 0, Zurch, Swtzerland Bucher C. & Burgound U. (990), A ast and ecent response surace approach or structural relablty problems, Structural Saety, 7, Bucher, C. & Most, T. (008), A comparson o approxmate response unctons n structural relablty analyss, Probablstc Engneerng Mechancs, 3, Burges, C.J.C. (998), A tutoral on Support Vector Machnes or Pattern Recognton. Data Mnng and Knowledge Dscovery : -67 Dtlevsen O., Madsen H.O. (999), Structural Relablty Methods, Wley Hurtado, J.E. & Alvarez, D.A. (003). A classcaton approach or relablty analyss wth stochastc nte element modellng, Journal o Structural Engneerng, 9, Hurtado, J.E. (004), An examnaton o methods or approxmatng mplct lmt state unctons rom the vewpont o statstcal learnng theory, Structural Saety, 6, Longeut-Hggns MS (963). The eect o nonlneartes on statstcal dstrbutons n the theory o sea wave, J Flud Mech, 7(3): Low Y.M. (03). A new dstrbuton or ttng our moments and ts applcatons to relablty analyss, Structural Saety, 4, -5 Madsen H.O., Krenk S. & Lnd, N.C. (986). Methods o Structural Saety, Prentce-Hall. Melchers R.E. (999), Structural Relablty, analyss and predcton, New York: Wley & Sons. Moarezadeh MR & Melchers RE (006). Nonlnear wave theory n relablty analyss o oshore structures, Probablstc Engneerng Mechancs, (), 99-. Vapnk, V. (995), The Nature o Statstcal Learnng Theory, New York: Sprnger-Verlag. 8