AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM USING THE FIRST-ORDER METHOD FOR STRUCTURAL RELIABILITY

Transcription

1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 07; 7():93-07 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM USING THE FIRST-ORDER METHOD FOR STRUCTURAL RELIABILITY M.A. Shayanfar *,, M.A. Barkhordar and M.A. Roudak The Centre of Ecellence for Fundamental Studes n Structural Engneerng, Iran Unversty of Scence and Technology, Narmak, Tehran 6846, Iran ABSTRACT Monte Carlo smulaton (MCS) s a useful tool for computaton of probablty of falure n relablty analyss. However, the large number of samples, often requred for acceptable accuracy, makes t tme-consumng. Importance samplng s a method on the bass of MCS whch has been proposed to reduce the computatonal tme of MCS. In ths paper, a new adaptve mportance samplng-based algorthm applyng the concepts of frst-order relablty method (FORM) and usng () a new smple technque to select an approprate ntal pont as the locaton of desgn pont, () a new crteron to update ths desgn pont n each teraton and (3) a new samplng densty functon, s proposed to reduce the number of determnstc analyses. Besdes, although ths algorthm works wth the poston of desgn pont, t does not need any etra knowledge and updates ths poston based on prevous generated results. Through llustratve eamples, commonly used n the lterature to test the performance of new algorthms, t wll be shown that the proposed method needs fewer number of lmt state functon (LSF) evaluatons. Keywords: relablty analyss; Monte Carlo smulaton; mportance samplng; frst-order relablty method; desgn pont. Receved: 5 February 06; Accepted: 0 July 06 * Correspondng author: The Centre of Ecellence for Fundamental Studes n Structural Engneerng, Iran Unversty of Scence and Technology, Narmak, Tehran 6846, Iran E-mal address: shayanfar@ust.ac.r (M.A. Shayanfar)

2 94 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak. INTRODUCTION In structural relablty analyss dfferent eamples of uncertantes are observed n materal, loads and geometrc propertes. In such crcumstances and to have a realstc understandng of a structure durng ts lfetme, uncertanty has to be taken nto account. To do so, Structural relablty theory s eploted [-5]. In the component relablty analyss, the man purpose s to evaluate the probablty of falure, P f, by a mult-dmensonal ntegral on the falure doman as where,,..., X X X X n T Pf f X ( X ) dx g( X ) 0 represents the vector of random varables and f ( X ) s the jont probablty densty functon (JPDF) of the vector of random varables. g( X ) s the LSF such that gx ( ) 0 defnes the safety doman and gx ( ) 0 defnes the falure doman. However, the evaluaton of the ntegraton of Eq. () s so much dffcult and complcated because of nvolvng multple ntegral and JPDF of random varables, especally for the large and comple structures wth low probablty of falure and mplct LSFs. In such a condton, other alternatves such as appromaton methods and MCS have attracted more attenton [6-4]. Appromaton methods such as frst- and second-order relablty method (FORM and SORM) are based on frst- and second- order appromaton of LSF at the desgn pont, respectvely. Desgn pont whch les on the lmt state surface and has the mnmum dstance from the orgn of standard normal coordnate system (U-space), defned by Hasofer and Lnd [5], has been also proved to be the most probable falure pont (standard normal coordnate system s reached usng the transformaton u ( ) ) [6]. Ths mnmum dstance s called relablty nde and s denoted by (t should be noted that LSF n standard normal coordnate system wll be denoted by g, hereafter). Appromaton methods can work approprately n many practcal eamples but they requre a dfferentable LSF whch s an unattanable condton n many cases. In smulaton methods, such as MCS, random samples are generated based on samplng densty functon and LSF s evaluated for the sample. Then, probablty of falure gven by MCS s consdered the proporton of the number of samples correspondng to falure,. e. negatve value of LSF, to total samplng number. MCS does not need the mathematcal form and dervaton of LSF but the large number of smulatons, needed to ncrease the accuracy, has become a bg computatonal burden. That s why many researches have been focused on developng MCS by reducng the number of requred random samples to make t more practcal. On ths way dfferent samplng methods such as Latn Hypercube samplng [7], Antthetc Varates [,8] and Importance samplng [9] can be named. Snce mportance samplng has been shown to be effectve among dfferent types of samplng, many researches has been focused on mprovement of smulaton wth ths technque of samplng. X ()

3 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 95 One useful way to ncrease the effcency of mportance samplng method s obtaned by means of the locaton of desgn pont n the process of relablty analyss. Many methods wth such an approach have been proposed that requre the desgn pont or the features of LSF [0-4]. Adaptve mportance samplng method s another method whch uses desgn pont based on the results of prevous determnstc analyses and wthout any need to the features of LSF and desgn pont [5-7]. In ths paper, a new method to mprove the effectveness of MCS has been presented. In ths method, whch falls nto the class of adaptve mportance samplng method and uses the concepts of FORM, a smple technque has been proposed to reach an approprate ntal desgn pont from a random pont. Ths ntal pont s then updated n each teraton, based on a smple proposed crteron. The samplng densty functon needed to generate random samples s chosen to have the same form as the probablty densty functon (PDF) of random varables, but wth dfferent mean and standard devaton n each teraton. The aforementoned technque, crteron and samplng densty functon as the consttuents of the proposed adaptve mportance samplng-based algorthm are shown to be effectve n mprovng the performance of MCS through numercal eamples.. IMPORTANCE SAMPLING The basc MCS computes the value of the mult-dmensonal ntegral of Eq. () by generatng random samples based on the dstrbuton of random varables. Generatng random samples on the whole random varable space wthout any focus or weght on dfferent regons of the space causes the basc MCS to need a huge number of samples. Ths makes the basc MCS tme-consumng and consequently mpractcal to many problems. Importance samplng method [] represents one of the best deas to decrease the number of random samples and ncrease the convergence speed. In mportance samplng method, the ntegral of Eq. () s rewrtten as fx ( X) Pf hx( X ) dx () h ( X) g( X) 0 X where hx ( X ) s samplng densty functon. Now random samples are generated based on h ( X ) and then probablty of falure s calculated from X P f f ( X ) n h ( X ) (3) n X I ( X ) X where the ndcator I( X ) for safe and falure smulaton s defned as

4 96 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak 0 gx ( ) 0 I( X ) gx ( ) 0 (3) Dfferent choces of hx ( X ) can sgnfcantly affect the number of requred random samples. It has been observed that locatng the samplng densty functon around desgn pont works properly. It was mentoned n the prevous secton that many methods applyng ths dea have been proposed, but they need the characterstcs of LSF or desgn pont whch are both naccessble n many cases. In such cases, adaptve mportance samplng-based methods, whch do not need these characterstcs and work wth knowledge obtaned n prevous determnstc analyses, can be good choces. In the followng secton, a new adaptve mportance samplng-based algorthm wth representaton of a smple technque, crteron and samplng densty functon s proposed. 3. PROPOSED ALGORITHM As was prevously mentoned, the proposed mportance samplng-based algorthm s made up of () a new technque to select an approprate ntal pont as the locaton of ntal desgn pont, () a new crteron to update the desgn pont n each teraton and (3) a new samplng densty functon to generate random samples. 3. A new technque to select an approprate ntal pont as the locaton of ntal desgn pont Intal desgn pont n smulaton methods plays a sgnfcant role. In fact, an approprate ntal desgn pont can decrease the number of requred random samples. If any random pont s selected as the ntal desgn pont, t can lead to a lengthy and tme-consumng analyss because ths pont determnes the locaton of samplng densty functon and drectly affects the sample generaton procedure. Ths problem becomes bgger when the ntal desgn pont goes further from the LSF [7]. That s why n the proposed algorthm, a specfc attenton s allocated to the ntal desgn pont. Suppose the random pont P s the frst generated random pont (ntal random pont). If ths ntal random pont s closely located around the LSF, t s accepted as ntal desgn pont, otherwse t s moved to P whch s to be closer to the LSF. Ths procedure wll be contnued untl the pont falls n the neghborhood of LSF wth the acceptable dstance. In fact, f g ( ) P c where c s a predetermned relatvely small value, t s consdered closely located around LSF and thus selected as the ntal desgn pont. However, f g ( ) P c, we move from P to P on the lne connectng P to the orgn of coordnate system (Fg. ). We call ths lne and the vector connectng the orgn of coordnate system, O, to P, drecton lne and drecton vector ( OP ), respectvely. The represented

5 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 97 eplanatons mean that the ntal pont just moves on the drecton lne (n the postve or negatve drecton of the drecton vector) untl the acceptable dstance s reached and then ths pont s selected as the ntal desgn pont. Fgure. Movement towards lmt state surface on the drecton lne and usng the aulary pont In order to buld a formula by whch the net pont, P, can be obtaned from the current pont, P, as t s seen n Fg., we move unt towards postve drecton of reach the followng new pont called aulary pont, OA OP A OP to OP (5) OP where. denotes norm of the vector. Now by evaluatng the functon g n ths pont, we can have an understandng of the behavor and rate of change n g. What we are lookng for s the step sze n OP OP OP OP (6) Such that the pont P s located on the lmt state surface. Snce the change of unt n the magntude of drecton vector OP results n the change of g( A) g( P), consderng

6 98 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak the lnear behavor, s calculated as follows. g( P ) g( P) g ( A ) g ( P) Snce P s to be located on the lmt state surface,. e. g( P ) 0, Eq. (7) can be rewrtten as g( P ) g ( P) g ( A ) Eq. (6), the elements of whch are calculated from Eqs. (8) and (5), gves the net pont P n terms of the current pont P. It should be noted that snce the above formulaton has been completed by the assumpton of lnear behavor of the LSF, f t s nonlnear g( P ) wll not be zero and therefore the teraton of Eq. (6) should contnue untl our condton s satsfed and the acceptable dstance s reached. Ths smple technque to brng the ntal random pont closer to LSF and selectng ths pont as the ntal desgn pont causes the samplng densty functon to be close to the border of safe and falure regon and thus, compared to MCS, more random samples wll fall nto the falure regon and ths s one of the reasons the proposed algorthm mposes less computaton cost. 3. A new crteron to update the desgn pont n each teraton In the prevous part, we dscussed that a good ntal desgn pont whch s closely located around the border of safe and falure regon can be a good start. However, snce t s just close to ths border and s not necessarly close to the real desgn pont, t s not necessarly the best pont. Thus, n order to make the algorthm more effcent, the ntal desgn pont needs to be updated. One possble crteron to update the desgn pont s to select a pont, n the net teratons, wth () smaller dstance to the orgn and () smaller absolute value of g (both compared to the prevous desgn pont) as the new desgn pont and not to change the desgn pont f ths pont does not have these two features [7]. Ths crteron may work n many problems but t has a bg drawback. In fact, f a desgn pont at a specfc step of the procedure, falls very close to the LSF such that at ths desgn pont g s very close to zero, t wll be very dffcult for other ponts to replace the prevous desgn pont although they are closer to the real desgn pont. In such cases the new pont may be closer to the real desgn pont and have the frst condton satsfed but snce the absolute value of g at ths pont s not smaller than that at the prevous desgn pont, although t may be suffcently small, the second condton s not satsfed and ths pont fals to become the net desgn pont. Thus, one potentally good desgn pont s mssed because of the defcency and nfleblty of the crteron. Our suggeston to avod the above problem and make the crteron more fleble n (7) (8)

7 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 99 acceptance of new potentally good ponts as desgn ponts s to select a pont wth () smaller dstance to the orgn compared to the prevous desgn pont and () smaller absolute value of g compared to a specfc number (we select for ths number) as the new desgn pont. As t can be seen, these two condtons are very smlar to the prevous two condtons. The only dfference s n the second condton where the absolute value of g s no longer compared to the prevous desgn pont. Ths seemngly small dfference lets potentally good canddates, whch are suffcently close and not necessarly closer than the prevous desgn pont to the border, be selected. Thus, usng our crteron to update the desgn pont, the above mentoned drawback of nfleblty can be removed. 3.3 A new samplng densty functon to generate random samples After selectng a pont as desgn pont (at the begnnng or durng the analyss), samples should be generated based on samplng densty functon. We determned the samples to have the same dstrbuton as varables but not necessarly wth the same means and standard devatons. At each step of the analyss, we consder the last desgn pont the mean pont of the samplng densty functon. Now, to complete the characterstcs of samplng densty functon and to have a unque functon, ts standard devaton should be specfed, too. Our suggeston for the standard devaton of samplng densty functon s the followng descendng lnear functon n 0 n where n s the total number of samples, s the number of sample and 0 s standard devaton of the random varables. and n are ntroduced as two coeffcents such that 0 and n 0 are standard devatons of samplng densty functon at the begnnng and end of the analyss, respectvely. We determne and n to be mn( ) (relablty nde untl teraton ) and, respectvely. Ths means for bgger relablty nde, standard devaton s selected bgger to cover further space from the mean pont. As t s seen, at the end of analyss, standard devaton reduces to 0. At ths spot, all consttuents of the proposed algorthm have been prepared. In the followng secton and through numercal eamples, the represented formulaton and suggestons are shown to be effcent n reducng the number of requred random samples. But before gong through numercal eamples, the summary of the proposed method based on prevous eplanatons, are epressed as the followng step-by-step algorthm:. Wrte the LSF n terms of the basc random varables,.e. g (,,..., n ) 0.. Use the transformaton u ( ) and wrte the LSF n terms of the varables of standard normal coordnate system,.e. g( u, u,..., un ) Generate a random pont ( P ) and calculate the value of LSF ( g ) and dstance to the center of standard normal coordnate system at ths pont. (9)

8 00 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak 4. If g( P) g( P) c (we set c n ths paper), P s selected as ntal desgn pont and f c use the teratve formula of Eq. (6) untl g ( ) P c. Then P s selected as ntal desgn pont. 5. Generate a random pont accordng to the random varable PDF wth the selected ntal desgn pont as ts mean pont and functon of Eq. (9) as ts standard devaton. 6. Compute the value of LSF ( g ) and dstance to the orgn of standard normal coordnate system at ths pont. 7. If the value of LSF s negatve, the value PDFrandom varables PDF samplng s added to the falure number ( falure number=falure number+pdf random varables PDF samplng ). 8. If the absolute value of LSF s smaller than and the dstance to the orgn s smaller than ths dstance for prevous ponts, the new pont and ts dstance to the orgn are selected as new desgn pont and new relablty nde, respectvely. 9. Repeat steps 5 to 8 untl all samples are generated. 0. Compute probablty of falure by dvdng the falure number by the total number of samples. 4. NUMERICAL EXAMPLES In ths secton, several eamples are selected from the lterature to demonstrate the strong performance of the proposed method. Eample In ths eample the LSF takes the followng form g ( ) 46.4 (0) where and follow normal dstrbuton wth statstcs , 709.7, and The proposed method starts wth the random pont z [0.84,0.65] T wth 0.38 n standard normal coordnate system. Snce LSF at ths pont s too large,. e. 7.33, whch means the pont s not closely located around LSF, the teratve process of Eq. (6) started to work and after 4 teratons brought the random pont to z [ , ] T where the value of LSF s 0.78 and s Locatng the samplng densty functon at ths ntal desgn pont, and puttng of Eq. (9) equal to the relablty nde (the mnmum dstance of all generated ponts at the spot), the algorthm contnued the analyss and after only 75 teratons probablty of 7 9 falure was calculated P f.469 0, whle usng MCS, t takes more than 0 samples. Ths shows the strong performance of the proposed method and also the mportance of the locaton of samplng densty functon and ntal desgn pont.

9 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 0 Eample The LSF of ths eample s epressed as ( ) g ( ) 0.( ).5 () where and both have standard normal dstrbuton. Startng wth the ntal random pont z [ , 0.89] T wth g 3.504, the proposed algorthm reaches the ntal desgn pont z [.57,.34] T wth g 0.7 after teratons. By means of ths ntal desgn pont, the proposed algorthm takes 43 teratons to gve Pf 0.004, whle ths number for MCS s 7096 teratons. Ths bg dfference s also a sgn of mprovement of the proposed algorthm compared to MCS. Eample 3 The LSF of ths eample s defned as follows ( ) g ( ) 0.5( ) 3 () where and both follow standard normal dstrbuton. After 3 teratons, the ntal random pont z [0.045, 0.946] T wth g was brought to the ntal desgn pont z [0.39,.98] T wth g In the proposed algorthm and usng the ntal desgn pont to locate the samplng densty functon, 39 teratons are requred to compute probablty of falure Pf However, MCS gves the answer after more teratons,. e. 94 teratons. Eample 4 The LSF consdered n ths eample has the followng form g ( ) (3) 3 wth standard normal random varables. In the proposed algorthm after 0 teratons Pf s reached, whle n MCS the number of requred teratons s 734. Ths result n the proposed algorthm, s obtaned after passng from the ntal random pont z [0.694,0.86] T wth g.638 to the ntal desgn pont z [.556,.954] T wth g 0.05 by teraton. Thus, t can be mentoned ths s another eample of the sutablty of our algorthm.

10 0 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak Eample 5 Consder the followng LSF g ( ) ( ) ( 0) 4 (4) where and both have normal dstrbuton wth the same statstcs 0 and 3. In the proposed algorthm 633 teratons are requred to compute Pf Ths number compared to the correspondng number n MCS,. e , shows the power of the proposed algorthm n reachng the answer. teraton, to brng the ntal random pont z [ ,0.7059] T wth g to the ntal desgn pont z [.974,.606] T wth g was a huge contrbuton n creatng ths dfference. Eample 6 The LSF of ths eample ncludng 6 random varables has the followng form g ( ) sn(00 ) (5) where to 6 are random varables wth statstcs as shown n Table. The proposed algorthm started the relablty analyss of the above LSF wth the ntal random pont z [ 0.709,0.993, 0.65,0.8005,0.9606,0.079] T wth g and after teraton could brng ths pont closer to lmt state surface and gve the ntal desgn pont z [.30,0.97,.9875,.5973,3.7,0.567] T wth g Ths movement towards the lmt state surface and gettng closer to ths surface caused the proposed algorthm to compute the probablty of falure Pf 0.07 after 340 teratons, whle MCS uses 7655 teratons to gve the answer. 6 Table : Statstcs of random varables n eample 6 Varable Mean Standard devaton Dstrbuton 0 Lognormal 0 Lognormal 3 0 Lognormal 4 0 Lognormal Lognormal 6 40 Lognormal Eample 7 The LSF of ths eample s defned as

11 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 03 9 g ( ) (6) where to 0 have standard normal dstrbuton. After teraton, the ntal random pont z [0.694,0.86, 0.746,0.868,0.647, , 0.443,0.0938,0.95,0.998] T wth g.3 changed to z [.544,.9874,.868,.045,0.648,.97,.0848,0.96,.407,.768] T as the ntal desgn pont wth g Ths helped the proposed algorthm to reach P usng 503 teratons, nstead of 7830 teratons used n MCS. f Eample 8 Consder the concal structure of Fg. subjected to a compressve aal load P and a bendng moment M [8]. The geometrcal and mechancal features of the structure are presented n Table and are defned as ndependent normal varables. The man phenomena that can nvolve the falure of the structure are () the loss of strength of the structure and () the bucklng of the structure due to nstablty. For the loss of strength, accordng to the large margn obtaned n the analyss, ths falure mode shall not be analyzed. It s only the bucklng of the structure that wll be analyzed under the combned solctatons. Accordng to NASA space vehcles desgn rules [8] the bucklng crteron s Fgure. The concal structure of Eample 8 P P cr M M (7) cr where P cr and M cr are crtcal aal load and bendng moment (for bucklng), respectvely. Accordng to NASA [8].

12 04 M.A. Shayanfar, M.A. Barkhordar and M.A. Roudak P M cr cr Et cos 3( ) Et r cos 3( ) where and enable to correlate theoretcal results wth epermental ones ( 0.33, 0.4 accordng to NASA [8]). Accordng to Eqs. (7)-(9), the LSF of the structure s g( ) 3( ) P M Et cos r 7 7 MCS needs more than 0 teratons to compute P f 5.3 0, whle the proposed algorthm uses 5785 teratons to gve the answer. Ths s done by passng from the ntal random pont z [0.376, 0.06, 0.976, 0.358, ,0.5886] T wth 5 g to the ntal desgn pont z [4.957,.863, ,.6335, 4.87,.093] T wth g after 4 teratons. Thus, ths s another eample showng the strength of the proposed algorthm n complcated eamples. Table : Statstcs of random varables n eample 8 Varable Mean Standard devaton Dstrbuton E (MPa) Normal t (m) Normal (rd) Normal r (m) Normal M (Nm) Normal P (N) Normal (8) (9) (0) The results of these eamples are summarzed n Table 3. These results ndcate that the proposed algorthm s very effcent and therefore t can be used nstead of MCS. As t can be seen n Table 3, the number of requred teratons to fnd probablty of falure s smaller than that n MCS. Last column of the table shows a huge reducton n determnstc analyses as a huge success of the proposed algorthm n makng the smulaton methods more practcal. Ths s generally because the ntal pont s more approprately selected and the concentraton of samplng s on more mportant regons,. e closer to lmt state surface and

13 AN ADAPTIVE IMPORTANCE SAMPLING-BASED ALGORITHM 05 desgn pont. On ths way, the represented technque, crteron and samplng densty functon of ths paper have gven a sgnfcant contrbuton to make the proposed algorthm more effcent. Eample Table 3: Comparson of results n eamples to 8 Iteratons Probablty of falure MCS Proposed algorthm Dfference CONCLUSION In ths paper, a new relablty algorthm, utlzng the concepts of mportance samplng and FORM, has been proposed. In ths algorthm, the computatonal cost s not large and there s no need to dervatve of lmt state functon. Ths s because of addng a technque, a crteron and a samplng densty functon whch all try to generate samples around the border of safe and falure regon. These propertes have led to the aforementoned relablty algorthm the power of whch n consderable reducton of the number of requred samples has been shown n llustratve numercal eamples. REFERENCES. Hohenbchler M, Gollwtzer S, Kruse W, Rackwtz R. New lght on frst- and secondorder relablty methods, Struct Saf 987; 4: Rackwtz R. Relablty analyss-a revew and some perspectves. Struct Saf 00; 3(4): Cho SK, Grandh RV, Canfeld RA. Relablty-Based Structural Desgn, Sprnger Verlag, London, Lu ZH, Zhao YG, Ang AHS. Estmaton of load and resstance factors based on the fourth Moment method. Struct Eng Mech 00; 36(): Kaveh A, Ilch Ghazaan M. Structural relablty assessment utlzng four metaheurstc algorthms, Int J Optm Cv Eng 05; 5(): 05-5.

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