Reliability Sensitivity Algorithm Based on Stratified Importance Sampling Method for Multiple Failure Modes Systems

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1 Chnese Journa o Aeronautcs 3(010) Chnese Journa o Aeronautcs Reabty Senstvty Agorthm Based on Strated Importance Sampng Method or Mutpe aure Modes Systems Zhang eng a, u Zhenzhou*,b, Cu e b, Song Shuang b a Schoo o Mechancs and Cv and Archtecture, Northwestern Poytechnca Unversty, X an 71007, Chna b Schoo o Aeronautcs, Northwestern Poytechnca Unversty, X an 71007, Chna Receved 30 November 009; accepted 10 Apr 010 Abstract Combnng the advantages o the strated sampng and the mportance sampng, a strated mportance sampng method (SISM) s presented to anayze the reabty senstvty or structure wth mutpe aure modes. In the presented method, the varabe space s dvded nto severa dsont subspace by n-dmensona coordnate panes at the mean pont o the random vector, and the mportance sampng unctons n the subspaces are constructed by keepng the sampng center at the mean pont and augmentng the standard devaton by a actor o. The sampe sze generated rom the mportance sampng uncton n each subspace s determned by the contrbuton o the subspace to the reabty senstvty, whch can be estmated by teratve smuaton n the sampng process. The ormuae o the reabty senstvty estmaton, the varance and the coecent o varaton are derved or the presented SISM. Comparng wth the Monte Caro method, the strated sampng method and the mportance sampng method, the presented SISM has wder appcabty and hgher cacuaton ecency, whch s demonstrated by numerca exampes. nay, the reabty senstvty anayss o ap structure s ustrated that the SISM can be apped to engneerng structure. Keywords: mutpe aure modes; reabty senstvty; Monte Caro smuaton; strated sampng method; mportance sampng method; strated mportance sampng method (SISM) 1. Introducton1 Reabty senstvty anayss has been used to nd the change rate o a mode output due to the changes o the mode nputs, whch s usuay soved by parta dervatve or numerca method. It can rank the dstrbuton parameters o the desgn varabes and gude the reabty desgn. Thereore, t s mportant to deveop an ecent method or assessng reabty senstvty [1-4]. There are two knds o methods n genera, whch are approxmate anayss method and numerca smuaton method, or the reabty senstvty anayss. Comparng wth the numerca smuaton method, the approxmate anayss method s reatvey smpe. When the perormance uncton s the near uncton *Correspondng author. Te.: E-ma address: zhenzhouu@nwpu.edu.cn oundaton tems: Natona Natura Scence oundaton o Chna ( , , ); Aeronautca Scence oundaton o Chna (007ZA5301); New Century Program or Exceent Taents o Mnstry o Educaton o Chna (NCET ); Natona Hgh-tech Research and eveopment Program (007AA04Z401) Esever td. Open access under CC BY-NC-N cense. do: /S (09) o norma random varabes, the approxmate anayss method can generate accurate reabty senstvty resuts, but or the hghy nonnear perormance uncton, ths method can ony gve an approxmate resut wth unknown precson. urthermore, the approxmate anayss method cannot be used or the structura system wth mutpe aure modes. Y. T. Wu [4] presented a reabty senstvty method based on the cumuatve dstrbuton uncton (C) o structura response varabe, wheren the Monte Caro smuaton method coud be used to compute the reabty senstvty. However, the drect Monte Caro smuaton method demanded enormous computatona cost or assessng the ow probabty events. Many varance reducton technques, such as the mportance sampng method [3-10], the strated sampng method [11-15], the ne sampng method [16-18], the drectona smuaton method [19-1] and the subset smuaton method [-4], were deveoped to mprove the ecency o the drect Monte Caro smuaton method. As an ecent agorthm wth reduced varance, the mportance sampng method s wdey used to cacuate the aure probabty n reabty engneerng. In Re.[6], the center o the tradtona mportance sampng s ocated at the desgn pont o the mt state

2 No.6 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) equaton, and up to 50% o sampe sze w a nto the aure regon when the degree o nonnearty o the mt state equaton s qute ow. Ths tradtona mportance sampng method needs to nd the desgn pont by the rst order reabty method (ORM) or other optmzaton methods rst, and t cannot be eectvey used or conductng the reabty anayss under mutpe aure modes. A new mportance sampng uncton s constructed by keepng the sampng center at the mean pont and augmentng the standard devaton by a actor o ( >1). The exampe gven by Re.[1] dspays that ths new mportance sampng uncton s sutabe to be apped to the cacuaton o the dynamca aure probabty o structures. In the strated sampng method, the varabe space s dvded nto severa dsont subspaces. Emphass can be attrbuted by mpementng more smuaton n the subspaces that contrbute more to the aure event. The concepts presented n Re.[4] are used to compute the transport propertes o gases, and the strated mportance sampng method can reduce the cacuaton eort n cacuatng the true transport coecents. Re.[15] extended the strated mportance sampng method to cacuatng the aure probabty o structura system wth seres mutpe aure modes. Ths artce presents a nove reabty senstvty method based on the strated mportance sampng method (SISM), whch s the combnaton o the strated sampng and the mportance sampng or structura system wth mutpe aure modes. The presented method can sgncanty reduce the estmate o the varance and mprove the sampng ecency, and t s especay sutabe to sove the hghy nonnear probems o structura system wth mutpe aure modes. The ormuae o the reabty senstvty estmate, the varance and the coecent o varaton are derved or the presented SISM. Ths artce s organzed as oows. The avaabe methods o reabty senstvty anayss, such as the drect Monte Caro smuaton method, the mportance sampng method and the strated sampng method, are ntroduced n Secton. Secton 3 expans the concept and the mpementaton o SISM-based anayss method n the norma varabe space. The easbty and ratonaty o the presented method are vered wth the numerca exampes n Secton 4. Secton 5 summarzes the man advantages o the presented method.. Avaabe Methods or Reabty Senstvty Anayss o Structura System.1. enton o reabty senstvty and ts souton based on Monte Caro method Snce the ndependent norma dstrbuton s a representatve dstrbuton o random varabe n reabty senstvty anayss, and the non-norma dstrbuton can be transormed nto equvaent ndependent norma dstrbuton [5], we many nvestgate the case o ndependent norma dstrbuton n ths artce. We assume that x=[x 1 x x n ] s an ndependent norma dstrbuton vector, and and are the mean and the standard devaton o x respectvey. In the structura system, we assume that the mt state equatons o aure modes are expressed by gm ( x ) 0 ( m 1,,, M ) (1) Thus, the aure regon m o g m (x) can be expressed by x: g ( x ) 0 ( m1,,, M) () m m When the aure modes are n seres, the aure regon can be dened by M (3) m1 When the aure modes are n parae, the aure regon can be dened by M m1 m m (4) And the aure probabty P o the structura system wth mutpe aure modes can be expressed by P{ x : x } ( x )d x I ( x ) ( x )d x (5) where (x) s the ont probabty densty uncton (P) o random vector x, the varabe space, and I (x) the ndcator o aure regon, and when x, I (x)=1, otherwse I (x)=0. By partay derentatng Eq.(5), the reabty senstvty o P wth respect to the dstrbuton parameter (=1,,, n) ( ncudng and ) o the basc varabe x,.e., can be obtaned as oows: I ( x) ( x) ( x)dx= ( x) I ( ) ( ) E x x ( 1,,, n) ( x) (6) where E[ ] denotes the expectaton vaue o operator. Obvousy, the reabty senstvty expressed by Eq.(6) can be easy cacuated through the Monte Caro smuaton method [3-4], n whch the expectaton vaue s estmated by the mean o the sampes generated rom the P (x)... Reabty senstvty anayss based on mportance sampng method The drect Monte Caro method has been used wdey n reabty senstvty anayss or ts hgh

3 66 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) No.6 precson. However, the huge computatona cost s an obvous drawback o ths method n assessng ow probabty events. By ntroducng an mportance sampng P, denoted by h(x), the mportance sampng method can mprove the ecency estmatng the reabty senstvty. Transormng Eq.(6) equvaenty, the reabty senstvty can be expressed by I ( x) ( x) h( x)dx ( 1,,, n) h( x) (7) The estmate o the reabty senstvty based on the mportant sampng method can be gven by N 1 I ( ) ( ) ˆ / x x ( 1,,, n) N 1 h( x ) (8) where x s the th sampe generated rom h(x). Comparng wth the Monte Caro method, the tradtona mportance sampng method has ower computatona cost, but t needs to search the desgn pont wth some other methods to construct the mportance sampng P..3. Reabty senstvty souton based on strated sampng method The dea o the strated sampng varance reducton technque s smar to that o the mportance sampng method. The varabe space n the strated sampng method s dvded nto severa dsont subspace (=1,,,), and satses the oowng reatonshps:, 1 ( ;, 1,,, n; 1,,, ) (9) where s the number o the dvded subspaces, and empty set. Accordng to the contrbuton o subspace to the aure probabty, the correspondng number o sampe sze s assgned to the subspace. Assumng that P (=1,,, ) s the probabty o, and P s the aure probabty o, then P can be expressed as P ( x)dx ( 1,,, ) (10) or, the sampng P (x) s seected as 0 x ( x) ( x) P ( x) ( )d x x x ( 1,,, ) (11) Thus, the aure probabty o can be approxmatey estmated by Pˆ N I ( ) ( ) x x ( x ) N 1 ( 1,,, ) (1) where x s the th sampe drawn rom wth (x) as the sampng P, N the tota number o sampes drawn wth (x), and I () an ndcator uncton o the aure regon n. The estmate o the aure probabty based on the strated sampng method can be expressed as the sum o Pˆ,.e. Pˆ ( x ) ( x ) N ˆ P ( x ) N I (13) By partay derentatng Eq.(13) wth respect to the dstrbuton parameter, the reabty senstvty estmate ˆP / based on the strated sampng can be estmated. As a varance reducton technque, the strated sampng method has ower computatona cost, but t needs seectng an approprate strated strategy. 3. Reabty Senstvty Souton Based on SISM or Structura System 3.1. Basc prncpe Combnng wth the advantages o the mportance sampng method and the strated sampng method mentoned above, an nnovatve method, SISM, s proposed to anayze the reabty senstvty o the structura system wth mutpe aure modes. Assumng that h(x) s the mportance sampng P o the whoe varabe space, whch can be dvded nto dsont subspaces (=1,,, ) by severa specc curved suraces. The probabty o subspace n the whoe varabe space can be cacuated by P h( x)dx ( 1,,, ) (14) Then h (x)(=1,,,) shown n Eq.(15) s seected as the mportance sampng P o. 0 x h ( x) h( x) P h( x) h( )d x x x ( 1,,, ) (15) By usng h (x) and the concept o the strated sampng method, SISM s proposed, then the aure probabty o the whoe varabe space can be estmated by N ( x ) I ( ) ˆ ˆ x P (16) Nh( ) x where x s the th sampe drawn rom wth h (x) as the mportance sampng P, and N the number o sampes drawn wth h (x). The varance o the aure probabty estmate based on SISM can be derved by

4 No.6 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) Var( Pˆ ) 1 1 N( N 1) N ( x ) I ( ) x 1 h ( ) x 1 N ( x ) I ( ) x N (17) 1 h ( ) x By partay derentatng Eq.(16) wth respect to the dstrbuton parameter, the reabty senstvty estmate Pˆ / based on SISM can be derved by Pˆ / 1 I ( ) ( ) N h ( x ) N x x 1 1 ( 1,,, n) (18) The varance and the varaton coecent o Pˆ / can be derved by Eqs.(19)-(0) respectvey. ˆ 1 Var( / ) N ( 1) 1 N N I ( ) ( ) x x 1 h ( ) x 1 N I ( ) ( ) N x x (19) 1 h ( ) x In the tradtona mportance sampng method, the desgn pont o the mt state equaton shoud be searched through ORM to construct the mportance sampng P. The mportance sampng P or a smpe random varabe wth snge aure mode s sketched n g.1(a). However, ths tradtona mportance sampng P s dcut to be constructed n the case o the mutpe aure modes. In Re.[1], a new knd o mportance sampng P s constructed by keepng the mean vector o the basc varabes nvarabe and augmentng the standard devaton by a actor o (>1). Ths new knd o the mportance sampng P wth a smpe random varabe o a snge aure mode s shown n g.1(b). By ntroducng parameter and augmentng the standard devaton o the mportance sampng P, more sampes w a nto the aure regon, and contrbutes sgncanty to the estmate o the aure probabty. Thus, the convergence o the estmate can be mproved. Ths new mportance sampng P does not requre the normaton o the desgn pont o the aure regon, thereore ts appcaton s as wde as that o the drect Monte Caro method. It can be used to anayze the probems o snge aure mode wth severa desgn ponts, seres mutpe aure modes, parae mutpe aure modes, and so on. Cov( ˆ / ) Var( Pˆ / ) Pˆ / (0) The mportant ssues o SISM needed to be researched ncude the seecton o the approprate mportance sampng P, the seecton o the approprate strated strategy to dvde the varabe space and the assgnment o sampe sze or the subspace. These ssues w be dscussed n the oowng subsectons. 3.. Seecton o approprate mportance sampng P or SISM The seecton o the approprate mportance sampng P h(x) s the key step or mprovng the ecency o SISM. Theoretcay, the best estmaton o the mportance sampng P h(x) s (x)/p. As P s unknown and shoud be soved, the other methods must be consdered or determnng the mportance sampng P. Commony, the mportance sampng P s chosen to be n a orm whch s smar to that o (x). g.1 Reatonshp between (x) and h(x). Assumng that the basc varabe x (=1,,, n) s an ndependent norma random varabe wth the mean vaue o and the standard devaton o,.e. x ~N(, ), the mportance sampng P h(x) constructed through augmentng the standard devaton a actor o by can be expressed by

5 664 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) No.6 h( x ) exp (1) ( ) ( ) n 1 ( x ) n n 1 ( ) Seecton o approprate strated strategy and assgnment o sampe sze o Specay detas aectng the convergence o SISM ncude the strated strategy o seectng approprate suraces to dvde the varabe space and the assgnment o the sampe sze o. Three common strated strateges are shown n g. [15]. n-dmensona hypersphere method. Moreover, t s dcut to drecty estmate the vaue o P n Eq.(14), whch s needed n h (x). The strated strategy o n-dmensona coordnate panes method s reatvey smpe, and P can be drecty estmated by n P h( x)dx. Thereore, h (x) o can be expressed by an expct expresson, and the strated strategy based on the n-dmensona coordnate panes method s empoyed n ths artce. When the tota number o sampes N s determned, the number o the sampes drawn rom h (x) o, N, satses the reatonshp o N /N = Var( Pˆ ) Var( Pˆ ). In order to mnmze the varance o the 1 estmate, Var ( Pˆ ) can be expressed by ˆ 1 Var( P ) N ( N 1) N ( x ) I ( ) x 1 h ( ) x 1 N N ( x ) I ( ) x h ( x ) 1 () g. Three common strated strateges o SISM. (1) n-dmensona coordnate panes method. The varabe space can be unormy dvded nto severa dsont subspaces by coordnate panes at the mean pont o the random vector. The tota number o the subspaces can be cacuated by = n. The subspaces o a -dmensona varabe space dvded by two panes are shown n g.(a). () n-dmensona poyhedron method. The varabe space can be dvded nto severa dsont subspaces by severa suraces o poyhedron. The smper the equaton orm o poyhedron s, the easer the sampng s. The subspaces o a -dmensona varabe space dvded by two poyhedron suraces are shown n g.(b). (3) n-dmensona hypersphere method. The n-dmensona varabe space can be dvded nto severa dsont subspaces by severa suraces o hypersphere. The subspaces o a -dmensona varabe space dvded by two hypersphere suraces are shown n g.(c). Addtona computatona cost s needed to determne the approprate orms and the number o suraces or the n-dmensona poyhedron method and the Thereore, the most approprate number o sampes o, N, can be expressed by Eq.(3) n case that the tota number s N. N N Var( Pˆ ) Var( Pˆ ) 1 ( 1,,, ) (3) rom Eqs.()-(3), Var ( Pˆ ) and N are nterreated, thus an teratve smuaton process s adopted to estmate the proper number o sampes o. The teratve smuaton process can be depcted as oows. Assumng that the nta vaues o ˆ (0) Var( P ) (=1,,, ) satsy the reatonshp o ˆ (0) Var( P ) 1 ˆ(0) ˆ(0) (1) Var( P ) Var( P ), the number N or the rst teratve step n the th subspace can be computed by Eq.(3). Accordng to Eq.(), ˆ ( k ) Var( P ) (=1,,, ; k1) or the kth teratve step can be computed by ( s ) ( s ) 1 1 ˆ ( k) Var( P ) k N ( s) ( s) ( s) ( ( x ) I ( )) ( ) h x x s

6 No.6 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) ( s ) k N k ( s) ( s) ( s) ( s) ( x ) I ( x ) h( x ) N ( s ) s1 1 s1 k k ( s) ( s) N N 1 (4) s1 s1 ( s where N ) ( s 1,,, k) s the number o sampes drawn rom the mportance sampng P o or the sth teratve step. The number be determned by ( k 1) N or the (k+1)th teratve step can ( k1) ( k1) ˆ( k) ˆ( k) 1 N N Var( P ) Var( P ) (5) where N (k+1) s the tota number o sampes drawn rom the whoe varabe space or the (k+1)th teratve smuaton step. The smuaton procedure o SISM can be summarzed as oows: (1) vde the whoe varabe space nto = n dsont subsapces (=1,,, ) at the mean pont usng the n-dmensona coordnate panes strated strategy. () Compute P by Eq.(14), and construct the mportance sampng P h (x) by Eq.(15). (1) (3) Assgn the sampe sze N n the rst teratve step or the th subspace. (0) (0) (0) Assumng Var( Pˆ ) Var( Pˆ ) Var( Pˆ ), the 1 number o sampes o th subspace drawn rom h (x) (1) or the rst smuaton step, N, can be assgned by Eq.(5). (4) Assgn the sampe sze ( k N 1) ( k 1) n the (k+1)th teratve step or the th subspace. Accordng to the normaton provded by kth step, ˆ ( k ) Var( P ) can be computed by Eq.(4), then the sampe sze o (k+1)th step or the th subspace, ( k 1) N, can be assgned by Eq.(5). (5) Cacuate the reabty senstvty estmate Pˆ, the varance Var( Pˆ ), and the coecent o varaton Cov ( Pˆ ) by Eqs.(18)-(0) respectvey. I the coecent o varaton Cov ( Pˆ ) s ess than an assumed crtca vaue, the teraton can be ended, otherwse, t s needed to reset k=k+1, and sht to Step (4). It shoud be noted that when the aure regon s axsymmetrc wth regard to the n-dmensona coordnate panes at the mean vaue pont, then P P P 1 and the numbers o sampes drawn rom each subspace are the same, namey N 1 = N = =N. In ths case, the ecency o SISM s equa to that o the mportance sampng method. 4. Exampes 4.1. Exampe 1: nonnear mt state uncton The non-near mt state uncton s g(x)= exp(0.x 1 +1.)x, where the basc random varabes x 1 and x are ndependent and have standard norma dstrbutons. Through augment the standard dervaton by a actor o 1 = =.0, the mportance sampng P h(x) can be wrtten as 1 x1 x h( x) exp ( ).0.0 The resuts computed by our methods are shown n Tabe 1. ˆP ˆP ˆP 1 1 Reabty senstvty Tabe 1 Resuts o Exampe 1 Monte Caro method/ Strated sampng method/ Method/ Sampng sze Importance sampng method/ SISM/ Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% ˆP Estmate Varaton coecent Reatve error/%

7 666 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) No.6 The resuts o Exampe 1 show that the presented method s sutabe or the reabty senstvty anayss o the structure wth a snge aure mode. The resuts cacuated by the presented method are n good agreement wth those cacuated by Monte Caro method, and the argest error o these resuts s 1.43%. The SISM has astest convergence rate among the our methods. Wth the same coecent o varaton o the estmate o reabty senstvty, the presented method needs mnmum sampe sze, and the number o sampes or ths method s 5.08% o that or Monte Caro method, 13.3% o that or strated sampng method, and 60% o that or mportance sampng method. 4.. Exampe : parae aure modes Equatons g 1 = x 1 5x 1 8x +16 and g = 16x 1 + x + 3 are the mt state unctons o a system, and the mt state o the system s g = max(g 1, g ), where x 1 and x are mutuay ndependent standard norma varabes. The parameters are seected as 1 = =.0. The resuts computed by our methods are shown n Tabe. ˆP ˆP ˆP ˆP 1 1 Reabty senstvty Tabe Resuts o Exampe Monte Caro method/ Strated sampng method/ Method/ Sampng sze Importance sampng method/ SISM/ Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% The resuts o Exampe show that the presented method s sutabe or the reabty senstvty anayss o the structura system wth parae aure modes. The resuts cacuated by the presented method are n good agreement wth those cacuated by Monte Caro method, and the argest error o those resuts s %. The SISM has the astest convergence rate among the our methods. Wth the same coecent o varaton o the estmate o reabty senstvty, the presented method needs mnmum sampe sze, and the number o sampes or ths method s 5.975% o that or Monte Caro method, 0.879% o that or strated sampng method, and % o that or mportance sampng method Exampe 3: sera aure modes A 9-box structure s composed o 64-bar and 4-pate eements, and t s used to smuate the arpane wng as shown n g.3. The matera o the 9-box structure s aumnum aoy. The orgna data are taken rom Re.[6]. Two sgncant aure modes o structure system are enumerated by the optmum crteron method. The mt state unctons g 1 and g o two sgncant aure modes and the mt state uncton g o the system are shown as oows: g mn( g1, g) g1 4.0R78 4.0R R77 P g 0.9 9R R77 P g.3 Sketch o 9-box structure. Assumng that a the basc ndependent random varabes are n norma dstrbuton, the mean vaue and the varaton coecent o the apped oad P are P = 150 kg and Cov(P)=0.5, respectvey. The mean vaues o R (=68, 77, 78) are R 68 R 77 R kg, and the varaton coecent o R are Cov(R )=0.1. The seected parameters are R.0 77 R 78 P. The resuts computed by our methods are shown n Tabe 3. R 68

8 No.5 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) ˆ R 68 ˆ R 68 ˆ R 77 ˆ R 77 Reabty senstvty Monte Caro method/ Tabe 3 Resuts o Exampe 3 Strated sampng method/ Method/ Sampng sze Importance sampng method/ SISM/ Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% ˆ R 78 ˆ R 78 ˆ P ˆ P Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% Estmate Varaton coecent Reatve error/% The resuts o Exampe 3 show that the presented method s sutabe or the reabty senstvty anayss o the structura system wth sera aure modes. The resuts cacuated by the presented method are n good agreement wth those cacuated by Monte Caro method, and the argest error o these resuts s 1.85%. The SISM has the astest convergence rate among the our methods. Wth the same coecent o varaton o the estmate o reabty senstvty, the presented method needs mnmum sampe sze, and the number o sampes or ths method s 19.66% o that or the Monte Caro method, % o that or the strated sampng method, and % o that or the mportance sampng method Exampe 4: engneerng appcaton o mpct mt state equatons An nner ap o an arcrat s a compex structure whch s composed o a sub-ap and a man-ap. The nte eement mode o the nner ap s shown n g.4. Resut anayzed by nte eement sotware shows that the strans at the Node and Node 9073 are most sgncant. Consderng the actor o stran aure, the mt state unctons g 1 and g o two sgncant aure modes and the mt state uncton g o the system are shown as oows: g mn( g1, g) g g1 ( E1, G1, ) g 36.5 g ( E1, G1, ) where E 1 and G 1 are the eastc moduus and shear moduus o the Node respectvey, E and G are the eastc moduus and shear moduus o Node 9073 respectvey, s the aerodynamc oad o wng, and g 1 (E 1,G 1,) and g (E,G,) are the response vaues o the strans n Node and Node 9073 respectvey. By ntroducng a dmensoness parameter, the aerodynamc oad = 0 (1+ ) w g.4 nte eement mode o nner ap.

9 668 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) No.5 dspay ts dsperson characterstc, and n whch 0 s the gven ts aerodynamc oad o wng. E 1, G 1, E, G and are mutuay ndependent norma varabes, and ther dstrbuton parameters are shown n Tabe 4. Ony the mportance sampng method and SISM are used to anayze the reabty senstvty o ths exampe, because o ther huge computaton cost, the Monte Caro method and the strated sampng method are not easbe to be used at ths tme. The resuts o Exampe 4 are shown n Tabe 5. Tabe 4 strbuton parameters o basc random varabes Random varabe Mean Varaton coecent Standard varaton E 1 /(kn m ) G 1 /(kn m ) E /(kn m ) G /(kn m ) Tabe 5 Resuts o Exampe 4 Method Reabty senstvty Importance sampng method SISM (Sampng sze s , (Sampng sze s , compute tme s s) compute tme s s) Estmate P ˆ Varaton coecent Reatve error/% Estmate P ˆ Varaton coecent Reatve error/%.878 The resuts cacuated by the SISM are n good agreement wth those cacuated by the mportance sampng method, and the argest error o the resuts s 3.676%, whch satses the requrement o the engneerng precson. Wth the same coecent o varaton o the estmate o reabty senstvty, the needed computng tme o the presented method s ony 54.57% or that o the mportance sampng method. 5. Concusons Combnng the advantages o the mportance sampng and the strated sampng, a new technque, SISM, s presented and used to anayze the reabty senstvty. The ormuae o the reabty senstvty estmate, the varance and the coecent o varaton are derved or the presented method. Comparng wth the strated sampng method and the mportance sampng method, the presented method can obtan more obvous varance reducton and ecency mprovement n cacuaton. The presented method s especay sutabe or sovng the reabty senstvty o the structura system wth mutpe aure modes and hghy nonnear mt state unctons. Reerences [1] Mechers R E, Ahammed M. A ast approxmate method or parameter senstvty estmaton n Monte Caro structura reabty. Computers and Structures 004; 8(1): [] Ahammed M, Mechers R E. Gradent and parameter senstvty estmaton or systems evauated usng Monte Caro anayss. Reabty Engneerng and Sys- System Saety 006; 91(5): [3] Wu Y T, Stakanta M. Varabe screenng and rankng usng sampng-based senstvty measures. Reabty Engneerng and System Saety 006; 91(6): [4] Wu Y T. Computatona methods or ecent structura reabty and reabty senstvty anayss. AIAA Journa 1994; 3(8): [5] Bucher C G. Adaptve sampngan teratve ast Monte-Caro procedure. Structura Saety 1988; 5(): [6] Mechers R E. Search-based mportance sampng. Structura Saety 1990; 9(): [7] Mechers R E. Importance sampng n structura system. Structura Saety 1989; 6(1): [8] Yaacob I. Observatons on appcatons o mportance sampng n structura reabty anayss. Structura Saety 1991; 9(4): [9] Au S K, Beck J. Important sampng n hgh dmensons. Structura Saety 003; 5(): [10] Wu B, Ou J P, Zhang J G. Importance sampng technques n dynamca structura reabty. Chnese Journa o Computaton Mechnacs 001; 18(4): [n Chnese] [11] Chares T. Renement strateges or strated sampng methods[j]. Reabty Engneerng & System Saety 006; 91(10): [1] Tamara J W. Use o sampng technques and reabty methods to assst n evauaton and repar o arge scae structures. Ph thess, New York: Schoo o Cv and Envronmenta Engneerng, Corne Unversty, [13] Hasan A K. Smuaton-based reabty assessment methods o structura systems. Ph thess, Coege Park, M: epartment o Cv and Envronmenta Engneerng, Unversty o Maryand, 001. [14] Bng G, Wang C. The use o strated mportance sampng or cacuatng transport propertes. Chemca Physcs etters 199; 188(3):

10 No.5 Zhang eng et a. / Chnese Journa o Aeronautcs 3(010) [15] Q, eng Y S. An mproved strated sampng Monte-Caro method to anayze the reabty o structura systems. Acta Aeronautca et Astronautca Snca 1995; 16(5): [n Chnese] [16] Song S, u Z Z. ne sampng reabty anayss or ow cyce atgue e o aeronautca engne dsc structure. Key Engneerng Materas 006; 34-35: [17] Pradwarter H J, Schueer G I, Koutsoureaks P S, et a. Appcaton o ne sampng smuaton method to reabty benchmark probems. Structura Saety 007; 9(3): [18] u Z Z, Song S, Yue Z, et a. Reabty senstvty method by ne sampng. Structura Saety 008; 30(6): [19] tevsen O, Mechers R E, Guver H. Genera mut-dmensona probabty ntegraton by drectona smuaton. Computers and Structures 1990; 36(): [0] Ne J S, Bruce R E. rectona methods or structura reabty anayss. Structura Saety 000; (3): [1] Ne J S. A new drectona method to assess structura system reabty n the context o perormance-based desgn. Ph thess, Batmore, M: epartment o Cv Engneerng, Johns Hopkns Unversty, 003. [] Au S K, Beck J. Estmaton o sma aure probabty n hgh dmensons by subset smuaton. Probabstc Engneerng Mechancs 001; 16(4): [3] Kataygots S, Cheung S H. Appcaton o spherca subset smuaton method and auxary doman method on a benchmark reabty study. Structura Saety 007; 9(3): [4] Song S, u Z Z, Qao H W. Subset smuaton or structura reabty senstvty anayss. Reabty Engneerng & System Saety 009; 94(): [5] Smth G N. Probabty and statstcs n cv engneerng. ondon: Wam Cons Sons & Co td, 1986: [6] Song B. A study on method or computng structura system aure probabty. Ph thess, X an: Schoo o Aeronautcs, Northwestern Poytechnca Unversty, [n Chnese] Bographes: Zhang eng Born n 198, he receved B.S. and M.S. degrees rom Northwestern Poytechnca Unversty (NWPU) n 004 and 007 respectvey. Now he works as a Ph.. canddate n NWPU. Hs man research nterest s the reabty engneerng o ght vehce. E-ma: nwpuwndy@ma.nwpu.edu.cn u Zhenzhou Born n 1966, she s the proessor o Northwestern Poytechnca Unversty. Her man research nterest s the reabty engneerng o ght vehce. E-ma: zhenzhouu@nwpu.edu.cn