A Sequential Optimization and Mixed Uncertainty Analysis Method Based on Taylor Series Approximation

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1 11 th World Cogress o Structural ad Multdscplary Optmsato 07 th -1 th, Jue 015, Sydey Australa A Sequetal Optmzato ad Med Ucertaty Aalyss Method Based o Taylor Seres Appromato aoqa Che, We Yao, Yyog Huag, Y Zhag College of Aerospace Scece ad Egeerg, Natoal Uversty of Defese Techology, Chagsha, 1007, Hua Provce, P. R. Cha, cheaoqa@udt.edu.c 1. Abstract I ths paper relablty-based optmzato (RBO uder both aleatory ad epstemc ucertates s studed based o combed probablty ad evdece theory. Tradtoally the med ucertaty aalyss s drectly ested optmzato whch s computatoally prohbtve. To solve ths problem, a effectve way s to decompose the RBO problem to separate determstc optmzato ad med ucertaty aalyss sub-problems by sequetal optmzato ad med ucertaty aalyss (SOMUA method, whch are solved sequetally ad alterately tll covergece. SOMUA trasforms the RBO problem to ts quas-equvalet determstc formulato based o the verse Most Probable Pot (MPP of ectve ad costrat fuctos each focal elemet. As the MPP detfcato calculato s comple, the computatoal cost grows rapdly wth the crease of focal elemets. To mprove the effcecy of SOMUA, ths paper t s proposed to use Taylor appromato to trasform determstc optmzato. The effcacy of the proposed method s demostrated wth two test problems. It shows that the computatoal cost ca be greatly reduced. However, the optmum may be very close to but ot as good as that of SOMUA, whch eeds further research.. Keywords: Relablty-based optmzato, Aleatory ucertaty, Epstemc ucertaty, Sequetal optmzato, Taylor seres appromato. Itroducto Due the estece of ucertates egeerg, relablty-based optmzato (RBO s wdely studed ad appled to ehace the system relablty. Geerally ucertates clude two categores: the aleatory type arsg from the heret system radomess ad the epstemc type due to subjectve lac of owledge [1]. I ths paper, combed probablty ad evdece theory s used to deal wth the med ucertates. Based o probablty ad evdece theory, the RBO problem uder med ucertates s formulated as fd μ m f s.t. { f ( pz,, f } Pf 1R (1 { g pz,, c} Pf co 1Rco L U μ where P f ad f co P are the target falure plausblty of the ectve ad costrat respectvely. The desg varable vector s subject to radom ucertates ad ts mea value s optmzed. The parameter vector p ad z are radom ad epstemc ucertaty vectors respectvely. rom ths formulato t s clear that at each search pot the falure plausblty of the ectve ad costrat must be aalysed whch eeds coduct the epesve med ucertaty aalyss. Thus f the med ucertaty aalyss s drectly ested the optmzato, the computatoal cost would be uaffordable. To allevate ths problem, Yao proposed a sequetal optmzato ad med ucertaty aalyss (SOMUA method to decompose the RBO problem to separate determstc optmzato ad med ucertaty aalyss sub-problems, whch are solved sequetally ad alterately utl covergece s acheved []. SOMUA frstly decomposes the total relablty target to each focal elemet of the epstemc ucertates. The each focal elemet the ucerta ectve ad costrat are trasformed to the quas-equvalet determstc formulatos by calculatg the verse Most Probable Pot (MPP correspodg to the target relablty target ths focal elemet. The MPP search s based o optmzato, whch duces etra calculato ad gets worse whe the umber of focal elemets grows bg. To solve ths problem, t s proposed to use Taylor appromato to trasform determstc optmzato. I each cycle, ucertaty aalyss s coducted at the determstc optmum to obta ts MPP each focal elemet. The the epstemc ucertates are assged the values of MPP uder the assumpto that the worst case wll happe wth these values. Sce the epstemc ucertates are fed, oly radom ucertates are left. Thus the ucerta dstrbutos of the ectve ad 1

2 costrat fuctos are also radom, whch ca be appromated by Taylor seres appromato methods. As ths method has the same structure as SOMUA to coduct the optmzato ad med ucertaty aalyss sequetally, t s amed as Taylor-SOMUA dcatg that the determstc trasformato s based o Taylor seres. The orgal SOMUA method s deoted as MPP-SOMUA emphaszg the MPP based determstc trasformato. The rest of the paper s structured as follows. rst, the prelmares of RBO ad the SOMUA method are troduced. The the proposed method Taylor-SOMUA s developed, followed by two test demostratos. ally some cocluso remars are preseted.. Prelmares.1. Relablty aalyss uder med ucertates The probablty space of a radom ucerta vector [ 1 N ] s descrbed by a trple (,,Pr, where s the uversal set of all possble values of, s a -algebra over, ad Pr s a probablty measure dcatg the probablty that the elemets of occur. The evdece space of a epstemc ucerta vector z ( z z y Nz s also descrbed by a trple ( C,, m, where C cotas all the possble dstct value set of z, 1 m s the basc probablty assgmet (BPA fucto whch maps C to [0,1] satsfyg the followg aoms: 1 AC, m( A 0; for the empty set, m( 0; for all the A C, ma ( 1. The set A whch satsfes ma ( 0 s called a focal elemet. s the set of all the focal elemets [, ]. or a system respose fucto g( z,, ts put cludes both the aleatory ucerta vector defed by,,pr ad the epstemc ucerta vector z descrbed by ( C,, m wth NC focal elemets. Deote the falure rego as {( z, g( z, a}. The precse probablty of falure s bouded by ts lower lmt called belef (Bel ad ts up lmt called plausblty ( defed as [, 5, 6] N C 1 N C Bel{( z, } mc ( Bel {( z, } ; Bel {( z, } Pr{ zc, g( z, a} ( 1 {( z, } mc ( {( z, } ; {( z, } Pr{ zc, g( z, a} ( Bel {( z, } ad {( z, } are called the sub-belef ad sub-plausblty of the focal elemet c (1 N. The methods to calculate Bel {( z, } ad {( z, } are referred to [7] ad [5]. C.. The SOMUA method Deote the cycle umber as 1. Drectly gore all the ucertates (the ucerta varables are assged wth fed values ad ru the determstc optmzato. Deote the optmum as ad ts ectve respose as f. Aalyze the plausblty of the ectve falure ad the costrat falure at the optmum uder med ucertates wth the med ucertaty aalyss method SLO-ORM-UUA [5]. or each focal elemet c(1 NC, the sub-plausblty ad co ca be frst calculated wth the MPP [, p ] ad [, p ]. The the total falure plausblty MPP MPP MPP ( co MPPco MPPco MPPco ad ca be calculated wth (. Calculate the target sub-plausblty T ad T co of the ectve falure ad the costrat falure for each focal elemet c (1 N by ( 1 ( T co = co P f co MPP MPP MPP ( ( ad ( 1 ( T co co co ( ( where ( 1 ( 1 C P f = ( ad. Idetfy the correspodg verse MPP of, whch are deoted as [, p ] ad [ MPPco, p MPPco MPPco ] for the ectve ad costrat th focal elemet respectvely. The the determstc optmzato problem for the 1 th cycle ca be formulated as ( 1 fd μ ( 1 ( 1 ( 1 ( 1 ( 1 ( 1 ( 1 ( ( m f ma f ( μ, f ( μ f μ s, p MPP MPP 1NC ( 1 ( 1 s.t. g ( μ c, 1 NC ( ( 1 ( 1 ( 1 ( 1 ( g ( μ gμ s co, p MPPco MPPco ( 1 ( 1 s μ MPP, s co μ MPPco L ( 1 U μ

3 Icrease the cycle umber 1 ad coduct the determstc optmzato of the et cycle. Repeat the precedg steps utl covergece s acheved. or more detaled troducto of SOMUA, please refer to []. 5. Taylor-SOMUA 5.1. The trasformato of determstc ectve ad costrat At the optmum of the th determstc optmzato, coduct the med ucertaty aalyss ad obta the MPP [ MPPco, p MPPco MPPco ]. Assg the fed value z MPPco to the epstemc ucertaty vector z uder assumpto that the falure wll occur wth bgger probablty whe the epstemc ucertaty vector s at ths value. The oly radom ucerta vectors ad p are left. Accordgly the costrat fucto respose ( (,, g pz MPPco s also radom. Deote g rad( p, g( pz,, MPPco. Assume the dstrbuto of g rad follows the ormal dstrbuto, the ts mea ad stadard devato ca be appromated based o the frst order Taylor seres epaso as follows: ( g g ( μ, μ Accordg to the target falure plausblty rad rad p (, P g rad g rad (, rad p 1 1 p p p (5 ( g dstrbuto fucto ad geerally as ( 1 ( 1 1 T co T co ( 1 ( 1 T co 0 T co ( 1 rad p T co rad = ( (, where s stadard ormal cumulatve, relablty costrat ca be formulated as ( g (, ( g ( p, c (6 Actually g rad ( p, may ot be ormally dstrbuted, thus t should be verfed whether the ormal dstrbuto assumpto s ratoal ad the accuracy of appromato formulato (6 s wth acceptable level. Deote ( ( ( 1 = (. Substtute [, p ] to (5 ad obta ( ( g ad ( g. If co rad co MPPco MPPco co rad ( ( ( co rad co co rad co rad g follows ormal dstrbuto, the the equato ( g ( g c should est. ( ( ( Thus deote = co ( g rad co co ( g rad c. If s smaller tha the predefed threshold, the the ormal dstrbuto assumpto s ratoal. Otherwse t s suggested to revse the stadard devato as ( 1 ( ( ( g c ( g / (7 co rad co rad co Ad use ths value stead of the oe estmated (5 for costrat trasformato (6. The trasformato of the ectve fucto s the same as the aforemetoed procedure, ad the determstc ectve s formulated as f ( μ,, MPP μp z ( ( m f ma ( 1 (,, 1 1 MPP P f pz f( pz,, MPP (8 NC ( T P 1 1 p 5.. The Taylor-SOMUA algorthm To sum up, the detaled procedure of the Taylor-SOMUA algorthm s as follows: Step 1: Deote 1. Drectly gore all the ucertates ad ru the determstc optmzato. Step : or the th cycle solve the determstc optmzato problem. Deote the optmum as μ ad the ectve value as f. Step : Coduct ucertaty aalyss at μ wth SLO-ORM-UUA method [5]. or each focal elemet c(1 NC, calculate the sub-plausblty ad co wth the MPP [ MPP, p MPP MPP] ad [ MPPco, p MPPco MPPco ]. Deote the ectve falure plausblty as ad costrat falure plausblty as co. If all the falure plausblty satsfes the relablty requremet, go to Step 5. Otherwse go to et step. Step : the value of epstemc ucertaty vector as z MPPco or z MPP ad substtute t to the costrat ad ectve fucto. Deote g ( p, g( pz,, ad f ( p, f( pz,,. Deote ( 1 ( co = ( ad co rad MPPco ( ( 1 = (. Deote rad MPP ( ( ( ( co co rad co co rad = ( g ( g c

4 ( ( ( ( ( ad = ( f rad ( f rad f. The the determstc optmzato problem of 1 th cycle s formulated as ( 1 L ( 1 U fd μ ( μ ( 1 ( 1 ( 1 m f ma f ( μ 1NC ( 1 ( 1 ( 1 ( 1 ( 1 ( 1 f ( μ f( μ, μp MPP ( T ( f rad ( ( 1 ( ( f 0, ( f rad c ( fj / ( ( ( ( 1 (,, MPP P f pz f( pz,, MPP f 0, ( f rad P 1 1 P (9 ( 1 ( 1 s.t. g ( μ c, 1 NC ( 1 ( 1 ( 1 ( 1 ( 1 1 g ( μ g( μ, μp, zm PPco ( T co co ( g rad ( ( 1 ( ( f co 0, co ( g rad c co ( g rad / co ( ( 1 g(,, MPPco pz P g(,, MPPco f co 0, co( g rad 1 pz P 1 p where the symbol the subscrpt represets the focal elemet de. Deote 1 ad go to Step. Step 5: Chec covergece. If the relatve chage betwee the optmums of two cosecutve cycles s smaller tha the threshold, ed the algorthm. Otherwse go to Step. 6. Tests 6.1. Test 1: a umercal problem fd m f (10 s.t. { f(, p f }, f(, p (.5 pz { g, p0} 0.01, g(, p ( z0.7 p, The optmzato varable s subject to ormal dstrbuto N(,1.0. The ucerta parameter p follows ormal dstrbuto N (.0,1.0. The BPA of the epstemc ucertaty z s as follows: c1 [ 1,0, m( c1 ; c [0,1], m( c (11 The optmzato result of Taylor-SOMUA s compared wth that of MPP-SOMUA ad the tradtoal ested method, whch are preseted Table 1. The covergece hstory s depcted gure 1. All of the three methods obta optmzato desgs whch satsfy relablty requremets. It ca be observed that the optmum of Taylor-SOMUA s slghtly bgger tha that of other two methods, but ts computatoal cost s the smallest, whch proves ts effcacy balacg the computato cost ad optmzato effect {g<0} Cycle umber Cycle umber (a The covergece hstory of the ectve ad ts (b The covergece hstory of the costrat falure plausblty falure plausblty gure 1: The optmzato covergece hstory of Test 1

5 Table 1: The optmzato results of Test 1 MPP-SOMUA Taylor-SOMUA Tradtoal ested method co Cycle umber Total umber of fucto calls Number of fucto calls used for determstc trasformato Test : The Gols s speed reducer desg optmzato problem T fd : [ ] 7 m: f s. t. { f f } 10%, { g 0} 1%, 1 11 f( ( ( ( ( 6 57 g1: 7.0 / ( 1 1 0, g : 97.5 /( 1 10, g :1.9 /( 6 10 g :1.9 5 / ( 7 1 0, g5 : A1 / B , g6 : A / B (1 g7 : 0.0 0, g8 : /, g9 : 1 / 1.0 g10 : ( / 1 0, g11 : ( / a 1 6 a A1 a10, B1 a6, A a10, B a7.6.6, ,17 1 8, ,.9.9, The optmzato varables ecept are radom. The mea values are optmzed ad the stadard devatos are [1um, 1um, 0um, 0um, 1um, 0um] for [ ] respectvely. or the four epstemc ucertates a 1, a, a, ad a, oe terval s cosdered for each ucertaty as follows: a1 [70.0,750.0], a [16.5,17.5], a[0.09,1], a [157,158] (1 The optmzato results of Taylor-SOMUA ad MPP-SOMUA are preseted Table. The covergece hstory of Taylor-SOMUA s depcted gure. The optmum of Taylor-SOMUA s slghtly bgger tha that of MPP-SOMUA, but ts computatoal cost s much less tha MPP-SOMUA, whch proves ts effcecy {g >0} 5 {g 6 >0} {g 8 >0} {g >0} Cycle umber Cycle umber (a The covergece hstory of the ectve (b The covergece hstory of the falure plausblty ad ts falure plausblty of the hard costrats gure : The optmzato covergece hstory of Test 5

6 Table : The optmzato results of Test Determstc optmum Taylor-SOMUA optmum MPP-SOMUA optmum Desg varables.5, 0.7, 17, 7.,.5050, 0.7, 7., 7.95,.5050, 0.7, 17, 7., 7.715,.50, , ,.99, Objectve {f >} {g5>0} Costrats {g6>0} {g8>0} {g11>0} 0 0 Cycle umber Total umber of fucto calls Acowledgemets Ths wor was supported part by Natoal Natural Scece oudato of Cha uder Grat No ad Coclusos I ths paper, the RBO method uder med ucertates s studed based o sequetal optmzato ad med ucertaty aalyss method. To allevate the computatoal problem of the orgal SOMUA method whch eeds MPP to trasform the determstc optmzato formulato, t s proposed to use Taylor appromato to trasform the determstc ectve ad costrat ths paper. The effcacy of the proposed method s demostrated wth two test problems. It shows that the computatoal cost ca be greatly reduced. However, the optmum may be very close to but ot as good as that of MPP-SOMUA, whch proves the effcacy of the proposed method balacg the computatoal effcecy ad optmzato effect. However, the applcablty of ths method hghly olear optmzato problems stll eeds further studes. 9. Refereces [1] J.C. Helto, J.D. Johso, Quatfcato of margs ad ucertates: alteratve represetatos of epstemc ucertaty, Relablty Egeerg ad System Safety, 96(9, , 011. [] W. Yao,.Q. Che, Y.Y. Huag, Z. Gurdal, M. va Toore, A sequetal optmzato ad med ucertaty aalyss method for relablty-based optmzato, AIAA Joural, 51(9, 66-77, 01. [] G. Shafer. A mathematcal theory of evdece. Prceto Uversty Press, Prceto, [] W.L. Oberampf, J.C. Helto, Ivestgato of evdece theory for egeerg applcatos, th No-Determstc Approaches orum, Dever, Colorado, 00(AIAA [5] W. Yao,.Q. Che, Y.Y. Huag, M. va Toore, A ehaced ufed ucertaty aalyss approach based o frst order relablty method wth sgle-level optmzato. Relablty Egeerg ad System Safety, 116(Auguest, 8-7, 01. [6]. Du, Ucertaty aalyss wth probablty ad evdece theores. The 006 ASME Iteratoal Desg Egeerg Techcal Cofereces & Computers ad Iformato Egeerg Coferece, PA, 006. [7]. Du, Ufed ucertaty aalyss by the frst order relablty method, Joural of Mechacal Desg, 10(9: 9101,