Possibility-Based Design Optimization Method for Design Problems with both Statistical and Fuzzy Input Data

Transcription

1 6 th World Congresses of Structural and Multdscplnary Optmzaton Ro de Janero, 30 May - 03 June 2005, Brazl Possblty-Based Desgn Optmzaton Method for Desgn Problems wth both Statstcal and Fuzzy Input Data K.K. Cho 1, u Du 1, Byeng Dong Youn 2, Davd Gorsch 3 (1) Center for Computer Aded Desgn, College of Eng., The nversty of Iowa, Iowa Cty, IA 52242,.S.A. (2) Dept of Mechancal Engneerng, College of Eng., nversty of Detrot Mercy, Detrot, MI 48219,.S.A. (3) AMSTA-TR-N (MS 263), S Army Natonal Automotve Center, Warren, MI ,.S.A. 1. Abstract The relablty based desgn optmzaton (RBDO) method s prevalng n stochastc structural desgn optmzaton by assumng the amount of nput data s suffcent enough to create accurate nput statstcal dstrbuton. If the suffcent nput data cannot be generated due to lmtatons n techncal and/or faclty resources, the possblty-based desgn optmzaton (PBDO) method can be used to obtan relable desgns by utlzng membershp functons for epstemc uncertantes. For RBDO, the performance measure approach (PMA) s well establshed and accepted by many nvestgators. It s found that the same PMA s very much desrable approach also for the PBDO problems. nlke the probablty theory n whch the statstcal nformaton for the nput data s well known and developed, n the possblty theory, the nput membershp functon, whch affects the outcome optmum desgn, s not unque. In ths paper we propose two steps to generate the membershp functon from the avalable data: generatng a temporary pdf usng the avalable data, and generatng a membershp functon from the temporary pdf. The less detaled nformaton s avalable for the nput data; the membershp functon that provdes more conservatve optmum desgn should be selected. In many ndustry desgn problems, we have to deal wth the statstcal random and fuzzy nput varables smultaneously. For the desgn problem wth both statstcal random and fuzzy nput varables, t s not desrable to use RBDO snce t could lead to an unrelable optmum desgn. Ths paper proposes to use PBDO for desgn optmzaton for such problems. For fuzzy varables, several methods for membershp functon generaton are proposed. For statstcal random varable, the membershp functon that yelds least conservatve optmum desgn s proposed by usng the possblty-probablty consstency theory and the least conservatve condton. The proposed approach for desgn problems wth mxed type nput varables s appled to some example problems to demonstrate feasblty of the approach. 2. Keywords: random varable, fuzzy varable, performance measure approach (PMA), nverse possblty analyss, maxmal possblty search (MPS) method, possblty-based desgn optmzaton (PBDO) 3. Introducton In recent years, aleatory and epstemc uncertantes are used n the structural analyss and desgn optmzaton. The relablty based desgn optmzaton (RBDO) method s prevalng n stochastc structural desgn optmzaton by assumng that the amount of nput data s suffcent enough to create accurate nput statstcal dstrbuton. When suffcent nformaton of nput data cannot be obtaned due to lmtatons n tme, human, techncal, and faclty resources, etc., the nput nformaton s not belevable. In ths case, the probablty method cannot be used for relablty analyss and desgn optmzaton. To deal wth the stuaton that nput uncertantes have nsuffcent nformaton, a possblty-based desgn optmzaton (PBDO) method should be used for structural desgn. For RBDO, the performance measure approach (PMA) s well establshed and accepted by many nvestgators [1-3]. In Ref. [4], the enrched performance measure approach (PMA+) has been proposed, where computatonal effcency and stablty have been mproved by makng four mprovements over the orgnal PMA: as a way to launch RBDO at a determnstc optmum desgn, as an effcent probablstc feasblty check, as an enhanced hybrd-mean value (HMV+) method (see Refs. [4,5]), and as a fast relablty analyss under the condton of desgn closeness. Hghly nonlnear and non-monotonc response examples are used to demonstrate the effcency and stablty of the PMA+ method for nverse relablty analyss and PMA-based RBDO method. For PBDO, lke RBDO, PMA wth an nverse possblty analyss s more approprate than other approaches, such as a possblty ndex approach. In Ref. [6], a PBDO method s proposed along wth a new numercal method, called maxmal possblty search (MPS), for nverse possblty analyss and employng the performance measure approach (PMA) that mproves numercal effcency and stablty n PBDO. The MPS method resolves dsadvantages of the vertex method and the multlevel-cut method such that t evaluates possblty constrants effcently and accurately for nonlnear structural applcatons. Monotonc and non-monotonc response examples are used to demonstrate the effcency and stablty of the MPS method for nverse possblty analyss and PMAbased PBDO method n Ref. [6]. However, n many ndustry desgn problems, we may have to deal wth the nput statstcal random and fuzzy varables smultaneously. Ths s a challengng problem snce basc defntons of probablty of falure and possblty of falure are fundamentally dfferent. It s currently not known whether these two defntons can be ntegrated to defne a unfed falure rate for combned statstcal random and fuzzy varables. For the desgn problem wth both statstcal random varable and fuzzy varable, t s not desrable to use RBDO snce t could lead to an unrelable optmum desgn. Ths paper proposes a method to solve the structural desgn optmzaton problem wth both statstcal random and fuzzy nput varables by utlzng the PBDO method. For nput statstcal random varable, the membershp functon that yelds least conservatve optmum desgn s selected by usng the possblty-probablty consstency theory and the least conservatve condton [7]. On the other hand, unlke the probablty theory n

2 whch the statstcal nformaton for the nput data s well known and developed, for the possblty theory, the nput membershp functon s not unque and can be generated by several dfferent ways. In ths paper we propose two steps to generate the membershp functon from the avalable data: generatng a temporary pdf usng the avalable data, and generatng a membershp functon from the temporary pdf. Ths paper nvestgates several methods to generate temporary pdf dependng on avalablty of data nformaton. Also ways to generate the membershp functons from full statstcal data or nsuffcent data are proposed n ths paper. The less detaled nformaton s avalable for the nput data; the membershp functon that provdes more conservatve optmum desgn should be selected. For PBDO, ths paper proposes the performance measure approach (PMA), whch s smlar wth the conventonal α-cut dea. However, the proposed maxmal possblty search (MPS) method s more effcent than the conventonal α-cut method. Two examples are used to test the proposed PBDO process for mxed nput varables by analyzng the obtaned optmum desgns for relablty by usng varous possble probablstc random nputs to assure the optmum desgn s relable. 4. Possblty Based Desgn Optmzaton (PBDO) For general engneerng applcatons, the PBDO model can be formulated as mnmze Cost( d) subject to Π ( G ( d ( )) > 0) α, = 1, 2,, np T n r d = [ d ] R s the desgn vector, = [ ] T n R where s the vector of fuzzy varables where the fuzzy varable has the membershp functon Π ( x ) and the maxmal grade [13] max{ Π ( x )} = d ; α t s a target possblty of falure; and n, nr, and np are the number of desgn varables, fuzzy varables, and possblty constrants, respectvely. All fuzzy varables consdered here are assumed to satsfy the unty, strong convexty and boundedness, and be mutually non-nteractve. The possblty constrants are descrbed by Π () αt for any falure event G ( d ( )) > 0. t (1) The PMA approach, whch s developed for desgn optmzaton, s appled to the PBDO model to formulate as mnmze Cost( d) where G Π s th possblty constrant. subject to GΠ ( d ( )) 0, = 1, 2,, np (2) The problem can be standardzed by the transformaton:, ( ) 1 d = Π (3) 1 Π, ( ) R > d where Π, ( x ) and Π, ( x ) are the left sde and rght sde of the membershp functon of the nput fuzzy varable, respectvely, and d R s the maxmal grade of ths membershp functon. Then, evaluaton of the possblty constrant requres solvng the followng optmzaton problem: maxmze G subject to 1 α t To solve ths problem, the proposed maxmal possblty search (MPS) method s Step 1. Set the teraton counter (0) k = 0 ( ) wth the convergence parameter performance G( u ) and the senstvty G( u ). et d = G( u ). Step 2. Compute the next pont as [ ] T n r = R. et k = k + 1. (0) ( k) ( k) ( k+ 1) ( k) u = sgn( d ) where π 1 π t 3 (0) = (4) ε = 10. Set j = 1. et u 0. Calculate the sgn( ) (sgn( ),sgn( ),,sgn( nr )) f t = αt and = 1 2 Step 3. Calculate the performance G( u ) and ts senstvty G( u ). et ( k) ( k 1) 2 ( k) ( k) ( k) ( 1) d = G( u ) + βd k where ( k) β = ( G( u ) / G( u ) ). If sgn( G( u )) = sgn( u ), t s the maxmum pont and stop. If G( u ) G( u ), let j = k and go to Step 2. Otherwse, go to Step 4. Step 4. Go to Step 5, wth u, G( u ) and G( u ). () l Step 5. et l = 0 and d = G( u ). Go to Step 6. ( k + 1) Step 7. Calculate the performance G( u ) and ts senstvty G( u ). If Step 6. Calculate the new pont u on the boundary of the doman from the start pont u along the search drecton d. et k = k + 1. () l 2

3 G sgn( ( u )) = sgn( u), for u = π t or u = π t x G ( u ) < ε, for πt < u < π t x then t s the maxmum pont and stop. Otherwse, go to Step 8. Step 8. se G( u ), G( u ), G( u ) and G( u ) to construct the thrd order polynomal P () t on the straght lne between u, and u where t s the parameter for the lne. Calculate the maxmum pont t for ths polynomal. et * the pont on the lne correspondng to t. et k = k + 1. * G ( k +1) u be Step 9. Calculate the performance G( u ) and the senstvty G( u ). Check the convergent crtera usng the equaton n Step ( l+ 1) ( k) ( l) 7. If convergent, stop. Otherwse, let the new conjugate drecton be d = G( u ) + βd where ( k) ( k 2) 2 β = ( G( u ) / G( u ) ). et j = k, l = l+ 1, and go to Step Desgn Problem Wth Mxed Input Varables If all nput varables are random, RBDO can be used for desgn optmzaton. If all nput varables are fuzzy, PBDO should be used to obtan more conservatve optmal desgn. However, n real ndustral applcaton, the problems often nclude both nput varables wth full statstcal nformaton and nput varables wthout enough data to be characterzed as random varables as well. RBDO s not sutable for these problems because t s dangerous for desgn to characterze stochastc propertes usng nsuffcent data. There are fundamental dffcultes to combne random varables and fuzzy varables together for desgn snce probablty theory and possblty theory have dfferent measures. A preferable method for these problems s transferrng the random varables nto fuzzy varables. From desgn pont of vew, even though dealng wth random varables as fuzzy varables loses some accuracy, the optmal desgn should be conservatve. For random varables, snce the suffcent data are avalable, the membershp functons for these varables, when vewed as fuzzy, can be generated usng the probablty possblty consstency prncple and the least conservatve prncple. After transferrng the random varables nto fuzzy varables, PBDO can be used for desgn optmzaton. The optmal desgn should be conservatve even though the optmum cost mght be somewhat hgher. 6. Membershp Functon Generaton The generaton of the nput membershp functons of the fuzzy varables usng the avalable lmted set of data s a very mportant step of the possblty analyss and PBDO. Several methods have been proposed dependng on the number and the knd of the data avalable. Ths paper ntroduces a two-step procedure: 1) generatng a temporary probablty densty functon of the fuzzy varable from the avalable data and 2) generatng the membershp functon of the fuzzy varable from the temporary probablty densty functon. Methods of generatng a temporary probablty densty functon from the raw data ncludes: 1. If the only avalable nformaton for the nput fuzzy varable s just the upper and lower bound, the temporary probablty densty functon can be constructed as the unform dstrbuton between these two bounds by aplace s Prncple of Insuffcent Reason [8]. For detals, the reader s referred to Ref. [9]. 2. If the only avalable nformaton for the nput fuzzy varable s the judgment of experts (subjectve) wth the most lkely value and the nterval correspondng to the certan confdence levels, the temporary probablty densty functon can be generated usng the framework of Program-Evaluaton and Revew Technque (PERT) analyss [10]. 3. If the fuzzy varable has a random nature wth the expermental dstrbuton type and the avalable data, the temporary probablty densty functon can be constructed by the parametrc method such as maxmum lkelhood estmate [11]. Notce that, snce the nput data s not suffcent, or the random nature of the varable s not belevable, the temporary probablty densty functon cannot be used drectly for a probablty-based method. 4. If the fuzzy varable has a random nature but the avalable data are not suffcent to assgn the probablty of elementary events, the temporary probablty densty functon can be estmated usng the non-parametrc method such as kernel-smoothng method [12]. Methods of generatng membershp functon of the fuzzy varable from temporary probablty densty functon ncludes: Method I: The membershp functon satsfes the probablty-possblty consstency prncple and the least conservatve prncple [7] and s symmetrc on the cumulatve dstrbuton functon. The probablty-possblty consstency prncple asserts the probablty of any event cannot exceed the possblty of ths event. Ths prncple confrms the possblty theory should always be more conservatve than the probablty theory. For a gven temporary probablty densty functon of a fuzzy varable, the membershp functon of ths fuzzy varable satsfyng the probablty-possblty consstency prncple s not unque. From desgn pont of vew, the more conservatve desgn s not always the better desgn. It s natural to choose the membershp functon, whch s conservatve (.e., satsfyng the probablty-possblty consstency prncple), such that t wll provde an optmum desgn. Ths necesstates the least conservatve prncple. For all the conservatve membershp functon of the fuzzy varable, the least conservatve one s the most desrable choce. Theorem: If the fuzzy varable has the temporary cumulatve dstrbuton functon F( x) and the consdered membershp 3

4 functon s symmetrc on ths temporary cumulatve dstrbuton functon, the membershp functon of ths fuzzy varable satsfyng the probablty-possblty consstency prncple and the least conservatve prncple s unque: 2 F( x) x { x: F( x) 0.5} Π ( x) = 1 2 F( x) 1 = 2 2 F( x) x { x: F( x) > 0.5} Proof: Frst, ths membershp functon satsfes the probablty-possblty consstency prncple. That s, for any event { x A R}, possblty of ths event s where xa, xa R and x condton A A A A Π( A) = sup{ Π ( x), x A} =Π ( x ) = 2 F( x ) = 2 2 F( x ) A A x. It s obvous that A (, x ) ( x, ), thus the probablty of ths event satsfes the consstency P ( A) = f( x)d x f( x)d x+ f( x)d x=π( A) x A xa A Second, ths membershp functon satsfes the least conservatve prncple. Assume there s a membershp functon Π ( x) satsfes the probablty-possblty consstency prncple and s symmetrc on the cumulatve dstrbuton functon F( x). It s suffcent to show that, for any xa x R, Π ( x) Π ( x). Assume F ( x ) < 0.5 (the proof s analogous when F ( x ) > 0.5), there exsts that 1 F ( x) = F ( x). Denote A = (, x) ( x, ), snce Π ( x) s symmetrc on F( x), x Π ( x) =Π ( x) =Π ( A ) P( A ) =Π ( x ) x x x > x such whch means that Π ( x ) s the least conservatve one among the membershp functons that s symmetrc on the cumulatve dstrbuton functon and satsfes the probablty-possblty consstency prncple. Method II: The membershp functon satsfes the system level probablty-possblty consstency prncple and the least conservatve prncple. For the desgn purpose, the probablty-possblty consstency prncple and the least conservatve prncple should be appled on the whole desgn space. Note that the margnal membershp functon s not dependent on other varables. If the dmenson of the desgn space s nf, the membershp functon s where F( x) Π ( x) = 1 2 F( x) 1 nf s the temporary cumulatve dstrbuton functon of the fuzzy varable. To prove ths, assume nf fuzzy varables [ ] T n f = R have the temporary jont probablty densty functon f ( x ) where f ( x ) and F( x ) are temporary probablty densty functon and cumulatve dstrbuton functon. After transformaton of V = 2 F( ) 1 the varables V = [ V ] T R nf are unformly dstrbuted on nterval [ 1,1]. Denotng A the outsde of the k- dmensonal box [ (1 α),1 α ], usng the probablty-possblty consstency and the least conservatve prncples, Π ( A) = 1 (1 α) nf. Thus the jont membershp functon of V s Π ( v) = 1 v and so the jont membershp functon of s nf Π ( x) = 1 2 F( x) 1. Assume nf fuzzy varables are non-nteractve, then the margnal membershp functon s Π ( x ) = max{ Π( x ), x fxed} and so Π ( x ) = 1 2 F ( x ) 1 nf. Method III: The method proposed by Savoa [13]. If the membershp functons generated usng these two prncples are not vewed as conservatve enough, the membershp functon proposed by Savoa [13] can be a good alternatve: Π ( x) = F ( x ) + f ( x)( x x ) + F ( x ) R R where f ( x ) s the temporary probablty densty functon and x and x R are selected such that f ( x) = f( xr) = f( x). 7. Examples In order to llustrate the PBDO wth mxed varables, frst a mathematcal problem s used wth three cases: (1) wth all varables random for whch RBDO s sutable, (2) wth all varables fuzzy for whch PBDO s used, and (3) wth mxed random and fuzzy varables for whch PBDO s used. The mathematcal problem s V nf 4

5 2 1 = Cost( d) = ( d + d 10) /30 ( d d + 10) /120 G ( ) 1 / ( ) = 1 + ( ) + ( ) G ( ) ( ) G ( ) = 1 80/( ) d T T ntal T = [0, 0] and d = [10, 10], d = [5, 5] 2 For frst case where two varables are random, the RBDO problem s to mnmze Cost( d) subject to PG ( ( d, ) > 0) Φ( β ), = 1,2,3 where N( d,0.3), = 1, 2, β = 2, = 1, 2,3 t t RBDO of ths problem s solved usng the PMA+ method n Ref. [5]. The RBDO hstory s shown n Table 1 and Fgure 1. Table 1. RBDO Hstory of Mathematcal Problem Iter. Cost d 1 d 2 G 1 G 2 G 3 NFE NA Det Opt Act. Act. Inact. 42 Fgure 1. RBDO Hstory of Mathematcal Problem For second case where two varables are fuzzy varables, wth membershp functons generated from the normal pdfs, the PBDO problem s to mnmze Cost( d) subject to Π ( G ( d, ) > 0) α, = 1,2,3 t where Π ( x ) = 1 2 F( x ) 1, = 1, 2, α = 0.023, = 1, 2,3 t The PBDO hstory s shown n Table 2 and Fgure 2. As expected, the PBDO result n Table 2 yelds a hgher optmum cost ( vs ) compared to the RBDO result n Table 1. Table 2. PBDO Hstory of Mathematcal Problem Iter. Cost d 1 d 2 G 1 G 2 G 3 NFE

6 Opt Act. Act. Inact. 52 Fgure 2. PBDO Hstory of Mathematcal Problem For thrd case where one varable s random whle another one s fuzzy (wth trangular membershp functon), after transferrng the random varable nto the fuzzy varable, the mxed varable problem can be formulated to mnmze Cost( d) subject to Π ( G ( d, ) > 0) α, = 1,2,3 where ~ Π ( d,0.9), Π ( x ) = 1 2 F ( x ) 1, t 1 tr α = 0.023, = 1,2,3 t The PBDO hstory for the mxed varable problem s shown n Table 3 and Fgure 3. The optmum cost ( 1.626) of the mxed varable problem s larger than those of RBDO and PBDO n Tables 1 and 2, respectvely. Table 3. PBDO Hstory of Mxed Varable Problem Iter. Cost d 1 d 2 G 1 G 2 G 3 NFE Opt Act. Act. Inact. 54 Fgure 3. PBDO Hstory of Mxed Varable Problem 6

7 To evaluate relablty of the optmum desgn obtaned from the RBDO, PBDO, and mxed varable problems, Monte Carlo Smulatons have been carred out for these optmum desgns. The Monte Carlo smulaton results are shown n Table 4, ncludng the system level relablty. Note that the requred component level relablty s less than 2.3% falure rate n all cases. Table 4 shows that, the RBDO result barely satsfes the component level relablty, wth a slght volaton for constrant G 1 (2.56%). On the other hand, the PBDO result satsfes both the component and system level target relablty. That s, the PBDO result s more conservatve than that of RBDO. For mxed nput varables, the MCS smulaton s carred out by treatng fuzzy varable 1 as a random varable, wth a temporary pdf that s unform dstrbuton correspondng to the trangular membershp functon. The MCS result shows the optmum desgn satsfes the system level relablty. However, note that, even though the optmum cost for the mxed varable problem s hgher ( 1.626) than that of PBDO ( 1.698), accordng to MCS, the falure rate (0.18%) s hgher than that of PBDO result (0.16%) snce the correspondng nput dstrbuton s more wdely vared for the mxed varable problem. Table 4. Falure Rate sng MCS for Mathematcal Problem Method G 1 G 2 G 3 System evel RBDO PBDO Mxed Var For the second problem, the vehcle sde mpact problem shown n Fgure 4 [14] s used. The desgn objectve s to mnmze the vehcle weght whle enhancng sde mpact crash performance for passenger protecton. The cost s Cost( d )= d d d d d d7 and constrants are G 1 (x) = ( ) G 2 (x) = 1.86+( ) G 3 (x) = 3.02+( ) G 4 (x) = ( ) G 5 (x) = ( ) G 6 (x) = 0.42+( ) G 7 (x) = 0.72+( ) G 8 (x) = 0.68+( ) G 9 (x) = 1.35+( ) G 10 (x) = 0.16+( ) Fgure 4. Vehcle Sde Impact Model Table 5. Input Varable for Sde Impact Problem Random Varable Std. Dstr. Desgn ower Intal pper Dev. Type Varable Bound Desgn Bound 1 (B-pllar nner) Normal (B-pllar renforce) Normal (Floor sde nner) Normal (Cross member) Normal (Door beam) Normal (Door belt lne) Normal (Roof ral) Normal (Mat. floor nner) Normal (Mat. floor sde) Normal (Barrer heght) 10.0 Normal 11 (Barrer httng) 10.0 Normal For the frst case where all varables are random, the RBDO problem s to 10 th and 11 th random varables are not regarded as a desgn varable. 7

8 mnmze Cost( d) subject to PG ( (, d) > 0) Φ( β ), = 1,,10 t where the random nput varables are gven n Table 5and β = 2, = 1,,10 RBDO of ths problem s solved usng the PMA+ method n Ref. [5]. The RBDO hstores are shown n Tables 6 and 7. Table 6. RBDO Hstory of Cost and Desgn Varables Iter. Cost d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d Det Opt Table 7. RBDO Hstory of Constrants and Number of Functon Evaluatons Iter. G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 NFE Det Opt. Act. Ina. Ina. Ina. Ina. Ina. Act. Ina. Act. Ina. 166 For the second case where all varables are fuzzy varables, wth membershp functons generated from the normal pdfs, the PBDO problem s to mnmze Cost( d) subject to Π ( G (, d) > 0) α, = 1,,10 t where Π ( x ) = 1 2 F ( x ) 1, = 1,,11, α = 0.023, = 1,,10 t The PBDO hstores are shown n Tables 8 and 9. Agan, the PBDO result n Table 8 yelds a hgher optmum cost (29.21 vs ) compared to the RBDO result n Table 6. Table 8. PBDO Hstory of Cost and Desgn Varables Iter. Cost d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d Det Opt Table 9. PBDO of Constrants and Number of Functon Evaluatons Iter. G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 NFE Det Opt. Act. Ina. Act. Ina. Ina. Act. Act. Ina. Act. Ina. 221 For the thrd case where frst 7 varables are random and the last 4 varables are fuzzy wth trangular membershp functon on the 6σ length nterval, after transferrng 7 random varables nto fuzzy varables, the mxed varable problem can be formulated to t 8

9 mnmze Cost( d) subject to Π ( G ( ;d) > 0) α, = 1,,10 where Π ( x ) = 1 2 F ( x ) 1, = 1,,7, t ~ Π ( d, 0.012), = 8, 9, ~ Π (0.0, 20.0), = 10,11, tr tr α = 0.023, = 1,,10 t The PBDO hstory for the mxed varable problem s shown n Tables 10 and 11. The optmum cost (35.73) of the mxed varable problem s larger than those of RBDO and PBDO n Tables 6 and 8, respectvely. Table 10. PBDO wth mxed varables hstory for cost and the desgn varables Iter. Cost d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d Det Opt Table 11. PBDO wth mxed varables hstory for constrants and functon evaluatons Iter. G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 NFE Det Opt. Ina. Ina. Ina. Ina. Ina. Ina Act. Ina Act Ina. 239 To evaluate relablty of the optmum desgn obtaned from the RBDO, PBDO, and mxed varable problems, Monte Carlo Smulatons have been carred out for these optmum desgns. The Monte Carlo smulaton results are shown n Table 12, ncludng the system level relablty. Note that the requred component level relablty s less than 2.3% falure rate n all cases. nlke the mathematcal problem, Table 13 shows that, the RBDO result does not satsfy even the component level relablty, wth falure rate of 5.84% for G 7. The reason s that G 7 has two MPP ponts. On the other hand, the PBDO result satsfes both the component and system level target relablty. That s, the PBDO result s much more conservatve than that of RBDO. For mxed nput varables, the MCS smulaton s carred out by treatng fuzzy varables 8 ~ 11 as random varables, wth temporary pdfs that are unform dstrbutons 6σ length nterval correspondng to the trangular membershp functon on the 6σ length nterval. The MCS result shows the optmum desgn satsfes the system level relablty. However, note that, even though the optmum cost for the mxed varable problem (35.73) s hgher than that of PBDO (29.21), accordng to MCS, the falure rate (4.95%) s hgher than that of PBDO result (0.48%) snce the correspondng nput dstrbuton s more wdely vared for the mxed varable problem. Table 12. MCS result for the sde mpact example Method G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 System evel RBDO PBDO Mxed Var Conclusons If suffcent nformaton s not avalable for the nput data, PBDO should be used nstead of RBDO. For the problem wth both random varables and fuzzy varables, RBDO s not desrable snce the stochastc nformaton for nsuffcent data s not accurate. Ths paper proposed membershp functon generaton methods from the avalable lmted data; and a maxmal possblty Search (MPS) method wth nterpolaton to resolve dsadvantages of vertex method and multlevel-cut method, such that t evaluates possblty constrants effcently, stably and accurately for nonlnear structural applcatons. For the problem wth mxed nput random and fuzzy varables, ths paper proposed to transfer all random varables nto fuzzy varables and then use PBDO. The PBDO optmal desgn s conservatve consderng the nsuffcent data. Some numercal examples llustrate the result. 9. Acknowledgement Research s supported by the Automotve Research Center sponsored by the.s. Army TARDEC. 9

10 10. References 1. Youn, B.D. and Cho, K.K. Selectng Probablstc Approaches for Relablty-Based Desgn Optmzaton. AIAA J 2003, 42(1), Youn, B.D., Cho, K.K., Gu,. and Yang, R.-J. Relablty-Based Desgn Optmzaton for Crashworthness of Sde Impact. J Struct Multdsc Optm 2004, 27(3), Youn, B.D., Cho, K.K. and Park, Y.H. Hybrd Analyss Method for Relablty-Based Desgn Optmzaton. J Mech Des, ASME 2003, 125(2), ; Proceedngs of 2001 ASME Desgn Engneerng Techncal Conferences: 27th Desgn Automaton Conference. Pttsburgh, PA 4. Youn, B.D., Cho, K.K. and Du,. Enrched Performance Approach for Relablty-Based Desgn Optmzaton. AIAA J 2005, 43(4), Youn, B.D., Cho, K.K. and Du,. Adaptve Probablty Analyss sng an Enhanced Hybrd Mean Value Method. J Struct. Mult. Opt. 2005, 29(2), Du,., Cho, K.K. and Youn, B.D. An Inverse Possblty Analyss Method for Possblty-Based Desgn Optmzaton. AIAA J (to be appear) 7. Nkolads, E., Cudney, H.H., Chen, S., Haftka, R.T., and Rosca, R. Comparson of Probablstc and Possblty Theory-Based Methods for Desgn aganst Catastrophc Falure under ncertanty, Proceedngs of 1999 ASME Desgn Engneerng Techncal Conferences: 11th Internatonal Conference on Desgn Theory and Methodology, September 12-15, 1999, as Vegas, Nevada, DETC99/DTM Savage,.J. The Foundatons of Statstcs. New York, Dover Publcatons, Sentz, K. and Feson, S. Combnaton of Evdence n Dempster-Shafer Theory. SAND nlmted Release, Prnted Aprl Vallappan, S. and Pham, T.D. Constructon the Membershp Functon of a Fuzzy Set wth Objectve and Subjectve Informaton. Mcro. Cvl Engr (8) Hoel, P.G. Introducton to Mathematcal Statstcs, 3rd ed. New York, Wley, Wand M.P. and Jones M.C., Kernel Smoothng. Chapman and Hall, ondon, Savoa, M., Structural Relablty Analyss through Fuzzy Number Approach, wth Applcaton to Stablty, Computers & Structures, Vol. 2002, (80) Gu,., Yang, R-J., Tho, C.H., Makowsk, M., Faruque, O., and, Y., Optmzaton and Robustness for Crashworthness of Sde Impact, Int. J. Vehcle Desgn, Vol. 26, No. 4,