A NEW STUDY ON RELIABILITY-BASED DESIGN OPTIMIZATION. Center for Computer-Aided Design. and. Department of Mechanical Engineering

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1 Tu, J. and Cho, K.K., A New Study on Relablty Based Desgn Optmzaton, ASME Journal of Mechancal Desgn, Vol. 2, No. 4, 999, pp A NEW STUDY ON RELIABILITY-BASED DESIGN OPTIMIZATION Jan Tu, Kyung K. Cho, and Young H. Park Center for Computer-Aded Desgn and Department of Mechancal Engneerng College of Engneerng The Unversty of Iowa Iowa Cty, IA ASME Journal of Mechancal Desgn Vol. 2. No. 4 pp December 999 Ph.D. Canddate, e-mal: jtu@ccad.uowa.edu Professor and Drector, e-mal: kkcho@ccad.uowa.edu, correspondng author Adjunct Assstant Professor, e-mal: ypark@ccad.uowa.edu

2 ABSTRACT Ths paper presents a general approach for the probablstc constrant evaluaton n the relablty-based desgn optmzaton (RBDO). Dfferent perspectves of the general approach are consstent n prescrbng the probablstc constrant, where the conventonal relablty ndex approach (RIA) and the proposed performance measure approach (PMA) are dentfed as two specal cases. PMA s shown to be nherently robust and more effcent n evaluatng nactve probablstc constrants, whle RIA s more effcent for volated probablstc constrants. Moreover, RBDO often yelds a hgher rate of convergence by usng PMA, whle RIA yelds sngularty n some cases. NOMENCLATURE X Random system parameter; X = [X ] T ( =, 2,,n) x Outcomes of the random system parameter; x = [x ] T ( =, 2,,n) x L, x U L Lower and upper tolerance lmts of the system parameter; x x x Φ( ) Standard normal cumulatve dstrbuton functon (CDF) G(x) System performance functon; system fals f G(x) < 0 F G (g) CDF of the system performance functon G(x); F G (g) = P(G(x) < g) np Total number of the probablstc constrants n the RBDO model P f Falure probablty; P f = F G (0) = P(G(x) < 0) P f Prescrbed falure probablty lmt β s Relablty ndex; β s = Φ (P f ) = Φ (F G (0)) β t Relablty target ndex; β t = Φ ( P f ) U g u g=0 u β= βt Target probablstc performance measure; P(G(x) < g ) = P f MPP of RIA correspondng to G(u) = 0 n the u-space; β s = u g=0 MPP of PMA correspondng to G(u) = g n the u-space; g = G( ) u β= βt

3 Introducton In engneerng desgn, the tradtonal determnstc desgn optmzaton model (Arora, 989; Haftka and Gurdal, 99) has been successfully appled to systematcally reduce the cost and mprove qualty. However, the exstence of uncertantes n ether engneerng smulatons or manufacturng processes calls for a relablty-based desgn optmzaton (RBDO) model for robust and cost-effectve desgns. In the RBDO model for robust system parameter desgn, the mean values of the random system parameters are usually used as desgn varables, and the cost s optmzed subject to prescrbed probablstc constrants by solvng a mathematcal nonlnear programmng problem. Therefore, the soluton from RBDO provdes not only an mproved desgn but also a hgher level of confdence n the desgn. To date, almost all researchers on RBDO (Enevoldsen, 994; Enevoldsen and Sorensen, 994; Chandu and Grand, 995; Frangopol and Corots, 996; Cho et al., 996; Yu et al., 997 and 998; Wu and Wang, 996; Grandh and Wang, 998) have used the relablty ndex evaluated n the tradtonal relablty analyss to prescrbe the probablstc constrant. In ths paper, the probablstc constrant evaluaton n RBDO s studed from a broader perspectve. It s shown that the target probablstc performance measure of the proposed performance measure approach (PMA) evaluated n an nverse relablty analyss s consstent wth the conventonal relablty ndex approach (RIA) n prescrbng the probablstc constrant for RBDO. Moreover, t s llustrated that the probablstc constrant can be effectvely evaluated from dfferent perspectves n a general approach where RIA and PMA are two specal cases. 2

4 Thus, dfferent perspectves of the general approach are consstent n prescrbng the probablstc constrant because any of them can suffcently dentfy the exact status of the probablstc constrant. However, they are not equvalent n solvng the RBDO problem. It s shown n ths paper that PMA s nherently robust for RBDO and s more effcent n evaluatng the nactve probablstc constrant. In contrast, RIA may yeld sngularty n many RBDO applcatons though t s more effcent n evaluatng the volated probablstc constrant. Thus, effcency and robustness n solvng RBDO problems can be acheved by usng PMA and RIA adaptvely dependng on the estmated margnal status of the probablstc constrant n the RBDO teratons. 2 Probablty Analyss of the System Performance Functon The uncertantes of an engneerng system are dentfed by the varatons of the random system parameter X = [X ] T ( =, 2,, n). The probablty dstrbuton of X s descrbed by ts cumulatve dstrbuton functon (CDF) F ( X x ) or probablty densty functon (PDF) f X ( x ), and s often bounded by the tolerance lmts of the system parameter (Da and Wang, 992; Ayyub and McCuen, 997). The system performance crtera are descrbed by the system performance functons. Consder a system performance functon G(x), where the system fals f G(x) < 0. The statstc descrpton of G(x) s characterzed by ts CDF F G (g) as z G( x) < g F G (g) = P(G(x) < g) =... f ( X x ) dx... dxn, z L U x x x () where f X ( x) s the jont probablty densty functon (JPDF) of all random system parameters and g s named the probablstc performance measure. The probablty analyss of the system performance functon s to evaluate the non-decreasng FG(g)~g 3

5 relatonshp, whch s performed n the probablty ntegraton doman bounded by the system parameter tolerance lmts gven n Eq. (). A generalzed probablty ndex β G, whch s a non-ncreasng functon of g, s ntroduced (Madsen et al., 986) as F G (g) = Φ( β G ) (2) whch can be expressed n two ways usng the followng nverse transformatons (Rubnsten, 98), respectvely, as β G (g) = Φ (F G (g)) g(β G ) = F G (Φ( βg)) (3a) (3b) Thus, the non-ncreasng β G ~g relatonshp represents a one-to-one mappng of F G (g)~g and also completely descrbes the probablty dstrbuton of the performance functon. Snce the system performance s often non-normal dstrbuton, the β G ~g relatonshp s generally nonlnear. 3 General Defnton of the RBDO Model In the robust system parameter desgn, the RBDO model (Enevoldsen and Sorensen, 994; Chandu and Grand, 995; Cho et al., 996; Wu and Wang, 996; Yu et al., 997 and 998; Grandh and Wang, 998) can generally be defned as mnmze Cost(d) (4a) subject to P f, j = P(Gj(x) < 0) P f, j, j =, 2,, np (4b) d L d d U (4c) where the cost can be any functon of the desgn varable d = [ d ] T [ µ ] T ( =, 2,, n), and each prescrbed falure probablty lmt P f s often represented by the relablty 4

6 target ndex as β t = Φ ( P f ). Hence, any probablstc constrant n Eq. (4b) can be rewrtten usng Eq. () as F G (0) Φ( β t ) (5) whch can also be expressed n two ways through nverse transformatons as β s = Φ (F G (0)) β t g = F G (Φ( βt)) 0 (6a) (6b) where β s s tradtonally called the relablty ndex and g s named the target probablstc performance measure n ths paper. To date, most researchers have used the relablty ndex approach (RIA) of Eq. (6a) to drectly prescrbe the probablstc constrant as β s (d) β t (7a) k k At a gven desgn d = [ d ] T k [ µ ] T, the evaluaton of relablty ndex β s ( d k ) for RIA s performed usng the well-developed relablty analyss (Madsen et al., 986) as z z < k L U β s ( d ) = (... f X ( x) dx... dx n ), x x x (7b) Φ G( x) 0 It s clear that Eq. (6b) can also be used to prescrbe the probablstc constrant and t s called the performance measure approach (PMA) as g (d) 0 (8a) and the evaluaton of target probablstc performance measure g ( the nverse relablty analyss (Tu and Cho, 997) as g k ( d ) = z z x G( x) < g d k ) n PMA s called F L U G (... fx( ) dx... dxn), x x x (8b) 5

7 4 Broader Perspectve of the Probablstc Constrant Evaluaton RIA and PMA are drectly derved from the general defnton of the probablstc constrant and they are consstent n prescrbng the probablstc constrant n RBDO. In fact, the probablstc constrant can be understood from an even broader perspectve, where RIA and PMA are just two specal cases. 4. Example Consder a system descrbed by two ndependent, unformly dstrbuted random system parameters, X ~ Unform[a, b ] ( =, 2), and ther PDFs are expressed as f X ( x ) = ( b a ), a x b, =, 2 (9a) where the mean values and varances of system parameters are expressed, respectvely, as z b µ = xf X ( x) dx = ( b ) a a 2, =, 2 (9b) z σ 2 b = ( x µ ) fx ( x) dx = (b a a) 2, =, 2 (9c) In the system parameter desgn, both µ and µ 2 are chosen as desgn varables, d = [d, d 2 ] T [µ, µ 2 ] T, and ther varances are constants as = = /3. Thus, the PDFs of system parameters can be expressed n terms of desgn varables as f X ( x )=/ 2, d d x +, =, 2 (9d) Snce X and X 2 are mutually ndependent, ther JPDF can be explctly expressed as fx( x ) = fx ( x) fx ( x2) = / 4, d x d +, =, 2 (9e) 2 Consder a probablstc constrant n the RBDO model that s defned as σ 2 σ 2 2 P(G(x) < 0) P f = 2.275% = Φ( β t ) (9f) where β t = Φ ( ) = 2. The system performance functon G(x) and ts CDF are 6

8 G(x) = x + 2x 2 0 F G (g) = 4 z G( x) < z... dx... g (9g) dx n, d x d +, =, 2 (9h) At three dfferent desgns, d = [3.7, 3.7] T, d 2 = [4.2, 4.2] T, and d 3 = [4.5, 4.5] T, the F G (g)~g relatonshp can be obtaned by performng the probablty ntegraton n Eq. (9h) repeatedly wth dfferent values of g. Then, the correspondng non-ncreasng β G ~g curves are obtaned usng Eq. (3a), whch are llustrated n Fg.. Fgure. General Interpretaton of Probablstc Constrant 4.2 General Interpretaton of the Probablstc Constrant Note that the comprehensve defnton of the probablstc constrant n Eq. (4b) ncludes two nequalty relatons. Conceptually, any probablstc constrant n Eq. (4b) (or Eq. (5)) can be represented by a set of three smple constrants, where two nequalty constrants are related to each other through an equalty constrant by Eq. (2), as β G β t g 0 F G (g) = Φ( β G ) (0a) (0b) (0c) 7

9 The lmt-state of Eq. (0a) s represented n Fg. by the vertcal lne at β t = Φ ( P f ), the lmt-state of Eq. (0b) s represented by the β G -axs, and the lmt-state of Eq. (0c) s represented by the non-ncreasng β G ~g curve. Thus, the β G -g space s naturally dvded nto four regons as Actve Pont: β G = β t and g = 0 (a) Infeasble Regon: β G β t and g 0 (b) Feasble Regon: β G β t and g 0 (c) Ambguous Regons: (β G β t ) g < 0 (d) The probablstc constrant s volated for desgn d as the correspondng βg~g curve 2 passes through the nfeasble regon. It s actve for desgn d as ts βg~g curve passes through the actve pont. And t s nactve for desgn d 3 as ts βg~g curve passes through the feasble regon. In other words, a gven desgn s nfeasble f the non-ncreasng β G ~g curve passes through the nfeasble regon, whle the desgn s feasble f the curve passes 2 through the feasble regon. For the actve probablstc constrant at desgn d, the only pont outsde the ambguous regons s the actve pont (βt, 0) because β s ( d 2 ) = βt and g 2 ( d ) = 0. That s, the probablstc constrant can be evaluated by fndng any pont on the βg~g curve that s outsde the ambguous regons. Thus, a sngle nequalty relaton can be used to represent the probablstc constrant, such as Eq. (7a) n the conventonal RIA or Eq. (8a) n the proposed PMA. On the β G ~g k curve for desgn d = [4, 4] T, as shown n Fg. 2, the pont (β s, 0) s dentfed n RIA by performng relablty analyss of Eq. (7b), and the pont (β t, g ) s dentfed n PMA by 8

10 performng nverse relablty analyss of Eq. (8b). The probablstc constrant s volated n RIA because β s =.52 < β t = 2 as well as n PMA because g = < 0. Fgure 2. Illustraton of Probablstc Constrant Evaluaton at d k = [4, 4] T 4.3 Sngularty of RIA n the Probablstc Constrant Evaluaton Note that any pont on the β G ~g curve that s outsde the ambguous regons, such as the pont (β a, g a ) n Fg. 2, can suffcently dentfy the status of the probablstc constrant, whle RIA and PMA are two extreme cases. However, RIA can yeld sngularty n the probablstc constrant evaluaton. For ths, consder desgn d 3 = [4.5, 4.5] T, whose non-ncreasng β G ~g curve s shown n Fg.. Note that the pont (β s, 0) does not exst because the falure probablty of the desgn s zero. Numercally, the relablty ndex β s ( ) approaches nfnty and thus RIA yelds sngularty. Ths happens because the system performance functon G(x) s postve everywhere n the correspondng probablty ntegraton doman of the desgn. If G(x) s negatve everywhere, the falure probablty of the desgn s one hundred percent and RIA yelds sngularty agan as the relablty ndex approaches negatve nfnty. In contrast, PMA s nherently robust because the pont (βt, g ) always exsts. d 3 9

11 5 A General Approach for the Probablstc Constrant Evaluaton A general approach for the probablstc constrant evaluaton can be establshed by fndng the pont (β a, g a ) between (β s, 0) and (β t, g ) so that a sngle nequalty relaton can be used to represent the probablstc constrant. For general evaluaton of the probablstc constrant gven n Eqs. (0a) to (0c), the Taylor seres expanson of Eq. (0c) at the pont (β a, g a ) can be obtaned n two ways by usng ts equvalent forms n Eq. (3a) and Eq. (3b), respectvely, as β G (g) = β g(β G ) = g a a n d βa ( g ga ) + n d g n! n= n d ga ( β + G β a ) n d β n! n= G n n (2a) (2b) By assumng g = 0 (RIA) n Eq. (0b) and substtutng Eq. (2a) nto Eq. (0a), an nequalty relaton can be obtaned to represent the probablstc constrant as β G (0) = β a n d βa ( ga ) + n d g n! n= n β t (3a) Smlarly, by assumng β G = β t (PMA) n Eq. (0a) and substtutng Eq. (2b) nto Eq. (0b), another nequalty relaton s obtaned to represent the probablstc constrant as g(β t ) = g a n d ga ( β + t β a ) n d β n! n= G n 0 (3b) Because hgh order dervatves n Eqs. (3a) and (3b) are generally dffcult to obtan n practcal applcatons, the mth-order approxmaton of the probablstc constrant s nstead used n two ways dependng on whether the pont (β a, g a ) s exemplfed by relablty or nverse relablty analyss, respectvely,.e., 0

12 m n d βa ( ga ) βa ( d ) + n d g n! n= n β, for gven g a = α g (4a) t g a m n d ga ( βt βa) ( d ) + n d β n! n= G n 0, for gven β a = ( α) β s + α β t (4b) where the adaptve factor s n 0 α, and m depends on the specfc approxmate probablty ntegraton method. For example, m = f the frst-order relablty method (FORM) s used and m = 2 f the second-order relablty method (SORM) s used. It s clear that Eq. (4a) becomes the conventonal RIA of Eq. (7a) f α = 0, and Eq. (4b) becomes the proposed PMA of Eq. (8a) f α =. The consstency of varous perspectves n the general approach s mantaned by usng a pont (β a, g a ) that can suffcently dentfy the lmt-state of the probablstc constrant, whch s ensured by the adaptve factor so that the pont s n between (β s, 0) and (β t, g k ). If the probablstc constrant s actve at desgn d, then the unque pont s (βt, 0) for arbtrary adaptve factor 0 α because β a ( d k ) = βs( d k ) = βt and g a ( d k ) = g k ( d ) = 0. Thus, varous perspectves of the general approach are consstent n prescrbng the probablstc constrant and they are exchangeable n RBDO teratons. 6 FORM for Approxmate Probablty Integraton Ether Eq. (4a) or Eq. (4b) can be used to prescrbe a probablstc constrant n the RBDO model. At desgn d k n the RBDO teratons, the evaluaton of Eq. (4a) requres relablty analyss and the evaluaton of Eq. (4b) requres nverse relablty analyss. In ether case, the multple ntegraton s nvolved and the exact probablty ntegraton s n general extremely complcated to compute,.e.,

13 z z G( x) < ga z z G( ) < β a ( d k ) = Φ (F G (g a )) = Φ (... f ( ) X dx... dx ) L U x n, x x x (5a) g a ( d k ) = F G (Φ( βa)) = FG (... f ( ) X dx... dx ) L U x n, x x x (5b) x g a The Monte Carlo smulaton (MCS) (Rubnsten, 98) provdes a convenent approxmaton for both relablty analyss and nverse relablty analyss because t drectly approxmates the β G ~g relatonshp. The mnmum MCS sample sze for fndng the pont (β a, g a ) s usually suggested as L = 0 P( G( x) g ) = 0 F ( g ) = 0 Φ( β a ) (6) a G a where L ncreases exponentally n terms of β a and becomes very large f the relablty target s hgh, e.g., L = 7692 for g a = 3. Thus, MCS becomes prohbtvely expensve for many engneerng applcatons. Some approxmate probablty ntegraton methods has been developed to provde effcent solutons (Bretung, 984; Madsen et al., 986; Kureghan et al., 987; Wu and Wrschng, 987; Tvedt, 990), such as FORM or the asymptotc SORM. FORM often provdes adequate accuracy and s wdely accepted for RBDO applcatons. The RIA and PMA can be used effectvely wth FORM n the probablstc constrant evaluaton. If the more accurate (and also more expensve) SORM s necessary n some engneerng applcatons, the ntermedate perspectve of the general approach becomes attractve. Ths paper focuses on RBDO usng FORM for approxmate probablty ntegraton. Thus, RIA and PMA, the two extreme cases of the general approach, are analyzed next and compared n RBDO applcatons. 2

14 6. General Interpretaton of FORM In FORM, the transformaton (Hohenbchler and Rackwtz, 98; Madsen et al., 986) from the nonnormal random system parameter X (x-space) to the ndependent and standard normal varable U (u-space) s requred. If all system parameters are mutually ndependent, the transformatons can be smplfed as u = Φ ( F X (x)), =, 2,, n (7a) x = (Φ (u)), =, 2,, n (7b) F X The performance functon G(x) can then be represented as G U (u) n the u-space. The pont on the hypersurface G U (u) = g a wth the maxmum jont probablty densty s the pont wth the mnmum dstance from the orgn and s named the most probable pont (MPP) u g=g a. The mnmum dstance, named the frst-order relablty ndex βa,form, s an approxmaton of the generalzed probablty ndex correspondng to g a as β a,form β a = β G (g a ) (8) Inversely, the performance functon value at the MPP u β= βa wth the dstance βa from the orgn s an approxmaton of the probablstc performance measure g a as g a,form = G U ( u β= βa ) ga = g(β a ) (9) Thus, the frst-order relablty analyss s to fnd the MPP on the hypersurface G U (u) = g a n the u-space, and frst-order nverse relablty analyss s to fnd the MPP that renders the mnmum dstance β a from the orgn. In two specal cases, the MPP u g=0 s found by performng frst-order relablty analyss n RIA, whle the MPP u β= βt s found by performng frst-order nverse relablty analyss n PMA. 3

15 6.2 Frst-Order Relablty Analyss In tradtonal frst-order relablty analyss (Madsen et al., 986), the frst-order relablty ndex β a,form s the soluton of a nonlnear optmzaton problem mnmze u (20a) subject to G U (u) = g a (20b) where the optmum s the MPP u and thus βa,form = g=g a. Many MPP search algorthms (such as HL-RF, Modfed HL-RF, AMVFO) and general optmzaton algorthms (such as SLP, SQP, MFD, augmented Lagrangan method, etc.) can be used to u g=ga fnd the MPP (Wu and Wrschng, 987; Wu et al., 990; Lu and Kureghan, 99; Wang and Grandh, 994; Wu, 994; Cho, et al., 996; Yu, et al., 997 and 998). 6.3 Frst-Order Inverse Relablty Analyss In frst-order nverse relablty analyss, the frst-order target probablstc performance measure g a,form s the soluton of a sphere-constraned nonlnear optmzaton problem (Tu and Cho, 997) mnmze G U (u) (2a) subject to u = β a (2b) where the optmum s the MPP u and thus ga,form(β a ) = G U ( ). β= β a u β= βa General optmzaton algorthms (such as SLP, SQP, and MFD) can be used to solve ths sphere-constraned optmzaton problem, whch s generally easer to solve than the optmzaton problem n Eqs. (20a) and (20b) due to the regular sphere constrant of Eq. (2b). In partcular, the advanced mean-value frst-order method (AMVFO) (Wu et al., 990; Wu, 994) can also be used effectvely n PMA for many engneerng applcatons. 4

16 6.4 Example Consder the same probablstc constrant defned n Secton 4., where the CDFs of the unformly dstrbuted system parameters are z x FX ( x) = fx ( x) dx = ( x a ) (b a), a a x b, =, 2 (22a) Snce mean values are chosen as desgn varables, d = [d, d 2 ] T [µ, µ 2 ] T, and varances are constants as = = /3, the CDFs of system parameters can be σ 2 σ 2 2 rewrtten n terms of desgn varables as F ( X x ) = (x d +) 2, d x d +, =, 2 (22b) The transformatons between the x-space and the u-space at desgn ndependent system parameters can be expressed as d k for two u = Φ ( F X (x)) = Φ ( (x d +) 2 ), =, 2 (22c) k k x = 2Φ(u ) + d, =, 2 (22d) and the performance functon s then transformed nto the u-space as k k G U (u) = 2Φ(u ) + 4Φ(u 2 ) + d + 2 d 2 3 (22e) k At desgn d = [4.0, 4.0] T, the contours of the performance functon,.e., G U (u) = g for dfferent g values, and the MPP locus n the u-space are llustrated n Fg. 3, where u g=0 u β β the MPP (or ) s found usng frst-order relablty analyss of RIA, and the u β β = s MPP (or ) s found usng frst-order nverse relablty analyss of PMA. The = t u g=g correspondng (βg~g) FORM curve s then compared wth the exact β G ~g curve n Fg. 4, where RIA dentfes the pont (β s,form, 0) and PMA dentfes the pont (β t, g FORM ). 5

17 Fgure 3. Illustraton of MPP locus n the u-space Fgure 4. Probablstc Constrant Evaluaton by FROM As dscussed n Secton 4.3, at desgn d 3 = [4.5, 4.5] T, the performance functon of Eq. (22e) s postve everywhere n the probablty ntegraton doman as 3 3 G(x) = G U (u) = 2Φ(u ) + 4Φ(u 2 ) > 0, d x d +, =, 2 (23) and thus frst-order relablty analyss by Eqs. (20a) and (20b) of RIA yelds no soluton. In contrast, the frst-order nverse relablty analyss of PMA can always be performed. 6

18 7 Computatonal Effcency n Probablstc Constrant Evaluaton If the Monte Carlo smulaton (MCS) s used for probablty analyss, the computatonal efforts requred to fnd pont (β s, 0) n RIA and pont (β t, g ) n PMA can be quantfed by the mnmum MCS sample sze L suggested n Eq. (6) as L RIA = 0 Φ( β s ) and L PMA = 0 Φ( β t ). That s, a. f the probablstc constrant s nactve, then β s > β t and L PMA < L RIA ; b. f the probablstc constrant s actve, then β s = β t and L PMA = L RIA ; c. f the probablstc constrant s volated, then β s < β t and L PMA > L RIA. It s ponted out that nverse relablty analyss s easer to solve than relablty analyss snce the sphercal constrant n Eq. (2b) s more regular compared to the general nonlnear constrant n Eq. (20b). In practcal applcatons, the computatonal efforts assocated wth RIA (usng frst-order relablty analyss) and PMA (usng frstorder nverse relablty analyss) cannot be easly quantfed, snce RIA and PMA are searchng for dfferent MPPs. However, t s generally easer to fnd the MPP that s closer to the orgn of the u-space (whch means searchng an MPP n a more restrctve soluton space as shown n Fg. 3). Thus, the estmatons of the computatonal efforts assocated wth RIA and PMA can also be establshed for three dfferent scenaros so that a. f β s > β t, then the MPP of PMA s closer to the orgn than the MPP of RIA u β= βt u g=0 (.e., ); u β= βs b. f β s = β t, then PMA and RIA search the same MPP as = ; u β= βt u β= βs c. f β s < β t, then the MPP of RIA s closer to the orgn than the MPP of PMA. 7

19 Therefore, PMA s not only nherently robust but s also more effcent for evaluatng nactve probablstc constrants, whle RIA s more effcent for volated probablstc constrants. Note that the computatonal dfference between RIA and PMA becomes sgnfcant f and are far apart n the u-space, whle the exact status of the u β= βt u g=0 probablstc constrant s unknown untl ether or s fnally found. Hence, t s u β= βt desred to adaptvely select RIA or PMA n the RBDO teratons dependng on the margnally estmated status of the probablstc constrant at the begnnng of the MPP search. u g=0 8 Dfference of PMA and RIA n RBDO In prevous sectons, t has been llustrated that the probablstc constrant n RBDO can be nterpreted from a broader perspectve n the general approach. These dfferent perspectves of the general approach are consstent n prescrbng the probablstc constrant, but they are dfferent n terms of robustness and computatonal effcency n probablstc constrant evaluaton. Furthermore, usng dfferent perspectves of the general approach n prescrbng the probablstc constrant actually yelds dfferent rates of convergence n solvng the RBDO problem. The RBDO problem s usually solved by search methods for constraned nonlnear optmzaton, such as SLP, SQP, and MFD. The search method starts wth an ntal desgn and teratvely mproves t wth the desgn change obtaned by solvng an approxmate subproblem defned by the lnearzed probablstc constrants. The dfference s that the lnearzed probablstc constrants from dfferent perspectves are not equvalent n predctng the desgn change. The RIA and PMA are compared here to llustrate ther dfferences n solvng the RBDO problem. 8

20 In RIA, the probablstc constrant of Eq. (7a) s lnearzed at desgn d k n defnng the search drecton determnaton subproblem as k β s ( d ) + d T βs( d k ) (d d k ) β t (24a) where β s ( d k T ) and d βs( d k ) are obtaned n the frst-order relablty analyss,.e., k β s ( d ) = (24b) u g=0 T k d β s ( d ) = T d u G G U U ( u ) g=0 ( u ) g=0 (24c) Smlarly, the probablstc constrant of Eq. (8a) n PMA s lnearzed as g k ( d ) + T d g ( d k ) (d d k ) 0 (25a) where g k ( d ) and T d g ( d k ) are obtaned n the frst-order nverse relablty analyss as g k ( d ) = GU( ) (25b) u β= βt T d g ( d k ) = T G d ( u = (25c) U β β ) t For comparson, the lnearzed probablstc constrants from Eq. (24a) of RIA and Eq. (25a) of PMA are rearranged, respectvely, as d T G U ( k u g =0) (d d ) u G U ( u g =0 ) (β s ( d k ) βt) (26a) T dg U ( = t k u β β )(d d ) GU( ) (26b) u β= βt If the probablstc constrant s actve at a gven desgn d k k, then =, βs( d ) u β= βt = βt, and G U ( ) = 0. Thus, Eqs. (26a) and (26b) become dentcal, whch means RIA u β= βt and PMA are the same n dentfyng the lmt-state of the probablstc constrant n the desgn space. On the other hand, Eqs. (26a) and (26b) are rather dfferent f the constrant u g=0 9

21 s volated or nactve. As a result, the desgn changes computed from them are dfferent and therefore the RBDO convergence rates are dfferent, too. 8. Example Consder the same system descrbed n Secton 4., where the desgn varable d = [d, d 2 ] T [µ, µ 2 ] T 2 2, and varances are constants as σ = σ 2 = /3. The RBDO problem s to mnmze Cost(d) = d + d 2 (27a) subject to P(G j (x) < 0) P f, j, j =, 2 (27b) d 0 & d 2 0 (27c) where the two system performance functons are defned as G (x) = x + 2 x 2 0 G 2 (x) = 2 x + x 2 0 (28a) (28b) and the prescrbed falure probablty lmts are P f, = 200%. & P f,2 = 300%. (.e., β t, = Φ (0.02) = & β t,2 = Φ (0.03) =.88). k At desgn d, the transformatons for the two system parameters are defned n Eqs. (22c) and (22d). Thus, the performance functons can be represented n the u-space as k k G U, (u) = 2Φ(u ) + 4Φ(u 2 ) + ( d + 2 d 2 3) (28c) k k G U, 2 (u) = 4Φ(u ) + 2Φ(u 2 ) + (2 d + d 3) 2 (28d) 8.. In RIA, the RBDO problem s to mnmze Cost(d) = d + d 2 (29a) subject to β s, (d) β t, = (29b) β s,2 (d) β t,2 =.88 (29c) d 0 & d 2 0 (29d) 20

22 k k k At desgn d = [ d, d ] T, two relablty ndexes are computed by performng two frst-order relablty analyses to and mnmze u + u 2 (30a) k k subject to G U, (u) = 2Φ(u ) + 4Φ(u 2 ) + ( d + 2 d 2 3) = 0 (30b) 2 2 mnmze u + u 2 (30c) k k subject to G U,2 (u) = 4Φ(u ) + 2Φ(u 2 ) + (2 d + d 2 3) = 0 (30d) In ths example, any general nonlnear programmng algorthm can easly solve the optmzaton problems n Eqs. (30a) to (30d). The SLP can be used to solve the overall 0 RBDO problem. Startng from an ntal desgn d = [4, 4] T, the SLP converges after fve teratons and ten relablty analyses. The RBDO hstory s lsted n Table. 0 Table. RBDO Hstory usng RIA ( d = [4.000, 4.000] T ) Iteraton k Cost d k d 2 k j = j = 2 k k k k βs, βt, β s,2 βs,2 β actve.88 actve β s, t, In PMA, the RBDO problem s to mnmze Cost(d) = d + d 2 (3a) subject to g j ( d) 0, j =, 2 (3b) d 0 & d 2 0 (3c) 2

23 0 0 0 At ntal desgn d = [ d, d ] T, two target probablstc performance measures are 2 computed by performng two frst-order nverse relablty analyses to 0 0 mnmze G U, (u) = 2Φ(u ) + 4Φ(u 2 ) + ( d + 2 d 2 3) (32a) 2 2 subject to u + u 2 = β = (2.054) 2 2 t, (32b) and mnmze G U,2(u) = 4Φ(u ) + 2Φ(u 2 ) + (2 d d 2 3) (32c) 2 2 subject to u + u 2 = β = (.88) 2 2 t,2 (32d) Note that the last terms n Eqs. (32a) and (32c) are not functons of u but are related only to the ntal desgn. Thus, the MPPs for these two nverse relablty analyses are the same for any ntal desgn, even though the values of the correspondng target probablstc performance measures are dfferent for usng varous d 0. The solutons of the nonlnear optmzaton problem of Eqs. (32a) and (32b) can be obtaned usng SLP, SQP, or MFD as, j= u β= βt = [ u, j=, j=, u 2 ] T [.278,.608] T (33a) d 0 2 g ( d 0 ) = 2Φ( u, j= ) + 4Φ( u, j= 0 2 ) + ( + 2 d 3) (33b) and the solutons of the nonlnear optmzaton problem of Eqs. (32c) and (32d) are, j=2 u = [ u, j= 2, j= 2, u 2 ] T [.478,.63] T (33c) β= β t d 0 2 g 0 2 ( d ) = 4Φ( u, j= 2 ) + 2Φ( u, j= ) + (2 + d 3) (33d) Thus, the probablstc constrants by PMA s lnear n terms of desgn varables as g ( d ) = g ( d 0 ) + (d d 0 0 ) + 2(d2 d 2 ) d + 2d (34a) g 2 ( d ) = g 2 ( d 0 ) + 2(d d 0 0 ) + (d2 d 2 ) 2d + d (34b) 22

24 Consequently, ths RBDO problem can be solved as a lnear programmng problem. opt The optmum d = [4.23, 4.230] T can be obtaned n one teraton from an arbtrary ntal desgn and only two frst-order nverse relablty analyses of Eqs. (32a) to (32d) are requred to compute g ( d 0 ) and g 2 ( d 0 ) n Eqs. (34a) and (34b). The RBDO results by PMA and RIA are compared n Table 2. RBDO Cost d opt Table 2. RBDO usng PMA and RIA d 2 opt Total Number of RBDO Iteraton Total Number of Relablty or Inverse Relablty Analyses PMA RIA Sngularty of RIA n RBDO In the prevous secton, t s shown that the convergence of RBDO s ndependent of the ntal desgn f PMA s used for the probablstc constrant evaluaton. In ths secton, t wll be shown that RBDO usng RIA s very senstve wth respect to the ntal desgn. For the gven ntal desgn d 0 = [3.5, 3.5] T, the RBDO hstory usng RIA by SLP s lsted n Table 3. The SLP fals n the frst teraton due to the sngularty of RIA at desgn d = [4.325, 4.625] T, where the frst-order relablty analyss for both probablstc constrants have no soluton. Ths s because the last terms n Eqs. (30b) and (30d) are postve at desgn d,.e., 2 µ + µ 2 3 = > 0 (36a) µ + 2 µ 2 3 = > 0 (36b) 23

25 Thus, the performance functons are postve everywhere n the probablty ntegraton domans of the desgn, and the correspondng falure probabltes are zeros. 0 Table 3. RBDO Hstory usng RIA ( d = [3.500, 3.500] T ) Iteraton k Cost d k d 2 k k k β s, β s, Dscusson The comprehensve probablstc constrant s represented by Eq. (8a) n PMA and s drectly measured by the target probablstc performance measure. In RIA, on the other hand, the probablstc constrant s measured by the relablty ndex, whch s often a nonlnear transformaton of the correspondng probablstc performance measure. In a case where the system has non-normally dstrbuted random system parameters and the probablstc constrants are for lnear performance functons, PMA yelds lnear constrants of desgn varables whle RIA yelds nonlnear constrants. It s expected that, for the general nonlnear performance functons n practcal applcatons, PMA yelds a hgher rate of convergence for RBDO than the conventonal RIA. 9 Summary It s clearly shown n ths paper that the well-accepted RIA represents only one perspectve of the probablstc constrant. From a broader perspectve, the general approach for the probablstc constrant evaluaton s developed, where RIA and the proposed PMA are two extreme cases. Although varous perspectves of the general approach are consstent n prescrbng the probablstc constrant, ther sgnfcant 24

26 dfferences n solvng the RBDO problem s llustrated. The PMA s nherently robust and more effcent n evaluatng nactve probablstc constrants, and t yelds a hgher overall RBDO rate of convergence. On the other hand, RIA s more effcent for volated probablstc constrants, but the sngularty behavor of RIA restrcts ts applcatons n broader engneerng desgn practces. The overall effcency of solvng the RBDO problem depends on the balance between the total number of overall teratons and the computatonal efforts n each teraton. In practcal applcatons, the RBDO problem can be solved robustly and more effcently by adaptvely choosng RIA and PMA dependng on the estmated margnal status of the probablstc constrant n the RBDO teratons. Acknowledgement: Research s partally supported by the Automotve Research Center sponsored by the U.S. Army TARDEC. REFERENCES Arora, J.S., 989, Introducton to Optmum Desgn, McGraw-Hll, New York, NY. Ayyub, B.M. and McCuen, R.H., 997, Probablty, Statstcs, & Relablty for Engneers, CRC Press, New York, NY. Bretung, K., 984, Asymptotc Approxmatons for Multnormal Integrals, Journal of Engneerng Mechancs, Vol. 0(3), pp Chandu, S.V.L. and Grandh, R.V., 995, General Purpose Procedure for Relablty Based Structural Optmzaton under Parametrc Uncertantes, Advances n Engneerng Software, Vol. 23, pp Cho, K.K., Yu, X., and Chang, K.H., 996, A Mxed Desgn Approach for Probablstc Structural Durablty, Sxth AIAA/USAF/NASA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton, Bellevue, WA. Da, S.H. and Wang, M.O., 992, Relablty Analyss n Engneerng Applcatons, Van Nostrand Renhold, New York, NY. Enevoldsen, I., 994, Relablty-Based Optmzaton as an Informaton Tool, Mech. Struct. & Mach., Vol. 22(), pp Enevoldsen, I. and Sorensen, J.D., 994, Relablty-Based Optmzaton n Structural Engneerng, Structural Safety, Vol. 5, pp

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