Dimension reduction method for reliability-based robust design optimization

Transcription

1 Comuters and Structures xxx (2007) xxx xxx Dmenson reducton method for relablty-based robust desgn otmzaton Ikjn Lee a, K.K. Cho a, *, Lu Du a, Davd Gorsch b a Deartment of Mechancal and Industral Engneerng, College of Engneerng, The Unversty of Iowa, Iowa Cty, IA 52241, Unted States b US Army RDECOM/TARDEC AMSRD-TAR-N, MS 157, 6501 East 11 Mle Road, Warren, MI , Unted States Receved 5 March 2007; acceted 30 Arl 2007 Abstract In relablty-based robust desgn otmzaton (RBRDO) formulaton, the roduct qualty loss functon s mnmzed subject to robablstc constrants. Snce the qualty loss functon s exressed n terms of the frst two statstcal moments, mean and varance, three methods have been recently roosed to accurately and effcently estmate the moments: the unvarate dmenson reducton method (DRM), erformance moment ntegraton (PMI) method, and ercentle dfference method (PDM). In ths aer, a relablty-based robust desgn otmzaton method s develoed usng DRM and comared to PMI and PDM for accuracy and effcency. The numercal results show that DRM s effectve when the number of random varables s small, whereas PMI s more effectve when the number of random varables s relatvely large. Ó 2007 Elsever Ltd. All rghts reserved. Keywords: Relablty-based robust desgn otmzaton (RBRDO); Dmenson reducton method (DRM); Performance moment ntegraton (PMI); Percentle dfference method (PDM); Senstvty analyss; Statstcal moment calculaton 1. Introducton In recent years, several aroaches to ntegrate robust desgn [1,2] and relablty-based desgn [3 5] have been roosed [6 8]. The relablty-based desgn otmzaton (RBDO) s a method to acheve the confdence n roduct relablty at a gven robablstc level, whle the robust desgn otmzaton (RDO) s a method to mrove the roduct qualty by mnmzng varablty of the outut erformance functon. Snce both desgn methods make use of uncertantes n desgn varables and other arameters, t s very natural for the two dfferent methodologes to be ntegrated to develo a relablty-based robust desgn otmzaton (RBRDO) method. * Corresondng author. E-mal addresses: lee@engneerng.uowa.edu (I. Lee), kkcho@engneerng.uowa.edu (K.K. Cho), ludu@engneerng.uowa.edu (L. Du), gorschd@tacom.army.ml (D. Gorsch). The roduct qualty n robust desgn can be descrbed by use of the frst two statstcal moments of a erformance functon: mean and varance [9]. Thus, t s necessary to develo methods that estmate the frst two statstcal moments of the erformance functon and ther senstvtes accurately and effcently. The statstcal moments can be analytcally exressed usng a mult-dmensonal ntegral. However, t s ractcally mossble to calculate the statstcal moments of the erformance functon usng the multdmensonal ntegral. Hence, there have been varous numercal attemts to estmate the moments more effcently: exermental desgn [10], frst order Taylor seres exanson [1,2,11], Monte Carlo smulaton (MCS) [12], mortance samlng method [13], and Latn hyer cube samlng method [14]. Monte Carlo smulaton could be accurate for the moment estmaton, however t requres a very large number of functon evaluatons. Therefore, n many large-scale engneerng alcatons, t s not ractcal to use Monte Carlo smulaton. The exermental desgn also needs a /$ - see front matter Ó 2007 Elsever Ltd. All rghts reserved. do: /

2 2 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx large amount of comutaton when the number of desgn varables s large. The frst order Taylor seres exanson has been wdely used to estmate the frst and second statstcal moments n robust desgn. However, the frst order Taylor seres exanson results n a large error esecally when the nut random varables have large varatons. Ths s because the frst order Taylor seres exanson does not use all nformaton of the robablty densty functons (PDF) of nut random varables. To overcome the shortcomngs exlaned above, three methods have been recently roosed: the unvarate dmenson reducton method (DRM) [15,16], erformance moment ntegraton (PMI) [8], and ercentle dfference method (PDM) [6,7]. In ths aer, RBRDO usng the unvarate DRM s roosed and the calculaton of the statstcal moments and ther senstvtes usng PMI s derved. In addton, the results of RBRDO usng the unvarate DRM are comared wth those desgns obtaned usng PMI and PDM. Both DRM and PMI are drectly estmatng the statstcal moments. On the other hand, n PDM, the robustness s acheved through a desgn objectve n whch the varaton of the desgn erformance s aroxmately evaluated through the ercentle erformance dfference between the rght and left tals of the erformance dstrbuton [6]. Thus, three methods can be comared n terms of how accurately these methods can fnd an otmum desgn to mnmze the varance of the erformance functon. Hence, n ths aer, three methods are evaluated by comarng the varances at the otmum desgns. PMI and DRM are also comared n terms of how accurately and effcently estmate the statstcal moments of the erformance functon. For the comarsons, several examles ncludng one-dmensonal and two-dmensonal erformance functons, and a large-scale engneerng roblem are used. These comarsons llustrate that the unvarate DRM s the most accurate and effcent method when the number of desgn varables s small and PMI s a better oton when the number of desgn varables s relatvely large. For the nverse relablty analyss of RBDO, the enrched erformance measure aroach (PMA+) [5] and ts numercal method, the enhanced hybrd mean value (HMV+) [4] are utlzed. where h(d) s the cost functon, G s the th constrant, and d s the desgn varable vector; and nc and ndv are the number of constrants and desgn varables, resectvely. The otmum desgn of the conventonal otmzaton roblem s the determnstc otmum that could be senstve to the varaton of nut desgn varables and other arameters. Due to the varaton of desgn varables and other arameters, the erformance functon h(d) also has varaton. Thus, n robust desgn, the robustness of a desgn objectve can be acheved by smultaneously otmzng the mean erformance l H and mnmzng the erformance varance r 2 H [6]. In other words, the goal of robust desgn s to fnd the most nsenstve desgn to the varaton of the desgn varables and other arameters. Snce robust desgn s fundamentally consderng the varatons of the desgn varables and other arameters, t s very natural to ntegrate robust desgn and relablty-based desgn n one formulaton. Ths desgn otmzaton s called relablty-based robust desgn otmzaton (RBRDO) and can be formulated to mnmze f ðl H ; r 2 H Þ subject to PðG ðx; dþ > 0Þ 6 Uð b t Þ; 1;...; nc d L 6 d 6 d U ; d 2 R ndv and X 2 R nrv ð2þ where f ðl H ; r 2 HÞ s the cost functon, d = l(x) s the desgn vector, X s the random vector, and G s the th robablstc constrant. Quanttes nc, ndv, nrv and b t are the number of robablstc constrants, desgn varables, random varables, and the th target relablty ndex, resectvely. Detaled exlanaton about RBDO can be found n [3 5]. In ths aer, the enrched erformance measure aroach (PMA+) [5] s ntroduced to erform nverse relablty analyss of the constrants. Fg. 1 comares a conventonal desgn otmzaton wth a RDO for a one-dmensonal erformance functon. Wth the same varablty of a desgn varable, the robust otmum shows less varaton of the erformance functon h(d) than the conventonal desgn otmum Three tyes of cost functon Snce the cost functon n Eq. (2) deends on l H and r 2 H for robust otmum desgn n RBRDO, t s a b-objectve 2. Fundamental concet of robust desgn 2.1. Relablty-based robust desgn In general, a conventonal (determnstc) desgn otmzaton roblem can be formulated to ð1þ mnmze hðdþ subject to G ðdþ 6 0; 1;...; nc d L 6 d 6 d U ; d 2 R ndv Fg. 1. Comarson of conventonal and robust desgn otmum [1].

3 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx 3 otmzaton roblem. The otmum of the b-objectve otmzaton deends on the weght on each term n the cost functon. However, snce the man goal of ths aer s not focused on determnaton of the weghts, nterested readers can refer to [17] for more detals. The cost functon f ðl H ; r 2 HÞ n Eq. (2) can be formulated n varous ways based on engneerng alcaton tyes [8,9]. The followng are three mortant cost functon tyes for relablty based robust desgn. (1) Nomnal-the-best tye 2 2 f ðl H ; r 2 H Þw l H h t r H 1 þ w 2 ; ð3þ l H 0 h t0 r H 0 where h t and h t0 are the target nomnal value and the ntal target nomnal value of the erformance functon h(x) resectvely, and w 1 and w 2 are weghts to be determned by the desgner. To reduce the dmensonalty roblem of two objectves, each term s normalzed by the ntal value l H 0 and r H 0. (2) Smaller-the-better tye 2 2 f ðl H ; r 2 H Þw l 1 sgnðl H Þ H r H þ w 2 : ð4þ l H 0 r H 0 (3) Larger-the-better tye f ðl H ; r 2 H Þw l 2 2 H 1 sgnðl H Þ 0 r H þ w 2 : ð5þ l H r H 0 3. Three methods for relablty based robust desgn As shown n Eqs. (3) (5), the man concern of RBRDO s how accurately and effcently the statstcal moments and ther senstvtes of the erformance functon h(x) can be estmated. Analytcally, the kth statstcal moment of the erformance functon can be obtaned usng the followng ntegraton: EðfhðXÞg k Þ fhðxþg k f X ðxþdx; where f X (x) s a jont robablty densty functon (PDF) of the random arameter X. As stated before, t s ractcally mossble to calculate the statstcal moments of the erformance functon usng Eq. (6) esecally when the dmenson of the roblem s relatvely large. For numercal evaluaton of Eq. (6), three methods have been recently roosed. These methods are brefly ntroduced n the followng sectons and comared. More mortantly, senstvty analyss of statstcal moments s derved and evaluated for accuracy. It should be noted that these three methods assume that nut varables are statstcally ndeendent of each other Dmenson reducton method The dmenson reducton method [15,16,18] s a newly develoed technque to calculate statstcal moments of ð6þ the outut erformance functon. There are several DRMs deendng on the level of dmenson reducton: (1) unvarate dmenson reducton, whch s an addtve decomoston of N-dmensonal erformance functon nto one-dmensonal functons; (2) bvarate dmenson reducton, whch s an addtve decomoston of N-dmensonal erformance functon nto at most two-dmensonal functons; (3) multvarate dmenson reducton, whch s an addtve decomoston of N-dmensonal erformance functon nto at most S-dmensonal functons, where S 6 N. In ths aer, the unvarate DRM s used for comutaton of statstcal moments and ther senstvtes. Comutatonal effcency of DRM s dscussed n Secton Basc concet of unvarate dmenson reducton method In the unvarate DRM, any N-dmensonal erformance functon h(x) can be addtvely decomosed nto one-dmensonal functons as hðxþ ff^hðxþ XN 1 hðl 1 ;...; l ; x ; l þ1 ;...; l N Þ ðn 1Þhðl 1 ;...; l N Þ where l s the mean value of a random varable X and N s the number of desgn varables. For examle, f h(x) =h(x 1,x 2 ),.e., N = 2, then the unvarate addtve decomoston of h(x) s hðxþ ff^hðxþ hðx 1 ; l 2 Þþhðl 1 ; x 2 Þ hðl 1 ; l 2 Þ: Usng the unvarate DRM, one N-dmensonal ntegraton n Eq. (6) becomes N one-dmensonal ntegratons, whch wll reduce the number of functon evaluatons sgnfcantly when the number of desgn varables s large. Ths reducton of the number of functon evaluatons s exlaned n Secton The one-dmensonal numercal ntegraton can be calculated usng the moment-based ntegraton rule (MBIR) [19], whch s smlar to Gaussan quadrature [20]. Accordng to MBIR, the kth statstcal moment of a one-dmensonal functon can be obtaned as EðfhðXÞg k Þ Xn 1 w h k ðx Þ; where w are weghts, x are quadrature onts (realzatons of a random varable X) and n s the number of weghts and quadrature onts. If PDF of the desgn varables s gven, then these weghts w and quadrature onts x can be obtaned usng MBIR. For the standard normal nut random varable wth three quadrature onts, the weghts and quadrature onts are shown n Table 1 [19]. Usng Eqs. (7) and (9), the mean value and varance of the erformance functon h(x) can be obtaned as ð7þ ð8þ ð9þ

4 4 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx Table 1 Weghts and quadrature onts for standard normal Quadrature onts Weghts x 1 x 2 x 3 w 1 w 2 w 3 ffffff 3 0 ffffff where n s the number of quadrature onts and N s the number of desgn varables. If the dstrbutons of all nut desgn varables are symmetrc, e.g., normal dstrbuton or unform dstrbuton, and the number of desgn varables s odd, then the requred number of functon evaluatons s reduces to FE ðn 1ÞN þ 1: ð13þ l H E½hðXÞŠ ( ) ff E XN hðl 1 ;...;l ;X ;l þ1 ;...;l N Þ ðn 1Þhðl 1 ;...;l N Þ ff Xn j1 1 X N 1 ð10þ w j hðl 1 ;...;l ;x j ;l þ1 ;...;l N Þ ðn 1Þhðl 1 ;...;l N Þ r 2 H E½ðhðXÞ l HÞ 2 ŠE½h 2 ðxþš l 2 H ( ) ff E XN h 2 ðl 1 ;...;X ;...;l N Þ ðn 1Þh 2 ðl 1 ;...;l N Þ l 2 H ff Xn j1 1 X N 1 w j h 2 ðl 1 ;...;x j ;...;l N Þ ðn 1Þh 2 ðl 1 ;...;l N Þ l 2 H ð11þ The estmaton of statstcal moments usng the unvarate DRM nvolves two aroxmatons. As shown n Eqs. (10) and (11), the unvarate DRM aroxmates the erformance functon h(x) usng the sum of one-dmensonal functons. If hðxþ P N 1 h ðx Þ where h (x ) s any functon of x only, then the aroxmaton s exact. However, f there are off-dagonal or mxed terms, then there s some error that results from aroxmatng off-dagonal terms usng sum of one-dmensonal functons. To reduce ths error, the bvarate DRM or multvarate DRM can be used. The second aroxmaton nvolves the numercal ntegraton usng the weghts and quadrature onts. Based on Gaussan quadrature theory [20], n quadrature onts and weghts gve a degree of recson of 2n 1. Hence, f three quadrature onts and weghts for each varable are used, the numercal ntegraton error for a quadratc erformance functon wll dsaear. If the erformance functon s hghly nonlnear, then three quadrature onts may not be suffcent to estmate the moments of the erformance functon. In ths case, the error can be reduced f the number of quadrature onts s ncreased Comutatonal effcency Even though the accuracy s the most mortant concern, t s also mortant to effcently estmate statstcal moments of the erformance functon for large-scale roblems. In general, when the outut moments are estmated usng the unvarate DRM and MBIR, the number of functon evaluatons requred s FE n N þ 1; ð12þ Therefore, when the number of desgn varables s large, the reducton becomes sgnfcant comared to the number of functon evaluaton n drectly ntegratng Eq. (6), whch s n N. However, although the reducton becomes sgnfcant when N s large, the number of functon evaluatons s stll ncreasng roortonally to the number of desgn varables as shown n Eq. (13). If bvarate DRM s used to estmate the frst and second outut moments, then the number of functon evaluatons wll ncrease exonentally to FE NðN 1Þ n 2 þ n N þ 1: 2 ð14þ For examle, f the number of desgn varables s 5 and the number of quadrature onts s 3, then the number of functon evaluatons by the unvarate DRM s 16 from Eq. (12) and the number of functon evaluatons by bvarate DRM s 106 from Eq. (14). Both of the numbers are less than 3 5 = 243, whch s the requred number of functon evaluatons for the numercal ntegraton of Eq. (6) by ncludng the mxed varable terms. However, the number of functon evaluatons by the unvarate DRM s sgnfcantly less than the number of functon evaluatons for bvarate DRM. For ths reason, the unvarate DRM s used to estmate statstcal moments n ths aer Senstvty of statstcal moments To obtan a robust desgn, not only the values of the frst and second statstcal moments but also the senstvtes of these moments are needed. Usng Eq. (6) and Rosenblatt transformaton [21] from the desgn sace (x-sace) to the standard Gaussan sace (u-sace), whch can be descrbed as F X (x) =U(u), senstvtes of the mean and varance of the erformance functon wth resect to the desgn varable l can be derved as ol H ðlþ o o ðrosenblatt transformatonþ hðxþf X ðxþdx hðxðu; lþþ/ U ðuþdu ohðxðu; lþþ / U ðuþdu ð15þ

5 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx 5 ohðxðu; lþþ ox ox ðu ; l Þ / ol U ðuþdu ðindeendencyþ or 2 H ðlþ o h 2 ðxþf X ðxþdx ol2 H o h 2 ðxðu; lþþ/ ol U ðuþdu ol2 H ðrosenblatt transformatonþ ðindeendencyþ oh 2 ðxðu; lþþ / ol U ðuþdu ol2 H oh 2 ðxðu; lþþ ox ð16þ ox ðu ; l Þ / ol U ðuþdu ol2 H where u s the standard normal varable. The nut varables are assumed to be ndeendent for the dervatons of Eqs. (15) and (16). To calculate oxðu;l Þ n Eqs. (15) and (16), Rosenblatt transformaton shown n Table 2 s used. For examle, f the nut varable s normally dstrbuted, then Table 2 shows that x can be exressed as x = l + r u. Snce r s fxed and u s ndeendent of an nut mean l, ox ðu ;l Þ 1 s obtaned. For Gumbel and unform dstrbuton, the same result oxðu;l Þ 1 s obtaned from Rosenblatt transformaton. For the Lognormal and Webull dstrbuton, oxðu;l Þ can be aroxmated to be 1. By usng the nverse transformaton from u-sace to x-sace, the assumton oxðu;l Þ (11), (15) and (16) can be further aroxmated by ol H ðlþ ff Xn X N ol k j1 1 or 2 H ðlþ ol k ff Xn w j ohðxþ ox k xðl1 ;...;x j ;...;l N Þ X N w j oh2 ðxþ ox j1 1 k ðn 1Þ oh2 ðxþ ox ol2 H : k xl ol k ff 1, and Eqs. (10) and xðl1 ;...;x j ;...;l N Þ ðn 1Þ ohðxþ ox ; k xl ð17þ ð18þ Snce the unvarate DRM does not use senstvtes of the erformance functon evaluated at the quadrature onts to estmate the moments, addtonal functon evaluatons are needed for the senstvty analyss usng Eqs. (17) and (18) Performance moment ntegraton (PMI) Dervaton of erformance moment ntegraton The mult-dmensonal ntegral n Eq. (6) for statstcal moments can be rewrtten usng Rosenblatt transformaton as Eðh k ðxþþ h k ðxþf X ðx; lþdx h k ðxðu; lþþ/ U ðuþdu ð19þ whch can also be wrtten n terms of the outut dstrbuton as Eðh k ðxþþ h k f H ðh; lþdh; h k ðxðu; lþþ/ U ðuþdu ð20þ where f H (h) s PDF of a erformance functon h(x). Snce the cumulatve dstrbuton functon (CDF) of the erformance functon can be exressed n terms of the standard normal CDF usng the followng transformaton F H (h) =U(t), Eq. (20) becomes Eðh k ðxþþ h k f H ðh; lþdh h k ðt; lþ/ðtþdt ð21þ where the arametrc varable t s the dstance from the orgn n u-sace to the most robable ont () as shown n Fg. 2. Hence, the mult-dmensonal ntegral can be rewrtten by a one-dmensonal ntegral. Smlar to the unvarate DRM, the erformance moment ntegraton (PMI) makes use of three quadrature onts and weghts to aroxmate the one-dmensonal ntegraton n Eq. (21). A dfference between the two methods s that quadrature onts of the unvarate DRM le on the x -axs, whereas quadrature onts of PMI le on the locus [3,22]. Therefore, the number of quadrature onts n the unvarate DRM Table 2 Probablty dstrbuton and ts transformaton between x and u-sace Parameters PDF Transformaton Normal l = mean; r = standard devaton f ðxþ ffffffff 1 x l 0:5 2 r e ½ r Š 2 X = l + ru 2 Log-normal r 2 ln 1 þ r l ; l lnðlþ 0:5r 2 f ðxþ ffffffffff 1 ln x l 0:5 2x r e ½ r Š 2 X exðl þ ruþ Webull l vc 1 þ 1 k ; r 2 v 2 C 1 þ 2 k C 2 1 þ 1 k f ðxþ k m ðx m Þk e ðx v Þk X v½ lnðuð UÞ a ÞŠ 1 k Gumbel l m þ 0:577 a ; r ffff 6a f ðxþ ae aðx mþ e aðx mþ X m 1 a ln½ lnðuðuþþš Unform a UðUÞ 1 ffffffff R U 2 e u2 2 du. l aþb 2 ; r b a ffffff f ðxþ 1 b a ; a 6 x 6 b 12 X = a +(b a)u(u)

6 6 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx Fg. 2. Aroxmaton of CDF Usng Locus [3]. ncreases as the number of desgn varables ncreases as shown n Eq. (12), whereas the number of quadrature onts n PMI does not change snce the ntegraton s erformed n the outut sace. Snce t follows the standard normal dstrbuton, the weghts and quadrature onts n Table 1 can be used to dscretze Eq. (21) as Eðh k ðxþþ h k ðt; lþ/ðtþdt ff 1 6 hk ðt; lþ ffff t 3 þ 1 6 hk ðt; lþ ffff t 3 : þ 4 6 hk ðt; lþ t0 ð22þ By changng the order of calculaton, Eq. (22) becomes Eðh k ðxþþ h k ðt; lþ/ðtþdt ff 1 6 fhðt; lþgk ffff t 3 þ 1 6 fhðt; lþgk ffff t hk ð ffff 4 3 ; lþþ þ 4 6 fhðt; lþgk t0 6 hk ð0; lþþ 1 6 hk ð ffff 3 ; lþ: ð23þ Usng the frst order relablty method (FORM) [23,24] and locus llustrated n Fg. 2, each term n Eq. (23) can be aroxmated as two functon values at two s and a functon value at the desgn ont. The functon values at s can be obtaned usng the nverse relablty analyss PMA to maxmze hðuþ; subject to kuk ffff ð24þ 3 The otmum result of Eq. (24) s denoted as h max ffff ffff b t, 3 whch can be used to aroxmate hð 3 ; lþ n Eq. (23). The term hð ffff 3 ; lþ n Eq. (23) can be aroxmated by the otmum result obtaned by mnmzng h(u) n Eq. (24) and denoted as h mn ffff b t. The term h(0; l) n Eq. (23) 3 can be aroxmated by h(l X ), whch s the erformance functon value at the desgn ont. Hence, usng these functon values and Eq. (23), the statstcal moments of a erformance functon can be calculated as Eðh k ðxþþ ff 1 6 hmn ffff k 4 þ 6 hk ðl X Þþ 1 6 hmax ffff k: ð25þ Consequently, the mean value and varance can be estmated by l H ff 1 6 hmn ffff þ 4 6 hðl XÞþ 1 6 hmax ffff ; r 2 H ff 1 6 ðhmn ffff b t Þ 2 þ h2 ðl X Þþ 1 6 ðhmax ffff b t Þ 2 l 2 3 H : ð26þ Thus, PMI s very effcent when the number of desgn varables s relatvely large Senstvty of statstcal moments Smlar to the senstvty calculaton n DRM, from Eqs. (21), (23) and (25), the senstvtes of the mean and varance of the erformance functon wth resect to a the desgn varable l can be derved as ol H o ff 1 6 ff 1 6 hðt; lþ/ðtþdt oh mn ffff þ 4 ohðl X Þ þ oh mn ffff ox 1 ohðxþ 6 ox þ 1 ohðxþ 6 ox ohðt; lþ /ðtþdt oh max ffff ox þ 4 ohðl X Þ þ xx mn xx max ox þ 4 6 ox ohðxþ ox oh max ffff ox xlx ox ð27þ

7 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx 7 or 2 H o ff 1 6 oðh mn ffff b t Þ 2 3 h 2 ðt;lþ/ðtþdt ol2 H þ 4 6 o h mn ffff 2 ff 1 6 ox þ 1 o h max ffff 2 6 ox 1 oh 2 ðxþ 6 ox xx mn þ 1 oh 2 ðxþ 6 ox oh 2 ðl X Þ þ 1 6 ox þ 4 oh 2 ðl X Þ 6 xx max ox ol2 H ox þ 4 oh 2 ðxþ 6 ox ox ol2 H oh 2 ðt;lþ 2 o h max ffff xlx ol2 H /ðtþdt ol2 H ð28þ Snce no exlct equaton s avalable for x, t s not ossble to analytcally obtan ox. However, the term ox can be aroxmated as followng. Usng Rosenblatt transformaton n Table 2, t s clear that x s a functon of u and l wrtten as x T ðu ; l Þ where T : x! u s the transformaton. Usng ou ff 0, ox can be aroxmated by ox ff 1. The verfcaton of the aroxmaton ox ff 1 for varous dstrbutons usng the followng examle hðxþ 1 80 wth the target X 2 1 þ8x 2þ5 relablty b t = 3 and fnte dfference method wth 1% erturbaton s gven n Table 3. By usng the aroxmaton ox ff 1, whch s a smlar ff 1 n DRM, senstvtes of the mean and var- to oxðu;l Þ ance of the erformance functon wth resect to l can be obtaned as ol H ff 1 ohðxþ 6 ox or 2 H ff 1 oh 2 ðxþ 6 ox xx mn xx mn þ 4 ohðxþ 6 ox þ 1 ohðxþ xlx 6 ox þ 4 oh 2 ðxþ 6 ox þ 1 oh 2 ðxþ xlx 6 ox xx max ; xx max ol2 H ð29þ Snce the senstvtes of the erformance functon on the rght hand sde of Eq. (29) are used durng the nverse relablty analyss descrbed n Eq. (24), no addtonal functon evaluatons are requred to calculate senstvtes usng Eq. (29) Percentle dfference method (PDM) Lke PMI, PDM uses the results of the nverse relablty analyss [6,7]. PMI utlzes the functon values at two s (h max ffff and h mn ffff b t Þ obtaned from the nverse relablty 3 analyss and the functon value at the mean l X to aroxmate the multdmensonal ntegraton n Eq. (6), whereas PDM uses the dfference between the functon values at two s to reresent the varaton of the erformance functon [6]. Hence, the RBRDO formulaton usng PDM s to mnmze subject to f ðhðl X Þ; h 1 h 2 Þ PðG ðx; dþ > 0Þ 6 Uð b t Þ; 1;...; nc d L 6 d 6 d U ; d 2 R ndv and X 2 R nrv ð30þ where 1 s a rght-tal ercentle, 2 s a left-tal ercentle and, n general, =1.When 1 = 0.95 and 2 = 0.05 [6,7], h 1 and h 2 n Eq. (30) are calculated from the nverse relablty analyss wth a target relablty ndex (b t = U ( 1 ) = 1.645), that s, h 1 h max b t 1:645 and h 2 h mn b t 1:645. As shown n Fg. 3, the dea of PDM s smle and could be vewed as meanngful, but t has rather serous shortcomngs. If the erformance functon s not monotonc, t may not be ossble to use h 1 h 2 as a measurement of robustness. In a non-monotonc erformance functon case, two s obtaned from the nverse relablty analyss may not aroxmate the left-tal and rght-tal ercentle accurately because the nverse relablty analyss searches s on the surface of the hyer-shere n Table 3 Verfcaton of assumton ox 1 ol 1 ff 1 usng hðxþ 1 80 X 2 1 þ8x 2þ5 Dstrbuton x ^x a ^x 1 x 1 ^l 1 l 1 N(5, 0.3) (5.7368, ) (5.7890, ) LN(5, 0.3) (5.7923, ) (5.8443, ) Webull(5, 0.3) (5.5289, ) (5.5799, ) Gumbel(5, 0.3) (6.2823, ) (6.3362, ) U(5, 0.3) (5.5044, ) (5.5546, ) a ^x means x obtaned from 1% erturbaton of l 1. Fg. 3. Basc concet of robust desgn usng ercentle dfference method [6].

8 8 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx u-sace. For examle, f h(x) =X 2 and X N(0,1) and the target relablty b t s 1.645, then two s become and.645. Thus, two ercentle erformances h 1 and h 2 are dentcal. In contrast to PDM, PMI and the unvarate DRM show the correct moment estmaton of the erformance functon h(x) =X 2. Thus, PDM-based RBRDO may dentfy a wrong global mnmum when there are several local mnma, as demonstrated n Secton 4.2. More sgnfcantly, there s no one ercentle that can be used n PDM to dentfy all local otma correctly as shown n Secton 4.2. Senstvty of the cost functon wth resect to a desgn varable l can be calculated usng a smlar rocedure as PMI ohðl X Þ ohðxþ ox xlx oh 1 oh 2 ff oh 1 ox ff ohðxþ 3.4. Comarson ox ox oh 2 ox ohðxþ ox ox xx 1 : xx 2 ð31þ Two crtera to dentfy whch method s effectve for robust desgn otmzaton (RDO) are comutatonal effcency and accuracy of the moment estmaton. In terms of comutatonal effcency, both PMI and PDM wll requre the same number of functon evaluatons f the same nverse relablty analyss method s used. In general, f the number of desgn varables s large, Eq. (12) shows that DRM requres more functon evaluatons than PMI and PDM. However, an advantage of usng the unvarate DRM s that the unvarate DRM does not requre senstvty nformaton (.e., no search) n estmatng the moments. Hence, the unvarate DRM can reduce the number of functon evaluatons durng lne searches. The objectve of the unvarate DRM and PMI s to aroxmate the mult-dmensonal ntegraton n Eq. (6). That s, both methods attemt to transform the multdmensonal ntegraton nto a readly comutable numercal ntegraton. However, PDM does not use any numercal ntegraton, nstead t uses the dfference of ercentle erformances. Thus, PDM may yeld wrong results when the erformance functon s non-monotonc. Both PMI and PDM may have a dffculty to fnd s when the erformance functon s non-monotonc and hghly nonlnear. On the other hand, DRM may accurately estmate the moments of the erformance functon regardless of the erformance functon tye. In terms of accuracy of the moment estmaton, the unvarate DRM yelds better results n most cases than PMI. If the erformance functon s hghly nonlnear, then the unvarate DRM wth three quadrature onts may not accurately estmate the second moment. In ths case, the error can be reduced f more quadrature onts are used n the unvarate DRM. However, PMI wth more quadrature onts than 3 may not necessarly yeld more accurate results. Ths s because functon values at quadrature onts, whch are obtaned usng FORM and search, are aroxmatons. More detals of comarson wth numercal examles are gven n the followng secton. 4. Numercal examles In ths secton, four cases of comarsons are carred out usng numercal examles. In Secton 4.1, the unvarate DRM and PMI are comared n terms of accuracy and effcency n estmaton of the moments and ther senstvtes of a erformance functon. PDM s excluded n Secton 4.1 snce t cannot estmate the moments of the erformance functon. In Secton 4.2, DRM, PMI, and PDM are comared usng a one-dmensonal fourth order olynomal for dentfcaton of correct robust otmum desgn. In ths one-dmensonal roblem, PMI and the unvarate DRM wth three quadrature onts can be consdered to be the same method. In Secton 4.3, comarson of three methods s carred out usng a two-dmensonal fourth order olynomal for desgn otmzaton. In Secton 4.4, a sde mact crashworthness examle s used for the comarson of DRM and PMI n terms of the number of the functon evaluatons n a large-scale engneerng roblem Comarson of PMI and DRM for comutaton of moments and senstvtes For the frst examle, the erformance functon s h 1 ðxþ 1 X 2 1 X 2 20 ; ð32þ where X N(5, 1) for = 1, 2. As shown n Table 4, both DRM and PMI rovde good estmaton of the mean value Table 4 Comarson of the frst and second moments of Eq. (32) Mean (l H ) Varance ðr 2 H Þ PMI DRM NI a PMI DRM NI h Error (%) No. of F.E b a NI means numercal ntegraton. b means 7 functon evaluatons and 7 senstvty calculatons.

9 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx 9 Table 5 Senstvty of mean value usng PMI and DRM for Eq. (32) PMI DRM Analytc Senstvty Error (%) Addtonal no. of F.E and standard devaton n comarson wth the exact numercal ntegraton results. The reason DRM has a larger error n estmaton of standard devaton s because the erformance functon n Eq. (32) has an off-dagonal term only. As mentoned n Secton 3.1.1, f the erformance functon has off dagonal terms only, then the unvarate addtve decomostons of the moments n Eqs. (10) and (11) may contan sgnfcant errors. For ths examle, PMI yelds reasonable estmaton of the moments because the desgn varables are normally dstrbuted, whch means that the nverse relablty analyss does not requre non-lnear transformatons from x-sace to u-sace, and the erformance functon s monotonc at the gven desgn. In the same token, the senstvtes n Tables 5 and 6 have smlar errors as Table 4. The total number of functon evaluatons for PMI to evaluate the mean and standard devaton s as shown n Table 4, where the frst 7 s the number of functon evaluaton for search and the second 7 s the number of senstvty calculaton for search. The number of Table 6 Senstvty of varance usng PMI and DRM for Eq. (32) PMI DRM Analytc Senstvty Error (%) functon evaluatons for DRM s 5. Snce the desgn varables are normally dstrbuted and the number of quadrature onts s odd, Eq. (13) s used for the total number of functon evaluatons. PMI does not requre addtonal functon evaluatons for the senstvty analyss of moments because PMI uses the senstvty nformaton n search. However, DRM does requre addtonal functon evaluatons for senstvty analyss, thus the total number of functon evaluatons needs to be doubled n DRM as shown n Table 5. Snce the frst examle contans an off-dagonal term only and the desgn varables are normally dstrbuted, the second examle s modeled as h 2 ðxþ 1 ðx 1 þ X 2 5Þ 2 ðx 1 X 2 12Þ 2 ; ð33þ where X Gumbel(5,1) for = 1, 2. The erformance functon n Eq. (33) contans both off-dagonal terms and dagonal terms, and the degree of the olynomal erformance functon s 2. Therefore, t can be exected that Table 9 Senstvty of varance usng PMI and DRM for Eq. (33) PMI DRM Analytc Senstvty Error (%) Table 7 Comarson of the frst and second moments of Eq. (33) Mean (l H ) Varance ðr 2 H Þ PMI DRM NI PMI DRM NI h Error (%) No. of F.E Table 8 Senstvty of mean value usng PMI and DRM for Eq. (33) PMI DRM Analytc Senstvty Error (%) Addtonal no. of F.E

10 10 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx DRM may yeld better results for ths examle. As exected, Tables 7 9 llustrate that DRM s accurate n estmaton of the moments and ther senstvtes. On the other hand, PMI yelds somewhat larger errors n estmaton of the moments and ther senstvtes. Ths s because the desgn varables follow Gumbel dstrbuton. In such a case, the nverse relablty analyss requres nonlnear transformatons from x-sace to u-sace, whch makes the erformance functon become hghly non-lnear and the FORM error become larger. Snce the Gumbel dstrbuton s not symmetrc, Eq. (12) s used for the total number of functon evaluatons for DRM Comarson of PMI, DRM and PDM for dentfcaton of robust otmum desgn In ths secton, three methods are comared n detal for roer dentfcaton of robust otmum desgn, usng a one-dmensonal examle. RDO can be formulated to mnmze r 2 H subject to 0 6 X 6 5 ð34þ where h 3 (X)=(X 4) 3 +(X 3) 4 +10andX N(l,0.4). Agan note that n one-dmensonal roblem, PMI and the unvarate DRM wth three quadrature onts can be consdered to be the same method. Fg. 4a llustrates the shae of the erformance functon and Fg. 4b llustrates the varances obtaned from DRM and PMI and ercentle dfferences from PDM. As mentoned before, n ths examle, PMI and DRM wth three quadrature onts can be consdered to be the same method snce the desgn varable s normally dstrbuted and there s no FORM error n a one-dmensonal functon. As shown n Fg. 4b, PMI and DRM wth three quadrature onts can aroxmate the varance of the erformance functon very well. On the other hand, PDM wth varous ercentles cannot estmate the moments. More sgnfcantly, the locaton of the otmum ont changes deendng on the ercentles used. In fact, there s no one ercentle that can be used to accurately dentfy the locaton of both local mnma smultaneously n Fg. 4b. Table 10 shows that the best ercentle should be located between 2r and 3r for the left local mnmum and the best ercentle should be located between 1.645r and 2r for the rght local mnmum. In Fg. 4b, Measure for varance s used nstead of varance. It s because PDM cannot estmate the varance of the erformance functon and uses ercentle dfferences as the measure for the varance. Another roblem of usng PDM for a hghly non-lnear erformance functon such as Eq. (34) s that PDM mght not be able to dentfy whch local mnmum s the global mnmum when there s more than one local mnmum. As shown n Table 10, the results of PDM wth three dfferent ercentles ndcate that the value of the cost functon at the left mnmum n Fg. 4b s less than the value at the rght mnmum, whch s wrong. Fg. 4. Shae and varance of erformance functon. Table 10 Poston and value of otmum usng three methods for Eq. (34) PMI and DRM PDM NI 3 ts 5 ts 1r 1.645r 2r 3r Left mn. x mn r 2 H or h 1 h Rght mn. x mn r 2 H or h 1 h

11 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx 11 Fg. 5. Accuracy of PMI and DRM wth fve quadrature onts. Fg. 6. Contour of erformance functon h(x). Table 10 also shows that PMI and DRM wth three quadrature onts yelds some errors n fndng the locaton of the otmum and estmatng the value of the otmum. Ths s because the erformance functon s a olynomal of degree 4, thus three quadrature onts may not be suffcent. In ths case, DRM and PMI wth fve quadrature onts are a good oton to acheve accuracy. The accuracy of DRM and PMI wth fve quadrature onts s llustrated n Table 10 and Fg Comarson of PMI, DRM, and PDM for RBRDO For the urose of comarson among the three methods, the cost functon of the smaller-the-better tye n Secton 2.2 s used and weghts are gven as w 1 = 0 and w 2 =1. Then, RBRDO can be formulated to [7]: mnmze subject to r 2 H Pð x 1 x 2 þ 6:45 > 0Þ 6 Uð b t Þ Pð1 6 x 6 10Þ P Uðb t Þ; 1; 2 ð35þ where h(x) =(x 1 4) 3 +(x 1 3) 4 +(x 2 5) , X N(l,0.4) for = 1, 2 and b t =3. Fg. 6 llustrates the contour of the erformance functon h(x) n the formulaton (35) and Table 11 shows the roertes of the random varables. As shown n Table 12, DRM wth 5 quadrature onts shows the best result n terms of locatng the mnmum varance and the estmaton of the varance has the smallest error. DRM wth 3 quadrature onts and PMI show error n estmaton of the varance snce the erformance functon s fourth order olynomal as exlaned n Secton PDM wth 1.645r shows better result than DRM wth 3 onts and PMI. However, as shown n Table 13, the otmum desgn vares deendng on the ercentles used. For ths roblem, a ercentle close to 2.0r shows the smallest varance of the erformance functon, whch does not mean that the ercentle (2.0r) s the best for all roblems as shown n the revous examle. Table 11 Proertes of random varables of Eq. (35) Random varable Std. dev. Dstr. tye d L d d U x Normal x Normal Table 12 Otmum desgn and cost comarson for Eq. (35) d 1 d 2 r 2 H or h 1 h 2 Analytc varance DRM (3ts) DRM (5ts) PMI PDM (1.645r) Table 13 Otmum desgn and cost of Eq. (35) wth varous ercentles Percentle d 1 d 2 Percentle dfference Analytc varance 0.5r r r r r r r RBRDO for sde mact crashworthness The RBRDO model of crashworthness for vehcle sde mact shown n Fg. 7 s formulated to Fg. 7. Vehcle sde mact roblem.

12 12 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx mnmze subject to 2 M r H w 1 þ w 2 M 0 r H 0 Pðabdomen load > 1:0kNÞ6Uð b t Þ Pðuer=md=lowerVC > 0:32 m=sþ 6 Uð b t Þ Pðuer=md=lower rb deflecton > 32 mmþ 6 Uð b t Þ ð36þ Pðubc symhyss force; F > 4:0kNÞ6Uð b t Þ Pðvelocty of B-llar at md-ont > 9:9mm=msÞ 6 Uð b t Þ Pðvelocty of front door at B-llar > 15:7mm=msÞ 6 Uð b t Þ d L 6 d 6 d U ; d 2 R 9 and X 2 R 11 ;b t 2 where M s the mass of the vehcle door and the erformance functon h(x)s the lower rb deflecton. The detaled equatons for the mass of vehcle door and constrants can be found n [25]. There are eleven random arameters and nne arameters out of the eleven random arameters are desgn arameters. The desgn arameters are the thckness (d 1 d 7 ) and materal roertes of crtcal arts (d 8, d 9 ) as shown n Table 14. Tables 15 and 16 show RBRDO results usng DRM wth 3 onts and PMI where equal weghts (w 1 = w 2 = 0.5) are used. Both methods show sgnfcant reducton n the robust objectve and very good accuracy n estmaton of the varance. However, a total number of functon evaluaton ( ) n PMI to estmate the varance s much less than ( ) n DRM wth 3 onts. When equal weghts (w 1 = w 2 = 0.5) are used, two otmum desgns seem to be a lttle bt dfferent, but ths dfference s due to the error of varance estmaton and characterstc of a b-objectve otmzaton. If the objectve s changed to mnmze the varance only, that s, w 1 = 0.0 and w 2 = 1.0, then two otmum results are almost dentcal as shown n Table 17. Table 14 Proertes of desgn and random arameters of Eq. (36) Random varable Std. dev. Dstr. tye d L d d U 1. B-llar nner (mm) Normal B-llar renforce (mm) Normal Floor sde nner (mm) Normal Cross member (mm) Normal Door beam (mm) Normal Door belt lne (mm) Normal Roof ral (mm) Normal Mat. B-llar nner (GPa) Normal Mat. floor sde nner (GPa) Normal Barrer heght (mm) Normal 10th and 11th random varables are not regarded as desgn varables 11. Barrer httng (mm) Normal Table 15 RBRDO results usng DRM for sde mact roblem Intal desgn Otmum desgn Mass Var. Analytc varance Mass Var. Analytc varance No. of F.E Table 16 RBRDO results usng PMI for sde mact roblem Intal desgn Otmum desgn Mass Var. Analytc varance Mass Var. Analytc varance No. of F.E Table 17 Otmum desgn comarson for sde mact roblem d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 Intal DRM a PMI a DRM b PMI b a For both DRM and PMI, w 1 = w 2 = 0.5 s used. b For both DRM and PMI, w 1 = 0.0, w 2 = 1.0 s used.

13 I. Lee et al. / Comuters and Structures xxx (2007) xxx xxx Dscusson and concluson Three methods (PMI, PDM, and unvarate DRM) are comared n terms of effcency and accuracy for comutaton of the statstcal moments and ther senstvtes. To comare the accuracy n estmaton of the statstcal moments of the erformance functon, two olynomal erformance functons wth two desgn varables are emloyed. In ths comarson, PDM s excluded snce PDM cannot estmate the moments of the erformance functon. The comarson shows that DRM can accurately estmate the statstcal moments of the erformance functon for the desgn varables wth both non-normal and normal dstrbutons. On the other hand, PMI can accurately estmate the statstcal moments of the erformance functon for the desgn varables wth normal dstrbutons. For nonnormally dstrbuted desgn varables, PMI shows some errors snce non-lnear transformatons make the erformance functon become hghly non-lnear. For RBRDO, a hghly nonlnear erformance functon was used for comarson uroses. Both the one-dmensonal and the twodmensonal examles show that, n most cases, PMI and DRM can dentfy the otmum desgn and estmate the cost functon accurately, whereas the otmum desgn of PDM vares deendng on the ercentle used, and PDM has dentfed a wrong global mnmum. To acheve better accuracy, DRM wth fve quadrature onts can be used. PMI and PDM yeld the same effcency f the same nverse relablty analyss s used to fnd s. Non-lnearty of the erformance functon affects the total number of functon evaluatons most sgnfcantly n RBRDO usng PMI and PDM. In estmaton of the statstcal moments usng DRM, the number of desgn varables affects the total number of functon evaluatons most sgnfcantly. Hence, f the number of desgn varables s large, t s recommended to use PMI, comared to DRM, for RBRDO. Acknowledgement Research s suorted by the Automotve Research Center that s sonsored by the US Army TARDEC. References [1] Kals M, Hacker K, Lews K. A comrehensve robust desgn aroach for decson trade-offs n comlex systems desgn. ASME J Mech Des 2001;123(1):1 10. [2] Su J, Renaud JE. Automatc dfferentaton n robust otmzaton. AIAA J 1997;35(6): [3] Du X, Chen W. A most robable ont-based method for effcent uncertanty analyss. J Des Manuf Automat 2001;4(1): [4] Youn BD, Cho KK, Du L. Adatve robablty analyss usng an enhanced hybrd mean value (HMV+) method. J Struct Multdscl Otm 2005;29(2): [5] Youn BD, Cho KK, Du L. Enrched erformance measure aroach (PMA+) for relablty-based desgn otmzaton. AIAA J 2005;43(4): [6] Du X, Sudjanto A, Chen W. An ntegrated framework for otmzaton under uncertanty usng nverse relablty strategy. ASME J Mech Des 2004;126(4): [7] Mourelatos ZP, Lang J. A Methodology for Tradng-off Performance and Robustness under Uncertanty, n: Proc of ASME Des Eng Tech Conf (DETC), Paer # DETC , [8] Youn BD, Cho KK, Y K. Performance moment ntegraton (PMI) method for qualty assessment n relablty-based robust otmzaton. Mech Based Des Struct Mach 2005;33(2): [9] Chandra MJ. Statstcal Qualty Control. Boca Raton, FL: CRC Press; 2001 [Chater 3]. [10] Taguch G, Elsayed E, Hsang T. Qualty Engneerng n Producton Systems. New York: McGraw-Hll; 1989 [Chaters 2 and 3]. [11] Buranatht T, Cao J, Chen W, A Weghted Three-Pont-Based Strategy for Varance Estmaton, n: Proc of ASME Des Eng Tech Conf (DETC), Paer # DETC , [12] Ln CY, Huang WH, Jeng MC, Doong JL. Study of an assembly tolerance allocaton model based on Monte Carlo smulaton. J Mater Process Technol 1997;70:9 16. [13] Rubnsten RY. Smulaton and Monte Carlo Method. New York: John Wley and Sons; [14] Walker JR. Practcal Alcaton of Varance Reducton Technques n Probablstc Assessments, n: Proc of the Second Int Conf on Radoactve Waste Management. Wnneg, Mantoba, Canada, 1986, [15] Xu H, Rahman S. A Moment-Based Stochastc Method for Resonse Moment and Relablty Analyss, n: Proc of 2nd MIT Conf on Comut Flud Sold Mech, Cambrdge, MA, July 17 20, [16] Xu H, Rahman S. A generalzed dmenson-reducton method for mult-dmensonal ntegraton n stochastc mechancs. Int J Numer Methods Eng 2004;61(12): [17] Marler RT, Arora JS. Survey of mult-objectve otmzaton methods for engneerng. Struct Multdscl Otm 2004;26(6): [18] Du X, Huang B. Uncertanty Analyss by Dmenson Reducton Integraton and Saddleont Aroxmatons, n: Proc of ASME Des Eng Tech Conf (DETC), Paer # DETC , [19] Xu H, Rahman S. A unvarate dmenson-reducton method for mult-dmensonal ntegraton n stochastc mechancs. Probab Eng Mech 2004;19(4): [20] Atknson KE. An Introducton to Numercal Analyss. New York, NY: John Wley and Sons; 1989 [Chater 5]. [21] Rosenblatt M. Remarks on A Multvarate Transformaton. Ann Math Statst 1952;23: [22] Youn BD, Cho KK. Adatve Probablty Analyss Usng Performance Measure Aroach, n: Proc of 9th AIAA/ISSMO Sym Multdsclnary Analyss and Otmzaton, Paer # AIAA , Atlanta, Georga, Setember 4 6, [23] Haldar A, Mahadevan S. Probablty, relablty and statstcal methods n engneerng desgn. New York, NY: John Wley and Sons; [24] Palle TC, Mchael JB. Structural relablty theory and ts alcatons. Berln, Hedelberg: Srnger-Verlag; [25] Youn BD, Cho KK, Yang R-J, Gu L. Relablty-based desgn otmzaton for crash-worthness of vehcle sde mact. Struct Multdscl Otm 2004;26(3 4):