ALTERNATIVE METHODS FOR RELIABILITY-BASED ROBUST DESIGN OPTIMIZATION INCLUDING DIMENSION REDUCTION METHOD

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1 Proceengs of IDETC/CIE 00 ASME 00 Internatonal Desgn Engneerng Techncal Conferences & Computers an Informaton n Engneerng Conference September 0-, 00, Phlaelpha, Pennsylvana, USA DETC00/DAC-997 ALTERATIVE METODS FOR RELIABILITY-BASED ROBUST DESIG OPTIMIZATIO ICLUDIG DIMESIO REDUCTIO METOD Ijn Lee, Kyung K. Cho, an Lu Du Department of Mechancal & Inustral Engneerng The Unversty of Iowa Iowa Cty, IA 54, USA lee@engneerng.uowa.eu, cho@engneerng.uowa.eu, luu@engneerng.uowa.eu ABSTRACT The objectve of relablty-base robust esgn optmzaton (RBRDO s to mnmze the prouct qualty loss functon subject to probablstc constrants. Snce the qualty loss functon s usually expresse n terms of the frst two statstcal moments, mean an varance, many methos have been propose to accurately an effcently estmate the moments. Among the methos, the unvarate menson reucton metho (DRM, performance moment ntegraton (PMI, an percentle fference metho (PDM are recently propose methos. In ths paper, estmaton of statstcal moments an ther senstvtes are carre out usng DRM an compare wth results obtane usng PMI an PDM. In aton, PMI an DRM are also compare n terms of how accurately an effcently they estmate the statstcal moments an ther senstvtes of a performance functon. In ths comparson, PDM s exclue snce PDM coul not even accurately estmate the statstcal moments of the performance functon. Also, robust esgn optmzaton usng DRM s evelope an then compare wth the results of RBRDO usng PMI an PDM. Several numercal examples are use for the two comparsons. The comparsons show that DRM s effcent when the number of esgn varables s small an PMI s effcent when the number of esgn varables s relatvely large. For the nverse relablty analyss of relablty-base esgn, the enrche performance measure approach (PMA+ s use. KEYWORDS Relablty-Base Robust Desgn Optmzaton (RBRDO, Dmenson Reucton Metho (DRM, Performance Moment Integraton (PMI, Percentle Dfference Metho (PDM, Senstvty Analyss. ITRODUCTIO In recent years, several approaches to ntegrate robust esgn [,] an relablty-base esgn [,4,5] have been propose [,7,8]. The relablty-base esgn optmzaton (RBDO s a metho to acheve the confence n prouct relablty at a gven probablstc level, whle the robust esgn optmzaton (RDO s a metho to mprove the prouct qualty by mnmzng varablty of the performance functon. Snce both esgn methos mae use of uncertantes n esgn varables an other parameters, t s very natural for the two fferent methoologes to be ntegrate to evelop relabltybase robust esgn optmzaton (RBRDO methos. The prouct qualty n robust esgn can be escrbe by use of the frst two statstcal moments of a performance functon: mean an varance [9]. Thus, t s necessary to evelop methos that estmate the frst two statstcal moments of a performance functon an ther senstvtes accurately an effcently. The statstcal moments can be analytcally expresse usng a mult-mensonal ntegral. owever, t s practcally mpossble to calculate the statstcal moments of the performance functon usng the mult-mensonal ntegral. ence, there have been varous numercal attempts to estmate the moments more effcently: expermental esgn [0], the frst orer Taylor seres expanson [,,], Monte Carlo smulaton (MCS [], mportance samplng metho [], an Latn cube samplng metho [4]. The Monte Carlo smulaton coul be accurate for the moment estmaton, however t requres a very large number of functon evaluatons. Therefore, n many large-scale engneerng applcatons, t s not practcal to use the Monte Carlo smulaton. The expermental esgn also nees a large amount of computaton when the number of esgn varables s large. The frst orer Taylor seres expanson has been wely use to estmate the frst an secon statstcal moments n robust esgn. owever, the frst orer Taylor Copyrght 00 by ASME

2 seres expanson results n a large error especally when the nput ranom varables have large varatons. Ths s because the frst orer Taylor seres expanson oes not use all nformaton of the probablty ensty functons (PDF of nput ranom varables. To overcome the shortcomngs explane above, three methos have been recently propose: unvarate menson reucton metho (DRM [5,], performance moment ntegraton (PMI [8], an percentle fference metho (PDM [,7]. In ths paper, relablty-base robust esgn optmzaton usng the unvarate DRM s propose an the calculaton of the statstcal moments an ther senstvtes usng PMI s erve. In aton, the results of RBRDO usng the unvarate DRM are compare wth the robust esgns obtane usng PMI an PDM. Both DRM an PMI estmate the statstcal moments. On the other han, n PDM, the robustness s acheve through a esgn objectve n whch the varaton of the esgn performance s approxmately evaluate through the percentle performance fference between the rght an left tals of performance strbuton []. Thus, three methos can be compare n terms of how accurately these methos can fn an optmum esgn to mnmze the varance of the performance functon. ence, n ths paper, three methos are evaluate by comparng the varances at the optmum esgn. PMI an DRM are also compare n terms of how accurately an effcently estmate the statstcal moments of the performance functon. For the comparsons, several examples nclung one-mensonal, two-mensonal performance functons an a large-scale engneerng problem are use. These comparsons llustrate that the unvarate DRM s the most accurate an effcent metho when the number of esgn varables s small an PMI s a better opton when the esgn menson s relatvely large. For the nverse relablty analyss of RBDO, the enrche performance measure approach (PMA+ [5] an ts numercal metho, enhance hybr mean value (MV+ [4] are utlze. Due to the varaton of esgn varables an other parameters, the performance functon h( X also has varaton. Thus, n robust esgn, the robustness of a esgn objectve can be acheve by smultaneously optmzng the mean performance μ an mnmzng the performance varance []. In other wors, the goal of robust esgn s to fn the most nsenstve esgn to the varaton of the esgn varables an other parameters. Snce robust esgn s funamentally conserng the varatons of the esgn varables an other parameters, t s very natural to ntegrate robust esgn an relablty-base esgn n one formulaton. Ths esgn optmzaton s calle relablty-base robust esgn optmzaton (RBRDO an can be formulate to mnmze f ( μ, subject to PG ( ( X ; 0 Φ( β, =,, nc ( t L U nv nrv, R an X R where f ( μ, s the cost functon, = μ(x s the esgn vector, X s the ranom vector, an G s the th probablstc constrant. Quanttes nc, nv, nrv an β t are the number of probablstc constrants, esgn varables, ranom varables, an the th target relablty nex, respectvely. Detale explanaton about RBDO can be foun n [,4,5]. In ths paper, the enrche performance measure approach (PMA+ [5] s ntrouce to perform the nverse relablty analyss of the constrants. Fgure compares a conventonal esgn optmum wth a robust esgn optmum for a one-mensonal performance functon. Wth the same varablty of a esgn varable, the robust optmum shows less varaton of the performance functon h( X than the conventonal esgn optmum.. FUDAMETAL COCEPT OF ROBUST DESIG. Relablty-Base Robust Desgn In general, a conventonal esgn optmzaton problem can be formulate to mnmze h( X subject to G ( X 0, =,, nc ( L U X X X, nv X R where h( X s the cost functon, G s the th constrant, an X s the esgn varable vector; an nc an nv are the number of constrants an esgn varables, respectvely. The optmum esgn of the conventonal optmzaton problem s the etermnstc optmum that coul be senstve to the varaton of nput esgn varables an other parameters. Fgure. Comparson of Conventonal an Robust Desgn Optmum []. Three Types of Cost Functon Snce the cost functon n Equaton ( epens on μ an for robust optmum esgn n RBRDO, t s a b- Copyrght 00 by ASME

3 objectve optmzaton problem. The optmum of the bobjectve optmzaton epens on the weght on each term n the cost functon. owever, the man goal of ths paper s not focuse on etermnaton of the weghts. Intereste reaers can refer to [7] for more etals. The cost functon f ( μ, n Equaton ( can be formulate n varous ways base on engneerng applcaton types [8,9]. The followng are three mportant cost functon types for relablty base robust esgn. ( omnal-the-best Type μ ht f( μ, = w( + w( μ h 0 t0 0 where an are the target nomnal value an the ntal h t h t0 target nomnal value of the performance functon h( X respectvely, an w an w are weghts to be etermne by the esgner. To reuce the mensonalty problem of two objectves, each term s normalze by the ntal value μ 0 an 0. ( Smaller-the-Better Type μ f( μ, = w sgn( μ ( + w( (4 μ 0 0 ( Larger-the-Better Type μ 0 f( μ, = w sgn( μ ( w( μ + (5. TREE METODS FOR RELIABILITY-BASED ROBUST DESIG As shown n Equatons (-(5, the man concern of RBRDO s how accurately an effcently the statstcal moments an ther senstvtes of the performance functon h( X can be estmate. Analytcally, the th statstcal moment of the performance functon can be obtane usng the followng ntegraton ({ ( } E h X { h( X} f ( x x ( = X where f X ( x s a jont probablty ensty functon (PDF of the ranom parameter X. As state before, t s practcally mpossble to calculate the statstcal moments of the performance functon usng Equaton ( especally when the menson of the problem s relatvely large. For numercal evaluaton of Equaton (, three methos have been recently propose. These methos are brefly ntrouce n the followng sectons an compare. More mportantly, senstvty analyss of statstcal moments s erve an evaluate for accuracy. It s note that these three methos assume that nput varables are statstcally nepenent of each other. 0 (. Dmenson Reucton Metho (DRM The menson reucton metho [5,,8] s a newly evelope technque to calculate statstcal moments of the output performance functon. There are several DRMs epenng on the level of menson reucton: ( unvarate menson reucton, whch s an atve ecomposton of - mensonal performance functon nto one-mensonal functons; ( bvarate menson reucton, whch s an atve ecomposton of -mensonal performance functon nto at most two-mensonal functons; ( multvarate menson reucton, whch s an atve ecomposton of -mensonal performance functon nto at most S-mensonal functons, where S. In ths paper, the unvarate DRM s use for computaton of statstcal moments an ther senstvtes. Computatonal effcency of DRM s scusse n Secton..... Basc Concept of Unvarate Dmenson Reucton Metho In the unvarate DRM, any -mensonal performance functon h( X can be atvely ecompose nto onemensonal functons as h( X h ( X h( μ,, μ, x, μ +,, μ = (7 ( h( μ,, μ where μ s the mean value of a ranom varable X an s the number of esgn varables. For example, f h( X = h( x, x, that s =, then the unvarate atve ecomposton of h( X s h( X h ( X h( x, μ + h( μ, x h( μ, μ (8 Usng the unvarate DRM, one -mensonal ntegraton n Equaton ( becomes one-mensonal ntegratons, whch wll reuce the number of functon evaluatons sgnfcantly when the number of esgn varables s large. Ths reucton of the number of functon evaluatons s explane n Secton... The one-mensonal numercal ntegraton can be calculate usng the moment-base ntegraton rule (MBIR [9], whch s smlar to Gaussan quarature [0]. Accorng to MBIR, the th statstcal moment of a one-mensonal functon can be obtane as n E({ h( X} = wh ( x (9 = where w are weghts, x are quarature ponts an n s the number of weghts an quarature ponts. If PDF of the esgn varables s gven, then these weghts w an quarature ponts x can be obtane usng MBIR. For the stanar normal nput ranom varable wth three quarature ponts, the weghts an quarature ponts are shown n Table [9]. Table. Weghts an Quarature Ponts for Stanar ormal Quarature Ponts Weghts Copyrght 00 by ASME

4 x x x w w w 0 Usng Equatons (7 an (9, the mean value an varance of the performance functon h( X can be obtane as μ Eh [ ( X] = j= = 4 E{ h( μ,, μ, X, μ,, μ n + ( h( μ,, μ } wh( μ,, μ, x, μ,, μ j j + ( h( μ,, μ μ μ E[( h( X ] = E[ h ( X] E{ h ( μ,, X,, μ = ( h ( μ,, μ} μ n j j wh ( μ,, x,, μ j= = ( h ( μ,, μ μ (0 ( The estmaton of statstcal moments usng the unvarate DRM nvolves two approxmatons. As shown n Equatons (0 an (, the unvarate DRM approxmates the performance functon h( X usng the sum of one-mensonal functons. If h( X = h( x where h( x s any functon of = x only, then the approxmaton s exact. owever, f there are off- agonal or mxe terms, then there s some error that results from approxmatng off-agonal terms usng sum of one-mensonal functons. To reuce ths error, the bvarate DRM or multvarate DRM can be use. The secon approxmaton nvolves the numercal ntegraton usng weghts an quarature ponts. Base on Gaussan quarature theory [0], n quarature ponts an weghts gve a egree of precson of n. ence, f three quarature ponts an weghts for each varable are use, the numercal ntegraton error for a quaratc performance functon wll sappear. If the performance functon s hghly nonlnear, that s, the orer of the performance functon s more than, then three quarature ponts may not be suffcent to estmate the moments of the performance functon. In ths case, the error can be reuce f the number of quarature ponts s ncrease... Computatonal Effcency Even though the accuracy s the most mportant concern, t s also mportant to effcently estmate statstcal moments of the performance functon for large-scale problems. In general, when the output moments are estmate usng the unvarate DRM an MBIR, the number of functon evaluatons requre s FE = n + ( where n s the number of quarature ponts an s the number of esgn varables. Specfcally, when the strbutons of all nput esgn varables are symmetrc, e.g. normal strbuton or unform strbuton, an the number of esgn varables s o, then the requre number of functon evaluatons s reuces to FE = ( n + ( Therefore, when the number of esgn varables s large, the reucton becomes sgnfcant compare to the number of functon evaluaton n rectly ntegratng Equaton (, whch s n. owever, although the reucton becomes sgnfcant when s large, the number of functon evaluatons s stll ncreasng proportonally to the number of esgn varables as shown n Equaton (. If bvarate DRM s use to estmate the frst an secon output moments, then the number of functon evaluatons wll ncrease exponentally to ( FE = n + n + (4 For example, f the number of esgn varables s 5 an the number of quarature ponts s, then the number of functon evaluatons by the unvarate DRM s from Equaton ( an the number of functon evaluatons by bvarate DRM s 0 from Equaton (4. Both of the numbers are less than 5 = 4, whch s the requre number of functon evaluatons for the numercal ntegraton of Equaton ( by nclung the mxe varable terms. owever, 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 use to estmate statstcal moments n ths paper... Senstvty of Statstcal Moments For the purpose of robust esgn, not only the values of the frst an secon statstcal moments but also senstvtes of these moments are neee. From Equatons (, (0 an (, an Rosenblatt transformaton from x-space to the stanar normal u-space [], senstvty of mean an varance of the performance functon wth respect to a esgn varable μ can be erve as ( μ = h( ( ; φu ( xuμ u u = h( ( ; φu ( xuμ u u (5 x( u; μ = h( ( ; φ U ( xuμ u u x 4 Copyrght 00 by ASME

5 ( μ = h ( ( ; φu ( xuμ u u = h ( ( ; φu ( xuμ u u x( u; μ = h ( ( ; φu ( xuμ u u x μ where u s the stanar normal varable. If an nput varable s normally strbute, then Rosenblatt transformaton shows that x can be obtane as x = μ + u. Snce s fxe x( u; μ an u s nepenent of an nput mean μ, can be approxmate as. By usng the nverse transformaton an x( u; μ assumpton, Equatons (5 an ( can be approxmate by n μ ( μ j h( h( w x ( x (7 x j x μ j= = x= ( μ,, x,, μ x= μ n ( μ j h ( x w μ j= = x j x= ( μ,, x,, μ h ( x ( x x=μ ( (8 Snce the unvarate DRM oes not use senstvtes of the performance functon evaluate at the quarature ponts to estmate the moments, atonal functon evaluatons are neee for the senstvty analyss usng Equatons (7 an (8.. Performance Moment Integraton (PMI The mult-mensonal ntegral n Equaton ( for statstcal moments can be rewrtten usng Rosenblatt transformaton as Eh ( ( X = h( x f ( x; μ x = X h ( xuμ ( ; φ ( u u U (9 whch can also be wrtten n terms of the output strbuton as Eh ( ( X = h( x( u; μ φ ( u u = h f ( h; μ h (0 U where f ( h s a probablty ensty functon (PDF of a performance functon h( X. Snce CDF of the performance functon can be approxmate n terms of the stanar normal CDF by FORM, usng the followng transformaton F ( h =Φt (, Equaton (0 becomes Eh ( ( X = h f ( h; μ h= h ( t; μ φ( tt ( where the parametrc varable t s the stance from the orgn n u-space to most probable pont (MPP as shown n Fgure. ence, the mult-mensonal ntegral can be approxmate by a one-mensonal ntegral. Smlar wth the unvarate DRM, the performance moment ntegraton (PMI maes use of three quarature ponts an weghts to approxmate the one-mensonal ntegraton n Equaton (. A fference between the two methos s that quarature ponts of the unvarate DRM le on the x -axs, whereas quarature ponts of PMI le on the MPP locus [,]. Therefore, the number of quarature ponts n the unvarate DRM ncreases as the number of esgn varables ncreases as shown n Equaton (. Fgure. Approxmaton of CDF Usng MPP Locus [] Snce t follows the stanar normal strbuton, the weghts an quarature ponts n Table can be use to scretze Equaton ( as Eh ( ( X = h( t; μ φ( tt ( 4 h (; t μ + h (; t μ + h (; t μ t= t= 0 t= By changng the orer of calculaton, Equaton ( becomes 5 Copyrght 00 by ASME

6 Eh ( ( X = h( t; μ φ( tt 4 {(; htμ} + {(; htμ} + {(; htμ} t= t= 0 t= 4 = h ( ; μ + h (0; μ + h ( ; μ Usng the frst orer relablty metho (FORM [,4] an MPP locus llustrate n Fgure, each term n Equaton ( can be approxmate as two functon values at two MPPs an a functon value at the esgn pont. The functon values at MPPs can be obtane usng the nverse relablty analyss PMA, to maxmze h( U (4 subject to U = The optmzaton result of Equaton (4 s enote as h( x 4 h( x h( x + + x x x mn max x=xmpp x=μx x=xmpp ( h ( x 4 h ( x h ( x h β =, max t whch can be use to approxmate h( ; μ. The term h( ; μ n Equaton ( can be approxmate as the optmal cost obtane by mnmzng h( U n Equaton (4 mn an enote as h. The term h (0; μ can be approxmate β = t as h( μ X, whch s the performance functon value at the esgn pont. ence, usng these functon values an Equaton (, the statstcal moments of a performance functon can be calculate as Eh ( ( X = h( t; μ φ( tt 4 { ht ( ; μ} + { ht ( ; μ} + { ht ( ; μ} t= t= 0 t= (5 4 = h ( ; μ + h (0; μ + h ( ; μ mn 4 max ( h + h ( μ + ( h βt = X βt = Consequently, the mean value an varance can be estmate by mn 4 max μ h + h( μ h βt = X + βt = ( mn 4 max ( h + h ( μ ( h μ βt = X + βt = Thus, PMI s very effcent when the number of esgn varables s relatvely large. x( u; μ By usng the assumpton that s use to obtan senstvtes n Equatons (7 an (8, senstvtes of the mean an varance of the performance functon wth respect to μ are obtane as + + x x x mn max x=xmpp x=μx x=xmpp μ (7 Snce the senstvtes of the performance functon on the rght han se of Equaton (7 are use urng the nverse relablty analyss, no atonal functon evaluatons are requre to calculate senstvtes usng Equaton (7.. Percentle Dfference Metho (PDM Smlar wth PMI, PDM also uses the results of the nverse relablty analyss [,7]. PMI utlzes the functon max values at two MPPs ( h an h mn β = t β t = obtane from the nverse relablty analyss an the functon value at the mean μ X to approxmate the multmensonal ntegraton n Equaton (, whereas PDM uses the fference between the functon values at two MPPs to represent the varaton of the performance functon []. ence, the RBRDO formulaton usng PDM s to mnmze f( h( μ X, hp h p subject to PG ( ( X ; 0 Φ( β, =,, nc (8 L U nv nrv, R an X R where p s a rght-tal percentle, p s a left-tal percentle an, n general, p + p =. When p = 0.95 an p = 0.05 [,7], h p an h p n Equaton (8 are calculate from the nverse relablty analyss wth a target relablty nex.45 max ( β t =.45, that s, h = h an. Fgure p β h t =.45 p = h mn β t =.4 5 llustrates basc concept of PDM. t Fgure. Basc Concept of Robust Desgn Usng Percentle Dfference Metho [] Copyrght 00 by ASME

7 The ea of PDM was smple an coul be vewe as meanngful, but t has rather serous shortcomngs. If performance functon s not monotonc, t may not be possble to use hp h p as a measurement of robustness. In a nonmonotonc performance functon case, two MPPs obtane from the nverse relablty analyss may not approxmate the left-tal an rght-tal percentle accurately because the nverse relablty analyss searches MPPs on the surface of the hyperu-space. For example, f hx ( = X an sphere n X ~ (0, an the target relablty s β t =.45, then two MPPs become.45 an.45. Thus, two percentle performances h p an h p are entcal. In contrast to PDM, PMI an the unvarate DRM show the correct moment estmaton of the performance functon hx ( = X. Thus, PDM-base RBRDO may entfy a wrong global mnmum when there are several local mnma, as shown n Secton 4.. More sgnfcantly, there s no one percentle that can be use n PDM to entfy all local optma correctly as shown n Secton Comparson Two crtera to entfy whch metho s useful for robust esgn optmzaton are computatonal effcency an accuracy of the moment estmaton. In terms of computatonal effcency, both PMI an PDM wll show the same number of functon evaluatons upon analyss f the same analyss metho s use. In general, f there are a large number of esgn varables, Equaton ( shows that DRM requres more functon evaluatons than PMI an PDM. owever, an avantage of usng the unvarate DRM s that the unvarate DRM oes not requre senstvty nformaton (.e., no MPP search n estmatng the moments. ence, the unvarate DRM can reuce the number of functon evaluatons urng lne searches. The objectve of PMI an the unvarate DRM s to approxmate the mult-mensonal ntegraton n Equaton (. That s, both methos attempt to transform the multmensonal ntegraton nto a realy computable numercal ntegraton. owever, PDM oes not use any numercal ntegraton, nstea t uses the fference of percentle performances. Thus, PDM may yel wrong results when a performance functon s non-monotonc. Both PMI an PDM may have a ffculty to fn MPPs when the performance functon s non-monotonc an the orer of the performance functon s greater than or equal to two. On the other han, DRM may accurately estmate the moments of the performance functon regarless of the performance functon type. In terms of accuracy of the moment estmaton, the unvarate DRM yels better results n most cases than PMI. If a performance functon s hghly nonlnear, then the unvarate DRM wth three quarature ponts may not be able to accurately estmate the secon moment. In ths case, the error can be reuce f more quarature ponts are use n the unvarate DRM. owever, PMI wth more quarature ponts than may not necessarly yel more accurate results. Ths s because functon values at quarature ponts, whch are obtane usng FORM an MPP search, are approxmatons. More etals of comparson wth numercal examples are gven n the followng secton. 4. UMERICAL EXAMPLES In ths secton, four comparsons are performe usng numercal examples. In Secton 4., PMI an the unvarate DRM are compare n terms of accuracy an effcency n estmaton of the moments an ther senstvtes of a performance functon. PDM s exclue n Secton 4. snce t cannot estmate the moments of a performance functon. In Secton 4., DRM, PMI, an PDM are compare usng a onemensonal fourth orer polynomal for entfcaton of correct robust optmum esgn. In ths one-mensonal problem, PMI an the unvarate DRM wth three quarature ponts can be consere to be the same metho. In Secton 4., comparson of three methos s carre out usng a twomensonal fourth orer polynomal for esgn optmzaton. In Secton 4.4, a se mpact crashworthness example s use for the comparson of DRM an PMI n terms of the number of the functon evaluatons n large-scale engneerng problem. 4. Comparson of PMI an DRM for Computaton of Moments an Senstvtes For the frst example, the performance functon s ( X X h X = (9 0 where X ~ (5, for =,. As shown n Table, both DRM an PMI prove goo estmaton of the mean value an stanar evaton n comparson wth the exact numercal ntegraton results. The reason DRM has a larger error n estmaton of stanar evaton s because the performance functon n Equaton (9 has an off-agonal term only. As mentone n Secton.., f the performance functon has off agonal terms only, then the unvarate atve ecompostons of the moments n Equatons (0 an ( may contan sgnfcant errors. Tab le. Comparson of the Frst an Secon Moments of Eq. (9 Mean ( μ Stanar Devato n ( PMI D RM I* PMI DRM I* h Err or, % o.of F.E. 7+7** * I means numercal ntegraton. ** 7+7 means 7 functon evaluatons an 7 senstvty calculatons. 7 Copyrght 00 by ASME

8 For ths example, PMI yels reasonable estmaton of the moments because the esgn varables are normally strbute, whch means that the nverse relablty analyss oes not requre non-lnear transformatons from x-space to u- space, an the performance functon s monotonc at the gven esgn. In the same toen, the senstvtes n Table an 4 have smlar errors as Table. The total number of functon evaluatons for PMI to evaluate the mean an stanar evaton s 7+7 as shown n Table, where the frst 7 s the number of functon evaluaton for MPP search an the secon 7 s the number of senstvty calculaton for MPP search. The number of functon evaluatons for DRM s 5. Snce the esgn varables are normally strbute an the number of quarature ponts s o, Equaton ( s use for the total number of functon evaluatons. PMI oes not requre atonal functon evaluatons for the senstvty analyss of moments because PMI uses the senstvty nformaton n MPP search. owever, DRM oes requre atonal functon evaluatons for senstvty analyss, thus the total number of functon evaluatons nees to be ouble n DRM. Table. Senstvty of Mean Value Usng PMI an DRM for Eq. (9 PMI DRM Analytc ( ( ( ( ( ( Senstvty Error, % Atonal 0 + o. of F.E. Tab le 4. Senstvty of Varance Usng PMI an DRM for Eq. (9 PMI DR M Analytc ( ( ( ( ( Senstvty Error, % Snce the frst example contans an off-agonal term only an the esgn varables are normally strbute, the secon example s moele as ( X + X 5 ( X X h ( X = (0 0 0 where X ~ Gumbel (5, for =,. The performance functon n Equaton (0 contans both off-agonal terms an agonal terms, an the egree of the performance functon s. Therefore, t can be expecte that DRM may yel better results for ths example. As expecte, Tables 5, an 7 llustrate that DRM s accurate n estmaton of the moments an ther senstvtes. owever, PMI yels somewhat larger errors n estmaton of the moments an ther senstvtes. Ths s because the esgn varables follow Gumbel strbuton. In such a case, the nverse relablty analyss requres non-lnear transformatons from X-space to U-space, whch maes the performance functon become hghly non-lnear an the FORM error become larger. Snce Gumbel strbuton s not symmetrc, Equaton ( s use for the total number of functon evaluatons for DRM. Tabl e 5. Comparson of the Frst an Secon Moments of Eq. (0 Mean ( μ Stanar Devaton ( PMI DRM I PMI DRM I h Error, % o.of F.E Tab le. Senstvty of Mean Value Usng PMI an DRM for Eq. (0 PMI DRM Analytc ( ( ( μ ( ( ( Senstvty Error, % Atonal 0 + o.o f F.E. Table 7. Senstvty of Varance Usng PMI an DRM for Eq. (0 PMI DRM Analytc ( ( ( ( ( ( Senstvty Error, % Comparson of PMI, DRM an PDM for Ientfcaton of Robust Optmum Desgn In ths secton, three methos are compare n etal for proper entfcato n of robust optmum esgn, usng a onemensonal ex ample. RBRDO can be formulate to mnmze 4 subject to h ( X = ( X 4 + ( X + 0 ( where X ~ ( μ,0.4. Fgure 4 (a llustrates the shape of the performance functon an Fgure 4 (b llustrates the varances obtane from DRM an PMI an percentle fferences from PDM. As mentone before, n ths example, PMI an DRM wth three quarature ponts can be consere to be the same metho snce the esgn varable s normally strbute an there s no FORM error n a one-mensonal functon. As shown n Fgure 4 (b, PMI an DRM wth three quarature ponts can approxmate the varance of the performance functon very well. On the other han, PDM wth varous percentles cannot estmate the moments. More sgnfcantly, the locaton of the optmum pont changes accorng to the percentles use. In fact, there s no one percentle that can be use to accurately entfy the locaton of both local mnma smultaneously n Fgure 4 (b. Table 8 shows that the best 8 Copyrght 00 by ASME

9 percentle shoul be locate between an for the left local mnmum an the best percentle shoul be locate between.45 an for the rght local mnmum. In Fgure 4 (b, Measure for Varance s use nstea of varance. It s because PDM cannot estmate the varance of the performance functon an uses percentle fferences as the measure for the varance. Another problem of usng PDM for a hghly non-lnear performance functon such as Equaton ( s that PDM mght not be able to entfy whch local mnmum s the global mnmum when there s more than one mnmum. As shown n Table 8, the results of PDM wth three fferent percentles ncate that the value of the cost functon at the left mnmum n Fgure 4 (b s less than the value at the rght mnmum, whch s wrong. to acheve accuracy. The accuracy of DRM an PMI wth fve quarature ponts s llustrate n Table 8 an Fgure 5. Table 8. Poston an Valu e of Optmum Usng Three Methos for Eq.( PMI an DRM PDM I pts 5 pts.45 x mn Left or Mn h h Rght Mn. p x mn p or h h p p (a Performance Functon ( h ( X (b Measure for Varance ( Fgure 4. Shape an Varance of Performance Functon Table 8 also shows that PMI an DRM wth three quarature ponts yels some errors n fnng the locaton of the optmum an estmatng the value of the optmum. Ths s because the performance functon s a polynomal of egree 4, thus three quarature ponts may not be suffcent. In ths case, DRM an PMI wth fve quarature ponts are a goo opton Fgure 5. Accuracy of PMI an DRM wth Fve Ponts 4. Desgn Comparson of PMI, DRM, an PDM for RBRDO For the purpose of comparson among the three methos, the cost functon of the Smaller-the-Better type n Secton. s use an weghts are gven as w = 0 an w =. Then, RBRDO can be formulate to [7] mnmze subject to P( x x Φ( βt P( x 0 Φ ( βt, =, ( 4 where ( x = ( x 4 + ( x + ( x an X j ~ ( μ j,0.4 for j =, an β t = Fgure llustrates the contour of the performance functon h( X n the formulaton ( an Table 9 shows the propertes of the ranom varables. 9 Copyrght 00 by ASME

10 Fgure. Contour of performance functon h( X As shown n Table 0, DRM wth 5 quarature ponts shows the best result n terms of locatng the mnmum varance an the estmaton of the varance has the smallest error. DRM wth quarature ponts an PMI show error n estmaton of the varance snce the performance functon s 4 th orer polynomal as explane n Secton... PDM wth.45 shows better result than DRM wth ponts an PMI. owever, as shown n Table, the optmum esgn vares accorng to the percentles use. For ths problem, a percentle close to.0 shows the smallest varance of the performance functon, whch oes not mean that the percentle (.0 s the best for all problems as shown n the prevous example. 4.4 RBRDO for Se Impact Crashworthness The RBRDO moel of crashworthness for vehcle se mpact shown n Fgure 7 s formulate to M mnmze ( w + w M o 0 subject to P(abomen loa.0 Φ( β t P(upper/m/lower VC 0. m/ s Φ( β t P(upper/m/lower rbeflecton mm Φ( β t ( P(pubc symphyss force, F 4.0 Φ( β t P(velocty of B-pllar at m-pont 9.9 mm/ ms Φ( β t P(velocty of front oor at B-pllar 5.7 mm/ ms Φ( β t L U 9, R anx R, βt = where M s mass of the vehcle oor an the performance functon h( x s the lower rb eflecton. The etale equatons for the mass of vehcle oor an constrants can be foun n [5]. There are eleven ranom parameters an nne parameters out of the eleven ranom parameters are esgn parameters. The esgn parameters are the thcness ( 7 an materal propertes of crtcal parts (, as shown n Table. 8 9 Table 9. Propertes of Ranom Varables of Eq. ( Ranom St Dstr. Varable Dev. Type L U x ormal x ormal Table 0. Optmum Desgn an Cost Comparson for Eq. ( or h h p p Analytc Varance DRM (pts DRM (5pts PMI PDM ( Table. Optmum Desgn an Cost of Eq. ( wth Varous Percentles Percentle Percentle Analytc Dfference Varance Fgure 7. Vehcle Se Impact Problem Table. Propertes of Desgn an Ranom Parameters of Eq. ( Ranom Varable St Dstr. Dev. Type L U. B-pllar nner (mm 0.00 ormal B-pllar renforce (mm 0.00 ormal Floor se nner (mm 0.00 ormal Cross member (mm 0.00 ormal Door beam (mm 0.00 ormal Door belt lne (mm 0.00 ormal Roof ral (mm 0.00 ormal Mat. B-pllar nner (GPa 0.00 ormal Mat. Floor se nner (GPa 0.00 ormal Barrer heght (mm ormal 0 th an th ranom. Barrer httng (mm ormal varables are not regare as esgn varables 0 Copyrght 00 by ASME

11 Table an 4 show RBRDO results usng DRM wth ponts an PMI when equal weghts ( w = w = 0.5 are use. Both methos show sgnfcant reucton n the robust objectve an very goo accuracy n estmaton of the varance. owever, a total number of functon evaluaton (54+54 n PMI to estmate the varance s much less than (09+95 n DRM wth ponts. Two optmum esgns when equal weghts ( w = w = 0.5 are use seem to be a lttle bt fferent, but ths fference results from the error of varance estmaton an characterstc of a b-objectve optmzaton. If the objectve s change to mnmze the varance only, that s, w =0.0an w =.0, then two optmum results are almost entcal as shown n Table 5. Table. RBRDO Results Usng DRM for Se Impact Problem Intal Desgn Optmum Desgn Analytc Analytc o. of Mass Var. Mass Var. Varance Varance F.E Table 4. RBRDO Results Usng PMI for Se Impact Problem Intal Desgn Optmum Desgn Analytc Analytc o. of Mass Var. Mass Var. Varance Varance F.E Table 5. Optmum Desgn Comparson for Se Impact Problem Intal DRM* PMI* DRM** PMI** * For both DRM an PMI, w = w = 0.5 s use. ** For both DRM an PMI, w = 0.0, w =.0 s use. 5. DISCUSSIOS AD COCLUSIO Three methos (PMI, PDM, an unvarate DRM are compare n terms of effcency an accuracy of computaton of statstcal moments an ther senstvtes. To compare the accuracy n estmaton of the statstcal moments of a performance functon, two polynomal performance functons wth two esgn varables are employe. In ths comparson, PDM s exclue snce PDM cannot estmate the moments of a performance functon. The comparson shows that DRM can accurately estmate the statstcal moments of the performance functon of esgn varables wth both non-normal an normal strbutons. On the other han, PMI can accurately estmate the statstcal moments of the performance functon of esgn varables wth normal strbutons. For non-normally strbute esgn varables, PMI shows some errors snce non-lnear transformatons mae the performance functon become hghly non-lnear. For RBRDO, a hghly nonlnear performance functon was aopte for comparson purposes. Both the one-mensonal an the two-mensonal examples show that, n most cases, PMI an DRM can entfy the optmum esgn an estmate the cost functon accurately, whereas the optmum esgn of PDM vares accorng to the percentle, an PDM has entfe a wrong global mnmum. To acheve better accuracy, DRM wth fve quarature ponts can be use. PMI an PDM yel the same effcency f the same nverse relablty analyss s use to fn MPPs. on-lnearty of the performance functon affects the total number of functon evaluatons most sgnfcantly n RBRDO usng PMI an PDM. In estmaton of the statstcal moments usng DRM, the number of esgn varables affects the total number of functon evaluatons most sgnfcantly. ence, f the number of esgn varables s large, t s recommene to use PMI, compare to DRM, for RBRDO.. ACKOWLEDGEMET Research s supporte by the Automotve Research Center that s sponsore by the U.S. Army TARDEC. 7. REFERECES. Kals, M., acer, K., an Lews, K., A Comprehensve Robust Desgn Approach for Decson Trae-Offs n Complex Systems Desgn, ASME Journal of Mechancal Desgn, Vol., o., pp. -0, 00.. Su, J., an Renau, J. E., Automatc Dfferentaton n Robust Optmzaton, AIAA Journal, Vol. 5, o., pp , Du, X., an Chen, W., A Most Probable Pont- Base Metho for Effcent Uncertanty Analyss, Journal of Desgn an Manufacturng Automaton, Vol. 4, o., pp. 47-, Youn, B. D., Cho, K. K., an Du, Lu, Aaptve Probablty Analyss Usng An Enhance ybr Mean Value (MV+ Metho, Journal of Structural an Multscplnary Optmzaton, Vol. 9, o., pp. 4-48, Youn, B. D., Cho, K. K., an Du, L., Enrche Performance Measure Approach (PMA+ for Relablty-Base Desgn Optmzaton, AIAA Journal, Vol. 4, o. 4, pp , Du, X., Sujanto, A., an Chen, W., An Integrate Framewor for Optmzaton Uner Uncertanty Usng Inverse Relablty Strategy, ASME Journal of Mechancal Desgn, Vol, o. 4, pp , Mourelatos, Z. P. an Lang, J., A Methoology for Trang-off Performance an Robustness uner Uncertanty, Proceengs of ASME Desgn Engneerng Techncal Conferences (DETC, Paper # DETC , Youn, B. D., Cho, K. K., an Y, K., "Performance Moment Integraton (PMI Metho for Qualty Copyrght 00 by ASME

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