Moving Least Square Method for Reliability-Based Design Optimization

Transcription

1 Movng Leas Square Mehod for Relably-Based Desgn Opmzaon K.K. Cho, Byeng D. Youn, and Ren-Jye Yang* Cener for Compuer-Aded Desgn and Deparmen of Mechancal Engneerng, he Unversy of Iowa Iowa Cy, IA E-mal: *Ford Research Laboraory Dearborn, MI Absrac A new relably-based desgn opmzaon (RBDO) mehod s presened, usng a movng lease square (MLS) mehod and performance measure approach (PMA). hs mehod wll provde so-called 6-sgma opmum desgn. Based on he leas square mehod, he MLS mehod s nroduced o beer approxmae mplc responses by mposng a varable wegh over a compac suppor. In addon, desgn sensvy daa s ncorporaed n he MLS mehod o demonsrae subsanal mprovemen n accuracy of he approxmae sensvy for RBDO. By akng advanage of he nverse PMA problem, a desgn of expermen (DOE) framework, ha s suable for relably analyss, s proposed o properly reproduce he man effecs and neracons of desgn parameers by combnng he axal sar (AS) and selecve neracon (SI) samplng n a sngle DOE buldng block. PMA s shown o be much more effecve han he relably ndex approach (RIA) when response surface mehodology (RSM) s used for RBDO. he proposed RSM s negraed wh he hybrd mean value (HMV) mehod of PMA o develop a robus and effcen RBDO process. A large-scale vehcle sde mpac model example problem s employed o demonsrae he new RBDO mehod. 2. Keywords: Relably-Based Desgn Opmzaon (RBDO), Response Surface Mehod (RSM), Movng Leas Square (MLS), Performance Measure Approach (PMA), Hybrd Mean Value (HMV) Mehod, and Crashworhness. 3. Inroducon When muldscplnary desgn opmzaon mehods are used, deermnsc opmum desgns frequenly push he lms of desgn consran boundares, leavng lle or no room for olerances (uncerany) n modelng and smulaon and/or manufacurng mperfecons. Consequenly, deermnsc opmum desgns obaned whou consderaon of uncerany can resul n unrelable desgns, ndcang he sgnfcance of he need for RBDO. Durng nal effors n RBDO developmen, RIA was mplemened by defnng a probablsc consran as relably [1]. However, RIA converges slowly, or even fals o converge, for a number of problems. o allevae hs dffculy, PMA has been nroduced by solvng an nverse problem for he frs-order relably mehod (FORM) [2]. Unsurprsngly, PMA s nherenly robus and more effecve, snce s easer o mnmze a complcaed cos funcon subjec o a smple consran expressed as he known dsance (.e., relably ndex) han o mnmze a smple cos funcon subjec o a complcaed consran. Subsequenly, a robus and effcen HMV mehod has been developed [3] for numercal soluon of he nverse PMA problem. However, even he proposed RBDO mehod does no provde an effecve mehodology o oban relably-based opmum desgn usng a lmed number of response analyses for large-scale models. o acheve hs objecve, a new RBDO mehod s proposed ha negraes RSM and he proposed RBDO mehod. Compared o deermnsc desgn opmzaon, he proposed RBDO requres he addonal accuracy of RSM, due o evaluaon of he probablsc consrans and her sensves. herefore, a new RSM, based on he MLS mehod [4] and a DOE specfcally suable for relably analyss, s proposed. he MLS mehod beer approxmaes he mplc response by mposng a varable wegh over a compac suppor. In he leraure [5], mos RSMs have been developed ulzng only response daa; lle aemp has been made o use boh response and sensvy daa o consruc approxmae responses. An RSM ha ulzes only response daa may be accepable as far as response values are concerned, however, he approxmae desgn sensvy obaned from he approxmae response may conan suffcen error as o be compleely neffecve for RBDO. If an accurae sensvy can be effcenly obaned, can be ncorporaed n he MLS mehod o subsanally mprove he accuracy of he approxmae responses, as well as he sensves requred for RBDO. PMA s shown o be much more effecve han RIA when RSM s used for RBDO, snce he sze of he DOE buldng block n PMA s clearly defned and snce PMA, unlke RIA, requres only he performance response for boh desgn opmzaon and relably analyss. A new DOE framework for RBDO s proposed, composed of AS and SI samplng, o properly reproduce he man effecs and neracons of desgn parameers, respecvely. Based on he mean value (MV) frs-order relably mehod, he neracon regon, parcularly for SI samplng, s selecvely chosen o nclude he MPP a whch a beer approxmaon s requred. he proposed RSM s negraed wh he HMV mehod of PMA for robus and effcen MPP search. An auomove sde mpac model [6] s used o demonsrae he proposed RBDO mehodology n Secon Relably-Based Desgn Opmzaon

2 In sysem parameer desgn, he RBDO model [3] can be generally defned as Mnmze Cos( d) subjec o PG ( ( X ) 0) Φ( β) 0, = 1, 2,, NP dl d du, d Rndv and X Rnrv where d = µ ( X) s he desgn vecor, X s he random vecor, and he probablsc consrans are descrbed by he performance funcon G (X) wh G (X)<0 ndcaes falure, her probablsc models, and her prescrbed confdence level β. Performance funcon falure s sascally defned by a cumulave dsrbuon funcon F G (0) as PG ( ( X) 0) = FG (0) = ( ) ( ) ( X) 0 fx x dx Φ G (1) β (2) In Eq. (2), f X (x) s a jon probably densy funcon, whch needs o be negraed. Some approxmae probably negraon mehods, such as he frs-order relably mehod (FORM) wh a roaonally nvaran relably measure [2], have been wdely used o provde effcen and adequaely accurae soluons. hrough nverse ransformaon, he probablsc consran n Eq. (2) can be furher expressed n wo dsnc forms as: β = ( Φ 1( F (0))) β (3) s Gp = F 1 ( Φ ( β )) 0 G (4) where βs and G p are respecvely referred o as he safey relably ndex and he probablsc performance measure for he h probablsc consran. Usng he relably ndex, Eq. (3) s hen employed o descrbe he probablsc consran n Eq. (1),.e., he so-called relably ndex approach (RIA). Smlarly, Eq. (4) can replace he probablsc consran n Eq. (1) wh he performance measure, referred o as he performance measure approach (PMA). 4.1 Frs-Order Relably Analyss n RIA In RIA, he frs-order safey relably ndex β s,form s obaned usng a FORM formulaed as an opmzaon problem, wh an mplc equaly consran n U-space defned as he lm sae funcon: mnmze U, subjec o G ( U ) = 0 (5) he opmum pon on he falure surface s referred o as he mos probable falure pon (MPFP) u search algorhm specfcally developed for FORM or general opmzer can be used o solve Eq. (5). G * G( U ) = 0. Any MPFP 4.2 Frs-Order Relably Analyss n PMA he frs-order relably analyss n PMA can be formulaed as he nverse of he frs-order relably analyss n RIA. he frs-order probablsc performance measure G p,form s obaned from a nonlnear opmzaon problem wh an n- dmensonal explc sphere consran n U-space, defned as mnmze G( U), subjec o U = β (6) he opmum pon on a arge relably surface s denfed as he mos probable pon (MPP) u* β = β. An advanced mean value (AMV) mehod can be used o numercally solve he nverse PMA problem. However, s found ha AMV mehod exhbs poor behavor for concave consran funcons, alhough s effecve for convex consran funcons. o overcome dffcules wh he AMV mehod, a conjugae mean value (CMV) mehod s proposed for he concave consran funcon n PMA [3]. he HMV mehod [3] combnes boh he CMV and AMV mehods - he CMV mehod s used for concave consran funcons and he AMV mehod s used for convex consran funcons. he HMV mehod has been shown o be very robus and effcen. here are hree major advanages n usng PMA as compared o RIA. Frs, s found ha PMA s nherenly robus and more effecve when he probablsc consran s eher very feasble or very nfeasble. hs s unsurprsng, snce s easer o mnmze a complcaed cos funcon subjec o a smple consran funcon wh known dsance (.e., relably ndex) han o mnmze a smple cos funcon subjec o a complcaed consran funcon. Second, and more sgnfcanly, PMA always yelds a soluon, whereas RIA may no yeld soluons for ceran ypes of dsrbuons, such as Gumbel or unform dsrbuons. hrd, PMA s more effecve han RIA when RSM s used for RBDO, snce he sze of he DOE block n PMA s clearly defned and snce, unlke RIA, PMA requres only he response surface of he performance measure for boh relably analyss and desgn opmzaon. 5. Response Surface Mehod for RBDO Alhough he proposed RBDO mehod represens a sgnfcan advancemen, RBDO requres very large compuaonal me for large-scale models wh a number of desgn consrans. Consequenly, negraon of he RSM wh he RBDO mehod s proposed n he new RBDO mehodology. he new RSM s nroduced usng he MLS mehod wh a DOE ha s specfcally developed for effcen and effecve relably analyss.

3 5.1 Movng Leas Square (MLS) Mehod wh Funcon Daa he MLS approxmaon can be formulaed as NB gˆ( d) = = 1 ( d) ( d) h h a ( d) a( d) (7) Lancaser and Salkauskas formulaed he local MLS approxmaon a d as [4] NB gˆ( dd, ) = = 1 ( d) ( d) = h h a ( dad ) ( ) (8) where NB s he number of erms n he bass, h ( d) are monomal bass funcons evaluaed a a se of gven sample pons d, and a (d) are her coeffcens, whch are funcons of he desgn parameer d. o compue he coeffcen vecor a(d), a weghed resdual s defned as NS J = [ ] 2 I = 1w( d di) gˆ ( d, di) g( di) 2 or J = NS ( Ha g) W( d)( Ha g) (9) = I= 1w( d di) h( di) a( d) g( di) where NS s he number of sample pons, w(d-d I ) s a wegh funcon wh a compac suppor, and g = [ g( d1) g( d2) g( dns )], H = [ h( d1) h( d2) h( dns )], W = dag [ w( d1) w( d2) w( dns )] (10) An approprae suppor sze a any daa pon d I s seleced so ha a suffcen number of neghborng daa pons s ncluded o avod a sngulary. A varable wegh over he compac suppor provdes local averagng o he response approxmaed by he MLS mehod. he mnmum of he weghed resdual J by J/ a=0 yelds he coeffcen a(d) n Eq. (7) as represened by ad ( ) = M 1( dbdg ) ( ) where M = HW( d) H, B = HW( d) (11) Subsung Eq. (11) no Eq. (7), he approxmaon gˆ( d) wh funcon nformaon can hen be expressed as gˆ( d) h ( dm ) 1( dbdg ) ( ) = 5.2 Movng Leas Square (MLS) Mehod wh Funcon and Sensvy Daa If can be obaned a an affordable cos, he sensvy of he response can be ncorporaed n he MLS mehod o subsanally mprove accuracy of he approxmae sensvy. he response and s sensvy provde (ND+1) NS daa, where ND s he number of elemens n he unon se of boh desgn and random parameers, and NS s he number of sample pons. In comparson, he response provdes only NS daa. hus, ncluson of he sensvy daa wll no only provde beer approxmaon of he response bu also provde he sensvy nformaon requred for RBDO. Recallng Eq. (7), he local MLS approxmaon a d can be formulaed as g( dd, ) = h ( dad ) ( ), g ( dd, ) = h ( dad ) ( ) (13) ˆ ˆ, j, j where g ˆ, j s a sensvy wh respec o he j h desgn parameer and h, j ( d) are monomal bass funcons evaluaed a a se of gven sample pons d. o compue he coeffcen vecor a(d), a weghed resdual s defned as ( ) [ ˆ ] J = J + J = Ha g W( d)( Ha g) g dg NS 2 ND NS 2 I= 1w( d di) g( d, di) g( di) j= 1 I= 1vj( d di) g ˆ, j( d, di) g, j( di) NS 2 ND NS I= 1w d di h di a d g di j= 1 I= 1vj d di h, j di a d g, j di = + = ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) where w(d-d I ) s a wegh funcon for he frs resdual funconal J g, v(d-d I ) s a wegh funcon for he second resdual funconal J dg, and g = [ g( d1) g( d2 ) g( d )], g, = [, ( d1 ), ( d2 ), ( d )] NS j g j g j g, g = [ g g,1 g, ] j NS ND H = [ h( d1) h( d2 ) h( d )], H, = [ h, ( d1 ) h, ( d2 ) h, ( d )] NS j j j j NS, H = [ H H,1 H, ] ND (15) W = dag w( d ) w( d ) w( d ), V = dag v( d ) v( d ) v( d ), W = dag W V V [ ] [ ] [ ] 1 2 NS 1 2 NS 1 ND Boh wegh funcons mus have he same compac suppor. he same concep presened n Secon 5.1 wh respec o deermnaon of an approprae suppor sze s employed here. he mnmum of he weghed resdual J by J/ a=0 yelds he coeffcen a(d) n Eq. (13) as represened by ad ( ) = M 1( dbdg ) ( ) where M = H W( d) H and B = H W( d) (16) he MLS approxmaon gˆ( d) wh boh funcon and sensvy nformaon can hen be expressed as ˆ( ) = ( ) 1( ) ( ) 5.3 Sensvy Approxmaon n Movng Leas Square (MLS) Mehod 2 (12) (14) g d h dm dbdg (17)

4 hs secon presens a sensvy approxmaon of he MLS mehod applcable o boh he MLS mehod wh funcon nformaon only, and wh boh funcon and sensvy nformaon. he noaon n developng a sensvy approxmaon follows ha of MLS mehod wh only funcon nformaon. he same dervaon of he sensvy approxmaon can be easly accomplshed for he MLS mehod wh funcon and sensvy nformaon. Eq. (7) s recalled o compue he sensves of a response approxmaon gˆ( d) generaed by he MLS mehod. o fnd he paral dervave of ĝ n he drecon of, say, d dfferenang boh sdes of Eq. (7) yelds g, =, +, ˆ ( d) h ( dad ) ( ) h ( da ) ( d) In general, h(d) s a smple funcon and s dervaves are easly compued. he dervaves of a(d) are more roublesome and mus be obaned by applyng Eq. (11). he paral dfferenaon of Eq. (11) wh respec o d yelds ( H W H) a+ ( H WH) a = H W g or a = M 1B ( g Ha) (19),,,,, I should be noed ha he marx H(d I ) s no a funcon of d. Replacng Eq. (19) by Eq. (18) yelds gˆ = h a+ h M 1B ( g Ha),,, Noe ha W, j =B, j =0 a sample pons, snce all frs-order dervaves of he funcons w (d) vansh a any sample pon for he MLS mehod. herefore, a he sample pons, Eq. (20) reduces o ˆ = h a. g,, 5.4 Desgn of Expermen (DOE) Unlke deermnsc desgn opmzaon, RBDO requres he addonal accuracy of he RSM, due o evaluaon of probablsc consrans and her sensves. When he FORM s used for relably analyss, accurae response and assocaed sensvy nformaon s requred a he mos probable pon (MPP), whch s a ceran dsance away from he mean value desgn pon. o mee hs requremen, he MLS mehod mus be negraed wh a DOE ha s specfcally sued for relably analyss. I s noed ha PMA s more effecve han RIA when RSM s used for RBDO, snce he sze of he DOE buldng block s clearly defned and a performance response can be conssenly used n boh relably analyss and desgn opmzaon. However, he sze of he DOE buldng block n RIA depends on how far he MPFP s locaed on a falure surface. As well, RIA requres wo dfferen responses, such as performance response and relably response [6]. o properly reproduce he man effecs and neracons of desgn parameers, a new DOE framework for RBDO s proposed composed of AS and SI samplng. he AS samplng s ncorporaed o represen he man effecs of he desgn parameers by selecng a cenral sample pon and samples along he desgn axes. o accuraely evaluae probablsc consrans and her sensvy, SI samplng selecvely chooses he sample pon a he corner of he neracon regon o nclude he MPP a whch a beer approxmaon s requred n he RBDO process, based on he MV frs-order relably mehod. ogeher, he AS and SI samplng mehods consue a sngle DOE buldng block. he sze of he DOE buldng block s defned by cβ, where β s a arge relably analyss and c s a parameer dependen on he nonlneary of he response, ypcally RBDO of Vehcle Sde Impac A large-scale applcaon of he new RBDO mehodology for vehcle desgn for sde mpac s llusraed n Fgure 1. Includng he fne elemen (FE) dummy model, he oal number of nodes and shell elemens n hs model s 96,122 and 85,941, respecvely. he nal laeral velocy of he sde deformable barrer s 49.89kph (30mph). he CPU me for one nonlnear FE smulaon usng he RADIOSS sofware s approxmaely 20 hours on an SGI Orgn he desgn objecve s o enhance sde mpac crash performance whle mnmzng vehcle wegh. (18) (20)

5 Fgure 1. Vehcle Sde Impac Model he sde mpac opmzaon problem can be defned as Mnmze Wegh( d) subjec o P(abdomen load 1.0 kn) 90% P(V*C 0.32 m/ s) 90% P(upper/mddle/lower rb deflecon 32 mm) 90% P(pubc symphyss force, F 4.0 kn) 90% P(velocy of B-pllar a mddle pon 10 mm/ ms) 90% P(velocy of fron door a B-pllar 15.7 mm/ ms) 90% dl d du, d R9 and X R11 wh 9 desgn and 11 random parameers are defned, as lsed n able 2. (21) able 1. Properes of Desgn and Random Parameers of Vehcle Sde Impac Model Random Varable Sd. Dev. Dsr. ype No of Desgn d L d d U 1 (B-pllar nner) Normal (B-pllar renforce) Normal (Floor sde nner) Normal (Cross member) Normal (Door beam) Normal (Door bel lne) Normal (Roof ral) Normal (Ma. Floor sde nner) Normal (Ma. Floor sde) Normal (Barrer hegh) 10.0 Normal 11 (Barrer hng) 10.0 Normal 10 h and 11 h random varables are no regarded as desgn varables. he opmal LHS wh a oal of 33 runs was used o generae a sample of desgn pons for consrucon of he sepwse regresson (SR) response surface [7]. he explc response used n he RBDO s summarzed n Ref. 6. In hs sudy, he explc approxmaons of responses are regarded as exac responses of vehcle sde mpac o demonsrae he new RBDO mehodology. Resuls n ables 2 and 3 correspond o RBDO whou RSM. he RSM-based RBDO wh response-only resuls are presened n ables 4 and 5. ables 6 and 7 presen he resuls for boh response and sensvy daa. As can be deermned from ables 2 o 7, he RSM-based RBDO resuls are farly accurae, compared o RBDO whou RSM. A he 90% relable opmum desgn, a wegh savngs of 20% s obaned. In comparson o an equvalen number of FEA analyses, he RSM-based RBDO s more effcen han he proposed RBDO whou RSM. However, he RSM-based RBDO whou sensvy yelds slghly dfferen resuls n consran feasbly, whch has only 2nd and 8h consrans acve, whereas ohers have 2 nd, 4 h, and 8 h consrans acve as shown n able 3 and 5. he dscrepancy n consran feasbly s due o dfference n 9 h desgn parameer. Neverheless, snce he wegh does no depend on 9 h desgn parameer, he opmum n able 4 s regarded as local opmum as long as he mnmum cos s mananed and he probablsc consrans n he RBDO are sasfed a he same me. able 2. RBDO Cos and Desgn Hsory whou RSM Ier. Cos d 1,X 1 d 2,X 2 d 3,X 3 d 4,X 4 d 5,X 5 d 6,X 6 d 7,X 7 d 8,X 8 d 9,X 9 X 10 X Op able 3. RBDO Probablsc Consrans Hsory whou RSM Ier. G p1 G p2 G p3 G p4 G p5 G p6 G p7 G p8 G p9 G p10 Analyss Equv. FEA

6 Op (1+11k 1 ) =365.4 able 4. RBDO Cos and Desgn Hsory Usng RSM wh Only Response Ier. Cos d 1,X 1 d 2,X 2 d 3,X 3 d 4,X 4 d 5,X 5 d 6,X 6 d 7,X 7 d 8,X 8 d 9,X 9 X 10 X Op able 5. RBDO Probablsc Consrans Hsory Usng RSM wh Only Response Ier. G p1 G p2 G p3 G p4 G p5 G p6 G p7 G p8 G p9 G p10 Analyss Equv. FEA = Op able 6. RBDO Cos and Desgn Hsory Usng RSM wh Boh Response and Sensvy Ier. Cos d 1,X 1 d 2,X 2 d 3,X 3 d 4,X 4 d 5,X 5 d 6,X 6 d 7,X 7 d 8,X 8 d 9,X 9 X 10 X Op able 7. RBDO Probablsc Consrans Hsory wh RSM wh Boh Response and Sensvy Ier. G p1 G p2 G p3 G p4 G p5 G p6 G p7 G p8 G p9 G p10 Analyss Equv. FEA (1+11k) =195.3 Op Concluson A new RBDO mehodology based on a proposed RSM (MLS mehod and AS+SI DOE) has been presened o mplemen an effcen RBDO process for large-scale models. he RSM-based RBDO mehod employs he PMA approach, snce n PMA he sze of he DOE buldng block s well defned and only performance response s requred. As llusraed usng a vehcle sde mpac model, he RSM based RBDO mehodology exhbs more effcen resuls han he proposed RBDO mehod whou RSM, whle mananng good accuracy. In addon, he ncluson of sensvy daa acceleraes he RBDO process as a resul of beer approxmaon of probablsc consrans and her sensves, yeldng he smalles number of employed RBDO eraons and equvalen FEAs. Hence, he RSM-based RBDO mehodology appears o provde a cos-effecve RBDO process for even large-scale models. 8. References 1. Cho, K.K., Yu, X., and Chang, K.H. (1996), A Mxed Desgn Approach for Probablsc Srucural Durably, Sxh AIAA/USAF/NASA/ISSMO Symposum on Muldscplnary Anal. and Op., Bellevue, WA, Madsen, H.O., Krenk, S., and Lnd, N. (1986), Mehods of Srucural Safey, Prence-Hall, Englewood Clffs, NJ. 3. Cho, K.K. and Youn, B.D., Hybrd Analyss Mehod for Relably-Based Desgn Opmzaon, 27 h ASME Desgn Auomaon Conference, Sepember 9-12, 2001, Psburgh, PA. 1 Rao of DSA compuaon and FEA compuaon, whch s assumed o be k=0.1.

7 4. Lancaser, P. and Salkauskas, K. Curve and Surface Fng; An Inroducon, Academc Press, London, Roux, W.J., Sander, N., and Hafka, R.. Response Surface Approxmaons for Srucural Opmzaon, In. J. Numer. Mehods Eng., 1998, 42, Gu, L., Yang, R.J., ho, C.H., Makowsk, M., Faruque, O., and L, Y. Opmzaon and Robusness for Crashworhness of Sde Impac, In. J. Vehcle Desgn, 2001, 26(4). 7. Draper, N. and Smh, H., (1966), Appled Regresson Analyss, John Wley & Sons, 1966.