Topology Optimization of Structures under Constraints on First Passage Probability

Transcription

1 Topology Opmzao of Srucures uer Cosras o Frs Passage Probably Juho Chu Docoral Sue, Dep. of Cvl a Evromeal Egeerg, Uv. of Illos, Urbaa-Champag, USA Juho Sog Assocae Professor, Dep. of Cvl a Evromeal Egeerg, Seoul Naoal Uv., Seoul, Korea Glauco H. Paulo Raymo Joes Char, School of Cvl a Evromeal Egeerg, Georga Isue of Techology, Alaa, USA ABSTRACT: A ew meho s propose o corporae he frs passage probably o sochasc opology opmzao usg sequeal compoug meho (Kag a Sog. Parameer sesves of he frs passage probably he probablsc cosra are erve o faclae he use of grae-base opmzer for effce opology opmzao. The propose meho s apple o bulg srucures subece o sochasc grou moo o f opmal bracg sysems whch ca ress fuure realzao of sochasc excaos whle achevg a esre level of relably. Opmal esg of a laeral loa-ressg sysem of a srucure s oe of he esseal asks srucural egeerg as s recly lke o bulg safey a operao. I parcular, relable operao a safey uer sochasc excaos by aural hazars such as earhquake, w loas are maor esg obecves. However, eermsc escrpo of fuure realzao of a raom process s frequely lme because oly a se of few me hsores are avalable. Therefore, a probablsc preco of srucural resposes base o raom vbrao aalyss s much eee he process for opmal esg. To aress hs ssue, he auhors performe a suy of opology opmzao of srucures uer sochasc excaos (Chu e al. revew. I he suy, raom vbrao aalyss by a scree represeao meho (Der Kuregha a srucural relably heory were egrae o opology opmzao framework. I ao, he auhors evelope he sysem relably-base opology opmzao framework uer sochasc excaos (Chu e al. o coser sysem falure eves wh sascal epeecy usg he marx-base sysem relably meho (Sog a Kag 9. The evelope meho helps sasfy probablsc cosras o a sysem falure eve, whch cosss of mulple lm-saes efe erms of ffere locaos, falure moes a me pos as opmzes a srucural sysem. Chu e al. ( revew has evaluae a saaeous falure probably of he srucure subece o raom excaos a a scree me po. However, a more praccal applcao egeerg ca be acheve f he falure probably s evaluae for exceeace eve over a me erval. Ths helps promoe he use of he propose sochasc opology opmzao framework for he esg of laeral loa-ressg sysem uer sochasc excaos. Thus, hs paper, a sochasc opology opmzao framework s propose o hale probablsc cosras o he frs passage probably.. RANDOM VIBRATION ANALYSIS USING DISCRETE REPRESENTATION METHOD.. Dscree represeao of sochasc process The scree represeao meho (Der Kuregha screzes a couous

2 sochasc process wh a fe umber of saar ormal raom varables. For example, a zero-mea Gaussa process f( ca be screze as: s T v ( f ( vs ( ( where s( eoes a vecor of eermsc bass fucos, whch s eerme from he specral characerscs of he process (Der Kuregha, a v s a vecor of ucorrelae saar ormal raom varables... Characerzao of lear sysem uer sochasc excaos The splaceme me hsory u( of a lear sysem subece o sochasc excaos, ca be eerme by usg Duhamel s egral a Eq. (,.e. u ( f(τ h( τ τ vs( h( τ τ s s T a v s va( (, a( s( h( ( where hs( s he u mpulse respose fuco of he egree-of-freeom of eres, a a( eoes a vecor of eermsc bass fucos. The, falure eves efe erms of resposes ca be escrbe he space of saar ormal raom varable v (Der Kuregha. v β(u, Safe oma αˆ ( MPP β(u, Falure oma ˆα ( u u( < u u( = Fgure : Geomerc represeao of saaeous falure a me a saar ormal space. For example, he saaeous falure eve,.e. he eve of a respose a a cera me = v u u( = exceeg a prescrbe hreshol u, s represee by he lear half space u u( = u a( T v as show Fgure. From he geomerc erpreao, a relably ex ca be compue as a close-form soluo,.e. β( u, u / ( ( * ˆ a α v ( where eoes he egave ormalze grae vecor of he lm-sae fuco evaluae a he mos probable falure po v*.. FIRST-PASSAGE PROBABILITY The frs passage probably s commoly ulze o f he probably of he falure eve escrbe wh a me erval (VaMarcke 975, Sog a Der Kuregha 6, Fumura a Der Kuregha 7. Oe of he avalable approaches for formulag he frs passage probably s efg he problem as a seres sysem problem such as: P( Esys P( xmax u( P u( u ( The frs passage probably he requres evaluao of compoe eves a each me po wh a erval. Moreover, a effce, relable a robus algorhm s requre o evaluae sysem falure probably wh sascal epeecy bewee he compoe eves fully cosere. To aress hese requremes, he sequeal compoug meho (SCM; Kag a Sog s aope hs suy.. SEQUENTIAL COMPOUNDING METHOD The sequeal compoug meho s a sysem relably meho ha compous compoe eves couple by uo or erseco sequeally ul a sgle compou eve represes he sysem eve. Wheever wo compoes are compoue, he probably of he ew compou eve s obae whle he correlao coeffces bewee he ew compou eve a each of he oher remag compoe eves are compue. For sace, compoug wo compoe eves a seres sysem ca be escrbe as PE ( E E PE ( E E (5 or

3 The relably ex βor of he compou eve Eor ca be eerme as follow: β [ PE ( E] [ (β,β ; ] (6 or, where Ф a Ф respecvely eoe he margal a b-varae cumulave srbuo fuco (CDF of saar ormal raom varable(s. v llusrae Fgure. More eals o a effce scheme o f ρ(or,k a compoug wo compoes couple by erseco ca be fou Kag a Sog (. The SCM s mplemee o compue he frs-passage probably he propose meho because he SCM ca prove he probably a parameer sesves of a sysem cossg of a large umber of compoes. ρ,k ρ,k β k β β ρ, g k = g g = = v ρ (or,k v. PARAMETETRIC SENSITIVITY OF SYSTEM RELIABILITY USING SCM Relably base esg/opology opmzao (RBDO/RBTO ca be effcely performe f parameer sesves of he probably ca be realy compue. I hs paper, he frs passage probably s cosere as a probablsc cosra opology opmzao. Therefore, sesvy aalyss of sysem relably cossg of may compoe eves s requre. Iegrag SCM wh he propose meho leas o sgfca reuco of compuaoal cos for he relably aalyss he evelope opology opmzao framework. g k = β k β or g or = Fgure : SCM proceure. sysem probably of hree eves; compou eve of E a E a a upae correlao coeffce. The correlao coeffces bewee he compou eve a he remag eves,.e. ρ(or,k, k =,, are eerme such ha ( z, z, z;,,, k,, k z ( β or, β k; (or, k u (7 where z eoes a vecor of saar ormal raom varables, φ s he o probably esy fuco of z, a Ωu represes he oma of he sysem eve as u ( Z β ( Z β ( Zk β k (8 ρ(or,k Eq. (7 s obae umercally by usg olear programmg. The sequeal compoug proceure of hree eves s v Psys/ Propose meho FDM Compoe umber, Fgure : Sesves of seres sysem wh compoes (uequal relably ces a equal correlao coeffces,.5: sesvy comparso bewee he propose meho a fe fferece meho, a compoe relably ces cossg of he seres sysem. To hs e, he auhors furher evelope he SCM o compue parameer sesves effcely (Chu e al. 5. Fgure shows umercal resuls of he sesvy aalyss for a seres sysem wh compoes.

4 5. TOPOLOGY OPTIMIZATION UNDER FIRST PASSAGE PROBABILITY Topology opmzao (Besøe a Sgmu ams o f he opmal maeral srbuos a esg oma Ω subece o racos a splaceme bouary coos whle sasfyg gve esg cosras. I hs paper, we coser a lear elasc a soropc cosue maeral wh a elascy esor D. Sol Isoropc Maeral wh Pealzao (SIMP; Besøe a Sgmu 999 moel s aope whch a smooh covex fuco : R [,] s efe by a power fuco represeao,.e. p ψ( x x (9 where p s a pealzao facor a x s a flere esy ρ e( wh a vecor of eermsc esg varables,. The SIMP moel expresses a elascy esor of a soropc maeral he sae of plae sress as: v ψ(ρ e( E D(ρ e( v ν ( ( v/ where s he Posso s rao, a E s he elascy esor of he sol maeral, where he esy s. The flere eleme esy ca be obae by usg a esy flerg meho such as he proeco echque (e.g. Gues e al., Sgmu 7 o avo checkerboar-paers a o acheve a mmum legh scale. By usg a lear ha kerel of raus r, he eleme esy ca be compue as a weghe average of he esg varables wh a fluece oma Ωe such as: ρ e( w / w ( e e Where w=(r r/r> s a wegh, a r s he sace bewee he ceros of eleme e a eleme, whch les wh he raus r of eleme e. I orer o avo sgulary of a sffess marx fe eleme aalyss, oe ees o se a lower bou o he eleme esy ρ e(.e., <ρm ρ e(. Usg he SIMP moel, he sffess marx of he e h eleme a s sesvy are obae as follows he elemebase compuaoal framework (Besøe a Sgmu : K (ρ ρ K, K (ρ / ρ pρ K ( p p e e e e e e e e e A formulao of opology opmzao uer sochasc excao wh he frs passage probably cosra ca be formulae as follows: m f ( ρ s. P g : u( u P( Esys :β,...,β Psys ρ ( wh M( ρ u(, ρ C( ρ u (, ρ K( ρ u(, ρ f(, ρ ( where eoes he oal umber of me pos urg sochasc excaos, M, C a K are he mass, ampg a sffess marces of he esg oma, respecvely, a u, u, u a f are he accelerao, velocy, splaceme a exeral force vecors a me, respecvely. We ome he epeece of flere eses ρ o he esg varables, ρ ρ (. 6. SENSITIVITY ANALYSIS: FIRST PASSAGE PROBABILITY Sesvy aalyss s a esseal proceure orer o use grae-base opmzao algorhms. A meho of ao sesvy aalyss for probablsc cosras s erve as follows. 6.. Ao sesvy aalyss Sesvy of he probablsc cosra o he frs passage probably s compue from he followg expresso. PE ( sys :β ( ρ,...,β ( ρ ( :β (,...,β ( e PEsys ρ ρ β ( ρ ρ ( β ρ β ( T ρ P c ρ

5 where P T eoes a flerg marx compue from Eq. (. c P( Esys :β ( ρ,...,β ( ρ/ β ca be obae as Chu e al. (5. The paral ervave β/ ρ s obae from he followg expresso. ak(, ρ u k(, β ( a ρ ρ k ρ (5.5 ρ ak(, ρ k Kaa-Tam PSD, Φ(ω ω=5.π ω (ra / s Noe of eres Fgure : Kaa-Tam power specral esy (PSD; Desg oma a loag cofgurao. Whe a uform me sep sze s use,.e.,,,..., a, Eq. ( ca be rewre as follows: PE ( sys T a(, ρ P a(, ρ ρ a(, ρ (, ρ a(, ρ ρ T a l (, ρ P la l (, ρ, m,..., lm ρ (6 where ζ f =. ζ f =. ζ f = l / l cu l / ak( l, k Frame eleme Desg oma: Quarlaeral eleme ρ (7 Pre-mulplyg he screze ao sysem from he goverg sysem equao Eq. ( wh a of mesoal ao varable vecor λ a ag o rgh-ha se erms of Eq. (6, he followg expresso s obae: 5 m TOP EL: m GROUND EL: m PE ( sys T a l (, ρ P la l (, ρ ρ lm (, T A( ρ u ρ λ u(, ρ A( ρ ρ ρ f (, ρ (.5γ η ρ f (, ρ f (, ρ η (.5γη ρ ρ B( ρ u(, ρ u(, ρ B( ρ ρ ρ E( ρ u(, ρ u (, ρ E ( ρ ρ ρ (8 More eals of he ao sesvy aalyss ca be fou Chu e al. ( revew. 7. NUMERICAL APPLICATIONS 7.. Comparso of he fe fferece meho a he ao meho The erve ao sesvy meho was compare wh he fe fferece meho (FDM o verfy accuracy a effcecy. For comparso, The sochasc sesmc excao f( s moele as a flere whe-ose process usg he Kaa-Tam fler moel wh he esy Φ (Fgure. The force vecor Eq. ( s replace wh a eral force vecor of f = M(ρlf( where he vecor l represes recoal srbuo of masses wh uy. The srucural colums represee by wo vercal les as show Fgure are moele by frame elemes whose eses rema uchage hroughou he opmzao process. Youg s moulus E =, MPa a mass esy =, kg/m are use as maeral properes for boh he quarlaeral a frame elemes. The ampg marx s cosruce usg a Raylegh ampg moel. Table summarzes he Kaa-Tam fler parameers of oma frequecy f a bawh ζ f, colum sze, me erval of eres, a he hreshol value u of he average rf rao a each me po. 5

6 Table. Parameers use for sesvy aalyss: esg oma, probablsc cosra a grou moo moel. f ζ f Ф Colum sze u m sec 5π. 5. x... A each me po, a compoe lm sae s escrbe as a eve ha he average er-sory rf rao evaluae a oes of eres Fgure excees he gve hreshol values as: T T ( a(, ρ Lef a(, ρ Rgh v Ef u (9 h where h s he sory hegh (m. Fgure 5 shows ha sesves calculae by he fe fferece meho a he ao meho mach well. The compuaoal cos of he ao sesvy meho s much less ha he fe fferece meho as show Fgure Fgure 5: Normalze sesves by ffere approaches. Fe fferece meho (FDM; Ao meho (AJM. 7.. Example : Probablsc cosra o frs passage probably The propose opology opmzao framework was apple o a mul-sory bulg for efyg opmal bracg sysem uer sochasc excaos. The esg oma for opology opmzao s show Fgure 7. Kaa-Tam parameers Seco 7. are use excep Ф =. The grou accelerao urao a a me sep are.s a.s, respecvely. A colum sze s.5m.5m a a hrea hol of er-sory rf rao u s.. A flerg raus r s.5m a a prescrbe esy.7 s apple uformly hroughou he mesh. The opology opmzao problem ca be formulae as: m Volume ( ρ s. P g : u( u P( Esys :β,...,β Psys wh M( ρ u(, ρ C( ρ u (, ρ K( ρ u(, ρ f(, ρ ( Normalze compuaoal me 5 - FDM AJM Number of Elemes Fgure 6: Normalze compuaoal me. 5 m Noe of eres LEVEL 5 EL: m LEVEL EL: 5 m LEVEL EL: m LEVEL EL: 5 m GROUND EL: m (c Fgure 7: Topology opmzao soluos o he mul-sory bulg example. esg oma; β sys =., P sys=.%; (c β sys=.5, P sys= 6.68%. Topology opmzao resuls correspog o ffere arge relably ces are show Fgure 7-(c. As he arge falure probably 6

7 ecreases, he coverge opologes become sgfcaly ffere especally a he lower level. I parcular, he erseco po of he bracg a he lower level moves up vercally a he hckess of he bracg creases a he lower level bu remas relavely sable a hgher levels. The covergece hsores of he obecve fuco a he sysem falure probably are show Fgure 8. Volume (m β sys P(E sys (c Fgure 8: Covergece hsory. volume; relably ex; (c falure probably. f( (m /s /L 8 6 β sys = β sys =. P sys =.668 β sys =.5 β sys = P =. sys No.erao Ial sysem Opmze sysem Tme (sec Fgure 9: Dyamc respose comparso of he problem show Fg. 7: Ipu grou moo accelerao; correspog yamc resposes of he al esg a he opmal esg. Fgure 9 shows me hsores of er-sory rf raos of he al esg a he opmal esg for a pu process raomly geerae from he Kaa-Tam fler moel. The opmze sysem shows mproveme yamc performace eve hough oly 58% of he al volume s use. 7.. Example : Mulple probablsc cosras o frs passage probably I hs example, wo probablsc cosras gve erms of rf raos are cosere Eq. ( wh he same moellg a opmzao parameers Seco 7.. The wo lm-sae fucos of rf raos are efe bewee he grou a he level; he a he r level, respecvely. E ach lm sae fuco s cosere a compoe falure eve. Thus, a opology opmzao framework wh wo compoe probablsc cosras s wre as: m Volume ( ρ s. P( E :β,...,β P (. sys a, a, sys sys b, b, Psys P( E : β,...,β (. wh M( ρ u(, ρ C( ρ u (, ρ K( ρ u(, ρ f(, ρ ( I ao, sysem falure probably cosra combg hose wo cosras Eq.( uer sascal epeece ca be cosere opmzao as: m Volume ( ρ 8 s. P g : u( u = PE ( sys : β a,,...,β a,,β b,,...,β b, Psys (. wh M( ρ u(, ρ C( ρ u (, ρ K( ρ u(, ρ f(, ρ ( Topology opmzao resuls a covergece hsores are show Fgure a Fgure. Fal opologes obae from he opmzao problem wh wo compoe probablsc cosras Eq. ( a ha wh he sysem cosra Eq. ( show smlar maeral srbuos. However, sgfcaly ffere opologes from wo approaches may be obae uer ffere opmzao a moellg parameers. The opmal volume from he sysem probablsc cosra s slghly hgher ha he oe from compoe probablsc cosras. 7

8 5 m Noe of eres LEVEL EL: m LEVEL EL: m 9. ACKNOWLEDGEMENT The auhors graefully ackowlege fug prove by he Naoal Scece Fouao (NSF hrough proec CMMI. We also ackowlege suppor from he Doal B. a Elzabeh M. Wlle eowme a he Uversy of Illos a Urbaa- Champag. Ay opo, fg, coclusos or recommeaos expresse here are hose of he auhors a o o ecessarly reflec he vews of he sposors. GROUND EL: m (c Fgure : Topology opmzao soluos o he mul-sory bulg example (β sys =., P sys=.%: esg oma; sysem cosra; (c compoe cosras. Volume (m β sys β sys (c 8 6 Compoe Sysem β sys β sys No.erao β sys =. & β sys =. β sys =. Fgure : Covergece hsory. volume; relably ces of compoe cosras; (c relably ex of sysem cosra. 8. CONCLUSIONS I hs paper, a ew opology opmzao framework s propose for srucures uer sochasc excaos uer cosras o frspassage probably. The propose opology opmzao frame work proves a effce way for srucural egeers o oba opmal esg soluos sasfyg probablsc cosras o sochasc respose he cocepual esg process of srucural sysems subec o sochasc excaos.. REFERENCES Besøe, M. P., a Sgmu, O. (. Topology opmzao heory, mehos a applcaos. NY, Sprger Verlag. Besøe, M.P., a Sgmu, O. (999. Maeral erpolao schemes opology opmzao. Archve of Apple Mechacs, 69(9, Chu, J., a Sog, J., a Paulo, G.H. (. Sysem relably base opology opmzao of srucures uer sochasc excaos. h Ieraoal Coferece o Srucural Safey & Relably, New York, NY. Chu, J., a Sog, J., a Paulo, G.H. (5. Parameer Sesvy of Sysem Relably Usg Sequeal Compoug Meho. Srucural safey, 55, 6 6. Chu, J., a Sog, J., a Paulo, G.H. (. Topology opmzao of srucures uer sochasc excao. Srucural a Mulscplary Opmzao. Uer revew. Der Kuregha, A. (. The geomery of raom vbraos a soluos by FORM a SORM. Probablsc Egeerg Mechacs, 5, 8 9. Fumura, K., a Der Kuregha, A. (7. Talequvale learzao meho for olear raom vbrao. Probablsc Egeerg Mechacs, (, Gues, J.K., a Prevos, J.H., Belyschko, T. (. Achevg mmum legh scale opology opmzao usg oal esg varables a proeco fucos. Ieraoal Joural for Numercal Mehos Egeerg, 6(, 8 5. Kag, W.H., a Sog, J. (. Evaluao of mulvarae ormal egrals for geeral sysems by sequeal compoug. Srucural Safey, (, 5. Sgmu, O. (7. Morphology-base black a whe flers for opology opmzao. Srucural a Mulscplary Opmzao, ( 5,. Sog, J., a Der Kuregha, A. (6. Jo frs-passage probably a relably of sysems uer sochasc excao. Joural of Egeerg Mechacs, (, Sog, J., a Kag, W.H. (9. Sysem relably a sesvy uer sascal epeece by marx-base sysem relably meho. Srucural Safey, (, VaMarcke, E.H. (975. O he srbuo of he frspassage me for ormal saoary raom processes. Joural of Apple Mechacs,, 5. 8