Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects

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1 Modelg Mulbod Dac ses Wh Uceaes. Pa I: Theoecal ad Copuaoal Aspecs Ada adu* Coa adu ad Mehd Ahada ga Polechc Isue ad ae Ues *Copue cece Depae Mechacal Egeeg Depae {csadu Absac Ths sud eploes he use of geealzed poloal chaos heo fo odelg cople olea ulbod dac sses he pesece of paaec ad eeal ucea. The poloal chaos faewo has bee chose because offes a effce copuaoal appoach fo he lage olea ulbod odels of egeeg sses of ees whee he ube of ucea paaees s elael sall whle he agude of uceaes ca be e lage (e.g. ehcle-sol eaco. The poposed ehodolog allows he quafcao of ucea dsbuos boh e ad fequec doas ad eables he sulaos of ulbod sses o poduce esuls wh eo bas. The fs pa of hs sud peses he heoecal ad copuaoal aspecs of he poloal chaos ehodolog. Boh ucosaed ad cosaed foulaos of ulbod dacs ae cosdeed. Dec sochasc collocao s poposed as less epese aleae o he adoal Gale appoach. I s esablshed ha sochasc collocao s equale o a sochasc espose suface appoach. We show ha ul-desoal bass fucos ae cosuced as eso poducs of oe-desoal bass fucos ad dscuss he eae of poloal ad gooec oleaes. Paaec uceaes ae odeled b fe-suppo pobabl deses. ochasc focgs ae dscezed usg ucaed Kahue-Loee epasos. The copao pape Modelg Mulbod Dac ses Wh Uceaes. Pa II: Nuecal Applcaos llusaes he use of he poposed ehodolog o a seleced se of es pobles. The oeall cocluso s ha despe s laos poloal chaos s a poweful appoach fo he sulao of ulbod sses wh uceaes. Kewods: ulbod sse ucea poloal chaos sochasc ODE sochasc DAE.

2 Ioduco Paccal echacal sses ofe opeae wh soe degee of ucea. The uceaes ca esul fo pool ow o aable paaees (e.g. aao suspeso sffess ad dapg chaacescs fo ucea pus (e.g. sol popees ehcle-ea eaco o fo apdl chagg focgs ha ca be bes descbed a sochasc faewo (e.g. ough ea pofle. Fo ealsc pedcos of he sse behao ad pefoace ulbod dac odels us accou fo hese uceaes. Cue ehods used o foall assess uceaes clude Moe Calo sulaos ad lea ad olea appoaos of he sse espose. Moe Calo sulaos ae cosl ad he accuac of he esaed sascal popees poes wh ol he squae oo of he ube of us. Peubao sascal leazao ad olea appoao ehods copue seeal oes of he ucea dsbuo of he soluo bu do o capue esseal feaues of he olea dacs (e.g. as eealed b powe specal des. Ths sud apples he geealzed poloal chaos heo o foall assess he ucea ulbod dac sses. We esgae he copuaoal aspecs of copoag aous pes of uceaes o ulbod dac sse odels ad llusae he ehodolog peseed o paccal eaples elaed o ehcle dacs ad obl off-oad codos. The appoach used s o eed he odel alog he sochasc deso o eplcl paaeeze he ucea dsbuo. Poloal chaos offes a effce copuaoal appoach fo he lage olea ulbod odels of egeeg sses of ees. Fo such sses he ube of ucea paaees s elael sall whle he agude of uceaes ca be e lage (e.g. ehcle-sol eaco. The ehods dscussed hs sud allow he quafcao of uceaes ad eable he sulaos of ulbod sses o poduce esuls wh eo bas sla o he wa he epeeal esuls ae ofe peseed. Moeoe he poposed ehodolog allows he quafcao of uceaes boh e ad fequec doas. Poloal chaos has bee successfull appled sucual echacs ad flud echacs sudes. To ou owledge hs s he fs applcao o ulbod dacs. I apples poloal chaos o dffeeal algebac equaos (DAEs fo cosaed sses. I poposes dec sascal collocao as a aleae o he Gale ehod ad esablshes s equalece o a suface espose appoach. I peses a copuaoall effce wa o ea poloal oleaes based o splg he uldesoal egals. The sud s ogazed wo pas. Pa I (he cue pape peses he heoecal ad copuaoal aspecs of he sud. Afe podg a bacgoud o ehods cuel aalable o ea uceaes ulbod dac sses ad he geealzed poloal chaos epaso he pape dscusses odelg of paaec ad eeal souces of ucea echacal

3 sses. Ne oduces he bass fucos used fo he epeseao of he sochasc deso. Ths s followed b peseg he sochasc oda dffeeal equao foulao of ulbod dacs ad he sochasc dffeeal algebac foulao. The pape coues wh he aalss of he ucea odel esuls boh e ad fequec doas. Fall he coclusos of hs sud ae peseed. The Legede Jacob ad Hee poloals ae ge he apped. Pa II s he copleea pape [] whch peses uecal esuls ad dscussos fo epeseae case sudes. Bacgoud I hs seco we eew he equaos goeg ulbod dacs dscuss he ehods cuel eploed o ea uceaes such sses ad ge a oeew of he poloal chaos appoach. Poloal chaos has bee used eesel o odel uceaes sucual echacs ad fluds bu o ou owledge has o bee peousl appled o ulbod dac sulaos. Foulaos of Mulbod Dac Equaos The dacs of a ulbod sse ca be descbed local o global coodaes Caesa o geealzed coodaes. The dacs of a ucosaed echacal sse [] ca be descbed b a se of sulaeous fs ode dffeeal equaos (ODE: F & & F( ; p ( ( d d Hee R ae he geealzed posos R ae he geealzed d e eloces & R ae he geealzed acceleaos ad p R s a eco of sse paaees. The do oao epeses deae wh espec o e. Usg he sae oao as Eq. ( oe ca epese a cosaed echacal sse Caesa coodaes [3] b a sse of de-3 dffeeal algebac equaos (DAE: & T M ( & F( + λ ( ( d c We cosde holooc poso cosas deoed b : R R ; d d d d d d M : R R s he geealzed ass a F : R R R R ae he c eeal geealzed foces ad oques λ R epese he Lagage ulples ad T λ epese he cosa foces. Paal deaes ae deoed b subscps; e.g. s he Jacoba of wh espec o. B dffeeag he holooc poso cosas wce wh espec o e oe obas he cosa equaos fo eloc ad acceleao: ( (3 ( & τ ( 3

4 Replacg he poso cosas Eq. ( b he acceleao cosas fo Eq. (3 leads o he de- DAE foulao whch s coee fo copuaoal puposes [3]: T M ( ( & F( & ( ( λ τ ( The sse Eq. ( s aheacall equale wh he oe ge b Eq. ( howee a uecal schee appled o Eq. ( wll lead o ceasg eos he poso ad eloc cosas []. To alleae hs df-off he uecal soluo ( ~ ~ s poeced a each e sep oo he coodae cosa afold b solg he followg olea sse fo : ( ~ ( ~ T M + ( ~ ( η (5 lal he copued eloc s poeced oo he eloc cosa afold b solg he followg lea sse fo : ( ( ~ T M + ( ( η ( c I Eq. ( η R s aohe se of Lagage ulples. The poeced alues ( ae he ew uecal soluo a. Mehods Aalable fo Teag Uceaes The adoal odelg appoach assues a deal pu wh pecsel defed paaees whch deee he alue of he oupu. Whe soe paaee alues o eeal focgs ae o ow o cao be accuael epeseed he pobablsc faewo s oe appopae. I hs faewo oe odels ucea pu paaees ad foulaes he dac odel o eflec he popagao of ucea he oupu. oe of he cool-used ehods ha ae adaped fo solg sses wh uceaes ae ow eewed. A e geeal appoach s o sole he Foe-Pla (FP equao whch goes he eoluo of ucea dsbuo ude sse dacs [5]. The ehod wos a hgh-desoal pobabl space ad s o a paccal copuaoal ool fo sses of ees. Moe Calo appoach has bee used eesel dac odels. A eseble of us s pefoed wh each ebe usg a dffee se of paaees daw fo he coespodg ucea dsbuo [7-9]. The sascal popees of he oupus ae obaed fo he eseble of sulao esuls. The appoach s copuaoall epese as he esao of he aace coeges wh he ese squae oo of he ube of us. Relaed oe ecoocal appoaches ae La Hpecube aplg [] ad Baesa Moe Calo [-]. The peubao (ses appoach uses fs ad hghe ode ses coeffces o dee low ode oes of he sulao ucea [3-5]. Ths appoach s useful whe uceaes ae sall ad behae le peubaos of he odel [7].

5 The Neua sees epaso of he goeg sochasc opeao s descbed [8]. Applcao of hs echque o lage olea sses s howee dffcul. Respose appoao ehods buld splfed odels o capue he elaoshp bewee he ucea pus ad ucea oupus. ascal leazao [9] ad quadazao [] poduce odels wh he coec sascal oes. Howee he do o pode a foao o powe specal des of he espose whch s poa o aalzg ucea echacal sses. Fe ode olea sees [] ad olea ehods [3] wee used o descbe he pu-oupu elao of olea sses wh sochasc ecaos. These ehods ae dffcul o appl o sses wh ulple degees of feedo. The deesc equale odel [5] ad he sochasc espose suface appoach [-9] use poloal chaos epeseaos of he pus ad oupus ad deee he coeffces of he oupu odel hough a collocao appoach based o a sall ube of odel us. ochasc aeagg [3-3] ad o-gaussa closue echques [333] pusue foao egadg he sascal oes of he esuls. ochasc aeagg has poe o be effece fo deg appoae soluos fo weel daped sses. Is applcao o olea dac sses whch ae ofe odeael o heal daped has poe dffcul. The appoach eploed b [353] he coe of fe elees fo sold echacs uses a specal appoao he ucea whch allows hgh ode epeseaos. The fudaeal deas se fo Wee's hoogeeous chaos heo [37]. Copehese wo was doe b Ghae ad co-woes fo odelg ucea ohe applcaos cludg olea baos fluds poous eda ec. [ ]. Kaadas ad co-woes oduced he cocep of geealzed poloal chaos ad suded eesel s use o odel uceaes fluds cludg applcaos such as adeco ad dffuso ubulece ad flow-sucue eacos [-55]. Keese peses a copehese eew of ece ehods fo he uecal soluo of sochasc paal dffeeal equaos [5]. Poloal chaos offes a acable copuaoal appoach fo he lage olea ulbod odels of egeeg sses of ees. Fo such sses he ube of ucea paaees s elael sall whle he agude of uceaes ca be e lage (e.g. ehcle-sol eaco. Geealzed Poloal Chaos Epaso Poloal chaoses [3757] ae geealzaos of poloals o he case whee he depede aables ae heseles easuable fucos ( hs pape ado aables. The fudaeal dea s ha ado pocesses of ees ca be appoaed (wh aba accuac b sus of ohogoal poloal chaoses of ado depede aables. These sus offe a copuaoall aace appoach o epeseg he sae of phscal sses opeag ude ucea. 5

6 ecod ode ado pocesses ae pocesses wh fe aace; fo a phscal po of ew he hae fe eeg. A secod ode ado pocess X ( θ ewed as a fuco of he ado ee θ ( < θ < ca be epaded es of ohogoal poloal chaoses as [353]: ( ( θ X ( θ c (7 ae geealzed Ase-Wee poloal chaoses of ode ( Hee ( K es of he ul-desoal ado aable ( K. Fo Gaussa ado aables he bass ae Hee poloals fo ufol dsbued ado aables he bass ae Legede poloals fo bea dsbued ado aables he bass ae Jacob poloals ad fo gaa dsbued ado aables he bass ae Laguee poloals [.7]. The bass chaos poloals fo a coplee ohogoal bass fo he Hlbe space of squae egable ado aables fo (8 Ths ohogoal elao holds wh espec o he eseble aeage e poduc f g f ( g( w( d (9 Hee w ( s he o pobabl des of he ado aables. The sees Eq. (7 coeges o a ado pocess L sese [57]. I pacce a ucaed epaso s used X c ( ( Ths eas ha we cosde a fe ube of ado aables ( K ad su poloals ol up o a aal ode P. The oal ube of es ( + P! (! P! ceases apdl wh he ube of sochasc paaees ad he ode of he poloal chaos P. Noe ha sascal bleazao fo eaple also cosucs a sse ha s cosdeabl lage ha he ogal []. I Eq. ( ad fo he eag of hs pape we dop he eplc depedece of he ado aable o he ee θ. Modelg ouces of Ucea Mechacal ses Dac sses ae ofe affeced b ulple souces of ucea. The a classes ae paaec ucea ad ucea eeal ecaos. I hs seco we dscuss seeal appoaches o odel hese uceaes. Paaec Ucea Ucea paaees echacal sses ae alues bewee well defed bouds. Cosequel he cao be accuael epeseed as oal ado aables sce he Gaussa dsbuo has fe suppo. We wll focus o pobabl deses wh fe suppo ael he ufo ad bea dsbuos.

7 These dsbuos defed o he fe eal [ ] hae he followg pobabl des fucos (PDF:. Ufo pobabl dsbuo:. Bea pobabl dsbuo: w ( ( w ( ( a + b + ( ( ( a ( + b whee Γ ( a + Γ b + Γ Γ ( a+ b+ Hee Γ ( deoes he gaa fuco. The bea dsbuo has wo paaees a ad b whch defe he shape of he dsbuo as llusaed Fg.. Fo a b he bea dsbuo educes o he ufo pobabl dsbuo. e d..9 ufo..9 bea(..9 bea(3 pdf (a Ufo dsbuo (b Bea dsbuo (c Bea dsbuo Fgue. Eaples of dsbuos wh fe suppo Paaec Repeseao of ochasc Focg A poa aspec he sud of ehcle behao oe ough ea s odelg he ucea he ea pofle. Moe geeall a poa aspec he sud of echacal sses s he epeseao of ucea eeal focgs. I hs seco he eeal focg depeds o (space sead of (e o uel epese ea aao. The dscusso howee s geeal ad ca be decl appled o a e depede sochasc focg fuco. The ea pofle s cosdeed a ado pocess z ( defed oe he spaal doa D. The ado ea hegh ca be epeseed as a ea hegh z ( plus a su of deesc shapes g ( ulpled b ado apludes z ( z( + g ( (3 The shape fucos ae leal depede e.g. ca be chose fo a L D.If he ohooal base of he se of squae egable fucos ([ ] 7

8 ado apludes ae assued o be depede decall dsbued ado aables wh zeo ea ad aaces σ he focg coaace fuco s: ( g ( lal f he ea pofle z ( θ has ow coaace R ( R( z( z( z( z( σ g ( he he ado pofle ca be epeseed b he Kahue-Loee (KL epaso [353-5]: ( z( g ( + z λ (5 Hee s a depede se of ado aables of ea ad aace. The shape fucos g ( ae he egeecos ad λ he egealues of he coaace fuco: R( g ( d λ g ( ( D A paccal epeseao of he sochasc pocess s obaed b ucag he KL sees Eq. ( o es based o he elae agude of he egealues such ha λ + << λ. The deao of he ea suface fo he ea pofle z( z( s cosdeed a wde-sese-saoa (W pocess. A W ado pocess has a cosa ea ad a coaace ha depeds ol o he dsace bewee wo pos: ( ad z( z( z( z( ( z( z R (7 The powe des specu of a saoa ado pocess.e. he powe des of E[( z( θ z( ] quafes he fequeces o whch he pocess flucuaes. The specu of a sochasc ea pofle quafes he oughess of he suface. The powe specu of a W ado pocess s he Foue asfo of s coaace fuco (Wee-Khche heoe [353] ( ω R( ˆ ω e d (8 D R The chaacesc fequeces of he sochasc focg ae ge b: R ˆ ( ω λ (9 whee λ ae he egealues Eq. (. Usg he powe specu appoach he sochasc focg (ea ca be odeled as follows: Measue he powe specu of he focg (ea aace Calculae he coaace fuco R (τ as he ese Foue asfo of focg (ea powe specu ad Calculae he lages egealues of he coaace ad he coespodg egeecos ad use he o buld he Kahue- Loee sochasc epeseao of he focg (ea suface. 8

9 Bass Fucos To cosuc he poloal chaos appoao a se of ohogoal poloal bass fucos s defed wh espec o he pobabl des fuco. Fo he oe-desoal ufo dsbuo he bass fucos ae he Legede poloals: + L ( L ( d fo ( ad fo he oe-desoal bea dsbuo he Jacob poloals: ( ( ( + Γ a + b + ( ( ( a ( + b J J d fo ( a+ b+ Γ a + Γ b + The Jacob ad Legede poloals up o ode e ae ge he apped. We ae he bea dsbuo wh a b fo a sec pobabl des fuco. I hs pape we esc he dscusso o ulple depede ado aables K wh a o dsbuo of pobabl ( ( ( ( ( w ( K w w Kw ( ( The bass fucos ae ohogoal he o pobabl space: + + K ( K ( K w( K dkd fo (3 A ul-desoal ohogoal bass s cosuced as follows. Le ( P be he fal of oe-desoal poloals ohogoal wh espec o { } ( he des w ad cosde bass fucos defed b eso poducs of such poloals: ( ( ( ( ( ( K P P KP ( ( The choce peseed Eq. ( eables he ealuao of -desoal scala poducs fo sepaae oe-desoal scala poducs. The ohogoal codo fo Eq. (3 becoes: ( ( ( ( P ( w ( d ol f fo all P (5 + Theefoe he eso poducs of ohogoal poloals Eq. ( fo a ohogoal se he -desoal space of fucoals. A geealzao of he Caeo ad Ma heoe [57] was eploed [] o coclude ha each pe of Ase chaos coeges o a L fucoal he L sese he coespodg Hlbe space. I pacula ul-desoal Legede ad Jacob poloal chaoses fo a coplee ohogoal bass of he space of - desoal squae egable ado aables. I hs pape fo splc we cosde ha all aables K ae ( ( ( decall dsbued.e. w w K w alhough he oe geeal case wh depede aables daw fo dffee dsbuos ca be eaed slal. Cosequel wha follows he bass fucos ae eso poducs of sae-fal poloals. I [35359] bass fucos ge b Eq. ( of ul-desoal ode up o P ae cosdeed. The oal ube of bass fucos s: 9

10 ( + P! + K + P! P! ( Ths ube ceases qucl wh he ube of depede ucea aables ad he ode of he poloal chaos P as show Table. Table. Nube of poloal chaos coeffces P P P Aohe appoach s o choose eso poducs of oe-desoal poloals up o ode P hs case he deso of he sochasc space s P+ P L P (7 Ths seg s oe aual fo he collocao appoach as dscussed below. ochasc ODE Foulao Cosde he ulbod dac sse he ODE foulao ge b Eq. ( wh ucea paaees. The ucea paaees ae (fucoal of ado aables ad ca be epeseed usg he poloal chaos epaso as: p p ( e (8 The sae aables of he ulbod dac sse ae also fucoals of he ado aables ha descbe he souces of ucea ad ae epeseed as: ( d ( ( ( ( ( (9 We use subscps o deoe he copoes alog he deesc (sse deso ad supescps o deoe copoes alog he sochasc deso. The supescp-ol oao wll be used o epese he eco of sochasc coeffces T [ ] fo all L d (3 Iseg Eq. (3 o Eq. ( leads o: & & F ; p fo d (3 The equaos fo he e eoluo of he specal poloal chaos coeffces ( ad ( ca be deed ehe he Gale o collocao faewos as eplaed below. Gale Appoach Equao (3 s poeced oo spa{ } K. pecfcall we ae he eseble aeage scala poduc of Eq. (3 wh each bass fuco.

11 Cosdeg he ohogoal elaos hs pocedue leads o he followg (copoe-wse odel: ( ( d p F fo ; & & (3 The odel fo Eq. (3 descbes he e eoluo of he poloal chaos coeffces ( ad (. I s es lage ha he ogal odel ( ad coupled hough he olea es. Noe ha he dffeeal equaos Eq. (3 fo poloal chaos coeffces ae b heseles deesc ad ca be soled usg adoal e seppg algohs. The copuaoal challeges se fo he desoal of hs sse ad fo he olea couplg es whch eque he ealuao of ulple egal es. I he sulao of echacal sses wo pes of oleaes ae pcall ecoueed: poloal ad gooec. Fo powe oleaes f ( he sochasc Gale foulao leads o egals of he fo K K K (33 Ths epesso oles -desoal egals of + poducs of bass fucos. ce each bass fuco ge b Eq. ( s a eso poduc of oe-desoal chaos poloals he -desoal egao educes o copug oe-desoal egals ( ( ( ( ( ( ( ( ( ( ( ( ( P P P P K K K (3 Each oe-desoal egal ca be ealuaed usg a Gaussa uecal quadaue (Gauss-Legede fo ufo dsbuo ad Gauss- Jacob fo bea dsbuo. The poducs Eq. (3 ca be pe-copued ad used houghou he egao of Eq. (3. Fo gooec oleaes howee he ul-desoal eseble e poducs cao be spl o poducs of oe-desoal egals. Isead hese e poducs hae o be ealuaed b a ul-desoal quadaue wh odes q µ Kµ ad weghs q ρ Kρ. The quadaue odes ae - desoal ecos ( l l l µ µ µ K. The e poducs ae appoaed as: ( ( ( l l l q l l f f µ µ µ ρ (35 Oe possble choce s he ola quadaue foulas [5]. To aod ul-desoal egals we ca epad gooec fucos Talo sees ( ( ( ( f a f (3

12 Fo agues epeseed b a poloal chaos epaso he Talo becoes ( ( ( f a f K K L (37 The he e poducs ca be epessed a sees ( ( + ( ( E f a f a f K L K K L L (38 whee he poducs E ca be copued as Eq. (3. Ohe appoaos of gooec oleaes (e.g. Pade ca also be eploed. Collocao Appoach The collocao appoach s oaed b he pseudo-specal ehods [58]. A aa of sochasc collocao fo ucea aalss was poposed [59] he coe of Gaussa uceaes wh Hee poloal chaos. The echque has bee appled successfull he sud of copessble flows wh ucea []. I ode o dee eoluo equaos fo he sochasc coeffces ( we pose ha Eq. (3 holds a a ge se of collocao ecos: [ ] T d all fo µ µ µ L (39 Ths leads o: ( ( ( ( p F ; µ µ µ µ & & ( Cosde he a A of bass fuco alues a he collocao pos: ( ( A A µ A ( The collocao pos hae o be chose such ha A s osgula. The: ( A ( Equao ( becoes: p A A A F A ; & & (3 Deoe he collocao pos he ado sse sae space b: ( ( ( ( p A P A A ( Wh hs oao he collocao sse (3 ca be we as: ( P F ; & & (5 Equao (5 shows ha he dec collocao appoach educes o depede soluos of he deesc sse (. A Eq. ( ges he al codos fo Eq. (5. Equao (5 shows ha collocao s fac a

13 espose suface appoach ehod. I he espose suface appoach [ - 9] he saes of he sse a he al ad fal es ae ge fe desoal appoaos show Eq. (. Afe egao he sochasc soluo coeffces ae ecoeed usg: ( T ( ( T ( T ( A ( T A ( The collocao appoach eques depede us each usg a dffee alue fo he ado aables µ. The coeffces of he poloal chaos epaso a he fal e ae ecoeed usg Eq. (. q We ow dscuss he choce of collocao pos. Le γ K γ [ ] be h he oos of he q ode oe-desoal poloal fo he fal used he cosuco of bass fucos (Legede o Jacob. ce he bass fucos ae eso poducs he -desoal collocao ecos ae chose o hae each copoe equal o oe of hese pos: T µ [ µ ] [ L µ d γ L γ ] fo (7 q Thee ae possble collocao ecos. The deso of he sochasc space wh depede souces of ucea ad wh chaos poloals of q ode up o P s salle ha he ube of possble collocao pos <. Oe has o choose a subse of he ecos of fo ge b Eq. (7 as dscussed [89].The aleae appoach s o cosuc he bass fucos as eso poducs of oe-desoal poloals up o ode P q as ge b Eq. (7. I hs case P+ ad he ube of collocao pos equals he base sze. Noe o he Relao bewee Collocao ad Gale Mehods Cosde he sochasc sse Eq. (37 obaed hough collocao. Cosde q he Gaussa ul-desoal quadaue wh odes µ Kµ ad weghs ρ Kρ q ad use he quadaue odes as collocao pos. Mulpl Eq. (39 b ρ ( ad su oe o oba: µ ( µ ( µ ρf ( µ ( µ ; p ( µ ( & ρ µ (8 Deoe b { } he eseble aeage ealuaed b uecal egao wh he aboe choce of quadaue + + f ( g( w( d ρ f ( µ g( µ { f } f g L g (9 Equao (8 ca be we as: { } & F ; p (7 3

14 If he quadaue ehod s suffcel accuae o pesee he ohogoal of bass fucos he he lef had sde su coas ol oe ozeo e ad & { } F ; p (8 B copag Eq. (5 ad Eq. (3 we see ha collocao ca be egaded as he Gale ehod wh he eseble aeages ealuaed uecall wh a specfc choce of uecal quadaue. We fs cosde he oe-desoal case whee P +. The odes of he Gaussa quadaue ae he oos of he ode ohogoal poloal wh espec o he ge pobabl des. Fo ufo dsbuo oe chooses he Gauss-Legede ad fo bea dsbuo he Gauss-Jacob pos. The quadaue odes ad weghs (e.g. fo ufo dsbuo ae: L ( ( L + µ ρ w( d (5 ' ( µ L ( Gaussa quadaue egaes eacl poloals up o ode ad he ohogoal elaos ae peseed heefoe (5 holds. Replacg he eac egals Eq. (3 b he uecal quadaue foulas Eq. (5 adds a uecal eo e o he gh had sde of he equao (lef had sde s egaed eacl. Fo a sooh fuco f []: + L L ( f ( w( d ρ f ( µ + f ( η η [ ] (53 A (! Hee A s he hghes ode coeffce L (. The quadaue ucao eo fo Eq. (53 deceases apdl wh ceasg ad does o affec he specal coegece ae of he Gale soluo. The ul-desoal case eques he applcao of a eso poduc Gaussa quadaue foula. Fo he bass fucos fo Eq. (7 he collocao pos ae eacl he quadaue odes ad he coclusos of he oe-desoal aalss ca be decl appled. Fo he bass fucos fo Eq. ( he collocao pos ae a subse of he Gaussa odes ad he uecal ohogoal elao leadg o he fo Eq. (5 a o hold. Noe o he Relao bewee Dec ad ochasc Collocao The sochasc collocao algoh of Mahel e. al. [] uses he cuulae dsbuo fuco (CDF of a ado aable o ap s dsbuo o he eal [ ] : ( P[ ] ( α α Υ Υ (5 A se of collocao pos α ae chose he eal [ ] (fo eaple he Gauss-Legede pos. Each copoe of he ado soluo a a ge e s epessed es of he aable α ( ( ( α ~ h ( α Υ (55

15 5 whee he bass h ae he Lagage epolao poloals based o he epolao gd α. The sochasc equao obaed b seg Eq. (55 Eq. ( s foulaed es of he sochasc aable α ad [] s soled usg a Gale foulao he α space ad ealuag he egals b quadaue. Iposg decl ha he sochasc equao holds a he collocao pos α leads o ( d p F fo ~ ; ~ ~ ~ ~ ~ & & (5 Foall he coespodg collocao pos he sochasc aable space ae obaed b eg he CDF of each ado aable ( ( ( ( d α µ α µ Υ (57 Noe ha Eq. (55 ca be elaed o he poloal chaos epaso as follows ( ( ( ( ( h h ( ~ ~ ~ µ α (58 A dffee se of collocao pos he sochasc aable space s (foall used fo each sse copoe. These collocao pos ae ecalculaed fo each e sep (e eal ] [ +. Eq. (58 s geeal ad shows ha he sochasc collocao ehod of [] s o equale o he espose suface ehod. The ehod eques he uecal appoao of he CDF ad s ese fo each sse sae aable. I [] hs s accoplshed b uecal epolaos bewee he ad he α spaces. Fo ulple souces of ucea he eso of CDF Eq. (5 leads o a ( -desoal afold he ado aable space; he collocao pos eed o be chose alog hs afold. I s o clea how o eed hs appoach o ulple souces of ucea. ochasc DAE Foulao We ow cosde ulbod sses he DAE foulao ge b Eq. (. To cosuc he coespodg sochasc sse ha descbes he eoluo of poloal chaos coeffces se Eq. (3 o Eq. ( o oba: + T F M λ & & (59 The sochasc foulao wll eed o ae he algebac cosas o accou. The sochasc eoluo equaos ca be foulaed usg ehe he Gale o he collocao appoach.

16 Gale Appoach I he Gale appoach Eq. (59 s poeced oo { } spa L. Tag he eseble aeage scala poduc of Eq. (59 wh each bass fuco ad cosdeg he ohogoal elaos leads o he followg sochasc sse: + F M T fo λ & & ( Equao ( s a de 3 DAE fo he sochasc coeffces wh d dffeeal ad c algebac equaos. The copuaoal challeges se fo he deso ad fo he olea scala poducs (ohe ha poloal whch ae epese o calculae. Collocao Appoach I he collocao appoach we pose he sse ge b Eq. (59 o hold eacl a he se of collocao pos µ Kµ. Wh he oaos of Eq. ( ad Eq. ( hs appoach leads o he followg foulao of he sochasc sse: ( ( ( ( Λ + T F M fo & & ( The de foulao s: ( ( ( ( ( Λ T F M τ & & ( The poeco o he poso afold eads: ( ( ( ( M T + ~ ~ ~ (3 The poeco o he eloc afold s: ( ( ( ( M M T Ν ~ ( Dec collocao educes o he espose suface appoach whee he DAE s egaed depedel es wh dffee al alues. The ehod s aace due o s e sple pleeao. Noe o he Foulao of Algebac Cosas B solg he algebac poso cosa he sae ca be locall paoed o depede ( dep ad depede ( d coodaes.

17 ( h ( (5 dep d The plc ad eplc foulaos of he algebac cosas ge b Eq. (5 ae aheacall equale fo he deesc sse. I he sochasc foulao: dep h d ( The uceaes he depede ad depede aables ae coelaed hough he algebac cosas. Oe would le o hae he plc ad eplc foulaos of he cosas equale he sochasc foulao as well. I he Gale foulao he cosas ae poeced o he subspace of ees o hold. The plc cosa foulao Eq. ( s geeal dffee ha he oe obaed b poecg he eplc cosas of Eq. (: dep h d (7 The plc Eq. ( ad eplc Eq. (7 foulaos of he algebac cosas ae equale he sochasc collocao appoach: A dep h A d dep h( d (8 Ucea Resuls Te Doa The sascs of a ode fo he oupu ucea ca be deed fo he poloal chaos epeseao. The aeage of he h odel copoe s h ge b he ode e he sochasc epaso ( ( ( (9 I geeal he ea s dffee ha he odel pedco usg ea alues fo he ucea paaees ad focg. The coaace a of he odel sae a a e s obaed as: R ( ( ( ( ( (7 Usg hese easues he odel oupu ca be sualzed wh a eo ba epeseao of he ucea. Fequec Doa The foao he fequec doa s esseal fo a hoough udesadg of he dac sse behao. Nolea espose pheoea ulbod dac sses clude o-gaussa espose ulple esoace fequeces ad hgh-eeg low-fequec specal coe. The powe specu of he deesc sse espose s obaed b Foue ˆ ω (7 ( ( 7

18 A epeseao of he ucea he powe specu s obaed b a Foue asfo of he e sees of each poloal chaos coeffce Foue Foue ( ( ω ( ( ˆ ( ω ( ˆ (7 The sascs ad he pobabl des of he powe specal des of he odel pedcos ca be deed fo Eq. (79. Coclusos Ths pape esgaes he poloal chaos ehodolog fo he sulao of ulbod sses wh uceaes. Uceaes esul fo pool ow o aable paaees (e.g. aao suspeso sffess ad dapg chaacescs fo ucea pus (e.g. sol popees ehcle-ea eaco o fo apdl chagg focgs ha ca be bes descbed a sochasc faewo (e.g. ough ea pofle. Poloal chaos epaso s used o dsceze ado pocesses b eedg he soluo alog he sochasc deso. The poloal chaos bass fucos ae cosuced as eso poducs of -D ohogoal poloals. Ths appoach s chose sce ulbod odels ae lage ad hghl olea fo egeeg sses of ees he ube of ucea paaees s elael sall whle he agude of uceaes ca be e lage (e.g. ehcle-sol eaco. I addo he appoach allows fo he copuao of he ucea odel oupus boh he e ad fequec doas. Mulbod dac sses ae cosdeed boh he oda dffeeal equao (ODE ad dffeeal algebac equao (DAE foulaos. The DAE odel appeas he sulao of cosaed ulbod sses. The cosuco of he eoluo equaos fo he sochasc coeffces s dscussed boh he Gale ad collocao faewos. The Gale appoach o foulag he sochasc ODE s well esablshed he leaue. Ths appoach s eeded hs sud o foulae he sochasc DAEs. The dec sochasc collocao appoach poposed hee s oaed b pseudo-specal ehods. I s show ha dec collocao s equale o he cosuco of a sochasc espose suface. ochasc collocao s appled o boh ODE ad DAE foulaos. A coplee heo fo he coegece of he dec collocao appoach s o aalable a hs e. Mulbod dac sses pcall dspla wo pes of oleaes poloal ad gooec. Eseble aeages eed he Gale appoach lead o he ealuao of ul-desoal egals. I s show ha he eso poduc aue of bass fucos aes he ealuao of eseble aeages copuaoall effce fo poloal oleaes. Howee fo gooec oleaes ul-desoal egao s equed ad hs ca be accoplshed usg uecal quadaue ules. ochasc collocao ol eques he ealuao of fucos a dffee collocao pos. I ca be easl appled o a pe of oleaes ad uses ol he deesc sulao code. 8

19 Ths sud cosdes uceaes fo boh paaec ad ucea eeal focg souces. The wdel-used Gaussa dsbuo has fe suppo ad s o well sued o odel uceaes echacal sses whee ucea paaees e bewee well defed bouds. We odel paaec uceaes usg ufo ad bea dsbuos. Eeal sochasc focgs ae odeled b ucaed Kahue-Loee decoposos. We popose o calculae he coaace fuco of he sochasc pocess fo he powe specu of he focg aace whch ca be deeed epeeall. I he copleea pape Modelg Mulbod Dac ses wh Uceaes. Pa II: Nuecal Applcaos [] we pese seeal applcaos of he copuaoal ools dscussed hee. Acowledgees The wo of A. adu was suppoed pa b NF hough he awads CAREER ACI-9339 ad ITR AP&IM 598. The wo of C. adu was suppoed pa b he AdaceT facul deelope ga 77 fo ga Tech NF ADANCE Awad 9. Apped. Legede Jacob ad Hee Bass Fucos The oalzed Legede bass fucos up o ode fo aable ae ge b equaos (A L ( ; L ( ; L ( ( + 3 ; L 3( ( ( ( 3 3 ; L ( ( L ; ( ( ; L ( ( L ( ( L (A ( ( L L 8 ( ( The oalzed Jacob poloal bass o [-] wh a b up o ode fo oe aable ae ge b equaos (A.. 3 J ( ; J ( ; J ( ( + 5 ; J 3( (

20 ( ( ; J ( ( J ( ( J ( ( J (A J 8( ( J 9 ( ( ( ( J The oalzed Hee poloals ae ge b equaos (A.3 H ( ; H ( 3 ; H ( ( ; H 3( ( H ( ( 3 ; H ( ( H ( ( ; H ( ( ( ( H 8 + (A ( ( H H 8 ( ( Refeeces 7 [] adu A. adu C. ad Ahada M. Modelg Mulbod Dac ses wh Uceaes. Pa II: Nuecal Applcaos ep. subed. [] Dof R.C. ad Bshop R.H. Mode Cool ses 9 h edo Pece Hall NJ. [3] Haug E.J. Copue Aded Keacs ad Dacs ol I: Basc Mehods All ad Baco Boso 989. [] Hae E. ad Wae G. olg oda dffeeal equaos II: ff ad dffeeal-algebac pobles secod esed edo pge Bel 99.

21 [5] Caughe T.K. Nolea Theo of Rado baos I Adaces Appled Mechacs ol. Acadec Pess New o 97. [] Debeg M.F. A eac oluo o a Cea No-lea Rado bao Poble I. J. of No-lea Mechacs ol. 7 No. pp [7] Rubse R.. ulao ad he Moe Calo Mehod Joh Wle New o 98. [8] paos P.D. ad Mgole M.D. Aa Moe Calo ulao Pobablsc ucual Aalss hoc ad bao Dges ol. pp [9] Eese G. equeal daa asslao wh a olea quas-geosophc odel usg Moe-Calo ehods o foecas eo sascs J. of Geophs. Res. 99(C5 99. [] McKa M.D. Beca R.J. ad Cooe W.J. A Copaso of Thee Mehods fo elecg alues of Ipu aables he Aalss of Oupu fo a Copue Code Techoecs (: [] Beg M.. Noble G.. Pe K. Dheu J.R. Mlfod J.B. Hale R.A. Foal Ucea Aalss of a Lagaga Phoochecal A Polluo Model Eo. c. Techol [] Beg M Mlfod J.B. Applcao of Baesa Moe Calo Aalss o a Lagaga Phoochecal A Qual Model Aos. E. ol. 33 Issue Ja.. [3] Bule D.M. The Ucea Ozoe Calculaos b a aosphec Phooches Model Geophs. Res. Le [] olas R.. Ucea ad ses sudes of saosphec phooches Poc. of he NATO Adaced ud Isue o Aosphec Ozoe: Is aaos ad Hua Iflueces eded b A.C. A Rep. FAA-EE U.. Dep. Of Tasp. Washgo D.C. 98. [5] Due A.M. The Decoupled Dec Mehod fo Calculag es Coeffces Checal Kecs J. Checal Phscs ol [] Nafeh A.H. Peubao Mehods Wel-Iescece Lodo 973. [7] Nafeh A.H. Ioduco o Peubao Techques Wel-Iescece Lodo 98. [8] Adoa G. ochasc ses. Acadec Pess New o 983. [9] Robes J.B. ad paos P.D. Rado bao ad ascal Leazao Joh Wle ad os New o 99. [] Falsoe G. ochasc Leazao of MDOF ses ude Paaec Ecaos I. J. of No-lea Mechacs ol. 7 No. pp

22 [] paos P. ad Ghae R. Bouda Elee Mehod Aalss fo Rado bao Pobles J. of Eg. Mechacs ACE ol. 7 No Feb. 99. [] a de Wouw N. Nee H. ad a Cape D.H. A olea ees Appoach o he Appoao of ochasc Nolea Dacs. Nolea Dacs ol. 7 pp [3] Ca G.Q. L.K. ad Elshaoff I. A New Appoae oluo Techque fo Radol Eced No-lea Oscllaos I. J. of No-lea Mechacs ol. 7 No. pp [] Taag M.A. Pa W. P R.G. ad McRae G.J. A effce ehod fo paaec ucea aalss of uecal geophscal odels J. Geoph. Res [5] McRae G. New Decos Model Based Daa Asslao MIT Checal Egeeg couse. [] Isuapall.. ad Geogopoulos P.G. Popagao of Uceaes Phoochecal Mechass hough Uba/Regoal cale Gd-Based A Polluo Models Poc. of he A&WMA 9h Aual eeg Tooo Caada Jue 997. [7] Isuapall.. ad Geogopoulos P.G. Copuaoall Effce Mehods fo Ucea Aalss of Eoeal Models Poc. of he A&WMA specal cofeece o Copug Eoeal Resouce Maagee A&WMA IP [8] Isuapall.. Ro A. ad Geogopoulos P.G. ochasc Respose uface Mehods (RMs fo Ucea Popoagao: Applcao o Eoeal ad Bologcal ses hp:// ssepo/ss.hl Feb [9] Isuapall.. ad Geogopoulos P.G. Deelope ad Applcao of Mehods fo Assessg Ucea Phoochecal A Qual Pobles Ie Repo pepaed fo he U..EPA Naoal Eposue Reseach Laboao ude Coopeae Ageee CR [3] aooch R.L. Topcs he Theo of Rado Nose Godo ad Beach New o 93. [3] Robes J.B. ad paos P.D. ochasc Aeagg: A Appoae Mehod of olg Rado bao Pobles I. J. of No-lea Mechacs ol. No. pp [3] Zhu W.Q. ochasc Aeagg Mehods Rado bao Appled Mechacs Reew ol. 38 No. 5 pp [33] Wu W.F. ad L.K. Cuula-eglec Closue fo No-lea Oscllaos ude Rado Paaec ad Eeal Ecao I. J. of No-lea Mechacs ol. 9 No. pp

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24 [9] Xu D. ad Kaadas G.E. A New ochasc Appoach o Tase Hea Coduco Modelg wh Ucea I. J. of Hea ad Mass Tasfe [5] Xu D. ad Kaadas G.E. Ucea Modelg of Buges Equao b Geealzed Poloal Chaos Copuaoal ochasc Mechacs Poc. of he h I. Cof. o Copuaoal ochasc Mechacs Cofu Geece Jue. Eded b P.D. paos ad G. Deodas 55- Mllpess Roeda 3. [5] Xu D. ad Kaadas G.E. Modelg Ucea Flow ulaos a Geealzed Poloal Chaos J. Cop. Phs [5] Xu D. ad Kaadas G.E. upeses due o Ucea Bouda Codos I. J. fo Nuecal Mehods Egeeg subed 3. [53] Xu D. Luco D. u C.-H. ad Kaadas G.E. ochasc Modelg of Flow-ucue Ieacos usg Geealzed Poloal Chaos J. Fluds Egeeg ol [5] Luco D. u C-H. ad Kaadas G.E. Geealzed Poloal Chaos ad Rado Oscllaos ubed: I. J. fo Nu.l Meh. Egeeg. [55] Luco D. Xu D. u C.-H. ad Kaadas G.E. Pedcabl ad Ucea CFD I. J. Nu. Meh. Fluds 38( Oc. 3. [5] Keese A. A Reew of Rece Deelopes he Nuecal oluo of ochasc Paal Dffeeal Equaos (ochasc Fe Elees Isue fü Wsseschaflches Reche Techsche Uesä Bauschweg Ocobe 3. [57] Caeo R.H. ad Ma W.T. The Ohogoal Deelope of o- Lea Fucoals ees of Foue-Hee Fucoals Aals of Maheacs ol. 3 No. Apl 9. [58] Tefehe L.N. ad Baw D. III pecal Mehods MATLAB IAM 998. [59] Mahel L. ad Hussa M.. A ochasc Collocao Algoh fo Ucea Aalss NAA/CR-3-53 Feb. 3a. [] Mahel L. Hussa M.. Zag T.A. Baalle F. Ucea Popagao fo Tubule Copessble Flow a Quas-D Nozzle Usg ochasc Mehods AIAA h AIAA Copuaoal Flud Dacs Cofeece Jue 3- Olado FL 3b. [] Aso K. A Ioduco o Nuecal Aalss secod edo Joh Wle & os New o 989.