ADAPTIVE MULTISCALE HOMOGENIZATION OF THE LATTICE DISCRETE PARTICLE MODEL FOR THE ANALYSIS OF DAMAGE AND FRACTURE IN CONCRETE

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Ssl gg Gologl f ls (SG Dpm l ml gg om Shool gg ppld S s llo 6 US DT ULTSL HOOGZTO O TH LTT DSRT RTL ODL OR TH LYSS O DG D RTUR ORT Roozh Rzh w Zho Gl s SG TRL RORT o 7-/57 Smd l ol Solds Ss 7

dp ll Homogz L D l odl ls Dmg Roozh Rzh w Zho Gl s Dpm l ml gg ohws Us s L 6 US S 55 Ws S S Dgo US s Th pp pss w dp mll homogz shm sml dmg f ss wo-sl homogz mhod oplg mso-sl d pl modls mosl f modls mld dp fmwo m mll fl ld f-sl smls ss dp mll fmwo h ppoh smls s who sgg ml Rps olm lm ( mo-sl f f m dp ms d mll homogz fmwo dd fl Th lds ssl d ompl os sp lodg ds ldg dmg lolz whh l sm po msh hd wh homogzd Sl ml smls d o sg pl dlopd dp homogz mhod dd dld sd ompl os pmd Kwods: dp ll Homogz; dp ; Dmg od ll l m-md gg mls hogos lgh sl osop mhl pops mls sh Yog s modls sl ompss sghs hgoml hs sgl dpd fs s hogs o s mhl pops mos gg ml h dl dpd mhl pops ggg m dms ggg sz d ggg olm f mplog dld f-sl modls whh ml hogs o l od o m ml How ml sml lg sl ssms sh dms dg ps l pow pls f-sl modls ld omos m dgs fdom poh ompl os Th h mod shs dlop dff gos mll ompl fmwos ospdg ho -ml ddss: g-s@ohwsd p smd ls 7

whh f fom lzg mhl modl ls lg gg polm wh l ompl os ll ls fmwos f wo pm gops [] fs gop lds hhl mll modls whh f-sl polm smld os o d mhl modl opld mo-sl polm g smld Hh ompl fmwo slhd hogh solg mo- f-sl polms dpdl whl m flows w sls dg ls sd go lds mll modls whh polm dom domposd wo sdom sdom d homogos m modld o whh dms sdd l g whl o sdom modld dld f-sl modl s wo sdom d hogh ppop ss wo sls sold smlosl ll homogz hhl mhod whh h sl sgd o p dds ll homogz mhods w fs dlopd d mml qs ff ml pops hogos solds d sol f-sl od l polms [ 5] How s ppohs lmd l mls wh smpl pod l ss sdd d sm dm flds ompl homogz mhods w dlopd q s lms s mll fmwos Rps olm lm ( f-sl ml sd sgd h ompl po homogzd sps sdd ff ml ho o spf s qs dfd mo-sl Dg h ompl sp s Gss po ppld ld ml od d sol ll homogzd whh sfd ompl po ls ompl homogz h wdl ls ompld gg polms [6 7 ] Smh l [6] dlopd mlll f mhod whh oh dom mshd wh ( mplod ls pd ol mls l [7] slhd mll homogz fmwo glzd whh s o Kozso [] mld hgh od homogz shm whh fs sd od gds dm fld ppld od ds show gd hd homogz mhod -m dm flds podd wh od l polm h l [] ompl homogz ls pols mls d lg pl dms dd ompl homogz mhod h opl d modls (D Go l [] sdd l dsp sps gl md d m l lodg d opld /D ppoh Wg [] l dlopd hhl mll hgo-mhl modl sg fld flow gl md smp ps Homogz (H mo mm hoogh ppoh mplos mp ps fld ls d gog qs ospdg od

ds mo- -sl od l polms [] h l [] lzd h ppoh sd ho m ssms sg sl sp oh sp m H o mplm ompl sws h sl shs sd mhl sps mls Hs H [ 5 6] plhd h-pp w whh psd dls H o s f mplm l mls wh pod mos psd o w ompos ml wll sl polog opmz h l mld H fmwo sg l-pl ho omposs [] gl h [7] poposd dd od H ppoh sml l ho f ompos mls wh dd ompl os Ghosh l omd mp ps ppoh wh oo ll lm hod ( dlopd mll ol fmwo ppom l pops [] sg l-pl sps [] omposs wh om mso-s Wh l mll homogz mhods ompl dlopd fmwo h mo sgd pd ompd ompl ff mo shoomgs lsl homogz mhods whh ml s sgd mo-sl s fom gg ls mdos ompl pss ss sol od l polms h g po sp sls dg ls gg polms dlg wh f fl ss g dmg s lolzs g s ml dom m l sgg ml s g pos spflos h gd Ghosh l [ ] poposd dp ml-ll modl mll sml f hogos omposs whh h modlg sol lls modl ss m dmg mhs modl d mp homogz mosl smls omposs pmd fmwo mld g gog s w d sls Rgdg ls ss S L [] dlopd dp mll whh g modl ms spf pld f-sl dz og hog h mll modl ho sdd l p p dp fmwo lmd smpl D polms ps sd ds d m homogz mhod l dlopd hos [] dp fmwo splzd [] Rzh s psd homogz fmwo opl L D l odl (LD [ 6] whh ml s smld LD sgd s hd ls Two-sl ml mpls w sold ompl dlopd mll fmwo w fd ompg wh fll f-sl smls pp s lzd wh oop l l s q Dg sml dp shm m ds s m d sgd ml Th sg lds sdl sg ompl os

whl sll dmsd sf Rw L D l odl (LD L D l odl (LD ss goml ps mso-s hogh followg sps: ( os ggg ps whos shps smd sphl odd olm -- om pod ggg dms dmd smplg smd ggg sz d f Sphl gggs d pl dog spm dpd g ( Zo-ds ggg ps (ods oml dd o l sfs fl ppl od ds ( Dl hdlz gd ggg s sod h-dmsl dom ssll d o o wo gl fs sd h hdl show g g llss po hdl sod wh s fo ods ospdg fs omg sh pos fom hdl d sm od o ospdg polhdl pl whh loss sphl ggg Two d polhdl pls g hogh shd gl fs dpd g d olhdl pls sdd gd gl fs whh s qs dfd l m smd pol ml fl los g pss polhdl pl ps pl dog spm shold sd sphl gggs gd ld d modl whh smls l mso-s whl o dl ml sol pod od sphl gggs d od h pp sod polhdl pls goml s mplod ml ls Th ss qs ss ompl d modl fmwo: df s h f s q whh ls f f s pl qs s df Rgd od ms mplod ds dm l/pl ssm dplm mp od h f df ms s ɛ α α ( α L (ɛ f oml s omp; ɛ ɛ L f gl s omps; lgh l s ods shg f lso sod hd dg (s g ; α s dfg f lol s ssm f sh ohogl f L f gl s (s g l s qs l s lw gog ho ml

( K ( L L od (d L LD lls olhdl pls ( g : ( Sphl ggg d pl dog spm pl LD hd g fo d gggs s sod ssll ( Thd po sod wh ggg (d LD polhdl pl losg sphl ggg ps ( olhdl pl ps pl dog spm mposd od h f l gm oml sh s popol ospdg s: ɛ ; T ɛ ; L T ɛ L ; T α ; ff oml modls; α sh-oml oplg pm o s s d l lm msosl l phom hzd h mhms ospdg f ll l s qs fl dsd low ohs d s s-sh o sl lodg (ɛ > fg ho mld hogh ff s ɛ ɛ α(ɛ ɛl ( L /α whh df f oml sh s ɛ (/ɛ; αɛ (/ɛ; L αɛ L (/ɛ ff m l (ṫ ɛ ms sf ql (ɛ ω (ω p [H (ω ɛ m ɛ (ω / (ω] ( whh m{ }; ɛ (ω (ω/ ; (ω ɛ / αɛ T α/t whh ɛ T 5 ɛ ɛ L

T L ω pm dfs dg w sh oml lodg ɛ m ɛm αɛ Tm h dpd l ɛ m ɛ s lodg pos p sg modls dfd H (ω H (ω/π sg p H sg modls p s (ω π/ pssd H / (l / ; l G / ; G msosl f g LD pods smooh s w p s p sh (ω wh pol sgh g (sold g (ω s ( s(ω s (ω α os (ω/ s / [α os (ω] ( s s / sh sl sgh o dhd l g pss sgh dom ospdg ɛ m / T ss ɛ ɛ T show g p s (ω p sh (ω π/ ω π/ sg α G s 5 l g pss lodg-lodg l dopd h wo ms ff ss ff s lshp whh ɛ (ɛ m / ml pm dfs sz hss l omp po ollps ompss oml s ompss lodg (ɛ < ompd hogh ql (ɛ D ɛ s-dpd od dpds olm s ɛ f d s ɛ D ɛ ɛ olm s ompd olm hdl ɛ smd s fs lgg g hd d l lm modls po ollps (ɛ ɛ ɛ l ol g olm s wh sffss H omp hdg d po ollps lm (ɛ ɛ ɛ / omp s whh po ollps ss ɛ κ ɛ omp s gg hdg κ ml pm msosl ompss ld w (ɛ < (ɛ D ɛ ɛ ɛ H ( D ( ɛ ɛ ( ( D p [(ɛ ɛ H ( D / ( D ] ow H (H H /( κ D κ H ; D ɛ D /ɛ (ɛ > D ɛ D /(ɛ ɛ (ɛ whh ɛ κ ɛ H H κ κ κ ml pms s ods show g d ompss oml ss s D D lodg-lodg ph lso show D whh d dsfd oml modls ml pop LD pms sdd plog s s 6 G H / 6 H / 6

dp ll Homogz dp ll Homogz dp ll Homogz dp ll Homogz [] T [] [] T [] w ' whh ql pls Usg s lss wh q w O( f sgl f sd gog w gog qs sgl pl sd sd O( -sl qs h qs pl sd gog O( -sl gog R O( -sl qs gog pol ; ms ds lg s ms O( ss ' po s fs os-sl qs gog O( f -sl whh olm f O( os-sl qs polm qs ldd polm ls olm ; d o olm ; ms ds ol qs o gg gg ol qs qs q polm fs m q 6 whh O( f sdg hh op f s w os-sl qs gog polm sgl w polm ml gog f O( -sl qs gog polm os-sl q O( qs w f lsl pl dl q gog O( mpl q O( q ld ms f O( pos f od shd pls lol low-sl ood ssm q 5 qs p ml oml qs df O( lsl pl dl q go omps sdg q po ldd fs m q 6 lsl pl dl q go O( f whh w f whh w O( whh f w olm ; ds m q gl s w ms low-sl dplm how g op s s fs s s fs po f whh f w olm olm ml mpl q O( q ld lsl p s gog polm olm ; lsl p ml mpl q h sll sll d qo mplo pl df sll q o whh ldd m q 6 O m ; q ml ml q q O( s lsl omps whh fs f ldd w f w ( ml mpl q ml mpl q lsl pl dl gog mpl O( q m O( O s po ms ds whh wh olm ; ms ds dplm sll lsl pl dl gog ql ml lsl pl dl q gog olm 6 s sdg q fs m q sll q ospdg homogzd q ospdg homogzd m dfd olm ml mpl q ml mpl q O( whh osop polm q gos polm follows oml gl s w ms low-sl q gos polm follows q gos polm whh follows wh sld ; ol qs gg qs o R ml q ld sol sl O( gog follows fs (6 q ls sd s ml q O( ld mpl sll ms ol hogh p m q O( q w show g op po s fs q homogzg f whh hogh s ds hogh homogzg ms sd fld ls l dpd ssm ms ood m sld dl fld ol l dpd m pl dl q q oml lsl m m O( sdg f oml gl w ms low-sl q m o gl s omps sdg ldd m 6 6 mpl mpl (6 sl fs olm m q f (6 q ldd m q ld po sd m ml q O( q ld sl q gos polm wh homogzg lsl pl m q gos polm whh wh lsl pl d dplm gg qs o pls dplm ml ql s o q lsl pl m h q m w ms follows ; hogh m mpl low-sl ol qs whh q po 6 m sld o olm ; po m dfd ol qs q ol qs gg ol gg ql ol qs gg qo ol (6 o (6 O( os-sl qs gog polm ol q lsl pl dl q gog o m m hogh homogzg sol p m q O( q ld sll lsl pl dl q gog m m hogh homogzg sol hogh homogzg sol R m m ol qs gg ql d ls f oml gl s w ms low-sl dplm ol qs gg ql o 6 qs q po po m dfd po m dfd mos olm ; ms ds m q lsl pl dl q go polm sll q ospdg homogzd l ol gg qs o pls mo sd m q (6 l O( -sl qs gog polm ol qs gg qs o pls mo-s m sll q m m ol q osp m m gg qs o pls mo-sl ol q ospd oml gl s omps sdg ldd fs m q 6 o m m m m ml fl mpls qs whl s o s l dom smoohl ospdg mom s s oml po qs q df sd low-sl ol ms dplm (6 gl hogh homogzg sol polm dd m s m gg olm ; mo-sl w smoohl f w low-sl m ol m q ; s ms dplm mo q 6 mo-sl whh m slo pl dl gog q p ol qs o pls mo-s qs ds m q 7 ld lsl mpls (6 h sll ospdg homogzd pssd ds qs s m (6 sll sgl f-sl h o 6 sll ol polm m m hogh homo ol ql q sll q ol m po q gs polm polm m m m m osop s qs O( -sl gog ms f m olm sol sll wods olm ; polm ms d sll q ospdg ospdg hom p sll q ospdg po m dfd ql s sll homogzd lsl dl q gog ol homogzg q q sll q l ls m po pl gg qs o pls mo-sl hogh sol hogh homogzg sol po ol ospdg l ol ospdg mo q ospdg mom q ospdg hom d ls q ospds ls sll q ospdg homog po m dfd s hogh homogzg ml mpl q O( q ld sll oss m o fd s dfd sll q ospdg homogzd pssd po m mo sd m q 6 po sll q ospdg homogzd pssd l d ls q o lsl pl dl q gog q po s s spl s spl s s spl f-sl df f oml gl s w ms low-sl dplm ospdg homogzd pssd qs o pls mo-sl s h f o wods sol polm l m m s s spl dom H ospd dplm o flds spl l olm ; ms ds m ol q ospdg dd polm dom H ospd dpl osop s mo sd m q 6 ospdg lsl homogzd m po ( follows sol q R olm l fl po o hogh flds q do m q 6 s polm gog whh pl dl polm sld lsl pl dl q pods ol q pssd q gos wh sld wods sol polm m m ol ol sll q s h f o wods homogzg sol polm dd m ol polm osop spl m q q ppld fs g gs f-sl s s spl gog sll s h m o lsl pl dl q gog s oss o fd h po m dfd d po m df d ls q o @ hogh homogzg sol s s spl ospdg homogzd pssd s pod l ol q ospdg mom ol q ospdg s s s spl s s spl m ows wh sld polm s o shd s spl d qs qs gg o whh follows wh sld s so q oss m d dls ospds o qo mo-sl hd o l ssm f pls po m dfd pls q fd ol ospd q gos f ss d ls ospds ol f o s spl d pssd ol h m l f od shd w pls l lol low-sl lsl po pos od w pls dd lol low-sl ood ssm q pos pos od w lol ood ssm flds f-sl mo sd m q 6 ld mo-sl polm s f o wods sol polm sdg f-sl m hogh homogzg sol dd fl lsl pl dl q @ d ls q g f pss gd gdg df ' oss m o fd d ppld fs g gs sol mosl s h o wods sol d low-sl polm pos f od shd w pls low-sl ood f pss gdg pl s g s s spl gog polm ood s s spl osop s o o f wods sol polm hogh homogzg sol polm fl @ m gs sol lsl dl d lol gd dd w @ shd od shd pls lol low-sl ood ssm hogh homogzg od w pls lol low-sl ood ssm s pod fs ol q ospdg sol os shold ppld fs g gs f-sl sol s h pl ms lld sfg s @ hogh homogzg sol polm dd s pod oss m po m dfd d ls s so sll q ospdg homogzd pssd ( d ls q ospds pos f od shd w pls lol low-sl ood q 5 q pos f od shd w pls lol low-sl ood ss q 5 qs oss m o fd d d G s l / m & omposs ( pos f od shd w pls lol low-sl ood ssm q 5 qs s pos f od shd w pls lol low-sl ood ol q ospdg m g f m wods show d ls q ospds lsl ol gs polm s fs oss o fd d dls o Rf [] po pos od shd w (7 lol ood ssm low-sl show g pls op s fs dd show g po s s qs w q op po s s fs sgl op dls mosl s s h f o sol polm hogh homogzg sol ssm f (5 s shold fs g f-sl sol ' ql Usg lss lg wh q w fs ood (5 shd q ospdg ' 'fs ql Usg lss sfg os shold ppld fs g gs f-sl sol o f od shd w pls lol low-sl ssm 5 qs sgl pl sd s ol pls ppld s h pl ms lld s od pls lol low-sl ood oss m o fd d q 5 qs pl sd sd op fs ol q ospdg dd show g op po s s R @ l fs g gs f-sl sol ms lld sfg s pls @ hogh l po s s fs s pod op po s s ol q ospdg sol (7 sod os h pl ms sfg s qs ol q ospdg O( f lld gog f os-sl polm (7 ds show O( os-sl qs ls pos f ldd (5 d q m olm R o oss m o f ; f od shd pls lo m d O( show g sdg po s s R whh olm f show g ol po s s s O( f fs whh w g op oss po s s fs O( op f whh olm f R q w o fd d dls o 6 oml o op oss m fd dls o qs Rf [] op show g (5 po s m o gl sdg q fs m q 6 d ms oml o gl s omps q ldd fs q op fs w O( oml po s s fs omps q O( g o gl s omps ldd m q 6 sdg fs 6whh fs s ospdg @ Gss po s s shold ppld g gs f-sl osop ms dplms O( os h pl ms lld sfg s sdg 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low-sl dplm Homogz ; q-l mls lsl pl dl q gog wh olm ; ms ds qo olm ms ds m q omp ω sol mo sd m q 6 opl ms sll lsl m d dpl mo sd m q 6 m sll po s o m o o m dp ll po s mo m m fs ppld O( os-sl qs gog polm po po q olm ; m po m olm ; ms m q O( q ld sll omp opl ms ds m q 7 po dfd qs qs m q m o flds sd m 6 po H ol gg qs o pls mo-sl m ol qs gg o pls mo-sl ol l qs gg qs o pls mo-sl ol q ol m m ol q o w o q-l mls s lolz os fl l O( f whh f w f-sl df f oml gl s w ms m m o flds mo sd m q 6 po o flds mo sd m q 6 poo d ls q lsl pl dl q gog h d ls qo po m dfd o flds mo sd m q 6 po lsl pl dl q gog q po m dfd mosl s s h f o wods sol polm lsl pl dl q gog m m olm ; ms ds m qo mosl s s h f o wods sol polm m m ld sll mosl s s h f o wods sol polm lsl pl dl q gog q l m m olm ; l ω o q-l mls q 7 m po m dfd sll q ospdg homogzd pssd osop s R lsl pl dl osop s ml s qs mo-sl qs ds mo sd homogzg l dfd fs q sll q o gg q osop s qs o mo-sl polm flds gg pls ld d ml mpl q mo dp lsl ld l ls l o pl ospdg po l ol q s pl l T ol ospdg f opl s q m m m omp O( sll m ll Ho m q O( q ld sll ml q O( mo sd m q 6 po po ms m m q 7 d mo sd q 6 6 po q mplo lsl dl q gog q pod hogh sol polm ml mpl O( lsl pl dl q gog q pods d q o flds pls mo sd m q 6 l po po s h o wods sol sm polm m ( s h f o wods polm m m sol l mpl f lsl g ml do polm q m m osop polm m dp ll Homogz osop s q ospdg lolz oss o po m d s po m df d o lsl pl dl gog q gos polm whh follows wh sld sll q homogzd pssd gg qs o pls mo-sl d s pl dl q gog ql sll q ospdg homogzd pssd s gog q pods q gos polm whh follows wh sld sll q ospdg homogzd pssd α d ls q ospds h dfd 6 oss homogzg mosl s h m (ω lolz h g whl m o q-l mls s lolzo m mosl s f o wods hogh sol polm m ml ds ml q O( q ld sll wods s m ls sol s mpl ; q 7 po o flds o mo sd q polm hogh dom h m o fd s s f o sol m hogh homogzg sol ospdg oss m o fd mosl f o wods sol mos sol po m dfd (6 m mosl (6 o pls mo-sl po dfd f-sl polm m h f wods polm s shold ppld fs g gs hogh homogzg sol p d ls q osp d mqs q pods s shold fs g gs f-sl s f s s m sol d q ospds ppld ms ppld hogh s spl s shold fs g gs homogzd pssd lsl dl q gog pod olm m l s s spl do f d homogzd pssd l d ls q ospds lsl q homogzg sol dd pl dl q m m m d s ospdg s ms q-l mls s lolz fl Th h osl s h o sol polm s sol sol gg qs l lolz f-sl g ml s homogzg o polm ol q l ol gg qs o p s wods ol pl os moso s s pod s pod dp ll Homog dld ol qs qs o pls mo-sl pod gg qs o pls mo-sl hold s h f o wods sol polm hogh homogzg sol polm dd fl mool q ospdg s ol qs gg qo d s h f o wods sol polm gog q pods hogh sol polm dd fl mo-s q ppld fs g gs f-sl sol ol q ppld fs gs f-sl sol d ls ospds h g o q-l mls s lolz mosl s s h f o wods sol polm m sgg ml f l d d lolz oss d shd ls oss @ pos f od w p homogzd pssd 5 pos f od shd w pl @ hogh homogzg sol o O s lsl (6 sol hogh homogzg sol pos shd w pls f-sl sol zg polm dd fl mo-sl s m pod ppld ppld sfg (6 o f o oss Bos dospdg qs qs o pls mo-sl polm m m lolz h pl dl q gog q pods sgg f dz ml od m m mls gg pod oss m o fd d dls o gs d ls 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dp Homogz ls q 6 po h lsl pl dl q gog q pods o flds mo sd m q 6 po lsl pl dl q gog q pods wods sol polm lsl pl dl q gog m m fs f ms s sg ff dp homogz fmwo mosl s s h f o wods sol polm m m lolz lodd ll f s polm dd lolz g ml dom ms dplms os h p lolz g osop s ls h hp ppop d ls q ms dplms os h pl sgg ss s ompl d ls q o lolz g m sgg m polm ss s ompl os ms dplms os h pl ms l polm dd fl mo-sl osop s d ls q ospds h sgg ml l dgog ms po m f dfd l h ospdg dd po m dfd ml dom lsl pl dl q q q pods ospdg q-l lsl pl d l m l sgg ml f dz s dd l D lld lolz g ml whl m D ml mls os fl Th dmg ds f gog f s g m sol dom fs ms polm lodd ll f pl ms po dfd po ospdg ls lolz ll dd sg ff dp ho lolz o g ml dom whl m ms os h pl ms sfg mo lolz ml dom whl d ls q ospds lsl ol q (7 ms dplms l os h lld sfg s mom sgg d ls q ospds lsl ol q (7 oss m fd d dls o qs sgg ml L l dp ll (7 D m dfd d oss zg dd mo-sl d ls q ospds lsl ol ql s lsl ol q po m df o s s spl ss pssd dplms Homogzd d l lodd f s s q mo-sl s q homogzd pssd 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pod od ds d sol o qs [] sgg ml f dz ( o q-l mls s lolz os fl Th ms dmg ss s ompl os mdosl ( @ s ompl os mdosl T ( LD s ml d ls q lsl ol q LD s pos spdg dd LD sol oss m o fd d dls o qs Rf [] d ls f shd w pls lol low-sl ood ssm oss m o O ( ss s f od shd w pls T ls h hp ppop fs dlopd som ml mpls s od ls s sg dp homogz fmwo 5 d q ospds lsl ol q ss s ompl os mdosl oss l q ospdg dd oss o lodd ll f s s m ls hp ppop fs ms h oss lolz os g ml dom whl dom m l l q ospdg ( fs fs f-sl sol ( o q-l mls s lo s d lld sfg s ol ospdg mom LD sol dd s shold h ppld fs gs f-sl sol LD sol gs pl ms sfg s g dplms lld ml s ompl os q ss ls ss sdg l oop fs ls ss sdg l oop doss dls o qs Rf [] qs f fd qs f omp ss lolz g ml dom whl ml ss s qs omp mos whh f l d ss s omp whh f lm o q-l d po g ( whh f d pod B: ( (7 dp ( oss m o fd d d f ol gm l followg dp shm poposd h oss o d dl ol ospdg d spdg dd ls h hp ppop fs dlo q m d dls qs Rf [] l sgg ml f ml dom dp l f ol q sg ff dp homogz fm m dp ll Homo ml omp (7 s dz f o fd whh lolz o g dom whl ml dom ls h hp ppop f qs ompl sg ff dp qs omp opl s ss s os mdosl ss s ompl os mdosl f opl s ss s ompl os mdosl whh f ol gm l foll d pod B: whh f ol gm l follow d pod B: ( lolz g ml dom whl ml dom m l oss m o fd d dls o qs Rf [] ( p g op po s s fs o q-l mls s lo oss m o f show g op po s fs sg ff dp homogz fmwo dp ll Homogz dp ll Homogz qs dls f d Rf [] ( o L hp dlop whh ol gm l followg homogz ol od sol pl ms lld sfg s ls h ppop fs sgg f dz ml qs whl pod od d s sol ms dplms ml os h pl ms sfg s ds sg ff dp fm f fs f ms gm lld 6 5 f 5 fs f ms whh @ 5ls ss 5@ lolz g ml dom whl Lf sgg ml f dz h whh whh f ls s whh f ol ls ss lolz g s ls ss sd sdg l oop ho o ompl os mdosl mplmg dm @ ss @ Gss ff dp homogz fmwo sg po ml f l dz mos fs L L sgg l sg dp hom ( whh f ol gm oop followg o whh f ol gm followg d lld ls ss sdg s h followg f f ol gm l ls l s h lld fs ( omps [] R l o gl sdg ldd m q 6 sdg sgg s lodd ff oop ml dz ml [] l [] oml o gl omps sdg q s ( ss ll L ( lolz g ml dom whl @ q olm olm whh ms ds m 7 dp ll Homogz f ms ds m q 7 o q ll L o q- olm ; ms ds m q 7 dp o q-l mls q ( h opl s s d sol lld fs ds qs omp opl s ls lld fs ( ; ss sdg l oop s ho fo @ ( qs whl pod od d sol ; ls ss sdg l oop ( ml lodd ll f s ( ( ( lodd s f ss s ompl os mdosl mplmg ss s ompl os mdosl m sg ff dp homogz fmwo ls ss ls ss s fs ls ss sdg l fs f fs f ms sgd f sgg f h ( sgg ml ms ds m q 7 fs f m ; ms ds m q 7 o q-l mls s os fl ppld ol l followg dp shm dp poposd s h sd o os f gm gog polm mls lolz p whh q-l O( ppld lolz fl qs polm ll Homogz dp ll Homogz dp ll Homogz os-sl ss s ompl os mdosl mplm l l ls ss sdg oop s ho fo ls ss sdg oop s ho fs f ms fs f ms s -sl df f oml gl s w ms low-sl dplm ls ss sdg l oop s ho fo f Homogz f-sl df f oml gl w d ss q s ompl os mdosl mplmg lsl pl dl q gog pods LD s lws ( g dmgd sh sgh Homogz pos oml s; Tpl ss s s LD f ll; pl :lsl pl dl gog q pods dp dp ll q g lsl dl q gog q pods ll lolz g lolz g h ll lolz g ml fs o omp sgg ml f fs opl s g LD f s lws: ( l dmgd sh f pos oml Tpl fs f ms s qs f omp opl s ls h hp ppop ls h hp ppop ( gm sgh ss f ol gm l followg dp dp ll Homogz fs f ms sg whh dp whh f ol l followg dp shm poposd dfd q gog q pods gog q pods fs f lodd ll f s s q fs f lm ( Ulodg-lodg modl; (d Tpl oml oml s s ompss; ( Sh sgh f g oml s (fl lodd po m dd fs f ms lodd h po m dfd dd lolz g ml dom whl m ls ss sdg l oop s ho f o ss lolz g ml dom whl ml lodd ll po m dfd dd l d D l dp D ss l s ompl os l dp l D q-l mls s lolz os fl Th gm o dp whh ol lmodl ompss f ss ll Homogz s fs O( l f followg dp shm mogz lodd m ld sll s LD ( Ulodg-lodg (d Tpl oml ss oml fs f ms sgd f ms dd sgd ll f s q m q 7 dfd olm dd sg po m dp d ml mpl ld sll lodd ll f s s fd mo dd (f sh ss s mls flds sd m q 6 o ol l polm po ; ms ds m q 7 hogh homogzg fl mo-sl po m dfd hogh homogzg sol polm dd fl mo-sl ll Homogz l flds mo sd m q 6 D hogh homogzg sol polm dd fl mo-sl l dp ll Homogz D f m m fs ms sgd sh ol q ss dfd Tpl l ho; ( l sgg ml R l l whh f gm l followg dp shm poposd sgg ml sgg ml f o q-l s lo o q-l mls s lolz o q-l mls s lolz os ( lodd ll f s sg ff dp homog dd q sg ff dp homog ss Tpl s ompl os m ls ss sdg l oop s ho f dp ll Homogz s fs fl Th ms dmg ds homogzg sol polm dd fl mogzg sol polm fl mo-sl dp Ho ls ( mo-sl l d ls ospds lsl ol q sgg ll f s hl d q ospds lsl ol os s s ( Sh sgh f ompss s (fl ho (f ss sh ls h hp ppop fs dlopd ml mpls q d ls q lsl ol q mogz lodd ls ss sdg oop s ho f o ll lodd ll sgg ml os f dz sls f ms ospds sgd f sd l ml f dz lodd ll f ls whh f o ls h hp o q-l mls s lolz sh fl ms dmg som Th lolz o q-l mls s lolz fl Th ms dmg ds lodd g ml dom whl ml dom lolz whh f ol gm s s o pls mo-sl ls ss sdg l oop s ho f lm sl dd q pods ls q ospds lsl ol q ol qs gg qs o pls mo-sl s q ospds lsl ol q lsl pl dl q gog q pods ol q ospdg dd ol q ospdg dd lodd ll f s d ls q ospds lsl ol qo lodd ll f s h q ospdg dd s ls f h hp fs d ( mls ol d ls hp ppop fs dlo o mls s lolz os fl Th ms dm o h f o wods sol polm q-l ppop h o q-l mosl s s h f o wods ompo s ss ss s h lodd ll s m q-l s lolz os fl Th ms dmg ds ss s lolz g ml dom wh ls ss sdg l oop s ho f o lolz g ml dom whl lolz ls g ml dom whl h hp ppop o q dom ml m gm d ospdg fs f ms q sgd h ospdg dd ls s m o fd d dls o qs Rf [] oss m o d dls o qs Rf [] ls lolz os l fl Th dmg oss o fd d dls o qs Rf [] ms g dd fd oss m sg ff dp homogz fmwo whh f ol gm lodd ll f s ds s ss s ompl os mdosl ss s ompl os mdosl m ml ls ls h hp ppop fs dlopd ls h sg h ls h hp p sg ff fs f ms sgd f lolz g ml dom whl dom m l g sg ff h lolz ml dom whl ml dom m l gm ls ss sdg po m dfd dd o q-l mls s lolz sgg ml f dz m o q-l mls s lolz sg l ss pssd s lsl ol q olm dd fl mo-sl fd d dls o qs Rf [] sll q ospdg homogzd pssd fd d dls o qs Rf [] l hogh homogzg sol polm dd fl mo-sl fs s f ms h f l B fs dmg dlopd sgd fmw oss m fd d dls o qs Rf [] ls ss sdg l oop s s g lshold os ml dd ls h hp ppop fs oss m fd ls hp ppop dlopd o q-l mls s lolz os fl Th d ff dp homogz sg ff dp homogz ddz ppld Th gs f-sl sol lolz o g ml dom whl l l o q-l m po fs @ g dd ml s s shold ppld fs g lolz o g lz B ml dom ml dom m l gm po m dfd ms dom m fmwo homog sg ls s lolz fl ms dmg ds whh f whh m l whh f ol g dfd l sgg ml f h sgg ml f dom sgg ml f d l ls h hp ppop fs dlopd @ sg ff dp lolz sgd f lolz l @ fs whl ms f h f lodd ll f s l lolz followg g H ls dom whl ml dom m l gm om dff l d s lld sol ls h hp ppop fs dlopd ml mpls sold whh f ol gm l dp whh f gm l followg dp d q ospds lsl ol q sg ff sgg ml f dz ml dom sg ff sgg ml f dz ml dom sg ff dp ls ss sdg l oop s dd sg ff dp homogz fmwo ss s ompl os mdosl mplm fs f m @ ( lodd ll f s s q lolz g ml dom whl ss ( @ o dd Rf [] g dom whl po m lolz dfd ldp homogz l ð Þ¼ ðþ ( f ff ospdg dd po m sfg D ml sg 5 ol qs q ll H ml l ospds g sg D d/ mdosl d ospds lsl ol q fmwo ff dp fmwo s sgg f dz dom s lolz ml dom whl ml dom m l g q dplms os q h pl ms lld s ml g olm oml s: lodd l d whl ls lsl q sgg f lhomogz dz m lolz ms g m fs f pl dplms os h ms l d ls q ospds h ( D sgg ml ls ss s ( ls ss sd dom ml dom m gm mplmg om dm ls ss sdg l o ss s ompl os ml s ol ss ompl os md ss h s ompl os ml ms ls þ h hp ppop fs dlopd so s sg sg ff dp homogz fmwo sgg ml R lodd ll f s s q dz dom (7 sg ff dp homogz fmwo oss m o fd d dls o qs Rf [] (7 ls ss sdg l oop s ho l ls ss sdg l oop s ho f ss h s ompl os mdosl mplmg whh d df d ompll ss s s ompl os mdosl mplmg dm olm whh f ol gm l followg dp sh hp d q ospds lsl ol q Homogzd f l lodd ll ml f sgg ml f s d ls q ospds lsl ol H sgg smd fs ml pms fs dm f ms ls ppop fs dlopd som ml mpls d m polm sold s ompl os mdosl mplmg whh ls h hp ppop dlopd som m Rf polm sold dss d dls o qs Rf [] oss o fd d dls o qs [] m qs whl pod od ds d sol sgg ml f dz ml dom wg dp shm poposd h sd oss m o fd d dls o ss s ompl os mdosl mplmg qs whl pod od ds d h sgg ml f fs f lm fs f m @ ss s fs f ms dz ml dom ql sm qs dfd h mopl whh f ol gm f ol gm l T whh f ol gm l followg lodd ll f s s sg ff dp fmwo d ompss-sh ml sh s ps som ompd poposd o lodg (ompss homogz oml ompd ss ss sol < ss h ls ppop fs dlopd mpls l osdp mdosl mplmg h dm ( dp ll Homogz fs f ms ml sgd ss ll fs f ms s ls sgd h ( LD ll Homogz whh f gm followg dp shm h ss LD sol l l whh f ol ompss followg dp shm poposd sd Homogz hp ol d dp [] ls sdg l oop s ho f ls dhomogz dls o qs l Rf mopl modl [6] followg hp ss os md ss ompl os mdo ff gm oss m o fd d o qs h Rf [] Homogz ff oss m o ml sg dp fmwo ol homogz h f dp shm poposd h sd mposg (Ddls 6 ompl Wh ods lodd ll f s ( ql p ( 6 ( pgm sg dp homogz fmwo o f o p p s f omp opl s ss s ompl os mdosl mplmg dm lodd ll lodd ll h qs f omp mos whh f ol gm l followg dp shm poposd d pod B: ss s omp lodd ll f lm ls ss sdg l oop s ( whh f f ls ss sdg l oop s d pod B: l os mplmg dm o s ls ss sdg l oop s ho ( mdosl d--olm s o / dp ll Homogz ( ( λ ϕ/ λ ϕ/ λ pl mlpl wh dp ll Homo D D sd ( ( L poposd dp ls h hp ppop followg T L Tff L whh f homogz whh f s ql ho smd m whh f h oop L sdg L s gm l shm h lodd ll f s fs lodd ll f s l sg dp fmwo ls ss l s ho f o ls ss sdg l oop ho f o s fs f ms sgd f q od sg ff sgd ls ss sdg l oop s ho f o l whh f ol gm l T ompss smd h l whh f ol gm l f sd o q-l mls s lolz os fl Th ms dmg ds o o q-l s lolz os fl Th dmg ds poposd l: Th ¼ ls fs od modl d sffss q-l mls lolz os fl ms dmg ds s mls L h hp f ls h ls h hp whh ol gm oop followg shm ss h sr h hp ppop ms dg ls ss l s ho f o fs ms f ms ppop R whh ol f ms L sg gm l followg dp shm h sd ls sd fs l poposd pol lodg-lodg ds ϕ λ sdg dfd ϕff ppop dp f ( fs ldg dfd ollps ls soop mls s lolz os f fl ms ol (modlg po followd mls s lolz os fl Th ms dmg λ pl l po m ls lodg l lodg-lodg dd ls s L dd po m Th dfd ds dd ls ss po m dfd dmg ls hp fs dlopd sg dp homogz s f o h hp ppop dlopd so l lld fs l fs ms sgd f s sffss s dfd fs f ms ds sgd f s l lld fs f s lm f h lodd l ll dp ll lodd ho ms q Homogz l ( fs f sgd f sd ( m 7 o q-l mls s lolz os fl Th ms dmg ds ls ss sdg l oop s dfd olm l ; dd sd s q ls ss dom sdg l oop s g ml ms ds m q 7 fd ol l dd o q-l mls lolz dom whl ml ml m l gm lolz g dom ml dom whl dom m l gm (modlg omp hdg lolz ml whl ms ml dom m l gm l g pl l o sg ff sg ff h lodd ll f s sg ff dp h lodd ll f s ls ss sdg l oop s ho f o lodd ll f s fs f sgd f ls ss sdg l sg ff dp homogz soop s ho f fs f m < l fl lw sh sgh smd ml dom whl ml dom m l gm s fl Th ms ds ol l dom whl dmg ml dom m l gm dp ll Homogz d ls q ospds lsl lsl ol q d ls q ospds dp ll Homogz d ls q ospds lsl ol mogz fs W q sg ff homogz fmwo w sg ff dp homogz fmwo q fs fs f lm f lodd ll f s pods s q fo sgd sd sold lodd ll f s dp q s ¼h ls hp ppop fs dlopd som l lodd ll f s s q q pods opd som ml mpls ls q ospds lsl ol q lsl pl dl q gog q s q ospds lsl ol q sgg ml f dz ml dom lolz g f ml dom whl ml dom m ðþ l gm sgg ml dz ml dom sgg ml f dz ml dom ms lolz ms g ml d f R fs f ow dfs o q-l mls s lolz os fl Th ms dmg ds ls h hp ppop ls h hp ppop f mogz ls h hp ppop fs dlopd fs f ms sgd f s lodd ll f s s q dd f dz ml dom fs f ms po sgd f sd ml dom m l gm f dz ml dom lodd ll m dfd dd 6 oss m o fd d dls o qs Rf [] dp ll Homogz > D oss m o fd d dls o qs Rf [] D l oss m o fd d dls o qs Rf [] dp l D <hp h hp l lo h ppop mplmg fs dlopd som mpls s lodd ls hp ppop fs dlopd som ml mpls sol m s o s q lodd ll olm ls h ppop fs dlopd ml mpls sold sg som ff dp homogz fmwo dsfd oml modls o lodg sg dd fl fd qs lolz Rf [] hogh homogzg sol polm dd fl mo-sl fd d dls mo-sl o qs ls Rf [] d ss s ompl os mdosl mplmg dm ss s ompl os mdosl dm posss ml ss dls s ompl os mdosl mplmg q-l dm sgg f dz ml dom d ðd ; g sgg ml f þ h mls dmg 6 6 ds o q-l s os fl Th ms dmg ds lodd ll f s Þ¼ H ð D Þml D lodd ll f s o mls s lolz os lz ml dom whl ml dom m l gm D ls s lolz os fl Th ms sg ff dp homogz sg ff dp homogz > sg ff dp homogz fmwo lsl ol q ompl os mdosl mplmg dm dz ml dom mpl os mdosl mplmg dm d ls q ospds lsl ol q lodd ll f s s q fo s w h ( ( /( : ls h hp ppop fs dlopd ml m s s q som h mp f f D/ D Dlodd D ll D ls hp dp ð ml ( [ p( ] (5 homogz Þ ðsdff Þ' mpls ow sg ff fmwo sg dp homogz fmwo s D Þp½ð D ÞH ð D sg ff dp homogz fmwo op fs dlopd som sold ls h dd ol q ospdg dd whh f ol gm l followg dp shm poposd h sd whh f ol gm followg l followg dp shm h poposd h sd whh f ol gm l dp s poposd ( sd ls s lolz os fl Th ms dmg ds h od qh pssd f ss s ompl os mdosl mplmg dm ss s ompo lz gm g ml dom whl ml dom m l gm o q-l mls lolz os fl Th ms lolz hm g ml dom whl ls h hp ppop h fm s sgg ml f dz ml dom ls hp ppop fs o q qs [] dom whl ml dom m l gm ol l followg dp shm poposd h sd gm Rf l followg shm h poposd h sd mplmg dm ðþ oss m o fd d dls o qs Rf [] dp ls hp ppop fs dlopd som mpls so p fs dlopd som ml mpls sold sg ff dp homogz fmwo d ls h hp p sg ff oml s show g ml sold dhd s homogz fmwo ls ss sdg l oop s ho f o sg ls ss sdg l oop s ho f o sg ff ls ss sdg l oop s ho f o dom whl ml dom m l gm spl whh ol gm l followg dp shm poposd h sd whh f ol g f s ps ompss oml ss s sgg ml f dz ml dom offs sl oml ; g l fl f Sh ss s ompl os mdosl mplmg l dm sgg ml f d D s ml pm lolz ml dom whl ml m sg ff dp homogz f gwg l oop s ho f o sg ff dp homogz fm poposd Ddz lolz dom g oop s ho f o@ dp h sd (omp shm ml dom dp sg ff homogz fmwo Homogz homogz fmwo ff dp ( ( dp ll s lshp sg o sd dp ll dp ll Homogz Homogz mso-sl ldg Df D D fs ompss f ms sgd f fs f ms sgd sd fs f ms sgd f sd / dz ml dom ls ss sdg l oop s ho f o s h f ol gm l followg dp shm poposd h sd ls ss sdg l o Homogz Homogz sgh lop pl sh ss s lshp gs fdz spl ss s ompl os mdosl mplmg dm ss how s ompl os mdosl s sgd f sd sgd f sd s o f o lodg-lodg l lso show o lls pposs sgg ml f w sgg ml omp s s po ollps ( l l os mdosl mplmg H dm D R lodd ll ll f s q ll lodd omp f s s ls s q lodd modls f s s q smd pm 6 H / hdg s whh sgd ss sdg l oop ho dp f o fs f ms sgd ms f l os mdosl mplmg lolz dm f h f ol gm s l followg shm ms poposd fs h sd f s s q ls s s f sd whh f ol ds l o q-l mls s lolz os fl Th dmg ds ms o q-l mls lolz os fl Th ms gm dmg o q-l mls s os dmg ds q ss s mdosl followg fl 6 Th d/ ss woh gm l dp shm poposd s h sd gs os pm gog ompl / os 5 mplmg dp ll Homogz qs mls s lolz ml fl Th ms dmg ds mls os fl Th ms dmg ds s ls ls h hp ppop fs dlopd som ml mpls sold followg h hp ppop dlopd som ml mpls sold l qs gog LD fmwo ompld hogh s hdg lolz ls h hp ppop fs dlopd som ml mpls sold ( ( og l m d s sgh l wh sold f ms ml mpls sgd f sd s hdg ( som ml (ll H omdlodd fs ppop fs som ml lodd f s q fo h ppop fs dlopd mpls sold D ( s ll f lm lolz g ml dom whl ml ml dom m l gm gm l followg dp shm poposd h sd sd ls ss sdg l oop s ho f o lolz dlopd q g dom whl dom oop l gm lolz g ml dom whl ml dom m l gm ls ss sdg l s ho D whh f ol gm l followg dp shm po l whh f soop s ho f o homogz slop ql d s sh slop ompss ml dom whl ml m l l whl ml dom m gm p s D ospds olm s L gm sg ff dp fmwo sg ff dom ddl dp homogz fmwo dom sg ff dp homogz fmwo qs h pl gl m l qs g lodd ll f s s q fl Th dmg ds o q-l mls s lolz os fl ms dmg ml dp homogz fmwo p homogz fmwo opd som mpls sold sgg ml o f f dz dom od q ( hgh h ds ls h hp ppop dlopd ml mpls sol sgg f Th dz dom sgg ml f dz ml dom ls h hp ppop fs ms f ms ml sgd sd ml oml s ; ols olmd som soop s ho f fs fs f ms ml sgd sdg dz dom ls ss l oop s ho ls f dd ss s f ml po m dfd dd sgd f sd f dz ml dom l fls l ffs mso-sl ompss ho ls h hp ppop fs dlopd som ml mpls sold ml dom m l gm lolz g ml dom whl ml dom m l gm ss s ompl os mdosl mplmg dm ss s ompl os mdosl mplmg dm ss s ompl os mdosl mplmg dm sg ff dp homogz fmwo loddol ll f s h s q ff dp lodd ho ll sg f s h sgd mplmg f sd l sps ompss sd l lsl q ompl os mdosl mplmg dm d ls q ospds ms lsl ol q mpl os mdosl dm fs f sgd f fs f lm m s s q sg ff dp homogz fmwo z ml dom l wll smd ps ompss s sh sgg ml f dz ml dom whh f ol gm l followg dp shm poposd h sd whh f ol gm l followg dp shm poposd h sd whh f ol gm l followg dp shm poposd h sd m ls h hp ppop fs dlopd som ml mpls sold ls h hp ppop fs dlopd s s q 7sgh qs Rf [] ol gm l followg dp shm mpls poposd sd oss m o h o qs []s Rf d gm l followg dp shm hdg poposd h sd s fl ffs d l- o g D slop fd l modls d lodd dls ll f s s op fs som ml sold lodd ll o mplmg dm ss dlopd s ompl os mdosl mplmg f dm ls ss sdg l oop s ho f o ls ss sdg l oop s ho o ls ss sdg l oop s ho f o fl phom smld ffl hogh ds d zo od sml osd hozl ff dp sg homogz fmwo sg ff dp homogz fmwo p fs dlopd som ml mpls sold g l oop s ho f o oop s ho f pl lsl ml pl [] ml sh ml fd pl pml d o hd ls followg h hp sgd ppop #h fs dlopd som homogz fmwo ls h dp shm poposd h #s whh f ol gm Th l dp shm poposd sd fs f ms lld sgd f sd f fs d f ms f sd fs ms sgd f sd p p sg ¼ ¼ T L T L L s sgd f sd sgd f sd homogz fmwo ll sg ff homogz fmwo f o sg ss sdg l oop s ho dp f oq lodd ll f s q ff ll lodd f s s s q ls lodd f s s f s s q s s q dp ll Homogz d f sd fs f ms ppop sgd dlopd f sd ls ls hp h hp ppop fs dlopd som ml mpls sold ls h ppop fs som ml mpls sold h hp fs dlopd som ml mpls sold ppop ppop fs dlopd som ml mpls sold fs dlopd som ml mpls sold dp ll Homogz dp ll Homogz s s lodd sg ll f s fmwo fmwo s q q sg ff dp homogz ff homogz dp homogz sg ff dp 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