Estimation of 3D Deformation and Rotation rate Tensor from volumetric particle data via 3D Least Squares Matching

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1 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, Esimion of D Deformion nd Roion re Tensor from olmeri prile d i D Les Sqres Mhing J. Kihofer, P. Wesfeld, O. Ps, T. Nonn, H. G. Ms nd C. Brüker : Insie of Mehnis nd Flid Dnmis, Uniersi of Freierg, Germn, jens.kihofer@imfd.-freierg.de : Insie of Phoogrmmerie, Uniersi of Dresden, Germn : Dne Dnmis A/S, Skolnde, Denmrk Asr This rile dissses he esimion of he deformion re nd roion re ensor from resls oined D Les Sqres Mhing (LSM), hih is pplied o olmeri oel spes for he llion of D eloi fields. I is shon h LSM lred ields he eloi grdien mri for ompion of he deformion nd roion re ensor. Vorii, sher nd srin re lled iho ppling enrl differenes shemes. I is shon h LSM is ell sied for he erion of he deformion prmeers in ses of smll o medim deformions. The r nd onsrins ill e poined o nling he oeffiiens of he ensors. Resls re presened for simled nd eperimenl dses.. Inrodion In he ls dedes seerl inesigions he een performed o eperimenll desrie he moion of flid. The fndmenl heorem Helmhol ss h eer infiniesiml moion of flid elemen n e deomposed in rnslion, roion nd deformion. Iniill, he mesremen ehniqe Prile Imge Veloimer (PIV) s sed for D inesigions ielding plnr field ih rnslionl eloiies in he Elerin frme []. Ths, onl one pr of he fndmenl heorem old e desried in D. Deelopmens like Snning PIV [], Hologrphi PIV [] nd Tomogrphi PIV [4] epnded he desripion of flid moion o D. Here, orrelion sed ehniqes, like D ross orrelion, re freqenl pplied in he pos proessing on gr le oel spes o er he ero order rnslionl eloi omponens negleing he higher order erms of roion nd deformion. The ssmpion is h he flo field is smooh nd no signifinl inflened roionl or sher displemens, hs ielding he ero-order rnslionl displemen field ih n ddiionl nerin in mesremen de o negleing he higher-order erms. Redion of he mesremen nerin n e hieed indo deformion ehniqes [5]. The higher-order moion erms re hen esimed finie differene shemes of he eloi field informion on disree grids (indirel onsiderion of he rnslionl eloiies of neighoring elemens). The ssmpion is h he higher order flid moion of n elemen is onl ffeed he rnslionl eloi omponens of he neighoring elemens. Closel reled o ross orrelion pprohes is Les Sqres Mhing (LSM), hih re oh region-sed [6]. LSM hs lred een sessfll sed in he nlsis of D prile imges. Neerheless, ross orrelion ehniqes operformed LSM ese of he shorer proessing imes. In onrs o orrelion sed ehniqes, Les Sqres Mhing (LSM) shifs, roes nd srehes flid re. For his prpose, he les sqres mhing lgorihm ieriel ompres gr le informion of n inerrogion re in he firs ime sep ih he gr le informion in he seond ime sep. This is n ierie les sqres proedre ppling n ffine rnsformion on he inerrogion res. In D his resls in si rnsformion prmeers nd in D his resls in ele rnsformion prmeers for eh inerrogion re. So, he dnge of LSM is h hile lling he ero order rnslionl eloiies, he firs - -

2 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, order erms of moion re onsidered inresing he r of he eloi field. Moreoer, he ffine rnsformion inldes prmeers like roion, sher nd srin of he inerrogion re. This pper shos ho o er hese prmeers from he ffine rnsformion nd o rnsform hem in he desripion of flid moion. For his prpose, his rile firs dissses he neessr priniples (fndmenl heorem of flid moion, Les Sqres Mhing) nd he onneion of oh prinipls o lle he deformion re nd roion re ensor i he resls oined LSM. A prmeri sd is performed on single olmeri e o sho he r nd he onsrins of he esimion of deformion nd roion re. A simled Hill pe ore shos he srengh of he mehod for idenifing regions in he flo of high flid mehnil ineres. A he end e ill sho resls for he enhmrk eperimen of ore ring.. Priniples This seion smmries he si priniples for ndersnding he esimion of he roion nd deformion re ensor i he dse oined D Les sqres mhing. For his prpose, he fndmenl heorem for flid moion is shorl desried. The eqilen desripion in erms of rnsforming one ssem (si e) ino noher ssem (deformed e) s oined D LSM is disssed nd oh mhemil desripions re ompred shoing h D LSM ields he eloi grdien mri nd onseqenl he roion nd deformion re ensor. Fig. Definiion of flid moion: E represens he flid elemen o sessie ime seps; P is poin inside he infiniesiml flid elemen; he eloi of P ( P ) is he sm of he rnslionl eloi nd he deformion eloi def. Fndmenl heorem of flid moion The fndmenl heorem Helmhol ss h he eloi poin d inside n infiniesiml flid elemen E n e deomposed in onsn rnslionl eloi () of he flid elemen nd in eloi def (d) resling from he deformion of he flid elemen s fnion of he disne d from he ener of he flid elemen s shon in figre. Mhemill, his is rien: ( d, ) (, ) ( d ) () The eloi d in he infiniesiml olme elemen resling from deformion is rien in def - -

3 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, erms of he eloi grdien mri V: d V d d def ) ( () I is generll h he Join of he eloi n e deomposed in smmeri ensor D nd n smmeri ensor R, here R desries he roion re ensor nd D he deformion re ensor. D R V () The deformion re ensor desries srin ij, ij nd sher ij, i j of he infiniesiml olme elemen resling from eloi grdiens in he inner of he olme. D (4) The roion re ensor desries he orii in eh spil direion,,. R (5) I is oios h llion of he deformion re nd roion re ensor for fll desriing flid moion needs he elion of he eloi grdien mri V.. D Les Sqres Mhing (D LSM) The si priniple of Les Sqres Mhing is he elion of rnsformion, hih rnsforms he se of ssem ino noher se of he sme ssem. For his prpose he ses of he ssem re ompred i n ierie les sqres proedre. In he ppliion of D LSM o olmeri prile d s oined Tomo PIV, he ssem is n inerrogion olme (oid) nd he se of he ssem is he gr le represenion of priles. The rnsformion is eled in n ierie les sqre djsmen proedre, here he gr les in se one re rnsformed ino he gr les in se o. A deiled desripion n e seen in [7]. The geomeril model is n ffine rnsformion for eh oid E rnsforming ll posiions se - -

4 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, one in he inner of he oid ino he posiions se. E d d d d d d d d d (6) The inde mens he se of he olme elemen ime sep nd he inde he se of he olme elemen ime sep. The prmeers, nd re he onsn rnslionl displemens lid for eh poin in he infiniesiml oid E. In he folloing he rnslionl displemens in he oid E ill e () (,, ) T direl ielding he rnslionl eloi of ll poins in he oid E. Δ ) ( (7) The deformion of he oid is desried he rnsformion mri T: E E T (8). Veloi grdien mri The eloi grdien mri is neessr for he ompion of he deformion nd roion re ensor. The resl from LSM is he rnsformion mri nd he displemen eor. So, for gien poin P (d,d,d) T in he inner of he oid E he posiion of he poin P is lled i he rnsformion mri ssming h he rnslionl displemen is ero: P T P (9) The eloi of he poin de o he deformion is lled : ( ) ) ( P I T P P T P P d def Δ Δ Δ, I () So, omprison of () nd () i is ler h he eloi grdien mri V of olme elemen E nd s onseqene he roion re R nd deformion re ensor D is deried sring he ideni ensor I from he rnsformion mri T nd folloing diision he seprion ime Δ. I T V Δ ) ( () - 4 -

5 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl,. Nmeril ssessmen In his seion e presen nmeril resls for he esimion of he roion re nd deformion re ensor oined LSM. The r of he esimion is nlsed i snhei genered oel spes. For his prpose prmeril sd is performed nlsing he deformion of ell defined e. The inerpreion of flid mehnil spes is shon he end i simled Hill pe ore.. Ce deformion The folloing prmeril sd ers inflenes on he esimion of he roion re nd deformion re ensor. Prmeers re nmer of priles, noise nd srengh nd nmer of deformion. The snhei se-p is shon in figre. A i elemen (666 oel) is genered ih he oordine ssem in he ener of gri. A pr of he elemen (444) is hen filled ih homogenos sied priles ( oel elemens) rndom posiions hing Gssin inensi disriion. The posiions of he priles re hosen ih s piel r nd represen he originl e iho deformions. The prile posiions re hen deformed i defined sherings nd srins nd folloing roions. The shifed prile posiions represen he deformed e. The ndeformed nd deformed es re hen nlsed i LSM o lle he rnsformion mri nd in he folloing proessing sep he eloi grdien mri. The ompion of he LSM prmeer is performed n dped ode gien Thoms Nonn from Dne Dnmis. Fig. Nmeril se-p for he prmeril sd of he e deformion As LSM is n ierie les sqres proedre, i is sensiie o sring les. The nlsis herein is performed iho he knoledge of sring les resling in higher nmer of neessr ierion seps o lle he rnsformion mri ihin predefined ondries. To mke he resl independen of he nmer of ierion seps, he ierie proedre is sopped fer ierions, hih is fr eond neessril o ierions for smll deformions []. In he folloing he le de is defined s he relie r of he mesred deformions, here σ l is he lled le ( i, ij ) nd σ is he iniil le ( i, ij ): de σ l σ σ % Figre on he lef shos he r of he sole le of roion s fnion of prile per piel (ppp). Prile per piel is defined genered nmer of prile posiions diided he - 5 -

6 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, sie of one side of he e (herein 44 pi²). The deformion onsiss of roions rond eh is nd srin nd sher in o plnes represening e ih lo mgnide in deformion lrge nmer of differen deformions. A lo prile densiies he error is er high p o 8 % onerging o % ihin slighl inresing ppp. Common D PIV imges he nmer prile densi of.... ppp. This is mh higher hn he osered order.5 ppp in figre. This is in greemen ih [8]. The sed D LSM pproh for Prile Trking Veloimer, hih mesres eloiies loer nmer prile densi. Neerheless, he folloing inesigions ill e performed. ppp, ese he im of his pper is he ppliion of LSM o Tomo PIV. B o poin o, LSM deliers onsn r of % eond.5 ppp. Fig. Ar in mesremen; lef: s fnion of ppp (deformion: φ κ 5 ; d., d.9, d ; d., d., e ); righ: s fnion of noise (deformion: ; φ κ ; d, d, d ; d, d, e ) Figre on he righ shos he deiion s fnion of noise. Noise is inlded in h he gr le of eh oel posiion is sperimposed rndoml sndrd deiion. The mimm sndrd deiion is defined eeen nd 55 mening noise is sndrd deiion of nd noise is sndrd deiion of 55. I is lerl reognile h de inreses o more hn % from noise of.. This h LSM is pplile, if he noise does no inrese % of he 55. The reson prol is h he shpes of he priles highes mgnide in he imges, herein sr o smer ih he srronding. The relie deiion in sole roionl srengh s fnion of inresing roionl srengh (inresing deformion) shos figre 4. The solid line shos he deiion for, nd he dshed line shos he deiion for, 5. The solid line remins nerl onsn p o roion in -direion of o 5. The deiion is pre lo (< %). Beeen 5 nd he deiion drmill inreses eing eond % o. If roions rond he oher es re inlded, o hrerisis n e onlded (dshed line). The deiion for smll roions is inresed ( 7%) nd he sdden inrese of he deiion is shifed o loer les of. This prolem is ell knon in lierre nd resls from he ssmpion of he ffine rnsformion. The rnsformion is liner, hih is lid for smll deformions. B hen he sole deformion is inresed, he rnsformion eomes non-liner

7 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, Fig 4. Ar in mesremen s fnion of (deformions: φ κ -solid- /5 -dshed-; d d, d ; d, d, d ) Figre 5 shos he inflenes on he r on sher nd srin. In oh lsses of deformion rends re isile. On he one hnd he deiion gin drmill inreses, hen erin deformion is eeeded. On he oher hnd he deiion inreses for loer deformion, if mliple deformions re inlded. Neerheless, he deiion of srin nd sher is in he order of mgnide of o % remining nerl onsn lso in he se if mliple deformions re inlded. Fig 5. Ar in mesremen; lef: s fnion of sher in (deformions: φ κ ; d d, d ; d -solid- /. -dshed-, d -solid- /. -dshed-); righ: s fnion of srin in (deformions: φ κ ; d d, d ; d -solid- /. -dshed-, d -solid- /. -dshed-) The prmeril sd of he e deformion shos h LSM n hndle mliple deformions p o erin srengh. The r is in he order of % if onl one lss of deformion is inlded. If mliple lsses of deformion re inlded, he r dereses o o %. Neerheless, srong deformions re no ple of he LSM lgorihm. The reson is prol in he ssmpion of n ffine rnsformion

8 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl,. Hill pe ore In he nmeril ssessmen e se he hill pe ore ring for lidion of he erion of he deformion nd roion re ensor from he LSM resls. The hill pe ore is hreried spheril sremlines, don like ore ore nd o sgnion poins (psrem nd donsrem). The e solion of he Nier Sokes eqions for he inner pr (r<r) is rien in lindril oordines: U R r.5 U Φ os( ) R U.5 U r sin( Φ ) R Φ U R is he rdil eloi, U Φ is he irmferenil eloi nd U is he rnspor eloi of he ore. For he oer pr of he ore (r>r) he solion is rien: U R R os( Φ ) r U U.5 U R sin( Φ ) r Φ As he LSM lgorihm is defined in Cresin oordines, he eloi field is rnsformed from lindril oordines ino Cresin oordines. The sie of he genered eloi field is oel. The Cresin grid is hen filled ih homogenos sied priles ( oel) ih Gssin inensi disriion, mening h 9% of he hole olme is filled ih priles. The s-piel displemen of he priles de o he eloi field is hen esimed ih Rnge-K sheme. Aferrds rndom noise ih sndrd deiion of.5 is sperimposed. The hosen ondiions (rndom prile posiions nd noise) re good pproimion for rel eperimenl ondiions. The sie of he nled oids is 555 oel ih n oerlp of 75%. This resls in 648 rnslionl displemens nd rnsformion ensors. The isliion of some resls shos figre 6. The figre shos he eloi eors in he smmer plne -, he D sremlines sring oe he psrem sgnion poin, nd isosrfes of he orii mgnide (gr), he posiie (ornge) nd he negie (le) srin. Moreoer, some nmeril resls of he deformion re nd roion re ensor re inlded. The genered sremlines lerl follo he e solion of he hill pe ore. The sremlines form spheril shpe. The eloi eors in he smmer plne idenif he eloi disriion of hill pe ore. No oliers re reognile. The isosrfes of orii nd srin elded from he rnsformion mri deried LSM idenif he min feres of he hill pe ore iho n frher pos proessing seps. The pper nd loer sgnion poins re highlighed. The olme elemens in he pper sgnion poin re ompressed in -direion nd epnded in - nd -direion ( <, >, >). In he loer sgnion poin he sene is inered. The isosrfe of he orii mgnide idenifies he don like ore ore. The nmeril resl for he roion re ensor in he smmer plne - he ore ore for < nd > is gien. The prefies of he enries for orii gree ih he phsil inerpreion

9 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl,. D..6 R, <.7.7 R, >.7.7. D..6 Fig 6. Visliion of some resls oined LSM pplied o he simled hill pe ore ring; isosrfe of orii mgnide (le) idenifies he don like ore ore, isosrfe of posiie (red) nd negie (le) srin res idenif sgnion poins; D sremlines sho spheril shpe nd he eloi eors in he smmer plne - hrerie he eloi field of he hill pe ore The nmeril resls sho h he esimion of he roion nd deformion re ensor i LSM deliers orre resls iho n ehsing pos proessing i enrl deriies. The e deformion shoed n r of o % for smll nd single deformions nd % for smll nd mliple deformions. The Hill pe ore shoed h phsil spes n simpl e elded from LSM. 4. Eperimen The eperimenl se-p is shon in figre 7. The lser em of oninos Argon-Ion lser Coheren Inno 7 ( W) psses n opil lens ssem o djs he desired hikness of he ligh shee. The roing mirror drm refles he lser em ino he direion of he osered olme nd generes sessiel prllel ligh shee plnes ih hikness of mm. The sdied flo is ore ring relling in n ogonl glss nk filled ih er. The ore is genered he ei of pison e ih dimeer of 5 mm. The nerll on seeding priles ( mirons) re injeed ino he ener of he ore generor. The prile imges re reorded ih hree mer ssem onsising of digil high speed mers Phoron APX RS ih resolion of 4 4 Piel² nd n nglr displemen of 45, -45 nd 9. The mers re eqipped ih eleenri lenses f-nmer 6 resling in prllel projeion for he osered olme. The side of he ogon opposie o he enrne side for he lser is oered ih ligh soring m h redes sr refleions. This is lso lid for he fes opposie he mers, hs giing perfel lk kgrond. The eperimens re performed ih reording re of frmes/s. Using snning plnes resls in seprion ime of ms for eh sseqen illminion of one sn plne. The imge sie of he mers nd he ligh shee hikness define he mesred olme, resling in o 9 9 mm³ for one ligh shee

10 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, Fig 7 Eperimenl se-p onsising of hree high speed mers APX RS eqipped ih eleenri lenses The olmeri reonsrion s ell s LSM is performed Dne Dnmis Sofre. The olmeri reonsrion ses on mliple projeie rnsformion of eh mer ie ino he deph ler of he osered olme nd minsore lgeri reonsrion ehniqe. For frher informion on he olmeri reonsrion see []. In ol reglr grid of oel ih gr le informion is genered for eh ime sep. The eloi field s ell s he prmeers of he ffine rnsformion is lled s presened in [7]. To smmrie, he LSM lgorihm is performed on 5³ oids resling in 5³ eloi eors nd rnsformion mries. The resl is islied in figre 8. The ore ring is lerl reognile. Beside he eloi eors he isosrfe of orii mgnide is inlded. To poin o, he orii mgnide is no lled i enrl differene shemes, i he prmeers of he ffine rnsformion. Fig 8 Vore ring: he eloi eors he niform lengh; he le isosrfe represens he orii mgnide shoing he ore ore 5. Conlsion The rile disssed he esimion of he roion nd deformion re ensor i he ffine prmeers resling from LSM. I is shon h he llion is srigh forrd iho sing n enrl differene shemes. The r is esimed i he deformion of ell defined e. The resl is h LSM is lid for smll mliple deformions nd for medim single deformions. The r is o % for single deformions nd nder % for mliple - -

11 5h In Smp on Appliions of Lser Tehniqes o Flid Mehnis Lison, Porgl, 5-8 Jl, deformions. I seems h LSM is no sensile o high deformion re, sensile o high deformions. The erion of he prmeers is performed on simled s ell s on n eperimenl ore ring, oh shoing h he phsil hrer of he ore ring is desried he prmeers of he ffine rnsformion. Fre ork hs o e performed in ses of lrge deformions. Here he liner ffine rnsformion needs o e repled non liner model. Referenes [] Adrin RJ (99) Prile-imging ehniqes for eperimenl flid mehnis. Ann Re Flid Meh :6-4 [] Ch Brüker 997 D snning PIV pplied o n ir flo in moored engine sing digil highspeed ideo. Mes. Si. Tehnol. 8 [] Sori J nd Akinson C (8) Tords C-D digil hologrphi flid eloi eor field mesremen-omogrphi digil hologrphi PIV (Tomo-HPIV). Mes. Si. Tehnol. 9 [4] Elsing GE (8) Tomogrphi Prile Imge Veloimer, Ph.D. Thesis, Deprmen of Aerospe Engineering, Delf Uniersi of Tehnolog [5] Srno F () Ierie imge deformion mehods in PIV, Mes. Si. Tehnol [6] Trope C, Yrin A, Foss JF (7) Springer Hndook of Eperimenl Flid Mehnis [7] Wesfeld P, Ms HG, Ps O, Kihofer J, Brüker C () -D Les Sqres Mhing for Volmeri Veloimer D Proessing, eped for pliion in he Proeedings of he 5 h Inernionl Smposim on Applioions of Lser Tehniqes o Flid Mehnis, 5-8 Jl, Lison [8] Ishik M, Mri Y, Wd A, Ighi M, Okmoo K, Ymmoo F () A noel lgorihm for prile rking eloimer sing he eloi grdien ensor. Ep. In Flids 9 [9] Kihofer J, Brüker C, Ps O (9) Tomo PTV sing D Snning Illminion nd eleenri Imging. Proeedings of he 8h Inernionl Smposim on Prile Imge Veloimer, Melorne, Asrli [] Ms HG, Wesfeld P, Pe T., Bokjer N, Kihofer J, Brüker C (9) Phoogrmmeri ehniqes in mli-mer omogrphi PIV. Proeedings of he 8h Inernionl Smposim on Prile Imge Veloimer, Melorne, Asrli - -