Physical Simulation Using FEM, Modal Analysis and the Dynamic Equilibrium Equation

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1 Physcal Smulaon Usng FEM, Modal Analyss and he Dynamc Equlbrum Equaon Paríca C. T. Gonçalves, Raquel R. Pnho, João Manuel R. S. Tavares Opcs and Expermenal Mechancs Laboraory - LOME, Mechancal Engneerng and Indusral Managemen Insue - INEGI, Poro, Porugal Faculy of Engneerng of he Unversy of Poro - FEUP, Poro, Porugal ABSTRACT: Ths paper presens a physcal approach o smulae objecs deformaon n mages. To physcally model he gven objecs he fne elemen mehod s used, and o mach he objecs nodes modal analyss s consdered. The desred dsplacemen feld s esmaed hrough he dynamc equlbrum equaon. To solve hs dfferenal equaon dfferen negraon mehods can be used. In hs paper we presen and dscuss he resuls obaned usng four numercal negraon mehods: cenral dfference, Newmark s and mode superposon alled wh he former wo. Some mprovemens are nroduced n hs work o allow he physcal smulaon even when no all of he objecs nodes are successfully mached. KEYWORDS: Deformable Objecs, Deformaon Smulaon, Dynamc Equlbrum Equaon, Fne Elemen Mehod, Modal Analyss. INTRODUCTION All real objecs are deformable; ha s why mos of hem canno be accuraely modelled f hey are consdered rgd bodes. Deformable objecs smulaon may be acheved usng deformable models, whch can be a challengng ask because dfferen applcaon areas have dfferen requremens: some requre accuracy, lke medcal mage analyss; ohers requre real me neracvy, lke vrual envronmens. Alhough many mehods have been proposed o accuraely smulae deformable objecs a neracve raes, few are currenly able o do so. In hs paper, we use he approach proposed by Terzopoulos e al. (987, 988) o do realsc deformaon smulaons consderng an elasc model based on he resoluon of he dynamc equlbrum equaon. Therefore, we consder: Sclaroff s (995) and Sclaroff & Penland s (995) soparamerc fne elemen o physcally model each objec; Shapro and Brady s (99) modal shape descrpon o mach he nodes (daa pons) of he objecs; and Penland and Horowz (99) decomposon of objec deformaon no rgd and non-rgd modes. In hs paper, we propose a soluon o apply hs physcal approach o objecs ha do no have all of her nodes successfully mached. Furhermore, we also verfy ha he negraon mehod used o solve he dynamc equlbrum equaon may nfluence he compuaon speed and he obaned smulaon. PHYSICAL MODELLING The approach consdered n hs work for physcal modellng can be appled o D/3D objecs represened n mages. To buld each objecs physcal model we employ he fne elemen mehod (FEM), a sandard engneerng echnque whose nerpolaon funcons are developed o allow connuous maeral properes along he model (Penland & Horowz 99). Namely, we employ Sclaroff s (995) soparamerc fne elemen ha uses a se of radal bass funcons ha allows an easy nseron of he daa pons; hus, Gaussan nerpolans are used and he model nodes do no need o be prevously ordered. In consequence, wh hs fne elemen, when a D or 3D objec s modelled s as f each of s feaure pons are covered by an elasc membrane or a blob of rubbery maeral, respecvely (Tavares, Sclaroff & Penland 995). Sarng wh a collecon of m sample pons X (x,y,z ) of he objec o be modelled, he nerpolaon marx H (whch relaes he dsances beween objec nodes) of Sclaroff s soparamerc fne elemen (Sclaroff 995, Sclaroff & Penland 995) s bul usng: g ( X ) X σ X = e, () Where σ s he sandard devaon (ha conrols he nodes neracon. Then, he nerpolaon funcons, h, are gven by:

2 h m ( X ) = a g ( X ) k = k k, () where a k are coeffcens ha sasfy h = a node and h = a he oher m- nodes. These nerpolaon coeffcens compose marx A and can be deermned by nverng marx G defned as: g G = g m ( X ) L g ( X ) M O m ( ) ( ) X L gm X m M. (3) Thus, for a D objec (for he 3D approach see Tavares, Sclaroff 995), marx H wll be: h L hm L H ( x) =, (4) L h L hm and he mass marx of he D Sclaroff s soparamerc elemen s defned as (Tavares, Sclaroff 995, Sclaroff & Penland 995): M M =, (5) M where M s a sub-marx m m defned as T - M = ρπσ A ΓA = ρπσ G - ΓG (because A s symmerc, A T = A), ρ s he mass densy, and he elemens of marx Г are he square roos of he elemens of marx G. On he oher hand, he D sffness marx s gven by: K K K =, (6) K K where K j are symmerc m m sub-marces dependng on consans ha are funcons of he vrual maeral adoped for he objec (Tavares, Tavares e al., Sclaroff 995, Sclaroff & Penland 995). In hs work we use Raylegh s dampng marx, C, whch combnes he mass and sffness marces wh consrans based upon he chosen crcal dampng (Bahe 996, Cook e al. 989). 3 MATCHING OBJECTS NODES To mach he nodes of he nal and arge objecs, each generalzed egenvalue/egenvecor problem s solved usng: K Φ = MΦΩ, (7) where Φ s he modal marx of he shape vecors φ (whch descrbe he modal dsplacemen (u,v) of each D node due o vbraon mode ), and Ω s he dagonal marx whose enres are he squared egenvalues ncreasngly ordered. Afer buldng he modal marx for each objec, he wo objecs nodes can be mached comparng he dsplacemen of each node n he respecve modal egenspace (Shapro & Brady 99). The man dea s ha low order modes of wo smlar shapes wll be very close even n he presence of affne ransformaon, non-rgd deformaons, local shape varaons or nose. Thus, o mach he nodes of he nal objec, I, wh he ones of he arge objec, T, an affny marx, Z, s bul whose enres are defned as: Z j I, T, j I, T, j = u u + v v, (8) where he affny beween nodes and j wll be (zero) f he mach s perfec, and ncreases as he mach worsens. In hs work, o fnd he maches wo search mehods are consdered: a local mehod and a global one. The local mehod was proposed by Sclaroff (995), Sclaroff & Penland (995) and Shapro & Brady (99), and consss n searchng each row and each column of he affny marx for her lowes values. Ths mehod has he man dsadvanage of dsregardng he objec srucure as searches for he bes mach for each node. On he oher hand, he global mehod proposed by Basos & Tavares (6) and Tavares & Basos (5) consss n descrbng he machng as an assgnmen problem, and solvng usng an approprae opmzaon algorhm. 4 DYNAMIC EQUILIBRIUM EQUATION To esmae he objecs deformaon, and hus o oban her ransonal shapes, aendng o physcal properes, we solve he second order ordnary dfferenal equaon commonly known as Lagrange s dynamc equlbrum equaon: & + CU& + KU R, (9) M U = for each me sep, where U, U & and U & are respecvely he nodal dsplacemen, velocy and acceleraon vecors, and R s he vecor of he loads. In hs paper, o solve he dynamc equlbrum equaon we used four negraon mehods: cenral dfference, Newmark s and mode superposon n conjuncon wh each of he former wo. 4. Cenral Dfference Mehod The cenral dfference mehod s a drec and explc negraon mehod, wh second order precson f he mass marx s dagonal (Bahe 996, Cook e al. 989). However, usng Sclaroff s soparamerc fne elemen he mass marx s no usually dagonal. The velocy and acceleraon vecors are approxmaed by: U& = ( U + U ), ()

3 & = ( U& + + U& ), () U makng he dynamc equlbrum equaon o suffer a half me sep delay n velocy, ransformng C U & n CU &. Thus, Equaon 9 may be approached by: MU = R + M K U + M C U &. () A dsadvanage of hs algorhm s ha has only frs order precson, because he vscous forces, CU &, are delayed half a me sep. Anoher dsadvanage of he cenral dfference mehod s ha s condonally sable, hus he me sep used mus be small or he sysem wll dverge (Bahe 996). 4. Newmark s Mehod In hs work we also consdered Newmark s mehod (orgnally proposed by Newmark 959) ha has been modfed and mproved by many researchers. Newmark s mehod s a drec negraon process ha consders: U U& + = U + U & + + χ U & + αu, (3) + [( δ ) U&& + δu& ] = U& +, (4) + & where χ and δ are chosen n order o conrol sably and accuracy (Bahe 996). Thus, he dynamc equlbrum equaon (for me + ) may be rewren as: δ + M + C + K U = R + χ χ M + δc U + M χ M χ χ + χ ( χ δ ) C U& + ( δ ) δ C U&. (5) Ths numercal mehod s uncondonally sable f χ δ.5. In hs work we used δ =.5 and χ =.5, whch means ha no numercal dampng was nroduced and Newmark s mehod was employed as a second order mplc scheme. 4.3 Mode Superposon Mehod The mode superposon mehod proposes he ransformaon of he modal dsplacemens, V, no he nodal dsplacemens, U, usng he egenvecors marx Φ: U = ΦV. Thus, usng he generalzed coordnaes he equlbrum equaon can be rewren as: & T T + Φ CΦV& + ΩV = Φ R, (6) V because Φ T K Φ = Ω and Φ T M Φ = I, where I s he deny marx. V & and V & are, respecvely, he frs and second order dervaves of he modal dsplacemen vecor. Ths negraon mehod obans new sffness, mass and dampng marces wh smaller bandwdh (Bahe 996), allowng he resoluon of he dynamc equlbrum equaon wh jus a par of he models vbraon modes. Ths nave reduces he nvolved compuaonal cos by gnorng he local componens of he ransformaon beween he shapes, essenally assocaed wh nose. However, n hs paper, we used he mode superposon mehod wh all he vbraon modes. As an ndrec negraon mehod, o proceed wh he resoluon of he dynamc equlbrum equaon, we have o apply a drec negraon mehod o solve he ransformed uncoupled equaons se. Thus, we consdered he wo mehods prevously descrbed: cenral dfference and Newmark s. 4.4 Implc Loads and Inalzaon For he mplc loads appled on each mached node, we consder ha each load s proporonal o he assocaed dsplacemen (Pnho & Tavares 4): ( ) k( X F X ) R, j, =, (7) where R() s he h componen of he load vecor, X F, s he coordnaes of node n he arge objec, X j, represens he coordnaes n objec j (.e. he objec obaned n he j h eraon) and k s a global sffness consan. However, some nodes of he nal objec may no be successfully mached wh any of he nodes n he arge objec. Thus, suppose ha B s an unmached node beween nodes A and C, mached wh nodes A and C of he arge objec, respecvely (Fg. ). If B s he h node n he j h shape, hen he h componen of he load vecor can be gven by: R () = k [ Wp ( X F, p X j, B )], (8) p (nodes beween A and C ) where W p s he wegh of node p, accordng o s machng affny wh node B provded by Equaon 8 hus, he hgher he machng affny, he lesser he wegh. For he 3D case, A and C are he closes mached nodes o B mached o A and C, respecvely (Fg. ). A and C le nsde a sphere cenred a pon, he rgd geomerc ransformaon of pon B (esmaed n hs work usng Horn s (987) mehod), and wh he dsance from o C as he radus. Hence, n Equaon 8, node p s each one of he unmached nodes lyng nsde ha sphere.

4 A B A A B A ferences beween he fnal resuls. C C C C Fgure. Esmaon of he mplc loads for unmached nodes n D cases (lef) and 3D cases (rgh). The soluon we adoped o esmae he nal dsplacemen s o consder n erms of he mplc loads (Pnho & Tavares 4): U U c u () = R() k () = f f k, (9) k = where U () represens he h componen of he nal nodal dsplacemens vecor, and c u s a consan o be defned by he user. The nal modal velocy vecor s consdered n erms of he nal modal dsplacemen vecor (Pnho & Tavares 4): () c U () U& =, () v where c v s anoher user defned consan. Fgure. From lef o rgh: he wo orgnal mages (he seleced sobar conours are poned ou); maches found. Table. Resuls obaned o acheve a nodal dsplacemen vecor norm lesser han 5x -4 pxels. Mehod Number Dsance o arge Tme of seps objec (% of nal (seconds) dsance) Cenral dfference Newmark s Mode superposon and cenral dfference Mode superposon and Newmark s EXPERIMENTAL RESULTS Consder he wo conours obaned from real mages of weaher sobars wh 7 and 39 nodes respecvely, n Fgure. Usng he descrbed machng approach wh global search and rubber as he vrual maeral, all he 7 nodes of he nal objec are successfully mached. To smulae he gven objecs deformaon, we consdered crcal dampng beween % and % and chose o sop he compuaon process when he Eucldean norm of he nodal dsplacemens vecor s lesser han 5x -4 pxels. Oher relevan consans were chosen as follows: =.5s, k=n/m, c u =, c v =. Under hese crcumsances, he seps and he me needed o complee he compuaon process by each one of he negraon mehods are ndcaed n Table, as well as he dsance beween he las esmaed objec and he arge one. (In hs work, we used a compuer wh an Inel Penum D a 3GHz and GB of RAM.) In Fgure 3 are he resuls obaned wh he Newmark mehod n he esmaon of he nvolved deformaon, whch are vsually smlar o he oher ones bu slghly faser (Table ). If, nsead of soppng he compuaon process when he nodal dsplacemens norm s lesser han 5x -4 pxels, we sop when he dsance beween an esmaed shape and he arge objec s, for example, 4% of he dsance beween he orgnal objecs, we have a decrease n number of seps as well as n me (Table ) and vsually here are lle df- Fgure 3. Resuls obaned wh Newmark s mehod. In black are he nal and arge objecs, and n grey he nermedae ones obaned afer 5, 3, 6, 56,, and all seps (from boom o op). Table. Resuls obaned o acheve a dsance lesser han 4% of he nal dsance. Mehod Number of seps Tme (seconds) Cenral dfference Newmark s Mode superposon and cenral dfference Mode superposon and Newmark s The fac ha we need more % of he seps and seconds more o reach a decrease of only.% n dsance s a consequence of he fac ha he nodal dsplacemens norm decreases as he number of seps ncreases. Ths means ha he bgges par of he nal mage dsplacemens happens durng he frs seps of he compuaon process. If we use he local search mehod for he modal machng, only of he 7 nodes of he nal objec are successfully mached wh nodes of he arge one (Fg. 4). Ths means ha Equaon 8 wll be used nsead of Equaon 7 o deermne he mplc loads of he 6 unmached nodes. Agan, o acheve a nodal dsplacemens norm lesser han 5x -4 pxels, he resuls obaned by he dfferen negraon mehods are very smlar, beng Newmark s mehod he fases one (Table 3). In Fgure 4 we can compare he resuls obaned

5 by he mode superposon wh Newmark s negraon mehod when only nodes of he nal objec are mached; and when all nodes are successfully mached. Table 3. Resuls obaned o acheve a nodal dsplacemen norm lesser han 5x -4 pxels consderng only nodes mached. Mehod Number Dsance o arge Tme of seps objec (% of nal (seconds) dsance) Cenral dfference Newmark s Mode superposon and cenral dfference Mode Superposon and Newmark s he objecs, 37 of he 47 nodes of he nal objec are successfully mached (Fg. 5). To smulae he gven objecs deformaon, we used crcal dampng beween % and % and chose eraons o sop when he nodal dsplacemens norm s lesser han 5x -4 pxels. Oher relevan consans used are: =.3s, k= 6 N/m, c u =, c v =. Under hese condons, he seps and he me each one of he negraon mehods needed o complee he compuaon process are dsplayed n Table 5, as well as he dsance beween he las esmaed shape and he arge objec. In Fgure 6, we can see some nermedae shapes esmaed wh Newmark s mehod, one of he fases ones. Fgure 4. From lef o rgh: he orgnal conours and he maches found; resuls obaned by he mode superposon wh Newmark s negraon mehod when only nodes of he nal objec are mached; and when all nodes are mached. As menoned n he prevous secon, he cenral dfference mehod needs a small me sep o preserve sably; bu Newmark s mehod used as a second order mplc scheme s uncondonally sable and we can use any me sep. Thus, changng he me sep o s and he sffness consan o 5 N/m, we can no use he cenral dfference mehod, bu we can acheve excellen resuls usng Newmark s mehod alone or combned wh he mode superposon mehod, Table 4. Table 4. Resuls obaned o acheve dfferen dsances usng Newmark s mehod or he mode superposon wh Newmark s mehod when all or only nodes are successfully mached. Dsance o arge All nodes mached nodes mached objec (% of nal Number Tme Number Tme dsance) of seps (seconds) of seps (seconds) For a 3D example, consder he frs objec dsplayed n Fgure 5 obaned from a real pedobarography mage (Tavares e al. ), and he second one obaned by a rgd ransformaon appled o he frs (5º roaon along he zz axs). In hs example s easy o evaluae he accuracy of he obaned resuls: he nermedae objecs should have he same shape bu be n dfferen posons. Usng he modal machng wh local search descrbed earler and rubber as he vrual maeral for Fgure 5. From lef o rgh and op o boom: pedobarography 3D objec (nensy surface); arge objec obaned by applyng a rgd geomerc ransformaon o he frs one; he 37 maches found beween he objecs nodes. Table 5. Resuls obaned o acheve a nodal dsplacemens norm lesser han 5x -4 pxels. Mehod Number Dsance o arge Tme of seps objec (% of nal (seconds) dsance) Cenral dfference Newmark s Mode superposon and cenral dfference Mode superposon and Newmark s Fgure 6. Resuls obaned wh Newmark s mehod afer 8,, 46 and 768 seps (from lef o rgh and op o boom). In he back s he arge objec for comparson. Changng he me sep o s and he sffness consan o 7 N/m, we can no use he cenral dfference mehod because he me sep s oo large, bu we acheve a dsance of 5% of he nal one n

6 only 39 seconds usng Newmark s mehod or he mode superposon wh Newmark s mehod. In Fgure 7, we can see he fnal resuls obaned wh hose mehods compared wh he arge objec he fac ha we can no see means he resuls are good. Fgure 7. Fnal esmaed objec obaned wh Newmark s mehod (or mode superposon wh he Newmark mehod) o acheve a dsance beween he fnal esmaed objec and he arge one lesser han 5% of he orgnal dsance. 6 CONCLUSIONS In hs paper we descrbed a physcal approach o smulae objecs deformaon n mages. In hs approach we used wo soppng crera o end he resoluon of he dynamc equlbrum equaon: one based n he nodal dsplacemens norm, he oher based n he dsance beween objecs. Ths las soppng creron allows a sgnfcan decrease on compuaonal cos whou a grea loss n accuracy. We proposed a soluon ha enables he applcaon of he used approach on objecs ha do no have all of her nodes successfully mached. Tha soluon consss on applyng o hose nodes mplc loads ha depend on her machng affnes. The expermenal resuls obaned n he machng process and n he esmaon of he nvolved deformaon, some presened n hs paper, are coheren wh he physcally expeced behavour of he objecs modelled, valdang he used approach. As he cenral dfference mehod has frs order precson and s sably s condoned by he chosen me sep; when he mode superposon mehod s employed wh he resuls are of second order bu he accuracy sll depends on a small me sep. However, f he mode superposon mehod s used wh Newmark s mehod, hen he resuls are equal o he ones obaned by Newmark s mehod bu can spend a few more seconds n he compuaon process. Hence, when comparng he resuls obaned by he used negraon mehods, we would recommend Newmark s mehod o solve he dynamc equlbraon equaon. Ths suggeson s based on he accurae resuls ha can be obaned and on he speed of he assocaed compuaon process. Alhough our resuls are que sasfacory, he compuaon process s no very fas when comes o 3D objecs. So, n he fuure, we could ry parallel mplemenaons and dfferen negraon mehods o solve he dynamc equlbrum equaon faser. Also, we can ry oher approaches o deermne he mplc loads n he unmached nodes. 7 ACKNOWLEDGMENTS The presened work was parally done n he scope of he projec Segmenaon, Trackng and Moon Analyss of Deformable (D/3D) Objecs Usng Physcal Prncples, wh reference POSC/EEA- SRI/55386/4, fnancally suppored by FCT - Fundação para a Cênca e a Tecnologa. The second auhor would lke o hank he suppor of he gran SFRH/BD/834/3 of he FCT. 8 REFERENCES Basos, L. and Tavares, J. 6. Machng of objecs nodal pons mprovemen usng opmzaon. Inverse Problems n Scence and Engneerng 4 (5): Bahe, K.-J Fne Elemen Procedures. New Jersey: Prence-Hall. Cook, R., Malkus, D. and Plesha, M Conceps and Applcaons of Fne Elemen Analyss. New York: John Wley and Sons. Horn, B Closed-Form Soluon of Absolue Orenaon usng Un Quaernons. Journal of he Opcal Socey of Amerca 4 (4): Newmark, N A Mehod of Compuaon for Srucural Dynamcs. ASCE Journal of he Engneerng Mechancs Dvson 85 (3): Penland, A. & Horowz, B. 99. Recovery of Nonrgd Moon and Srucure. IEEE Transacons on Paern Analyss and Machne Inellgence 3 (7): Pnho, R. & Tavares, J. 4. Morphng of Image Represened Objecs Usng a Physcal Mehodology. 9h ACM Symposum on Appled Compung, Ncosa, 4-7 March 4. Sclaroff, S Modal Machng: a Mehod for Descrbng, Comparng, and Manpulang Dgal Sgnals. PhD Thess. Massachuses Insue of Technology. Sclaroff, S. & Penland, A Modal Machng for Correspondence and Recognon. IEEE Transacons on Paern Analyss and Machne Inellgence 7 (6): Shapro, L. S. & Brady, J. M. 99. Feaure-based correspondence: an egenvecor approach. Image and Vson Compung (5): Tavares, J. M.. Análse de Movmeno de Corpos Deformáves usando Vsão Compuaconal. PhD Thess. Faculdade de Engenhara da Unversdade do Poro. Tavares, J. M., Barbosa, J. & Padlha, A.. Machng Image Objecs n Dynamc Pedobarography. RecPad' - h Poruguese Conference on Paern Recognon, Poro, - May. Tavares, J. M. and Basos, L. 5. Improvemen of Modal Machng Image Objecs n Dynamc Pedobarography usng Opmzaon Technques. Elecronc Leers on Compuer Vson and Image Analyss 5 (3): -. Terzopoulos, D., Pla, J., Barr, A. & Flescher, K Elascally deformable models. In M. C. Sone (ed.) Proceedngs of he 4h Annual Conference on Compuer Graphcs and neracve Technques. SIGGRAPH '87, Anahem, 7-3 July 987. New York: ACM Press. Terzopoulos, D., Wkn, A. & Kass, M Consrans on deformable models: recoverng 3D shape and nongrd moon. Arfcal Inellgence 36 (): 9-3.