A Unified Approach for Subspace Simulation of Deformable Bodies in Multiple Domains

Transcription

1 A Unfed Approach for Subspace Smulaton of Deformable Bodes n Multple Domans Xaofeng Wu Rajadtya Mukherjee Huamn Wang he Oho State Unversty (a) Constraned deformaton (b) Deformaton after cut (c) Unconstraned deformaton Fgure 1: A hammock example. Under the vertex-based parttonng framework we present a unfed subspace smulaton system to anmate a mult-doman deformable body n real tme wthout any couplng constrant. Our experment shows that ths system can effectvely avod the lmtatons of many exstng mult-doman subspace smulators and t can flexbly handle a varety of deformaton cases. Abstract Mult-doman subspace smulaton can effcently and convenently smulate the deformaton of a large deformable body by constranng the deformaton of each doman nto a dfferent subspace. he key challenge n mplementng ths method s how to handle the couplng among multple deformable domans so that the overall effect s free of gap or lockng ssues. In ths paper we present a new doman decomposton framework that connects two dsjont domans through couplng elements. Under ths framework we present a unfed smulaton system that solves subspace deformatons and rgd motons of all of the domans by a sngle lnear solve. Snce the couplng elements are part of the deformable body ther elastc propertes are the same as the rest of the body and our system does not need stffness parameter tunng. o quckly evaluate the reduced elastc forces and ther Jacoban matrces caused by the couplng elements we further develop two cubature optmzaton schemes usng unform and non-unform cubature weghts. Our experment shows that the whole system can effcently handle large and complex scenes many of whch cannot be easly smulated by prevous technques wthout lmtatons. CR Categores: I.3.7 [Computer Graphcs]: hree-dmensonal Graphcs Anmaton. Keywords: subspace smulaton doman decomposton fnte element method nonlnear elastcty cubature approxmaton. e-mal: {wux mukherjr whmn}@cse.oho-state.edu 1 Introducton he fundamental dea behnd subspace deformaton also known as dmensonal model reducton or reduced-order deformaton s to constran the deformaton of an object nto a precomputed subspace. he computatonal complexty of ths method s O(r 4 ) when usng cubc polynomal [Barbč and James 2005] or even O(r 3 ) when usng cubature approxmaton [An et al. 2008] n whch r s the number of deformaton modes n the subspace bass. Snce r can be sgnfcantly smaller than the number of vertces subspace smulaton can be several orders of magntude faster than full-space smulaton. Unfortunately to handle detaled local deformaton of a large object subspace smulaton must use a suffcently large r whch compromses ts effcency and makes t less attractve. One soluton to ths effcency ssue known as mult-doman subspace smulaton s to dvde an object nto multple domans and smulate the deformaton of each doman n ts own subspace. If d s the number of domans and each subspace contans r modes then the computatonal cost can be as low as O(dr 4 ) or even O(dr 3 ) theoretcally whch s much lower than the cost of usng dr modes for the whole object. Most exstng mult-doman technques choose to partton mesh elements.e. tetrahedra nto dsjont domans. he challenge s how to handle the couplng among these domans each of whch has ts own local deformaton and rgd moton. If the nterface between two domans s small and the doman connectvty contans no loop the couplng process can be smplfed by gnorng the deformaton of the nterface and attachng one chld doman to ts parent doman as Barbč and Zhao [2011] showed. Unfortunately such an assumpton s not always plausble such as the hammock example shown n Fgure 1. o elmnate the gap between two adjacent domans a straghtforward dea s to enforce hard constrants on the nterface vertces usng Lagrangan multplers [Huang et al. 2006]. Snce these constrants may conflct wth the doman subspaces they can cause the lockng ssue that suppresses the local deformaton effect. Instead of usng hard constrants Km and James [2011] used sprng forces as soft constrants to connect two domans. her result depends on the choce of the sprng stffness: f the stffness s hgher the gap becomes smaller but the deformaton s more locked; and f the stffness s lower the deformaton s less locked but the gap becomes larger. her orgnal mplementaton also consders subspace local deformaton only assumng that the rgd moton of each doman has

2 Deformable Interface Rgd Moton Nonlnear Elastcty Specal Lmtatons [Huang et al. 2006] Yes Yes No Mesh-dependent complexty [Barbč and Zhao 2011] No Yes Yes Loop-free domans [Km and James 2011] Yes No Yes Stffness parameter tunng [Yang et al. 2013] Yes Yes No Restrcton on the number of modes Our method Yes Yes Yes See Secton 7 able 1: he lmtatons of mult-doman subspace smulaton technques. By usng the vertex-based mesh parttonng strategy and ntegratng rgd and non-rgd motons of all of the domans nto a sngle solve our mult-doman subspace smulaton system effectvely avods many lmtatons of the exstng technques. been gven by some earler smulaton. Recently Yang and colleagues [2013] developed a boundary-aware method to construct lnear deformaton modes so that both the gaps and the lockng ssues can be avoded. hs method requres the number of deformaton modes n each bass to be at least sx tmes the number of ts nterfaces. Snce these modes are specfcally desgned they cannot be easly reused when the domans are assembled nto other shapes. he method also has dffculty n handlng large nonlnear elastc behavors even after usng modal warpng [Cho and Ko 2005]. In general exstng technques have varous lmtatons as shown n able 1. How to flexbly and accurately handle the couplng among multple domans stll remans as an open problem. In ths paper we propose to partton the vertces not the elements nto multple domans. he beneft of ths new doman decomposton framework s straghtforward: the domans are now ndrectly coupled by the elastc forces of the couplng elements connectng the domans so there s no need to use addtonal couplng constrants. o develop a practcal mult-doman subspace smulaton system under ths framework we made the followng contrbutons. A sngle elastc model. We show that the same model can be used to handle the elastc deformaton of the whole object ncludng the domans and the couplng elements. We can easly ncorporate the elastc forces of the couplng elements called the couplng forces nto the smulaton of each doman to acheve the couplng effect wthout parameter tunng. A unfed mult-doman system. It s straghtforward to separate the rgd moton of a sngle doman from ts subspace deformaton and solve them smultaneously as a sngle dynamcal system. But t s not straghtforward to do so for multple domans that are connected by couplng forces. In ths work we formulate a unfed system that contans rgd motons and subspace deformatons of all of the domans and we show ths s needed to reduce artfcal dampng. Cubature approxmaton. A computatonally expensve step n subspace smulaton s the evaluaton of the reduced elastc forces and ther Jacoban matrces. Here we present two cubature approxmaton schemes to effcently calculate the reduced forces and ther Jacoban matrces. We show that by usng dfferent weghts for dfferent force components one of the schemes can acheve better approxmaton results. Our experment reveals the capablty of our system n handlng large and complex deformable bodes many of whch cannot be easly smulated by prevous mult-doman subspace smulaton technques. Despte beng more versatle our system can stll run at a hgh frame rate (from 16.7 to 100 FPS). he system has no strct requrement on the subspace bass so t s compatble wth better bass constructon methods and elastc models n the future. 2 Other Related Work Smulaton of deformable objects. he smulaton of deformable objects has been an mportant research topc n computer graphcs snce the early work by erzopoulos and colleagues [1987]. A varety of technques have been developed to smulate cloth [Baraff and Wtkn 1998; Cho and Ko 2002; Chen et al. 2013] elastc rods [Bergou et al. 2008; Umetan et al. 2014] and volumetrc deformable bodes [eran et al. 2003; Müller et al. 2005; Wang et al. 2010]. Although the performance of graphcs hardware has been sgnfcantly mproved n recent years t s stll computatonally expensve to smulate a large deformable object whch may take hours or even days. hs restrcts the smulators from usng hgh-resoluton meshes when they are appled n realtme applcatons such as artstc modelng vrtual surgeres and electronc games. Subspace smulaton of deformable bodes. Orgnally desgned to solve problems n other engneerng felds the subspace smulaton approach [Sfaks and Barbc 2012] has ganed popularty n computer graphcs recently. he early subspace smulator developed by Pentland and Wllams [1989] uses modal analyss to buld lnear vbraton modes as the subspace bass. Hauser and colleagues [2003] studed how to ntegrate collsons and other constrants nto modal analyss. Cho and Ko [2005] developed the modal warpng method to make lnear modes more sutable for handlng large deformatons by removng ncorrect vertex deformaton components caused by lnear elastcty. o more accurately model nonlnear elastcty under large deformaton Barbč and James [2005] constructed nonlnear deformaton modes from modal dervatves. An and collaborators [2008] proposed to calculate the reduced elastc force n the subspace by cubature approxmaton. her method reduces the computatonal complexty of subspace smulaton to O(r 3 ) n whch r s the number of modes n the subspace bass. Harmon and Zorn [2013] presented a bass augmentaton scheme to help subspace smulaton capture local deformaton caused by collson contact. Recently Hahn and colleagues [2014] used a pose-varyng bass to smulate clothng wth wrnkle detals. When smulatng deformable objects n the subspace an nterestng queston s whether collsons can be handled n the subspace as well. Based on the fact that two prmtves of the same object cannot be n contact wthout suffcent deformaton the collson cullng method developed by Barbč and James [2010] used subspace coordnates to check the exstence of potental self collsons. eng and colleagues [2014] explored the use of self contact patterns n subspace smulaton of artculated bodes and formulated a posespace cubature scheme to quckly resolve collsons wthout checkng colldng prmtves. In ths work our research s focused on mult-doman subspace smulaton and our system can beneft from the use of exstng and future subspace collson handlng methods. Subspace flud smulaton. Graphcs researchers have also nvestgated the use of subspace smulaton n flud anmaton. reulle and colleagues [2006] showed that both the velocty feld and the boundary couplng force can be reduced for subspace smulaton. Wcke and collaborators [2009] later mproved the usablty of ths method by decomposng the volume nto domans and enforcng velocty constrants at the doman nterface. Gven hghresoluton flud smulaton data Km and Delaney [2013] bult a reduced model to effcently generate anmaton results under new dy-

3 namc condtons. Recently Ando and colleagues [2015] combned a reduced pressure projecton solver wth a boundary conformng bass to speed up the pressure projecton step wthout volatng the free-surface boundary condton. 3 Background Doman D Doman D j In ths secton we wll present the background knowledge of deformable body smulaton n both the full space and the subspace. Full-space smulaton. Gven a deformable body wth N vertces we can formulate the equaton governng ts moton as Mü + D u + f nt (u) = f ext (1) where u R 3N s the stacked vertex dsplacement vector M R 3N 3N and D R 3N 3N are the mass and dampng matrces and f nt R 3N and f ext R 3N are the stacked nternal and external force vectors. o stably ntegrate u over tme by Equaton 1 many full-space smulators use the mplct Euler method whch requres solvng a large sparse lnear system wth 3N degrees of freedom. Subspace smulaton n an nertal frame. he basc dea behnd subspace smulaton s to constran the dsplacement vector u nto a subspace spanned by a set of r representatve dsplacement vectors {u 1 u 2... u r } also known as deformaton modes. hese vectors can be assembled nto a 3N r matrx U as the bass for subspace smulaton. So we can convert the dsplacement vector nto u = Uq n whch q R r contans the reduced coordnates of u n the subspace. Followng the method presented n [Barbč and James 2005] we use generalzed egenvalue decomposton to construct U such that U s mass orthogonal: U MU = I n whch I s the r r dentty matrx. By combnng u = Uq and U MU = I wth Equaton 1 we now obtan the governng equaton n the subspace: q + U DU q + U f nt (Uq) = U f ext. (2) Usng mplct tme ntegraton we can formulate Equaton 2 nto a dense r r lnear system. Snce r can be sgnfcantly smaller than 3N subspace smulaton can be orders of magntude faster than full-space smulaton. Subspace smulaton n a non-nertal frame. As Barbč and Zhao [2011] ponted out Equaton 2 s not sutable for smulatng the rgd moton of a deformable object snce ncorporatng rgd modes nto the bass U wll cause U to be tme-dependent and we cannot afford updatng U over tme. A more practcal way to handle the rgd moton s to smulate t separately from subspace deformaton by rgd body dynamcs as dd n [Barbč and James 2005; Kaufman et al. 2008]. Because non-rgd deformaton s now defned n a non-nertal frame we must consder four fcttous forces appled at each vertex x : f cor = 2 m ω (U q) f ne = m v f eul = m ω x (q) f cen = m ω ω x (q) whch are the Corols force the nertal force the Euler force and the centrfugal force respectvely. Here v and ω are the lnear and angular veloctes of the non-nertal frame m s the vertex mass U s the sub-matrx of U correspondng to vertex x and x (q) s the vertex poston n the non-nertal frame. Note that Equaton 3 requres all of the vectors to be defned n the non-nertal frame. By summng up all of the four forces and stack them nto a vector f fc R 3N we can formulate subspace smulaton of the non-rgd deformaton n the non-nertal frame by: q + U DU q + U f nt (Uq) = U ( R f ext + f fc ( v ω ω q q) ) (4) (3) : Doman elements E : Couplng elements E j : Doman elements E j : Vertex partton X : Vertex partton X j Fgure 2: A 2D beam. We segment ts vertces and elements nto two domans whch are connected by couplng elements. where R s a tme-varyng rotaton matrx from the global frame to the non-nertal frame. Note that R s not needed n front of f fc snce t has already been defned n the non-nertal frame. o effcently project the fcttous forces nto the subspace: U f fc we adopt the fast sandwch transform method 1 developed by Km and James [2011]. Bascally ths method uses pre-computed matrx blocks to drectly calculate U f fc based on the fact that these forces can be treated as lnear functons of R q and q. 4 Mult-Doman Subspace Smulaton In ths secton we wll nvestgate how to effcently smulate both rgd and non-rgd motons of a deformable body made of multple domans. o begn wth we wll present our new vertex-based mesh parttonng strategy n Subsecton 4.1. We wll then dscuss how to ncorporate couplng forces nto subspace smulaton and rgd body smulaton of each doman n Subsecton 4.2. Fnally we wll study numercal ntegraton and lnear solve n Subsecton Mesh Parttonng Gven a mesh M = (E X) n whch E s the set of ts elements and X s the set of ts vertces we choose to partton X nto d dsjont sets { X = d} wth X = d =1 X and X X j = for any j. hs parttonng can be automatcally generated by mesh segmentaton algorthms or manually created from user nput. Gven {X } we can further partton the elements E. Let E be a subset of E whose elements have vertces comng from X only. We defne D = (E X ) as the -th doman as shown n Fgure 2. If the vertces of an element belong to more than one doman ths element s a couplng element that connects these domans. he elements connectng doman D and doman D j form an nterface layer E j. We name D and D j as neghbors f E j. o smplfy our dscussons later we assume that an element does not connect more than two domans. It should be straghtforward to extend our methods to handle elements connectng three or more domans as well. able 2 lsts some of the symbols to be used n ths paper. 4.2 Inter-Doman Couplng Smlar to many exstng subspace smulators our system smulates rgd body dynamcs separately from non-rgd local deformaton. he man queston s: how can we ntegrate the couplng force nto the smulaton of these two components for each doman? 1 Smlar performance can be acheved by usng precomputed quanttes too as Barbč and Zhao [2011] proposed.

4 Symbol D E X E j e x l R v ω U l f l F j f j τ j Defnton he th doman of the mesh he element set of doman D he vertex set of doman D he couplng elements connectng D and D j he k th element of E j he l th vertex of element e n the global frame he rotaton matrx of doman D Doman D s lnear and angular veloctes he bass for vertex V l he couplng force on V l caused by e he reduced couplng force of D caused by E j he net couplng force on doman D caused by E j he net couplng torque on doman D caused by E j able 2: Symbols. hs table summarzes some of the symbols. Couplng force n subspace smulaton. Our system does not explctly model the deformaton of each couplng element. Instead the motons of the adjacent domans ndrectly cause the deformaton whch trggers the couplng force exerted on the domans next. Let E j be an nterface layer connectng doman D and doman D j. Its couplng force appled on D can be formulated n the subspace: F j = (U l ) R f l (5) e E j x l X where f l s the force of element e appled on ts vertex x l n the global frame R s the rotaton matrx of D and U l s the sub-matrx of doman D s bass U that corresponds to vertex x l. Intutvely Equaton 5 frst rotates the couplng force to the doman s non-nertal local frame and then reduces t to the doman s subspace. We wll use F and f to denote the forces n the subspace and the full space respectvely. By addng all of the couplng forces appled on doman D we can extend Equaton 4 to govern the non-rgd deformaton of D as q + U D U q + U f nt (U q ) + F j = Fsub (6) n whch F sub s defned as: F sub = U E j ( R f ext + f fc ( v ω ω q q ) ). (7) he terms wth subscrpt ndcate that they belong to doman D. Couplng force n rgd body smulaton. For nteractve applcatons we cannot assume that the rgd moton of a doman has already been gven as n [Km and James 2011]. So we need to know how the couplng force affects the rgd moton as well. Accordng to rgd body dynamcs [Baraff 1997] we have the equatons descrbng the lnear and rotatonal motons of doman D : m v + D v v = f ext e E j x l X fl = fext I ω + D ω ω = τ ext f j e E j x l X τl = τext τ j (8) n whch m and I are doman D s mass and nerta matrx τ ext s the total external torque τ l s the couplng torque caused by the couplng force f l and D v and D ω are two dampng matrces. For Raylegh dampng the dampng matrces can be computed from m I and the Jacoban matrces. For smplcty we use f j and τ j to denote the net couplng force and torque caused by the elastc forces of the couplng elements n E j. otal energy wo-step 10-4 s One-step 10-4 s wo-step 10-3 s One-step 10-3 s wo-step 10-2 s One-step 10-2 s me (s) Fgure 3: he energy dsspaton process of a 3D beam after ntal deformaton. hs plot vsualzes how the total energy dsspates when usng dfferent ntegraton schemes and tme steps wthout dampng. It ndcates that the one-step ntegraton scheme can preserve the energy well even when usng a large tme step. By combnng Equaton 6 wth Equaton 8 we now obtan a dynamcal system that governs both non-rgd and rgd motons of the domans whch are parttoned from the orgnal mesh. 4.3 Numerc Integraton Gven the dynamcal system presented n Subsecton 4.2 we can now study ts numercal ntegraton schemes and matrx solvers. wo-step ntegraton scheme. Snce we separate the rgd body moton of each doman from ts non-rgd moton a natural way to smulate a body wth multple domans s to solve rgd and non-rgd motons separately. Specfcally we propose a two-step ntegraton scheme to solve body motons wthn one tme step. Frst we assume that a deformed doman s rgd and we use the backward Euler method to ntegrate Equaton 8. After that we treat the local frame of each doman to be statc and we use the backward Euler method to ntegrate Equaton 6. Unfortunately our experment shows that ths two-step ntegraton scheme can cause very large artfcal dampng on the rgd moton unless a suffcently small tme step s used. Fgure 3 shows how the total energy dsspates when we use ths scheme to smulate a twodoman beam example. As the tme step gets larger the artfcal dampng effect becomes more obvous and the energy dsspaton rate s sgnfcantly ncreased. We beleve ths problem s fundamentally due to the fact that the two-step scheme gnores the nterplay between the rgd moton and the non-rgd moton wthn one tme step. he magntude of ths gnored term scales wth O( t 2 ) whch gradually vanshes when the tme step t becomes smaller. o solve ths problem we tested many deas such as asynchronous ntegraton and exchangng rgd and non-rgd nformaton multple tmes wthn one tme step. None of these deas s effectve as our experment shows. One-step ntegraton scheme. o use larger tme steps wthout artfcal dampng we propose to ntegrate rgd and non-rgd motons of all of the domans nto a unfed system and solve t by a sngle backward Euler step. Snce the fcttous forces contan nonlnear terms we ntegrate them explctly and treat the overall external force F sub appled n the subspace of doman D as constant. Let prmed symbols represent the states at the next tme t + t and non-prmed symbols represent the states at the current tme t. By lnearzng the elastc forces n Equaton 6 we obtan the

5 equaton for smulatng non-rgd deformaton as: ( ) I + U ( td + t 2 K )U q + t E j = t ( F sub U fnt E j F j) + q F j Q j Q j (9) where I R r r s the dentty matrx K R r r s doman D s tangent stffness matrx at tme t and Q j = [ q v ω q j v j ] ω j R 12+r +r j s the stacked velocty vector of doman D and doman D j ncludng the reduced deformaton veloctes the lnear veloctes and the angular veloctes. Here r s the number of modes n the subspace of Doman D. By applyng the same dea to lnearze Equaton 8 we obtan the lnear equaton for smulatng the rgd moton: m (v v ) + t( E j I (ω ω ) + t( E j ( f ) j Q j Q j + f j) + Dv v = tf ext ( τ ) j Q j Q j + τ j) + Dω ω = tτ ext. (10) Detals about Equaton 9 and 10 are n the appendx. Once we formulate the non-rgd deformaton and the rgd moton of each doman usng the two equatons we combne them nto a sngle lnear system and update the domans through a sngle lnear solve. Fgure 3 shows that ths one-step ntegraton scheme suffers less from the artfcal dampng ssue even when usng a large tme step. Matrx solver. he one-step ntegraton scheme results n a lnear system AQ = b where A R (6d+ r ) (6d+ r ) s a block-wse sparse symmetrc matrx. Fgure 4 shows a parttoned armadllo example and ts correspondng block matrx structure n whch the rows and the columns are n the order of D 1 D 2... D d. Intutvely each dagonal block represents a doman and two symmetrc off-dagonal blocks ndcate an nterface layer between two domans. Our experment shows that the nterplay between rgd motons and non-rgd motons can cause A to become ll-condtoned. hs nterplay s represented by off-dagonal matrx blocks that exchange nformaton between rgd motons and non-rgd motons. hs nformaton exchange process demands more teratons whch can be manfested as a hgh condton number. For example the condton number of the octopus example shown n Fgure 9 can be as hgh as 10 8 to In contrast f we choose the two-step ntegraton scheme nstead the matrx s decoupled nto a block dagonal matrx and the condton number drops to 10 3 to 10 4 mmedately. hs explans why the two-step scheme cannot beneft from the use of multple teratons because t s equvalent to a block descent solve sufferng from the slow convergence ssue. hs ssue exsts even f we use other teratve solvers such as conjugate gradent and generalzed mnmal resdual. So we choose to use sparse block Cholesky decomposton to solve the lnear system drectly. o reduce the number of new non-zero blocks ntroduced by the decomposton process we re-order the domans so that the decomposton order follows the doman connectvty. In an deal case when the doman connectvty has no loop no new block wll be ntroduced and the computatonal cost of the drect solve s lnear to the number of domans as n [Barbč and Zhao 2011]. A smlar dea was also used n [Hecht et al. 2012] for fast Cholesky re-factorzaton. o apply Cholesky decomposton the matrx must be symmetrc postve defnte. Usng the nvertble FEM method [Irvng et al. 2004; eran et al. 2005] we ensure that the stffness matrx s symmetrc and sem-postve defnte. heoretcally the matrx may stll have zero egenvalues but they are extremely rare as eran and colleagues [2005] ponted out and we dd not notce any falure n our experment. Fgure 4 vsualzes the domans of an armadllo example and ts matrx layouts (a) An armadllo wth unconstraned moton (b) he matrx layout n our system (c) he matrx layout after symamd Fgure 4: An armadllo example. We segment ths model nto 11 domans. As a result the system matrx has 11 blocks n ts rows and columns as shown n (b). After usng our doman re-orderng method Cholesky decomposton creates a matrx wth 33K nonzeros. In contrast after usng the matlab command symamd Cholesky decomposton creates a matrx wth 39K nonzeros shown n (c). 5 Cubature Approxmaton o mprove the system performance we must know how to quckly evaluate the reduced elastc forces and ther Jacoban matrces n the subspace. In general there are two effcent technques to accomplsh ths task: cubc polynomal proposed by Barbč and James [2005] and cubature approxmaton developed by An and colleagues [2008]. he cubc polynomal method s exact but t has a hgh computatonal cost O(drmax) 4 where r max s the maxmum number of modes used n the subspaces. So we choose cubature approxmaton nstead whose computatonal complexty s O(drmax). 3 o determne the cubature samples and weghts for the elastc forces nsde of a doman we follow the non-negatve-least-squares-based optmzaton approach presented n [An et al. 2008]. We wll dscuss how to generate the tranng and testng data for cubature optmzaton n Secton 6. Our focus n ths secton s on cubature optmzaton and approxmaton of the couplng forces. Unform weght cubature optmzaton. he deformaton of a couplng element causes not only the reduced couplng force n each doman s subspace but also the lnear force and the torque that affect the rgd moton of each doman. Snce all of these forces must be evaluated we propose to use cubature approxmaton for all of them and we handle them together n cubature optmzaton. For smplcty we defne the overall force vector generated by an nterface layer E j as: F = F j F j F j j F j j f j f j f j j f j j τ j τ j τ j j τ j (11) j

6 Valdaton error Non-unform weght Unform weght (a) Usng cubature approxmaton (b) Usng exact evaluaton Number of cubature samples Fgure 5: he errors of usng dfferent cubature schemes. hs plot shows that usng non-unform weghts provdes more accurate approxmaton than usng unform weghts. However ther dfference becomes less sgnfcant as the total number of samples ncreases. Fgure 6: A caterpllar example. Our cubature approxmaton scheme accurately estmates the reduced forces wthn the domans and at the nterfaces. So the result of usng cubature approxmaton n (a) s comparable to the result usng exact evaluaton n (b). whch contans all of three forces appled on both doman D and doman D j. We normalze each force component to avod any bas durng the optmzaton process. Once we obtan a large set of these hgh-dmensonal force vector data we use the optmzaton framework [An et al. 2008] to generate cubature samples and weghts. Snce these forces are collected nto a sngle vector and optmzed together they wll be evaluated wth the same cubature weghts. Fgure 5 shows the approxmaton errors usng dfferent numbers of samples when we use ths optmzaton strategy. Non-unform weght cubature optmzaton. We can make cubature approxmaton more accurate by usng dfferent samples and weghts for dfferent forces. If we use dfferent samples for dfferent forces we need to evaluate more samples n total whch becomes less effcent n practce. So we choose to stll use the same samples but dfferent weghts for dfferent forces. Specfcally we frst run the unform weght optmzaton strategy to obtan cubature samples and unform weghts as before. We then allow the weghts to vary by runnng four addtonal optmzaton processes for F j F j j (f j f j j ) and (τ j τ j j ) respectvely. Here we stll use the same cubature weghts for f j and f j j and τ j and τ j j to ensure lnear and angular momentum conservaton n rgd body moton. Note that the use of non-unform weghts wll cause the lnear system presented n Secton 4.3 to be asymmetrc so we replace Cholesky decomposton by LU decomposton n our drect solver accordngly. Snce the solver contrbutes a small porton of the overall computatonal cost the use of an asymmetrc matrx has lmted nfluence on the system performance. Fgure 5 compares the approxmaton errors of usng unform and non-unform cubature weghts. In general the non-unform weght scheme s more accurate especally when the number of samples s small. Fgure 6 shows the result of usng cubature approxmaton s vsually ndstngushable from the result of exact evaluaton. 6 Implementaton Here we wll provde some mplementaton detals of our system. Subspace bass constructon. A nce feature of our system s that t does not have any strct requrement on the subspace bass. In fact the bass of each doman can be generated usng many exstng technques ncludng moton capture data pre-computed smulaton data and modal analyss. In our current mplementaton we use lnear modal analyss and modal dervatves proposed n [Barbč and James 2005] to generate both lnear and nonlnear deformaton modes n the bass. Our system allows the bass to be generated separately for each doman. In our experment t took (a) Rgd nterface (b) Non-rgd nterface Fgure 7: Comparson of rgd and non-rgd nterfaces. Our system allows the nterface to be naturally deformed as shown n (b). less than fve mnutes to fnsh constructng subspace bases for all of the domans. In contrast Km and James [2011] constructed the bass for the whole mesh frst and then dvded each deformaton mode nto separate ones for the domans. her method becomes computatonally expensve f the mesh s too large. he method also has dffculty n factorng out the rotatonal components n the resultng bass of each doman. Fnally ther bases may cause the system matrx to be nearly sngular as our experment shows. Cubature optmzaton. Snce the scope of our method s not lmted to sknned characters we can not sample skeleton poses to generate pose data for cubature optmzaton as dd n [Km and James 2011]. Instead we apply random forces on each doman to get randomly deformed confguratons for a non-sknned mesh. We then use the nvertble FEM method [Irvng et al. 2004; eran et al. 2005] to generate full-space smulaton data. Gven the tranng data we perform ntra-doman cubature optmzaton n the same way as descrbed n [An et al. 2008]. We process nterface cubature optmzaton dfferently as dscussed prevously n Secton 5. In our experment the tranng data took 2 to 10 mnutes to generate and the cubature optmzaton step took 5 to 30 mnutes. 7 Results We tested our system on an Intel Core GHz processor. We use the OpenMP lbrary to parallelze our smulaton steps. All of our examples are smulated at the tme step t = 1/90s. Currently our examples use the St. Venant-Krchhoff materal model whose stffness s parameterzed by two Lamé coeffcents. Fgure 7b shows our system can smulate natural deformaton of the whole object ncludng the nterface between two domans. In contrast f we do not allow the nterface to deform the mult-doman problem can be smplfed but the result looks strange as Fgure 7a shows. Here we generate the result n Fgure 7a by enforcng the rgdty constrant on the nterface durng bass reconstructon.

7 Name and # of Interface Cubature ets Doman Doman Cholesky Full-space otal Statstcs Modes DoFs ets (nterface / total) Dynamcs Couplng Factorzaton Converson Cost Armadllo / ms 3.4ms 0.9ms 4.1ms 16.7ms 62K nodes 241K tets / ms 4.8ms 2.0ms 5.0ms 30.3ms 11 domans 10 nterfaces / ms 6.2ms 3.2ms 6.2ms 45.1ms Octopus / ms 3.3ms 1.0ms 2.4ms 16.1ms 24K nodes 84K tets / ms 5.7ms 2.3ms 3.3ms 38.4ms 17 domans 16 nterfaces / ms 7.9ms 3.8ms 4.4ms 59.8ms Hammock / ms 1.7ms 0.3ms 1.4ms 10.0ms 14K nodes 42K tets / ms 2.0ms 0.7ms 2.1ms 14.4ms 5 domans 4 nterfaces / ms 3.1ms 1.3ms 2.4ms 23.9ms Caterpllar / ms 1.0ms 5.6ms 2.8ms 11.4ms 31K nodes 98K tets / ms 1.3ms 1.2ms 4.2ms 19.8ms 8 domans 7 nterfaces / ms 2.2ms 2.2ms 5.0ms 32.7ms able 3: Mesh and performance statstcs. hs table lsts the number of modes per doman the total degrees of freedom the number of nterface tetrahedra the number of nterface cubature tetrahedra the total number of cubature tetrahedra the computatonal cost of each smulaton step and the total computatonal cost per tme step. # of Frame Rate Name Modes sngle doman no nter. cubature ours Armadllo Octopus Hammock Caterpllar able 4: Frame rates. hs table lsts the smulaton frame rates. Performance evaluaton. able 3 summarzes the statstcs of the 3D models and ther performances n our experment. It shows that the doman dynamcs step whch constructs the unfed system for lnear solve s typcally the computatonal bottleneck. When a model (such as the armadllo example) contans many nterface tetrahedra the use of nterface cubature s necessary to reduce couplng force and matrx evaluaton costs. In contrast f a model (such as the caterpllar example) does not contan many nterface tetrahedra the performance mprovement provded by nterface cubature becomes small as able 4 shows. able 4 also llustrates that the smulaton speed s much lower f the system uses a sngle doman wth the same number of degrees of freedom. hs s because the matrx would become dense whch s more computatonally expensve to construct and solve. Note that our system s compatble wth parallel computng and t has a good potental to run faster on GPU. An nterface doman n the full space. If the nterface between two subspace domans contans a thck layer of elements we can treat t as a doman as well and smulate t n the full space. o do that we smply ncorporate ts mplct tme ntegraton nto the lnear system and handle the couplng between the nterface doman and the subspace doman n the same way as before. Snce the nterface doman s defned n the global frame there s no need to smulate ts rgd moton separately. One use of these nterface domans s to allow user to edt the mesh. For example user can cut the mesh n the nterface domans as shown n Fgure 1b and 9c. hs cannot be done f the doman s smulated n the subspace. Comparson to sprng-based couplng. An nterestng queston s what f we couple the domans drectly by sprngs not by 10 5 Sprng-based method (faled) Our method Ground truth Fgure 8: A 3D beam example parttoned nto two domans. hs example compares the results of sprng-based couplng elementbased couplng (as our method) and full-space smulaton (as ground truth). Our method allows all of the elements to use the same stffness so t avods the couplng problems that may occur n stffness parameter tunng. couplng elements. o answer ths queston we mplemented a sprng-based couplng method n our system smlar to [Km and James 2011]. We also smulated the ground truth n full-space. Fgure 8 compares the results of these methods. It ndcates that the result qualty of the sprng-based couplng method really depends on the sprng stffness parameter. If the stffness s too low the sprngs cannot keep the two domans well attached and f the stffness s too hgh lockng and artfcal dampng ssues wll become obvous. he stffness value depends on the mesh resoluton and the tme step so t needs to be tuned for dfferent nterfaces and smulaton cases. In contrast our couplng method allows all of the elements to use the same stffness and t does not need to tune the stffness of the couplng elements separately although the stablty ssue stll occurs f the stffness s too hgh. 8 Conclusons We present a new mult-doman subspace smulaton technque based on the vertex-based mesh parttonng strategy. Our research shows that ths technque can overcome the lmtatons of many prevous technques and our system can be bult upon the same elastc model wthout addtonal hard or soft couplng constrants. Our experment demonstrates the flexblty of our system n handlng large and complex deformable bodes and ts performance s comparable to those of the exstng subspace smulaton technques.

8 (a) he domans (b) Unconstraned deformaton (c) Deformaton after cut Fgure 9: An octopus example. Our system effcently smulates ths octopus example n real tme whch contans 17 domans. Graph. (SIGGRAPH Asa) 27 5 (Dec.) 165:1 165:10. Ando R. hürey N. and Wojtan C A dmenson-reduced pressure solver for lqud smulatons. Computer Graphcs Forum (Eurographcs) Baraff D. and Wtkn A Large steps n cloth smulaton. In Proceedngs of the 25th Annual Conference on Computer Graphcs and Interactve echnques ACM New York NY USA SIGGRAPH Fgure 10: Dscontnuty artfact. hs example demonstrates the dscontnuty artfact at the nterface due to large deformaton. Lmtatons and future work. Our system does not solve the fundamental dscontnuty ssue among multple domans smulated n ther own subspaces. Because of ths the surface may be uneven and the couplng elements may be nverted when the body undergoes large deformaton as shown n Fgure 10. One soluton to ths problem s to extend the boundary-aware dea [Yang et al. 2013] to nonlnear deformaton modes whch however s not so straghtforward and needs further research n the future. he couplng force n our system needs addtonal cubature samples for evaluaton whch ncreases the overall computatonal cost. Our plan s to mplement the whole system on GPU so that forces and Jacoban matrces can be quckly evaluated n parallel. Our system separates rgd motons from non-rgd motons. Although t provdes a practcal soluton to dynamc smulaton t s not strctly correct and the result can be dfferent from the result of full-space smulaton. Currently our system ntegrates nonlnear fcttous forces explctly and mplements mplct tme ntegraton by a sngle Newton teraton. As a result the system can become unstable when t handles complex hyperelastc materals or fast rotatons. We can ntegrate fcttous forces mplctly and use multple Newton teratons n the future f the computatonal resource allows. Fnally our system does not consder collsons yet and t s nterestng to know how self collsons can be handled n mult-doman subspace smulaton. 9 Acknowledgments We thank Nvda and Adobe for fundng and equpment support. hs work was supported n part by NSF grant IIS References An S. S. Km. and James D. L Optmzng cubature for effcent ntegraton of subspace deformatons. ACM rans. Baraff D An ntroducton to physcally based modelng: Rgd body smulaton - unconstraned rgd body dynamcs. In An Introducton to Physcally Based Modellng SIGGRAPH 97 Course Notes 97. Barbč J. and James D. L Real-tme subspace ntegraton for St. Venant-Krchhoff deformable models. ACM rans. Graph. (SIGGRAPH) 24 3 (July) Barbč J. and James D. L Subspace self-collson cullng. ACM rans. Graph. (SIGGRAPH) 29 4 (July) 81:1 81:9. Barbč J. and Zhao Y Real-tme large-deformaton substructurng. ACM rans. on Graphcs (SIGGRAPH 2011) :1 91:7. Bergou M. Wardetzky M. Robnson S. Audoly B. and Grnspun E Dscrete elastc rods. ACM rans. Graph. (SIG- GRAPH) 27 3 (Aug.) 63:1 63:12. Chen Z. Feng R. and Wang H Modelng frcton and ar effects between cloth and deformable bodes. ACM rans. Graph. (SIGGRAPH) 32 4 (July) 88:1 88:8. Cho K.-J. and Ko H.-S Stable but responsve cloth. ACM rans. Graph. (SIGGRAPH) 21 3 (July) Cho M. G. and Ko H.-S Modal warpng: Real-tme smulaton of large rotatonal deformaton and manpulaton. IEEE rans. Vs. Comp. Graph (Jan.) Hahn F. homaszewsk B. Coros S. Sumner R. W. Cole F. Meyer M. DeRose. and Gross M Subspace clothng smulaton usng adaptve bases. ACM rans. Graph. (SIGGRAPH) 33 4 (July) 105:1 105:9. Harmon D. and Zorn D Subspace ntegraton wth local deformatons. ACM rans. Graph. (SIGGRAPH) 32 4 (July) 107:1 107:10. Hauser K. K. Shen C. and O Bren J. F Interactve deformaton usng modal analyss wth constrants. In Graphcs Interface

9 Hecht F. Lee Y. J. Shewchuk J. R. and O Bren J. F Updated sparse Cholesky factors for corotatonal elastodynamcs. ACM rans. Graph (Sept.) 123:1 123:13. Huang J. Lu X. Bao H. Guo B. and Shum H.-Y An effcent large deformaton method usng doman decomposton. Comput. Graph (Dec.) Irvng G. eran J. and Fedkw R Invertble fnte elements for robust smulaton of large deformaton. In Proceedngs of SCA Kaufman D. M. Sueda S. James D. L. and Pa D. K Staggered projectons for frctonal contact n multbody systems. ACM rans. Graph. (SIGGRAPH Asa) 27 5 (Dec.) 164:1 164:11. Km. and Delaney J Subspace flud re-smulaton. ACM rans. Graph. (SIGGRAPH) 32 4 (July) 62:1 62:9. Km. and James D. L Physcs-based character sknnng usng mult-doman subspace deformatons. In Proceedngs of SCA Müller M. Hedelberger B. eschner M. and Gross M Meshless deformatons based on shape matchng. ACM rans. Graph. (SIGGRAPH) 24 3 (July) Pentland A. and Wllams J Good vbratons: Modal dynamcs for graphcs and anmaton. SIGGRAPH Comput. Graph (July) Sfaks E. and Barbc J FEM smulaton of 3D deformable solds: A practtoner s gude to theory dscretzaton and model reducton. In ACM SIGGRAPH 2012 Courses SIGGRAPH 12 20:1 20:50. eng Y. Otaduy M. A. and Km Smulatng artculated subspace self-contact. ACM rans. Graph. (SIGGRAPH) 33 4 (July) 106:1 106:9. eran J. Blemker S. Hng V. N.. and Fedkw R Fnte volume methods for the smulaton of skeletal muscle. In Proceedngs of SCA eran J. Sfaks E. Irvng G. and Fedkw R Robust quasstatc fnte elements and flesh smulaton. In Proceedngs of SCA erzopoulos D. Platt J. Barr A. and Flescher K Elastcally deformable models. In Proceedngs of the 14th Annual Conference on Computer Graphcs and Interactve echnques ACM New York NY USA SIGGRAPH reulle A. Lews A. and Popovć Z Model reducton for real-tme fluds. ACM rans. Graph. (SIGGRAPH) 25 3 (July) Umetan N. Schmdt R. and Stam J Poston-based elastc rods. In ACM SIGGRAPH 2014 alks ACM New York NY USA SIGGRAPH 14 47:1 47:1. Wang H. O Bren J. and Ramamoorth R Multresoluton sotropc stran lmtng. ACM rans. Graph. (SIG- GRAPH Asa) 29 6 (Dec.) 156:1 156:10. Wcke M. Stanton M. and reulle A Modular bases for flud dynamcs. ACM rans. Graph. (SIGGRAPH) 28 3 (July) 39:1 39:8. Yang Y. Xu W. Guo X. Zhou K. and Guo B Boundaryaware multdoman subspace deformaton. IEEE rans. Vs. Comp. Graph (Oct) A Dervaton of Equaton 9 Snce only nternal elastc forces are ntegrated mplctly we can dscretze Equaton 6 n tme by pluggng n q = ( q q)/ t: ( q q)/ t + U D U q + U f nt (U q ) + F j = F sub. (12) By lnearzng the nternal forces at tme t we can get the approxmate nternal force at the next tme nstant t + t: f nt (U q ) = fnt (U q ) E j (U q ) + fnt q (q q ) = f nt (U q ) + fnt (U q ) u u q (q q ) = f nt (U q ) + tk U q. (13) Wth smlar lnearzaton and dervaton we get the equaton descrbng the reduced couplng force as: F j = F j + F j Q j Q. (14) j E j After mergng Equaton 12 to 14 we obtan Equaton 9. B E j Equatons for Implct Integraton Let x l be the l-th vertex of element e n the global frame and x l be ts poston n the local frame of doman D. he matrx-vector product term n Equaton 9 can be evaluated as: = E j e E j x l 0 x l 1 X F j q q + F j v v + F j ω ω + F j q j q j + F j v j v j + F j e E j x l 0 X x l 1 X j t(r U l 0 t(r U l 0 ω j ω j fl0 ) (R x l U l 1 1 q + v + ω R x l 1 ) fl0 ) (R x l j U l 1 1 q j + v j + ω j R j x l 1 + ) (15) n whch the term f l 0 / xl 1 s one of the 3 3 sub-blocks of the element stffness matrx correspondng to element e. he matrxvector product terms n Equaton 10 can be expanded n a smlar fashon. Matrx symmetry. Although t s not ntutve the system matrx assembled from Equaton 9 and 10 s ndeed symmetrc. hs s because every term n these equatons has a correspondng symmetrc term n one of the other equatons. For example the term E j F j / v j n Equaton 15 s one sub-block of the system matrx A whose ts transpose s: F j v = j E j e E j x l 0 X x l 1 X j = e E j x l 0 X x l 1 X j t(r U l 0 x l 0 fl0 ) x l 1 fl1 t R U l 0 = E j f j j q. (16) Here the symmetry of element e s tangent stffness matrx.e. ( f l 0 / xl 1 ) = f l 1 / xl 0 s used n ths dervaton. Equaton 16 ndcates that E j F j / v j and E j f j j / q are two symmetrc blocks. Smlar relatonshps can be derved for other terms.