Detail Preserving Continuum Simulation of Straight Hair

Transcription

1 To appear n the ACM SIGGRAPH conference proceedngs Detal Preservng Contnuum Smulaton of Straght Har Aleka McAdams Andrew Selle Kelly Ward Eftychos Sfaks Joseph Teran Unversty of Calforna, Los Angeles Walt Dsney Anmaton Studos Abstract Har smulaton remans one of the most challengng aspects of creatng vrtual characters. Most research focuses on handlng the massve geometrc complexty of hundreds of thousands of nteractng hars. Ths s accomplshed ether by usng brute force smulaton or by reducng degrees of freedom wth gude hars. Ths paper presents a hybrd Euleran/Lagrangan approach to handlng both self and body collsons wth har effcently whle stll mantanng detal. Bulk nteractons and har volume preservaton s handled effcently and effectvely wth a FLIP based flud solver whle ntrcate har-har nteracton s handled wth Lagrangan selfcollsons. Thus the method has the effcency of contnuum/gude based har models wth the hgh detal of Lagrangan self-collson approaches. CR Categores: I.3.7 [Computer Graphcs]: Three-Dmensonal Graphcs and Realsm Anmaton I.6.8 [Smulaton and Modelng]: Types of Smulaton Anmaton Keywords: har smulaton, contnuum models 1 Introducton Smulatng har on a vrtual character remans one of the most challengng aspects of computer graphcs. As har s an ntegral part of creatng many vrtual characters, the problem s especally mportant. Unfortunately, the massve number of hars nteractng and colldng makes ths task especally challengng. Many approxmatons for smulatng har exst, but they typcally fal to provde the amount of detal that real har exhbts. Several applcatons, such as feature flms, am to capture the hgh degree of complexty caused by several thousand nteractng har strands. Even though ndvdual har dynamcs scale well to multple hars (as each har s dynamcally decoupled), accurately smulatng many hars nteractng wth each other remans a challenge. Instead, numerous approaches have been developed to manage the complexty of many hars nteractng by smulatng a smaller set of gude hars (typcally no more than several hundred) that nteract wth large repulson forces, nterpolatng a larger number of hars for renderng. Ths leads to very effcent smulaton tmes, but a lmted amount of har detal s captured (gnorng stray hars such as the so-called flyaways ) because, essentally, each gude har represents hundreds (or even thousands) of actual hars. Alternatvely, there have been several methods that treat every smulated har as part of a flud-lke contnuum volume. These approaches naturally model har nteracton wthout explct collsons. However, the downsde s that ntrcate features of ndvdual hars are lost because each har s enslaved to the contnuum. Fgure 1: (Left) Real har exhbtng ntrcate webbng as result of complex collson/contact. (Rght) Our method creatng a smlar effect. Our technque factors har computaton nto two parts: a coarse, hghly coupled volumetrc behavor, whch s effcently modeled by a contnuum; and a fner, more locally coupled Lagrangan partcle smulaton of har. Unlke prevous contnuum-based approaches that only smulate gude hars that do not nteract drectly, our method smulates many hars (several thousand) that are allowed to collde drectly (as well as through the volume). We handle self-collsons more effcently than fully Lagrangan collson models because the volume does most of the work towards resolvng collsons. Thus, our method can capture the ntrcate detals of many ndvdual strands whle effcently mantanng the overall har volume. 2 Related Work Har modelng s an actve area of research that can be dvded nto the followng categores: har shape modelng, har strand dynamcs, har complexty management and nteracton, har shadng and self-shadowng. Statc har shape modelng, such as [Yu 2001; Km and Neumann 2002; Choe and Ko 2005], har shadng and shadow computaton are beyond the scope of our paper, so we do not dscuss them further. A more detaled dscusson on these topcs can be found n the recent survey paper [Ward et al. 2007]. Strand Dynamcs. Many har smulaton technques consder dfferent geometrc and consttutve models for modelng ndvdual har strands or clumps of har startng wth mass sprng systems [Rosenblum et al. 1991] and projectve dynamcs [Anjyo et al. 1992]. Mass/sprng smulated lattce deformers can be added to model torson [Plante et al. 2002; Selle et al. 2008]. Rgd body channg methods are also commonly used [Brown et al. 2004; Choe et al. 2005; Hadap 2006]. Technques based on elastc rod theory have become popular startng wth [Pa 2002] and contnung wth [Grégore and Schömer 2006; Bertals et al. 2006]. [Spllmann and Teschner 2007] also uses elastc rod theory but adaptvely changes the number of ponts that defne the har curve. Most recently, [Bergou et al. 2008] provdes a novel modelng of twst as devaton from a canoncal frame as well as a method to evolve ths separately quasstatcally. Work n strand dynamcs has been extensve, and our technque can be used wth the practtoner s model of choce, but for smplcty we use the mass/sprng model. Har Interacton and Complexty. The smplest (but most expensve) approach to har complexty management s smulatng and renderng every har [Rosenblum et al. 1991; Anjyo et al. 1992; 1

2 To appear n the ACM SIGGRAPH conference proceedngs Fgure 2: Our method models the har both as curves (left) and as a volumetrc velocty feld (rght). Selle et al. 2008]. Whle ths theoretcally yelds the most detal, t s the most ntractable, especally f self-collsons are consdered. Thus most practtoners use a clumpng or contnuum level of detal scheme that trades accuracy for computatonal tractablty: Contnuum models. [Hadap and Magnenat-Thalmann 2001] uses a Lagrangan flud (smoothed partcle hydrodynamcs) technque to model flud forces on a mass/sprng system. Whle ths effcently handles bulk behavor, detaled har behavor s lost. In addton, snce smoothed partcle hydrodynamcs (SPH) s used, forcng flud nteracton through sprng lke forces, ths approach s effectvely as numercally dffcult as standard self-repulson sprng forces. [Bando et al. 2003] also uses a contnuum approach whle dscardng partcle connectvty for real-tme applcatons. Unfortunately, ths results n vsual artfacts n renderngs because of the lack of connectvty n addton to the contnuum s loss of hgh fdelty detal. [Petrovc et al. 2005] presented a contnuum approach that rasterzed har to an Euleran velocty feld and level set wth the goal of reducng computatonal complexty, rather than mprovng on fdelty. Dynamcs on the grd was lmted to repeated averagng (an approxmaton of vscosty) to obtan frcton effects. Unlke our method, they dd not consder the ncompressblty or other flud behavors that could be mparted on the velocty feld. In addton, the goal of the work was to model relatvely smple harstyles on stylzed characters, so the detal lost by usng a coarse grd was acceptable for ther applcaton. Clumped models. There are many technques that smulate har nteractons through a sparse set of dsjont gudes that collde wth one another [Bertals et al. 2006; Hadap 2006; Gupta et al. 2006; Chang et al. 2002]. A greater set of rendered har strands are then ether nterpolated between these gudes or clumped to the gude strands. [Plante et al. 2002] used a lattce whose vertces were smulated wth a mass/sprng model to defne many more nterpolated hars. [Choe et al. 2005] syntheszed addtonal hars from a sparse set of smulated gude hars wth a statstcal model. [Bertals et al. 2006] used a combnaton of nterpolaton and clumpng to generate rendered strands. Many clumped technques also use adaptvty to merge or splt groups of har to reduce computaton where and when the requred detal s low [Bertals et al. 2003; Ward et al. 2003; Ward and Ln 2003]. The use of sparse clumps often allows the use of cheap repulson penaltes as opposed to geometrc collsons because gudes are expected to have thckness, and n Secton 4.2 we wll dscuss how ths breaks down wth thousands of gudes. 3 Method Overvew Our approach combnes a Lagrangan har solver wth an Euleran flud smulator (see Fgure 2). We use a mass/sprng model for the har partcles, connectng each partcle n sequence wth a sprng and every other partcle wth a bendng sprng. Torson could be added to the mass/sprng model, or a dfferent strand model (such as an artculated rgd body or Cosserat model) could be used. Each Fgure 3: Outlne of the Lagrangan (top) and Euleran (bottom) components of our method. tme ntegraton step of the dynamc system proceeds as follows (see Fgure 3): 1. Save collson-free poston and velocty 2. Velocty step v n+1 = v n + ta (x n, v n+1 ) 3. Use the volume technque to modfy v n+1 velocty ˆv n+1 (Substeps 3a, 3b, 3c) 4. Modfy ˆv n+1 for collsons to get v n+1 5. Compute fnal poston x n+1 = x n + tv n+1 nto a corrected where x n, v n s the tme t n poston and velocty of the th partcle, respectvely, and t s the tme step. Note that step 4 s dscussed n Secton 4.2 and mplemented usng [Brdson et al. 2002]. a (x n, v n+1 ) s separated nto a lnear dampng part and non-lnear elastc part as n [Brdson et al. 2003] to preserve elastc modes. Step 3 rasterzes the har volume to a grd representng the har densty and velocty, modfes the velocty feld to handle bulk self-nteracton and then apples the modfed nformaton back to the partcle veloctes (Secton 4.1). 4 Solver 4.1 Volume A volume method s an deal way to handle self-nteracton of many hars because when many hars are near each other, ther contact and collsons allow a propagaton of any force through hars n contact. Ths behavor s smlar to the behavor of a flud, whch conserves momentum and mass. For computatonal effcency, we draw on ncompressble flud smulaton technques to model our har volume. Densty modelng s stll possble as dscussed later n Secton The two common approaches used to model flud volumes are Lagrangan technques (smoothed partcle hydrodynamcs, vortex partcle methods) and Euleran technques (common pressure/velocty ncompressble solvers such as [Stam 1999]). Euleran technques tend to be more effcent because nearest neghbor searches are unnecessary. The downsde of Euleran approaches s that they are most natural on unform grds that lmt the amount of detal avalable. Even partcle based methods actually lmt detal, because hgh degree unstructured stencls also create numercal smoothng. For our purposes, we wll use an Euleran approach and compensate for ths lack of detal by usng hgh resoluton Lagrangan collson handlng technques to capture hgh fdelty detal. An ncreasngly popular approach for flud advecton s the FLIP method (ntroduced to graphcs by [Zhu and Brdson 2005]) whch 2

3 To appear n the ACM SIGGRAPH conference proceedngs Fgure 4: A demonstraton of volumetrc densty control. (Left) hgh densty and low volume. (Mddle) moderate densty and moderate volume. (Rght) low densty and hgh volume. replaces the tradtonal Euleran advecton equaton of velocty n vt + (v )v = 0 wth a Lagrangan step xn+1 = xn + tv. Ths velocty s rasterzed to a grd and made dvergence free usng the Chorn projecton method [Chorn 1967]. The dvergence free velocty feld s compared to the orgnal grd velocty feld, and ths dfference s nterpolated to the partcles and appled as an mpulse. Ths can be thought of as a partcle flud method that uses ncompressble technques to provde an mplct modelng of ncompressble behavor rather than an explct one, provdng a convenent way to communcate between the grd and partcles. In fact, [Losasso et al. 2008] demonstrate a hybrd grd/partcle flud technque that uses FLIP as a couplng mechansm between the grd and the partcles. Gven the nput canddate veloctes v?n+1 and postons xn from step 2, we follow steps smlar to [Losasso et al. 2008], though t s mportant to note our canddate postons contan the effects of elastcty. Fgure 5: A volumetrc separaton constrant can be defned to control the amount of stckng. (Left) no separaton condton s appled. (Rght) an easy to satsfy separaton condton s specfed to produce mmedate decouplng. where I(x, v) s trlnear nterpolaton at locaton x of a vector feld v and ξ controls the amount of FLIP vs. PIC. In our smulatons, we typcally used a value of ξ = Densty Control The rasterzed cell weghts are used to defne a densty whch can be targeted to a user defned densty as n [Losasso et al. 2008] by usng a dvergence source term n the Posson equaton. Near collson objects ths method can lose effectveness because knematc velocty constrants nterfere wth dvergence. Thus, we also modfy the fxed veloctes on Neumann faces f the face s weght s less than the densty target. We smply add to the constraned velocty n the collson body s normal drecton. Ths s analogous to a penalty repulson n Lagrangan dynamcs, but handlng t n the Posson equaton means t wll be made consstent globally. The results of densty control can be seen n Fgure 4. At each tmestep, we resze our grd to ft the volume of har, keepng the grd spacng constant. We then rasterze canddate tme tn+1 segments at ther canddate locatons defned by forward Euler, nstead of partcles drectly, to ensure good coverage of the volume. Consder a segment wth partcles (, j) S at ts canddate po?n+1 n,xj + tvj?n+1 )). Gven stons (.e. (x0, x0j ) = (xn + tv a pont x n space whch has dstance dj (x) to the segment, defne a weght wj (x) = max (0, r dj (x)), where r s a userdefned radus of nfluence for each segment. We typcally used r = 3 x/2 n our smulatons where x s the grd sze. Then the rasterzed velocty at any pont s P (,j) S wj (x)[(1 αj (x))v + αj (x)vj ] P v(x) = (,j) S wj (x) Separaton Control A problem wth volume/contnuum approaches to har s that nearby hars are forced to behave smlarly. Ths s desred when har s under compresson because t forces the velocty to be zero at the center, preventng nterpenetraton. However, when two regons of har have dsparate velocty felds, t may be desrable to control the amount of stckng ths artfcal couplng can create (Fgure 5). In fact, ths problem also appears n flud technques when sold objects are coupled to the flud. where αj (x) s the nterpolaton fracton of the closest pont on the segment. Veloctes and weghts are computed on the faces (for MAC veloctes) and cell centers for densty control and separaton condton computaton. Once the veloctes and weghts are rasterzed, we must setup a?n+1 Posson system 2 p = vgrd to project the grd veloctes to be dvergence free n step 3b. Any cell wth a weght lower than a threshold s set as a Drchlet p = 0 condton, and any face that s nsde a knematc collson body s gven a zero Neumann boundary condton and the velocty on that face s set to the object velocty. A dvergence source term s set to target densty (see Secton 4.1.1). We solve the system wth precondtoned conjugate gradent and then compute the dvergence free velocty feld n+1?n+1 vgrd = vgrd p. Addtonal vscosty could also be added at ths stage to create addtonal frcton (varable vscosty s also a possblty, see [Carlson et al. 2002; Rasmussen et al. 2004]). To prevent unwanted stckng, we compute a har separaton condton durng the rasterzaton process. Consder a face of the grd havng the two ncdent cell veloctes v1 and v2. A face s consdered separatng f v1 n v2 n < γ, where n s the vector pontng from cell 1 to cell 2 and γ s a scalar separaton parameter. Ths means that the doman of the grd should be decoupled at ths face, and cell 1 should not see the pressures on cell 2 and vce versa. The row of the matrx of each cell s modfed to see the other cell as a ghost Drchlet p = 0 cell. Ths s accomplshed smply by zerong aj and aj n the matrx (preservng symmetry). Unfortunately, performng ths change means we cannot project the 1 face velocty because the gradent stencl x (p2 p1 ) s no longer defned. Thus, nterpolaton of veloctes to partcles n cell 1 or 2 for FLIP s not defned so these partcles are not changed durng the FLIP update. Even so, ther collsons are resolved by Lagrangan self-collsons. See Fgure 5 rght for results of ths approach. Fnally, n step 3c, FLIP s used to apply ths velocty back to the th partcle usng the formula ˆ n+1?n+1 n n+1 v n+1 = ξ v?n+1 + (I(xn, vgrd vgrd )) +(1 ξ)i(x, vgrd ) 3

4 To appear n the ACM SIGGRAPH conference proceedngs c Fgure 6: A hand-anmated character walkng wth a harstyle smulated by our method. The rghtmost mage s a closeup. Images Dsney. All Rghts Reserved. 4.2 Lagrangan Collsons son et al. 2002] for edge/edge pars. However, the placement of our volume handlng before applyng self-collsons not only makes fndng a fnal collson free confguraton possble, but t actually precondtons the collson step effectvely replacng the repulson step of Brdson. Ths allows us to prevent poor confguratons before they create vsual artfacts n the self-collson and rgd group steps whch s essental n har where the lack of stablzng pont/face nteractons make collsons potentally more damagng. In fact, n many ways t s a better precondtoner than proxmtybased repulsons, because our volume formulaton consders veloctes as well as postons. We llustrate the effectveness of a volumetrc response to stackng collsons n Fgure 7. Once step 3 s done, we have a velocty feld that roughly consders self-collson and global collsons; however, t msses fne detals because t has been computed at the coarser resoluton of the grd. If we had only a few hundred gudes, repulsons mght be a good opton for removng the remanng collsons, but at the denstes of our examples, they become mpractcal. Ths s because repulsons are proxmty based, so as densty ncreases, the fact that thckness must decrease means that the repulsons look smaller compared to the veloctes. Thus we turn to swept self-collsons to handle our fne collsons. Geometrc collsons have been studed extensvely for cloth smulaton because they are essental for preventng vsual artfacts. The current state of the art s based on [Brdson et al. 2002] whch uses a three stage process to ensure no collsons are mssed. Frst, contact s precondtoned usng penalty-based repulsons that are small enough to prevent vsual artfacts. Second, self-collsons are appled to stop as much nterpenetraton as possble. In ths step, collsons are detected by coplanarty tests between nearby pont/face and edge/edge pars, and mpulses are appled to colldng pars. As new collsons may have been created by these mpulses, ths step s terated over untl all collsons are resolved. Thrd, rgd groups (mpact zones) are used as a fnal safety net, postcondtonng the collsons. Subsequent papers have typcally focused on mprovng the latter two components. For example [Sfaks et al. 2008] replaces the second step wth a globally coupled collson scheme and [Harmon et al. 2008] mproves rgd mpact zones. 5 Results We have demonstrated our technque wth a range of examples n our prototype system; one can easly mplement t wth a standard Lagrangan har smulator and Euleran flud solver. Examples are rendered wth Renderman and standard har shadng models. It would be feasble to apply the recent subsurface scatterng and shadowng acceleraton technques of [Moon et al. 2008; Znke et al. 2008; Bertals et al. 2005], some of whch mght beneft from our rasterzed volume. Whle we do not generate any addtonal strands at render tme, tradtonal nterpolaton or clumpng technques can naturally be utlzed f more strands are desred for renderng detal. A typcal example uses a grd wth 603 cells n the volumetrc step; however, ths number s dynamcally updated as the volume of har changes shape. We also demonstrate complex contact and collson usng a brad pattern consstng of 1500 hars wth approxmately 100 partcles each n Fgure 9. The three har sectons begn n a very loose confguraton wth the top partcle of each har fxed. As gravty pulls the hars downward, a brad pattern emerges due to our Euleran/Lagrangan collsons. Fgure 6 shows our technque appled to an anmated character walkng wth 10,000 smulated hars. Our last example (Fgure 8) compares our method to Lagrangan and Euleran collsons alone. A bundle of 1200 hars s draped across a perpendcularly hangng bundle of 1200 more hars (240,000 partcles). Whereas purely Lagrangan collsons create hghly actve collson mpulses due to nadequate har couplng, purely Euleran collsons are overly damped and fal to resolve the collsons. Our result shows a moderate amount of couplng from the volume together wth fne detals obtaned wth Lagrangan collsons. Fgure 1 shows that our method compares favorably to real har behavor. For cloth, mprovng collsons through better post-condtonng s a useful technque because falures n repulsons and self-collsons typcally do not result n sgnfcant vsual artfacts. In har, however, a deluge of repulsons and collsons appled usng relatvely unstable edge/edge nteractons results n confguratons that would be dffcult to correct wth a better rgd group technque. [Selle et al. 2008] mentons many of these dffcultes and n fact they resorted to turnng off collsons n contact cases to prevent these ptfalls. In our mplementaton, we follow the collson algorthm n [Brd- Fgure 7: Collsons under stackng. A comparson of geometrc collsons wth (rght) and wthout (left) volumetrc precondtonng. A comparson of runtmes for the three technques shows the purely Euleran technque has the lowest average runtme per frame 4

5 To appear n the ACM SIGGRAPH conference proceedngs Our Hybrd Method Lagrangan Collsons Euleran Collsons 1.14 mn/frame 3.65 mn/frame 7.97 mn/frame 9.34 mn/frame N/A mn/frame N/A 3.55 mn/frame 5.67 mn/frame Both Lagrangan collsons and our method successfully stop the fallng bundle whle the Euleran collsons do not. Our method exhbts a more natural couplng whch stll exhbts hgh fdelty detal. Geometrc collsons tme Volume substep tme Total tme Fgure 8: A cylndrcal bundle of hars s dropped on another bundle of suspended hars wth a tmng table (2400 hars). (5.6m). Although our method computes both volumetrc and Lagrangan collsons, t s stll sgnfcantly faster (8m) per frame than Lagrangan collsons alone (13.6m), showng the effectveness of the Euleran dvergence-free solve as a collson precondtoner. A further breakdown of tmng can be found n Fgure 8, and we note that the remander of tme s spent on tme ntegraton. In addton, the resoluton of the volume creates some numercal vscosty and n partcular angular velocty dsspaton. Ths can be controlled by reducng the use of the volume (at the expense of less effcency) or by ncreasng the resoluton of the grd. Addtonally, f the grd s too coarse, peces of har that become severely tangled may not be able to separate. Smlarly snce frcton s modeled partally by vscosty (numercal and modeled), t s somewhat naccurate. So n the future we would lke to nvestgate usng an octree grd, to allow dfferent resolutons n dfferent parts. For example, usng the work of [Ward et al. 2003] n a large volume of har, har that s not vsble may not requre a hghly detaled velocty feld. Creatng a level set by applyng a fast marchng method to the prevous tmestep s densty volume could derve a refnement crteron. Parallelzaton of the grd rasterzaton, Posson solve, and FLIP applcaton would also allow hgher resolutons. We expect these algorthms to map well to multcore archtectures and graphcs hardware. We are also nterested n applyng ths technque to cloth as a replacement to the Brdson repulsons to better precondton the cloth collsons. We note that smulaton tme was about 15 mnutes (26.9% Lagrangan collsons, 33.9% volumetrc, 39.2% mass/ sprng tme ntegraton) per frame for the character n Fgure 6 on a sngle machne whch s an mprovement over the 16-way parallel runtmes of [Selle et al. 2008]. The tme step was determned by the mass sprng Courant condton, though we found n many examples that the volumetrc step provded some extra stablty, allowng us to relax the tme step restrcton. 6 Dscusson Whle we beleve our technque makes hgh-fdelty nteractons tractable, there are some lmtatons. We have not tred ths method on non-straght har as straght har tends to exhbt the most complcated stackng confguratons. Brdson s collson handlng algorthm n generalzed coordnates may be dffcult to apply because mpulses are gven n maxmal coordnates, and the assocated generalzed coordnate response can be ambguous. The FLIP mpulse wll be smlarly dffcult to apply. Thus, n future work we plan to experment wth the maxmal coordnate mass-sprng torson model of [Selle et al. 2008]. We note, however, that the brad Fgure 9 smulaton (1500 hars, 150 partcles each) n Fgure 9 shows both large and small scale nteractons, showng promse for applcatons of our method. 7 Concluson We have presented a technque that hybrdzes Lagrangan and Euleran har smulaton technques. Inspred by recent FLIP and SPH flud technology, we show that our model can be useful as a way of controllng the ntegraton of volume based forces. In addton we show how the volume can ease collson dffcultes wth har by actng as an mproved precondtoner. We have shown that the factorzaton of har nteracton nto a coarse, globally-coupled phenomenon and a hghly detaled Lagrangan vew s an effectve strategy. Ths mproves effcency by allowng both types of behavors to be captured n the most natural way possble. Acknowledgements A. McAdams s supported n part by NSF DMS E. Sfaks and J. Teran are supported n part by DOE 09-LR BERA, NSF DMS , NSF CCF , ONR N , and an Intel Larrabee Research Grant. We would lke to acknowledge Maryann Smmons, Arthur Shek, and Joe Marks for useful dscussons and support. Addtonally, we apprecate the Walt Dsney Anmaton Studo artsts provdng us wth our anmated character example. 5

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