Cloth Simulation. Burak Ertekin i MSc Computer Animation and Visual Effects

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1 Cloth Smulaton Burak Ertekn MSc Computer Anmaton and Vsual Effects November 17, 2014

2 1 Abstract In ths paper, we explan how to make a Cloth Smulaton usng Mass Sprng Model. We explaned the nternal dynamcs of the sprng model together wth the ntegraton methods. Explct Euler ntegraton method s used to obtan new postons and veloctes at each tme step. In the end, we talked about collson detecton methods that are effcent and powerful enough to use wth ths model. Our am was to mplement a smple yet realstc cloth smulaton. 2 Introducton Computer Anmaton s started to beng used more frequently n ths past decade. The moton pcture ndustry s usng anmaton technologes to create more nfluental and fascnatng work. Computer games gets more realstc wth faster and effcent anmaton solutons. Cloth Smulaton s an essental element n computer anmaton. We are usng cloth-lke materal everyday. It s crucal to mplement ths feature n our anmatons n hghly detaled and speed-effcent manner. The progress of cloth smulaton started wth the dea of creatng elastcally deformable objects. And today we have hghly nteractve, real-tme solutons. Mass sprng system s the most common approach n soft body dynamcs. Other approaches such as, fnte element smulaton, energy mnmzaton methods exsts. Even though some of these approaches gve physcally more accurate results, t s tme neffcent n terms of computaton. Therefore mass sprng systems are wdely used and developed. In the mplementaton, we are usng mass sprng system as models. Wth Euler Integraton and a smple collson detecton algorthm, we am to anmate a fne cloth smulaton. In ths paper, frstly we wll explan what s a mass sprng system. We wll get n detals wth the forces used n the system. Then we wll pass on ntegraton methods. Manly we wll talk about Euler Integraton methods however we wll also menton about other ntegraton methods. Later on, we wll see dfferent collson detecton and response methods. Fnally we wll menton on some nterestng future work on cloth smulaton. 3 Related Work Tradtonally, models n computer graphcs were geometrcally or algebracally defned prmtves. Terzopoulos et al. suggested smulatng physcal propertes such as tenson and rgdty to model deformable objects[1]. Ths would gve the statc shapes of the models. And by ncludng other physcal propertes such as mass and dampng, they suggested smulatng dynamcs of the objects. Terzopoulos and Cargnan et al. dscretzes cloth as rectangular mesh[2] whle Baraff et al. models the cloth as a trangular mesh of partcles[3]. Wang et al. stated the advantage of trangular mesh over rectangular mesh[4]. 1

3 Terzopoulos derved energy functons usng a contnuum formulaton n ther mplementaton. However they dd not use any dampng force n the nternal sprng force equaton, only Hooke s Law: kẋ was used. Cargnan added a force whch damps cloth stretch and shear but not bend. Cloth s a materal wth strong resstance to stretchng motons whle t has very lttle resstance to bendng and shearng motons. As Baraff mentoned, ths results n a stff underlyng dfferental equaton of moton. Snce explct methods are bad choce to solve stff equatons because of the necessty of small tme steps, Baraff demonstrated mplct methods overcomng the computaton lmts of explct methods[3]. Desburn et al. presented another method: a rapd approxmaton of the mplct ntegraton[5]. Kang et al. came up wth another approxmaton method for ntegraton[6]. Volno et al. proposed an adaptaton of the mplct Mdpont method.[7] Collsons are a major bottleneck n cloth smulaton as sad by Brdson et al.[8]. Several dfferent approaches has been made on ths topc however these approaches dffer consderng other methods mplemented on the model. Volno et al. proposed the correcton of the partcle acceleraton after a collson[9]. Ths acceleraton s used as a collson force by the model and ntegrate wth the other exstng force of the partcle. Baraff suggested alterng postons of the cloth partcles[3]. Brdson et al. combned geometrc collsons wth repulson force. Selle et al. used parallel Gauss-Sedel collson/repulson response[10]. Neal presents CollsonGrd system whch uses heght representaton of terran to approxmate the collsons[11]. 4 Mass Sprng Model Mass Sprng System s an approach to model flexble and deformable objects. In ths approach, we have a set of ponts connected by weghtless elastc sprngs formng the model. The forces on these sprngs are represented by Hooke s Law, that s: F = kx. Where k s the sprng constant. In ths approach, we can specfy k accordng to our needs n the model. It s common to set a damper wth each sprng to create a better control of the moton[12]. A smple dampng force on the -th mass pont can be represented as: F d = k d v (k d > 0) Where k d s the dampng factor and v s the current velocty of -th mass pont[6]. We use Hooke s Law and ths smple dampng formula to obtan our nternal sprng forces n our mplementaton. And we obtan ths fnal equaton: F = k s x k d v (k d > 0) We have three dfferent sprngs n a mass sprng system: structural, shear and flexon sprngs. Interactons between dfferent parts of the cloth are formed by structural sprngs. The resstance aganst bendng s modeled by flexon sprngs. And shearng sprngs are used to model shearng resstance. Rectangular mesh has a small resstance to the dagonal bendng. Wang suggested addng 2

4 dagonal flexon sprng for ths problem[4][13]. In order to reduce the number of sprngs for each mass, Wang elmnated the shearng sprngs n one drecton and proposed usng dagonal flexon sprng. Wth ths they obtaned a rectangular trangle mesh. Instead, they use an equlateral trangle mesh. Wth ths new structure, they only use two types of sprngs: structural and flexon snce trangular mesh can resst shearng tself. Even though t has the same computaton, t has more resstance aganst bendng, stretch and compresson[4]. 5 Integraton 5.1 Explct Euler Integraton In our mplementaton, we use Newton s equaton of moton and ntegrate t wth explct Euler method to obtan the new poston and velocty for our system. Explct Euler ntegraton scheme can be shown as: v t+h x t+h = v t + F t h m = x t + v t+h h Where v t denotes the velocty of the -th mass pont at tme t whle F t s the force actng on the mass pont and x t denotes the locaton[6]. Wth ths method we can easly calculate x t+h. Even though explct Euler ntegraton s a very smple scheme, t creates numercal nstablty unless the tme step n smulaton s small enough. As Desburn stated, the tme step dt whch must be nversely proportonal to the square root of the stffness[5]. 5.2 Implct Euler Integraton Implct ntegraton schemes offers stable anmatons wth large tme steps wthout problem. Therefore mplct methods are generally used for cloth smulatons, t provdes faster computaton. Implct Euler ntegraton scheme can be shown as: v t+h x t+h = v t + F t+h h m = x t + v t+h h The only dfference s F t t+h s replaced wth Ft. In ths ntegraton method, the soluton s not straghtforward. To solve a dfferental equaton lke dy dt = f(y ) numercally, the dervatve s not f(y t ) throughout the tme nterval, but the dervatve s a weghted average of dervatve f(y t ) at the begnnng of the nterval and f(y t+h ) at the end of the nterval. The update formula s: Y t+h = Y t + h[(1 λ)f(y t ) + λf(y t+h )] λ s between 0 and 1. When λ s 0, the update formula becomes the explct Euler method. When λ s 1, the formula s the mplct Euler method. And ths substantates the mplct formula above[6]. 3

5 5.3 Other Integraton Methods Implct methods gves you more stable results however these results are necessarly not more accurate. We can obtan more accuracy by usng more complex ntegraton methods such as Rosenbrook method. Although we need to solve four tmes a lnear system of each ntegraton step, our computaton tmes becomes longer than mplct methods[7]. Volno suggested mplct Mdpont method, however n the case of nonlnear problems, they had numercal nstablty. Runge-Kutta ntegraton method s a good opton for numercal accuracy however t creates dscontnutes n moton n case of collson. Snce we are nterested n cloth smulaton, t s unwse to use ths method n our model[5]. 6 Collson Detecton A realstc collson detecton mechansm s crucal for any physcally plausble anmaton system. It s hard to mplement a collson detecton algorthm whch works for any possble case n cloth smulaton. However t s easy to detect the deformatons on a cloth lke object because of ts thnness; the objects most lkely perce through the cloth and create an unrealstc look. Moreover, n cloth smulaton all the partcles has the potental to collde wth each other and the envronment n any gven tme step[8]. In order to have more realstc smulaton, we have to ncrease partcle count however ths wll also ncrease the complexty of our algorthm. On the other hand, self-collsons are very mportant for cloth smulaton because we can form folds and wrnkles by cloth/cloth nteractons. In our smulaton we use a smple method of poston alteraton as our collson detecton/response soluton. Even though ths soluton has some problems n some cases, t generally works fne wth low-resoluton work. As Baraff mentoned, cloth/cloth collson are detected by checkng the partcles and the edges of cloth trangles. They overcame the cloth/cloth collson problem by nsertng a strong dampng force to push the cloth apart. As for the cloth/object ntersectons, they test each ndvdual cloth partcle wth the faces of the object. The faces of the object are grouped to a herarchcal boundng box tree. The problem wth poston alteraton s that partcle s neghbours affected by ths alteraton[3]. To overcome ths problem, Baraff suggested to add an arbtrary correcton value y whch s used only to move to desred locaton durng the backward Euler step[3]. x = h(v 0 + v ) + y Brdson proposed to use an mpulse needs to be appled to two ponts n the cloth: I ˆn and I ˆn where I s the magntude and ˆn s the drecton. For pont/trangle case, where x 1 x 2 x 3 are nteror ponts of trangle wth barycentrc coordnates w 1, w 2, w 3 nteractng wth pont x 4, the adjusted mpulses are 2I Ĩ = 1 + w1 2 + w2 2 + w2 3 4

6 v new = v + w (Ĩ/m)ˆn = 1, 2, 3 v 4 new = v 4 (Ĩ/m)ˆn For edge/edge case; when a pont wth poston a along the edge x 1 x 2 nteracts wth another pont wth poston b along the edge x 3 x 4, the adjusted mpulses are 2I Ĩ = a 2 + (1 a) 2 + b 2 + (1 b) 2 v 1 new = v 1 + (1 a)(ĩ/m)ˆn v 3 new = v 3 (1 b)(ĩ/m)ˆn v 2 new = v 2 + a(ĩ/m)ˆn v 4 new = v 4 b(ĩ/m)ˆn Weghtng the mpulses n ths way ntroduces approprate torques for offcenter nteractons as well as gvng contnuty across trangle boundares, and converges to the expected formulas when the nteror ponts approach mesh nodes [8]. Geometrc self-collsons whch occur n foldng and contact stuatons can be really expensve. To resolve ths problem, we use self-repulsons. It s smlar to collsons, we have both pont/trangle and edge/edge nteractons. We have nelastc collson mpulse to stop approachng nteractons pars and elastc repulson mpulse to push nteracton pars further apart[10]. Inelastc repulson mpulse s I c = mv n /2 where v n s the normal velocty and m s the mass of partcles. Elastc repulson of a sprng wth a stffness k s ( ( )) 0.1d I r = mn tkd, m t v n where d = h (x 4 + w 1 x 1 w 2 x 2 w 3 x 3 )ˆn and where w are the barycentrc weghts of the free pont x 4 projected to the trangle, x {1,2,3} are the trangle s pont locatons, h s the repulson thckness and ˆn s the trangle normal[8][10]. Another approach to collson detecton s to use a heght feld representaton of terran to approxmate collson detecton and response[11]. The CollsonGrd system defnes the geometry of objects n sngle axal drecton. By projectng the poston of each partcle on the CollsonGrd plane and blnearly nterpolate the nearest four data ponts to determne the heght of the surface. The value s compared to heght of the partcle along the collson axs to determne f there s a collson. As for collson response, ts resolved manually by changng poston and velocty of partcles[11]. 7 Concluson and Future Work Cloth Smulaton s a very fascnatng topc n computer anmaton for a long tme. We tred to explan the basc way to create a cloth. However the area of research for ths topc s extensve. Renderng, collson detecton and response, ntegraton methods, fabrc, shearng and tearng can be told as the 5

7 man topcs[14][15]. Due to ts partcle based structure, the computaton costs ncreases drastcally when we ncrease the partcle count. In order to obtan more realstc look on the smulaton, we need to use more partcles n the smulaton. In addton to the exstng structure; when there s a tearng happens on the cloth, new partcles are formed. Therefore, there s always a new demand for new methods n cloth smulaton to create more realstc and nteractve work. References [1] D. Terzopoulos, J. Platt, A. Barr, and K. Flescher, Elastcally deformable models, SIGGRAPH Comput. Graph., vol. 21, no. 4, pp , Aug [Onlne]. Avalable: [2] M. Cargnan, Y. Yang, N. Magnenat-Thalmann, and D. Thalmann, Dressng anmated synthetc actors wth complex deformable clothes, n Proceedngs of the 19th Annual Conference on Computer Graphcs and Interactve Technques, SIGGRAPH 1992, J. J. Thomas, Ed. ACM, 1992, pp [Onlne]. Avalable: [3] D. Baraff and A. Wtkn, Large steps n cloth smulaton, n Proceedngs of the 25th Annual Conference on Computer Graphcs and Interactve Technques, ser. SIGGRAPH 98. New York, NY, USA: ACM, 1998, pp [Onlne]. Avalable: [4] X. Wang and V. Devarajan, 2d structured mass-sprng system parameter optmzaton based on axsymmetrc bendng for rgd cloth smulaton, n Proceedngs of the 2004 ACM SIGGRAPH Internatonal Conference on Vrtual Realty Contnuum and Its Applcatons n Industry, ser. VRCAI 04. New York, NY, USA: ACM, 2004, pp [Onlne]. Avalable: [5] M. Desbrun, P. Schröder, and A. Barr, Interactve anmaton of structured deformable objects, n Proceedngs of the 1999 Conference on Graphcs Interface 99. San Francsco, CA, USA: Morgan Kaufmann Publshers Inc., 1999, pp [Onlne]. Avalable: [6] Y.-M. Kang, J.-H. Cho, H.-G. Cho, and C.-J. Park, Fast and stable anmaton of cloth wth an approxmated mplct method, n Proceedngs of the Internatonal Conference on Computer Graphcs, ser. CGI 00. Washngton, DC, USA: IEEE Computer Socety, 2000, pp [Onlne]. Avalable: [7] P. Volno and N. M. Thalmann, Implementng fast cloth smulaton wth collson response, n Proceedngs of the Internatonal Conference on Computer Graphcs, ser. CGI 00. Washngton, DC, USA: IEEE Computer Socety, 2000, pp [Onlne]. Avalable: 6

8 [8] R. Brdson, R. Fedkw, and J. Anderson, Robust treatment of collsons, contact and frcton for cloth anmaton, ACM Trans. Graph., vol. 21, no. 3, pp , Jul [Onlne]. Avalable: [9] P. Volno, M. Courchesne, and N. Magnenat Thalmann, Versatle and effcent technques for smulatng cloth and other deformable objects, n Proceedngs of the 22Nd Annual Conference on Computer Graphcs and Interactve Technques, ser. SIGGRAPH 95. New York, NY, USA: ACM, 1995, pp [Onlne]. Avalable: [10] A. Selle, J. Su, G. Irvng, and R. Fedkw, Robust hgh-resoluton cloth usng parallelsm, hstory-based collsons, and accurate frcton, IEEE Trans. Vs. Comput. Graph., vol. 15, no. 2, pp , [Onlne]. Avalable: [11] K. Neal, Collson approxmaton for real-tme cloth smulaton, n ACM SIGGRAPH 2004 Posters, ser. SIGGRAPH 04. New York, NY, USA: ACM, 2004, pp. 89. [Onlne]. Avalable: [12] R. Parent, Computer Anmaton, Second Edton: Algorthms and Technques, 2nd ed. San Francsco, CA, USA: Morgan Kaufmann Publshers Inc., [13] X. P. Insttut and X. Provot, Deformaton constrants n a mass-sprng model to descrbe rgd cloth behavor, n In Graphcs Interface, 1996, pp [14] M. S. Souza, A. von Wangenhem, and E. Comunello, Fast smulaton of cloth tearng, SBC, vol. 5, no. 1, [15] N. Metaaphanon, Y. Bando, B. Chen, and T. Nshta, Smulaton of tearng cloth wth frayed edges, Comput. Graph. Forum, vol. 28, no. 7, pp , [Onlne]. Avalable: 7

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