Fluid Animation with Dynamic Meshes

Transcription

1 Computer Graphcs Proceedngs, Annual Conference Seres, 2006 Flud Anmaton wth Dynamc Meshes Bryan M. Klngner Bryan E. Feldman Nuttapong Chentanez James F. O Bren Unversty of Calforna, Berkeley Fgure 1: Top: A paddle mxes smoke n a tank. Bottom: A cross-secton of the smulaton meshes used for each frame. Abstract Ths paper presents a method for anmatng flud usng unstructured tetrahedral meshes that change at each tme step. We show that meshes that conform well to changng boundares and that focus computaton n the vsually mportant parts of the doman can be generated quckly and relably usng exstng technques. We also descrbe a new approach to two-way couplng of flud and rgd bodes that, whle general, benefts from remeshng. Overall, the method provdes a flexble envronment for creatng complex scenes nvolvng flud anmaton. Keywords: Natural phenomena, physcally based anmaton, computatonal flud dynamcs. CR Categores: I.3.5 [Computer Graphcs]: Computatonal Geometry and Object Modelng Physcally based modelng; I.3.7 [Computer Graphcs]: Three-Dmensonal Graphcs and Realsm Anmaton; I.6.8 [Smulaton and Modelng]: Types of Smulaton Anmaton. 1 Introducton Although systems for physcally based flud anmaton have developed rapdly n recent years and can now relably generate producton-qualty results, they stll have some lmtatons. Smulaton domans can change substantally from step to step because of deformng boundares, movng obstacles, and evolvng flud moton, yet current systems based on fxed grds are not deally suted to handle these stuatons. We propose a method to smulate fluds wth such rapdly changng domans by generatng a new tetrahedral smulaton mesh at each tme step. When generatng the mesh, we use the poston and shape of boundares as well as crtera based on the vsually mportant parts of the flud and velocty feld to construct a szng feld that dctates the desred edge length for tetrahedra throughout the doman. We then use an effcent and relable meshng algorthm adapted from [Allez et al., 2005] to produce a mesh that s refned accordng to ths feld. We use unstructured tetrahedral meshes because they conform to curved and rregular boundares better than axs-algned grds wth the same number of grd elements and allow for precse control of refnement throughout the doman. We transfer the physcal propertes of the smulaton from the old mesh to the new mesh usng a generalzaton of the sem-lagrangan velocty advecton technque that ntroduces no addtonal smoothng. We then perform a mass conservaton step that has been extended to allow a new, sngle-step soluton of two-way couplng between flud and rgd bodes. Overall, ths approach provdes a flexble framework for flud smulaton that opens the door to many features. We have mplemented the system and tested t n a varety of scenaros such as the one shown n Fgure 1. We have found that the combnaton of unstructured tetrahedral domans and dynamc remeshng creates a versatle envronment for the creaton of complex and vsually nterestng flud anmatons. 2 E-mal: {klngner feldman nchentan job}@eecs.berkeley.edu From the ACM SIGGRAPH 2006 conference proceedngs. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. ACM SIGGRAPH 2006, Boston, MA c Copyrght ACM Background The anmaton of fluds through physcal smulaton has become an mportant tool n the vsual effects ndustry. One approach that has been popular n recent years makes use of a spatal dscretzaton based on regular, fxed, hexahedral grds. Some examples of ths approach can be found n [Foster and Metaxas, 1996], [Foster and Metaxas, 1997], [Stam, 1999], [Yngve et al., 2000], [Fedkw et al., 2001], [Foster and Fedkw, 2001], [Enrght et al., 2002], [Carlson et al., 2002], [Feldman et al., 2003], and [Goktekn et al., 2004]. The most

2 Computer Graphcs Proceedngs, Annual Conference Seres, 2006 commonly used storage scheme for these approaches s the staggered grd scheme. Ths method offsets storage of dfferent quanttes on the grd, and was frst descrbed by [Harlow and Welch, 1965]. Efforts have been made to enhance these methods to allow for better conformance to rregular boundares such as the free surface of lquds, complex obstacles, or rregularly shaped domans. [Losasso et al., 2004] descrbed an octree-based method that retans many of the advantages of regular grds whle allowng computatonal effort to be focused n partcular parts of the smulaton doman; ths enables detaled trackng of movng boundares such as lqud surfaces. Both [Carlson et al., 2004] and [Guendelman et al., 2005] have demonstrated methods for two-way couplng of obstacles to flud. Unstructured tetrahedra have also been used for flud smulaton wthn the graphcs communty. Two examples of ths are [Feldman et al., 2005a] and [Elcott et al., 2005]. The frst method uses a velocty-based approach whle the second uses a vortcty-based formulaton. It s a blend of deas from these two papers, along wth a generalzaton of the sem-lagrangan velocty advecton technque for movng meshes descrbed n [Feldman et al., 2005b] that forms the heart of our method. The dea of movng meshes ndependent of a fxed or partcle-centrc coordnate system s not a new one; arbtrary Lagrangan-Euleran (ALE) methods were desgned for just ths purpose. They have proven useful n the smulaton of hghly deformable elastc materals. ALE was frst descrbed n [Hrt et al., 1974], where t was used wth fnte dfferences to solve compressble flud problems. [Donea et al., 1977] went on to apply ALE n a fnte element settng. An excellent survey of the development of ALE methods appears n [Donea et al., 2004]. Examples wthn the graphcs lterature that feature movng meshes wthout remeshng nclude [Shah et al., 2004] and [Rasmussen et al., 2004], both of whch translate the grd to follow the vsually mportant porton of the flud. Another approach to handlng changng domans s to dspense wth the mesh altogether, nstead usng Lagrangan partcles for smulaton of fluds. A few examples of ths approach are [Terzopoulos et al., 1989], [Desbrun and Can, 1996], [Can and Desbrun, 1997], [Stora et al., 1999], [Müller et al., 2003], [Premože et al., 2003], and [Müller et al., 2004]. These meshless methods are partcularly well suted to changng domans because ponts can move freely wthout concerns about mesh qualty. Because we regenerate a new smulaton mesh at each tme step, the vablty of our method hnges on fast, hgh-qualty, relable tetrahedral mesh generaton. Whle a hstory of unstructured mesh generaton s outsde the scope of ths paper, [Owen, 1998] and [Teng and Wong, 2000] provde good surveys of the feld. For our mesh generator we selected the approach descrbed n [Allez et al., 2005]. Ths nnovatve method produces meshes whch conform to domans of arbtrary topology quckly and relably. Also, t allows for the local edge length of the tetrahedra to be specfed arbtrarly throughout space, whch allows us to easly perform adaptve mesh refnement from step to step. The meshes produced by ths technque are Delaunay, whch provdes mproved gradent estmaton and allows us to sgnfcantly smplfy some of the expressons that arse when nterpolatng velocty values stored on the mesh. 3 Methods The key contrbuton of our method s to demonstrate the freedom granted by remeshng at each smulaton tme step. The core of our system s based on the smple, effcent methods for dscretzng the nvscd Euler equatons on tetrahedral meshes descrbed n [Elcott et al., 2005] and [Feldman et al., 2005a]. We have made a few modfcatons n order to combne the best aspects of both approaches that are descrbed below. Once we have a good dscretzaton, we need a way to propagate nformaton from one mesh to the next. [Feldman et al., 2005b] detals a generalzaton of the standard sem- Lagrangan velocty advecton technque that allows smulaton state to be transferred between deformng domans wthout ncurrng addtonal smoothng. We demonstrate that ther approach can easly be appled to transfer nformaton between two arbtrary, topologcally unrelated meshes, whch s requred to acheve more general evoluton of the smulaton doman from step to step. Fnally, we need to quckly and relably generate a new tetrahedral mesh for each tme step that suts the current smulaton condtons, such as conformance to boundares and obstacles as well as any desred refnement. Although methods have long exsted to mesh arbtrary domans, most are relatvely slow n comparson to smulaton runnng tmes or don t relably termnate under realstc condtons. The avalablty of effcent, versatle meshng algorthms such as [Allez et al., 2005] has made the generaton of a new mesh at each tme step practcal. Any changes that were requred to make these peces work together harmonously are dscussed below. Also, we descrbe a new, sngle-step method to acheve two-way couplng between obstacle and flud moton. 3.1 Dscretzaton We use a staggered flud state storage scheme that stores pressures at tetrahedron crcumcenters and face-normal veloctes, the component of velocty n the drecton of the face normal, at the face crcumcenters. Smlar schemes have been used n [Botta and Hempel, 1996], [Elcott et al., 2005] and [Feldman et al., 2005a]. These methods are a generalzaton of the staggered grd scheme orgnally proposed by [Harlow and Welch, 1965]. Ths staggered method s used to dscretze the nvscd Euler equatons: u t = (u ) u p ρ + f ρ subject to the mass conservaton constrant for ncompressble fluds: u = 0. (2) In these equatons, u s the flud velocty, t tme, p pressure, ρ densty, and f any external forces. The symbol denotes the vector of dfferental operators = [ / x, / y, / z] T. We account for the changes n the mesh over a tme step drectly durng sem-lagrangan advecton (see Secton 3.2) Dscrete Dervatve Operators Dvergence and gradent operators are needed as part of the mass conservaton step. We make dscrete estmates of these dervatves followng the formulaton presented n [Losasso et al., 2004] and [Elcott et al., 2005]. The dvergence of a tetrahedron s computed as an area weghted sum of the tetrahedron s face normal veloctes. The gradent at a face crcumcenter n the drecton of the face s normal s computed usng fnte dfferences. The dfference n crcumcenter pressures adjacent to a face s dvded by the dstance between these crcumcenters. In Delaunay meshes, the lne connectng adjacent tetrahedra crcumcenters passes through the crcumcenter of the face between them and s n the drecton of that face s normal. Ths property of Delaunay meshes motvates our storage scheme at crcumcenters because the gradent estmate s equvalent to the gradent of a pecewse lnear functon that nterpolates the crcumcenter values. (1) 821

3 ACM SIGGRAPH 2006, Boston, MA, July 30 August 3, Velocty Interpolaton The staggered scheme stores only the component of velocty n the face normal drecton. For both the sem-lagrangan step and to advect smoke partcles for renderng, a full velocty vector must be found at arbtrary postons n the mesh. We nterpolate velocty vectors from face normal veloctes usng the two-step method developed n [Elcott et al., 2005]. Frst, a velocty vector, u t, s computed at each tetrahedron crcumcenter, then we nterpolate wthn Vorono cells usng u t values at the cell vertces. Velocty u t for tetrahedron t s found by solvng the small lnear system N tu t = z t where N t s a matrx contanng 4 rows of the face normals of t and z t s a vector of the 4 face normal veloctes assocated wth t. For a dvergence-free feld, ths soluton has the remarkable property that nterpolatng back to the face crcumcenters exactly recovers the orgnal face-normal veloctes. Thus nterpolatng the u t veloctes also exactly nterpolates the face-normal velocty components, and does not ncur the error one would otherwse expect from a twostep nterpolaton method. To fnd a velocty at an arbtrary pont we nterpolate wthn the Vorono cell usng the tetrahedra veloctes assocated wth the cell. Ths nterpolaton s based on the method of [Warren et al., 2004], whch presents a way to nterpolate wthn a general convex polytope. They nterpolate the value at the pont x as a weghted sum of the polytope s node values where node t s unnormalzed weght s computed as w t(x) = N t f σ t n f x + d f. (3) Here, σ t s the set of polytope faces that ntersect at node t. The denomnator s the product of dstances from x to the faces n σ t computed usng the face normals,n f, and plane offsets, d f. N t s the determnant of a matrx of face normals n σ. Weghts from all nodes are normalzed to sum to 1 before use n the weghted sum. To smplfy ths computaton we take advantage of two propertes: 1) n a Delaunay mesh, edges are n the drecton of the Vorono cell s face normals and 2) the volume of tetrahedron t s 1/6 E t where E t s a matrx formed from the three vectors of edges emanatng from a common node of t. After some manpulaton, whch s omtted for brevty, Equaton (3) appled to node weghts wthn a Vorono cell can be smplfed to w t(x) = 6Vol(t) 3 =1 (p pv) (ct x) (4) where w t(x) s the weght assocated wth the node at tetrahedra t s crcumcenter, Vol(t) s the volume of tetrahedron t, p v s the poston of the node assocated wth the Vorono cell, p are postons of the other nodes of t, c t the crcumcenter of t, and x the nterpolaton poston. A smlar observaton appears n [Ju et al., 2005], and we fnd that wth t the velocty nterpolaton s qute effcent. All quanttes appearng n Equaton (4) are already stored for use n other parts of the tmestep, savng the need to compute the terms n Equaton (3). When advectng large numbers of partcles, veloctes at nodes of tetrahedra can be frst be found usng Equaton (4) and then quckly nterpolated n a lnear fason over the tetrahedra to advect the partcles. 3.2 Generalzed Sem-Lagrangan Step The smple and stable sem-lagrangan method has become the standard tool for advecton of the velocty feld for graphcal applcatons [Stam, 1999]. The basc dea of the method y y Current step x Prevous step x x (t) Fgure 2: A two-dmensonal representaton of the generalzed sem-lagrangan advecton step. We trace back from the poston where a velocty s stored n the new mesh, x = (x, y), nterpolate the velocty usng the old mesh and velocty feld, and update the velocty n the new mesh. s that we can fnd a velocty that wll advect to a pont by tracng back from that pont and nterpolatng the old velocty feld. Ths method does not rely on veloctes beng stored at any partcular place, as long as the velocty can be nterpolated throughout space. We can extend ths technque naturally to meshes whch change arbtrarly at each tme step as n [Feldman et al., 2005b]. Ths extenson does not ncur any addtonal smoothng compared to usng sem-lagrangan advecton wth statc meshes. Suppose at tme t veloctes are stored at locatons x (t) (n our case, the face crcumcenters), and we want to fnd the velocty at a partcular face locaton x (t). We trace back from x (t) through the velocty feld of the prevous tme step to a pont x, whch has no necessary correspondence to any feature of the old mesh. Then, we update the velocty at x (t) to the value nterpolated from the old velocty feld at x. Because the veloctes from the prevous step are stored on a dfferent mesh, we have to trace back and nterpolate usng ths prevous mesh (see Fgure 2). 3.3 Remeshng The doman boundares, obstacles, and smoke are free to move and change from step to step of the smulaton. By regeneratng the mesh at each tme step we can ensure that our doman conforms well to boundares and s refned n vsually mportant areas. We accomplsh ths by usng the varatonal tetrahedral meshng algorthm presented n [Allez et al., 2005]. Ths method allows for generaton of tetrahedral meshes that conform well to an arbtrary nput surface mesh, have no restrctons on topology (.e., allow nested vods), and allow for szng of tetrahedra throughout the doman based on arbtrary crtera. Our mplementaton dffers from the orgnal algorthm n a couple of detals. As n the orgnal method, refnement of the mesh s controlled by a szng functon µ(x) that, for any pont x n the smulaton doman, returns the desred local edge length of the tetrahedra. Whle the orgnal algorthm bulds ths szng functon by fndng the mnmum combnaton of local feature sze and dstance to a boundary pont x 822

4 Computer Graphcs Proceedngs, Annual Conference Seres, The flud velocty s dvergence free and the rgd body velocty s rgd. 3. The lnear and angular momentum of the combned system s conserved. Fgure 3: Left: a vsualzaton of the szng feld for a rectangular doman wth an rregular obstacle at the top and a plume of smoke at the bottom. Rght: the resultng smulaton mesh. Obstacle faces are colored green. from x, we nstead formulate t as follows: µ(x) = k 0 + mn (k d d(x), k s (1 s(x)), k ω (1 ω(x))) (5) In ths equaton, k 0 s an offset value that controls the mnmum value of the szng feld, and hence the mnmum local edge length of tetrahedra. d(x) s the dstance to the closest obstacle or boundary whch demands refnement, s(x) s a functon of the densty of smoke partcles, and ω(x) s a functon of the vortcty of the velocty feld. The parameters k d, k s, and k ω respectvely control the weght each of these functons has on the szng feld. These three factors are the same as those used for octree refnement n [Losasso et al., 2004]. The overall goal of the szng feld s to focus computatonal effort n the most vsually mportant parts of the scene, that s, near closed boundares, where the velocty feld vares most, and where smoke s vsble. Fgure 3 shows an example of a szng feld and the resultng mesh. Fgure 4 demonstrates the benefts of refnement near areas of hgh vortcty and smoke densty. Ths meshng method s teratve, so the mesh from the prevous smulaton tme step can be used as an ntal guess for the node placement n the mesh at the next smulaton tme step. Because there s, n general, strong temporal coherence between steps of the smulaton, the szng feld does not change too much and so the nodes from the prevous step are often a good ntalzaton. Before the algorthm proceeds, the ntal node placement s corrected to match the szng feld of the current step. One other modfcaton we made to the algorthm s that, when optmzng the node postons, we move nodes to the average of the barycenters of the surroundng tetrahedra nstead of the crcumcenters. We have found that whle ths tends to slghtly decrease the average qualty of tetrahedra n the mesh, t often leads to substantal mprovements n the qualty of the worst elements of the mesh, whch are of more concern for numercal smulaton. Of course, remeshng takes tme, so t s mportant to consder the mpact t has on overall smulaton performance. The tme spent generatng meshes for each smulaton step vares, but generally accounts for less than a quarter of the overall smulaton tme. In Secton 4 we show tmng nformaton for several examples. In [Carlson et al., 2004] these condtons are enforced sequentally. Whle for many cases ths produces results that look very good, under some stuatons artfacts can be created because enforcng one of the condtons n general wll break a prevously enforced one. Examples of such artfacts mght be flud leakng through sold boundares or poor performance n pston-lke stuatons. Our mplementaton dffers from [Carlson et al., 2004] n a couple of ways, but most sgnfcantly we enforce these condtons smultaneously wthn the mass conservaton step. In general, the mass conservaton step solves for pressures that accelerate the velocty feld to be dvergence free. In prevous works, ncludng those wth two-way couplng, the mass conservaton step treats faces to behave as flud or explctly prescrbes ther veloctes. For flud faces, the pressure accelerates the velocty proportonal to the gradent of the pressure whle for prescrbed faces, the pressure does not effect the flud. For a more complete dscusson of flud/prescrbed-velocty mass conservaton see [Fedkw et al., 2001]. We extend mass conservaton to nclude a dynamc, rgd body. To do so, we solve for acceleraton of the flud and the rgd body, gnorng pressure for both. We then solve for a pressure term that satsfes boundary and ncompressblty constrants to fnd the fnal acceleratons. The rgd body acceleratons can be computed by creatng a matrx R that s multpled by a vector of the pressures that surround a rgd body. R can be formed by a seres of matrx multplcatons: R = b 1.. b k [ M I 1 ] [ A1b T 1 A k b T k ] (6) where b = [ n T (r n ) T], n s the normal of the th face, r s the vector from the rgd objects center of mass to poston of the th face, and A s the area of that face. The rghtmost matrx fnds the net force-torque couple actng on a rgd body by summng up the contrbuton due to pressure forces actng on rgd body mesh faces. The force-torque couple s converted to a lnear and angular acceleraton of 3.4 Two-way Couplng and Mass Conservaton The moton of flud and rgd bodes that mutually effect each other can be complex and vsually appealng. The nteracton occurs as a consequence of the condtons that: 1. The veloctes n the normal drecton are the same at the nterface of the flud and the rgd body surface. Fgure 4: A comparson between unform and selectvely refned smulaton meshes. Left: a frame from a smulaton usng approxmately unformly szed tetrahedra. Rght: the same frame usng approxmately tetrahedra refned near areas of hgh vortcty and smoke densty. The refned mesh preserves the fne detal n the velocty feld and near the vsble smoke, enhancng vortex acton and natural movement. The runtmes of the two are equvalent. 823

5 ACM SIGGRAPH 2006, Boston, MA, July 30 August 3, 2006 Fgure 5: Red partcles are transfered from the left tank to the rght by squeezng and releasng the central bulb. The blue valves are coupled to the flud smulaton and prevent backflow. the body by the mddle (6 6) block matrx. M s a dagonal matrx wth the mass of the rgd body on the dagonals and I s the nerta matrx. The leftmost matrx n the multplcaton returns the acceleraton of the flud-rgd faces n the drecton of the face normal due to the lnear and angular acceleraton of the rgd body. By constructon, acceleratons generated by ths matrx behave rgdly. Computng pressure acceleratons of both the flud and flud-rgd faces can be expressed as a matrx A multpled by a vector of all the pressures. A row of A that corresponds to a face wth flud on both sdes contans the same entres as the standard gradent matrx multpled by 1/ρ. A row of A that belongs to a face at the flud-rgd nterface has element values obtaned from the correspondng row of R. The elements of ths row are placed at columns correspondng to the pressures that surround the rgd-body. Wth A bult, mass conservaton ncludng two way couplng proceeds much n the same way as n the all-flud case, wth A replacng the role of the dscrete gradent matrx. For a gven vector of pressures, p, the ntermedate velocty feld, z, s accelerated to the end-of-step velocty, z, by z = z + tap. For the flud faces, z s found by applyng all terms of Equaton (1) except the pressure term. For the flud-rgd faces, z s found usng a rgd body smulator wthout pressure forces appled. To fnd a partcular pressure that accelerates z such that z s dvergence free we solve the lnear system tdap = Dz. (7) Ths lnear system can be solved effcently usng PCG snce the the matrx DA, whch replaces the dscrete Laplacan from the all flud case, s also a postve-defnte symmetrc matrx. Usng the same machnery, we can also nteract wth constraned rgd bodes. Ths smply requres fndng an R matrx that correctly computes face acceleratons due to pressure. For example, one could easly alter R such that the body was constraned to just rotate about the orgn by replacng b n Equaton (6) wth b = [ (r n ) T] and usng only the I 1 block for the center matrx. Ths dea could be extended further to nclude even artculated bodes. 4 Results and Dscusson We mplemented the method descrbed above n matlab 1 and C, makng use of Pyramd [Jonathan Shewchuck, personal communcaton] for Delaunay trangulaton and pxe 2 for all renderngs. Typcal smulaton tmes for meshes wth 100,000 tetrahedra were about 1 mnute per frame. Table 1 compares remeshng and smulaton tmes for several of the examples presented n ths paper. The mages n Fgure 1 show smoke n a tank mxed by the scrpted moton of a paddle. Refnement of the smulaton mesh near the paddle ensures good conformance to ts curved surfaces that produce nterestng vortex effects n the smoke Remeshng tme Total tme Percent per frame (mean) per frame (mean) remeshng Fgure sec 64.8 sec 20.3% Fgure sec 44.5 sec 18.7% Fgure sec 35.8 sec 16.1% Fgure sec 796 sec 39.3% Table 1: A comparson of remeshng and smulaton tme for selected examples. In Fgure 5, a pump transfers partcles from the left tank to the rght tank as the bulb n the mddle s squeezed and released. The blue valves on ether sde of the bulb prevent backflow. The moton of these valves s not scrpted. Instead, they are modeled as rgd bodes constraned to rotate about an axs and ther moton s caused by two-way nteracton wth the flud. Fgure 6 demonstrates the two-way nteracton of the Stanford bunny wth smoke cannons. On the left s a lghter bunny whch s tossed about by the force of the cannons and also affects the moton of the smoke. On the rght s a heaver bunny that drops quckly to the ground. In Fgure 7, smoke moves through an array of obstacles n a hgher resoluton mesh of over 500,000 tetradra. Although qualty of the mesh elements does not suffer at ths level of refnement, the proporton of tme spent meshng ncreases to 39.3%. The moton of the smoke at the hgher resoluton s more lvely and exhbts more fne-scale detal. A vortcty enhancement method, such as those n [Fedkw et al., 2001] and [Selle et al., 2005] could be used to further enhance the flud moton but we do not fnd such enhancement necessary and so have not mplemented t. We have presented a system for performng flud anmaton usng unstructured tetrahedral domans that can change arbtrarly at each tme step. Although our current mplementaton models completely flud-flled domans, we beleve t would be well-suted for use wth surface trackng technques for lqud smulaton. Acknowledgments We thank the other members of the Berkeley Graphcs Group for ther helpful crtcsm and comments. Ths work was supported n part by Calforna MICRO and , and by generous support from Apple Computer, Pxar Anmaton Studos, Autodesk, Intel Corporaton, Sony Computer Entertanment Amerca, and the Alfred P. Sloan Foundaton. Klngner and Feldman were supported by NSF Graduate Fellowshps. References Allez, P., Cohen-Stener, D., Yvnec, M., and Desbrun, M Varatonal tetrahedral meshng. In the Proceedngs of ACM SIG- GRAPH 2005, Botta, N., and Hempel, D A fnte volume projecton method for the numercal soluton of the ncompressble naver-stokes equatons on trangular grds. Frst Internatonal Symposum on Fnte Volumes for Complex Applcatons, (July),

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