Projective Fluids. Abstract. 2 Related Work. 1 Introduction. Marcel Weiler Graduate School CE, TU Darmstadt

Transcription

1 Marcel Weler Graduate School CE, TU Darmstadt Projectve Fluds Dan Koscher Graduate School CE, TU Darmstadt Jan Bender Computer Anmaton Group, RWTH Aachen Unversty Fgure 1: Left: Breakng dam scenaro wth 1.94 mllon partcles, featurng obstacles wth complex sold boundares. Rght: Flud and cloth are smulated together n the Projectve Dynamcs framework. Abstract We present a new method for partcle based flud smulaton, usng a combnaton of Projectve Dynamcs and Smoothed Partcle Hydrodynamcs (SPH). The Projectve Dynamcs framework allows the fast smulaton of a wde range of constrants. It offers great stablty through ts mplct tme ntegraton scheme and s parallelzable n large parts, so that t can make use of modern mult core CPUs. Yet exstng work only uses Projectve Dynamcs to smulate varous knds of soft bodes and cloth. We are the frst ones to ncorporate flud smulaton nto the Projectve Dynamcs framework. Our proposed flud constrants are derved from SPH and seamlessly ntegrate nto the exstng method. Furthermore, we adapt the solver to handle the constantly changng constrants that appear n flud smulaton. We employ a hghly parallel matrx-free conjugate gradent solver, and thus do not requre expensve matrx factorzatons. Keywords: Projectve Dynamcs, SPH, fluds, mplct ntegraton Concepts: Computng methodologes Physcal smulaton; 1 Introducton Snce Projectve Dynamcs was proposed by Bouazz et al. [2014], t has receved wde nterest from the computer graphcs communty. Its robustness and effcency make t a desrable method for the smulaton of constraned partcle systems. Our goal s to smulate fluds. In computer graphcs, SPH has become predomnant for the partcle based, Lagrangan smulaton of fluds [Ihmsen et al. 2014b]. For our smulatons, we propose a new 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. Copyrghts for components of ths work owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. c 2016 ACM. MG 16, October 10-12, 2016, Burlngame, CA, USA ISBN: /16/10 DOI: flud constrant derved from SPH, that adheres to the constrant structure requred by Projectve Dynamcs. Moreover, we adapt parts of the solver, to mprove the performance of Projectve Dynamcs n scenes where constrants change frequently. In flud smulaton ths s the case. The orgnal Projectve Dynamcs method reles on Cholesky factorzaton to accelerate the soluton of large sparse lnear systems.changes n the constrants are ncorporated nto the factorzaton by sparse Cholesky updates, but ths procedure s only practcal when the number of changes per tme step s low. To gan the flexblty requred by fluds, we employ a matrx-free conjugate gradent (CG) solver for optmzaton over the partcle postons durng the global step. Meanwhle, our method stll adheres to the Projectve Dynamcs framework and we can handle other constrants n a unfed solver as well. We nhert the stablty for stff constrants and large tme step szes, whle our solver s able take advantage of mult core systems snce the conjugate gradent method s easy to parallelze. 2 Related Work The physcally-based smulaton of fluds has been an mportant and actve research topc for several decades. As most flud smulaton technques are based on the Naver-Stokes equaton several methods for a dscretzaton were developed. Especally, Lagrangan dscretzatons based on SPH have become ncreasngly popular for nteractve free-surface flows. We would lke to refer to reader to the work of Brdson [2008] for a general overvew of flud smulaton technques whle we recommend the state-of-the-art report of Ihmsen et al. [2014b] to gan nsght nto recent developments n SPH approaches n the feld of computer graphcs. SPH Flud Smulaton Buldng on the poneerng work on SPH smulaton of Monaghan [1992], Müller et al. [2003] ntroduced the partcle-based concept for flud smulaton to the computer graphcs communty. The method was later extended to a spatally adaptve SPH dscretzaton by Adams et al. [2007]. Due to the fact that a wde range of lquds s ncompressble, e.g. water, several approaches to elmnate or reduce compresson wthn the flud were developed. Becker and Teschner [2007] presented a method that guarantees a maxmum compresson based on precomputed, scenaro-dependent stffness coeffcents. As ths method penal-

2 zes compresson usng forces and as t s based on explct tmentegraton the maxmum stable tme step can become arbtrarly small for large stffness coeffcents. The frst step towards mplct solvers for mantanng ncompressblty was taken by Solenthaler and Pajarola [2009] proposng a predctve-correctve scheme that teratvely computes partcle pressures. Followng the general concept of mantanng ncompressblty usng mplct solves, methods based on holonomc constrants [Bodn et al. 2012], a dscretzaton of the pressure Posson equaton [He et al. 2012; Ihmsen et al. 2014a], poston based dynamcs [Mackln and Müller 2013] or power dagrams [de Goes et al. 2015] were presented n the followng years. Recently, a method mantanng not only constant densty but also a dvergence-free velocty feld was proposed by Bender and Koscher [2015]. Besdes the development of pressure solvers, mathematcal models were developed n order to model complex physcal phenomena whch nclude the smulaton of multphase flows wth hgh densty contrast [Solenthaler and Pajarola 2008], robust sold flud couplng [Aknc et al. 2012], versatle surface tenson [Aknc et al. 2013], robust thn features [He et al. 2014] and hghly vscous lquds [Peer et al. 2015; Takahash et al. 2015; Bender and Koscher 2016]. Poston Based and Projectve Dynamcs The concept of Poston Based Dynamcs (PBD) was frst ntroduced by Müller et al. [2007] for the smulaton of cloth and cloth balloons. Over the years, the constrant based concept of PBD evolved from a smple and robust smulaton method for cloth to a fully fledged framework supportng a varety of constrants developed n the followng years. Poston-based methods range from approaches for har, fur and rods [Bender et al. 2015] over cloth and solds [Bender et al. 2014a; Deul et al. 2014] and even to fluds [Mackln and Müller 2013], just to menton a few. Moreover, two-way couplng between any of the approaches can be easly realzed [Bender et al. 2015]. For a complete survey on poston-based approaches we would lke to refer the reader to the state-of-the-art report of Bender et al. [2014b]. Besdes all the advantages, PBD cannot accurately handle soft constrants as they are usually enforced through early termnaton of the teratve constrant solver pror to convergence. Whle the resultng dynamc behavor s stll vsually appealng, materal propertes, e.g. stffness, are hard to control as they solely rely on the tme step sze and the solver s resdual. In order to solve that problem, Bouazz et al. [2014] presented the Projectve Dynamcs framework for general constrants buldng upon the work for optmzaton ntegrator based smulaton of mass-sprng systems of Lu et al. [2013]. Through reformulaton of the mplct Euler scheme as optmzaton problem and smplfcaton of elastc energes to quadratc dstance functons, they smulate elastc objects usng a local/global alternatng mnmzaton technque. Whle the local step s ntended to satsfy all underlyng constrants ndependently by poston alteraton followng the PBD concept, the global step can be nterpreted as smart averagng of the computed goal postons whch also accounts for nerta terms. Ths results n a stable and robust smulaton of soft constrants for elastc objects. Recently, the convergence of both PBD and Projectve Dynamcs was mproved by Wang et al [2015] usng a Chebyshev sem-teratve approach whle Naran et al. [2016] generalzed Projectve Dynamcs usng an alternatng drecton method of multplers for optmzaton n order to smulate general nonlnear consttutve models. In contrast to prevous work, we present a novel, fully mplct approach based on Projectve Dynamcs for the smulaton of fluds. We formulate an SPH based constrant that seamlessly ntegrates nto the framework and effcently solve the lnear equaton system n the global step usng a matrx-free conjugate gradent solver. We demonstrate the robustness, stablty and effcency of our approach on complex scenaros wth thousands of partcles. 3 Projectve Fluds Solver Our flud smulaton method s based on Projectve Dynamcs and SPH. In ths secton we descrbe how we extend Projectve Dynamcs to handle our flud constrants. 3.1 Projectve Dynamcs Implct Euler Tme Integraton Projectve Dynamcs uses mplct tme ntegraton to evolve a system of partcles over tme, subject to constrants C(x) = 0. Gven a vector x (n) R 3m of m stacked partcle postons at tme step n and ther veloctes as v (n) R 3m, the state of the system at the next tme step s calculated as x (n+1) = x (n) + tv (n+1) v (n+1) = v (n) + tm 1 ( f ext + f nt (x (n+1))), where M s the mass matrx, f ext s the sum of external forces lke gravty, f nt (x) s the sum of nternal (constrant) forces and t denotes the tme step sze. The nternal forces try to mnmze the constrant potentals Z (x), whch are ntroduced n the followng secton, such that f nt(x) = Z(x). Martn et al. [2011] show that Equaton (1) can be transformed nto an optmzaton problem: x (n+1) = argmn x 1 ( M 1 2 x s (n)) t 2 F (1) Z (x), (2) where. F denotes the Frobenus norm and s (n) = x (n) + tv (n) + t 2 M 1 2 fext s a term descrbng the partcle movement wthout nternal forces. The values of x that mnmze Equaton (2) are the partcle postons for the next tme step. Quadratc Constrant Potentals In general, the constrant potentals Z (x) are defned as a materal model Ψ ( ), appled to a stran E (x), such that Z (x) = Ψ (E (x)). Ψ ( ) can be hghly nonlnear, whch makes Newton s method the frst choce to fnd the mnmum of Equaton (2). Unfortunately, ths nvolves fndng the Hessan of a system of nonlnear equatons n every teraton, whch s very expensve. Bouazz et al. [2014] therefore replace the hghly nonlnear potentals wth specally desgned quadratc potentals. They notce that the set of rest states of a constrant, the so-called constrant manfold, s ndependent of ts potental. For each constrant they ntroduce an auxlary varable p. p contans the postons of the partcles n the constrant, whch are projected onto the constrant manfold. The resultng constrant potental s a quadratc dstance measure between the current and the projected postons: Z (x, p) = w 2 Dx Pp 2 F + δc(p). (3) D and P are constant matrces and w 0 s a weght for the constrant. δ C(p) s an ndcator functon that s zero f p les on the constrant manfold and otherwse. The ndcator s used to formalze the requrement that p should le on the constrant manfold. Alternatng teratve solver Wth the new constrant potentals from Equaton (3), the potental that has to be mnmzed becomes 1 ( M 1 2 x s (n)) 2 2 t 2 F + w 2 DSx Pp 2 F + δc (p) (4)

3 wth S beng a selector matrx that selects the partcles belongng to constrant. The optmzaton now has to be performed over the postons x and the auxlary varables p. By mnmzng Equaton (4) for x and p ndependently n turn, the global mnmum s found. Ths approach s a form of block coordnate descent and wll converge for any pseudo-convex potental [Tseng 2001]. In the local step, the partcle postons x are kept fxed, whle the mnmzaton for p s performed by projectng the auxlary varables onto the respectve constrant manfold. Snce each constrant s treated ndependently, ths step can be easly parallelzed. Durng the global step, the p are kept fxed. Equaton (4) then becomes a smple quadratc potental and can be solved by fndng the pont where ts dervatve becomes zero. Ths leads to the system of lnear equatons ( M t + ) w S T 2 D T D S x = M t 2 s(n) + w S T D T P p. (5) The matrx on the left hand sde of the equaton s constant as long as the constrants do not change. In ths case, performng Cholesky factorzaton n a preprocessng step allows for an effcent solve durng the smulaton. 3.2 The Flud Constrants In typcal SPH flud smulatons, a partcle s evolved n tme, subject to the constrants that the nternal pressure at ts locaton should be zero. The relaton between pressure p (x) and flud densty ρ (x) at the partcle s modeled by an equaton of state (EOS). There are dfferent EOS n use, we chose p (x) = ρ(x) ρ 0 1, where ρ 0 s the flud s rest densty. Requrng p (x) = 0 formalzes the constrant that the flud densty should reman constant. Snce t only depends on the partcle postons, ths constrant has exactly the form that s requred by Projectve Dynamcs. We, therefore, defne our flud constrants as C (x) = ρ(x) ρ 0 1. (6) To estmate ρ at each partcle, we use SPH. At pont x the densty ρ can be nterpolated from the n neghborng partcles usng ρ = n m jw j, (7) where m j s the mass of partcle j and W j = W (x x j, h) s a kernel functon wth h beng the smoothng length of the kernel. In practce, kernels are chosen to be sphercal and have compact support. Ths way, only the n neghborng partcles nsde the support radus have to be consdered n the nterpolaton. Furthermore, the kernel needs to be normalzed and ts shape should be close to a Gaussan [Monaghan 1992] to offer best nterpolaton results. Snce only the n partcles nsde the support radus contrbute to the densty, only these partcles are part of a flud constrant n our solver. Thus, from equatons (6) and (7) we get the flud constrant ( C (x) = 1 n ) m jw j 1. (8) ρ 0 To avod the problem of partcle defcences at frees surfaces, we restrct the constrant to postve values. Ths approach s equvalent to clampng negatve pressures to zero, a common soluton n SPH free surface flow, see e.g. [Ihmsen et al. 2014a], [Bender and Koscher 2016]. Note that snce partcle neghborhoods change, our constrants may contan dfferent partcles n each tme step. 3.3 Constrant Projecton Now that we have establshed our flud constrant, we need to fnd a way to project t onto ts constrant manfold. Ths means we want to fnd auxlary varables p = S x + p, for each constrant, such that C (p ) = 0. In the followng paragraphs we drop the subscrpt for better readablty. We solve ths problem by lnearzng the constrant and teratvely approachng ts rest state n a Newton fashon, an dea also known from Poston Based Dynamcs [Müller et al. 2007] and dscussed n depth n [Bender et al. 2014b]. As a frst guess, we set p (0) = S x. In the n-th teraton, the lnearzaton yelds C(p) C(p (n) ) + pc(p (n) ) T p (n). (9) D Alembert s prncple restrcts p (n) to the constrant gradent: p (n) = λ pc(p (n) ), (10) where the scalar λ s a Lagrange multpler. By nsertng Equaton (10) nto Equaton (9) and solvng for λ, we get λ = C(p (n) ) pc(p (n) ) T pc(p (n) ). (11) To calculate the approxmate poston correcton we need the constrant gradent. Takng the dervatve of Equaton (8) wth respect to p reveals pc(p (n) ) = 1 ρ 0 n wth whch the poston correcton becomes p (n) = 1 ρ 0 n m j pw j, m jλ pw j. Because of the lnearzaton performed n Equaton (9), C(p (n) + p (n) ) wll n general not be zero. Therefore, the correct projected poston s found by teratve updates p (n+1) = p (n) + p (n). The teraton contnues untl C(p (n) + p (n) ) < ɛ, wth ɛ beng a small constant. In our experments, we found that on average three to four teratons were suffcent to reach ɛ = Only n rare cases more than ten teratons were requred. Algorthm 1 llustrates the teratve projecton procedure. Note that n Mackln et al. [2013] take a smlar approach, but n ther work the constrants are not ndependently projected onto ther rest states. Instead, after each solver teraton x s updated so that ntermedate partcle postons are known to all constrants. 3.4 Lnear System Solve After all auxlary varables have been projected onto ther correspondng constrant manfold, we have to solve Equaton (5) for x n the global step. We chose the matrces D and P to

4 Algorthm 1 The constrant projecton algorthm. 1: functon PROJECTCONSTRAINT(p) call wth p = S x 2: C calcconstrantpotental (p) 3: whle C > ɛ do 4: C calcconstrantgradent (p) 5: f C s 0 then 6: break already fond a mnmum 7: p p C C C 2 apply poston correcton 8: C calcconstrantpotental (p) 9: return p be the dentty matrces. Experments wth dfferental coordnate matrces, as proposed by Bouazz et al. [2014], have not shown a sgnfcant performance ncrease. The system matrx thus becomes A = M + t 2 wst S, and the rght hand sde becomes b = M s (n) t 2 wst p. Projectve Dynamcs owes much of ts performance to the fact that the system matrx s constant and can be factorzed n advance. Ths allows the global step to be solved very effcently. In our flud smulaton on the other hand, the constrants change n every tme step as partcles flow past each other and ther neghborhoods change. The system matrx has to be updated n each step, and so does the Cholesky factorzaton. A step that has prevously been consdered preprocessng has now become performance-crtcal. When only a small number of constrants change, sparse Cholesky updates can be used to avod a complete refactorzaton, but n our flud smulaton almost all constrants change each tme step. In practce ths s unfeasble, and so we decded to use a matrx-free conjugate gradent (CG) solver nstead. CG s a popular teratve algorthm for solvng large, sparse systems of lnear equatons. For an excellent dervaton of the algorthm we refer the reader to [Shewchuk 1994]. CG s used to solve sparse, lnear systems of the form Ax = b for a square, symmetrc, postve defnte matrx A. The man advantage of CG for our smulaton s, that t only references A through ts multplcaton wth a vector.we make use of ths fact by provdng only the result of the matrx-vector multplcaton and not the matrx A tself. Smlarly, we can calculate the rght hand sde vector wthout buldng any matrx for b. Our matrx-free algorthm s llustrated n Algorthm 2. Note that we parallelze over the constrants and, therefore, need to apply updates to the result vectors atomcally. Snce the number of concurrent threads on a modern CPU s sgnfcantly lower than the number of constrants, threads only rarely block each other. 3.5 Inherted Trats Our solver nherts several mportant trats from Projectve Dynamcs. The frst notable property s that there are no hard constrants n Projectve Dynamcs. Ths leaves our flud slghtly compressble. Snce we use mplct tme ntegraton we can combat ths problem wth hgh stffness parameters w. Even wth large tme step szes, we do not have to worry about stablty. We adapt the tme step sze durng our smulaton accordng to the CFL condton. 4 Implementaton Detals We mplemented our smulaton framework n C++. We employ the Egen 3.2 lbrary for basc lnear algebra computatons and use Intel Threadng Buldng Blocks for parallelzaton. By nature, matrxfree CG conssts only of vector operatons whch can be trvally parallelzed. Parallelzaton of the local step s straghtforward as well, snce the constrants n Projectve Dynamcs are vewed as Algorthm 2 The functons to calculate the product r = Σx and the rght hand sde vector b n a matrx-free fashon. 1: functon MATRIXFREEATIMES(x) 2: r 0 to accumulate system matrx tmes vector 3: for each C do n parallel 4: for j 1 to number of partcles n C do 5: l global ndex of partcle j 6: atomc r l r l + w x l 7: for 1 to number of partcles do n parallel 8: r r + M x t 2 9: return r 10: functon MATRIXFREEB(x) 11: b 0 to accumulate rght hand sde vector 12: for each C do n parallel 13: k number of partcles n C 14: p empty vector of sze 3k 15: for j 1 to k do 16: l global ndex of partcle j 17: p j x l 18: p PROJECTCONSTRAINT(p) 19: for j 1 to k do 20: l global ndex of partcle j 21: atomc b l b l + w p j 22: for 1 to number of partcles do n parallel 23: b b + M s (n) t 2 nerta term 24: return b ndependent. Each constrant s projected onto ts constrant manfold and ther nfluence on the rght hand sde of the system of lnear equatons s updated atomcally. To accelerate the search for neghborng partcles, we use the parallel spatal hashng algorthm proposed by Ihmsen et al. [2011]. Collsons wth statc objects were modeled usng sold boundary partcles. To get a good dstrbuton of partcles on an arbtrary object, Posson-Dsk samples are created on the surface of an nput mesh. These partcles are then used n the densty estmaton of SPH. We use the corrected densty computaton proposed by Aknc et al. [2012] to account for the nherently rregular samplng and to not have to sample the object s volume. For the densty calculaton we use the cubc splne kernel proposed by Monaghan et al. [1992]. It s cheap to compute and works well nsde the flud, but SPH wll nherently underestmate the flud densty at free surfaces. Ths leads to negatve pressure values and causes partcles near the surface to clump together. To crcumvent ths problem, we clamp the pressure to non-negatve values, an approach that s popular n computer graphcs [Ihmsen et al. 2014b]. After each smulaton step artfcal vscosty s appled. We use the XSPH varant of the vscosty term proposed by Schechter et al. [2012]. Ths formulaton was already successfully used by Bender and Koscher [2016]. 5 Results In ths secton we present the results of our smulatons and dscuss ther meanng. The scenaros we chose here each concern the reproducton of desred effects, or propertes of our solver. A common test case n flud smulaton for computer graphcs s the dam break. Fgure 2 shows our verson of the scenaro. The breakng and overtakng waves that can be observed here are a desrable result and typcal for lquds. A closely related scenaro s the double breakng dam. In Fgure 3 we show the smulaton of two flud blocks that start wth a dagonal offset. On collson, the smulated lqud clearly forms the expected splashes and thn sheets.

5 Fgure 2: A breakng dam scenaro. A block of water flows under gravty and shows typcal breakng waves. # flud avg. tme avg. tme Scene partcles per step step sze three dragons s s double breakng dam s s breakng dam s s nlet s s cloth s s Table 1: Tme measurements for dfferent smulatons. Fgure 3: A dagonal double dam break scenaro n a rectangular doman shows typcal splashes and thn sheets. Adaptve t Whle the employed mplct tme ntegraton s uncondtonally stable, the tme step sze s stll lmted by the CFL condton. Flud behavor can only be correctly recreated when partcles travel less than ther radus n one tme step. Otherwse partcle collsons may be mssed. Ths means that the tme step sze has to be lowered when the flud flows faster. When the flud s movng slowly on the other hand, we can be generous wth the tme step sze used n our smulaton. In all vdeos we allow a maxmum tme step sze of t max = 0.01s and adapt t dependng on the maxmum partcle velocty n the scene. The average tme step szes for our dfferent scenaros are shown n Table 1. 6 Concluson and Future Work Fgure 4: Water spurts from an nlet nto a box. Even at hgher veloctes and constrant stffness values our solver remans stable. The scenaro depcted n Fgure 4 s a test for the stablty of our method. Snce Projectve Dynamcs only smulates soft constrants, we use hgh stffness values n the range of w = 10 6 to counter compresson of the flud. Nevertheless we are able to stably smulate ths scenaro wth an average tme step sze of more than 6ms (see Table 1). Stll the maxmum densty never rses more than 0.01% over the rest densty durng the whole smulaton. Snce we are usng the boundary handlng method proposed by Aknc et al. [2012], we are able to handle collsons wth arbtrary sold objects. An example of the nteracton between flud and complex boundares s shown n Fgure 1, left. The rght sde shows an expermental unfed smulaton of our fluds wth other Projectve Dynamcs constrants. Furthermore, the artfcal vscosty term shows that our framework can easly be combned wth other SPH methods. Tmngs Table 1 llustrates the performance of our solver on the dfferent scenaros descrbed above. All measurements were run on a computer wth two Intel Xeon E processors wth 12 cores each, clocked at 2.7GHz, and 64GB of RAM. Our solver performed a maxmum of 30 local/global teratons, whle constrant projecton and conjugate gradent solver ran untl convergence. In ths paper we have presented a new method for Lagrangan flud smulaton, that uses Projectve Dynamcs to resolve our SPH based pressure constrants. In contrast to other mplct SPH solvers we mantan the ablty to smulate compressble fluds. Stll we can keep the densty devaton below 0.01% by usng hgh stffness values, whle our mplct tme ntegraton allows us to use large t. Snce n ts core our solver stll operates n the Projectve Dynamcs framework, flud and other constrants can be solved n a unfed manner. We employ a matrx-free CG solver to effcently handle constrants that change n every tme step. Due to the hghly parallel nature of CG and Projectve Dynamcs, our algorthm can make use of modern mult core CPUs. Matrx free CG also shows promsng performance on the GPU [Weber et al. 2013]. Consequently, we are consderng an mplementaton on graphcs hardware for further speedup. In future, we are plannng to scrutnze the performance of our method n complex scenaros wth many dfferent knds of constrants. Frst experments wth the smulaton of nteractons between flud and soft bodes have shown the potental of ths approach. The reproducton of addtonal flud propertes, lke surface tenson, s somethng we want to nvestgate as well. Acknowledgments The work of the authors s supported by the Excellence Intatve of the German Federal and State Governments and the Graduate School of Computatonal Engneerng at TU Darmstadt. The dragon model s courtesy of the Stanford Computer Graphcs Lab.

6 References ADAMS, B., PAULY, M., KEISER, R., AND GUIBAS, L. J Adaptvely Sampled Partcle Fluds. ACM Trans. on Graphcs 26, 3, 48. AKINCI, N., IHMSEN, M., AKINCI, G., SOLENTHALER, B., AND TESCHNER, M Versatle Rgd-Flud Couplng for Incompressble SPH. ACM Trans. on Graphcs 31, 4, 62:1 62:8. AKINCI, N., AKINCI, G., AND TESCHNER, M Versatle Surface Tenson and Adheson for SPH Fluds. ACM Trans. on Graphcs 32, 6, 1 8. BECKER, M., AND TESCHNER, M Weakly Compressble SPH for Free Surface Flows. In ACM SIGGRAPH / Eurographcs Symposum on Computer Anmaton, 1 8. BENDER, J., AND KOSCHIER, D Dvergence-Free Smoothed Partcle Hydrodynamcs. In ACM SIGGRAPH / Eurographcs Symposum on Computer Anmaton, 1 9. BENDER, J., AND KOSCHIER, D Dvergence-Free SPH for Incompressble and Vscous Fluds. IEEE Trans. on Vsualzaton and Computer Graphcs. BENDER, J., KOSCHIER, D., CHARRIER, P., AND WEBER, D Poston-Based Smulaton of Contnuous Materals. Computers & Graphcs 44, 1, BENDER, J., MÜLLER, M., AND MACKLIN, M A Survey on Poston-Based Smulaton Methods n Computer Graphcs. Computer Graphcs Forum 33, 6, BENDER, J., MÜLLER, M., AND MACKLIN, M Postonbased smulaton methods n computer graphcs. In Eurographcs 2015 Tutorals, Eurographcs Assocaton. BODIN, K., LACOURSIÈRE, C., AND SERVIN, M Constrant fluds. IEEE Trans. on Vsualzaton and Computer Graphcs 18, BOUAZIZ, S., MARTIN, S., LIU, T., KAVAN, L., AND PAULY, M Projectve Dynamcs: Fusng Constrant Projectons for Fast Smulaton. ACM Trans. on Graphcs 33, 4, BRIDSON, R Flud Smulaton for Computer Graphcs. A K Peters / CRC Press. DE GOES, F., WALLEZ, C., HUANG, J., PAVLOV, D., AND DES- BRUN, M Power Partcles: An ncompressble flud solver based on power dagrams. ACM Trans. on Graphcs 34, 4, 50:1 50:11. DEUL, C., CHARRIER, P., AND BENDER, J Poston-based rgd body dynamcs. Computer Anmaton and Vrtual Worlds 27, 2, HE, X., LIU, N., LI, S., WANG, H., AND WANG, G Local Posson SPH for Vscous Incompressble Fluds. Computer Graphcs Forum 31, HE, X., WANG, H., ZHANG, F., WANG, H., WANG, G., AND ZHOU, K Robust Smulaton of Sparsely Sampled Thn Features n SPH-Based Free Surface Flows. ACM Trans. on Graphcs 34, 1, 7:1 7:9. IHMSEN, M., AKINCI, N., BECKER, M., AND TESCHNER, M A Parallel SPH Implementaton on Mult-Core CPUs. Computer Graphcs Forum 30, 1, IHMSEN, M., CORNELIS, J., SOLENTHALER, B., HORVATH, C., AND TESCHNER, M Implct Incompressble SPH. IEEE Trans. on Vsualzaton and Computer Graphcs 20, 3, IHMSEN, M., ORTHMANN, J., SOLENTHALER, B., KOLB, A., AND TESCHNER, M SPH Fluds n Computer Graphcs. Eurographcs (State of the Art Reports), LIU, T., BARGTEIL, A. W., BRIEN, J. F. O., AND KAVAN, L Fast Smulaton of Mass-Sprng Systems. ACM Trans. on Graphcs 32, 6, 214:1 214:7. MACKLIN, M., AND MÜLLER, M Poston Based Fluds. ACM Trans. on Graphcs 32, 4, 1 5. MARTIN, S., THOMASZEWSKI, B., GRINSPUN, E., AND GROSS, M Example-based Elastc Materals. ACM Transactons on Graphcs 30, 4, 72:1 72:8. MONAGHAN, J Smoothed Partcle Hydrodynamcs. Annual revew of astronomy and astrophyscs 30, 1, MÜLLER, M., CHARYPAR, D., AND GROSS, M Partcle- Based Flud Smulaton for Interactve Applcatons. In ACM SIGGRAPH / Eurographcs Symposum on Computer Anmaton, MÜLLER, M., HEIDELBERGER, B., HENNIX, M., AND RAT- CLIFF, J Poston Based Dynamcs. Vsual Communcaton and Image Representaton 18, 2, NARAIN, R., OVERBY, M., AND BROWN, G. E ADMM Projectve Dynamcs: Fast Smulaton of General Consttutve Models. In ACM SIGGRAPH / Eurographcs Symposum on Computer Anmaton, 1 8. PEER, A., IHMSEN, M., CORNELIS, J., AND TESCHNER, M An Implct Vscosty Formulaton for SPH Fluds. ACM Trans. on Graphcs 34, 4, SCHECHTER, H., AND BRIDSON, R Ghost SPH for Anmatng Water. ACM Trans. on Graphcs 31, 4, 61:1 61:8. SHEWCHUK, J An Introducton to the Conjugate Gradent Method Wthout the Agonzng Pan. Tech. rep. SOLENTHALER, B., AND PAJAROLA, R Densty Contrast SPH Interfaces. In ACM SIGGRAPH / Eurographcs Symposum on Computer Anmaton, SOLENTHALER, B., AND PAJAROLA, R Predctvecorrectve Incompressble SPH. ACM Trans. on Graphcs 28, 3, 40:1 40:6. TAKAHASHI, T., DOBASHI, Y., FUJISHIRO, I., NISHITA, T., AND LIN, M Implct Formulaton for SPH-based Vscous Fluds. Computer Graphcs Forum 34, 2, TSENG, P Convergence of a Block Coordnate Descent Method for Nondfferentable Mnmzaton. Journal of Optmzaton Theory and Applcatons 109, 3, WANG, H A Chebyshev Sem-teratve Approach for Acceleratng Projectve and Poston-based Dynamcs. ACM Trans. on Graphcs 34, 6, 246:1 246:9. WEBER, D., BENDER, J., SCHNOES, M., STORK, A., AND FELL- NER, D Effcent GPU Data Structures and methods to Solve Sparse Lnear Systems n Dynamcs Applcatons. Computer Graphcs Forum 32, 1,