Efficient Neighbor Search for Particle-based Fluids

Transcription

1 Journal of the Appled Mathematcs, Statstcs and Informatcs (JAMSI), 2 (27), No. 3 Effcent Neghbor Search for Partcle-based Fluds JURAJ ONDERIK Comenus Unversty Slovaka, Bratslava and ROMAN ĎURIKOVIČ The Unversty of Sant Cyrl a Metod, Trnava, Slovaka Abstract Lagrangan partcle-based anmaton s a popular strategy for smulatng complex phenomena as fluds. Due to ts nherent mesh-less nature the set of neghbor partcles wthn a specfed range must be effcently found. In ths paper we propose Cell Indexng a novel approach for searchng approxmate neghbor partcles necessary for effcent flud smulaton usng SPH. Instead of storng partcles nto a fxed 3D grd or hash map, we encode ther coordnates and ndex nto a to key. The lst of keys s then sorted usng lnear tme radx sort. A smple traversal usng H-mask (see subsecton (4.2)) can quckly accumulate approxmate neghbors wthout problematc cache msses of Spatal Hashng, large memory requrements of full 3D grds or O(n log n) tme complexty of kd-trees. Furthermore we can acheve sub-cell precson 1 by usng larger H-masks, whle havng only constant factor slowdown. Usng H-mask can substantally ncrease the precson of Spatal Hashng or 3D grds, however more cache msses or larger memory requrements arse. We have demonstrated our approach wthn a standard SPH based flud smulaton. Mathematcs Subject Classfcaton 2: I.3.5, I.3.7 Addtonal Key Words and Phrases: Neghbor Search, Cell Indexng, H-Mask, Mult-Phase Smoothed Partcle Hydrodynamcs 1. INTRODUCTION Physcally based anmaton of complex natural phenomena s a very actve topc among the whole computer graphcs research. Smulaton methods for rgd and deformable solds are been coupled wth varous flud smulaton methods. For both Euleran [Guendelman et al. 25; Carlson et al. 22; Losasso et al. 26] and Lagrangan approaches [Keser et al. 25; Solenthaler et al. 27] unfed soldflud smulaton technques has been proposed. Stll a number of ssues related to stablty, accuracy, realstc boundary condtons, performance etc. arse n any flud smulaton approach. Recent graphcs hardware allows huge parallelzaton of many tme consumng problems, thus smulaton algorthms has to be adapted. Generally flud smulaton technques solvng full 3D Naver-Stokes equatons can emal: juraj.onderk@fmph.unba.sk emal: roman.durkovc@fmph.unba.sk Comenus Unversty Slovaka, Bratslava

2 be categorzed as Euleran and Lagrangan. In Euleran approaches governng equatons are evaluated on a fxed mesh (usually a 3D grd), whereas Lagrangan methods can be both mesh-based or mesh-less. In both Lagrangan strateges mesh or partcles are not fxed the to doman, but are advected wth the flud. Ths can smplfy governng equatons (see secton (3)) and allows vrtually unlmted smulaton space. Besde these advantages, t s usually complex to extract smooth boundary representaton of nterfaces, correctly handle nterface tenson n small features as bubbles and droplets, perform complex remeshng (mesh-based technques) or alternatvely effcently fnd neghbor (closest) partcle pars (mesh-less technques). Besde full Naver-Stokes smulatons, several shallow water technques exsts, where flud nterface s smulated usng wave equatons. However, n combnaton wth Naver-Stokes equaton they can acheve qute realstc results.[thürey et al. 26; Irvng et al. 26] In ths paper we focus on Lagrangan partcle-base smulaton of vscous fluds usng Smoothed Partcle Hydrodynamcs (SPH). (see secton (3)) It s a mesh-less approach, where all physcal quanttes are sampled on partcle locatons. The nfluence of each partcle s only local 2 thus t s essental to fnd the set of neghbor partcles. Snce ths hghly affects further calculatons of flud dynamcs t becomes usually the bottleneck of the overall anmaton. In secton (4) we have extended the well known Spatal Hashng for varyng partcles support length (see subsecton (4.1)) and developed a novel neghbor search algorthm fxng several ssues of prevous methods. 2. RELATED WORK It s beyond the scope of ths artcle to gve an extensve overvew of the flud smulaton problematcs, therefore we focus here only to related Euleran and meshless Lagrangan works. 2.1 Euleran Grd-based Methods Snce ntroducng (by Foster and Metaxas) the smulaton of full 3D Naver-Stokes equatons to graphcs communty, an actve research started to mprove ther famous Euleran-based MAC-grd method[foster and Metaxas 1996]. 3 Stam proposed the popular extenson of the basc MAC by usng a sem-lagrangan ntegraton scheme and teratve solver of the pressure equaton.[stam 1999] Fedkw et al. developed a Partcle Level Set method for accurate nterface trackng.[enrght et al. 22; Enrght et al. 25] Advanced smulaton of meltng and flowng of hghly vscous, non-newtonan fluds (e.g. wax) were presented n [Carlson et al. 22]. Later Carlson et al. used dstrbuted Lagrangan multplers [Carlson et al. 24] for anmatng the nterplay between rgd bodes and vscous ncompressble fluds. Drect two way couplng between mesh-based thn solds and Euleran-based fluds was done n [Guendelman et al. 25; Losasso et al. 26]. Hong and Km [Hong and Km 23] use a Volume Of Flud (VOF) ndcator functon to smulate a two-phase flud flow and bubbles wth surface tenson forces. 2 Wthn a support dstance from partcles locaton 3 Nevertheless a full 3D flud smulaton n CFD engneerng has been already well establshed

3 Losasso et al. extended the partcle Level Set Method nterface trackng of multple nteractng fluds [Losasso et al. 26] and presented a cuttng-edge flud smulator. A generalzed soluton to modelng hydraulc eroson s presented by Beneš et al. n [Benes et al. 26]. Recently Chentanez [Chentanez 27] developed a method for anmatng ncompressble lquds wth detaled free surfaces usng Lattce-Based Tetrahedral Mesh. Octrees has been used to speed-up grd-based flud smulatons [Losasso et al. 24]. Further combng 2D heght-felds wth a full 3D Naver-Stokes smulaton near the nterface allowed effcent anmaton of large bodes of water.[irvng et al. 26]. Recently controllng flud moton whle preservng detals was successfully appled to both grd-based Lattce-Boltzmann Method (LBM) and Lagrangan Smoothed Partcle Hydrodynamcs (SPH).[Thürey et al. 26] 2.2 Lagrangan Partcle-based Methods Partcle-based smulatons were ntroduce to graphcs communty by Reeves. These anmatons of smple fountans and sprays were represented by uncoupled partcles, whch was nsuffcent for realstc flud smulaton. Later Desbrun et al. [Desbrun and Can 1996] started usng Smoothed Partcle Hydrodynamcs 4 (SPH) n computer graphcs to couple partcles and smulate anmatng hghly deformable bodes ncludng vscous fluds. Müller et al. further popularze SPH technque by anmatng a pourng water nto a glass at nteractve rates [Müller et al. 23]. By averagng vscosty, they extended the model to mult-phase fluds. [Müller et al. 25]. Snce the pressure force s computed locally usng explct ntegraton, ther model s sutable only for compressble and near ncompressble fluds. For stff fluds ncompressblty s acheved usng very small tme steps. 5 Premože et al. ganed ncompressblty by solvng the pressure globally (usng Posson equaton) wth an teratve lnear equaton solver. [Premože et al. 23] Ths technque s referred as Movng Partcle Sem-mplct (MPS). Clavet et al. descrbe a SPH based nteractve technque for smulatng vscoelastc and plastc fluds by connectng partcles wth temporary sprngs [Clavet et al. 25]. Unfed partcle-based SPH model for couplng solds and fluds has been presented n [Keser et al. 25] and later n [Solenthaler et al. 27]. Recently a successful approach of couplng SPH and Partcle Level Set has been done by Losasso.[Losasso 27]. Promsng results are shown by Harada when movng the entre SPH calculaton to GPU and precalculate the nfluence of sold boundary partcles. 6. Ther smulatng of 6k partcles run at 17 frames.[harada et al. 27a; 27b]. Further smplfcatons of SPH has been proposed by Adams et. al [Adams et al. 27] by adaptvely samplng partcles and allowng them to have varous szes. For sparse partcles Kpfer et al. developed novel technque for computng neghbor partcles [Kpfer and Westermann 26]. 4 orgnally developed by Lucy and Monaghan 5 Manly due to stablty ssues 6 Commonly sold-flud boundary condtons are solved by fxng partcles to the sold volume near the surface. Ths ncrease the number of partcles and lower down performance

4 3. MULTI-PHASE SMOOTHED PARTICLE HYDRODYNAMICS Inspred by Monte-Carlo ntegraton Smoothed Partcle Hydrodynamcs (SPH) was ntally developed by Lucy and Monaghan for smulatng flow of nterstellar gas wthn Astrophyscs[Monaghan 25]. Unlke tradtonal mesh-based methods (FEM, FDM) where physcal quanttes are evaluated on a fxed Mesh (grd), SPH belongs to mesh-less methods, where calculated values are approxmated from neghborng ponts (partcles). Partcles have no fxed topology, thus are sutable for complex dynamc phenomena as fluds. Due to the partcle-based Lagrangan nature of SPH mass conservaton s trvally satsfed and convecton of the substance s nherent. 3.1 Smoothng wth Partcles Gven a set of partcles (nterpolaton ponts) r carryng values A = A(r ) of feld quantty A we can approxmate A(r) at arbtrary pont r by a convoluton wth a radal symmetrc smoothng kernel W (r, h) as [Monaghan 25; Müller et al. 25] A(r) A W = A(r )W ( r r, h)dr V j A j W (r r j, h) (1) r j and further replaced by summaton over neghbor partcles havng fnte volume V j = m j /ρ j. Mass of partcles s a constant property, thus volume s defned as the raton between mass m j and densty ρ j. The convoluton smoothng kernel must have the followng propertes W (r, h)dr = 1 lm W (r, h) = δ(r) (2) h r where h s the kernels support length,.e. the dstance to whch partcle affects other partcles. Notce n lmt case h kernel W (r, h) must be the Drac delta functon δ(r). Makng the kernel second order dfferentable we can further express gradent A(r) and laplacan 2 A(r) of feld functon A(r) at arbtrary pont as A(r) = j V ja j W (r r j, h) A(r) = j V ja j W (r r j, h) 2 A(r) = j V ja j 2 W (r r j, h) Ths propertes greatly smplfy further calculatons of Lagrangan-based flud dynamcs. 3.2 Evaluatng Flud Propertes usng SPH The well known Naver-Stokes governng equatons of a smple Newtonan sothermal and ncompressble flud can by expressed n Euleran form by two conservaton laws,.e. the conservaton of mass (contnuty equaton) and the conservaton of momentum (momentum equaton) ρ t = ρ v (contnuty eq.) ( ) v ρ t + v v = p + µ 2 v + ρg (momentum eq.) (3) (4)

5 Lagrangan (partcle based) formulaton can be obtaned usng the materal dervatve Dq Dt = q t + v q.7 as Dρ Dt = ρ v ρ Dv Dt = P + µ 2 v + ρg (= F press + F vsco + F ext ) Assumng constant mass of all partcles total mass of flud s always conserved. we can thus completely omt contnuty equaton from further calculatons. Vewng the momentum equaton as Newton s second law (f = ma),.e. we only need to calculate forces actng on partcles. Notce the rght hand sde of the momentum equaton can be thus expressed as a sum of pressure F press, vscosty F vsco and external F ext force felds. Ths external force feld further contans all other force felds actng on flud (e.g. nterface tenson F nt or gravty F grav ) Assumng a fnte volume of partcles, we get the relaton between total force f actng on -th partcle and respectve force feld. Smply by ntegratng the force feld over the partcles volume, settng F = F(r ) and usng SPH approxmaton (equaton (3)) we get f = F(r)dr V F W (r r j, h) = V F W (r r, h) = F = V F (6) r r Flud Propertes. Assumng ρ j = m j /V j densty can be approxmated usng SPH as well. Further pressure P occurred at -th partcle s descrbed by Tat s equaton [Becker and Teschner 27] 8 ρ(r ) = (( ) γ ) ρ m j W (r r j, h) P = k gas 1 (7) ρ j Usng blndly the SPH approxmaton for pressure force f press and multphase vscosty force f vsco volates the acton-reacton prncple, snce the forces are not symmetrc (f f j ). Varous symmetrzaton approaches exsts among the SPH lterature (see [Rtche and Thomas 21; Monaghan 25]), however we adapt here to smplfed formulaton by Müller [Müller et al. 25], 9 where the pressure and vscosty s only averaged. 1 Notce partcle volume V = m /ρ s usng approxmate densty ρ computed n equaton (7). r (5) f press f vsco C nt = V F press = j P + P j V V j W press (r r j, h) 2 = V F vsco = µ + µ j V V j (v j v ) 2 W vsco (r r j, h) (b) 2 j = V j cj nt 2 W poly (r r j, h) (c) j (a) (8) 7 Snce partcles move wth the flud the convectve term v q s not present 8 An smpler alternatve s the deal gas equaton P = k(ρ ρ ) 9 See [Desbrun and Can 1996] for alternatve formulatons 1 Ths gves more stable smulaton, wth the cost of small energy dsspaton

6 C nt Interface tenson force f nt acts along the normalzed gradent n = of the C nt nterface color feld C nt (r) and s proportonal to ts curvature κ = 2 C nt. Interface color feld values C nt at partcle locatons are approxmated from partcles nterface color values c nt usng SPH (see equaton (8) (c)). Interface tenson force s thus defned as f nt = σ nt κ n = σ nt 2 C nt C nt C nt Smoothng Kernels. In our computatons we use the followng smoothng kernels adapted from [Müller et al. 23] and ts dervatves (see appendx (6)). W poly (r, h) = 315 { (h 2 r 2 ) 3 r h r = r 64πh 9 otherwse W press (r, h) = W vsco (r, h) = 15 πh πh Smulaton Overvew { (h r) 3 { r 3 2h 3 + r2 h 2 + h 2r 1 r h r = r otherwse r h r = r otherwse Our smulaton loop contans three man steps, namely the Neghbor Search, Force Computatons and the Tme Integraton. We use explct tme ntegraton, thus the overall smulaton can be categorzed as explct and force-based. (see algorthm (1)). As shown n the algorthm searchng for neghbor partcles (.e. buldng the lst of close partcle pars) s done n SearchNeghbors(h, H), where h s support length and H s the subdvson factor (see secton (4)). Next the force computaton s done by teratng over all partcles (lnes 2-15). Frst densty ρ s accumulated from neghbors (lnes 4-6), then pressure P s calculated (lne 7) and fnally total force actng on partcle s accumulated (lnes 8-14). Tme ntegraton s performed usng standard second order Leap-frog scheme (lnes 16-19). Ths looks smlar to the smple frst-order explct Euler scheme, however when the velocty s ntalzed 11 correctly as v = v (t 1 2 t) the code perfectly mmc the followng Leap-frog update rules (see equaton (11)) (9) (1) v (t t) v (t 1 2 t) + tf (t) m r (t + t) r (t t) + tv (t t) (11) Due to the explct nature of our method, only small tme steps are allowed to mantan stablty and accuracy. Formally, -th partcle can be ntegrated wth a maxmal tme step t accordng to the Courant condton of convergence 12 t t c = α h ( t = mn t c, α h ) (12) c v 11 The ntalzaton routne trval and s omtted here 12 Combatng numercal errors tme step should be even smaller.

7 In: support length h, subdvson factor H and delta tme t functon SPH(h, H, t) 1: Neghbours SearchNeghbors(h, H) 2: foreach P n Partcles do 3: ρ ; C ; 2 C ; f f ext /* ntalze */ 4: foreach P j n Neghbours(P ) do /* accumulate densty */ 5: ρ ρ + m j W poly (r r j, h) 6: end (( ) γ ) P k gas ρ ρ 1 /* calculate pressure */ 7: 8: 9: 1: 11: 12: 13: 14: 15: 16: 17: 18: 19: end foreach P j n Neghbours(P ) do /* accumulate forces */ P f f V V +P j j 2 W press (r r j, h) /* (= f press ) */ µ f f + V V +µ j j 2 (v j v ) 2 W vsco (r r j, h) /* (= f vsco ) */ C C + V j cj nt W poly (r r j, h) /* (= C nt ) */ 2 C 2 C + V j cj nt 2 W poly (r r j, h) /* (= 2 C nt ) */ end C nt C nt f f σ nt 2 C nt /* (= f nt ) */ end foreach P n Partcles do /* Leap-Frog */ v v + t f m r r + tv end Algorthm 1: Our flud smulaton step usng SPH where c 13 s speed of sound n the flud and α.3 s the Courant number. Stablty (for fast movng partcles) can be further ncreased by choosng t as n equaton (12). Choosng the global tme step as t = mn { t } s safe, but obvously neffcent. Snce usually only a few partcles wll need ths mnmal tme step, we can rather ntegrate each partcle usng ts own tme step t. Smlarly as Desbrun [Desbrun and Can 1996] to synchronze ntegraton of partcles we choose a user defned smulaton frame tme t max and fnd smallest postve number n satsfyng t = t max /2 n t and set partcles tme step to t. When the tme step changes, we must further correct the poston of the partcle as r r + ( t,new )2 ( t,old )2 8m f (13) 13 Materal property,.e. user defned constant

8 Boundary Condtons. In SPH context sold-flud boundary condtons are usually modeled by smply attachng ghost partcles 14 near the boundary nsde the sold object. Snce ts complex deformatons ths s a natural choce for deformable and meltng solds. However for pure rgd objects one can precompute the nfluence of ghost partcles (wth respect to bodes frame) to a dstance feld and further speed up smulaton omttng from them further calculatons. [Harada et al. 27a]. Interface Extracton. Realstc lookng vsualzatons can be acheved by extractng a smooth nterface representaton. Besde usng Marchng Cubes to extract the zero-level of the mplct densty functon ρ(r) = several attempts has been done. We refer nterested reader to [Adams et al. 27] for further detals. 4. NEIGHBOR SEARCH Smlarly to n-body problem, searchng for close partcle pars s a crucal problem n almost any partcle-based flud smulaton. Snce ths usually becomes the bottleneck of the smulaton tme, effcent algorthms are necessary. In the SPH context each partcle affects only a set of neghbor partcles whch le wthn ts support dstance h. Formally we defne for each partcle p ts set of neghbor partcle ndces N (h) as N (h) = { j r r j h} (14) Besde the nave and neffcent O(n 2 ) all-par-test, several algorthms usually based on spatal subdvson and fast approxmate neghbor search has been proposed [Teschner et al. 23; Kpfer and Westermann 26; Keser 26; Cohen et al. 1995]. Here we extend the usual Spatal Hashng (subsecton (4.1)) technque for faster SPH smulaton and propose a novel approach Cell Indexng (subsecton (4.2)) based on ndexng non-empty cells n a vrtual subdvson grd. 4.1 Extended Spatal Hashng A common approach to optmze the all-par-test s to subdvde the smallest enclosng axs algned boundng box (AABB) nto a 3D grd of cells wth sze h. For each partcle wth poston r = (x, y, z) we use ts relatve coordnates to AABBs mnmal corner c mn = (x mn, y mn, z mn ) and calculate ts (postve) cell coordnates cell(x, y, z, h) = (, j, k) (see (15)). Into each cell (, j, k) we could store ndces of partcles wth the same cell coordnates. Dependng on the AABB and cell sze rato, such voxelzaton obvously lead to huge memory consumpton. 15 Teschner et. al [Teschner et al. 23] overcome ths problem by ntroducng spatal hash functon hash(, j, k) whch naturally maps partcles cell coordnates to a bucket n a hash map. Ths enables grd sze to be vrtually unlmted and does not store empty cells. The hash functon s defned as ( ) x x mn y y mn z z mn cell(x, y, z, h) =,, = (, j, k) h h h (15) hash(, j, k) = ( p 1 xor j p 2 xor k p 3 ) mod M 14 For statc objects ther postons are not ntegrated 15 storng empty cells s not optmal

9 where p 1 = , p 2 = and p 3 = are large prme numbers and M s the sze of the hash map. Usng cell and hash functons nserton of a partcle s O(1) thus buldng the data structure for approxmate neghbor search s lnear. To fnd neghbors N (h) for -th partcle we need examne only partcles from all (26 n 3D) surroundng cells. Assumng the partcle dstrbuton s approxmately even, each cell contans only a constant number of partcles and thus overall neghbor search s near lnear. However n the SPH framework partcles can cluster 16. Ths leads usually to larger bucket szes 17 or slower hashng. To reduce ths problem we can naturally hash partcles nto smaller cells but then we need to examne partcles from more neghborng cells. Formally let h = h/(2h + 1) be the subdvded cell sze and H the subdvson factor. When calculatng approxmate neghbors of partcle n cell (, j, k) we need to examne partcles from all ( + s, j + t, k + u) cells, where (s, t, u) M H (h). The set H-mask M H (h) contans relatve coordnates of all neghbor cells whch are not completely outsde the support of processed cell. (see fgure (1) (d)) M H (h) = {(s, t, u) (h s, h t, h u) < h s, t, u { H,, +H}} (16) Fnally notce, that larger masks 18 completely contans smaller masks,.e. G < H M G (h) M H (h). Ths allows smulatons where partcles have dfferent support length. Indeed, assumng cell sze s h and partcles support s h we select H-mask for ths partcle accordng to H = h/h Hashng naturally ntroduce cache msses, thus examnng all M H (h) can even slow down the process 2. Wth larger H sze of hash map (M) must ncrease to avod numerous collsons. It can be estmated expermentally. 4.2 Cell Indexng Inspred by the staggered grds [Kpfer and Westermann 26] we have developed a novel approach for searchng approxmate neghbor partcles n a lnear tme, tryng to avod several dsadvantages of the spatal hashng. Smlarly to other grd methods we frst calculate the AABB of all partcles and use further only ther relatve coordnates to AABBs mnmal corner c mn. For each partcle we defne t s key(n,, j, k) as key(n,, j, k) = n + 2 I + 2 I+J j + 2 I+J+K k where 2 I > N 2 J > B x /h 2 K > B y /h 2 L > B z /h where n s ndex of partcle, N s the number of partcles, B x, B y and B z are dmensons of boundng box, (, j, k) = cell(x, y, z) are cell coordnates and I, J, K and are arbtrary constants 21 dependng on the sze of the AABB and cell sze h. 16 smulatng compressble (near ncompressble) fluds wth strong pressures 17 We assume all buckets have fxed sze as speed-up 18 wth larger subdvson factor H 19 Naturally ths raton must be reasonable small 2 Storng all ndces of H-neghbor cells leads to more memory usage and slowdown 21 usually 16 or 32 (17)

10 Notce, that functon key(n,, j, k) encode ts parameters to a unque key. Thus gven a key q we can compute unque (n,, j, k). 22 1N partcle key key order N+2 N+1 N+4 partcle ndex N-1 a b Same (,j) partcle ndex 2N 2N+2 3N 4N dfferent (,j) key order j key(m, m ) key(n, n ) m n key(m, m,j m ) key(n, n,j n ) j m j n non-empty cells partcle ndex c d N-1 ( +1,j -1) 2-mask partcle ndex -1 N+4 current mask ( -2,j -1) +1 Support regon j Fg. 1. Cell Indexng prncples. In (a) 1D case s shown. The left most column represents 1D space dvded nto cells. Dashed columns symbolze ndces of partcles. Each row thus contan all partcles wth equal -coordnate. Dotted lnes show the orderng of sorted keys. In (b) 2D extenson s llustrated. Notce the thrd dmenson depcts partcle ndex, thus has no geometrcal meanng. In (c) 1D searchng prncple s shown. The search follows key orderng, thus (n 1D) the -coordnate as well. Smple 1-mask (cells wth thck outlne) s show and respectve non-empty cells 1,, +1 (lght-gray cells). In (d) H-mask s extended to 2D showng ts non-empty cells and ther correspondng locaton n the mask. Snce the presented 2-mask s (n 2D) equal to the full 5x5 mask, the benefts of H-mask s not obvous from the pcture, however for larger H (or n 3D) one can clearly see, that some corner cells of the full (2H+1)x(2H+1) square mask wll be omtted n respectve sphercal H-mask. 1D Case. Suppose a 1D case show n fgure (1) (a), where the keys reduce to key(n, ) = n + 2 I. Gven two arbtrary partcles m and n wth keys key(m, m ) key(n, n ) ther cell coordnates also satsfy m n and vce-versa. After sortng 22 Partcle ndex s n = q mod 2 I, cell coordnates are = (q/2 I ) mod 2 J,

11 all keys, we can thus effcently access partcles wth ncreasng cell coordnates. Gven a key q ndex of assocated partcle s m = q mod 2 I. When buldng the neghbor set N(h) for partcle p whch les n cell we need to examne only partcles n cells 23 1,, + 1. Ths can be smply acheved by storng ndces K( 1 ), K() and K( +1 ) 24 of frst 25 keys n non-empty cells 1 < < +1. We examne all keys (assocated partcles) n cell 1 only when 1 = 1 startng wth key at ndex K( 1 ). Smlarly we examne cell +1 only when +1 = + 1. We are processng partcles n -th cell untl the respectve key maps coordnates of ths cell,.e. for each key q must hold q/2 I = q K() /2 I. Assumng the maxmum number of partcles nsde one cell s bound, ths traversal s O(n). However when smulatng partcles wth SPH ths upper bound s not guaranteed, but fortunately n average case there are only few partcles n one cell. To overcome partcle clusterng problem 26 we can choose the cell sze h = h/(2h+1), where H s a subdvson factor dependng on the flud propertes. When searchng for neghbors we need now to examne cells H < H+1 < < < +H 1 < +H and thus store 2H + 1 key ndces K( H ), K(),, K( +H ), what slows down the traversal by a constant factor O(H) per teraton. The algorthm thus stays lnear O(HN). 2D and 3D Case. The extenson to 2D and 3D s not obvous (see fgure (1) (b)). In 2D when key(m, m, j m ) key(n, n, j n ) then j m j n whle the order of m and n s non-decreasng only f j m = j n (j-coordnates are equal). When calculatng neghbors for partcle n cell (, j) we need to store all 9 key ndces K( s, j t ) s, t 1,, +1 and examne non-empty cell ( s, j t ) only when ( s, j t ) = ( + s, j + t). Furthermore n 3D we need 27 ndces K( s, j t, k u ) s, t, u 1,, +1 and examne non-empty cell ( s, j t, k u ) only when ( s, j t, k u ) = ( + s, j + t, k + u). We could generalze ths approach for cell subdvson takng s, t, u H,, +H or even better usng H-mask where (s, t, u) M H (h). The traversal s therefore n average case lnear wth complexty O( M H (h) N). Sortng Keys. To make the overall neghbor search algorthm lnear, we must sort keys n lnear tme. As ponted by Terdman [Terdman 2] even a set of floatng pont numbers can be quckly sorted n O(n) usng radx sort. Snce we have encoded ndces of partcles nto ther keys 27 we can use radx sort and have correct ndces after sortng. 5. IMPLEMENTATION AND RESULTS As shown n fgure (2) three dfferent smulatons of the classcal Dam-break test has been performed. Increasng the stffness and surface tenson coeffcent, flud behaves less compressble thus more realstc. In all three scenaros we used 16 partcles and set the rest densty ρ o = 1kg/m 3, mass m =.12kg, support length h =.5m vscosty µ = 5Ns/m 2 and used a fxed tme steppng wth 23 Assumng sze of cell s h 24 K() s ndex of key n lst of all keys 25 key n cell wth the smallest partcle ndex 26 Too many partcles n one cell 27 Keys are usually 64-bt numbers, 16 bts for each (n,, j, k) component

12 delta tme t =.2s. The surface tenson coeffcent σ vared from.6 (top), 1.4 (md) to 2.2 (bottom). We set stffness k gas to 2Nm/g (top), 4Nm/g (md) and 7N m/g (bottom). Boundary condtons are handled by nelastc partcle-plane collson resoluton wth wet frcton. Smple partcle-to-plane projecton s used to prevent overshootng. State varables are ntegrated usng explct Leap-Frog ntegraton scheme wth fxed tme steppng. The smulaton has been performed on Moble P4 1.7 GHz wth GeForce 4 Go. Fg. 2. The classcal Dam-break test wthn our SPH smulaton envronment. The three dfferent test smulatons demonstrate varous flud behavor wth respect to changng stffness and surface tenson parameters. Top row descrbes the behavor of more compressble flud wth less attracton between partcles, mddle row corresponds to hgher constants and fnally the last row represents a stffer flud wth stronger surface tenson. Notce the snapshots are not taken n equal tme steps. To test our approach we run each test case wth three dfferent neghbor search methods, namely the nave O(n 2 ) all-pars-test, Spatal Hashng and our Cell Indexng. 28 and measured the average tme spent for searchng neghbors and ntegraton of one tme step. These results are summarzed n table (I). 28 Other (tree) methods are left a the subject of future work

13 All-Pars Spatal Hashng Cell Indexng Dynamcs Test Case 1 1,95 s,42 s,25 s,25 s Test Case 2 1,97 s,45 s,27 s,24 s Test Case 3 1,93 s,43 s,27 s,28 s Table I. Dam-break tests. 16 Partcles. Average tme spent n one smulaton tme step n each test case. 6. CONCLUSION AND FUTURE WORK We have proposed and demonstrated Cell Indexng as a novel approach for searchng approxmate neghbor partcles wthn a SPH base flud smulaton. Due to cache msses, Spatal Hashng has been slghtly outperformed by our method, wthout large memory requrements of Full 3D vozelzaton. Our approach s nherently lnear, thus wll probably outperform other herarchcal methods 29 for larger data sets. We have ntroduced a H-mask to acheve speed-up by searchng on a subcell resoluton. Ths seamlessly fts nto Cell Indexng a can even extend Spatal Hashng. 3 Due to the mmense computatonal power of current graphcs hardware, we wll nvestgate n the future the possblty of mplementng our approach on GPU. Further we wll try to nvolve spatal and temporal coherence nto our method and allow t to use multresolutonal flud as been done n[keser et al. 26] The nherent compressblty of SPH can be decreased by solvng the pressure mplctly. Therefore we wll explore methods as MPS[Premože et al. 23] and solve the pressure equaton teratvely. Appendx - Smoothng Kernel Dervatves W poly (r, h) = πh 9 2 W poly (r, h) = πh 9 W press (r, h) = 45 πh 6 2 W vsco (r, h) = 45 πh 6 { (h 2 r 2 ) 2 r { (h 2 r 2 )(7r 2 3h 2 ) { (h r) 2 r r { (h r) r h r = r otherwse r h r = r otherwse < r h r = r otherwse r h r = r otherwse (18) 29 They usually need O(n log n) to rebuld. 3 However more cache msses or larger memory requrements arse

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