Implementing Lattice Boltzmann Computation on Graphics Hardware

Transcription

1 To appear n The Vsual omputer Implementng Latte oltzmann omputaton on Graphs Hardware We L, Xaomng We, and re Kaufman enter for Vsual omputng (V) and epartment of omputer Sene State Unversty of New York at Stony rook Stony rook, NY bstrat LM s a physally-based approah that smulates the mrosop movement of flud partles by smple, dental and loal rules. We aelerate the omputaton of the LM on general-purpose graphs hardware, by groupng partle pakets nto textures and mappng the oltzmann equatons ompletely to the rasterzaton and frame buffer operatons. We apply stthng and pakng to further mprove the performane. In addton, we propose tehnques, namely range salng and range separaton, that systematally transform varables nto the range requred by graphs hardware and thus prevent overflow. These approahes an be extended to a ompler that automatally translates general alulatons to operatons on graphs hardware. Keywords: smulaton. Introduton Graphs hardware, Latte oltzmann Method, flow Smulatons of flud behavor are n great demand n flm makng, as well as n vsual smulatons suh as anmaton desgn, texture synthess, flght smulaton, and sentf vsualzaton. In omputatonal Flud ynams (F), flud propertes, suh as densty and veloty are typally desrbed by the Naver-Stokes (NS) equatons, whh have nonlnear terms makng them too expensve to solve numerally n real tme. Instead of alulatng the marosop equatons, we an smulate the lnear, mrosop LM [, 9, ] to satsfy the NS equatons. The flud flow onssts of many tny flow partles and the olletve behavor of these mrosop partles results n the marosop dynams of a flud. Sne the marosop dynams of flud s nsenstve to the underlyng detals n mrosop physs [?], ths gves us the possblty of usng a smplfed mrosop knet-type model to smulate ts movements. The LM an be understood as a ellular utomata representng dsrete pakets movng on a dsrete latte at dsrete tme steps. The alulaton s performed on a regular grd. t eah grd ell, there are varables ndatng the status of the grd pont. ll the ells modfy ther status at eah tme step based on lnear and loal rules. lthough faster than other solutons to the NS equatons, the omputaton of the LM s stll slow. The alulatons at eah pont are smple, but there s usually a large amount of ells. Therefore, a pratal use of the LM typally demands parallel superomputers [9, ]. ommodty graphs hardware an perform pxel-orented operatons very effently. Not only the operatons are ppelned n dedated hardware, but there are usually up to four olor hannels and multple pxel ppelnes that essentally provde parallel proessng. The speed of graphs hardware doubles approxmately Emal:{lwe,wxaomn,ar}@s.sunysb.edu every sx months whh s muh faster than the mprovng rate of PU. Inspred by the performane of graphs hardware and the resemblane n the omputaton pattern between the LM and the rasterzaton stage, we propose to aelerate the LM on ommodty graphs hardware by storng pakets of the LM as textures and translatng the oltzmann equatons nto renderng operatons of the texture unts and the frame buffer. We further apply stthng to redue the overhead of texture swthng and pakng to redue the memory requrement as well as to explot parallelsm of the four olor hannels. In addton, we present range salng that transforms the range of arbtrary varables and any ntermedate results to fulfll the requrement of graphs hardware. The tehnques guarantee no overflow no matter how the varables are evaluated whle the salng fators are hosen to take the full preson of the hardware. We use LM equatons as an example to show how the salng s appled and observe only up to % of error. In addton, the range separaton proposed n ths paper overomes the non-negatve lmts requred n ertan stages of the graphs ppelne. lthough we fous on the LM and ts applatons to vsual smulaton of fluds and smoke, our approah s extendable to any ellular-automata-typed alulatons. We expet that the tehnques proposed n the paper, suh as range salng and range separaton, be developed nto a ompler that automatally generates renderng nstrutons equvalent to varous general omputatons wth the range of all varables and ntermedate and fnal results properly transformed. The rest of the paper s organzed as follows. Frst, we revew related work. In Seton 3 we present range salng and range separaton whh are rtal omponents of our approah n aeleratng general omputatons on graphs hardware. In Setons 4 and 5, after a bref ntroduton of the theory of the LM, we present our methods of mappng the LM equatons as mult-pass rasterzaton operatons and how the range salng s appled to the LM. In Seton 6, we gve the expermental results. Related Work Graphs hardware has been extended to varous applatons beyond ts orgnally-expeted usage. Examples nlude matrx multplaton [], 3 onvoluton [6], morphologal operatons suh as dlaton and eroson [7], omputaton of Vorono dagrams and proxmty queres [4, 5], voxelzaton [], algebra reonstruton [], and volumetr deformaton [6]. Graphs arhteture, suh as OpenGL, an be treated as a general SIM omputer [3]. Varous omputatons are mplemented as the operatons of the texture mappng unt and the frame buffer. Fnal results are obtaned n one or more renderng passes. oth Peery et al. [3] and Proudfoot et al. [4] have developed languages for programmable proedural shadng systems as well as omplers that automatally generate nstrutons orrespondng

2 To appear n The Vsual omputer to renderng operatons on graphs hardware. However, to apply these deas to other applatons, the lmted value range and auray of graphs hardware have to be onsdered. Trendall et al. [7] gave several formulas for saled and based funtons whose value ranges are wthn the lmts and appled the method to the omputaton of nteratve austs. Some of the above applatons sale and shft the varables n ther omputatons so that they ft nto the value range of the graphs hardware. Ther salng and shftng parameters are hosen ether trvally or emprally. The range salng proposed n ths paper provdes a systemat way for mappng general omputatons onto graphs hardware whh guarantees that all the nputs and outputs as well as the ntermedate results are not lamped by the hardware, n addton to explotng the hardware preson as muh as possble. There are a few papers on aeleratng flow vsualzaton on graphs hardware. Hedrh et al. [3] explot pxel texture to ompute lne ntegral onvoluton, whh s a tehnque for vsualzng vetor data. Jobard et al. [8] translate texture adveton omputatons to frame buffer operatons to aelerate the vsualzaton of the moton of flows. Weskopf et al. [9] extend Jobard et al. s work to 3 flows. They also take advantage of the newly avalable OpenGL extensons, namely offset texture and dependent texture. ll these tehnques are for the vsualzaton of flud wth gven veloty felds. In ontrast, ths paper fouses on the smulaton, spefally the generaton of the felds, suh as veloty, that are requred for the vsualzaton. y employng smlar vsualzaton tehnques, our approah an map the whole smulaton and vsualzaton onto the graphs hardware wthout the need of transferrng data bak and fore between the host memory and the graphs memory. Harrs et al [?] mplement oupled map latte (ML), an varaton of ellular automata, on graphs hardware, whh has smlar motvaton and applatons to ths paper. We expet our work on LM eventually result n a ompler of general omputatons to graphs hardware, therefore one of the fous of the paper s how to systematally mappng general equatons to renderng operatons onsderng the range and preson lmtaton of the graphs hardware. esdes, we propose tehnques suh as texture pakng and stthng to further aelerate the exeuton on hardware. 3 General omputatons on Graphs Hardware We use the rasterzaton unts (e.g., Nvda s regster ombners) and the frame buffer n graphs hardware to mplement addton, subtraton and multplaton. vson and other more omplated alulatons are replaed wth lookup tables storng preomputed values. Input values are stored n textures and output values are ether oped from the frame buffer to textures or wrtten dretly to the textures wth the render-to-texture extenson. Values n graphs hardware are lamped to ether [, ] or [, ], dependng on the stage of the graphs ppelne. Therefore, we need to transform the value ranges of all the nputs and outputs. Trendall et al. [7] also mapped the lower bound of the range to by basng for better auray. In ontrast, we generally avod ntrodung any bas durng the mappng. (The reason s explaned n seton 3.). That s, we only apply salng to hange the numeral ranges. Range salng an be onsdered as a smulaton of floatng pont on fxed-pont hardware, whereas floatng pont texture and frame buffer have been suggested for future hardware [5]. However, even f the support to floatng pont were avalable n the rasterzaton stage of the graphs hardware, t would be sgnfantly slower and ould take more texture memory than ts fxed-pont ounter part. The range salng proposed n ths paper makes no assumpton of the preson of the hardware. We always prefer the rasterzaton unts beause they are more flexble than the frame buffer. However, dstrbutng ertan operatons to the frame buffer redues the number of renderng passes and the need of opyng the frame buffer ontents nto textures. 3. Range Salng The salng fators should be arefully seleted so that no lampng error ours and the omputaton explots the full preson of the hardware. For any nput or output funton f(x), we dvde t wth ts maxmal absolute value and obtan a saled funton f(x), suh that: f max f(x) = f(x) () where f max = max x( f(x) ) whh we refer to as the left-hand salar. Obvously, f(x) [, ]. We then use f(x) throughout the hardware ppelne. We should also make sure that durng the omputaton of f(x), no ntermedate result s lamped. We denote U(f) as the maxmal absolute value of all ntermedate results durng the evaluaton of f(x), no matter what omputaton order s taken f the omputaton ontans multple operatons. It s easy to see that U(f) f max. If we multply /U(f) on the rght-hand-sde before omputng, we guarantee that no overflow ours. We refer to U(f) as the rghthand salar. If a funton s(x) s a weghted sum of several other funtons, s(x) = kf(x), we ompute the saled funton s(x) as follows: s(x) = U(s) s max k f max U(s) f (x) () The left-hand salar and the upper bound of the ntermedate values are omputed as: s max = max( k f max ( x, f (x) > ), k f max ( x, f (x) < )) (3) U(s) = max( k U(f )( x, f (x) > ), k U(f )( x, f (x) < )) (4) That s, eah f(x) s saled by k f max /U(s) before summaton. We know k f max /U(s). The sum s then multpled by a onstant U(s)/s max. Note that U(s)/s max. To multply wth a fator larger than, we have two hoes: () utlzng the output sale mappng of the regster ombners; () applyng multplaton and addton or usng dot produt. s max s obtaned by summng the maxmal values of postve and negatve entres separately and seletng the sum wth the larger absolute value. U(s) s omputed smlarly. Sne s(x) may not reah the omputed maxmal value s max, t s better to replae s max wth the atual largest absolute value s max suh that s max < s max. To measure s max, we mplement a software-only verson of the same alulatons on the PU and feed nto varous nput values. If p(x) s the produt of several funtons, p(x) = k f(x). We have p(x) = U(p) p max k f max U(f ) f (x) (5)

3 To appear n The Vsual omputer where p max = k f max (6) U(p) = k U(f ) (7) If the funton g(x) s an nput for a renderng pass, we speulate that U(g) = g max. pparently, f all f (x) are nputs, p max = U(p) = k f max. In eah renderng pass, the hardware performs a mx of addtons (subtratons) and multplatons. We frst obtan the absolute maxmal values of nputs ether by pror knowledge or measurement of software smulaton on the PU. Then, before evaluatng a funton f(x), we ompute ts left-hand and rght-hand salars by groupng ts rght-hand sde nto sums and produts and by reursvely applyng Equatons 3, 4, 6 and 7. Next, we dvde the rght-hand sde by U(f) and dstrbute U(f) to eah nput funtons aordng to Equatons and 5. ll the sale oeffents of the nputs are omputed n software. 3. Range Separaton onventonal graphs hardware only supports a numeral range of [,]. Reent OpenGL extensons, suh as Nvda s regster ombners, expand the range to [-, ] n the rasterzaton stage, whereas the fnal ombner and the frame buffer are stll lmted to [, ]. Hene, negatve values have to be transformed before they are sent to the fnal ombner or the frame buffer. soluton s to apply bas and salng to transform the range to [, ] before enterng the fnal ombner stage and the frame buffer. However, t s then very dffult to utlze the fnal ombner or the frame buffer for multplaton or addton on the based varables. onsequently, the number of renderng passes and the tmes of bakng-up the ontents of the frame buffer are lkely to nrease. We propose to separate the postve and negatve ranges and avod basng as an alternatve soluton. Smlar to other approahes, we sale all the varables to ft nto the range of [-, ] as the frst step. If all the values of a varable are onstantly nonnegatve or non-postve, t s trval to map them to [, ]. To handle a varable ontanng both postve and negatve values, we dvde the range of [-, ] nto two parts, [-, ] and [, ]. For arbtrary funton f, we have: f = [f] + + [f] = [f] + [ f] + (8) where [ ] + and [ ] denote the lampng to [, ] and [-, ], respetvely. Obvously, both [f] + and [ f] + ontan only or postve values. Let f and g be two funtons. ddton (subtraton), multplaton (salar and dot produt) and dvson are then performed as follows: f + g = ([f] + + [g] + ) ([ f] + + [ g] + ) (9) fg = [f] + [g] + + [ f] + [ g] + ([f] + [ g] + + [ f] + [g] + ) () f g = fg g () Note that n Equaton, g an be omputed as: g = ([g] + ) + ([ g] + ) () Other omputatons are mplemented wth lookup table, hene are treated as nputs Fgure : The 3Q9 latte geometry. The veloty dretons of the 8 movng paket dstrbuton are shown as arrows. Range separaton ntrodues more alulatons than the unseparated verson, but reall that salng and basng requres addtonal operatons as well. In prate, we hoose ether basng and salng or range separaton dependng on whh an be exeuted n fewer renderng passes and nvolves fewer texture unts. We make a hoe onsderng also the fat that range separaton provdes one addtonal bt of preson. 4 Latte oltzmann Method Now, let s revew the prnples of the latte oltzmann method [, 9, ] and see what omputatons are needed for the smulaton of the LM. The LM onssts of a regular grd and a set of paket dstrbuton values. Eah paket dstrbuton f q orresponds to a veloty dreton vetor e q shootng from a node to ts neghbor. The ndex q desrbes the -dmensonal sub-latte where q s the sub-latte level and enumerates the sub-latte vetors. Fgure depts a sngle node of the 3Q9 model (9 pakets n 3 spae), whle the left part of Fgure shows four grd nodes of the Q9 (9 pakets n spae). The arrows n the fgures represent the e q vetors. The LM updates the paket dstrbuton values at eah node based on two rules: ollson and propagaton. ollson desrbes the redstrbuton of pakets at eah loal node. Propagaton means the pakets move to the nearest neghbor along the veloty dretons. These two rules an be desrbed by the followng equatons: ollson : f new q ( x, t) f q( x, t) = Ω q (3) propagaton : f q( x + e q, t + ) = f new q ( x, t) (4) where Ω s a general ollson operator. Sne omponents of e q an only be hoosen from {-,, }, the propagaton s loal. The densty and veloty are alulated from the paket dstrbutons as follows: ρ = f q (5) v = ρ q f q eq (6) q The ollson operator s seleted n a way that mass and momentum are onserved loally. Suppose that there s always a loal 3

4 To appear n The Vsual omputer equlbrum partle dstrbuton f eq q dependent only on the onserved quanttes ρ and v, then the ollson step s hanged to: fq new ( x, t) f q( x, t) = τ (fq( x, t) f eq q (ρ, v )) (7) where τ s the relaxaton tme sale. f eq q s deded by the followng equaton: f eq q (ρ, v ) = ρ( q + q < e q, v > + q < e q, v > + q < v, v >) (8) < x, y > denotes the dot produt between two vetors x and y. The onstants q to q depend on the employed latte geometry. The smulaton of the LM then proeeds as follows: () ompute densty aordng to Equaton 5; () ompute veloty (Equaton 6); (3) ompute equlbrum dstrbuton (Equaton 8); (4) update dstrbutons by Equaton 7 and go bak to step (). More detals on the LM model an be found n [8]. 5 Mappng LM to Graphs Hardware 5. lgorthm Overvew To ompute the LM equatons on graphs hardware, we dvde the LM grd and group the paket dstrbutons f q nto arrays aordng to ther veloty dretons. ll the paket dstrbutons wth the same veloty dreton are grouped nto the same array, whle keepng the neghborng relatonshp of the orgnal model. Fgure shows the dvson of a model. We then store the arrays as textures. For a model, all suh arrays are naturally, whle for a 3 model, eah array forms a volume and s stored as a stak of textures. The dea of the stak of textures s from texture-based volume renderng, but note that we don t need three replated opes of the dataset. boundary ondtons and update the dstrbuton textures. The updated dstrbuton textures are then used as nputs for the next smulaton step. dst. densty veloty oundary ondton equl. dst. Propagaton new dst. Fgure 3: The data flow of the hardware aelerated LM omputatons. To redue the overhead of swthng between textures, we stth multple textures representng paket dstrbutons wth the same veloty dreton nto one larger texture. The left part of Fgure 5 shows an example, n whh, every four sles are stthed nto a larger texture. The ppelne depted n Fgure 3 s then operated on the stthed textures. 5. Propagaton ordng to Equaton 4, eah paket dstrbuton havng non-zero veloty propagates to the neghborng grd every tme step. Sne we group pakets based on ther veloty dretons, the propagaton s aomplshed by shftng dstrbuton textures n the dreton of the assoated veloty. We deompose the veloty nto two parts, the veloty omponent wthn the sle (n-sle veloty) and the veloty omponent orthogonal to the sle (orthogonal veloty). The propagaton s done for the two veloty omponents ndependently. To propagate n the dreton of the n-sle veloty, we smply translate the texture of dstrbutons approprately, as shown n Fgure 4. = Veloty dreton Tme t Tme t New sle Fgure : vson of the Q9 model. Only 4 grds out of 9 are shown. Pakets are grouped aordng to ther veloty dretons. ll the other varables, the densty ρ, the veloty v and the equlbrum dstrbutons f eq q are stored smlarly n textures. We projet multple textured retangles wth the olor-enoded denstes, velotes and dstrbutons. For onvenene, the retangles are parallel to the vewng plane and are rendered orthogonally. Therefore, the texture spae has the same resoluton as the mage spae and the nterpolaton mode s set to nearest-neghbor. s shown n Fgure 3, the textures of the paket dstrbutons are the nputs. ensty and veloty are then omputed from the dstrbuton textures. Next, the equlbrum dstrbuton textures are obtaned from the denstes and the velotes. ordng to the propagaton equaton, new dstrbutons are omputed from the dstrbutons and the equlbrum dstrbutons. Fnally, we apply the Unstthed sles Stthed sles Fgure 4: Propagaton of the paket dstrbutons along the dreton of the veloty omponent orthogonal to the sles. If we don t stth multple sles nto one texture, the propagaton n the dreton of the orthogonal veloty s done smply by renamng the dstrbuton textures. eause of the stthng, we need to apply translaton nsde the stthed textures as well as opyng sub-textures to other stthed textures. Fgure 5 shows the out-of sle propagaton for stthed sles. The ndexed bloks denote the sles storng paket dstrbutons. The retangles n thker lnes mark the sub-textures that are propagated. For example, the subtexture omposed of sles to 3 s shfted down by the sze of one sle n the Y dmenson. Sles 4 and 8 are moved to the next textures. Note that n tme step t +, a new sle s added owng to the 4

5 To appear n The Vsual omputer nlet or the boundary ondton, whle a blok () has moved out of the framework and s dsarded. Veloty dreton dst. bouned ;;;;;;; dstrbutons propagated ;;;; ;;; dstrbutons dst leavng ;;;; dst.bouned propagated ;;;;;;; dstrbutons dstrbutons ;;;; New dst. Tme t Tme t+ Wall Open Fgure 5: Propagaton of the paket dstrbutons along the dreton of the n-sle veloty and applaton of boundary ondton. The retangular edges n blue thk lnes are the boundares. 5.3 oundary ondton Paket dstrbutons on the boundary should be handled dfferently from the nternal ones. general approah s to ompute the new dstrbutons for the boundary dstrbutons, then set the new values nto the dstrbutons textures. The omputaton an be done wth ether the PU or the graphs hardware. oune-bak boundary ondton an be easly handled by the graphs hardware. eause the partles are grouped aordng to ther veloty dretons, we smply opy the boundary paket dstrbutons to the texture of the opposte veloty dreton. Smlar to propagaton, the boune-bak s treated for n-sle veloty and orthogonal veloty separately. For a boundary fae unparallel to the sles, the nterseton of the fae wth a sle s a lne segment (or a urve, f we allow non-planar boundary fae). We set the dstrbutons next to the nterseton by drawng texture strps whh are just one texel wde. Fgure 4 shows the boune-bak from the left and the top walls. Note that the dstrbutons leavng one sle beome the new dstrbutons of the sle wth the opposte veloty dreton. For a boundary fae parallel to the sles, usually a texture needs to be updated. In Fgure 5, the blok marked new sle s obtaned from the sle at the same poston but wth an opposte dreton, or t s set to the nlet dstrbutons f the sle s adjaent to an nlet fae. 5.4 Pakng In our prelmnary work [8] of hardware aelerated LM, we stored only the dstrbutons of the same dreton n a sngle texture. ue to the restrton of the urrent graphs hardware and the onsderatons of effeny, every dstrbuton texture s n the format of RG. Hene, eah f q s replated 4 tmes nto the RG hannels, and the operatons over the dstrbutons are duplated as well. In ths paper, we pak four f qs from dfferent dretons as an RG texel. That s, a sngle texture s omposed of four dstrbuton arrays wth dfferent veloty dretons. Ths pakng sheme redues the memory requrement of dstrbutons to nearly /4 of the desgn wthout pakng. To ompute densty ρ (refer to Equaton 5), we use a dotprodut to add the dstrbutons stored n dfferent olor hannels, as shown n Fgure 6. The pakng essentally redues the number of operatons to one quarter for the omputaton of densty. Multple dstrbuton textures are added together wth the OpenGL extensons of mult-textures and the regster ombners. In addton, we also utlze the addtve blendng of the frame buffer to make t unneessary to bakup the ntermedate ontents by opyng the frame buffer to a texture or swthng to dfferent frame (pxel) buffer. The alulaton of velotes s a lttle more omplated owng to the pakng, sne eah dstrbuton needs to be multpled wth ts own dreton vetor e q and the RG omponents an t be read ompletely ndvdually n the urrent mplementaton of the regster ombners. Therefore, we need to dot-produt the dstrbutons n the RG hannels wth (,, ) or (,, ) to separate the dstrbutons. The value n the blue hannel s extrated wth an alpha ombner. s shown n Fgure 7, we add eght dstrbuton sles (stored n two textures) weghted by the orrespondng e q wth sx ombner stages. Here we apply a trk on the fnal ombner stage so that t adds four nputs. Note that the nputs and to the fnal ombner stage are saled by and s set to.5, whle the other two nputs of the fnal ombner stage are sent to Spare and Seondary. gan, we use addtve blendng of the frame buffer to add onseutve outputs from the fnal ombner to avod opyng the frame buffer. Range separaton are appled to e q so that negatve values are not lamped. That s, we ompute v + and v separately and later add them together. 5.5 Salng of the LM Equatons In ths seton, we show how to apply the range transformaton desrbed n Seton 3 to the LM equatons. ssume fq max s the left-hand salar of the paket dstrbutons and the equlbrum paket dstrbutons of sub-latte q. We defne the saled dstrbutons fq and the saled densty ρ as: ρ = f q = f max q f q (9) ρ ρ = fq max max ρ f max q () Sne all the f q are postve nputs, ρ max = U(ρ) = f max q q. We also defne: ρ = ρmn () ρρ max where ρ mn s the lower bound of the densty and q ρ [, ]. ordng to Equaton and the symmetry of the LM, the rght-hand fator of the saled veloty U( v ) s: U( v ) = ρ max mn b q f max q { e q[b] > } () where b s the dmenson ndex of vetor e q. Note that U( v ) and v max are salars nstead of vetors. Then, the saled veloty s omputed as: v v = v = U( v ) fq max ( max v max ρ U( eq) v )ρ mn fq (3) Wth suh range salng, Equatons 7 and 8 beome: f q( x + e q, t + ) = fq( x, t) τ ( fq( x, t) q f eq q ) (4) 5

6 To appear n The Vsual omputer Tex Label dst. Tex Label dst. General ombner RG General ombner + RG General ombner RG X X.5 Fnal ombner + (-) + RG Tex Label dst. Tex Label 3 dst. General ombner + lpha General ombner + lpha General ombner + lpha Spare + Seondary Fgure 6: The onfguraton of the regster ombners for omputng densty from paked dstrbutons. denotes onstants dependent on q and the salng fators Tex Label General ombner General ombner General ombner General ombner dst. Prmary Label Label Fog Tex Label dst. RG General ombner 3 RG e q e q e q e q + RG General ombner 4 + RG lpha X General ombner 4 lpha e q e q e q e q + RG General ombner 5 + RG X.5 Fnal ombner + (-) + RG Spare + Seondary Fgure 7: The onfguraton of the regster ombners for omputng veloty from paked dstrbutons. 6 Expermental Results f eq q q ) ρ( qρmax fq max U(fq eq = U(f eq ) + qvmax ρ max U(f eq q ) 4 q(v max ) ρ max U(f eq q ) 4 q(v max ) ρ max U(f eq q ) < e q, v > + < e q, v > + < v, v >) (5) For 3Q9 model, q, q, q and q, hene: U(f eq q ) = max((ρmax q + qv max ρ max + 4 q(v max ) ρ max ), ( qv max ρ max 4 q(v max ) ρ max )) (6) Note that n Equaton 5, we saled the vetors before the dot produts. The salng fator s hosen to be a power of two so that t s easy to mplement t n hardware. We have mplemented our tehnques on an Nvda GeFore4 T 46 ard that has 8M of memory. For omparson, we also mplemented the LM n software on a P wth a.6ghz P4 proessor and 5M R memory. We use the 3Q9 model throughout the experments. 6. uray major onern about usng graphs hardware for general omputaton s the auray. Most graphs hardware supports only 8 bts per olor hannel. There have been a few lmted supports of 6-bt textures but are too restrted for a relatvely omplated applaton suh as the LM smulaton. Fortunately, the varables of the LM fall nto a small numeral range whh makes the range salng effetve. esdes, the property of the LM, that the marosop dynams s nsenstve to the underlyng detals of the mrosop physs [?], relaxes the requrement on the auray of the omputaton. Fgures 8a and 8b shows two olor-enoded veloty sles ex- 6

7 To appear n The Vsual omputer trated from a 3 LM smulaton. Fgure 8a s omputed by the PU wth floatng-pont auray and Fgure 8b s obtaned wth the hardware approah desrbed n the paper. Fgure 8 shows the exaggerated error mage wth all pxel values saled up by. ll the values are transformed to the range of [, 55] for dsplay. Vsually, t s dffult to see any dfferene between Fgure 8a and 8b. tually, the maxmal pxel-wse dfferene s less than % after one step of smulaton. rthm sale for both the axes. Note that stthng s very effetve for grds equal to or smaller than 83. tme(se) E+ 9.E+.3E+.E+.4E+.8E- 7.E-.7E- 9.5E- 8.8E-.6E-.3E- 3.6E-3 4.3E-..3E-. 3.6E-3..E+3.E+4.E+5 3.4E- 8.3E-.7E+ 4.E-.E-.E+6.E+7 Software No Stthng Stthng.E+8 ell Fgure : Tme per step of the LM omputon wth graphs hardware (wth or wthout stthng) and software. (a) (b) () The speedup fator of the hardware aelerated method aganst the software approah s more learly shown n Fgure. The proposed method s at least 5 tmes faster than ts software ounterpart exept for the 63 model. Note that the memory requrement of a 563 szed model s muh larger than the memory on the graphs board. tually, only the dstrbutons for a 3Q9 model need = 39M. Our mplementaton stll ahves.7 seond per step for a 563 model, whh s aeptable for an nteratve applaton. Fgure 8: Veloty sle omputed by (a) software, (b) graphs hardware. () The dfferene mage between (a) and (b) saled up by a fator of ten. 6. Performane Stthng smaller textures nto bgger ones sgnfantly redues the overhead of texture swthng. efore testng the performane of our hardware mplementaton of LM, we frst determne how bg the stthed textures should be. Fgure 9 shows the relatonshp between the area of the stthed textures and the number of ells that the hardware an handle per seond. For our hardware onfguraton, a texture of the sze of 5 5 performs the best. In fat, for textures smaller than 56 56, the omputaton tme s nearly ndependent on the sze of the textures. Note that we restrt the dmensons of the textures to be powers of two, although we ould explot the non-power-of-two-texture extenson to aheve even better results. speedup fator No Stthng Stthng E E+4.E E+6.E+7.E+8 ell k ells/s Fgure : Speedup fator of the hardware aelerated method to the software method. 5 5.E+.E+3.E+4.E+5.E+6.E pplaton area We vsualze the smulaton results by ether dretly showng the olor-enoded veloty feld (as n Fgure 8) or by njetng partles nto the system from an nlet. These partles are onsdered massless, that s, they do not affet the flow alulaton. The partles are adveted aordng to the veloty of the grd ell. The number of partles depends on the densty of the flud. We then render the sene wth texture splats [8]. Fgure a shows an mage of smoke emanatng from a hmney and then blown by the wnd. The left sde of the grd s assgned a speed along the X axs to model the effet of wnd. We also norporate an upward fore due to the dfferene n the temperature feld. Fgure b shows the result of hot steam rsng up from a teapot and ts spout aordng to a veloty feld smulated wth our LM approah and hardware aeleraton. oth the smoke and the steam are smulated on a 33 grd. Fgure 9: Number of ells proessed per seond as a funton of the area of the stthed textures. Note that the X axs s n Logarthm sale. Fgure ompares the tme (n seonds) per step of the hardware LM wth a software mplementaton. The statsts does not nlude the tme for renderng. The Stthng urve refers to the performane after stthng small ones nto 5 5 textures, whle No Stthng does not. Note that the hardware aelerated tehnque wns n speed for any sze of the model, exept that for the 63 grd, the No Stthng method s slower than software. However, smply by stthng the sxteen 6 6 textures nto one textures gans a speedup fator of. Fgure s n loga- 7

8 To appear n The Vsual omputer In ths paper, we presented the algorthms for mplementng the Latte oltzmann Model on ommodty graphs hardware. Expermental results show that the LM an be smulated on urrent low-ost omputers n real tme for a grd sze of up to 64 3 and nteratvely for a grd sze of 8 3. lthough we foused on the LM, our tehnques an be extended to other omputatons. It s also possble to generalze our methods nto a framework of aeleratng a large varety of applatons on onventonal graphs hardware and ts future enhaned versons. One of our planned dreton s a development envronment nludng a language desrbng general parallel omputatons and a ompler that automatally translates ode wrtten n the language nto avalable operatons on graphs hardware. We would also lke to develop a debugger for onvenently nspetng the ntermedate results of the graphs ppelne. knowledgement Ths work has been supported by an ONR grant N434. We thank Klaus Mueller and Suz Yoakum-Stover for helpful dsussons. Referenes [] S. hen and G.. oolean. Latte boltzmann method for flud flows. nnu. Rev. Flud Meh., 3:39 364, 998. [] S. Fang and H. hen. Hardware aelerated voxelzaton. omputers & Graphs, 4(3):433 44, June. (a) Smoke emanatng from a hmney and then blown up by the wnd. [3] W. Hedrh, R. Westermann, H.-P. Sedel, and T. Ertl. pplatons of pxel textures n vsualzaton and realst mage synthess. In 999 M Symposum on Interatve 3 Graphs, pages 7 34, prl 999. [4] K. Hoff, T. ulver, J. Keyser, M. Ln, and. Manoha. Fast omputaton of generalzed vorono dagrams usng graphs hardware. In Proeedngs of SIGGRPH 99, pages 77 86, ugust 999. [5] K. Hoff,. Zaferaks, M.. Ln, and. Manoha. Fast and smple geometr proxmty queres usng graphs hardware. In M Symposum on Interatve 3 Graphs, pages 45 48, Marh. [6] M. Hopf and T. Ertl. eleratng 3d onvoluton usng graphs hardware. In IEEE Vsualzaton 99, pages , Otober 999. [7] M. Hopf and T. Ertl. eleratng morphologal analyss wth graphs hardware. In Workshop on Vson, Modellng, and Vsualzaton VMV, pages ,. [8]. Jobard, G. Erlebaher, and M. Y. Hussan. Hardwareaelerated texture adveton for unsteady flow vsualzaton. In IEEE Vsualzaton, pages 55 6, Otober. (b) Hot steam rsng up from a teapot and ts spout. Fgure : pplatons of the hardware aelerated LM. 7 susson [9].. Kandha. Large Sale Latte-oltzmann Smulatons. Ph thess, Unversty of msterdam, eember 999. [] E. S. Larsen and. Mllster. Fast matrx multples usng graphs hardware. The Internatonal onferene for Hgh Performane omputng and ommunatons,. []. Muders. Three-mensonal Parallel Latte oltzmann Hydrodynams Smulatons of Turbulent Flows n Interstellar ark louds. Ph thess, Unversty at onn, ugust 995. [] K. Mueller and R. Yagel. On the use of graphs hardware to aelerate algebra reonstruton methods. In SPIE Medal Imagng onferene, 999. [3] M. S. Peery, M. Olano, J. rey, and P. J. Ungar. Interatve mult-pass programmable shadng. In Proeedngs of M SIGGRPH, pages 45 43, July. 8

9 To appear n The Vsual omputer [4] K. Proudfoot, W. R. Mark, S. Tzvetkov, and P. Hanrahan. real-tme proedural shadng system for programmable graphs hardware. In Proeedngs of M SIGGRPH, pages 59 7, ugust. [5] T. Purell, I. uk, W. Mark, and P. Hanrahan. Ray trang on programable hardware. In Proeedngs of M SIGGRPH, ugust. [6]. Rezk-Salama, M. Sheuerng, G. Soza, and G. Grener. Fast volumetr deformaton on general purpose hardware. In Pro. SIGGRPH/Eurographs Workshop on Graphs Hardware,. [7]. Trendall and. J. Stewart. General alulatons usng graphs hardware wth applatons to nteratve austs. In Renderng Tehnques : th Eurographs Workshop on Renderng, pages 87 98, June. [8] X. We, W. L, K. Mueller, and. Kaufman. Smulatng fre wth texture splats. In Proeedngs IEEE Vsualzaton,. [9]. Weskopf, M. Hopf, and T. Ertl. Hardware-aelerated vsualzaton of tme-varyng and 3 vetor felds by texture adveton va programmable per-pxel operatons. In Workshop on Vson, Modelng, and Vsualzaton VMV, pages ,. 9