Graphical Models. Reducing numerical dissipation in smoke simulation q. Zhanpeng Huang a,, Ladislav Kavan b, Weikai Li c, Pan Hui c, Guanghong Gong a

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1 Graphcal Models 78 (2015) Contents lsts avalable at ScenceDrect Graphcal Models journal homepage: Reducng numercal dsspaton n smoke smulaton q Zhanpeng Huang a,, Ladslav Kavan b, Weka L c, Pan Hu c, Guanghong Gong a a Department of Automaton Scence and Electrcal Engneerng, Behang Unversty, Bejng, Chna b Computer and Informaton Scences Department, Unversty of Pennsylvana, PA, USA c Department of Computer Scence and Engneerng, Hong Kong Unversty of Scence and Technology, Hong Kong artcle nfo abstract Artcle hstory: Receved 3 March 2014 Receved n revsed form 18 November 2014 Accepted 28 December 2014 Avalable onlne 7 January 2015 Keywords: Smoke and gaseous phenomena Physcally based methods Numercal scheme Numercal dsspaton Numercal dsspaton acts as artfcal vscosty to make smoke vscous. Reducng numercal dsspaton s able to recover vsual detals smeared out by the numercal dsspaton. Great efforts have been devoted to suppress the numercal dsspaton n smoke smulaton n the past few years. In ths paper we nvestgate methods of combatng the numercal dsspaton. We descrbe vsual consequences of the numercal dsspaton and explore sources that ntroduce the numercal dsspaton nto course of smoke smulaton. Methods are nvestgated from varous aspects ncludng grd varaton, hgh-order advecton, subgrd compensaton, nvarant conservaton, and partcle-based mprovement, followed by dscusson and comparson n terms of vsual qualty, computatonal overhead, ease of mplementaton, adaptvty, and scalablty, whch leads to ther dfferent applcablty to varous applcaton scenaros. Ó 2015 Elsever Inc. All rghts reserved. 1. Introducton Smoke s desrable n vsual effect and vdeo game ndustres. It s also one of challengng problems n computer graphcs due to ts complexty and turbulence. To obtan realstc smoke and gaseous phenomena, physcally based methods wth Naver Stoke Equatons (NSEs) have been explored to model underlyng flud dynamcs. Although numercally ntegratng NSEs have been studed n computatonal flud dynamcs (CFD), computer graphcs researches focus on smplfed dscretzaton and numercal schemes when vsual qualty matters most. Smplfcatons make physcally based methods possble for smoke smulaton but ntroduce the numercal dsspaton. The numercal dsspaton ncreases flud vscosty to make t appear more vscous than ntended. It degrades the vsual q Ths paper has been recommended for acceptance by Peter Lndstrom. Correspondng author. appearance by smearng out fne detals and dampng down the moton quckly. The numercal dsspaton has been recognzed to have substantal vsual consequences to the smoke smulaton. Many sources ntroduce numercal dsspaton to the course of the smoke smulaton. Coarse spatotemporal dscretzaton produces numercal truncaton errors, whch s proven to have a form of vscosty [1]. As flud quanttes are only defned on dscrete locatons such as grd ponts and partcles, nterpolaton schemes are requred to calculate values at undefned postons, whch s equvalent to smoothng operatons that produce the numercal dsspaton. The sem-lagrangan method [2] s wdely used for the smoke smulaton attrbuted to ts uncondtonal stablty and ease of mplementaton, but t generates a large amount of the numercal dsspaton n backward tracng and advecton subroutnes. Many advanced methods are constructed based on the sem-lagrangan method to guarantee the uncondtonal stablty. However, they also nhert the dsadvantage of massve numercal dsspaton /Ó 2015 Elsever Inc. All rghts reserved.

2 Massve effort has been devoted to combat the numercal dsspaton from dfferent aspects. Some methods are developed to elmnate sources of the numercal dsspaton. For nstance, t s straghtforward to reduce the numercal dsspaton by ncreasng spatal resoluton and reducng tme step. However, both approaches ncrease computatonal overhead. Adaptve mesh [3], rregular mesh [4,5], and dynamcal mesh [6] are proposed to reduce the numercal dsspaton wthout sgnfcantly ncreasng computaton. Rather than drectly reducng the numercal dsspaton, several methods generate artfcal detals to compensate for vsual loss usng vortcty confnement [7,8] and subscale turbulence models [9,10]. Grd-based methods requre resamplng flow feld, whch s equvalent to the low-pass flter to smear out hgh-frequency components. Partcle-based methods only carry quantty but do not dsspate quantty, whch does not suffer from the numercal dsspaton problem. However, partcle methods have problems such as partcle redstrbuton. Hybrd partcle and grd methods [11,12] are proposed to leverage advantages of partcle and grd to reduce numercal dsspaton. In ths paper, we nvestgate the numercal dsspaton n smoke smulaton n terms of where t comes out, what mpact t has, and how to combat t. The rest of paper s organzed as follows: we gve a bref ntroducton to the basc smoke smulaton n Secton 2 and address the sources of the numercal dsspaton n Secton 3; n Secton 4 we nvestgate and compare methods of combatng numercal dsspaton from dfferent aspects, followng by a concluson n Secton Background Smoke and other gaseous phenomena are normally smplfed to be ncompressble and homogenous, whch does not decrease the applcablty to model basc dynamcal mechansms. The NSEs to model smoke are derved as: ¼ mr2 u rp q þ f r u ¼ 0 ðaþ ðbþ ð1þ where u s velocty, p and q denote pressure and flud densty respectvely. m s knematc vscosty to measure how vscous the flud s and f represents the resultant external force. The two equatons ndcate that the flud should conserve both momentum and mass. The frst equaton s derved from Newton s second law wth left-hand term presentng acceleraton and rght-hand terms the net force exerted on flud. NSEs are too complcated to solve for analytcal soluton drectly. The NSEs usually break down nto smple terms ncludng advecton, pressure, dffuson, and external force [2]. The smple terms can then be easly solved ndvdually. If we defne the terms as operators denoted by A; P; D, and F, the operator S to solve NSEs can be wrtten as [13]: S ¼ P F D A where Z. Huang et al. / Graphcal Models 78 (2015) ð2þ 8 A ¼ ðurþq >< D mr2 u F ¼ >: P þ q ¼ 0; so that r u ¼ 0 ðdþ ðbþ ðcþ ð3þ where q can be velocty, temperature, or any other flud quantty. 3. Numercal dsspaton as artfcal vscosty Numercal solutons are dfferent from exact soluton due to numercal truncaton errors. The truncaton errors nclude addtonal hgh-order terms whch nfluence flud moton and appearance. We start wth the smple onedmensonal advecton to analyze the mpact on ¼ 0; u > 0 ð4þ If we dscretze t usng forward Euler for the tme dervatve and frst-order backward dfference for the space dervatve we can get: q nþ1 q n þ u qn q n 1 ¼ 0 Dt Dx q nþ1 We can rearrange t to get ¼ q n Dt qn q n 1 u Dx Recallng the Taylor seres for q n 1 gves q n 1 ¼ n 2! n ð5þ ð6þ Dx 2 2 þ OðDx3 Þ ð7þ Substtutng t nto above equaton and dong the cancelaton gves q nþ1 ¼ q n n n u þ q u þ OðDx 2 2 Deletng the second-order truncaton error and rewrtng t gets q nþ1 q n þ n ¼ Dx q 2 ð9þ Whch s the forward Euler n tme appled to the ¼ 2 ð10þ The Laplacan of q n one dmenson s r 2 q 2 q=@x 2. Defnng m 0 ¼ udx and substtutng t nto the ¼ m0 r 2 q ð11þ

3 12 Z. Huang et al. / Graphcal Models 78 (2015) The modfed advecton equaton ncludes an addtonal vscosty-lke term. It s called artfcal vscosty as t s purely of numercal orgn wthout any physcal meanngs. Even though the flud s nvscd, the artfcal vscosty performs lke physcal vscosty to make the flud vscous. 4. Combatng numercal dsspaton 4.1. Grd varaton The aforementoned artfcal vscosty ndcates the numercal dsspaton s proportonal to grd spacng Dx. In theory, we could reduce the numercal dsspaton by refnng the grd spacng. However, refnng grd dramatcally ncreases computatonal cost. Takng the wdely used MAC grd for example, f the grd resoluton ncreases by c tmes n three dmensons, the number of grd cells ncreases up to c 3 tmes. As the grd spacng decreases by c, tme step has to reduce to 1=c to meet Courant Fredrchs Lewy (CFL) condton (even for the uncondtonally stable sem-lagrangan method, the tme step s usually set to be several tmes of CFL number n practce). The total computatonal cost rses up to c 4 tmes, whch makes the grd refnement method scale poorly and ll-suted for practcal use. To allevate the problem, Berger and Olger [14,15] ntroduced adaptve mesh refnement (AMR) nto CFD to dscretze local regons wth dfferent grd resolutons. Losasso et al. [3] proposed an octree structure to adaptvely resolve rregular boundares wth hgh-resoluton grd whle usng low-resoluton grd at empty space. The method dramatcally mproves vsual effects around rregular boundares wth a small ncrement of computatonal cost. Fg. 1 shows vsual comparson between the basc method and the octree-based method wth smlar grd resolutons. As the grd s adaptvely refned around rregular boundary, the method captures more vsual detals than tradtonal unform grd method. However, t ncreases the complexty of grd structure. The addtonal computatonal cost pays off only when grd refnements are lmted to a few local regons. In addton, non-unform grd constructs a nonsymmetrc lnear system to solve for pressure, whch may cause potental numercal nstablty. As hexahedral grd does not algn to horzonal and vertcal boundares, t cannot resolve rregular boundares that do not algn to coordnate axes. Tetrahedral grd s proposed to represent rregular boundares wth much less numercal errors. Elcott et al. [4] dscretzed and solved NSEs on tetrahedral grd. The method can present complcated boundares but has problem of energy dsspaton, whch can be modfed usng mplct tme-reversble velocty ntegraton on the tetrahedral grd [16]. Tetrahedral grd has ntrnsc complcated structure to ncrease the computatonal cost. A hybrd method s proposed to combne the hexahedral and tetrahedral structures [5]. The tetrahedral grd s used to resolve rregular boundares whle the hexahedral grd s used at empty regons wthout obstacles. Fg. 2 gves vsual comparson of smoke smulatons usng dfferent dscretzaton schemes. As most nterestng features are generated around rregular boundares, the hybrd grd s able to obtan smlar vsual qualty as tetrahedral method near boundares. As the hybrd grd uses unform grd mesh n most empty spaces, t reduces the total computatonal cost comparable wth the unform grd method. Instead of the statc grd structure, several works reduce computatonal overhead by translatng and reconstructng grd mesh accordng to boundary condtons. Shah et al. [6] tracked smoke movement wth movable and scalable grd mesh. Empty regons wthout smoke are not dscretzed n order to decrease computatonal cost. To guarantee vsual consstence between smulaton doman and empty regons, Cohen et al. [17] used a smple partcle system outsde of the smulaton doman. Velocty at doman boundary s assgned as ntal condtons of the smple partcle system. Zhu et al. [18] extended the unform grd to create a large far-feld grd. The unform grd mantans fne resoluton to capture detaled features n nterestng regons whle extended coarse grd s used to obtan large feld vew of regons that does not requre fne presentaton. Fg. 3 compares three dfferent grd confguratons. In left fgure, the grd s translated along bulk moton of smoke. Only regons flled wth smoke are computed to reduce computatonal overhead. The smoke regon expands as smoke moves around, whch can be parttoned nto two subregons as llustrated n the mddle fgure. The smoke subregon s calculated usng physcal methods to capture major body of smoke whle the less mportant subregon s modeled wth smple non-physcal methods to reduce computaton. In order to smulate smoke from a Fg. 1. Vsual comparson of smoke smulatons usng octree grd (left) [3] structure and MAC (rght) [7].

4 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 2. Smoke smulatons usng (a) tetrahedra mesh, (b) hybrd mesh, and (c) hexahedra mesh [5]. Fg. 3. Left: The fully bounded translatng grd [6]; mddle: partally bounded translatng grd [17]; rght: fully bounded translatng grd wth dfferent resolutons [18]. far dstance, the rght fgure extends coarse cells around fne grd to obtan a large-feld vew whle mantanng fne resoluton at local regons of nterest. Other than translatng and extendng the grd mesh, Feldman et al. [19] and Klngner et al. [20] dynamcally deformed tetrahedral grd mesh to adapt to varable boundares and movng obstacles. All adaptve grd methods have common drawbacks of hgh complcated structure and hgh computatonal cost. A full descrpton of truncated error ncludes both Dx and Dt terms. Reducng tme step also reduces numercal dsspaton at the cost of ncreasng computatonal overhead. In smoke smulaton, we have to carefully choose the tme step that does not produce strange results and breaks the numercal stablty Hgh-order backward tracng To get the prevous poston where current quantty ends up at nqured grd pont at the next tme step, the basc advecton employs the forward Euler scheme to trace the trajectory n reverse to calculate prevous poston, leadng to certan numercal dsspaton. Some hgh-order schemes such as Runge Kutta can be used to mprove 4.2. Advecton mprovement The sem-lagrangan advecton s predomnant attrbuted to ts smplcty and uncondtonal stablty. The method s wdely used as the basc blocks to construct other advanced advecton schemes. The basc sem- Lagrangan advecton s gven as: u 0 ðx; t þ DtÞ ¼u n ðx u n ðx; tþdt; tþ ð12þ whch means that the velocty at locaton x at next tme step s the velocty at locaton ðx u n ðx; tþdtþ at current tme step t. As current velocty s always bounded to prevous veloctes, t guarantees uncondtonal stablty. Fg. 4. 2D smoke smulatons usng the sem-lagrangan advecton coupled wth the forward Euler (left) and the second-order Runge Kutta (rght) methods.

5 14 Z. Huang et al. / Graphcal Models 78 (2015) the accuracy. Brdson [1] recommended a second-order Runge Kutta for backward poston tracng. Fg. 4 shows the vsual results usng the frst-order forward Euler and the second-order Runge Kutta. The Runge Kutta scheme produces more curly detals at local regons. The backward tracng only ntroduces a fractonal part of numercal dsspaton to the entre smulaton. Boostng accuracy of the backward tracng wth hgh-order schemes lmts vsual mprovement to the fnal results. In addton, the hgh-order schemes requre more computatonal cost. The second-order Runge Kutta s regarded as a default scheme to balance numercal accuracy and computatonal cost Hgh-order nterpolaton Most lkely the prevous poston s not on grd ponts where quanttes are defned and stored. We have to nterpolate nearby grd ponts to obtan a good approxmaton. The lnear nterpolaton s predomnant due to ts smplcty and stablty, but t suffers from massve numercal dsspaton because the lnear nterpolaton s smlar to the low-pass flter. Catmull Rom nterpolaton has secondorder numercal accuracy, but t lkely overshoots locally and drops nto an unstable feedback loop to make smulaton blow up fnally. Smply preventng overshoot produces addtonal numercal dsspaton n the vcnty of local mnma and maxma [1]. Fedkw et al. [7] clamped slopes to zero where slope sgns are flpped. The modfed scheme s locally monotone to guarantee stablty and second-order accuracy. However, the method s excessve snce t flattens overshoots whch may be just fne. Huang et al. [21] proposed to adaptvely flatten overshoots that break global bounds n order to mantan dversty of flud features. As t does not requre full-scale suppresson, the scheme also reduces computatonal overhead. A vsual comparson of smoke usng lnear, modfed, and adaptve Catmull Rom nterpolatons s shown n Fg. 5. There are numerous hgh-order nterpolatons n CFD [22], but they are not wdely used n smoke smulaton for several reasons. As all quanttes on grd ponts requre nterpolaton, a small computatonal ncrement of nterpolaton wll sgnfcantly ncrease total computatonal overhead. In [7], tme cost of the monotonc scheme s about 18 tmes larger than the lnear nterpolaton. Even though the adaptve scheme reduces computatonal cost, t s about 3 4 tmes larger than the lnear scheme [21]. In addton, the hgh-order nterpolatons requre more ponts to construct stencl, whch makes t dffcult to handle nner boundary condtons. Wde stencl s also complcated to handle adaptve and non-unform grds. Hgh-order nterpolatons requre lmters to avod oscllatons, new extrema, and possble nstablty [23]. Smlar to backward tracng, nterpolaton s not the major contrbuton to the total numercal dsspaton, but t lkely ncreases total computatonal overhead. We have to choose nterpolaton scheme based on numercal accuracy versus runtme performance trade-off curve Hgh-order advecton The basc sem-lagrangan advecton s popular for smoke smulaton. However, excessve numercal dsspaton makes t ll-suted for smulaton that requres hgh vsual qualty. Km et al. [24,25] appled the Back and Forth Error Compensaton and Correcton (BFECC) to the sem-lagrangan Courant Isaacson Rees (CIR) to obtan second-order accuracy both n tme and space. Forward and backward sem-lagrangan schemes are compared to estmate numercal error of a sngle sem-lagrangan advecton, whch s then used to compensate a thrd forward sem-lagrangan advecton. Fg. 6 shows vsual comparsons usng the sem-lagrangan and the BFECC advectons. The BFECC generates much more detaled features that the basc sem-lagrangan scheme, especally vortcal features that are swept out by the numercal dsspaton. The backward advecton mples that the equaton s able to evolve backward, whch s napproprate for parabolc and non-reversble partal dfferental equatons. The BFECC requres great computatonal cost as t calculates the sem-lagrangan advecton three tmes. Selle et al. [23] proposed a modfed MacCormack scheme whch requres much less computatonal cost. In the BFECC method, the error s treated as a current tme quantty so that t s advected forward to the next tme step for compensaton. Snce no strong proof shows that t s a quantty related to the current step, the error can also be used to correct the frst forward sem-lagrangan advecton. Fg. 7 llustrates the vsual comparson of smoke usng the basc sem-lagrangan and the modfed MacCormack methods. The MacCormack scheme generates more detals than the sem-lagrangan method. Comparng vsual mprovements usng hgh-order nterpolatons and ntegrators n Fgs. 4 and 5, boostng numercal accuracy of advecton schemes s able to obtan greater mprovement. It also reduces computatonal overhead as t does not requre a thrd advecton. Snce the BEFCC and the modfed MacCormack are both constructed based on the sem-lagrangan scheme, the two Fg. 5. Smoke ball collson smulatons usng lnear scheme (left), mono Catmull Rom (mddle), and adaptve Catmull Rom nterpolatons (rght) [21].

6 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 6. Smoke smulatons wth the sem-lagrangan (top) and the BFECC methods (bottom) [25]. Fg. 7. Smoke smulatons wth the sem-lagrangan (left) and the modfed MacCormack methods (rght) [17]. methods are uncondtonally stable. However, the back and forth round step makes them napproprate to work near nterface of multphase flows. It may not conserve momentum f the frst forward advecton goes across to the other sde flud wth dfferent densty and then comes back wth then same velocty. The same problem happens near boundares of obstacles and computatonal doman. It s sutable to replace them wth the sem-lagrangan scheme near these regons. In the sem-lagrangan method tme step s normally set to be less than 5 tmes of CFL condton number, so t s safe to turn off the BEFCC and the modfed MacCormack at 5Dx off the nterface and boundares. Other than advectng velocty, Molemaker et al. [26] proposed the QUICK scheme to advect velocty fluxes. QUICK [27] s a thrd-order accurate explct scheme. It produces numercal dsspaton only at fnest spatal scale that can be resolved by the grd. Fg. 8 llustrates the vsual results wth dfferent advectons. The QUICK scheme generates much more detals than the basc sem-lagrangan method, especally on hgh-resoluton grd.

7 16 Z. Huang et al. / Graphcal Models 78 (2015) The frst-order forward Euler s unstable for the QUICK [28]. It can be replaced wth other temporal schemes wth large stablty domans such as second-order Adams Bashforth (AB2) [29]. As the QUICK s condtonally stable only when t meets CFL condton, t greatly ncreases computatonal cost. Attrbuted to ts regular memory access pattern, t normally requres less computatonal cost than other sem-lagrangan based schemes wth careful memory access. Apart from advectng velocty flux, several methods use constraned nterpolaton profle (CIP) [30,31] to advect both the velocty and ts dervatves, whch can boost numercal accuracy to thrd order. However, orgnal CIP requres a compact stencl and t s condtonally stable wth tght CFL condton. The method does not consder dervatves at farthest cell corner, so t only guarantees numercal accuracy when backward-traced pont s not far from the start corner. To mprove the numercal stablty, Song et al. [32] proposed the monotonc CIP (MCIP) by suppressng potental overshoots along all dmensons. The excessve suppresson obtans uncondtonal stablty but ntroduces some numercal dsspaton. The dmensonsplttng strategy also dramatcally ncreases computatonal cost [33]. As splttng computaton s related to axs drectons, t normally produces some numercal dsspaton. The problem s even serous n shear motons [34]. Km et al. [35] proposed an unsplt sem-lagrangan CIP (USCIP) to guarantee uncondtonal stablty. Two addtonal symmetrcal terms are added to consder off-axs movements such as shear and rotaton to avod unnecessary wggles. It s able to reduce the numercal dsspaton wth relatve low computatonal overhead. Fg. 9 compares vsual results by usng varous advecton schemes. The BFECC and the MCIP obtan more compellng vsual results than the basc sem-lagrangan schemes, whle the USCIP method produces hgher vsual qualty than other methods. CIP-based schemes advect both velocty and dervatves. Ther computatonal cost s comparable wth the BFECC method, but much hgher than the basc sem- Lagrangan method. As USCIP does not requre dmenson splttng, t requres less computaton than the BFECC and the USCIP methods. To llustrate how the numercal dsspaton dsperses and smears out detals, we compared several uncondtonal stable advectons usng the standard Zalesak dsk test [36]. The benchmark s a grd wth a slotted dsk centered at ð50; 75Þ of 15 cells n radus, 5 cells n wdth, 25 cells n length of slot and 5 cells of an upper brdge. A constant vortcty velocty feld s gven as [37]: u ¼ðp=314Þð50 yþ v ¼ðp=314Þðx 50Þ ðaþ ðbþ ð13þ so that the slotted dsk fnshes one revoluton every 628 tme steps. Fg. 10 shows results of one revoluton wth dfferent advecton schemes. An accurate advecton scheme s able to translate and rotate nterface wthout dstorton and deteroraton, especally at nterface wth hgh curvature. The basc sem-lagrangan method has too much numercal dsspaton to mantan basc shape of the dsk. The BFECC and MacCormack schemes keep generally shape but smooth slots and corners. The MCIP scheme conserves the dsk shape wth certan eroson at nner and outer corners, whle USCIP almost suffers no dstorton and area loss. Table 1 lsts area loss of the dsk wth dfferent schemes and grd confguratons. The loss column represents percentage of the area loss. Negatve value means the fnal dsk s smaller than the orgnal dsk whle postve value ndcates the dsk grows. Wth the same grd confguraton, hgh-order schemes such as the USCIP suffer from much less area loss than low-order schemes such as the sem- Lagrangan. Wth the same advecton, the area loss also decreases as grd resoluton ncreases. As hgh-order schemes are able to suppress numercal dsspaton even at relatve low-resoluton grds, the mprovement gan s not sgnfcant by ncreasng grd resolutons when usng hgh-order schemes. Whle frst-order sem-lagrangan method has too much numercal dsspaton on coarse grd, usng fne grd sgnfcantly reduces total numercal dsspaton. We can also fnd that the amount of loss wth the sem-lagrangan method drops relatvely much faster than that of hgh-order schemes, whch ndcates that low-order schemes are much more senstve to the grd resoluton than hgh-order schemes. Table 2 compares dscussed advectons n terms of numercal stablty, numercal accuracy, and computatonal cost. Snce the backward tracng and nterpolaton schemes are orthogonal, aforementoned hgh accurate trackng schemes n Secton 4.2.1, hgh-order nterpolatons n Secton and hgh-order advectons n Secton can be ntegrated together to further mprove the numercal accuracy. In addton, other smoke quanttes ncludng densty, temperature, and concentraton can also be advected wth above methods to reduce total numercal dsspaton. However, f hgh-order schemes are excessvely used for all quanttes, hgh-frequency components of quanttes wll accumulate smultaneously to produce hghly sharp Fg. 8. Smulatons usng the sem-lagrangan advecton (left) and the QUICK advectons wth low- and hgh-resoluton grd confguratons [26].

8 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 9. Smoke smulatons wth varous advecton schemes [35]. Fg. 10. Zalesak dsk after one revoluton advected by dfferent schemes [38]. Table 1 Area loss of dsk wth dfferent advecton and grd confguratons (ntal area = 5.82e 2). Sze Sem-Lag. MacCormack BFECC QUICK USCIP Area Loss (%) Area Loss (%) Area Loss (%) Area Loss (%) Area Loss (%) e e e e e e e e e e e e e e e e e e e e Table 2 A lst of several advectons for smoke smulaton. Advecton Numercal stablty Numercal accuracy Computatonal cost Representatve reference Sem-Lagrangan Uncondtonal stable Frst-order Low [2] BFECC Uncondtonal stable Second-order Hgh [24] MacCormack Uncondtonal stable Second-order Medum [23] QUICK Condtonal stable Thrd-order Hgh [26] MCIP Uncondtonal stable Thrd-order Hgh [32] USCIP Uncondtonal stable Thrd-order Medum [35] nterface. For nstance, Fg. 11 shows two 2D smoke smulatons usng dfferent advecton confguratons. Left mage shows result usng a combnaton of the modfed MacCormack scheme for velocty and the sem-lagrangan for densty; rght mage uses the modfed MacCormack scheme for both velocty and densty. Although the rght one has much less numercal dsspaton for velocty and densty, they accumulate to produce clear sharp nterface, whch makes t artfcal aganst the real smoke Sub-grd compensaton The numercal dsspaton ncreases the vscosty to make smoke nadequate of hghly energetc and turbulent appearance. Rather than drectly reducng the numercal dsspaton by changng dscretzaton schemes and smulaton algorthms, many works focus on how to add small-scale features back wth heurstc models Vortcty confnement Fedkw et al. [7] ntroduced the vortcty confnement method [39] nto smoke smulaton. The method njects back a certan amount of energy to recover turbulent features by addng a tuneable body force. The force s perpendcular to the drecton of vortcty maxmum magntude to enhance local rotaton. It s desgned to be lnear proportonal to the grd spacng so that the modfed NSEs degenerate to the orgnal NSEs as the addtonal term vanshes n the lmt. Selle et al. [11] coupled the vortcty confnement wth vortcty equaton. The vortcty confnement force can be syntheszed drectly from current vortcty feld. The left two mages of Fg. 12 compare vsual results usng the basc and the vortcty confnement methods. Injectng some vortcal force back nto flow sgnfcantly mproves vsual effects, but t also ntroduces some random artfcal turbulence at local regons. The vortcty confnement s an nexpensve approach to recover small features elmnated by the numercal

9 18 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 11. Usng hgh-order schemes to advect all quanttes produces sharp nterface. Left smulaton advects velocty wth the MacCormack and densty wth the sem-lagrangan scheme; rght one advects both velocty and densty wth the MacCormack scheme. dsspaton. However, the method cannot determne how much energy should be njected back wthout causng nstablty. Fg. 12 gves comparsons by njectng dfferent amount of vortcty confnement forces. The smulaton becomes quas-unstable and degenerates to a random turbulent chaos when too much force s added. The problem les n the fact that the coeffcent s constant over the entre smulaton doman. To solve the problem, He et al. [8] presented an adaptve vortcty confnement coeffcent accordng to helcty other than user-defned constant value. Fg. 13 llustrates the vsual comparson between the constant and varable vortcty confnement methods. They further developed a robust second vortcty confnement method [40] to guarantee the numercal stablty even wth large confnement coeffcents. Vortcty confnement methods add knetc energy to dlute energy dffuson caused by the numercal dsspaton, but they cannot fully compensate for excessve numercal dsspaton [26]. It s napproprate for low vscous flud that nerta s so well conserved that flud moton can propagate for a sgnfcant dstance wthout beng damped and broken down Subscale turbulence model Many other methods use hgh-level turbulence models to compensate for sub-grd detals loss due to the numercal dsspaton. The basc mean flow s smulated on coarse grd whle turbulent detals are syntheszed usng procedural synthess methods. The methods can be descrbed as [41]: u ( NSðUÞSTðu 0 Þ ð14þ where NS s a basc flud smulator to calculate mean velocty U on the coarse grd, ST s a syntheszer to generate turbulent feld u 0 and s the ntegraton operaton to couple the two felds together. Early methods [42 45] used Kolmogorov spectrum to generate pseudo-random turbulence n frequency space. These methods have a common drawback of generatng turbulence at ncorrect regons. Recently curl operaton [46] s appled on Perln [9,47] and wavelet vector noses [48] to generate turbulent components at varous frequences and scales. Energy cascade s modeled to nclude spatal dstrbuton and turbulence dynamcs usng local assembled wavelets [48], lnear k e equaton [47], oneequaton [9], and complete tow-equaton of k e [49,10]. Instead of drectly layerng turbulence components on the basc flow, Zhao et al. [41] modeled fluctuatons as controllable turbulence force to agtate the basc flow. The method guarantees temporal consstence wth the basc NSEs smulators. Rather than calculatng the basc flow, Gregson et al. [50] employed a mult-scale trackng method to reconstruct temporally coherent velocty feld from pror low-resoluton densty mages. However, the Fg. 12. Smoke smulatons wth dfferent amount of vortcty confnement forces. From left to rght the coeffcent s 0 (wthout vortcty confnement), 0.25, 0.5 and 2.0 respectvely [11].

10 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 13. Smulatons usng the vortcty confnement methods wth constant (rght) [8] and varable coeffcents (left) [7]. method requres fne tunng to obtan good balance between reconstructed detals and nose levels. Fg. 14 gves vsual comparsons between dfferent turbulence enhancement methods. It shows that more turbulent detals are generated when more turbulence components are added to the basc flow. However, smoke appears chaotc f too much turbulence components are added. In order to obtan realstc detaled features, tral and error effort are requred to determne the amount of turbulence components. Procedural turbulence methods decouple hghfrequency components from the basc flow and use syntheszed small-scale turbulence to compensate for fne feature loss caused by the numercal dsspaton. The methods enable anmaton artsts to generate a fast basc smulaton and then add detals wthout changng gross moton, whch helps the artsts to shorten the turnaround. However, as the fnest scale s determned by local velocty and temporal averagng rather than the grd resoluton, Usng a much hgh-resoluton grd s able to produce more realstc vsual results than the non-physcal synthess methods. Addtonally, the temporal averagng operatons may smooth out vsual appearance to some extent Invarants conservaton Another baselne of reducng the numercal dsspaton s to conserve basc flud nvarants. In rgd and deformable body smulatons, great mprovements can be acheved by followng the lnear and angular momenta conservaton prncples. Smlar strategy can be appled n flud smulaton by conservng nvarants ncludng vortcty, crculaton, and energy. Elcott et al. [4,51] proposed to conserve crculaton along arbtrary smplcal mesh. As vortex s an mportant vsual cue of smoke, the method uses the back-trackng ntegrals to solve the vortcty formulaton of NSEs. The method preserves dscrete crculaton on grd mesh to acheve no vortcty numercal Fg. 14. Vsual effects of dfferent turbulent enhancement methods. (a) Orgnal coarse smulaton; (b) wavelet subgrd turbulence; (c) controllable and ntermttent turbulence method; (d) Add vortcty confnement to (a); (e) wavelet turbulence to (d); (f) (h) controllable and ntermttent turbulence method wth q = 0.8, 0.2 and 0.1 respectvely, where q s control parameter to calculate resultant velocty feld wth qu þð1 qþu 0 [41].

11 20 Z. Huang et al. / Graphcal Models 78 (2015) dsspaton. As shown n Fg. 15, the conservaton method s able to produce smlar vsual results as real smoke whle the basc sem-lagrangan method loses most fne features due to the excessve numercal dsspaton. Snce the method s constructed based on the sem- Lagrangan advecton, t s also uncondtonal stablty. In addton, the vortcty-based smulaton requres no pressure calculaton. Intrnsc storage wthout reference to global and local coordnate system constructs a very sparse lnear system, whch greatly decreases computatonal cost comparable wth the sem-lagrangan method. However, the method preserves the crculaton but does not conserve energy. Although a L 2 projecton [51] can be employed to preserve the total vortcty energy, the global operaton propagates nfluence to the entre doman. In addton, the projecton nduces artfcal vsual effects around boundares. In flud smulaton, the energy loss relates to tme step, grd spacng, and total frame numbers. The factors are coupled n the smulaton course, makng t dffcult to elmnate energy dsspaton by tunng each sngle term. Besdes, the computatonal cost s senstve to the spatal resoluton and the tme step. In order to preserve energy wthout sgnfcantly ncreasng computatonal cost, Mullen et al. [16] proposed a fully Euleran ntegraton scheme ndependent of spatotemporal resolutons. The energy conservaton scheme [52,53] and the non-dsspatve advecton ensure no energy loss on arbtrary smplcal grds. In addton, t guarantees uncondtonal numercal stablty. Fg. 16 shows energy dsspaton of dfferent schemes over tme. The energy s tghtly related wth the smoke moton. Hgh energetc smoke generally obtans rch and long-tme actve detals. As shows n the fgure, the basc sem-lagrangan method dsspates most energy over tme, whle the energy conservaton method almost dffuses no energy, whch ndcates t does not have the numercal dsspaton problem. The Euleran ntegraton preserves energy exactly over tme. However, t nevtably suffers from certan numercal dffuson due to the low-pass flterng operatons of resamplng on the dscrete grd, whch s a common drawback of grd-based methods Partcle based methods Pure Euleran methods start wth a feld sampled on grd ponts and end wth advected feld resampled on the grd at each step. The Euleran methods may reduce numercal dsspaton to some extent, but suffer from the fundamental problem of feld resamplng: the scale of Fg D vortces mergng comparsons between the real expermental result (a), the conservaton method (b), and the basc sem-lagrangan method (c) [4].

12 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 16. Most schemes dsspate a sgnfcant amount of energy over tme whle the fully Euleran scheme preserves energy exactly [16]. fnest features s lmted by the Nyqust frequency of the grd. Even for a pure translaton velocty feld, the hghest spatal frequency that can be relably advected has perod of 4Dx [1]. A good Euleran scheme may flter out hghfrequency components as a smoothng operaton whle a bad Euleran scheme may further generate vsual artfacts. In contrast to the Euleran methods, the Lagrangan methods employ dscrete partcles to carry varables along wthout dsspaton. There s no flterng and dsspaton loss n advecton. From ths pont of vew, Lagrangan partcle methods are perfect for advecton to elmnate numercal dsspaton. In fact, there are already some pure partcle methods for flud smulaton, such as smoothed partcle hydrodynamcs (SPH) [54] and some pure vortex partcle methods [55]. Several vortex partcle methods employ geometry structures to descrbe vortex structure ncludng flament [56,57], rng [58,59] and sheet [60] to capture varous subtle features such as leapfroggng vortex rngs and vortex sheddng, whch are dffcult to smulate usng the grd-based methods. However, apart from ther specfed problems such as partcle redstrbuton, pure partcle methods cannot enforce ncompressble condton as effcent and accurate as the grd methods. Besdes, the partcle-based methods requre sophstcated partcle dstrbuton to obtan good performance and qualty. A preferable remedy s to add Lagrangan machnery to the Euleran scheme n a hybrd way. Selle et al. [11] combned Lagrangan vortex partcle and Euleran grd together to overcome the weakness of both methods. The vortex partcles are used to carry and preserve vortcty wthout dsspaton loss whle the Euleran grd s employed to calculate velocty and vortcty confnement force [7] to recover swrlng detals. As the vortex partcles are only used to ncrease vsual detals, no partcle dstrbuton s requred. Yoon et al. [61] proposed a smlar scheme wth combnaton of the vortex partcles and Euleran grd. Basc flow s calculated on relatve coarse grd to reduce the computatonal cost. Partcle vortcty s transferred to a hgh-resoluton vortcty feld to obtan fne detals. The method uses a kernel functon to calculate vortcty feld on hgh-resoluton grd ponts, whch cannot guarantee the vortcty feld free of dvergence. In addton, as all vortex partcles make ther contrbutons to vortcty forces, t lkely syntheszes strong vortcty forces to produce vsual artfacts. The vortex partcles are sutable to preserve swrlng features but requre addtonal computatons to nduce velocty from vortcty. A straghtforward modfcaton s to drectly store velocty on partcles. However, t s far from enough as pressure projecton requres velocty nformaton of the whole doman, whch means we cannot just consder ndvdual partcles but account for partcle partcle nteractons on top of advectons. Partcle-n-cell (PIC) [62] s a good strategy to couple partcle and grd whle solvng the pressure projecton problem. Velocty and other quanttes are defned and advected on partcles to elmnate the numercal dsspaton, but they are calculated and adjusted by pressure gradents on the grd ponts. The method reduces the numercal dsspaton of advecton but ntroduces even more to the total numercal dsspaton of the smulaton because quanttes are averaged to the grd ponts from the partcles and then nterpolated back to the partcles from the grd ponts. Zhu and Brdson [12] proposed the flud mplct partcle (FLIP) method to reduce the total numercal dsspaton. The FLIP nterpolates the changes rather than the quantty values on the

13 22 Z. Huang et al. / Graphcal Models 78 (2015) Fg. 17. Smoke smulatons wth sem-lagrangan (left), PIC (mddle), and FLIP (rght) schemes [63]. Fg. 18. The applcablty of dfferent types of methods n several typcal flud smulaton scenaros. The applcablty s classfed nto four levels wth weak, moderate, sutable, and strong from nsde to outsde. grd ponts. It adds the changes to the quantty value other than replacng t. The changes are smoothed but not accumulated to acheve almost no numercal dsspaton. Fg. 17 shows the vsual comparson of smoke usng dfferent methods. The FLIP method obtans much more vsual detals than both the basc sem-lagrangan method and the PIC method. As only changes are transferred n the FLIP method, the velocty fluctuatons on partcles may average down to zero on the grd ponts and then show up as unexpected perturbatons at other tme steps, leadng to certan nose. Snce PIC has no such problem, a better soluton s to combne them together wth a tunable regularzaton parameter to decay the nose and suppress the numercal dsspaton. 5. Concluson In ths paper, we dscussed the numercal dsspaton problem for smoke smulaton n computer graphcs communty. The numercal dsspaton s wdely regarded to have vsual consequences of vsual results. Many methods have been developed to solve the problem. The methods reduce numercal dsspaton or compensate for vsual loss from varous aspects. Most methods fall nto three categores: Decreasng, Conservng, and Increasng. Table 3 compares methods n the categores n terms of computatonal cost, vsual qualty, mplementaton, adaptablty, and scalablty. The adaptablty ndcates whether the method s adaptve to dfferent applcaton scenaros. The scalablty measures how t s easy to ntegrate the method wth exstng applcatons. The followng lst gves the common characterstcs of methods n each category, whch has ts advantages and dsadvantages to make them sutable for dfferent applcaton scenaros ncludng hghqualty vsual effect, real-tme performance, anmaton control, dynamc doman, obstacle-coupled nteracton, and large-scale smulaton, as llustrated n Fg. 18. Decreasng : Solvng the problem by reducng the amount of numercal dsspaton. The grd varaton methods n Secton 4.1 and the advecton mprovement methods n Secton 4.2 fall nto ths category. The decreasng methods normally use fne-resoluton dscretzaton and hgh-order numercal schemes to reduce the total amount of the numercal dsspaton. As grd

14 Z. Huang et al. / Graphcal Models 78 (2015) Table 3 Method comparson n terms of computatonal cost, vsual qualty, mplementaton, adaptablty, and scalablty. Category Method Computatonal cost Vsual qualty Ease to mplement Adaptablty Scalablty Decreasng [3] Low Moderate Moderate Moderate Moderate [4] Hgh Hgh Moderate Hgh Low [5] Moderate Hgh Moderate Hgh Low [6] Low Low Easy Low Low [17] Low Moderate Easy Moderate Low [18] Moderate Hgh Moderate Hgh Moderate [19] Hgh Hgh Hard Hgh Low [20] Hgh Hgh Hard Hgh Low [1] Low Low Easy Hgh Hgh [7] (mono. nterp.) Moderate Moderate Easy Hgh Hgh [21] Low Moderate Easy Hgh Hgh [24,25] Hgh Moderate Easy Hgh Hgh [23] Moderate Moderate Easy Hgh Hgh [26] Moderate Hgh Easy Hgh Hgh [32,33] Hgh Moderate Moderate Moderate Moderate [35] Moderate Hgh Moderate Hgh Hgh Conservng [4,51] Hgh Hgh Hard Moderate Low [16] Hgh Hgh Hard Hgh Low [12] Moderate Hgh Easy Hgh Hgh [11] (vort. part.) Moderate Moderate Moderate Moderate Hgh [61] Hgh Moderate Moderate Moderate Hgh Increasng [7,11] (vort. confn.) Moderate Moderate Moderate Hgh Hgh [8,40] Moderate Moderate Moderate Hgh Hgh [46] Moderate Moderate Easy Hgh Hgh [47] Moderate Moderate Moderate Moderate Moderate [48] Hgh Moderate Moderate Moderate Moderate [9] Hgh Moderate Hard Moderate Hgh [10,49] Hgh Hgh Hard Moderate Hgh [41] Hgh Hgh Moderate Hgh Hgh [50] Hgh Hgh Hard Low Low resoluton s drectly related to the fnal vsual effect, grd varaton methods have strong applcablty n applcatons wth hgh-qualty effect requrement. The grd-based methods are flexble to handle boundary and obstacle problems, makng grd varaton methods sutable for applcatons wth dynamc doman and obstacle-coupled nteracton. However, great efforts are requred to mplement spatal dscretzaton and guarantee the numercal stablty on rregular grd structures. The heavy computatonal cost also degrades the runtme performance, makng them ll-suted for realtme applcatons. As computaton s exponentally proportonal to the doman dmensons, tremendous computatons and memory storage are requred for large-scale smulatons. Besdes, most calculatons and storage strategy tghtly depend on the grd structures. Changng grd structures requres great changes n the smulaton, whle the advecton mprovement methods are generally orthogonal to other subroutnes n the ppelne, the modfcatons wll not mpact other subroutnes, so the advecton mprovement methods are much easer to ntegrate wth exstng applcatons than grd varaton methods. In addton, the advecton mprovement methods are compatble wth the grdbased methods. They are feasble to handle the dynamcal doman and obstacle boundary problems wth the help of grd-based methods. Comparng wth the grd varaton methods, advecton mprovement methods mprove the vsual qualty wth relatve less computatonal overhead. They are preferable for real-tme applcatons that do not requre very hgh-qualty results. Conservng : Solvng the problem by conservng ndrect flud quanttes. The nvarants conservaton methods n Secton 4.4 and the partcle methods n Secton 4.5 are two typcal conservng methods, whle they preserve quanttes n dfferent ways. The nvarants conservaton methods ntegrate ndrect flud quanttes (e.g. crculaton and energy) on specfc structures (e.g. smplcal mesh) to avod smulatng drect quanttes that greatly mpact flud motons. As drect quanttes such as velocty are not drectly nvolved n calculatons that produce the numercal dsspaton, the numercal dsspaton s sgnfcantly reduced, makng them especally sutable for applcatons wth hghqualty requrement. Especally, as no energy s dsspated for the energy conservaton methods, smoke remans energetc to guarantee long-tme performances. The nvarants conservaton methods are usually constructed on the grd structures, so they may nhert the advantages of the grd varaton methods. However, they need sophstcated strateges to guarantee conservaton, whch mpedng ther adapton and stablty to other applcaton scenaros. The partcle methods preserve quantty by leveragng the conservaton characterstc of the partcle. The partcle-based methods have totally dfferent smulaton ppelnes from grd-based methods, leadng to dfferent applcabltes. The partcle-based methods are dffcult to handle boundary problems such as obstacle nteracton and dynamc doman. Hybrd partcle and grd methods allevate the problems. The partcle based methods do not requre some computatonal ntensve subroutnes,

15 24 Z. Huang et al. / Graphcal Models 78 (2015) whch reduces the computatonal cost to make them sutable for realtme applcatons. Comparng to the grd pont number of grd-based methods, partcle number s more relaxant to the doman dmensons, makng partcle methods applcable to large-scale smulatons wth less computatonal cost. Increasng : Solvng the problem by ncreasng addtonal components. The compensaton methods n Secton 4.3 are typcal ncreasng methods as they add components back nto flow to synthesze detaled features. The vortcty confnement methods are usually drectly ntegrated nto the grd methods as an addtonal term. They need no modfcaton to orgnal grd structure and do not mpact orgnal smulaton. As computatonal cost s nexpensve, the vortcty confnement methods are able to ncrease dversty of flow moton wth much few efforts. They have the same applcablty as the grd-based methods. Most sub-grd compensaton methods employ the grd-based methods for the basc flow smulatons, so they have smlar abltes to the grd-based methods. However, the requrement to guarantee consstence between the basc flow and the sub-grd synthess flow degrades ther abltes. Especally, computatonal cost ncreases sgnfcantly when complcated turbulence models are used on sub-grd structure to generate turbulent detals. A great advantage of the ncreasng methods s that the synthess procedure s decoupled from the basc smulaton. The synthess procedure can be treated as a postprocessng wthout nfluencng the gross moton, whch makes the ncreasng methods sutable for anmaton control. However, As turbulent detals are generally generated usng synthess methods, the detals may appear artfcal and less comparable wth the grd-based methods. In most tme, we struggle to elmnate the numercal dsspaton, whle occasonally we leverage t to guarantee the numercal stablty. As smoke s typcally smplfed as nvscd flud, there s no vscous frcton at the dsspatve range of energy cascadng. If there was no numercal dsspaton actng as artfcal vscosty, energy would ple up unbounded at the fnest scales (wth hghest wavenumber as k ¼ 2p=Dx) and cause numercal nstablty [49]. Addtonal dsspatons are requred to mantan numercal stablty. The popular sem-lagrangan scheme s so much dsspatve over a wde range of energy spectrum that t degrades the vsual qualty. As we have observed n the full Euleran ntegraton method [16], vsual results may look unnatural f there s no numercal dsspaton. Addng a certan amount of numercal dsspaton wll make t a lttle vscous and appear plausble as real smoke. References [1] R. Brdson, Flud Smulaton for Computer Graphcs, AK Peters, Ltd., [2] J. Stam, Stable fluds, n: Proc. SIGGRAPH 99, 1999, pp [3] F. Losasso, F. Gbou, R. Fedkw, Smulaton water and smoke wth an octree data structure, n: Proc. SIGGRAPH 04, vol. 23(3), 2004, pp [4] S. Elcott, Y. Tong, E. 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