Anmatng Bubble Interactons n a Lqud Foam Oleksy Busaryev Oho State Unversty Tamal K. Dey Oho State Unversty Huamn Wang Oho State Unversty Zhong Ren Zhejang Unversty Fgure : Coke foam. By representng foam geometry usng a weghted Vorono dagram, our partcle-based algorthm can effcently provde bubble features n exstng lqud anmaton. Ths example contans up to 00K bubbles and each frame takes less than 20 seconds to smulate. Abstract Bubbles and foams are mportant features of lqud surface phenomena, but they are dffcult to anmate due to ther thn flms and complex nteractons n the real world. In partcular, small bubbles (havng dameter <cm) n a dense foam are hghly affected by surface tenson, so ther shapes are much less deformable compared wth larger bubbles. Under ths small bubble assumpton, we propose a more accurate and effcent partcle-based algorthm to smulate bubble dynamcs and nteractons. The key component of ths algorthm s an approxmaton of foam geometry, by treatng bubble partcles as the stes of a weghted Vorono dagram. The connectvty nformaton provded by the Vorono dagram allows us to accurately model varous nteracton effects among bubbles. Usng Vorono cells and weghts, we can also explctly address the volume loss ssue n foam smulaton, whch s a common problem n prevous approaches. Under ths framework, we present a set of bubble nteracton forces to handle mscellaneous foam behavors, ncludng foam structure under Plateau s laws, clusters formed by lqud surface bubbles, bubble-lqud and bubble-sold couplng, burstng and coalescng. Our experment shows that ths method can be straghtforwardly ncorporated nto exstng lqud smulators, and t can effcently generate realstc foam anmatons, some of whch have never been produced n graphcs before. Enrchng lqud anmaton wth bubbles and foams can sgnfcantly mprove ts realsm. However, bubbles and foams are dffcult to smulate due to ther dfferent physcal propertes n the real world, such as surface tenson of the lqud, lqud volume percentage, and bubble szes. For example, small bubbles n a mcrofoam of coffee latte are creamy and lqud-lke, whle bubbles n a soap foam are larger and more transparent. Bubbles also have complex nteracton behavors, ncludng clusterng, coalescng, deformng, and nteractng wth lquds or solds. Physcal nature of these phenomena has attracted attenton of mathematcans, physcsts and computer scentsts. A consderable amount of research [Brakke 992; Gardner et al. 2000; Weare and Hutzler 200; Km et al. 2007] was done to form a mathematcal descrpton of foam geometry and to model dynamc foam propertes, such as surface evoluton and topologcal changes n the foam structure. Beng physcally accurate, these models are often not drectly applcable to handle a large number of bubbles, because of ther computatonal cost. Keywords: Lqud, foam, bubble nteracton, weghted Vorono dagram, natural phenomena, surface tenson. Lnks: e-mal: e-mal: DL PDF Web Vdeo {busaryev,tamaldey,whmn}@cse.oho-state.edu zhongren@sggraph.org Introducton Alternatvely, partcle-based approaches [Ku ck et al. 2002; Greenwood and House 2004; Cleary et al. 2007] have been proposed n graphcs to effcently smulate lqud bubbles and foams, assumng that bubble surface deformaton s less notceable n a dense foam. Ths s a vald assumpton for small bubbles, not only because of ther small scales, but also because of the larger surface tenson effect that can quckly restore bubble shapes from deformaton. In order to model nter-bubble dynamcs, these methods typcally treat each bubble as a sphere, and then they apply nteracton forces whenever two spheres ntersect. Whle ths approach s suffcent for standalone bubbles, t fals to properly capture connectvty when bubbles form clusters and s of lmted use n modelng surface-tenson-based nteracton among bubbles on a lqud surface. How to mantan volume conservaton for each ndvdual bubble s another challengng problem for these technques, snce foam geometry and bubble shapes are not explctly represented. To solve these problems, we propose a smple and effcent partclebased algorthm that can smulate realstc bubble nteractons n complex foam scenaros, as shown n Fgure. The basc dea behnd ths method s to represent bubbles usng weghted ponts and gather them nto a weghted Vorono dagram. Our work shows that the algorthm can beneft from usng ths dagram n three ways.
Frst, t explctly reveals bubble connectvty. From the weghted Vorono dagram, we can drectly nfer whch two bubbles share the lqud flm, and whch bubbles are potental canddates for tensonbased nteracton. The Vorono dagram allows us to accurately model the nstablty of multple-bubble ntersectons that s quckly resolved nto trple (n 2D) or quadruple (n 3D) ntersectons n the real world. Secondly, t provdes a smple and natural approxmaton to the actual foam geometry, and t can be drectly used for renderng. Fnally, the weghted Vorono dagram can be used to explctly calculate bubble volumes, whch s a crucal component n the volume correcton process. Our system ppelne starts each tme step wth formulatng bubble nteracton forces, based on the bubble connectvty nformaton provded by the weghted Vorono dagram. The forces are then used n an mplct ntegrator to evolve bubble postons and veloctes over tme. After that, t calculates the volume of each bubble usng the Vorono cell, and compensates volume changes by adjustng the Vorono weght. Fnally, the system handles burstng and coalescng effects by performng topologcal changes, and t reconstructs the weghted Vorono dagram for the next tme step. In summary, we propose a novel partcle-based algorthm to smulate bubble nteractons n a lqud foam, by makng the followng contrbutons: ) a weghted Vorono representaton that models bubble connectvty and foam geometry; 2) a set of bubble nteracton forces that can produce varous nteracton behavors; and 3) a volume correcton method for partcle-based bubbles. We llustrate our method wth anmatons of typcal real-world foam scenaros, ncludng clusterng, stackng, stckng to lqud or sold surfaces, burstng and coalescng. We also ncorporate ths method nto a partcle level set lqud smulator, and test ts capablty of addng realstc bubbles and foams n lqud anmaton. 2 Prevous Work Foam Physcs. Real-world foams exhbt sgnfcantly dfferent structures and dynamc behavors due to ther physcal propertes. A dry foam, n whch the lqud volume s typcally less than percent of the whole volume, can form a specfc structure under laws dscovered by the Belgan physcst Joseph Plateau. Taylor [976] later proved that ths structure mnmzes the bubble surface area under volume constrants. Dfferent from the dry foam, a wet foam has more complex structures due to ts lqud volume, and t has been studed n multple ways [Weare et al. 993; Herzhafta et al. 2005; Pazza et al. 2008]. Foam structures can also be classfed accordng to ther bubble szes, such as monodsperse foam [Kraynk et al. 2003] that contans unformly szed bubbles and polydsperse foam [Kraynk et al. 2004], n whch the bubble sze can vary. Compared wth foam structure, foam dynamcs s a less studed problem due to the dffculty n observng dynamc behavors, ncludng dranage, rheology, coarsenng, and mergng. A comprehensve s- tudy on both foam structure and foam dynamcs can be found n Weare and Hutzler s book [200]. In ths paper, we focus on smulatng rheologcal and mergng behavors of polydsperse foams, n both dry and wet cases for graphcs applcatons. Due to the smlarty between the weghted Vorono dagram and foam geometry, physcsts studed usng the dagram to model statc foam structures [Redenbach et al. 202] n the past. Although they also used the dagram to ntalze foams n dynamc smulaton, they chose to solve bubble dynamcs usng more sophstcated models [Kraynk et al. 2004; Brakke 992] nstead, snce only a small set of bubbles were consdered n ther typcal examples. In contrast, we found the weghted Vorono dagram can be used drectly n dynamc smulaton as well, and t provdes a good approxmaton to foam structure for a great number of bubbles. Foam Smulaton. The smulaton of bubbles and foams nvolves two aspects: the deformaton of ndvdual bubble surfaces, and the nteractons among bubbles, lquds and solds. An early example of deformable bubbles s proposed by Brakke [992]. Durkovc [200] used a sprng mesh to represent bubble surfaces and approxmated nteracton forces usng an ntermolecular Van der Waals force model. To facltate topologcal changes, Km et al. [20] developed a 2D algorthm usng the mmersed boundary method. Kelager [2009] ntroduced the ghost bubble technque to the vertex-based dry foam smulaton. Implct representatons, such as Volume-of-Flud (VOF), can also be used to smulate deformable bubbles, as Hong et al. [2003] and Mhalef et al. [2006] showed. Zheng et al. [2006] proposed a regonal level set method to mplctly model lqud foams as mult-manfold surfaces. Km et al. [2007] addressed the volume loss n the regonal level set method usng a volume control technque. Whle these methods can handle surface deformaton of ndvdual bubbles, they need consderable computatonal tme to handle a large number of bubbles. By gnorng surface deformaton, partcle-based technques are specfcally developed for handlng bubble nteractons wth the envronment. Duran [995] frst proposed to use a mass-sprng model to anmate bubble nteractons n 2D. Kück et al. [2002] extended ths dea to 3D, and also developed a way to render Plateau borders and curved flms between contactng bubbles. Greenwood and House [2004] ncorporated the Kück model nto a partcle-levelset-based lqud smulator. Bubble nteractons may also be approxmated by Smoothed Partcle Hydrodynamcs (SPH) as n [Cleary et al. 2007; Thürey et al. 2007; Hong et al. 2008], though they typcally do not consder nternal foam geometry. Whle we also represent bubbles as partcles n our system, we mprove smulaton accuracy by the use of weghted Vorono dagrams, and we can handle more bubble nteracton behavors. Flud Smulaton. Flud smulaton s an mportant research topc n computer graphcs, and varous technques were proposed for anmaton purposes, ncludng Euleran approaches [Foster and Metaxas 996; Stam 999; Enrght et al. 2002; Chentanez and Müller 20], Smoothed Partcle Hydrodynamcs (SPH) [Müller et al. 2003; Adams et al. 2007], Lagrangan-based methods usng meshes [Bargtel et al. 2007; Thürey et al. 200; Wcke et al. 200], and smplfed or hybrd algorthms [Losasso et al. 2004; Bargtel et al. 2006; Wang et al. 2007; Losasso et al. 2008]. Some of these approaches [Brochu et al. 200; Sn et al. 2009] also leveraged Vorono dagrams for smulaton. Snce our system provdes addtonal foam features to lqud anmaton, t can be straghtforwardly ncorporated nto any lqud smulator. 3 Bubbles and Foams A sphercal bubble s a three-dmensonal ball B(x, r) of radus r centered at the pont x. These sphercal bubbles come together to form a foam. Therefore, the space of the foam s the pontwse u- non of a set of balls: B. Each sphercal bubble B n a foam takes a share of ths space, whch s called the bubble of B. The center and the radus of a bubble are smply taken as those of ts sphercal counterpart. The boundary of a bubble s a set of curved surface patches called flms. These flms are formed n a way so that the total surface area over all bubbles s mnmzed under volume constrants, as Taylor [976] and Sullvan [998] ponted out. In partcular, the geometry of dry foam, whose lqud volume s typcally less than percent, can be descrbed by Plateau s laws:. Flms meet n threes along Plateau borders at angles of 2π/3. 2. Plateau borders meet n fours at angles of cos ( /3). 3. Each bubble flm has constant mean curvature.
r 2π/3 2π/3 l j r j r j weghted dstance of any pont x R 3 from p P s gven by d 2 (x, p) = x p 2 r 2. The weghted Vorono dagram and ts d- ual weghted Delaunay trangulaton are then defned n the same way as the orgnal ones, except that they replace Eucldean dstances wth squared weghted dstances. Fgure 2b shows a weghted Vorono example wth ts Delaunay trangulaton. (a) Two bubbles restng n the Plateau equlbrum. The nterfacal flm between bubbles s a sphercal patch wth radus r j. (b) A weghted Vorono dagram (dotted), Delaunay complex (sold edges), space-fllng dagram (blue curves), and alpha complex (dark edges and trangles). Fgure 2: Bubble and foam confguraton. Any dry foam geometry not followng Plateau laws s unstable, and the surface tenson force wll rearrange those bubbles back nto the Plateau equlbrum. Gven a par of contactng bubbles as shown n Fgure 2a, the nterfacal flm between them s not necessarly planar and t can be modeled as a patch of sphere whose radus r j s calculated as: r j = r j r, n whch r and r j are the rad of the two bubbles respectvely and r r j. We can verfy that ths confguraton corresponds to the Plateau equlbrum. We also notce that r j becomes larger when the two rad get closer. In partcular, the flm s planar when the two rad are dentcal. For multple ntersectons, flms between bubbles may not be sphercal. Instead they can be approxmated by a mean curvature proportonal to the pressure dfference across the flm. When pressure dfference s nonzero, the flm s curved towards the bubble wth a lower pressure. 3. Foam Representaton We defne a foam as a tessellaton of the space occuped by a unon of balls. A well known data structure called space-fllng dagram exsts n computatonal geometry, whch also tessellates such a s- pace. Observng ths analogy, we represent a foam as the space fllng dagram. Smply stated, a space fllng dagram s the ntersecton of a unon of balls wth the weghted Vorono dagram (also called power dagram) whch s a generalzaton of the well known Vorono dagram [Aurenhammer 987]. Although t does not exactly model curved nterfacal flms n the real world, t provdes a sound approxmaton to bubble structures and foam geometry. A Vorono dagram s suffcent n representng monodsperse foams, n whch bubbles have the same sze. To further handle polydsperse foams wth dfferent bubble szes, we need the weghted Vorono dagram. Specfcally, an orgnal sphercal bubble B = B(x, r ) s represented as the ntersecton between B and the weghted Vorono cell of x wth weght r. A detaled defnton of these concepts s gven below. Vorono Dagram and Delaunay Trangulaton. Let P be a pont set n R 3. For any pont p P, the Vorono cell V p of p s defned as the locus of ponts n R 3 havng p as ther nearest neghbor n P: V p = {x R 3, for q P : x p x q }. Each Vorono cell s convex and ts boundary conssts of lower-dmensonal convex faces. The collecton of Vorono cells and ther faces forms a cell complex tessellatng R 3, called the Vorono dagram of P. Its dual complex s called the Delaunay trangulaton of P. Weghted Counterparts. Vorono dagrams and Delaunay trangulatons can also be constructed usng non-eucldean metrcs. Specfcally, we can assocate each pont p P wth a realvalued weght r (r 0). Ths s equvalent to representng each weghted pont p as a ball wth radus r centered at p. The squared Space-Fllng Dagrams. After we construct the Vorono dagram by treatng each sphercal bubble as a weghted pont, we represent the bubble of B = B(x, r ) n a foam as the ntersecton of B wth the Vorono cell for x. Flms between two bubbles are defned as the ntersecton of balls wth Vorono polygons. Such decomposton of the unon of balls nto convex cells s known as a space-fllng dagram. It should be noted that we do not need an explct representaton of the foam geometry durng smulaton. Instead, we use the connectvty nformaton among bubbles, whch s captured by the dual of the space fllng dagram called the alpha complex [Edelsbrunner 200]. For example, Fgure 2b shows the dark edges and shaded trangles n a 2D alpha complex. The edges are dual to Vorono edges whch correspond to the common ntersecton of two bubbles and the trangles are dual to the Vorono vertces whch correspond to the common ntersecton of three bubbles. The edges of the alpha complex provde nformaton about whch bubbles are n contact. 4 Foam Dynamcs Varous factors contrbute to the moton of bubbles. For example, two bubbles n contact can experence both repulson and attracton forces, whch try to brng them back to the Plateau equlbrum. Meanwhle, bubbles on a lqud surface are lkely to form clusters, caused by weak nteracton forces even before they contact. In ths secton, we present a set of bubble forces to model varous foam effects n our system. These forces can be roughly grouped nto three categores based on the type of ther nteractons: bubble-bubble nteracton, bubble-lqud nteracton, and bubble-sold nteracton. The nteractons between two bubbles can be trggered ether by a strong or a weak nteracton force as we wll see later. The strong nteracton happens only between bubbles that are n contact whle the weak nteracton happens only between bubbles that are Vorono neghbors but are not n contact. We dstngush the latter from the former by testng Delaunay edges that are not alpha complex edges. 4. Bubble-Bubble Interacton Bubbles n a dry foam (such as soap foam), whose lqud volume s typcally less than percent, form a specfc geometrc structure that mnmzes the total surface area under volume constrants. Formulated n the 9 th century by the Belgan physcst Joseph Plateau, Plateau s laws descrbe ths structure usng three condtons as we descrbed before. Soap bubbles not n ths confguraton are unstable, and the surface tenson force quckly rearranges them back nto the Plateau equlbrum. In partcular, when two separate bubbles touch each other, a strong attracton force tres to move them closer nto the Plateau confguraton. Meanwhle, f bubbles are closer than necessary, the force wll become repulsve due to both surface mnmzaton and volume constrants. Gven two sphercal bubbles wth rad r and r j respectvely, we can reach the Plateau equlbrum when the dstance l j between two bubble centers satsfes: l 2 j = r 2 + r 2 j r r j. Fgure 2a shows such an example, n whch bubble flms form 2π/3 angles accordng to Plateau s laws. Ths condton also provdes a reasonable approxmaton for multple bubble cases. However, t becomes less vald for wet foams, n whch the extra lqud n nterfacal flms can affect the surface tenson. Unfortunately, the geometrc structure of wet foam s more
c a f e g d b φ > 0 φ < 0 Fgure 3: Lqud bubbles (c and d), surface bubbles (f and g), and ar bubbles (a, b, and e). They are classfed accordng to ther sgned dstances to the lqud surface. complex, and no clear conclusons have been made to ths problem yet. Here we ntroduce a wetness coeffcent λ ( λ ) to account for dfferent wetness condtons: l 2 j = r 2 + r 2 j + (3λ )r r j. () For example, settng λ = 0 allows us to mmc a completely dry foam, where bubbles form a polyhedral structure. Makng λ close to models a wet foam, where bubbles have almost sphercal shape and merely touch each other n the equlbrum state. Compared to the statc foam geometry, bubble-bubble nteracton n dynamc envronment s more dffcult to measure and study. Smlar to [Kück et al. 2002; Greenwood and House 2004], we use a smple sprng model to handle the force between two contactng bubbles. Let x and x j be the centers of two ntersectng bubbles neghborng n the weghted Vorono dagram, we defne a strong nteracton force appled on x usng the restng length l j : f snt = k j ( x x j l j x x j x x j n whch k s a constant stffness coeffcent. 4.2 Bubble-Lqud Interacton ), (2) To smulate bubble-lqud nteracton, we classfy bubbles nto three types based on ther postons wth respect to a lqud volume. Let ϕ be the sgned dstance functon to a lqud surface, a bubble centered at x wth radus r s named as a lqud bubble, f ϕ(x ) < r. When r ϕ(x ) r, we defne the bubble as a surface bubble, f and only f t s not solated from the lqud surface by other bubbles. The rest of bubbles are called ar bubbles. Fgure 3 shows an example, n whch bubbles e, f, and g all satsfy the condton: r ϕ(x ) r. However, bubble e s stll an ar bubble, snce t s separated from the lqud by bubble f and g. We detect ths separaton by samplng ϕ over the bubble s sphercal boundary. If there exsts a negatve sample that s not covered by other bubbles, t sgnfes a surface bubble, otherwse t s an ar bubble. Snce ar bubbles do not drectly nteract wth the lqud, we wll only consder lqud bubbles and surface bubbles next. Lqud Bubble. A lqud bubble wthn a lqud volume s subject to two forces n our system. Smlar to [Cleary et al. 2007] and [Hong et al. 2008], we frst defne a lqud drag force dependng on the bubble velocty relatve to the flud: f drag = c drag r 2 (u(x ) v ) u(x ) v, (3) n whch c drag s a drag coeffcent, u(x ) s the lqud velocty at the bubble center x, and v s the bubble velocty. The other force s the buoyant force that lfts the bubble up to the surface. Assumng lqud bubbles are ncompressble n the lqud volume, we have: f buoy = V ρg = 4 3 πr3 ρg. (4) where g s the gravty acceleraton, ρ s the lqud densty, and V s the bubble volume. (a) A surface bubble n equlbrum f lad (b) An mmersed surface bubble f lad (c) A lfted surface bubble Fgure 4: A surface bubble n dfferent states. Its equlbrum state can be reached by the use of a lqud adheson force. f wnt (a) Bubbles before beng attracted f wnt (b) Bubbles after beng attracted Fgure 5: Bubble clusterng caused by surface tenson. We use a weak nteracton force to obtan ths effect. Surface Bubble. Surface bubble moton s hghly nfluenced by surface tenson. Whle modelng surface tenson drectly on bubbles are dffcult and computatonally expensve, we defne two nteracton forces here, both of whch are formulated to acheve specfc surface bubble behavors. Lke other surface tenson effects, surface bubbles move n a way such that the lqud surface area can be mnmzed. Therefore, a surface bubble s able to reach an equlbrum on the lqud surface, under both the buoyant force and the surface tenson force, as Fgure 4a shows. When the bubble gets mmersed (n Fgure 4b), the buoyant force becomes larger whle the surface tenson force becomes weaker, so the total force wll push the bubble to ts equlbrum state. Meanwhle, a lfted surface bubble (n Fgure 4c) receves a larger surface tenson force than the buoyant force, so t wll be pulled back. Snce t s dffcult to accurately calculate both forces n ths case, we defne a lqud adheson force f lad to help the surface bubble stay on the lqud surface: f lad = σ lad ϕ(x ) ϕ(x ), (5) n whch σ lad s an adheson coeffcent, and ϕ(x ) s the lqud surface normal at x. Surface tenson can also cause two separated bubbles to move toward each other, n order to mnmze the overall lqud surface area as Fgure 5 shows. So for any two non-contactng surface bubbles that are neghbors n the weghted Vorono dagram, we defne a weak nteracton force usng the approxmated sne value of the surface nclnaton angle θ j : f wnt j = α sn θ j x j x x j x, sn θ j = r j x x j, (6) n whch α s a force magntude parameter. It should be noted that nter-bubble nteractons, ncludng both strong and weak nteracton, rely on the topology of the weghted Delaunay trangulaton (or ts dual weghted Vorono dagram). Delaunay edges that belong to the alpha complex ndcate strong nteracton forces for contactng bubbles, whle the remanng Delaunay edges help us formulate weak nteracton forces for surface bubbles. 4.3 Bubble-Sold Interacton In order to model the hydrophlcty of certan sold objects, we use an adheson force to prevent bubbles from leavng sold surfaces: f sad = σ sad ψ(x ) ψ(x ), (7) n whch σ sad s an adheson coeffcent and ψ(x ) gves the sgned dstance from the bubble center x to the sold. The adheson force acts on each bubble that has ever touched the sold. In addton, we
defne a sold attracton force to move surface bubbles closer to the sold, due to a smlar surface mnmzaton reason we dscussed before n Subsecton 4.2: n whch β s a sold attracton coeffcent. 5 Foam Smulaton f sat = β ψ/ ψ, (8) Based on foam representaton and foam dynamcs presented n Secton 3 and 4 respectvely, we develop a foam smulator to update bubbles and foams n each tme step. It frst uses an mplct ntegrator to evolve bubble postons and veloctes, accordng to foam dynamcs. It then apples a volume conservaton constrant to compensate the bubble volume loss durng smulaton, especally when bubbles are heavly squeezed. Fnally, we perform topologcal updates on the foam structure and reconstruct the weghted Vorono dagram, to ensure that t remans vald for further smulaton. 5. Tme Integrator We use the mplct solver proposed by Baraff and Wtkn [998] to evolve bubble partcles over tme. Gven bubble postons {x t } and veloctes {v t } at tme t, the solver uses the backward Euler method to compute bubble veloctes {v t+ } at the next tme step: ( M t f t / v t 2 f t/ x ) v t+ = Mv t + tf t, (9) Relatve Error (a) Wthout volume correcton 00% 75% 50% 25% (b) Wth volume correcton wth volume correcton no volume correcton 0% 0 0 20 30 40 50 60 70 80 90 Frame Index Fgure 6: A 2D cluster. Ths example vsualzes a cluster of unform bubbles wth (a) and wthout (b) volume correcton. The plot shows that the maxmum relatve error of bubble volumes can be greatly reduced, after usng our volume correcton method. n whch M s a mass matrx, t s the tme step, and f, x and v are vectors of bubble forces, postons, and veloctes respectvely. The force vector f s made of the gravty force, the ar dampng force, and the total nteracton force. The gravty force f grav = m g on bubble s ndependent of bubble poston and velocty, so ts Jacoban matrces f grav / x and f grav / v are both zero matrces. We defne the dampng force usng two dampng coeffcents (c vs and c lap ) and a normalzed Laplacan matrx L: f damp = c vs v c lap Lv. Its Jacoban matrx can be wrtten as: (c vs + c lap )I, for = j ( c j = c lap N N ) /2 j I, for j and N j (0) 0, otherwse n whch c j s a 3 3 sub-matrx, and N s the -rng contact neghborhood of vertex, gven by the weghted Vorono dagram. The overall nteracton force s the sum of three nteracton forces dscussed n Secton 4. Most nteracton forces are comparably small, so they can be treated explctly by smply gnorng ther Jacoban matrces. The only excepton s the strong bubble nteracton force modeled by a stff sprng n Equaton 2. Smlar to the formula proposed by Cho and Ko [2002], we use Taylor expanson to fnd the Jacoban matrx of f snt respect to x. We also drop the geometrc term when the sprng s compressed, to ensure numercal stablty. The lnear system n Equaton 9 s guaranteed to be symmetrc postve defnte. We solve t usng the Precondtoned Conjugate Gradent (PCG) method wth an ncomplete Cholesky precondtoner. Snce nteracton forces n our method depend on bubble connectvty that may vary over tme, the whole system s not uncondtonally stable. Fortunately, our experment shows that the system can robustly handle large tme steps ( t [0.005s, 0.02s]) for most examples wthout any oscllaton artfacts. 5.2 Volume Correcton The weghted Vorono dagram changes when bubble move. As a result, some bubbles represented by the cells of the space-fllng dagram may experence consderable volume changes. To preserve the bubble volume, we perform a volume correcton step. We consder two ways to adjust the bubble volume: to change the bubble poston x, or to change ts weght r. By modfyng the weght alone we can mplctly ncrease or decrease the volume of an ndvdual bubble. However, t may affect the volume of ts neghbors. One may try to terate the process to rectfy the volumes of all bubbles. But, ths does not work because the total volume of the unon of all bubbles changes only by the modfcatons of the bubbles on the boundary of the unon. The ntended changes for nteror bubbles are not reflected n ths sum. We need poston changes along wth weght changes to overcome ths constrant. Unfortunately, explct adjustment of postons wthout causng artfacts n bubble dynamcs s dffcult. We solve ths dlemma by adjustng weghts alone and then lettng the smulaton adjust the postons. The restng length l j between two neghborng bubbles depends on ther rad. If r ncreases relatve to r j, the bubble at x j s pushed away from x decreasng the overlap and thus ncreasng the bubble s volume. Fgure 6 shows the effect of ths volume correcton. The maxmum relatve error s 25 percent after volume correcton n contrast to a 75 percent error wthout t. Smaller tme steps and multple teratons can be used to reduce ths error further, f necessary. Another potental method to perform volume correcton s to ntroduce t as a force; however, ths approach s complcated by necessty of computng the volume dervatves. In order to know how much volume needs to be compensated for each bubble, we store ts ntal volume V 0. At each tme step, we calculate the volume V of ts Vorono cell n the spacefllng dagram, usng the method proposed by Cazals et al. [20]. We then multply the current bubble radus r by a factor of + γ((v 0 /V ) /3 ) t, where γ s a parameter to control the correcton amount. Snce we do not consder bubble surface deformaton caused by lqud, we do not dfferentate lqud bubbles, surface bubbles, and ar bubbles n volume calculaton. However, bubbles attached to a sold boundary have ther volumes occuped by a sold, so we compensate ther calculated volume V by a factor of 2. Fg-
Tme per Frame (s) 20 Anmatng Bubbles Computng Dagram 5 0 5 0 Number of Interbubble Sprngs Number of Vorono Polygons Number of Bubbles M 750K 500K Fgure 8: Soap foam. Soap bubbles ple up n a water contaner. 250K 0K 0 200 400 600 800 000 Frame Index 200 400 600 Fgure 7: Tmng results and scene complexty over 700 smulaton frames n the Coke foam example for 00K bubbles. ure 6 plots out maxmum relatve volume errors over 00 smulated frames, wth and wthout usng our volume correcton method. 5.3 Burstng and Coalescng Lqud dranage may cause foam topology to change over tme. In partcular, bubbles on the boundary of a foam cluster may burst, whle bubbles nsde the foam may merge wth other bubbles. To model ths process, we mantan an age varable e for each bubble. At each tme step, we randomly remove bubbles based on ther ages and rad. The probablty for bubble to get removed s defned as: P = e τr e, () n whch τ s the burstng speed coeffcent. Intutvely, Equaton means younger and smaller bubbles have better chances to survve, whle older and larger bubbles are lkely to dsappear. When deletng a bubble nsde of a foam, we dstrbute ts volume to ts closest neghbor and we set the neghbor s poston as the weghted average of ther orgnal postons before deleton. 6 Results The supplementary vdeo submtted along wth ths paper demonstrates our anmaton results. We mplemented our foam smulaton model and tested t on an Intel Core 5-2500K 3.3GHz CPU. To demonstrate nteracton wth lqud flows and sold surfaces, we coupled the foam model wth a partcle level set flud smulator [Enrght et al. 2002], runnng on a 643 unform Cartesan grd. We used adaptve tme steps n bubble smulaton, whch can vary from 0.02s to 0.005s, dependng on flud flows and bubble veloctes. Our system uses standard unts. The gravty acceleraton s 9.8m/s2, and the lqud densty s 03 kg/m3. A lst of parameters and ther ranges s summarzed n Table for convenence. We use the Coke foam example to study the complexty and computatonal cost of our system as Fgure and Fgure 7 show. In ths example, 00K bubbles are added nto the scene, and >500K bubblebubble sprngs are created to handle strong nteracton forces. Bubble szes vary from 0.5mm to 3mm. The total number of polygons n the weghted Vorono dagram s more than M. Ths example shows that the computatonal cost ncreases wth the scene complexty as expected. The smulaton tme spent on anmatng bubbles s approxmately one thrd of the tme spent on constructng the Vorono dagram n each tme step. For 00K bubbles, our system smulates each rendered frame n 3 to 7 seconds (excludng flud smulaton tme). It may seem that a dynamc Vorono dagram (a) Transparent bubbles (b) Opaque bubbles Fgure 9: Pourng water. Water bubbles emerge on the lqud surface, but quckly burst due to ther nstablty n the real world. We model ths effect by usng a larger burstng speed coeffcent. constructon would be more effcent, but a good practcal soluton for updatng a 3D Vorono dagram dynamcally s stll elusve unfortunately. In practce, t may suffce to compute a local Vorono dagram for a suffcently large neghborhood of each bubble. Our anmatons were rendered at 30Hz, usng the GPU-based mcro-polygon ray tracng method [Hou and Zhou 20]. We modeled bubbles as sem-transparent objects wth thn flms separately from the scene, n order to avod a large number of reflecton and refracton rays f dong ray tracng on bubbles drectly. We then ncorporated transparent appearances of bubbles nto the scene, by modfyng correspondng rays. To speed up the renderng process, we also mantaned a separate boundng volume herarchy for bubbles, snce ther ntersectons wth rays should be determned frst. Bubbles n a soapy lqud made by dsh detergent or hand soap tend to ple up, creatng dense and complcated foam structures. In ths example, we created such a scene wth up to 4K bubbles, as Fgure 8 shows. The bubble sze vares from 0.25 to cm, and the water contaner s 6cm 6cm 6cm. At the end of the anmaton, bubbles can form at least ten layers n the foam. To prevent foam ples from sudden moton, we reduced the burstng coeffcent τ, especally for those close to sold walls. We also vared the wetness coeffcent λ accordng to the dstance from the lqud surface, so bubbles on the top are drer than lqud surface bubbles. Each frame of ths example took 0.6 to 2.0s to smulate. Soap Foam. We smulated another Coke foam example that contans up to 6K bubbles wth varyng szes, as shown n Fgure and 0. The largest bubble n ths example has a sze of.25mm, whle the smallest bubble s 0.375mm bg. It shows how the bubble sze can affect bubble behavors, such as bubble-bubble attracton and burstng. We use the same wetness coeffcent λ = 0.4 for all bubbles, snce they do not ple up. For testng purpose, we also anmate the same scene by only usng strong nteracton forces for Coke Foam.
Notaton λ k cdrag σlad α σsad β cvs, clap γ τ Name Wetness coeff. Stffness coeff. Drag coeff. Lqud adheson coeff. Bubble attracton coeff. Sold adheson coeff. Sold attracton coeff. Dampng coeff. Volume correcton coeff. Burstng speed coeff. Usage To determne the restng dstance between two bubbles (n Equaton ) To model the strong nteracton force (n Equaton 2) To model the drag force for a lqud bubble (n Equaton 3) To model the lqud adheson force for a surface bubble (n Equaton 5) To model the weak nteracton force caused by tenson (n Equaton 6) To model the sold adheson force for any bubble (n Equaton 7) To model the sold attracton force caused by sold (n Equaton 8) To dsspate the knetc energy over tme (n Equaton 0) To determne the amount of volume compensaton n each tme step To specfy how fast bubbles burst (n Equaton ) Unt N/m kg/m3 N/m kg m/s2 N/m kg m2 /s2 m s Range [0, ] [.0, 8.0] [0.05, 0.5] [0.0, 20.0] [0.2, 0.6] [5.0, 30.0] [2.0, 6.0] [0 5, 0 4 ] [0.025, 0.2] Table : Parameters used n our system. not uncondtonally stable due to ts dependency on bubble connectvty, although t barely affects the system performance. 8 Concluson and Future Work In ths paper, we showed that a weghted Vorono dagram can be used to approxmate the actual foam geometry n a sound manner, for both dry and wet foams contanng small bubbles. Based on ths representaton, we demonstrated that a partcle-based system can effcently and realstcally handle bubble nteractons, even wthout explctly modelng surface deformaton or surface tenson. Our experment further tested ts compatblty wth exstng lqud smulators, and revealed ts capablty of generatng natural bubble effects, such as clusterng, stackng, burstng, coalescng, and bubble-lqud and buld-sold nteractons. (a) Bubbles (b) Floatng partcles Fgure 0: Coke foam. Images n (a) show that wth our nteracton forces, coke bubbles form clusters and are attracted to the mug walls. Wth nteracton forces lmted to basc collson responses, we can also smulate them as floatng plastc partcles n (b). collson responses. Ths allows us to smulate bubbles as f they were floatng plastc partcles. The smulaton tme of each frame vares from 0.5 to 2.8s. Pourng Water. Our system s also able to smulate bubbles n a complex lqud scene, such as the pourng water example shown n Fgure 9. There were 3.4K bubbles (wth szes of 0.75mm to 2.5mm) generated from the escaped ar partcles n the partcle level set flud smulator. Bubbles traveled wth the water flow and emerged on the lqud surface due to the buoyant force. We ncreased the burstng speed coeffcent τ n ths example to account for the nstablty of water bubbles n the real world. Each frame n ths example took 0.3 to 5.2s to smulate. 7 Lmtatons Lke other partcle-based approaches, our method does not handle bubble surface deformaton and t s not sutable for large bubbles (whose dameters are greater than cm). For numercal stablty, we formulate bubble nteracton forces usng a lnear stffness model. However, nteracton forces n the real world can be hghly nonlnear, and they can be affected by many other condtons that our system does not model so far, such as dranage and coarsenng effects. When dealng wth bubble-lqud nteractons, we do not consder how bubble motons can affect the lqud flow. Accurately preservng volumes n our system requres a smaller tme step, whch ncreases the computatonal cost. Fnally, our mplct ntegrator s Besdes solvng those lmtatons lsted n Secton 7, we plan to accelerate our system by the use of GPU-based algorthms. Snce we defne most parameters n our system based on effects rather than physcs, fndng optmal values for them becomes a challengng and tme-consumng task n practce. We are nterested n carryng out both expermental and numercal study on ths ssue n the future. Acknowledgments We thank Qmng Hou, Mngmng He, Kun Zhou and the Graphcs and Parallel Systems Lab at Zhejang Unversty for ther support and helpful suggestons n renderng. Ths work was supported n part by the NSF grant CCF 0830467 and the NSF of Chna grant No. 6003048. We also thank NVIDIA for addtonal support through equpment and fundng. References Adams, B., Pauly, M., Keser, R., and Gubas, L. J. 2007. Adaptvely sampled partcle fluds. ACM Transactons on Graphcs (SIGGRAPH) 26 (July). Aurenhammer, F. 987. Power dagrams: propertes, algorthms and applcatons. SIAM Journal on Computng 6, 78 96. Baraff, D., and Wtkn, A. 998. Large steps n cloth smulaton. In Proc. of SIGGRAPH 98, E. Fume, Ed., Computer Graphcs Proceedngs, Annual Conference Seres, ACM, 43 54. Bargtel, A. W., Goktekn, T. G., O bren, J. F., and Stran, J. A. 2006. A sem-lagrangan contourng method for flud smulaton. ACM Transactons on Graphcs 25 (January), 9 38. Bargtel, A. W., Wojtan, C., Hodgns, J. K., and Turk, G. 2007. A fnte element method for anmatng large vscoplastc flow. ACM Transactons on Graphcs (SIGGRAPH) 26 (July).
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