Volume Preserving Viscoelastic Fluids with Large Deformations Using Position-based Velocity Corrections

Transcription

1 Noname manuscrpt No. (wll be nserted by the edtor) Volume Preservng Vscoelastc Fluds wth Large Deformatons Usng Poston-based Velocty Correctons Tetsuya Takahash Yoshnor Dobash Isse Fujshro Tomoyuk Nshta Receved: date / Accepted: date Abstract We propose a partcle-based hybrd method for smulatng volume preservng vscoelastc fluds wth large deformatons. Our method combnes Smoothed Partcle Hydrodynamcs (SPH) and Poston-based Dynamcs (PBD) to approxmate the dynamcs of vscoelastc fluds. Whle preservng ther volumes usng SPH, we explot an dea of PBD and correct partcle veloctes for vscoelastc effects not to negatvely affect volume preservaton of materals. To correct partcle veloctes and smulate vscoelastc fluds, we use connectons between partcles whch are adaptvely generated and deleted based on the postonal relatons of the partcles. Addtonally, we weaken the effect of velocty correctons to address plastc deformatons of materals. For one-way and two-way flud-sold couplng, we ncorporate sold boundary partcles nto our algorthm. Several examples demonstrate that our hybrd method can suffcently preserve flud volumes and robustly and plausbly generate a varety of vscoelastc behavors, such as splttng and mergng, large deformatons, and Barus effect. Keywords Flud smulaton Vscoelastcty Deformaton Volume preservaton Velocty correcton Tetsuya Takahash UEI Research / Keo Unversty E-mal: tetsuya.takahash@ue.co.jp Yoshnor Dobash Hokkado Unversty / UEI Research E-mal: doba@me.st.hokuda.ac.jp Isse Fujshro Keo Unversty E-mal: fuj@cs.keo.ac.jp Tomoyuk Nshta UEI Research / Hroshma Shudo Unversty E-mal: tomoyuk.nshta@ue.co.jp 1 Introducton Partcle-based methods have been popular technques to smulate fluds, rgd bodes, deformable objects, and ther nteractons. These methods use a smple concept of partcle nteractons, whch can be easly extended to smulate varous types of objects, and were adopted for a versatle smulaton framework [16]. For conceptual smplcty and versatlty, we heren focus on partclebased methods. Snce vscoelastc materals play an mportant role n representng common materals (e.g., egg whte, gels, toothpaste, and slme) and producng vsually attractve characters and effects, smulatng vscoelastc materals has been requred for moves and vdeo games. In such entertanments, exaggerated representatons whch real materals do not exhbt, e.g., very large deformatons, are frequently requred to make characters and effects appear nterestng and mpressve. In computer graphcs, partcle-based methods for smulatng vscoelastc fluds have been proposed. However, prevous partcle-based, Smoothed Partcle Hydrodynamcs (SPH) methods whch depend on physcallybased vscoelastc models [17, 6] cannot smulate hghly deformable objects because, at a partcle poston, physcal contrbutons from farther partcles are lkely to be smaller owng to the decreasng property of fnte support kernels, and consequently, vscoelastc materals fal to restore ther orgnal shapes when they undergo sgnfcant deformatons. Ths problem can be addressed by combnng SPH and a geometrc method [28, 7]. However, these hybrd methods have another problem that they suffer from volume loss of materals, whch leads to unphyscal flud behavors, because the geometrc methods adopted by Takamatsu and Kana

2 2 Tetsuya Takahash et al. Fg. 1 Vscoelastc fluds smulated wth our hybrd method. (Left) Mergng spheres wth dfferent vscoelastcty values colldng wth a statc sold dragon. (Mddle) A vscoelastc ball colldng wth a sold cube. (Rght) Barus effect of a vscoelastc materal (see 4.4). [28] and Clavet et al. [7] make an ncompressble flud solver fal to preserve flud volumes. To solve the two problems above smultaneously, we propose a partcle-based hybrd method for smulatng vscoelastc fluds, whch can preserve flud volumes and enable large deformatons of the fluds. Our method uses a combnaton of SPH and Poston-based Dynamcs (PBD) [20, 4]. Whle usng SPH to preserve flud volumes, we explot PBD for velocty correctons whch acheve vscoelastc effects nvolvng large deformatons wthout negatvely nfluencng the volume preservaton. To correct partcle veloctes, we use a set of connectons between partcles, whch are adaptvely updated based on the postonal relatons of the partcles. We ncorporate sold boundary partcles nto our algorthm to smulate adheson of vscoelastc fluds to sold objects. Our key contrbuton les n velocty correctons whch allow for combnng a geometrc method wth SPH to smulate vscoelastc behavor wth large deformatons whle preservng flud volumes, snce the SPH-based methods [17, 6] cannot handle large deformatons and the prevously proposed hybrd methods [28,7] cannot preserve the volume of vscoelastc fluds. Fg. 1 llustrates characterstc behavors of vscoelastc materals smulated wth our hybrd method. Ths paper s an extended verson of our prevous paper [26]. In ths paper, we focus on the postonbased velocty correcton scheme. As new addtonal materals, we nclude a comparson of our method wth the SPH-based method [17] and propose a boundaryhandlng method for flud-sold couplng. 2 Related Work Vscoelastc fluds Mller and Pearce [18] and Terzopoulos et al. [30] proposed a sprng-based model that computes repulson and attracton forces among partcles to smulate vscoelastc materals. Ther method was adopted by Steele et al. [25] and Tamura et al. [29]. Clavet et al. [7] combned ths sprng-based method wth SPH to smulate materals wth elastcty, plastcty, and vscosty, adoptng a predcton-relaxaton scheme. A smlar sprng-based method was also proposed by Takahash et al. [27], who used PBD to smulate fluds wth vscosty and elastcty n a unfed framework. Takamatsu and Kana [28] combned Shape Matchng (SM), whch was orgnally proposed by Mu ller et al. [21] to smulate deformable objects, wth SPH to fast and robustly smulate vscoelastc fluds. Mu ller et al. [22] proposed an elastcty term whch uses Movng Least Square (MLS) to smulate elastoplastc objects. Solenthaler et al. [24] adopted the formulaton of ths elastcty term and computed t usng SPH nstead of MLS to allow for robustly smulatng flud wth varous propertes under the condton of collnear or coplanar partcle dstrbutons. The method of Solenthaler et al. [24] was extended to handle rotatonal motons of elastc materals [2]. Mao and Yang [17] ntroduced a vscoelastc force term, called nonlnear corotatonal Maxwell model, nto the Naver-Stokes equatons to smulate vscoelastc fluds. A method smlar to the method of Mao and Yang [17] was proposed by Chang et al. [6]. Gerszewsk et al. [9] proposed a new formulaton for smulatng elastoplastc materals, whch uses affine transformatons to compute the gradent of deformatons. Ths formulaton was solved by Zhou et al. [31] n an mplct manner to efficently and robustly perform smulatons wth larger tme steps. The method of Gerszewsk et al. [9] was extended by Jones et al. [14]. Jones et al. [14] used MLS to robustly smulate elastoplastc materals wth spatally and temporally varyng masses. Volume preservaton In partcle-based smulatons, volume preservaton has been acheved by mnmzng densty fluctuatons from the flud rest densty by en-

3 Volume Preservng Vscoelastc Fluds wth Large Deformatons Usng Poston-based Velocty Correctons 3 forcng flud ncompressblty [13] snce meshes whch are often used for volume preservaton of deformable objects are napproprate because of frequent topology changes. After Müller et al. [19] presented the basc SPH method, whch suffers from unacceptable densty fluctuatons and volume loss because of the dependence on an equaton of state (EOS), Becker and Teschner [3] proposed Weakly Compressble SPH and allevated large densty fluctuatons by replacng the EOF n [19] wth a stffer EOS called Tat equaton, requrng exceedngly smaller tme steps for stable smulaton. Solenthaler et al. [23] proposed a predctvecorrectve scheme, called Predctve-Correctve Incompressble SPH (PCISPH) to use larger tme steps whle satsfyng flud ncompressblty. A smlar predctvecorrectve scheme called local Posson SPH was proposed by He et al. [10]. Bodn et al. [5] proposed a system of velocty constrants to mprove the accuracy of PCISPH [23], whle achevng ncompressble flows. To further accelerate enforcng flud ncompressblty by allowng us to use larger tme steps than those used n PCISPH [23], Mackln and Müller [15] adapted PBD for flud smulaton, and Ihmsen et al. [12] proposed Implct Incompressble SPH (IISPH), whch computes pressure values usng sem-mplct ntegraton. A method that enforces flud ncompressblty usng sem-mplct ntegraton, called Incompressble SPH (ISPH) was also proposed by Cummns and Rudman [8]. 3 Proposed Method The key of our method s to separately acheve volume preservaton and large deformatons of vscoelastc materals. We rely on a partcle-based ncompressble flud solver to preserve flud volumes whle explotng an dea of PBD [20,4] to correct partcle veloctes for vscoelastc effects wthout negatvely affectng convergence of densty fluctuatons n the flud solver. We brefly descrbe computatonal process and mportant features of partcle-based ncompressble flud solvers n 3.1 and explan our velocty correcton scheme, whch enables large deformatons of vscoelastc materals whle preservng ther volumes n 3.2. In 3.3, we summarze our algorthm, gvng a lst of smulaton procedures and mplementaton detals. 3.1 Partcle-based Incompressble Flud Solvers To preserve flud volumes, we use a partcle-based ncompressble flud solver, such as ISPH [8], PCISPH [23], and IISPH [12]. In these SPH-based ncompressble flud solvers, fluds are dscretzed by partcles (partcle has poston x, velocty v, and mass m ), and pressure p s teratvely corrected to obtan pressure force F p whch acheves small devatons of densty ρ to the rest densty ρ 0 less than a certan crteron, e.g., 0.01%. We brefly explan the flow of computatons for the solvers. For detaled smulaton procedures and dscussons, refer to the papers of partcle-based ncompressble flud solvers [8, 23, 12]. We frst compute ntermedate velocty v for partcle by v = v t + t Fother, (1) m where t denotes tme, t tme step, and F other all forces except pressure and vscoelastc ones. Next, we use an ncompressble flud solver to obtan pressure p, whch s clamped as p = 0 f p < 0 not to cause attracton forces between partcles, through teratve correctons wth ntermedate velocty v and compute pressure force F p by ( F p = m p m j ρ 2 j + p j ρ 2 j ) W j, (2) where j denotes a neghborng flud partcle and W j a kernel wth a kernel radus h. Then, we compute velocty v for ncompressble flows: v t+1 = v + t Fp m. (3) Fnally, we update partcle poston x : x t+1 = x t + tv. (4) There are two mportant requrements to note; frst, partcle velocty v needs to be updated wth pressure force F p wth Eq. (3) to ntegrate partcle poston x wth Eq. (4), achevng ncompressble flows. However, the method of Takamatsu and Kana [28] does not satsfy ths requrement because ther method mxes veloctes derved from SM, whch does not ensure ncompressble flows, and SPH, and thus resultng veloctes cannot acheve ncompressble flows. Second, partcle poston x must be fxed after neghbor search steps untl pressure force F p s computed wth Eq. (2). As explaned n [23,12], neglectng changes of partcle postons causes erroneous sets of neghborng partcles and ther nterpartcle dstances and leads to naccurate estmates of physcal quanttes, whch can make the flud solver fal to converge or delay the convergence of the teratons. Snce the method of Clavet et al. [7] uses a predcton-relaxaton scheme, whch change partcle postons, ths requrement s not satsfed.

4 4 Tetsuya Takahash et al. 3.2 Velocty Correcton Scheme Fg. 2 A ball wth dfferent vscoelastcty values dropped onto the ground. Partcles are color coded based on ther velocty correcton coefficents c (low: cyan and hgh: dark cyan). Influence for vscoelastc effects Satsfyng the two requrements mentoned above, we correct partcle veloctes to acheve vscoelastc effects wth velocty correcton vector v (see 3.2.1). We obtan ntermedate velocty v by modfyng Eq. (1): Fother v = vt + t + v. (5) m In [26], although three velocty correcton schemes (shape matchng, sprng-based, and poston-based) were proposed to compute velocty correcton vector v, we heren use the poston-based scheme because of the followng advantages over the other schemes; frst, the poston-based scheme requres fewer neghbor partcles than the shape matchng scheme and uses parwse connectons whch can smplfy boundary condtons wth other objects unlke the shape matchng scheme whch needs to construct partcle clusters ncludng farther partcles. Second, the poston-based scheme nherts propertes of PBD [20, 4] and thus s robust and easy to tune parameters as compared to the sprng-based scheme. SPH-based method Our method rj h xj Poston-based Velocty Correcton To compute velocty correcton vector v, we construct a set of parwse connectons for neghborng partcles when an object s created, wth ther ntal nterpartcle dstance rj, whch s ntalzed as rj = xj, where xj = x xj. Hereafter, we call partcles connected wth other partcles as connected partcles. We correct veloctes of connected partcles only when ther nterpartcle dstance xj s larger than ther ntal nterpartcle dstance rj (flud expanson), namely rj < xj, because flud compresson, whch generally occurs when xj < rj, s resolved to preserve flud volumes wth our flud solver. To compute v for those connected partcles such that rj < xj, we defne a dstance functon Dj as Dj = max( xj rj, 0). Then, we compute v wth a velocty correcton coefficent c (0 l c ), where l s the lower lmt of c : Fg. 3 Comparson of the nfluence for vscoelastc effects between an SPH-based method and our method. Our velocty correcton vector s bascally obtaned by dvdng x by t. By usng Dj, the velocty correcton vector becomes zero when xj < rj. We addtonally ntroduce velocty correcton coefficents c and cj to control the stffness of vscoelastc materals; lower (hgher) c leads to softer (stffer) materals (see Fg. 2). Our velocty correctons enable large deformatons that cannot be generated wth SPH-based methods, e.g., [17, 6], because the nfluence of the SPH-based methods for vscoelastc effects decreases as nterpartcle dstances get larger and fnally becomes zero owng to fnte support kernels when h < xj whle the nfluence of our method ncreases when rj < xj (see Fg. 3). Sf xj c + cj v = Dj, t j 2 m + xj (6) where Sf denotes the number of connected flud partcles to partcle. To derve Eq. (6), we adopted an dea of poston correcton used n PBD [20,4], where poston correcton vector s defned as f x = S j xj ( xj rj ). m + xj Coefficent Relaxaton When large stress s appled to vscoelastc materals, they cannot completely return to ther orgnal shapes (ths s known as plastc deformaton). To smulate ths behavor, we weaken the effect of c f l < c, takng nto account the fact that connected partcles do not restore ther postonal relaton when ther connecton s extended larger than a certan dstance. Specfcally,

5 Volume Preservng Vscoelastc Fluds wth Large Deformatons Usng Poston-based Velocty Correctons Boundary-handlng for Solds We smulate nteractons of vscoelastc fluds and solds by ncorporatng sold boundary partcles nto our algorthm. Connectons between flud and sold partcles are updated followng Algorthm 1. We compute v by extendng Eq. (6) wth sold partcles: Sf Fg. 4 (Top) Two mergng vscoelastc balls. (Bottom) A vscoelastc ball thrown toward a column, splttng nto three lumps. c + cj xj Dj v = t j 2 m + xj Ss mk xk c + ck Dk, t 2 m + mk xk (8) k we update c by f ct+1 = max(ct t S w + wj { σj = j 2 γ +γ σj, l ), 1 f 2 j < 0 otherwse Dj h (7), where w (0 w ) controls the weakenng speed of c, and γ (0 γ ) s a yeld crteron. σj works so that the weakenng happens only when normalzed extenson γ +γ Dj /h s larger than 2 j. where k denotes a neghborng sold partcle and Ss the number of connected sold partcles to partcle. Snce force-based rgd body smulators are often adopted, we compute vscoelastc force Fv to exert the antsymmetrc effect of flud partcles to sold partcle, preservng ther momentum: f Fv S m xj c + cj = 2 Dj. t j 2 m + xj (9) Computng velocty correcton vector v for oneway sold-to-flud couplng s straghtforward. Assumng mk = n Eq. (8) for a sold partcle k, whch s fxed or moves along a determned path, we obtan Sf Connecton Control We adaptvely generate and delete partcle connectons dependng on the postonal relatons of partcles to allow vscoelastc fluds to merge wth others and splt nto several lumps. For mergng, we generate a new connecton between two partcles wth α (0 < α ) f α +α xj /h < 2 j, and f there s no connecton between them. For splttng, on the other hand, we remove partβ +β cle connectons wth β (0 < β ) f 2 j < xj /h. We show our connecton control algorthm n Algorthm 1 for clarty. Mergng and splttng of vscoelastc materals are demonstrated n Fg. 4. c + cj xj v = Dj t j 2 m + xj Ss c + ck xk Dk. t 2 xk (10) k Extendng Eq. (7) wth sold partcles, we weaken the effect of c as f ct+1 = max(ct t( S j s wj σj + S wk σk ), l ), (11) k w +w where wj = 2 j. Fg. 5 llustrates the effects of the flud-sold couplng wth an example of a cube thrown toward a wall. Algorthm 1 Connecton control algorthm 1: for all flud partcle do 2: for all neghborng partcle j do α +α 3: f xj /h < 2 j then 4: f there s no connecton then 5: generate a new connecton between and j 6: for all connected partcle j do β +β 7: f 2 j < xj /h then 8: delete the connecton between and j 3.3 Smulaton Algorthm and Implementaton We gve an outlne of our procedures n Algorthm 2. In our mplementaton, we use IISPH [12] as an ncompressble flud solver and set a convergence crteron as 0.01%. For flud-sold couplng of pressure and vscosty, we employ the method of Aknc et al. [1]. We use fnte support kernels as proposed by Mu ller et

6 6 Tetsuya Takahash et al. Table 1 Smulaton condtons and performance. N f and N s denote the number of flud and sold partcles, respectvely. ttotal s total smulaton tme per frame. Fg. 5 A cube thrown toward a wall wthout (left) / wth (rght) flud-sold couplng. al. [19]. To accelerate neghbor search steps, we used a varant of the z-ndex sort algorthm presented n [11]. There are sx adjustable parameters c, l, w, γ, α, and β that we ntroduced to cover varous phenomena of vscoelastc fluds, and we bascally use values smlar to c = 0.001, l = , w = 0.001, γ = 1.01, α = 1.0, and β = 2.0, adjustng them accordng to scenaros. Fgure # 2 (left) 2 (mddle) 2 (rght) 4 (top) 4 (bottom) 5 (left) 5 (rght) 6 (top) 6 (bottom) 7 (mddle) 7 (rght) 9 (mddle) 9 (rght) N f /N s 24.5k/38.6k 24.5k/38.6k 24.5k/38.6k 45.2k/60.3k 22.6k/67.4k 13.8k/27.0k 13.8k/27.0k 7.1k/21.8k 7.1k/21.8k 8.2k/19.2k 8.2k/19.2k 14.0k/27.5k 14.0k/27.5k up to 195.1k/168.1k 33.4k/65.7k 73.5k/108.8k ttotal (s) Algorthm 2 Smulaton algorthm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: for all partcle do fnd neghborng partcles for all partcle do update partcle connectons followng Algorthm 1 f partcle s a flud partcle then obtan v wth Eqs. (5), (8), and (10) relax c wth Eq. (11) compute partcle pressure p usng IISPH for all flud partcle do compute Fp wth Eq. (2) update solds wth Eq. (9) 11: for all flud partcle do 12: ntegrate v wth Eq. (3) 13: ntegrate x wth Eq. (4) 4 Results We mplemented our method n C++ and parallelzed t usng OpenMP 2.0. All the smulatons were performed on a PC wth a 4-core Intel Core GHz CPU and RAM 16.0 GB. We used a physcally-based renderer Mtsuba to render all the scenaros. Our velocty correctons generally occupy only 1.3% of whole smulaton tme. Smulaton condtons and performance are lsted n Table Large Deformaton We compare results generated wth our method and the SPH-based method usng a non-lnear corotatonal Maxwell model proposed by Mao and Yang [17] to show that our hybrd method can handle large deformatons of a vscoelastc materal, whch cannot be smulated wth the SPH-based method [17]. Though the method of Mao and Yang [17] resampled partcles to solve an ssue of partcle dsorder, we dd not do that n our experment to show that our method can generate large deformatons wthout the need of specal treatments. Fg. 6 llustrates a vscoelastc sphere dropped onto the ground. The vscoelastc ball smulated wth the method of Mao and Yang [17] exhbts slght elastc deformatons. After the degree of the deformatons exceeds a certan threshold, the ball fals to restore ther orgnal shape and breaks nto many lumps whch consst of farly close partcles because partcles close to others exert strong forces to preserve ther postonal relatons (see Fg. 3). The method of Mao and Yang [17] needs to use strong vscoelastc forces to make the vscoelastc ball exhbt slght elastc deformatons at the expense of numercal stablty. On the other hand, our method can handle large deformatons, restorng ts orgnal shape wthout resamplng partcles nor ntroducng numercal nstablty. 4.2 Volume Preservaton We compare our method and the exstng hybrd methods [28, 7], usng a scenaro of a vscoelastc ball compressed by a movng board along a vertcal path. Comparson to [28] Frst, we llustrate the dfference between our method and the method of Takamatsu and Kana [28], who combned SPH and SM. Snce the basc SPH [19] adopted by Takamatsu and Kana [28] as ther underlyng flud solver cannot preserve flud vol-

7 Volume V (m3) Volume Preservng Vscoelastc Fluds wth Large Deformatons Usng Poston-based Velocty Correctons Takamatsu and Kana Our method Reference Ϭ Ϭϵϱ Ϭ Ϭϴϱ Ϭ Ϭϳϱ Ϭ Ϭϲϱ Ϭ Ϭϱϱ Ϭ Ϭϰϱ Ϭ Ϭϯϱ Ϭ ϬϮϱ Ϭ Ϭϭϱ Ϭ Fg. 6 Comparson for large deformaton. (Top) the method of Mao and Yang [17]. (Bottom) our method. Fg. 7 Comparson for volume preservaton. (Left) ntal state. (Mddle) the method of Takamatsu and Kana [28]. (Rght) our method. umes, we use IISPH [12] nstead of the basc SPH [19] and consder the method of [28] as a hybrd of IISPH and SM. In ths comparson, we use 100 pressure teratons n IISPH for both methods because the method of [28] s lkely to fal to preserve flud volumes even f more pressure teratons are appled. Fg. 7 shows the comparson, where the left s the ntal state, and the mddle (rght) s the result of the method [28] (our method), and Fg. 8 llustrates the volume V (whch s estmated by summng up the vol ume of all flud partcles as V = V = m /ρ ) occuped by the ball for the method of Takamatsu and Kana [28], our method, and reference (ntal volume). The volume of the ball on the mddle decreases and ncreases dependng on the movng board. By contrast, the ball on the rght preserves ts volume, whch s farly close to the reference. Addtonally, when beng compressed, the ball on the rght exhbts horzontally spreadng motons, whch cannot be generated wth the method of [28]. 7 ϱϭ ϭϭϭ Frames ϭϱϭ ϮϬϬ Fg. 8 Profles of the volume V occuped by a ball for Fg. 7 Fg. 9 Comparson for volume preservaton. (Left) ntal state. (Mddle) the method of Clavet et al. [7] (Rght) our method. Comparson to [7] Second, we compare our method and the method of Clavet et al. [7], who combned SPH wth sprng systems usng a predcton-relaxaton scheme. Smlar to the above prevous method, snce Clavet et al. [7] used the basc SPH [19], we use IISPH as ther flud solver for the same reason. Fg. 9 demonstrates the comparson, where the left s the ntal state, and the mddle (rght) s the result of the method [7] (our method). Snce errors ntroduced by poston changes durng the teraton are relatvely small and t s dfficult to clarfy vsual or volume dfferences between the prevous method [7] and our method, we compare these methods n terms of the number of teratons requred to satsfy the convergence crteron of 0.01%, as shown n Fg. 10, where profles of teraton numbers for Clavet. et al [7] nspr and our method npos, and ther dfference nspr npos are llustrated. Although our method and the method of Clavet et al. [7] can genmoreover, our method s advantageous over the method erate smlar results, the method of Clavet et al. [7] s of [28] n terms of performance (see Table 1), and there lkely to requre more teratons to enforce flud ncomare two man factors; frst, the method of Takamatsu pressblty than our method. Ths s notceable espeand Kana [28] needs more neghborng partcles to stacally when the ball s compressed and sgnfcantly deblze smulatons than our method, and processng these formed by the board because large deformatons cause partcles s an addtonal cost. Second, ther method [28] strong attracton forces and ntroduce larger postonal uses Sngular Value Decomposton, whch s tme conerrors (see Dfference n Fg. 10). The extra teratons sumng, to obtan rotatonal matrces n SM whle our of the method of Clavet et al. [7] lead to slghtly hgher velocty correctons need smple computatons only. computatonal cost than our method (see Table 1).

8 8 Tetsuya Takahash et al. Clavet et al. Our method Dfference Iteraton number ϭϯϱ ϭϭϱ ϵϱ ϳϱ Fg. 13 Barus effect of a vscoelastc materal. ϱϱ ϯϱ ϭϱ Ͳϱ Ϭ ϱϭ ϭϭϭ ϭϱϭ ϮϬϬ Frames Fg. 10 Profles of teraton number requred to satsfy the convergence crteron of 0.01% for Fg. 9. of vscoelastc flud swells wder than the dameter of an openng when the stream goes through the openng because of vscoelastc forces whch restore flud s orgnal shape after the deformatons at the openng. Although our velocty correctng method s based on approxmated dynamcs of vscoelastc flud, we can plausbly generate Barus effect. 5 Dscussons and Lmtatons Fg. 11 Spheres wth dfferent vscoelastcty values successvely thrown toward a sold dragon. Partcles are color coded based on ther velocty correcton coefficents c (low coefficent: cyan and hgh coefficent: dark cyan). Fg. 12 A vscoelastc ball thrown toward a sold cube. 4.3 Interactons Fg. 11 llustrates several spheres wth dfferent vscoelastcty values successvely thrown toward a statc sold dragon. Smlar to Fgs. 4 (top) and 5 (rght), our method can easly handle mergng of fluds and adheson of fluds to a sold by updatng parwse nterpartcle connectons. Fg. 12 demonstrates two-way nteractons of a vscoelastc materal and a sold. The sold cube s gradually pulled toward a wall by the vscoelastc materal that connects to both of the sold cube and the wall. Ths effect cannot be smulated only wth the prevously proposed boundary-handlng scheme for pressure and vscosty [1]. 4.4 Barus Effect Fg. 13 demonstrates Barus effect of a vscoelastc ball. Barus effect (also known as de swell, exclude swell, and Merrngton effect) s a phenomenon that a stream Our method combnes PBD wth SPH to take advantage of geometrc methods, such as numercal stablty and capablty of handlng large deformatons. Addtonally, unlke SPH-based methods, our method has another beneft that tme steps are not generally restrcted by vscoelastcty values, namely velocty correcton coefficent c n our method; and thus our method can use larger tme steps and be faster than SPH-based methods. In contrast to postve aspects, however, there are a few negatve ssues on the adaptaton of PBD. Frst, our method explots PBD to approxmate vscoelastc behavors nstead of usng physcally-based models. Because of ths approxmaton, our smulaton results are less accurate than SPH-based methods whch use a physcally-based model for vscoelastc effects. Second, we ntroduce several parameters whch lack physcal meanngs nto our method to cover varous effects of vscoelastc fluds. These parameters need to be adjusted dependng on each scenaro through experments to generate desrable behavors of materals. Thrd, snce results of our poston-based velocty correcton can be affected by tme steps smlar to other geometrc methods, our method would produce dfferent behavor of vscoelastc fluds dependng on tme steps. When materals are drven to extreme stuatons (e.g., a ball forcbly and sgnfcantly deformed by a heavy object), partcle penetratons can be observed because of nterpartcle connectons whch exert larger velocty changes for farther connected partcles to enable large deformatons. Although ths can be addressed by adjustng the nfluence of velocty correctons, such an adjustment could make vscoelastc materals fal to restore ther orgnal shapes from hghly deformed shapes. Most vscoelastc materals undergo phase and property changes, e.g., turnng nto a stff sold, as tme

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