Wang XK, Ban XJ, Zhang YL et al. Surface tenson model based on mplct ncompressble smoothed partcle hydrodynamcs for flud smulaton. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 32(6): 1186 1197 Nov. 2017. DOI 10.1007/s11390-017-1793-0 Surface Tenson Model Based on Implct Incompressble Smoothed Partcle Hydrodynamcs for Flud Smulaton Xao-Kun Wang, Member, CCF, Xao-Juan Ban, Member, CCF, Ya-Lan Zhang, S-Nuo Lu, and Peng-Fe Ye School of Computer and Communcaton Engneerng, Unversty of Scence and Technology Beng, Beng 100083, Chna Beng Key Laboratory of Knowledge Engneerng for Materal Scence, Beng 100083, Chna E-mal: {wangxaokun, banx}@ustb.edu.cn; {zhang.yalan, 787666673, 1434497449}@qq.com Receved June 23, 2017; revsed September 28, 2017. Abstract In order to capture stable and realstc mcroscopc features of flud surface, a surface tenson and adheson method based on mplct ncompressble SPH (smoothed partcle hydrodynamcs) s presented n ths paper. It gves a steady and fast tenson model and can solve the problem of not consderng adheson. Molecular coheson and surface mnmzaton are consdered for surface tenson, and adheson s added to show the mcroscopc characterstcs of the surface. To smulate surface tenson and adheson stably and effcently, the surface tenson and adheson model s ntegrated to an mplct ncompressble SPH method. The expermental results show that the method can better smulate surface features n a varety of scenaros compared wth prevous methods and meanwhle ensure stablty and effcency. Keywords computer anmaton, flud smulaton, mplct ncompressble smoothed partcle hydrodynamcs (IISPH), surface tenson 1 Introducton Surface tenson s one of the most common and mportant physcal characterstcs to reveal mcroscopc effects of fluds. Wth the ncreasng demand for detals and realstc effects of flud smulaton, the smulaton of surface tenson s becomng more and more attractve. Surface tenson s produced by the coheson among the flud molecules. Because molecules n flud nteror are acted by forces from all drectons, they reach mechancal equlbrum. Whle the resultant force of flud molecules on the surface s not zero, t ponts to flud nteror and generates the surface tenson. In addton, accordng to the Laplacan law, surface tenson s capable of mnmzng surface area. For example, the droplets shrnk nto a sphere when they are not subected to external force. Usng SPH (smoothed partcle hydrodynamcs) method to smulate surface tenson realstcally, however, s stll a challengng problem. The computed densty of partcles at the flud-ar nterface s lower than ts real value, whch s caused by lackng neghbor partcles. It results n the generaton of negatve pressure and causes partcle clusterng. In addton, the smulaton of mcroscopc characterstcs takes a large amount of calculaton, and has the problem of tme step restrcton and numercal nstablty. For these reasons, we propose a surface tenson method based on IISPH (mplct ncompressble SPH) whch mproves the computatonal effcency and stablty, and obtans a good surface tenson and adsorpton effect. Nowadays, surface tenson models are manly dvded nto two categores: macroscopc model and mcroscopc model. The macroscopc model s also known as contnuous surface force model (CSF) [1], whch s a knd of color feld method. The value of the color feld changes sharply at two-phase nterfaces [1-4]. The Regular Paper Specal Secton of CAD/Graphcs 2017 Ths work was supported by the Natonal Natural Scence Foundaton of Chna under Grant Nos. 61572075 and 61702036, the Natonal Key Research and Development Program of Chna under Grant Nos. 2016YFB0700502 and 2016YFB1001404, and the Fundamental Research Funds for the Central Unverstes of Chna under Grant No. 2302017FRF-TP-17-012A1. Correspondng Author 2017 Sprnger Scence + Busness Meda, LLC & Scence Press, Chna
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1187 macroscopc model frst nterpolates the value of partcles color feld by SPH formulas. Next, the normal vector of the surface s calculated accordng to the color feld gradent. Then surface curvature and surface tenson could be constructed. But the surface s normal vector usually has devatons due to the SPH gradent formula, especally when the surface has sharp corners and few partcles. Ths results n a large computatonal error n surface curvature. Meanwhle the macroscopc model creates an asymmetrc force that s appled to flud partcles, whch could not ensure momentum conservaton. The mcroscopc model generates surface tenson by the coheson among partcles whch s mtatng molecular attracton [5-7]. Compared wth the macroscopc model, the mcroscopc model avods the calculaton of color feld s second dervatve and surface curvature whch are very senstve to partcles derangement. Moreover, the mcroscopc model s smple to mplement and more effcent. Nugent and Posch [5] used van der Waals equaton of state to calculate attracton pressure, and extended the calculaton range of attracton to four hours to obtan stable droplet effect. Tartakovsky and Meakn [7] proposed a molecular coheson model to produce surface tenson n numercal smulaton. It controls the attracton and repulson between partcles by cosne functon. Becker and Teschner [6] employed SPH kernel nstead of cosne functon to make range of attracton wthn smoothng length h. Moreover, the coheson models proposed by both Tartakovsky and Meakn [7] and Becker and Teschner [6], showed repulsve force n short range and attracton n long range. However, the mcro and macro models cannot well smulate large surface tenson and the surface curvature mnmum effects at the same tme. In addton, they may cause problems such as partcle clusterng and momentum volaton. Aknc et al. [8] presented a coheson-repulson model that can better deal wth those problems. They also consdered the adheson between flud and sold n ther model. Because the stablty and effcency s stll unsatsfactory, we combne the mplct ncompressble SPH method (IISPH) wth the mproved surface tenson and adsorpton force on the bass of Aknc et al. s [8]. Our method can better smulate flud surface s mcroscopc features and meanwhle mprove the numercal stablty and computatonal effcency. 2 Related Work Snce the surface tenson has an mportant effect on the detals of flud smulaton, many studes have explored the surface tenson methods that are approprate for flud smulaton. In SPH flud smulaton scope, early methods mnmze the surface curvature usng forces to acheve the surface tenson effects [2-3]. However, these methods have some problems. For nstance, the curvature calculaton s very senstve to partcle dsorder, and the external force appled to flud partcles s asymmetrcal and does not meet momentum conservaton. To ths end, researchers proposed only usng coheson of adacent flud partcles at the molecular level to solve the problems of early surface tenson model [6-7]. However, only usng coheson between flud partcles cannot ensure the surface mnmzaton. What s worse, t results n unreal flow patterns when the surface tenson s large. Clavet et al. [9] proposed a method that produces surface tenson by attracton. The proposed double densty relaxaton method can acheve strong surface tenson effects. But t s not sutable for smulatng low vscous lquds. Yu et al. [10] proposed a surface tenson method based on SPH. The curvature s estmated on the surface mesh and the surface tenson s appled to the adacent flud partcles surrounded by grds. But ths surface trackng method may not be able to detect solated flud blocks so that the surface tenson effect cannot be exhbted n these areas. Besdes, the surface tenson effect depends on the resoluton of trackng grds. Aknc et al. [8] skllfully created surface tenson by partcles nteracton. Ther method can handle large surface tenson whle mantanng momentum conservaton. It generates repulson for partcles too close and prevents partcle cluster problem of free surface wthout addtonal operaton. The adsorpton force makes fluds attracted by other substances. Steele et al. [11] proposed a Lagrangan method for vscous fluds that acheves the flud-tosold adsorpton effects. They used dstance-dependent forces to defne the adsorpton propertes of dfferent type substances. But ths method has dffcultes n smulatng hgh vscous fluds for usng lnear densty cores and strct ant-penetraton constrants. Subsequently, Clavet et al. [9] modeled the vscoelastc SPH flud s adheson through a dstance-based attracton term. Schechter and Brdson [12] realzed the flud-tosold adheson effect by calculatng ghost velocty at each sold partcle, whch s computed usng the velocty of sold and the tangental component velocty of the
1188 J. Comput. Sc. & Technol., Nov. 2017, Vol.32, No.6 nearest flud partcle. Then the XSPH method based on artfcal vscosty s used to calculate adheson. In the method of He et al. [13], the effect of adheson s acheved by edge effect of dfferent slp condtons wth velocty constrant. The method proposed by Aknc et al. [8] acheves a physcally reasonable flud-sold adheson wthout addtonal treatments (e.g., GhostSPH). And forces are symmetrcally appled to the adacent par of flud-flud partcles as well as flud-boundary partcles to ensure momentum conservaton. Recently, Yang et al. [22] proposed a refned surface tenson model usng parwse forces, whch accurately captures surface tenson wthout extra forces or constrants. 3 SPH Framework IntheLagrangandescrpton [3], flowcontrolledpartal dfferental equatons of Naver-Stokes for fluds can be expressed as dρ dt = ρ v, (1) ρ Dv Dt = p +ρ g +µ 2 v, (2) where v s the velocty, ρ s the densty, p s the pressure, µ s the vscosty coeffcent, and g represents the external force feld. (1) s the mass conservaton equaton and (2) s the momentum conservaton equaton. The theory of SPH [3] s to utlze the form of dscrete partcles to characterze the successve felds and use ntegraton to approxmate the felds. For partcle at locaton x, A(x ) = m A ρ W (x x, h), (3) where m and ρ represent the partcle mass and denstyrespectvely, W (x x, h) sthesmoothngkernel and h s the smoothng radus. Applyng (3)tothe denstyofapartcleatlocaton x yelds the summaton of densty ρ = f P = m W (x x, h). Thus, forces between partcles ncludng pressure f P and vscous force f v can be represented as ) W, f v = µ m ( P ρ 2 + P ρ 2 m v ρ 2 W. We employ Tat equaton [6] to calculate the pressure, that s, p = ρ 0c 2 (( ) γ S ρ 1), γ where ρ 0 = 1000s the rest densty ofthe flud, γ = 7 s the stffness parameter and c S s the velocty of sound. We use the equaton n [6] to compute vscous force. 4 Surface Tenson and Adheson Model 4.1 Surface Tenson In partcle-based flud smulaton, the surface tenson model usually adopts the method based on color feld, any partcle s assgneda color feld value c, and the same flud partcles have the same color of the feld value. Takngc ntothe SPHnterpolatonformula, we can get the color feld nterpolaton formula as follows: c = m c W. (4) ρ The normal vector of surface calculated by n = c, can make the normal vector of the surface pont to the nteror of the flud. And surface curvature can be measured by the dvergence of the normal vector. Its form s shown as follows: ρ 0 κ = 2 c. (5) n The surface tenson of flud can be structured by the normal vector and the surface curvature s shown as follows: F s = σκn = σ 2 c n n. Although the model acheves the effect of surface tenson through the color feld easly, t has some shortcomngs. Frstly, less neghbor partcles can cause calculaton error of 2 c. Besdes, the calculaton of the second dervatve s senstve to the partcle s dsorder. Tartakovsky et al. [7] proposed a coheson model to control the attracton and repulson between the partcles by the cosne functon. However, ths method causes clusterng phenomenon. Becker et al. [6] used SPH kernel functon nstead of cosne functon to create attracton. It s sutable for free surface wth hgh curvature, and can calculate the surface tenson effcently. But due to the lack of repulsve force, t wll cause severe clusterng phenomenon. (4) and (5) only consder the coheson, and can nether show the effect of surface tenson truly nor guarantee the mnmum of surface area. Therefore, we use a
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1189 surfacetensonmodelthatssmlartoaknc et al. s [8]. It consders molecular coheson-repulson as well as surface mnmzaton. Frst, the coheson force model consders the effect of attracton and repulson, smlar to the ntermolecular forces, makng the partcles attract each other when they are far away, or repel each other when they are too close. Ths avods force becomng too rgd, and t avods unstable problems as well. Its form s shown as follows: F c = αm m (x x )d( x x ), (6) where s the neghbor partcle of, m s the partcle s qualty, x s the partcle s poston, α s the surface tenson coeffcent, and d s the splne functon. It can be seen that the splne functon d determnes the nature of the surface tenson F c, and ts acton s smlar to that of the kernel functon. Becker et al. [6] used the SPH nuclear functon to construct the surface tenson, whose effect s not smlar to that of the ntermolecular force. Therefore f we use (6) to calculate the surface tenson, we need to construct splne functon d to show molecular nter-atomc forces. For comparson, we choose splne functon proposed by Aknc et al. [8] 4.2 Surface Tenson Modfcatons The above surface tenson model can construct the effect smlar to molecular nter-atomc forces do. But n order to smulate the mcroscopc features of flud surface better, and to show the area mnmzaton effect of flud, we stll need to add the correcton term to surface tenson. In order to avod calculatng the surface curvature n an explct way, the normal vector s expressed as follows, whch s smlar to the normal vector calculaton method accordng to the color feld: n = κ m W ( x x ), ρ where κ s the zoom factor. The surface tenson tends to reduce the curvature, and make dfferent dscrete samplng ponts have consstent drecton. Besdes, the curvature s proportonal to normal dfference, and we construct modfcatons as follows: F f = βm (n n ), where β s the correcton coeffcent. As we can see, the correcton term ncreases wth the ncrease of the curvature. Its value s 0 at the flat areas and the nteror of flud. It avods the standardzaton of the normal n and the explct calculaton of curvature. From the above, the revsed surface tenson can be expressed as: ( ) F cf = γ F c +F f, (7) where γ s the surface tenson control coeffcent, and we adopt γ = 2ρ 2 0 /(ρ2 +ρ2 ). γ > 1 represents beng near the flud surface, and γ 1 ndcates beng nsde the flud. Therefore, we can use γ to enlarge the surface tenson of the partcles whose neghbors are nsuffcent. 4.3 Adheson Between Flud and Sold Dfferent from surface tenson, adsorpton force s generated by molecular nteractons of dfferent materals. In ths subsecton, the adsorpton force model s manly amed at the effect between flud and sold. In ths paper, we process the flud-sold couplng smulaton n the followng steps: 1) samplng rgd body surface as boundary partcles, 2) usng the boundary treatment formulas proposed by Aknc et al. [15] to calculate flud densty, n whch the boundary partcles are consdered, and 3) dervng each force formula. The flud-sold couplng method can avod the adheson effect of flud partcles at the rgd boundary effectvely. Thus ths subsecton uses an adsorpton force calculaton formula whch can be appled to the flud-sold couplng method drectly [8]. F a = ηm ψ bk (x x k )y( x x k ), (8) k where k s a boundary partcle, x s the poston of the partcle, η s an adsorpton parameter, and ψ bk s the volume of the boundary partcle. y s the splne functon. We construct y as follows: y(r) = 0.01 h 5 { ( r 3 4 h) 2 + h 2 16, f h 2 < r h, 0, otherwse. Due to that usng the boundary treatment methods can solve the adheson and clusterng effect n border of flud partcle effectvely, the above type ust mposes the adsorpton effect on the partcles from h/2 to h to attract each other. You can see that the adsorpton s symmetrcal, namely F a k = Fa.
1190 J. Comput. Sc. & Technol., Nov. 2017, Vol.32, No.6 5 Surface Tenson and Adheson Based on IISPH Smlar to the method proposed by Aknc et al. [8] our surface tenson and adsorpton force model can mnmze the surface area, prevent clusterng, and realze the surface tenson and adsorpton effect realstcally. However, f the surface tenson or the adsorpton force of the partcles s the man force, tme step wll be lmted. An example s water droplets. Aknc et al. used the predctve-correctve n compressble SPH method to smulate flud, and used the adaptve tme step methods [15] to mprove the algorthm effcency, but there are stll some problems such as numercal nstablty, low effcency and so on. Therefore, n order to mprove the numercal stablty and computatonal effcency of the surface mcroscopc characterstcs smulaton, we propose the surface tenson and adsorpton algorthm based on IISPH n ths secton. The framework based on SPH flud smulaton has two classes. One class s based on the equaton of state, such as weakly compressble SPH (WCSPH), PCISPH [6,16]. The other knd s based on the pressure proecton method, such as ncompressble SPH (ISPH) [17-20]. The dea of ISPH method can be dvded nto the followng steps: 1) usng the force n addton to the pressure to predct the ntermedate velocty of partcle, 2) solvng the pressure Posson equaton, and 3) usng the pressure to solve other physcal quanttes. But ths knd of method has hgh computatonal cost. Therefore, Ihmsen et al. [21] proposed the IISPH method. They combned the contnuty equaton dscretzed by SPH wth the symmetrc SPH pressure to get the pressure Posson equaton (PPE). Then they used relaxaton Jacob teraton to solve the pressure. IISPH can use a large tme step and can make the densty devaton less than 0.01%. Usng an mplct method n the pressure teraton, IISPH mproves the effcency and stablty of the smulaton sgnfcantly. The IISPH method wll frstly dscrete the flud contnuty equaton Dρ/Dt = ρ v nto ρ (t+ t) ρ (t) t = m (v (t+ t) v (t+ t)) W (t). (9) The speed dfference v (t+ t) v (t+ t) n (9) depends on the pressure force F p (t) at tme t, and the pressure force s dependent on pressure p(t). Usng sem-mplct Euler ntegraton to update speed, we can express the speed n (9) as follows: v (t+ t) = v (t)+ t Fo (t)+fp (t) m, where F o (t) means the forces n addton to the pressure force, ncludng surface tenson, adsorpton force, vscous force, gravty, etc. Accordng to F o (t), we forecast ntermedate speed of flud partcles as follows v (t+ t) = v (t)+ t Fo (t) m. (10) Accordng to the form of(9), the ntermedate speed can deduce ntermedate densty of flud partcles as follows: ρ (t+ t) = ρ (t)+ t m (v (t+ t) v (t+ t)) W (t). (11) We make ρ (t+ t) = ρ 0, and use the ntermedate densty ρ (t+ t) nstead of ρ (t). Then we take them nto (9), whch can be expressed as ρ 0 ρ (t+ t) t = ( m v (t+ t) v (t+ t)) W (t). After reducng, the dfference of the densty s: ( t 2 F p m (t) Fp (t) ) W (t) m m = ρ 0 ρ (t+ t). (12) Next, we need to solve the above formula and get pressure value n a certan range of densty fluctuatons. Takng the pressure term nto the above formula, we can get the followng system of lnear equatons: a p = ρ 0 ρ (t+ t). (13) Itcanbeseenthatonlypressurevaluesareunknown n ths system. (13) can be solved by relaxaton Jacob teraton, namely p l+1 = (1 ω)p l + ω 1 ρ 0 ρ (t+ t) a p l, (14) a where l s the number of teratons and ω s the coeffcent of relaxaton.
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1191 In order to calculate (14), we need to determne a and a p l. Thus the dsplacement of pressure s expressed as where t 2Fp (t) m = t 2 = t 2 ( = d p + d = t 2 m ( p ρ 2 m ρ 2 + p W ρ 2 ) p + t 2m ρ 2 W )p W d p, (15) m ρ 2 W, d = t 2m ρ 2 W. We substtute (15) nto (12), and k dentfes neghbors of ρ 0 ρ (t+ t) = p m (d d ) W + m a and p l+1 a = d p d p k can be expressed as follows: d k p k W. m (d d ) W, (16) p l+1 = (1 ω)p l +ω 1 a ( ρ 0 ρ (t+ t) ( m d p l d p l ) d k p l k ) W. (17) k The surface tenson and adsorpton force algorthm based on IISPH s shown n Algorthm 1. The algorthm s calculaton process n each tme step s shown as follows: 1) searchng neghbor partcles of each flud partcle to get ts neghbor partcle collecton N (t); 2) calculatng the densty ρ (t) of each flud partcle and the resultant forces except pressure F o (t) = F cf (t)+f a(t)+fv (t)+g, and calculatng surface tensonf cf (t) andadhesonf a (t) accordngto(7)and (8) respectvely; 3) computng v, d accordng to (10) and (15) respectvely; 4) calculatng the ntermedate densty ρ (t) and a of each flud partcle accordng to (11) and (16) respectvely, and computng the ntal pressure as well; usng Jacob teraton to solve the pressure Posson equaton and calculatng d p l accordng to (15); computng ρ l+1 usng densty formula, and calculatng p l+1 and updatng t accordng to (17) untl the average densty devaton ρ err avg s less than the threshold η; 5) updatng the speed v (t+ t) and poston x (t+ t) of each flud partcle. Algorthm 1. Surface Tenson and Adheson Algorthm Based on IISPH 1: whle anmatng do 2: for each partcle do 3: Search neghbor partcles of, get N (t) 4: for each partcle do 5: Calculate the densty ρ (t) 6: Calculate the resultant force n addton to the pressure force F o (t) 7: Calculate ntermedate speed v 8: Calculate d 9: for each partcle do 10: Calculate ntermedate densty ρ (t) 11: Set the ntal value p 0 = 0.5p (t t) 12: Calculate a 13: l = 0 14: whle ρ err avg > η or l < 2 do 15: for each partcle do 16: Calculate d p l 17: for each partcle do 18: Calculate ρ l+1 19: Calculate p l+1 20: p (t) = p l+1 21: Calculate ρ err = ρ l+1 22: Calculate ρ err avg = 1 n 23: l = l +1 24: for each partcle do ρ 0 ρ err 25: v (t+ t) = v + tfp (t) m 26: x (t+ t) = x (t)+ tv (t+ t)
1192 6 J. Comput. Sc. & Technol., Nov. 2017, Vol.32, No.6 and use Blender to mplement offlne renderng. All expermental scenaros parameters n ths secton are shown n Table 1. Implementaton and Results Ths secton verfes the effectveness of the proposed surface tenson and adsorpton force method. The program operaton platform s the Intelr Xeonr E52687W v4 (8 core, 3.0 GHz, 20 MB cache) wth 72 GB memory. The surface constructon algorthm and the smulaton algorthm are mplemented n C++ language usng mult-threadng technology. The smulaton algorthm uses space background grd hash lookup to search neghbor partcles. We use the ansotropy method for surface reconstructon, employ OpenGL 3D graphcs lbrary to acheve real-tme dsplay smulaton, (a) Table 1. Settng and Statstcs of Two-Phase Breakng Dam Smulaton Item Smulaton doman sze Flud densty Smooth and kernel functon Smooth radus Wdth of the flud partcles Value 8m 8m 8m 1 000 kg/m3 Cubc splne functon 0.2 m 0.1 m Fg.1 shows the experment s partcles fgure of a square water flow on the tablet. At the begnnng, the (b) (c) Fg.1. Square water flow on the tablet. (a) No surface tenson and adsorpton. (b) Aknc et al. s method[8]. (c) Our method.
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1193 fallng water contacts wth the tablet gradually, and then the flud tles along the tablet gradually. Because of the surface tenson and the functon of adsorpton force, flud tends to become statc and form sngle-layer flud on the tablet. Fg.1(a) s the results of the IISPH method wthout surface tenson and adsorpton. It can be seen that f we gnore the functon of surface tenson and adsorpton force, partcles are n a relatvely decentralzed state, and the mcroscopc characterstcs of the flud are worse than those n Fgs.1(b) and 1(c). Fg.1(b) sthe resultsofakncet al. [8], andfg.1(c) s our results. It can be found that under the same condton, f partcles collde volently, our method s more stable than the method of Aknc et al. [8], and constrnged much faster (t can be seen from Fgs.8(b) and 8(c)). It shows that when dealng wth volent scenes, our smulatng model s stable and fast. In addton, the shape of the flud under steady state s manly decded by the surface tenson and the adheson that make the curvature of dfferent drectons become consstent, and mnmze the surface area. It can be seen n the fgure that our method s better than the method proposedby Aknc et al. [8] The effect of mnmzng the surface s better, and the whole smulaton process s more stable. Fg.2 shows the renderng results of square water flow on the tablet. They correspond to the partcle dagram n Fg.2. It can be seen from the fgure that flud looks more loose wthout surface tenson (Fg.2(a)). (a) (b) (c) Fg.2. Renderng results of square water flow on the tablet. (a) No surface tenson and adsorpton. (b) Aknc et al. s method [8]. (c) Our method.
1194 J. Comput. Sc. & Technol., Nov. 2017, Vol.32, No.6 Especally on the flud edge, the mcro effect s very poor and does not conform to the flud characterstcs n the realty. But after addng adsorpton and surface tenson force, detaled effect on flud edge s mproved sgnfcantly, as shown n Fgs.2(b) and 2(c). Comparng Fg.2(b) wth Fg.2(c), we can fnd that flud surface smulated by ourmethod s more smooth (shown by the last two pctures of the frst row), and the surface mnmzaton effect s much better (the last two pctures of the thrd row and the fourth row). Besdes, the stablty of our method s better (the last two pctures of the second row). The experment shown n Fg.3 s small peces fallng nto the water. The water crown phenomenon wll appearnthsprocess. It canbeseenfromfg.3(a)that the water block does not produce deformaton wthout the surface tenson. When collson occurs, the partcles are n a dsperse state wthout mutual constrants, because ofthe lack ofsurface tenson. It does not conform to the realty scene. After addng the surface tenson, the flud surface effect s mproved markedly, as shown nfgs.3(b)and3(c). Itcanbeseenfromthefrstrowof Fg.3 that the water block transforms to mnmze surface area due to the effect of surface tenson. Surface tenson makes the water block ntalzed as a square tend to sphere n free-fall process. In addton, due to the constrant of the surface tenson, flud partcles attract each other, and are controlled by attracton or repulson to keep the balance. Table 2 shows the comparson of expermental results ncludng square water flow on the tablet and small peces of wood fallng nto the water. It can be seen that our method can shorten the operaton tme of smulaton and mprove the effcency obvously under the same scene parameters. (Please refer to the vdeo rst.wmv provded onlne as supplementary materal at the webpage of the paper n http://www.sprnger.com/ournal/11390.) (a) (b) (c) Fg.3. Small cube flud fallng nto the water. (a) No surface tenson and adsorpton. (b) Aknc et al. s method [8]. (c) Our method.
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1195 Table 2. Settng and Statstcs of Two-Phase Breakng Dam Smulaton Scenaro Method Number of Partcles Computng Tme (s) Square water flow on the tablet Aknc et al. s method [8] 5491 331 Our method 5 491 224 Small peces fallng nto the water Aknc et al. s method [8] 32751 794 Our method 32 571 442 Fg.4 shows the experment of the water mpact plate. Ths method can realze the effect of water flow by addng the sngle layer flud at dfferent tme ponts, and does not show the expermental tme. Smlar to the prevous two experments, the experment compares Fg.4(a) wth Fg.4(b) and Fg.4(c) to show that the mcroscopc characterstcs of the flud surface are more realstc when the surface tenson and adsorpton force of the flud are added. Compared wth Aknc et al. s method [8], the flud stablty s better and the flud splash effect s not ntense (the last two pctures of the second row and the thrd row). In addton, t can also be further verfed by the last two pctures of the fourth row that the flud regon mnmzaton effect of our method s better and has a greater flud adheson area compared wth Aknc et al. s method [8]. (Please refer to the vdeo rst.wmv provded onlne as supplemental materal at the webpage of the paper n http://www.sprnger.com/ournal/11390.) (a) (b) (c) Fg.4. Splled water on the board. (a) No surface tenson and adsorpton. (b) Aknc et al. s method [8]. (c) Our method.
1196 J. Comput. Sc. & Technol., Nov. 2017, Vol.32, No.6 7 Conclusons We proposed a model of surface tenson and adheson based on IISPH for flud smulaton. Ths method mproved the surface tenson method proposed by Aknc et al. [8] It s mplemented by combnng our surface tenson model wth the mplct ncompressble SPH method (IISPH). Our method can smulate the flud surface tenson and attracton more vvdly, and mprove the numercal stablty and computatonal effcency. The expermental results showed that the method can smulate the surface tenson and the adsorpton of the flud better n a varety of scenaros. Compared wth Aknc et al. s method [8], our method has hgher effcency and better stablty, and can show mcro characterstcs, such as mnmzng the flud surface, n a better way. In addton, realzed by combng the mplct ncompressble SPH, the surface tenson and adsorpton can keep a good performance n a larger and more ntense scene. Furthermore, t can be easly added to other SPH methods. There stll has the problem n smulatng multphase fluds f the surface tenson s the domnant force actng on partcles. So future work wll be makng t sutable for multphase fluds. References [1] Brackbll J U, Kothe D B, Zemach C. A contnuum method for modelng surface tenson. Journal of Computatonal Physcs, 1992, 100(2): 335-354. [2] Morrs J P. Smulatng surface tenson wth smoothed partcle hydrodynamcs. Internatonal Journal for Numercal Methods n Fluds, 2000, 33(3): 333 353. [3] Müller M, Charypar D, Gross M. Partcle-based flud smulaton for nteractve applcatons. In Proc. the 2003 ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton, July 2003, pp.154-159. [4] Müller M, Solenthaler B, Keser R et al. Partcle-based flud-flud nteracton. In Proc. the 2005 ACM SIG- GRAPH/Eurographcs Symposum on Computer Anmaton, July 2005, pp.591-594. [5] Nugent S, Posch H A. Lqud drops and surface tenson wth smoothed partcle appled mechancs. Physcal Revew E, 2000, 62(4): 4968-4975. [6] Becker M, Teschner M. Weakly compressble SPH for free surface flows. In Proc. the 2007 ACM SIG- GRAPH/Eurographcs Symposum on Computer Anmaton, Aug. 2007, pp.209-217. [7] Tartakovsky A, Meakn P. Modelng of surface tenson and contact angles wth smoothed partcle hydrodynamcs. Physcal Revew E, 2005, 72(2): 254-271. [8] Aknc N, Aknc G, Teschner M. Versatle surface tenson and adheson for SPH fluds. ACM Transactons on Graphcs, 2013, 32(6): Artcle No. 182. [9] Clavet S, Beaudon P, Pouln P. Partcle based vscoelastc flud smulaton. In Proc. the 2005 ACM SIG- GRAPH/Eurographcs Symposum on Computer Anmaton, July 2005, pp.219-228. [10] Yu J, Wotan C, Turk G et al. Explct mesh surfaces for partcle based fluds. Computer Graphcs Forum, 2012, 31(2): 815-824. [11] Steele K, Clne D, Egbert P K et al. Modelng and renderng vscous lquds. Computer Anmaton & Vrtual Worlds, 2004, 15(3/4): 183-192. [12] Schechter H, Brdson R. Ghost SPH for anmatng water. ACM Transactons on Graphcs, 2012, 31(4): Artcle No. 61. [13] He X, Lu N, Wang G et al. Staggered meshless sold-flud couplng. ACM Transactons on Graphcs, 2012, 31(6): 439-445. [14] Lu G R, Lu M B. Smoothed Partcle Hydrodynamcs: A Meshfree Partcle Method. World Scentfc, 2004 [15] Aknc N, Ihmsen M, Aknc G et al. Versatle rgd-flud couplng for ncompressble SPH. ACM Transactons on Graphcs, 2012, 31(4): Artcle No. 62. [16] Solenthaler B, Paarola R. Predctve-correctve ncompressble SPH. ACM Transactons on Graphcs, 2009, 28(3): 341-352. [17] Bodn K, Lacoursere C, Servn M. Constrant fluds. IEEE Transactons on Vsualzaton & Computer Graphcs, 2012, 18(3): 516-526. [18] Cummns S J, Rudman M. An SPH proecton method. Journal of Computatonal Physcs, 1999, 152(2): 584-607. [19] Premžoe S, Tasdzen T, Bgler J et al. Partcle-based smulaton of fluds. Computer Graphcs Forum, 22(3): 401-410. [20] Losasso F, Talton J O, Kwatra N et al. Two-way coupled SPH and partcle level set flud smulaton. IEEE Transactons on Vsualzaton & Computer Graphcs, 2008, 14(4): 797-804. [21] Ihmsen M, Cornels J, Solenthaler B et al. Implct ncompressble SPH. IEEE Transactons on Vsualzaton & Computer Graphcs, 2014, 20(3): 426-435. [22] Yang T, Ln M C, Martn R R et al. Versatle nteractons at nterfaces for SPH-based smulatons. In Proc. ACM Sggraph/Eurographcs Symposum on Computer Anmaton, July 2016, pp.57-66. Xao-Kun Wang receved hs Ph.D. degree n computer scence and technology from Unversty of Scence and Technology Beng, Beng, n 2017. Currently, he s a postdoctoral researcher and lecturer at Unversty of Scence and Technology Beng, Beng. Hs research nterests nclude computer graphcs and vrtual realty.
Xao-Kun Wang et al.: Surface Tenson Model Based on IISPH for Flud Smulaton 1197 Xao-Juan Ban receved her Ph.D. degree n computer scence and technology from Unversty of Scence and Technology Beng, Beng, n 2003. She s a professor at Unversty of Scence and Technology Beng, Beng. Her research nterests nclude artfcal ntellgence, human-computer nteracton, and vrtual realty. S-Nuo Lu receved her B.S. degree n computer scence and technology from Unversty of Scence and Technology Beng, Beng, n 2016. She s currently a Ph.D. canddate n computer scence and technology of Unversty of Scence and Technology Beng, Beng. Her research nterests nclude computer graphcs and vrtual realty. Ya-Lan Zhang receved her B.S. degree n computer scence and technology from Unversty of Scence and Technology Beng, Beng, n 2014. She s currently a Ph.D. canddate n computer scence and technology of Unversty of Scence and Technology Beng, Beng. Her research nterests nclude computer graphcs and vrtual realty. Peng-Fe Ye receved hs B.S. degree n computer scence and technology from Unversty of Scence and Technology Beng, Beng, n 2015. He s currently an M.S. canddate n computer scence and technology of Unversty of Scence and Technology Beng, Beng. Hs research nterests nclude computer graphcs and vrtual realty.