A Fast Algorithm for the Discrete Element Method by Contact Force Prediction

Transcription

1 A Fast Algorit for te Discrete Eleent Metod by Contact Force Prediction C. Tooro 1 Departent of Resources and Environental Engineering, Waseda University K. Oaya and J. Sadai Departent of Geosyste Engineering, Graduate Scool of Engineering, Te University of Toyo Abstract Te discrete eleent etod (DEM) taes enorous calculation tie because it requires a very sall tie step, one sall enoug to represent te large frequency in te contact dynaic odel. In general, te equations of otion of particles are solved using te second-order Adas-Basfort etod, wic estiates te values of contact force in te following calculation tie by linear extrapolation, or by ulti-step etods suc as te predictor-corrector etod. Inspired by tese two conventional etods, we propose a Contact Force Prediction Metod tat aes a larger tie step possible. Our etod uses te predicted values of contact force at every contact point, wic are exact solutions or nuerical solutions of differential equations tat represent two particle contacts. It as been confired experientally tat te proposed etod gives reasonable results of pacing and discarge siulations, and accelerates DEM calculation 38 ties. Keywords: Discrete Eleent Metod, Calculation Tie, Tie Step, Nuerical Integration, Fast Algorit 1. Introduction Discrete Eleent Metod (DEM) siulations on a nuber of granular systes ave reported positive results since tis etod can estiate any effects at particle level 1). However, it requires enorous coputation tie because it needs a sufficiently sall tie step to follow ig fluctuations in te contact dynaic odel between particles. To overcoe te deficit of conventional DEM, speeding up DEM is proposed in tis study. Tere are several approaces to speeding up DEM, suc as prooting efficiency in te algorit to detect contact between particles, increase in te tie step and liitation of te nuber 3-4-1, Oubo, Sinjuu-u, Toyo, , Japan 1 Corresponding autor TEL: FAX: E-ail: tooro@aoni.waseda.jp Tis report was originally printed in J. Soc. Powder Tecnology, Japan, 4, (3) in Japanese, before being translated into Englis by KONA Editorial Coittee wit te perission of te editorial coittee of te Soc. Powder Tecnology, Japan. of particles to calculate. In tis study, we report a etod to increase te tie step. In DEM calculation, te tie step ust be set saller tan te particle size and te density becoes saller or te spring constant becoes larger, in order to obtain a stable nuerical solution. Several etods to increase te tie step ave been reported, suc as etods to set te spring constant saller )-4) and te particle size larger 5) tan te real value. Tis etod is an effective etod of increasing te tie step, but it ust be used after careful exaination of te influence on te penoenon to wic DEM calculation is applied, because delays in contact detection cannot be avoided. Te purpose of tis study is to develop an original DEM in wic a large tie step is possible by iproving te algorit needed to obtain te nuerical solution to te contact dynaic equation between particles. We propose a Contact Force Prediction Metod in wic predicted values of contact force at every contact point are calculated by a ore stable etod tan te conventional DEM and used to solve particle velocity and location. Te pacing (Fig. 1) and discarge (Fig. ) syste is cosen ere as an exaple 18 KONA No.3 (5)

2 of a penoenon in wic contact between any particles is iportant, and it is sown tat te proposed etod is effective for calculating te pacing ratio or te discarge rate.. Delays in contact searc and stability of calculation Wen te tie step is increased, DEM calculation ay becoe incapable of reproducing a real penoenon for te following reasons: one is delays in te contact detection and anoter is te error of te differential etod. In te following, te effects of tese factors wit te conventional etod are exained. Fig. 1 Fig. An exaple of pacing siulation.3 Hole. An exaple of discarge siulation.1 Outline of DEM Contact force f cij in DEM is calculated by projecting on te noral direction (f nij ) and te tangential direction (f sij ) as sown below: f cij (t)f nij (t)f sij (t), (1) were (t) eans a function of tie t. Te noral direction force is calculated fro te following equations: f nij (t) nij d a 1 nij(t)n ij (t) nij d a nij(t)v nij (t) () d nij (t)(r i r j ) x i (t)x j (t) (3) v ij (t)v i (t)v j (t) (4) v nij (t)v ij (t) n ij (t)n ij (t), (5) were is te spring constant, is te daping constant, r is te radius of te particle, d is te overlap between particles, n is te noral unit vector, v is te velocity vector, and x is te location vector. Subscript i denotes te pysical values of particle i; j denotes particle j; ij denotes between particles i and j, n denotes te noral direction value, and r denotes te relative value. a 1 and a are constants tat depend on te type of contact dynaic odel: a 1 1 and a for te linear spring odel and a and a.5 for te Hertz odel 6). Te tangential direction force is calculated fro te following equations: f sij (t) sij d a 3 nij(t)d sij (t)s ij (t) sij d a 4 nij(t)v sij (t) (6) d sij (t) v sij (t)dt (7) v s ij (t) sij (t) (8) v sij (t) v sij (t)v ij (t)v nij (t)r i i (t)r j j (t)n ij (t), (9) KONA No.3 (5) 183

3 were s is te tangential unit vector and is te angular vector velocity. a 3 and a 4 are constants tat depend on te type of te contact dynaic odel: a 3 a 4 for te linear spring odel and a 3.5 and a 4.5 for te Mindlin odel 7), 8). Te total tangential force is liited by te Coulob frictional liit. n n A B C f sij (t) ij f nij (t) s ij (t), (1) were is te friction constant. In te linear spring odel, it is difficult to teoretically decide te spring constant value except wen te particle sape is a dis, but it is usually used because of its convenience in te calculation 9). Fig. 3 3-particle contact proble. An error fro delays in contact detection Contact detection between particles ay produce a axiu delay tat equals te tie step. Tis delay can be ignored wen te tie step is sall, but te following two probles occur wen te tie step is large. One is te large contact force between particles due to te large strain, wic is not practical, because two particles tat are in contact wit eac oter continue to coe close during a delay in contact detection witout receiving contact force fro eac oter. A etod wereby te calculation tie is returned to te starting point wen two particles ae contact wit eac oter and were particle locations are recalculated, wic overcoes tis proble, as already been reported 5), 1). In tis study, tis proble is solved using te contact force prediction etod described below. Anoter proble is tat several particles ae contact wit te sae particle at sae tie. Two contacts wit a sall tie lag ay be isjudged as occurring at alost te sae tie because of te large tie step. Tis penoenon canges te beavior of particles after tey ae contact because te consuption of te collision energy is different. As an exaple, we consider a difference between a real penoenon and calculation result using te tree particles below. As sown in Fig. 3, tere are tree particles, A, B and C, witout rotation in a onediensional line, and te particles of bot ends, A and C, collide at te sae velocity wit te central particle, B, wose initial velocity is zero. Fig. 4 sows te calculated velocity of eac particle after contact wen te beginning of te contact between particle C and B is after t * fro te beginning of contact between particle A and B. Altoug natural gas ydrate pellets 11), 1) ave been cosen in tis siulation, paraeter values used in te siulation can Noralized Velocity, n/n() Table C B A Fig. 4 Noralized Tie Step, t * () Noralized velocity of te 3-particle proble Paraeter settings for DEM siulation of ice pellets Properties Unit Value Density [g/ 3 ] 8 Linear Spring Constant [N/].31 6 Young s Modulus [N/ ] Poisson s Ratio [].3 g [] 1. Friction Coefficient [].3 be substituted for te pysical properties of ice pellets 13), as sown in Table 1. Te particle diaeter is 1, and a linear odel is used. Te contact tie between particles in tis siulation is seconds, and te tie step is set up in 1/1 of te contact tie, in wic te calculation error becoes sufficiently sall. t * in Fig. 4 is te noralized 184 KONA No.3 (5)

4 value of t, te difference between te beginning of te contact between particles A and B and te beginning of te contact between particles C and B, divided by t c, te contact tie between two particles: t * t. (11) t c Te velocity is also noralized by te velocity of particle A and C before tey ae contact. Fig. 5 is te siulation result of te noralized velocity error of particle A calculated fro Fig. 4, wen te contact detection between particle A and B is sifted by te tie step in te syste in wic te contact between A and B and te contact between C and B tae place at te sae tie. It can be concluded tat DEM wit a large tie step is te etod tat allows te error sown in Fig. 5 copared wit DEM wit a sall tie step so tat te effect of te contact between te oter particles can be ignored. Terefore, te appropriate nuerical solution for DEM wit a large tie step is not a igly precise etod, but its calculation is sufficiently fast and stable for a large tie step. NO NO Contact Detection YES Calculation of Contact Force Calculation of Estiated Value All Contact Points are finised? YES All Particles are finised? YES Calculation of Particle Moveent NO Fig. 6 Algorit of calculation of contact force and oveent in te conventional etod and te new etod Error of Velocity of Particle A, (nn)/n () Fig. 5 Noralized Tie Step, t * () Error of velocity of particle A.3 Stability of calculation Fig. 6, except its sadowed area, is te general calculation procedure for DEM. Te explicit etod is used in DEM and particle i s velocity v i (t t), angular velocity i (t t), location x i (t t) and angular location i (t t) after tie step t are calculated using contact force f i (t) and torque T i (t). In te following, several solutions wit te conventional DEM are copared using te linear spring odel in te syste in wic a particle witout rotation aes contact wit a wall in one diension. Altoug several nuerical etods of solving te differential equation tat represents te contact between two particles can be considered for te conventional DEM, te easiest wit first-order accuracy is te explicit Euler etod 14). On te oter and, te second-order Adas-Basfort 15), Leapfrog 16) and predictor-corrector etods 1) are used wit te conventional DEM. Altoug te predicted value of te contact force in te second-order Adas-Basfort etod is calculated by linear extrapolation, tere is te proble of second-order accuracy not being strictly guaranteed unless appropriate processing is perfored at te discontinuous points, suc as te beginning and end points of te contact. Neiter is second-order accuracy guaranteed wit te Leapfrog etod, unless te value of te velocity during te contact is recalculated using an appropriate etod, because te particle velocity is defined by only te iddle point of a tie step. Moreover, altoug te correction process is usually repeated until an error KONA No.3 (5) 185

5 decreases enoug in te predictor-corrector etod, it is coon to liit te correction process to 1 or ties to sorten te coputational tie in conventional DEM. It is necessary to store all contact states until te calculation of predicted values for te velocity and location are finised because predicted values are calculated after all contact forces are calculated wit tis etod. Tis processing increases coputational tie. Table copares te 4 above-entioned etods. One-diensional collision of one particle is assued, and subscript i and n are oitted. In addition, subscript is defined wit t/ t and t; tie step, is siplified wit. Te contact force is calculated assuing a linear spring odel as follows: f d n. (1) Te analysis contents are explained as follows. In te case of te explicit Euler etod, te explicit Euler etod is expressed in te atrix description as follows: n 1 A n (13) d 1 d 1 A 1. (14) Te deterinant of A is as follows: A 1, (15) were is te ass of te particle. Fro Equation (15), it is found tat te eigenvalue of atrix A ay be larger tan 1. Tis sows te possibility of deviating fro a stability doain of a calculation wit te explicit Euler etod. Te calculation stability in Table is calculated in te sae way as wen. t * in Table is defined as follows: t * t, (16) t c were t c is te contact tie wen. t c p (17) Te conservation of energy in te explicit Euler etod is exained wen as follows: Table Coparison of 4 conventional etods (t/ t, t) Metod Equations Calculation Stability at Energy Conservation at Calculation Tie Explicit Euler -Order Adas- Basfort n 1 n n 1 n f d 1 d n f f 1 d 1 d 3 1 n n 1 Unstable Unstable (nd ) No Good (n x 1 n 1 x ) (n 1d 1) No Good Leap Frog Predictor- Corrector n 1/ n 1/ f ˆn 1 n dˆ1d d 1 d n 1/ f f n n 1 n 1 n 1 ( ˆf 1 f ) d 1 d 1 (ˆn 1 n ) t * p BNM 1 t * p x1 x 4 4 Good (n n 1 d d 1 ) (n d) (n d 1 n 1 d ) 4 (n 1d 1) No Good KONA No.3 (5)

6 n 1d 1 1 (n d )n d, (18) Equation (18) sows tat energy is not strictly conserved, but tere is a conservative quality for a conversion wen te eigenvalue of atrix A is 1. More concretely, tis eans tat te calculated energy value sows te inute fluctuations up and down around te true energy value and does not greatly decrease or increase. Te energy conservation in Table is exained in te sae way as wen. Good sows te case were te energy is not conserved but te eigenvalue is 1. Nuerical equations in ters of te energy conservation in Table sow te error of energy conservation, wic is defined as follows: e(n 1d 1)(n d ). (19) n 1 and d 1 are defined as follows: 3 1 n 1 n n 1. () 3 1 d 1 d d 1 Te calculation tie in Table sows te tie needed to calculate a one-diensional collision of a particle wen te calculation tie in te explicit Euler etod is 1. Adscript in te second-order Adas-Basfort etod sows te calculation tie increases wen appropriate processing is perfored at discontinuous points, suc as te beginning and end points of te contact to guarantee second-order accuracy. Wit te Leapfrog etod, te following forulae for te beginning and end points are used to guarantee second-order accuracy: n 1/ n f ; (1) / n n 1/ f. () / However, Equation () cannot be used for te explicit etod wen, so ore appropriate processing suc as te ulti-step etod is needed to guarantee second-order accuracy. Adscript wit te Leapfrog etod sows te calculation tie increases for te reasons given above. However, it is necessary to note tat te contact detection olds ost of te calculation tie rater tan te calculation of contact force in a state of crowded particles. Altoug te Leapfrog etod as te above-entioned proble, Table sows tat it is te superior of te tree etods wit second-order accuracy in calculation stability and energy conservation. Wit te above-entioned consideration, a onediensional collision of one particle wit a wall is assued. It is necessary for te tie step to set up a saller value in te case of collision of any particles, because te frequency of te contact dynaic odel becoes larger wen te nuber of contact points increases. In te following section, we propose a etod wereby a stable nuerical solution can be found for a large tie step in a syste of ultiple particle collisions. 3. Contact force prediction etod 3.1 Outline of contact force prediction etod A predicted value of te contact force at te stage were two particles collide is solved by tis etod. ˆf i (t t) ˆf cij (t t)ˆf oi (t t), (3) j were ˆ denotes a predicted value. We refer to tis etod in wic a predicted value of te contact force is used in tis way as te contact force prediction etod in tis study. Fig. 6 sows te calculation procedure for tis etod. As copared to te conventional calculation procedure, te procedure for predicting te contact force at every contact point, te sadowed area in Fig. 6, is added. Tus, it is necessary for te calculation tie needed to obtain te predicted value to be sall. Te eaning of using a predicted value of a twoparticle collision for a ulti-particle collision is exained ere fro te viewpoint of nuerical analysis. As an exaple of a ulti-particle collision, we assued a one-diensional collision of any particles witout rotation. Te dynaic equations of particle i and j in a one-diensional collision syste are as follows: dn i (t) i f ij (t) f i (t) (4) dt dn j (t) j f ij (t) f j (t), (5) dt were te second ter of te rigt-and side in eac equation is a suation of forces acting on particles i and j except for te contact force between particles i and j. For exaple, acieving second-order accuracy for te velocity of particle i depends on te following KONA No.3 (5) 187

7 equation: dn ( t) d n i (t t)n i (t) t i (t) n i (t), (6) dt dt were te ters above tird-order accuracy for t are oitted. Equation (4) is substituted for Equation (6) as follows: t n i (t t)n i (t) f ij (t) ˆf ij (t t) f i (t) ˆf i (t t), i (7) were df ˆf ij (t t)f ij (t) t ij (t) (8) dt df ˆf i (t t)f i (t) t i (t). (9) dt On te oter and, te following equation relates to Equation (8). df ij (t) dt dn ijn j (t)n i (t) ij j (t) dn i (t) dt dt f ij (t) f j (t) f ij (t) f i (t) ijn j (t)n i (t) ij.(3) Fro Equation (8) and Equation (3), it is found tat second-order accuracy is not acieved unless te influence of te oter contact forces, f i (t) and f j (t), is considered in te predicted value. However, it is not necessary to acieve second-order accuracy wit tis etod because DEM wit a large tie step j i perits an error of a certain order for te calculation of te contact force, as entioned in Section. Equation (7) sows tat first-order accuracy is acieved even if te influence of te oter contact forces, f i (t) and f j (t), is ignored. In te following, a etod wereby te influence of oter contact forces is ignored for te predicted value is considered. Altoug various etods can be used to obtain te predicted value, four etods wereby stable solutions can be acieved in a large tie step are selected and sown in Table 3. Te anner of consideration is siilar to tat used in Table, and te stability of calculation and energy conservation are exained wen. Exactly Good in ters of energy conservation eans tat te particle energy is conserved strictly according to te exaination siilar to Equation (18). Te odified Runge-Kutta etod wit second-order accuracy is as follows: n 1/ n f (31) / d 1/ d / n 1 n d 1 d n (3) f 1/ (33) 1 (n 1 n ). (34) Tis differs fro te conventional Runge-Kutta etod wit second-order accuracy in tat Equation Table 3 Coparison of 4 etods of calculating te estiated value of force (t/ t, t) Metod Calculation Stability at Energy Conservation at Application to te Hertz and Mindlin Model Calculation Tie Iplicit Trapezoidal Stable Exactly Good NO 1. Modified -Order Runge-Kutta t * p n1 n 4 4 OK 1.77 Good 4-Order Runge-Kutta t * M p 3 4 (nd ) 6 4 No Good OK.61 Exact Solution Stable Exactly Good NO KONA No.3 (5)

8 (34) is te trapezoid etod. Altoug te calculation wen is unstable wit te conventional Runge- Kutta etod wit second-order accuracy, it is interesting tat tis etod is stable even wit suc little odification. Exact Solution in Table 3 eans a etod wereby an exact solution to te differential equation of te contact dynaic odel is used for te predicted value. Stable in ters of te stability of calculation in Table 3 eans tat a stable solution for te predicted value of te contact force can be acieved wit a large tie step. However, tere is anoter liit for te tie step in DEM calculation, as entioned in Section.. Te tie step as to be set to less tan te contact tie t c, i.e., t * 1. Fro Table 3, te iplicit trapezoid etod and te exact solution etod are superior to te oter two etods in ters of te large tie step and energy conservation, but tese etods cannot be applied to te non-linear spring odel. Te etod tat can be applied to te non-linear spring odel and perits a large tie step is te fourt-order Runge- Kutta etod. Te calculation tie is sallest wit te iplicit trapezoid etod and te exact solution etod. However, te effect of te tie step tat can be set larger is bigger for te total speedup of DEM calculation tan te calculation tie of te predicted value is sall. Tis is because te calculation tie of te contact detection is larger tan tat of te calculation of te contact force, as entioned above. A predicted value is calculated by projecting on te noral direction and te tangential direction siilar to te contact force. ˆd nij (t t) and ˆd sij (t t) define te predicted value of te particle location in te noral and tangential direction, respectively; ˆn nij (t t) and ˆn sij (t t) define te predicted particle velocity in te noral and tangential direction, respectively. Tese values can be obtained by eiter etod, as entioned above, and te predicted value of te contact force in te noral direction, ˆf nij (t t), is obtained as follows: ˆf nij * 1 nij ˆda nij(t t) nij ˆda nij ˆn nij(t t), (35) were a 1 and a are siilar to te constants in Equation (). Wit te conventional DEM, tension in te noral direction is not peritted by te no-tension joint, so te predicted value of te contact force in te noral direction is also not peritted as follows: wen ˆf * nij, ˆfnij (t t) ˆf * nij wen ˆf * nij, ˆfnij (t t). (36) Te predicted value in te tangential direction, ˆf sij (t t), is obtained as follows: wen t ˆt cij, ˆf sij (t t) 3 sij ˆda nij(t t) ˆd sij (t t) 4 sij ˆda nij(t t) ˆn sij (t t) (37) wen t ˆt cij, ˆf sij (t t) 3 sij ˆda nij(t ˆt cij ) ˆd sij (t ˆt cij ) 4 sij ˆda nij(t ˆt cij ) ˆn sij (t ˆt cij ), (38) were a 3 and a 4 are siilar to te constants in Equation (6). ˆt cij is te tie wen te two-particle collision finises, i.e., ˆf nij, * and is obtained by te following linear approxiation equation, for exaple: f ˆt cij nij (t) t. (39) f nij (t) f nij * Moreover, te predicted value of te contact force in te tangential direction is liited by te Coulob frictional liit: ˆf sij (t t) ij ˆf nij (t t). (4) 3. Application exaples of te contact force prediction etod Soe exaples of DEM calculation using te aboveentioned original etod are introduced ere. In tis study, te pacing and discarge syste are used as an exaple of ulti-particle collision. Te properties of te particles ave already been sown in Table 1. Te particle size is 5, te nuber of particles is, and te calculation space is a quadratic colun wose botto is.3.3. Te contact tie between two particles in tis calculation is.1 4 seconds. Te periodic boundary condition is used in all side walls in order to oderate te wall effect. Te linear spring odel is used for te contact force odel, and te spring constant is deterined fro te contact tie by te Hertz spring odel. In all calculations, te exact solution etod is used to obtain te predicted value of te contact force wit te contact force prediction etod Pacing siulation Te pacing siulation results produced by te free fall of particles are sown ere. Particles fall freely fro a eigt of 1. Fig. 7 sows te tie cange of te energy suation of all particles in te case of te contact force KONA No.3 (5) 189

9 Kinetic Energy, (J) Fig prediction etod or te conventional etod, respectively. n t is te nuber of partitions in te contact tie as follows: t c Tie, (s) nt1.5 (New) nt3 (New) nt3 (Conv.) nt6 (Conv.) nt3 (Conv.) n t, (41) t were t t is te contact tie needed for one particle to contact a wall. Altoug te calculation is dispersed in n t 3 wit te conventional etod, it is not dispersed in n t 3 wit te proposed etod. Fig. 8 sows te coparison between te conventional etod and te proposed etod for te calculation results of te pacing ratio. Te calculated pacing ratio becoes larger as te tie step becoes Pacing Ratio, () Kinetic energy of total particles in pacing siulation Conv. New Fig. 8 n t () Result of pacing siulation larger. It is assued tat tis is because te overlap at te beginning of contact between two particles is aintained. How fine te tie step ust be set wit te proposed etod depends on ow uc calculation error is peritted. In Fig. 8, altoug te pacing ratio does not depend on te tie step in n t 3 wit te conventional etod, n t 3 wit te proposed etod. In oter words, te proposed etod can set te tie step 1 ties larger tan tat of te conventional etod. Te calculation tie of 1 step wit te proposed etod is 1. ties larger tan wit te conventional etod. Tus, te proposed etod accelerates DEM calculation 8 ties wen te tie step is set 1 ties larger tan conventionally. Incidentally, tere ave been various considerations regarding te rando pacing of ono-dispersed particles in te long ter 17), 18). Te rando pacing ratio sould be between te closest pacing ratio,.74, and te pacing ratio in a siple cube lattice, teoretically.5. Moreover, experiental results of ono-dispersed sperical particles by Westan and Wite 17) sow tat teir pacing ratio is between.553 and.63. Terefore, te pacing ratio.57 obtained in tis study is te appropriate value. 3.. Discarge siulation In order to siulate a syste in wic particle oveent is ore dynaic tan in te pacing syste, a.- diaeter circular outlet is opened at te botto of te paced particles bed, and te particles are allowed to fall freely fro te outlet to te floor, a distance of under 1. Siilar to Fig. 7, Fig. 9 sows te tie cange of te energy suation of all particles. Altoug te calculation is dispersed in n t 3 wit te conventional etod, it is not dispersed in n t 3 wit te proposed etod. Fig. 1 sows a coparison between te conventional etod and te proposed etod for te calculation results of te nuber of discarged particles fro te outlet. As for te discarge beavior, a big difference between te conventional etod and te proposed etod is not found, even wit te large tie step. It is believed tat tis is because te calculation error is canceled out due to te oveent of all particles. In Fig. 9, te inetic energy is not dispersed in n t 6 wit te conventional etod and n t 1.5 wit te proposed etod, so te proposed etod can set te tie step 4 ties larger tan tat of te conventional etod. Wen te calculation tie of 1 step is considered, te proposed etod accelerates DEM calculation 3 ties. 19 KONA No.3 (5)

10 Kinetic Energy, (J) Fig. 9 Nuber of Discarged Particles, () Conclusion Tie, (s) nt1.5 (New) nt3 (Conv.) nt3 (Conv.) nt6 (Conv.) nt3 (Conv.) Kinetic energy of total particles in discarge siulation 1 3 Fig. 1 Tie, (s) nt1.5 (New) nt3 (New) nt3 (Conv.) nt6 (Conv.) nt3 (Conv.) Result of discarge siulation DEM in wic a large tie step is possible is developed by iproving te algorit to solve te contact force between two particles. In general, te equations of otion of particles are solved by te second-order Adas-Basfort etod, wic estiates te values of te contact force in te following calculation tie by linear extrapolation, or by ulti-step etods suc as te predictor-corrector etod. Inspired by tese two conventional etods, we propose an original etod called te contact force prediction etod, wic uses te predicted value of te contact force at eac contact point of two particles. As for te etod of obtaining te predicted value of te contact force wit te proposed etod, four etods in wic a large tie step can be set are exained, and teir caracteristics are sown. Moreover, te pacing and discarge syste is cosen as an exaple of te ulti-particle collision, and te proposed etod accelerates DEM calculation 38 ties. In tis paper, soe calculation results of ono-dispersed particles are introduced as an iinent exaple, but we ave already confired tat te contact force prediction etod is useful for two sizes of particle at a nuber of approxiately 1 illion. Under te present paraeters, it is expected tat te nuber of particles calculated in DEM will be liited to 1 illion, even if several tecniques to accelerate DEM calculation, suc as te etod proposed ere, are applied. Tus, an additional acceleration, suc as parallel coputation or cobination wit a continuous etod, is needed for application to te general scale syste. However, it is confired tat te contact force prediction etod proposed in tis study is useful for DEM acceleration. Noenclature f : Force Vector of Particle (N) fˆ : Estiated Value of f (N) f c : Contact Force Vector of Particle (N) fˆc : Estiated Value of f c (N) f o : Force Vector of Particle except for Contact Force (N) fˆo : Estiated Value of f o (N) g : Acceleration of Gravity Vector ( /s) : Tie Step in Table and 3 (s) i : Particle Index () I : Moent of Inertia of Particle (g ) j : Particle Index () : Spring Coefficient (N/ or N/ 1.5 ) : Tie Index () : Mass of Particles (g) n : Unit Vector fro te Center of Particle to Contact Point () n t : Nuber of Steps in Contact Tie () r : Radius of Particle () s : Tangential Unit Vector () t : Tie (s) t c : Contact Tie (s) T : Torque Vector of Particle (N ) v : Velocity Vector of Particle (/s) n : Velocity of Particle (/s) KONA No.3 (5) 191

11 nˆ : Estiated Value of v (/s) x : Position Vector of Particle () d : Displaceent () dˆ : Estiated Value of d () t : Tie Step (s) t * : Noralized Tie Step () tˆc : Estiated Value of Contact Tie (s) t : Tie Difference of Contact Start Points (s) t * : Noralized Tie Difference of Contact Start Points () g : Constant Related to Restitution Coefficient () : Viscosity Coefficient (g/s or g/s/.5 ) : Angular Position Vector of Particle (rad) : Friction Coefficient () : Angular Velocity Vector of Particle (rad/s) Subscripts i : Particle i ij : Between Particle i and j j : Particle j n : Noral Direction s : Tangential Direction References 1) Cundall, P. A.: Rational Design of Tunnel Supports: A Coputer Model for Roc Mass Beavior Using Interactive Grapics for te Input and Output of Geoetrical Data, Tecnical Report MRD--74 Missouri River Division, US Ary Corps of Engineers. ) Kawaguci, T., Tanaa, T. and Tsuji, Y.: Nuerical Siulation of Fluidized Bed Using te Distinct Eleent Metod (te Case of Spouting Bed), Trans. Japan Soc. Mec. Eng. (Series B), , , (199). 3) Asaura, K., Harada, S., Funayaa, T. and Naajia, I.: Siulation of Descending Particles in Water by DEM, Sigen to Sozai, 11, 19-4, (1996). 4) Watanabe, H.: Critical Rotation Speed for Ball-Milling, Powder Tecnol., 14, 95-99, (1999). 5) Natsuyaa, S. and Horio, M.: Study of Dynaical Analysis for Scale Up of Fluidized-bed of Solid Dosage Fors by DEM Siulation, Kagau Souci, 9, (1). 6) Tioseno, S. P. and Goodier, J. N.: Teory of Elasticity, 3rd Edition, 38, McGraw-Hill Boo Copany, (197). 7) Mindlin, R. D.: Copliance of Elastic Bodies in Contact, J. Appl. Mecanics, 16, 59-68, (1949). 8) Mindlin, R. D.: Elastic Speres in Contact Under Varying Oblique Forces, J. Appl. Mecanics,, , (1953). 9) Taguci, Y.: Dynaics of Granular Matter fro te Pysical Point of View (I), J. Soc. Powder Tecnol., Japan, 3, 4-46, (1995). 1) Xu, B. H. and Yu, A. B.: Nuerical Siulation of te Gas-Solid Flow in a Fluidized Bed by Cobining Discrete Particle Metod wit Coputational Fluid Dynaics, Ce. Eng. Sci., 5-16, , (1997). 11) Oui, T.: Expectation of Gas Hydrate Transport Tecnology, J. Japanese Assoc. Petroleu Tecnol., 66, , (1). 1) Naajia, Y., Kawagoe, Y. and Taaoi, T.: Transporting Syste of Natural Gas Hydrate in Pellet For, Proceedings of JSME Transportation and Logistics 1, Kawasai, , (1). 13) Maeno, N. and Fuuda, M.: Seppyou no Kouzou to Bussei, 1, Koon Soin, (1986). 14) Hitotsuatsu, S.: Suuci Kaisei, 13, Asaura Soten, (198). 15) Te Society of Powder Tecnology, Japan: Funtai Siulation Nyuon, p.9, Sangyo Toso, (1998). 16) Moaer, M., Sinbrot, T. and Muzzio, J.: Experientally Validated Coputations of Flow, Mixing and Segregation of Non-Coesive Grains in 3D Tubling Benders, Powder Tecnol., 19, 58-71, (). 17) Cuberland, D. J. and Crawford, R. J.: Te Pacing of Particles, p.14-39, Elsevier, (1987). 18) Mogai, T. (ed.): Dositu Riigau, p , Giodo Suppan, (1969). 19 KONA No.3 (5)

12 Autor s sort biograpy Ciaru Tooro Ciaru Tooro is Researc Associate at te Faculty of Science and Engineering of Waseda University. Se received er B.E. fro te Dept. of Science and Engineering in 1998 at Waseda University, and er M.E. fro te Dept. of Geosyste Engineering in at te University of Toyo. Se received er Doctor of Engineering in 3 fro te University of Toyo. Her researc interests are resources processing and environental tecnology. Katsunori Oaya Katsunori Oaya is Researc Associate at te Departent of Geosyste Engineering, Graduate Scool of Engineering, te University of Toyo. He obtained is B.Sc. and M.Sc. degrees fro te University of Toyo in 1974 and 1976, respectively. Oaya s current researc interests are in te andling of particles and grains (grinding, classification, rando pacing, segregation, dispersion, etc.) and siulations using DEM (Discrete Eleent Metod). Jun Sadai Jun Sadai received a aster degree in geosyste engineering in 1987 and a Doctor of Engineering degree in geosyste engineering in 199, bot fro te Departent of Geosyste Engineering, Graduate Scool of Engineering, te University of Toyo. Te tee of is tesis was A Study of Control in Mineral Processing Plants. He joined Nippon Steel Corporation (NSC) in 199 and wored as a researcer and a senior researcer until. His researc subjects in NSC were odeling, siulation and te control of industrial plants, especially of steel aing plants and energy plants. He oved to te Departent of Geosyste Engineering, Graduate Scool of Engineering, te University of Toyo, as a lecturer in. His researc interests include natural gas ydrate (NGH), transportation and/or storage of NGH, application of NGH to industrial processes as well as te odeling, siulation, control and optiization of processes. Ms. Ciaru Tooro was is doctor course student fro 3, and now is a co-worer. He is a eber of te Mining and Materials Processing Institute of Japan, te Society of Instruent and Control Engineers, a eber of te board of Japan Federation of Ocean Engineering Societies. KONA No.3 (5) 193