Fast Calculation for Particle Interactions in SPH Simulations: Outlined Sub-Domain Technique

Transcription

1 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 Fast Calculaton for Partcle Interactons n SPH Smulatons: Outlned Sub-Doman Technque Buntara Sthenly Gan and Naohro Kawada Abstract A smple and easy algorthm s presented for a fast calculaton of kernel functons whch requred n flud smulatons usng the Smoothed Partcle Hydrodynamc (SPH) method. Present proposed algorthm mproves the Lnked-lst algorthm and adopts the Par-Wse Interacton technque, whch are wdely used for evaluatng kernel functons n flud smulatons usng the SPH method. The algorthm s easy to be mplemented wthout any complextes n programmng. Some benchmark examples are used to show the smulaton tme saved by usng the proposed algorthm. Parametrc studes on the number of dvsons for sub-domans, smoothng length and total amount of partcles are conducted to show the effectveness of the present technque. A compact formulaton s proposed for practcal usage. Keywords Technque, Flud smulaton, Smoothng Partcle Hydrodynamc (SPH), Partcle nteracton S I. INTRODUCTION MOOTHED Partcle Hydrodynamc (SPH) method whch was frstly ntroduced [,] for modelng astrophyscal phenomena s one amongst many partcle methods that has been used for smulatng the physcal behavor of flud and contnuum sold bodes. Recent progress n usng SPH method has been appled n the felds of flud and sold nteracton [3,4], mult-phase fluds and free surface flows [5]. In the SPH method, the so-called smoothng functon or kernel functon whch s based on partcle approxmaton plays a very mportant role n carryng out the ntegraton of governng partal dfferental equatons wthn the supportng doman. One of the mportant ssues for mplementng the SPH method usng the partcle approxmaton s how to perform effectvely the evaluaton of kernel functons based on a set of partcles scattered n an arbtrary manner. A lot of efforts have been done to mprove the governng equatons used n the smulatons and varaton of kernel functons as summarzed n [6], however there was a few research work whch has been done n enhancng the technque to carry out the nteracton among partcles n the supportng doman. Classcally, the partcles searchng algorthm s performed at a partcle to fnd other partcle nsde ts supportng doman wthn the entre smulaton doman. Buntara Sthenly Gan s wth the Archtecture Department, College of Engneerng, Nhon Unversty, Koryama, Japan (Correspondng author s phone: ; fax: ; e-mal: buntara@ce.nhon-u.ac.p) Naohro Kawada s wth the Graduate School n the Archtecture Department, College of Engneerng, Nhon Unversty, Koryama, Japan The searchng process s necessary at every tme step, and thus the computaton effort n ths searchng method s very tme consumng, and s not feasble for problems wth vast amount of partcles. The Lnked-lst technque was ntroduced n [7] before the SPH method was nvented, snce then; the technque s stll wdely beng used to perform the SPH method. The Lnked-lst algorthm uses unform meshes for bookkeepng wth the sze of h. Where, h s the radus of compact support doman of the kernel functon. Thus, all partcles n the neghborng sub-domans can then contrbute to the propertes of partcles n the sub-doman. An mprovement was made n [8], where the cylndrcal sub-doman s used as a bookkeepng devce to smulate shocks n accreton dsks. However, the crcular doman wll loose ts capablty to cover arbtrary smulaton doman whch s not crcular, n general problems. Unlke the rectangular sub-doman, the crcular sub-doman leaves the four corners of ts boundng rectangle untouched, thus overlappng between crcular sub-domans for bookkeepng can not be avoded, whch results n less effectveness of ths technque. The Herarchcal Tree codng has been also wdely used [9-], however due to the complexty n mplementng the algorthm; ths technque was not ganng any popularty n practce. In [], the applcaton of Herarchcal Tree codng was facltated wth parallel programmng to boost ts performance; nevertheless, the efforts merely ncrease more complextes for adoptng the algorthm proposed. In ths study, a smple and easy algorthm based on creaton of fxed sub-domans and ts outlnes s presented for a faster calculaton of kernel evaluatons. The present proposed technque based on the same concept wth the Lnked-lst, but the sub-doman wdth need not have to the sze of h. In most of smulaton problems, ths h sze s very small compared to the sze of the whole smulaton doman. Parametrc studes conducted n ths study have shown that too small sub-doman dvsons sze could result n an ncrease of computaton tme consderably. For an easy use, the sze of sub-doman s determned from equally dvdng the sze of the entre doman of smulaton by a constant number. The proposed technque s then, further facltated by the Par-Wse Interacton method [,3-4] to regster all partcles wthn the outlned sub-doman whch contrbute to the partcles nsde the sub-doman where the kernel functons are beng evaluated. II. SPH FORMULATION FOR NAVIER-STOKES EQUATIONS SPH can be consdered as a knd of nterpolaton method for 8

2 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 nteractons of arbtrary partcles n a support doman nsde the flud smulaton system [5]. In present study, SPH method s used for solvng the Naver-Stokes equatons problems. Fg. shows a typcal kernel functon W. The kernel functon shown s used n ths study was taken from the cubc splne famly whch s known as B-splne functon () as gven n [6]. Fg. Support doman of the kernel functon W of partcle 3 3 ξ + ξ 0 ξ < 3 W ( ξ, d) = α 6 ( ξ ) ξ < () 0 ξ Here, ξ = r / d, where r s the dstance between two partcles; d = κh,where κ s a constant; h s the smoothng 3 length; and α = d, α = 60 7πd, α 3 = πd are gven for one-, two- and three-dmensonal problem, respectvely. The partcle approxmaton of densty n the conservaton of mass governng equaton can be expressed as follow: whch are summarzed n [6]. Regardless of the governng equatons beng used n the SPH smulatons, the present technque s generally applcable and effectve for evaluatng the kernel functons. III. OUTLINED SUB-DOMAIN TECHNIQUE The present proposed algorthm mproves the Lnked-lst algorthm by allowng an arbtrary wdth for the fxed sub-domans and outlnng the sub-doman wth the κh wdth to guarantee all nteractng partcles are taken nto account n the calculatons. By dvdng the entre sze of smulaton doman nto equal sze of sub-domans, tme whch s requred for storng partcle data become longer but the tme whch s requred for searchng the neghborng partcle nsde the sub-doman s reduced whch results n consderable executon tme savng. To llustrate the present outlned sub-doman technque, a schematc two dmensonal arbtrary doman as shown n Fg. s used. The entre smulaton doman s frstly dvded nto smaller squares sub-domans and ther outlned squares. The wdth of the outer square s determned from the sze of the nner square added at both wdth and heght sdes by d whch s equal to the radus of kernel functon W n the support doman. There are overlappng areas between the adacent outer squares, hence the coverage of compact support of kernel functon from a partcle at the edge of nner squares s guaranteed. Durng the bookkeepng process, all the partcles n the nner squares are also regstered for the outer squares. Dρ Dt N = = m v W x () where, v ρ s the densty of partcle, m s the mass of partcle, = v v s the relatve velocty between partcles and. For the conservaton of momentum and energy, the partcle approxmaton of momentum and energy governng equatons takng artfcal vscosty nto account are gven as follows: Dv Dt α N α α σ σ = m + + = ρ ρ W x (3) Fg. A -D schematc representaton of outlned sub-doman technque De Dt N P P = m + + v ρ ρ = W x µ + ε ρ ε α α where, σ, σ are the stresses for partcles,, P, P are the pressures at partcles,, ε s the vscous stran rate for partcle, µ s the dynamc vscosty for partcle, and the artfcal vscosty [7,8]. There have been many varatons of the governng equatons (4) Here, for each partcle nsde the nner square that s ndexed acts as a center for evaluatng the governng equatons of (-4) to nteract wth the surroundng partcles nsde the outer square those are ndexed. Hence, the searchng for nteractng partcles wll only necessary sought nsde the outer square whch s requrng less tme rather than searchng for partcles n the entre doman of smulaton.the addtonal tme requred n the present technque s to regster all the partcles nsde the predefned sub-domans and ther outlned area to account partcles at the utmost boundary of the sub-domans for 9

3 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 calculaton of kernel functons. Regstraton of partcles to ther sub-domans and outlned area s conducted after each tme step calculaton fnshed. I II II IV. PARAMETRIC STUDY USING BENCHMARKS In the followng study, the tmng evaluatons were performed on a personal computer wth Core Duo E7500 CPU, Clock rate of.93 Hz, FSB speeds of 066 MHz and 4 GB Memory of RAM. FORTRAN compler was used to edt, modfy, comple, debug and run the source codes avalable from [6]. A. Shock tube -D problem The shock tube problem s one dmensonal benchmark whch was comprehensvely smulated by many researchers usng SPH method [,9]. The shock tube s a long straght tube flled wth gas, whch s separated by a membrane nto two equal parts n whch each part s ntally n equlbrum state of constant pressure, densty and temperature. When the membrane s taken away nstantaneously, a shock wave, a rarefacton wave and a contact dscontnuty wll be produced. The ntal condtons of the smulaton are smlar wth [], then ntroduced by [4] whch were taken from [0], for x 0 ( ρ =, ν = 0, e =. 5, p =, x = ) and for x > 0 ( ρ = 0. 5, ν = 0, e =. 795, p = , x = ). Here, ρ, p, e and ν are the densty, pressure, nternal energy and velocty of the gas, respectvely. x s the dstance between two partcles. A constant tme step of 00 s s used for runnng 000 steps calculaton. A constant smoothng length h = and multpler factor κ = are used n ths smulaton. Fg. 3 shows the scheme of outlned sub-doman technque whch s appled to the shock tube -D problem. Table I shows three dfferent total numbers of partcles used n the SPH smulaton. The parametrc study s conducted n whch the same percentage dstrbuton of partcles n two equal part of the tube was of 80% and 0% for both hgh densty and low densty regons. To keep a constant densty mantaned n the tube, the length of the tube s ncreased along wth the ncreasng number of partcles. For each case of the three shock tubes shown n Table I, the total length of the tube s dvded by vertcal sub-domans whch result n, 4, 8, 6, 3, 64, 8 and 56 dvsons. d L =l / M d l / l / Fg. 3 Outlned sub-doman scheme n the shock tube problem TABLE I PARAMETRIC STUDY FOR SHOCK TUBE -D PROBLEM Number of Partcles Hgh Low densty densty Total Length Executon Tme Rato Number of Dvson Fg. 4 Results of executon tme for shock tube -D problem Number of Partcles Fg. 4 depcts the ratos of tmng evaluaton of the present proposed technque compared wth the results from the orgnal technque wthout mprovement, for a varety number of partcles and dvson of vertcal sub-domans. From Fg. 4, the present proposed outlned sub-doman technque shows less computng tmes for larger number of partcles used n the smulaton. For 00 total number of partcles used n the smulaton, tme for regsterng the partcles became longer than for evaluatng the kernel functons, thus less effectveness were resulted. However, the present technque shows ts effectveness for larger number of partcles used n the smulaton. In case, the number of partcle s 0,000; the reducton n executon tme of less than 0% can be acheved when the dvsons of sub-domans n between 64 and 8 are used. B. Shear drven cavty -D problem The classc shear drven cavty -D problem s the flud flow wthn a closed square generated by movng the top sde of the square at a constant velocty whle the other sdes reman fxed. The flow wll reach a steady state and form a recrculaton pattern. In the smulaton, the dmenson the knetc vscosty and densty are ν =0-6 m /s and ρ =0 3 kg/m 3 respectvely. The top sde of the square moves at a velocty of V=0-3 m/s, thus the Reynolds number for ths case s one. A constant tme step of 5x0-5 s s used. A constant smoothng length of.5x0-5 s used. Table II shows four dfferent total numbers of partcles used n the smulaton to conduct a parametrc study on the present proposed technque. For each case of the four shear drven cavty -D problems shown n Table II, the smulaton doman s dvded equally by vertcal and horzontal sub-domans of, 4, 8, 6, 3, 64 and 8 dvsons. Fg. 5 shows the scheme how the outlned sub-doman technque s appled to the shear drven cavty -D problem. TABLE II PARAMETRIC STUDY FOR SHEAR DRIVEN CAVITY -D PROBLEM Number of Partcles nsde the square Sde Length Computaton Steps 30

4 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 I II II IV V TOP parameters n the -D problem, the flow wll reach a steady state and form a recrculaton pattern. In the smulaton, the dmenson the knetc vscosty and densty are ν =0-6 m /s and ρ =0 3 kg/m 3 respectvely. The top sde of the cube moves at a velocty of V=0-3 m/s n the dagonal x-z drectons. A constant tme step of 5x0-5 s s used. A constant smoothng length of.5x0-5 s used. Fg. 7 shows the scheme how the outlned sub-doman technque s appled to the shear drven cavty 3-D problem. Table III shows four dfferent total numbers of partcles used n the smulaton to show the effectveness of the present proposed technque. For each case of the four 3-D shear drven cavty problems shown n Table III, the smulaton doman s dvded by sub-domans of, 4, 8 and 6 equal dvsons n all Cartesan axes drectons. TABLE III PARAMETRIC STUDY FOR SHEAR DRIVEN CAVITY 3-D PROBLEM Fg. 5 Outlned sub-doman scheme n the shear drven cavty -D problem 00 Number of Partcles nsde the cube Sde Length Computaton Steps I II II IV Executon Tme Rato 0 0. Number of Partcles 50X50 00X00 00X00 000X Number of Dvson Fg. 6 Results of executon tme for shear drven cavty -D problem Smlar tendency wth the results obtaned from the shock tube -D problem s also observed. The present proposed outlned sub-doman technque shows sgnfcant tme reductons for larger number of partcles used n the smulaton. As shown n Fg. 6, for small number of partcle, the tme for regsterng the partcles became longer than evaluatng the kernel functons, thus less effectveness wll be resulted. The present technque shows much tme reducton for larger number of partcle used, as shown n the number of partcle of used n the smulaton. The computaton tme can be reduced to less than % when the dvsons of sub-domans about 3 are used. On the contrary, further ncreasng of number dvsons wll result n longer executon tme. C. Shear drven cavty 3-D problem The shear drven cavty -D problem s extended to a 3-D closed cube problem by movng the top sde of the cube at a constant dagonally 45 degree x-z drecton of velocty generated whle the other sdes reman fxed. By usng the same Fg. 7 Outlned sub-doman scheme n the shear drven cavty -D problem In smlar tendences wth the results obtaned from the prevous studes, the present proposed outlned sub-doman technque s showng less executon tme for larger total number of partcles used n the smulaton. As shown n Fg. 8, for less number of partcles, the tme requred for regsterng nteractng partcles became longer than evaluatng the kernel functons, thus less effectveness wll be resulted. y z x 3

5 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 Executon Tme Rato Number of Dvson Number of Partcles 30X30X30 40X40X40 50X50X50 60X60X60 Fg. 8 Results of executon tme for shear drven cavty 3-D problem The present technque shows less executon tme for larger number of partcle used, as shown for the number of partcle of the computaton tme can be reduced to less than 3%. Ths value s acheved when the dvsons of sub-domans about 8 s used. However, further ncreasng the number dvsons wll result n longer executon tme. V. PROPOSAL FOR PRACTICE From the results of parametrc studes, the effectveness of the present technque n reducng the executon tme depends on the dmensonal type problem, total number of partcles n the smulatons and number of dvsons adopted n the smulatons as shown n Fgs. 4, 6 and 8. The most reduced executon tme ratos resulted from parametrc studes are collected and summarzed n Table IV. Here, an addtonal non-dmensonal rato of L/h s ntroduced to represent physcal characterstc of the smulaton. In Table IV, L s the sze of sub-doman; h s the smoothng length; N s the total number of partcles used n the smulaton. For practcal purpose, a smple formulaton that relates all the parameters n the smulaton s to be sought. A unque relatonshp between the total number of partcles, radus of smoothng length and number of dvson s formulated. Fg. 9 shows a logarthmc relatonshp between the L/h rato \ D and ND whch s defned as ND = N, where D s the dmenson of the smulaton type. By usng the lnear regresson analyss to the data plotted n Fg. 7, a straght lne relatonshp can be formulated as follow. ( ). L / h 4 ND = 0. 5 (5) In term of total number of partcles used n the smulaton, N, the expresson (5) can be rewrtten as follows. N 4 D ( L h). / D D = ND = 0. 5 (6) Depends on the problem type and number of partcles used n the smulatons, the reducton of executon tme vares from one-hundredth to one-tenth rato can be acheved. TABLE IV THE MOST REDUCED EXECUTION TIME FROM PARAMETRIC STUDIES Shock Tube -D Shear Drven Cavty -D Shear Drven Cavty 3-D ND =(Total Number of Partcle, N ) /Dmenson Total Number of Partcles L/h rato ND II, III 0, ,000 I II III IV I II III IV ND = 0.5 (L/h ).4 Coeffcent of Determnaton, R = 0 00 Dvson and Smoothng Lengths rato, L/h rato Fg. 9 Results of parametrc study wth the most reducton tme rato Unlke the other technques proposed n the past, expresson n (6) s very smple and provdes a handy formulaton for mplementaton, thus recommended for practcal smulaton purpose. VI. APPLICATION In order to verfy the effectveness of the present proposed technque, a -D type flud smulaton problem s presented. The proposed formula n (6) was used to determne the dvson number of the sub-domans. The orgnal shock tube problem whch was used as the benchmark problem s modfed by dvdng the long straght tube flled wth gas nto four equal parts. The tube s separated by membranes at three locatons. All parts of the tube are ntally n equlbrum state of assumed constant pressure, densty and temperature. When all the membranes are taken away nstantaneously, shock waves, rarefacton waves and contact dscontnutes wll be produced. Fg. 0 shows the scheme of outlned sub-doman technque appled to the modfed shock tube -D problem. Consderng the flow of gas nsde the tube s adabatc n smooth regons, the functonal entropy can be set as a constant, thus the relatonshps between densty and pressure follows the sentropc law. 3

6 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 The ntal condtons of the smulaton are gven as: for x 0. 5 ( ρ =. 0, ν = 0, e =. 5, p =. 0, x = ); for 0. 5 < x ( ρ = 0. 5, ν = 0, e =. 895, p = , x = ); for < x ( ρ = 0. 8, ν = 0, e =. 87, p = 0. 73, x = ); and for x > ( ρ = 0. 3, ν = 0, e =. 544, p = 0. 85, x = ). Here, ρ, p, e, and ν are the densty, pressure, nternal energy and velocty of the gas, respectvely. x s the dstance between two partcles. A constant tme step of 0005 s s used for 3000 steps calculaton. There were 0400 partcles used n the smulaton. The radus smoothng length used n the smulaton was determned from the largest radus of smoothng length, d = h = whch s two tmes of dstance between partcles n the least densty regon, x = h = By usng (5), the dvson length L can be calculated; hence number of dvson of the tube length calculated s 80. Fgs. -4 show the mxture processes of dfferent pressure regons at varyng smulaton tmes along the x-axs after all the membranes that separated densty vared regons are taken nstantaneously at the same tme. The CPU tme requred for conductng the 000 steps smulaton was recorded to be 86 s whch s only about 7.% of the tme requred when no sub-domans along the length that was s. From the comparson of both executon CPU tmes, the present proposed technque shows ts effectveness n reducng the tme requred for conductng the modfed Shock Tube -D smulaton after 6 s X coordnate at 0 at 3 at 6 Fg. dstrbutons along the tube from the modfed shock tube -D smulaton results at 6 s after 9 s X coordnate at 0 at 3 at 6 at 9 Fg. 3 dstrbutons along the tube from the modfed shock tube -D smulaton results at 9 s x = Membranes x = 0. 5 x = x = x = Fg. 0 Modfed Shock Tube -D problem..0 at 0 at 3 at 6 at 9 at after 3 s X coordnate at 0 at 3 Fg. dstrbutons along the tube from the modfed shock tube -D smulaton results at 3 s 0. after 0. s X coordnate Fg. 4 dstrbutons along the tube from the modfed shock tube -D smulaton results at 0. s VII. CONCLUSION From parametrc studes, t can be concluded that the present proposed technque shows ts effectveness n the optmum ranges of number dvson whch gves the most reducton n executon tme, especally when larger number of partcles s used n the SPH smulaton. An easy and handy formulaton for practcal purpose for conductng a smulaton usng SPH method s proposed. Equal dvson of sub-domans; and the longest smoothng length 33

7 Internatonal Journal of Cvl and Envronmental Engneerng 6 0 selecton for the outlnes; are the two smple algorthms whch are seamlessly mplement able to the exstng SPH smulaton codes. For future works, applcaton of parallel programmng to the bookkeepng and evaluaton of kernel functons n sub-domans could further boost the performance of the present proposed technque n reducng the executon tme for smulaton usng SPH method. REFERENCES [] L. B. Lucy, Numercal approach to testng the fsson hypothess, Astronomcal Journal, vol. 8, pp , 977. [] R. A. Gngold and J. J. Monaghan, Smoothed partcle hydrodynamcs: theory and applcaton to non-sphercal stars, Monthly Notces of the Royal Astronomcal Socety, vol. 8, pp , 977. [3] H. H. Bu, K. Sako, and R. Fukagawa, Numercal smulaton of sol-water nteracton usng smoothed partcle hydrodynamcs (SPH) method, Journal of Terramechancs, vol. 44(5), pp , 007. [4] S. Potapov, B. Maurel, A. Combescure, and J. Fabsk, Modelng accdental-type flud-structure nteracton problems wth the SPH method, Computers & Structures, vol. 87(-), pp , 009. [5] N. Grener, M. Antuono, A. Colagross, D. Le Touze, and B. Alessandrn, An Hamltonan nterface SPH formulaton for mult-flud and free surface flows, Journal of Computatonal Physcs, vol. 8(), pp , 009. [6] G. R. Lu and M. B. Lu, Smoothed partcle hydrodynamcs: a meshfree partcle method. World Scentfc Publshng Co. Pte. Ltd., 007. [7] R. W. Hockney, S. P. Goel, and J. W. Eastwood, A 0000 partcle molecular dynamcs model wth long range forces, Chemcal Physcs Letters, vol. (3), pp , 973. [8] G. Gerard and D. Molten, Parallelzaton of a smoothed partcle hydrodynamc code for smulaton of shocks n accreton dsks, Memore Soceta Astronomca Italana, vol. 4, pp , 003. [9] A. W. Appel, An effcent program for many-body smulatons, SIAM Journal on Scentfc and Statstcal Computng, vol. 6(), pp , 985. [0] L. Hernqust, Herarchcal N-body methods, Computer Physcs Communcatons, vol. 48, pp. 07-5, 988. [] L. Hernqust and N. Katz, TreeSPH - A unfcaton of SPH wth the Herarchcal Tree method, The Astrophyscal Journal Supplement Seres, vol. 70, pp , 989. [] M. S. Warren and J. K. Salmon, A portable parallel partcle program, Computer Physcs Communcatons, vol. 87, pp , 995. [3] R. W. Hockney and J. W. Eastwood, Computer smulatons usng partcles, Adamhlger, New York, 988. [4] H. Rffert, H. Herold, O. Flebbe, and H. Ruder, Numercal aspects of the smoothed partcle hydrodynamcs method for smulatng accreton dsks, Computer Physcs Communcatons, vol. 89, pp. -6, 995. [5] J. J. Monaghan, Why partcle methods work (hydrodynamcs), SIAM Journal on Scentfc and Statstcal Computng, vol. 3, pp , 98. [6] J. J. Monaghan and J. C. Lattanzo, A refned partcle method for astrophyscal problems, Astronomy and Astrophyscs, vol. 49, pp , 985. [7] J. C. Lattanzo, J. J. Monaghan, H. Pongracc, and M. P. Schwartz, Controllng penetraton, SIAM Journal on Scentfc and Statstcal Computng, vol. 7(), pp , 986. [8] J. J. Monaghan, On the problem of penetraton n partcle methods, Journal of Computatonal Physcs, vol. 8, pp. -5, 989. [9] J. J. Monaghan and R. A. Gngold, Shock smulaton by the partcle method of SPH, Journal of Computatonal Physcs, vol. 5, pp , 983. [0] G. A. Sod, A survey of several fnte dfference methods for systems of hyperbolc conservaton laws, Journal of Computatonal Physcs, vol. 7, pp. -3,