Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS)

Transcription

1 Amercan Journal of Computatonal athematcs, 04, 5, Publshed Onlne December 04 n ScRes. Extenson of Smoothed Partcle Hydrodynamcs (SPH), athematcal Background of Vortex Blob ethod (VB) and ovng Partcle Sem-Implct (PS) Hrosh Isshk Insttute of athematcal Analyss (IA), Osaka, Japan Emal: sshk@dab.h-ho.ne.p Receved October 04; revsed 3 November 04; accepted 4 November 04 Copyrght 04 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract SPH has a reasonable mathematcal background. Although VB and PS are smlar to SPH, ther mathematcal backgrounds seem fragle. VB has some problems n treatng the vscous dffuson of vortces but s known as a practcal method for calculatng vscous flows. The mathematcal background of PS s also not suffcent. Not wth standng, the numercal results seem reasonable n many cases. The problem common n both VB and PS s that the space dervatves necessary for calculatng vscous dffuson are not estmated reasonably, although the treatment of advecton s mathematcally correct. Ths paper dscusses a method to estmate the above mentoned problem of how to treat the space dervatves. The numercal results show the comparson among FD (Fnte Dfference ethod), SPH and PS n detal. In some cases, there are bg dfferences among them. An extenson of SPH s also gven. Keywords SPH, PS, VB, Concentraton of ass or Vortcty, Estmaton of Space Dervatves. Introducton The author has once shown how to obtan the space dervatve n an rregular mesh usng the movng least square method []. A smlar problem s dscussed from a dfferent vewpont n SPH. Gngold & onaghan [] and Lucy [3] have developed Smooth Partcle Hydrodynamcs method (SPH). In SPH, the contnuous mass dstrbuton s approxmated by the fnte number of partcles. Namely, the contnuous quanttes are represented by How to cte ths paper: Isshk, H. (04) Extenson of Smoothed Partcle Hydrodynamcs (SPH), athematcal Background of Vortex Blob ethod (VB) and ovng Partcle Sem-Implct (PS). Appled athematcs, 5,

2 H. Isshk the fnte number of the dscrete quanttes. At the same tme, the space dervatves of the dstrbuted quanttes are expressed algebracally by the dscrete quanttes. Then, they transform the contnuous system nto a dscrete system convenent for the numercal soluton of the ntal and boundary value problem. Vortex Blob ethod (VB) s one of the practcal numercal tools for hgh Reynolds number flows [4]. The abducton s approxmated reasonably. However, the dffuson of vortces can t be approxmated precsely. We have shown that ths problem can be solved, f we apply the deas developed by SPH. ovng Partcle Sem-mplct method (PS) uses a smlar dscretzaton of the ntal and boundary value problem as SPH and VB. However, the mathematcal background s not suffcent. Not wth standng, the numercal results seem reasonable n many cases [5]. Ths suggests us that we may fnd the mathematcal background [] [6]. We have shown an nterestng relatonshp between SPH and PS as far as the gradent operator s concerned. The numercal results show the comparson among FD, SPH and PS n detal. In some cases, there are bg dfferences among them. A classfcaton of numercal solutons s shown n Fgure. The weak solutons: FE, FV, IR and GIR are Fnte Element ethod, Fnte Volume ethod, Integral Representaton ethod and Generalzed Integral Representaton ethod, respectvely. The strong solutons: FD, SPH, PS, LB and Col are Fnte Dfference ethod, Smooth Partcle Hydrodynamcs, ovng Partcle Sem-Implct ethod, Voltex Blob ethod, Lattce Boltzmann ethod and Collocaton ethod, respectvely. The general characterstcs of the varous numercal solutons are shown n Table. Snce SPH, VB and PS belong to the strong solutons and are mesh-less methods, the low computatonal cost becomes very mportant n some cases.. Dscretzaton Used n SPH and the Extenson α 3 Let x = x eα = x e+ x e + x e3 = x+ y+ zk be a poston vector, where the suffx n Greek letter s used to refer to the component of the coordnates, and the summaton conventon s used for the Greek suffxes. We defne a functon η ( x ) as an ntegral of w( x x ) ξ( x ) ρ( x ) wth respect to a volume V, where w( x ), ρ x are a weght functon, an auxlary functon and the densty of a flud, respectvely: ξ ( x ) and The functon η ( x ) and = ( ) d η x x x ξ x ρ x V. () V w ξ x can be scalars, vectors and tensors. If we ntroduce an dscretzaton of the volume V and the densty ρ as V = V, = 0,,, N, () Fgure. Classfcaton of numercal solutons. Table. General characterstcs of weak and strong solutons ( : very good, : good, : not good). Week method Strong method Stablty Precson Computatonal cost (memory, tme) 45

3 H. Isshk where the suffx n Roman letter s used to dscrmnate the pont, and V. We have an approxmaton of Equaton (): d and Furthermore, f we assume for the weght w ρ ( x)d V =, = 0,,, N, (3) = w( ) ξ s the mass of the volume element η x x x x. (4) w ( x )d V = (5a) V w = 0 for x > σ. (5b) If σ tends to 0, w( x ) tends to δ ( x ), where δ ( x ) s Drac s delta functon, namely:, when 0. Usng Equaton (6), we have a followng approxmaton: From Equatons (4) and (7), we obtan w x δ x σ (6) = ( ) ( ) ( ) d V = η x x x ξ x ρ r ξ x ρ x. (7) V w = w( ) If we assume ξ ( x ) =, then, we have from Equaton (8) ξ x ρ x x x ξ x. (8) = ( ) ρ( ) = ( ) ρ x w x x x V w x x. (9) V d Substtutng x = x, we obtan w ρ x = x x. (0) Hence, we have w( 0) = w( 0d ) V ρ x ρ x ρ x. () Ths means that Equaton (9) s a plausble approxmaton. Substtutng Equaton (9) nto Equaton (8), we obtan w( x x) ξ ( x) w( x x) ρ ( x) ξ ξ x = = w x x x. () Settng x to x n Equaton (), we have usng Equaton (0) ξ( x ) = w x x x ρ ( x ) ξ. (3) Operatng x α on both sdes of Equaton (8), we obtan ( x) ( x) w( ) ξ ρ ρ( x) + ξ( x) = x x ξ( x ). (4) x x x α α α 46

4 H. Isshk If we assume We derve w d ( ) = w( ) x x x x. (5) w x x w x x α α x x x x = = w α α ( x x ) x d x x x x x, (6) ( x) ( x x ) ρ w x x = α = x α α w ( ) α x x x x x. (7) Substtutng Equatons (9), (6) and (7) nto Equaton (4), we obtan ξ ( x) x Settng x to x n Equaton (8), we have ξ x α ( x) α α α x x = ρ w ( x x ) ( ξ( x) ξ( x )) ( x) x x α α x x = ρ ρ x w ( x x ) ( ξ( x ) ξ( x )) ( x ) x x ( x) α α α w ( x x ) x x = where we assume w ( 0) = 0. Now, we derve formulas for second dervatves. Frstly, from Equaton (4), we have ( x x ) x x ξ x ξ x ρ x ξ x ρ x ρ x ρ( x) ξ( x) x x x x x x x x = w x x ( x ).. (8). (9), (0) α β β α α β α β α β ξ () From Equatons (6) and (7), we obtan ( x x ) w β x x = w α β α ( x x ) x x x x x β x x w ( x x ) x x x x x x x x = w + ( x) w( x x ) ρ = x x x x α β α β α α β β αβ α α β β δ, 3 x x x x x x ( x x )( x x ) x x x x = w x x + w ( x x ) α α β β αβ α α β β δ, 3 x x x x x x () (3) αβ αβ αβ where δ s Chronecker s delta: δ = and δ = 0 when α = β and α β, respectvely. Substtutng Equatons (9), (7), () and (3) nto Equaton (), we derve 47

5 H. Isshk ( x x ) ξ x w ξ x ρ x ξ x ρ x ρ x ρ( x) = ξ( x) ξ( x) x x x x x x x x x x α β α β β α α β α β ( x x )( x x ) x x x x = w x x + w ( x x ) x α α β β αβ α α β β δ ξ 3 x x x x x x x x x x + w ( x x ) ( ξ( x) ξ( x) ) w ( x x ) ρ ( x) x x x x β β α α x x x x + w w ρ x x x x x ( x x ) ( ξ( x ) ξ( x )) ( x x ) α α β β ( x x )( x x ) x x x x ξ x w x x + w ( x x ) α α β β αβ α α β β δ 3 x x x x x x ( α α )( β β ) α α β β x x x x αβ δ x x x x = w x x + w ( x x ) 3 ξ x ξ x x x x x x x ( ) ( β β )( α α ) ( α α )( β β x ) x x xk + x x x x k + w x xk w x x ( ξ( x) ξ( x) ) k. ρ x x x x x Settng x to k k ( x x ) x n Equatons (), (3) and (4), we have δ = w x x + w ( x x ) x x α α β β α α β β w x x x x αβ x x x x α β 3 x x x x x x ( α α )( β β ) α α β β x x x x αβ x x x x ρ x δ = w + w ( ) α β x x x x x x ( x) ξ α x x β ρ 3 x x x x x x ( x ) k (4), (5) ( α α )( β β ) α α β β x x x x αβ δ x x x x = w w x x + x x ( ξ( x) ξ( x )) 3 x x x x x x k k k, (6) ( β β )( α α ) ( α α )( β β x ) x x xk + x x x x k + w x x w x x ( ξ( x ) ξ( x )), ρ x x x x x where we assume w ( 0) s fnte. From Equaton (9), we obtan for scalar φ and vector a x x φ = φ φ ρ [ ] [ ] w ( x x ) ( ( x ) ( x )) ( x ) x x ( x ) ( ) x a x a x w ( x ) x x a = x x ρ From Equaton (7), we obtan for scalar φ (7), (8). (9) 48

6 H. Isshk φ ( x) ρ( x ) k d ( x x ) w ( x x ) ( φ( x ) φ( x )) ( α α )( α α ) α α α α x x x x d x x x x = w w x x + x x ( φ( x) φ( x )) 3 x x x x x x ( α α )( α α ) ( α α )( α α x ) x x xk + x x x x k + w x x w x x ( φ( x ) φ( x )) ρ x x x x x k k k = w + x x ( x ) x x xk + w x xk w x x ( φ( x) φ( x) ) k, ρ x x x x x k k where d s the number of the dmenson. 3. Applcaton to Vortex Blob ethod (VB) The basc equatons for an ncompressble vscous flow are gven by Naver-Stokes equaton [7]: or ρ u + t ρ = + µ where µ s the coeffcent of vscosty, and the contnuty equaton: The pressure p satsfy (30) u u p u, (3) du ρ = p + µ u, (3) dt u =0. (33) The vscous flow s determned by Equatons (3) or (3), (33) and (34). If we ntroduce vortcty ω, the basc equatons can be rewrtten as Equaton (36) can be wrtten as p = ρ u u. (34) ω = u, u= 0, (35) ω ρ + ρu ω = ρ ω u+ µ ω. (36) t dω = ω u + ν ω, (37) dt where ν = µ ρ s the knematc vscosty. If ω s determned, the velocty u s obtaned by an ntegral representaton [7]: ( ) S G( x ξ t) ω( ξ t) Vξ + u ( x ξ t) ε u x, t = G x, ξ, t u ξ, t n G x, ξ, t n u ξ, t ds V ξ ξ ξ,,, d,,, where ε =,, 0 when x V, x S, x V S, respectvely, and (, ) δ (, ) (38) xg x ξ = x ξ, (39a) 49

7 H. Isshk G ( x ξ) x ξ ( x ξ ) n D, = π ln n D ( 4π) x ξ n 3D ( x ξ) x ξ n D xg( x, ξ) = G( x, ξ) = G ( x, ξ) e = ( π)( x ξ) x ξ n D 3 ( 4π)( x ξ) x ξ n 3D The second term on the rght hand sde of Equaton (38) s usually called Bo-Savart law. Vortex Blob ethod (VB) uses Equatons (37) and (38) and dscretzes the vortex feld as an assembly of concentrated vortces. The poston of the concentrated vortces are determned by ( t) d x a, dt (, t) (39b) (39c) = ua, (40) where a s the Lagrangan coordnates. It becomes very mportant how to obtan ω u and ω. In ths problem, we use the vortcty ω corresponds to the densty ρ n Secton. Then, we consder nstead of Equaton () Equaton (3) s replaced by Then, we have nstead of Equatons (8) and (9) = ( ) d η x x x ξ x ω x V. (4) V w ω ( x )d V =Ω, = 0,,, N. (4) w ξ x ω x = x x ξ x Ω. (43) w( ) ω( ) V w V d ω x = x x x = x x Ω. (44) In all equatons n Secton, f we replace ρ and wth ω and Ω, respectvely, we obtan the dfferental formulas n Secton athematcal Background of ovng Partcle Sem-Implct ethod (PS) From Equatons (9), (8) and (30), we have ( x t) α β β u, x x α α w ( u (, t) u (, t) ) β = x x x x x ρ ( x ) x x [ ] x x p = ( p( x ) p ( x )) w ( x x ) ρ x x x = + d ( x, ) ρ ( x ) ( x x ) ( x x ) ( ( x, ) ( x, )) k, (45), (46) α α α u t w w u t u t x x ( x ) x x xk + w x xk w x x ( φ( x) φ( x) ) k. ρ x x x x x k If the weght wr satsfes wr d d ~ wr and δ ( δ ) ( d ) π wrr dr dr r + =, (48) 0 (47) 40

8 H. Isshk then, we obtan where c( d ) s a constant. Then, we have wr ~ c d r, (49) wr cd wr cd ~ ~ wr and wr 3. dr r r dr = r = r (50) d d Substtutng Equatons (49) and (50) nto Equatons (45), (46) and (47), we obtan ( x t) α β β u, x x α α w ( u (, t) u (, t) ) β x x x x x ρ ( x ) x x [ ] x x p ( p( x ) p ( x )) w( x x ) ρ x x x d 3 ( x, ) ρ ( x ) ( x x ) ( ( x, ) ( x, )) u t w u t u t α α α x x, (5), (5) ( x ) x x xk α α + w k w ( u u ) k. ρ ( ) x x x x x x x k x x x x Now, we consder a weght k wr wth a small parameter 0 < ε = r and constant α : wr r e α = r+ ε + ε re 0 ( re r), where r = x. The weght wr has an asymptotc form: Ths asymptotc form s smlar to e ( 0 r r ) re re wr α ~ re when r re. r r α ε ε e r α = = r r ε < + + e k r defned by Equaton (4) n Ref. [5]. If e wr satsfes Equaton (49). In ths case, Equaton (5) has smlar forms as gven by Equaton () n Ref. [5], f we assume =, = 0,,. The author has once dscussed ths problem from a dfferent angle [] and had the concluson that the dscrete dfferental operators used n PS, especally, Laplace operator don t have the strct background from the mathematcal vewpont. Those operators should be consdered to be a knd of expermental formulas that s derved numercally. If we apply them to rregular grds, the results vary rregularly. We should say the operator estmate the dervatves statstcally. Qute naturally, the results are not unque. It may gve a good estmate n one tme and a wrong result n the other tme. Hence, f we need to verfy the accuracy, we should obtan several numercal results of the same problem usng the varous dscretzatons of the regon and take the statstcal average. Although we do not deny ths knd of approach, we wsh to ensure the relablty. For the purpose, we need much dscusson. Recently, a new paper [6] dscusses the accuracy of Laplace operator n PS n detal. Hopefully, we wsh to prove mathematcally, f possble, that, f we decrease the grd sze zero, then, the numercal error approaches stably to zero. e r s bg, 5. Numercal Verfcaton of Dscretzed Dfferental Operators of SPH wth Gaussan Kernel Numercal calculatons were conducted to verfy the valdty of the dscretzed dfferental operators such as (53) (54) (55) 4

9 H. Isshk Equatons (8) and (30). For smplcty, one-dmensonal (D) cases were consdered. The frst and second de- φ x : rvates of a scalar functon were calculated usng Equatons (8) and (30), respectvely. φ x = sn π xl n 0 x L (56) The regon s dvded nto N ntervals. Frst, we consder the unform mesh. Let dx and x ( = 0,,, N ) be the length of the nterval and the mdpont of the nterval, respectvely: 0.5dx0 for = 0 x =, dv = dx and = ρdv = ρd x. x + 0.5( dx + dx) for =,, As the weght functon wr, we use Gaussan functon: (57), (58) The weght functon wr satsfes wr r = exp. (59) πγ γ 0 w x dx =. (60) The D verson of Equatons (8) and (30) are gven by dφ x x w x x x x dx = ρ x x ( φ φ) ( x ), (6) = ( )( ) d φ x w x x x x ρ φ φ dx ( x ) x x xk + w x xk w x x ( φ( x) φ( x) ) k. ρ x x x x x k k (6) 5.. Unform Densty and Regular esh The densty ρ and the length of element dx are gven by where the computatonal regon s defned as 0 < x< L Verfcaton of Gradent Operator From Equatons (6), (63) and (45), we have ρ = = 0,,, N, (63a) dx= LN= 0,,, N, (63b) dφ x x w ( x x ) ( φ( x) φ( x) ) dx =. (64) x x We used parameters: L = 4, N = 80 and γ = dx = The numercal results are shown n Fgure. Although there exst large errors at the boundares as shown n Fgure (a), the numercal results agree very well wth the exact except the neghborhood of the boundares. If we extend the regon beyond the boundares, the estmatons wthn the orgnal regon are mproved as shown n Fgure (b) Verfcaton of Laplace Operator From Equaton (6), we have 4

10 H. Isshk d φ dx ( x) Fgure. A comparson between calculated and exact values of dφ/dx ((a) Not usng extenson of regon; (b) Usng extenson of regon). ρ( x ) = w ( x x ) ( φ( x ) φ( x )) x x x x + w x x w x xk x x k. ρ k ( φ φ) ( x ) k x x x x k The numercal results are shown n Fgure 3. If we extend the regon beyond the boundares, the estmatons wthn the orgnal regon are mproved. 5.. Non-Unform Densty and Regular esh The densty ρ and the length of element dx are gven by (65) ρ = + x L = 0,,, N, (66a) d x= LN, = 0,,, N, (66b) respectvely. The other parameters are same as n Secton 5... The numercal results are shown n Fgure 4. The numercal result agrees well wth the exact result Unform Densty and Non-Unform esh The densty ρ and the length of element dx are gven by ρ = = 0,,, N, (67a) d (.05) 0,,,, = N L L x = α = N α (.05 ) = L, (67b) N N respectvely. The other parameters are same as n Secton 5... The numercal results are shown n Fgure 5. The numercal result agrees well wth the exact result Non-Unform Densty and Non-Unform esh The densty ρ and the length of element dx are gven by = 0 ρ = + x L = 0,,, N, (68a) 43

11 H. Isshk Fgure 3. A comparson between calculated and exact values of d φ/dx ((a) Not usng extenson of regon; (b) Usng extenson of regon). Fgure 4. A comparson between calculated and exact values ((a) Gradent operator dφ/dx; (b) Laplace operator d φ/dx ). Fgure 5. A comparson between calculated and exact values ((a) Gradent operator dφ/dx; (b) Laplace operator d φ/dx ). 44

12 H. Isshk = N L L dx = α (.05) = 0,,, N, α (.05) = L (68b) N N respectvely. The other parameter are same as n Secton 5... The numercal results are shown n Fgure 6. The numercal result agrees well wth the exact result. 6. Comparson between SPH and PS 6.. D Formulas for Dscrete Gradent and Laplacan Operator For convenence, we summarze the D dscrete dfferental operators used n SHP and PS as follows. From Equatons (6) and (6), we have for SHP = 0 x x w x x x x dφ ( ) dx = φ φ ρ ( x ) x x ( x) w ( x x )( ( x) ( x) ) φ ρ = φ φ k ( x ) x x xk + w x xk w x x ( φ( x) φ( x) ) k, ρ x x x x x From Refs. [5] and [8], we have for PS where k, (69a) (69b) = ρ ( x )dx. (69c) dφ φ φ = ( x x) w( x x ), (70a) dx n x x d φ = dx λ n λ = ( φ φ) w( x x ), (70b) n = w( x x ), (7a) ( ) ( x ) x x w x x w x. (7b) Fgure 6. A comparson between calculated and exact values ((a) Gradent operator dφ/dx; (b) Laplace operator d φ/dx ). 45

13 H. Isshk 6.. Effects of esh and Weght on Estmaton of Gradent and Laplacan of a Gven Functon We summarze the meshes and weghts used n the followng n Table and Table 3, respectvely. The parameters used n Sectons are gven below. L = 4, N = 40, ρ =, γ = dx for SPH0, h = 8dx for SPH and SPH, and dx for PS, N = 8 for SPH0, SPH and SPH, and for PS. end _ cor In Sectons , the computatonal results are shown n Fgures 7-. The results are summarzed Table 4 and Table 5 n Secton Regular esh As shown n Fgure 7, there are no bg errors n all methods SPH0, SPH, SPH and PS. The accuracy of SPH0 s very hgh. Table. Classfcaton of mesh. Name dx L Regular dx = N Algebrac L L dx α = N = + N 00, dx = L = 0 Geometrc dx L N α.05 =, = N = 0 dx = L L = +, N Random dx α ( 0.065drand ) L Snusodal dx = α 0.5sn 4π + N N, = N = 0 = N = 0 dx = L dx = L Table 3. Classfcaton of weght. Name Weght SPH0 Gauss w( x) SPH Lucy w( x ) SPH Opt.kernel PS w( x ) x = exp, x < πγ γ 3 5 x x + 3, x σ = 4 σ σ σ 0, x > σ 3 x x 6 + 6, 0 x < 0.5h h h 3 4 x w x =, 0.5 h x < h 3h h 0, h x h x, 0 x < h = 0, h x 46

14 H. Isshk Table 4. Summary on dφ/dx ( : good, : not good, : bad). Regular Algebrac Geometrc Random Snusodal SPH0 SPH SPH PS Table 5. Summary on d φ/dx ( : good, : not good, : bad). Regular Algebrac Geometrc Random Snusodal SPH0 SPH SPH PS 6... Algebrac esh As shown n Fgure 8, a bg error occurs n Geometrc esh As shown n Fgure 9, a bg error occurs n d φ dx of PS. d φ dx of PS Random esh As shown n Fgure 0, the errors are rather small n all methods SPH0, SPH, SPH and PS not only for dφ dx but also for d φ dx. The errors n PS are surprsngly small Snusodal esh As shown n Fgure, the errors are rather bg n all methods SPH0, SPH, SPH and PS not only for dφ dx but also for d φ dx except dφ dx n PS. The errors n SHP0 are smaller than those n the other methods Summary of Results From the above mentoned numercal results, the followng summares are obtaned Applcaton to Soluton of Intal Value Problem D Flud oton wthout Pressure and Vscosty The moton of vast number of partcles dstrbuted n space under the acton of the gravtatonal force may be treated as a flud moton wthout pressure and vscosty [9] [0]: ρ ρu + = 0, (7a) t x u u + u = Π t x x, (7b) Π = 4πGρ, (7c) x where ρ, u and Π s the densty, velocty and gravtatonal potental. G s the gravtatonal constant. The soluton of the problem defned above by Euleran method s gven as follows: ) At tme t, assume ρ, u and Π are gven. ) ρ t and u t are obtaned from Equatons (7a) and (7b), and Π s obtaned from Equaton (7c). 47

15 H. Isshk dφ dx d φ dx SPH0 SPH SPH PS Fgure 7. Regular mesh. 48

16 H. Isshk dφ dx d φ dx SPH0 SPH SPH PS Fgure 8. Algebrac mesh. 49

17 H. Isshk dφ dx d φ dx SPH0 SPH SPH PS Fgure 9. Geometrc mesh. 430

18 H. Isshk dφ dx d φ dx SPH0 SPH SPH PS Fgure 0. Random mesh. 43

19 H. Isshk dφ dx d φ dx SPH0 SPH SPH PS Fgure. Snusodal mesh. 43

20 H. Isshk 3) ρ, and u at tme t+ dt s calculated. 4) Repeat ths process. In the soluton by Lagrangan method, frst, the equatons n Euleran form are transformed nto Laglangan form: Dx u Dt =, (73a) Du u u + u = Π Dt t x x, (73b) D ρ ρ u u ρ + = ρ Dt t x x, (73c) Π = 4πGρ. (73d) x The followng procedures gve the soluton of the problem defned above by Lagrangan method: ) At tme t, assume ρ, u and Π are gven. ) Dx Dt, Dρ Dt and Du Dt s obtaned from Equatons (73a), (73b) and (73c), and Π s obtaned from Equaton (73d). 3) x, ρ and u of the materal pont at tme t+ dt s calculated. 4) Repeat ths process. If x s obtaned, then, dx and ρ s obtaned as shown below: and dx = ( x+ x) + ( x x ) = ( x+ x ) (74a) ρ =. (74b) dx Hence, from the theoretcal vewpont, ths problem can be solved wthout usng the gradent operator. However, we use the gradent operator to obtan ρ usng the contnuty equaton. ) Trapezodal Dstrbuton of the Intal Densty The ntal condtons are gven by The boundary condtons are specfed as The computatonal condtons are as follows: x = x0 = LN, (75a) 0 + ( x 0.35L) when 0.5L x < 0.35L L when 0.35L x < 0.65L ρ = 0 ( x 0.65L) when 0.65L x < 0.75L L otherwse, (75b) u = 0. (75c) ρ = ρ, ρ = ρ ; (76a) 0 N N u = u, u = u. (76b) 0 N N L = 4, N = 4, ν = 0.0, dt = , t = 30000dt, G = 0.005, γ = dx for SPH0, h = dx for PS, N _ = 8 for SPH0 and for PS. The numercal results are shown n Fgure. The FD (Fnte Dfference ethod) uses the central dffer- end end cor 433

21 H. Isshk ρ n x 0 ρ n x FD SPH0 PS Fgure. Trapezodal dstrbuton of the ntal densty. ences for the frst and second space dervatves, and the precson of FD s consdered hgh. The Euleran soluton s used for FD, and the Lagrangan soluton s used for SPH0 and PS. The dstrbuton pattern of PS s slghtly dfferent from those of FD and SPH0. ) Rectangular Dstrbuton of the Intal Densty wth G = 0.00 The ntal condtons are gven by x= x0 = ( + 0.5) LN, (77a) when x < 0.5L ρ = when 0.5L x 0.75L when 0.75 L x, (77b) 434

22 H. Isshk The boundary condtons are specfed as u = 0. (77c) ρ = ρ, ρ = ρ ; (78a) 0 N N u = u, u = u. (78b) 0 N N The computatonal condtons are as follows: L = 4, N = 4, ν = 0.0, dt = , t = 30000dt, G = 0.00, γ = dx for SPH0, h = dx for PS, N _ = 8 for SPH0 and for PS. The numercal results are shown n Fgure 3. The FD uses the central dfferences for the frst and second space dervatves, and the precson of FD s consdered hgh, f we neglect the spurous oscllaton. The dstrbuton pattern of PS s slghtly dfferent from those of FD and SPH0. end end cor ρ n x 0 ρ n x FD SPH0 PS Fgure 3. Rectangular dstrbuton of the ntal densty wth G =

23 H. Isshk 3) Rectangular Dstrbuton of the Intal Densty wth G = The ntal condtons are gven by The boundary condton are specfed as Computatonal condton x = x0 = LN, (79a) when x < 0.35L ρ = when 0.35L x < 0.75 L, when 0.35L x (79b) u = 0. (79c) ρ = ρ, ρ = ρ ; (80a) 0 N N u = u, u = u. (80b) 0 N N L = 4, N = 4, ν = 0.0, dt = , t = 30000dt, G = 0.005, γ = dx for SPH0, h = dx for PS, N _ = 8 for SPH0 and for PS. The numercal results are shown n Fgure 4. The FD uses the central dfferences for the frst and second space dervatves, and the precson of FD s consdered hgh, f we neglect the spurous oscllaton. The dstrbuton pattern of PS s dfferent from those of FD and SPH D Flud oton wthout Pressure but wth Vscosty In the present secton, the vscosty s ntroduced: end ρ ρu + = 0, t x end cor u + u u = + ν u, Π t x x x Π = 4π Gρ, (8c) x where ν s the knematc vscosty. Snce the dscrete Laplacan operator of SPH0 generates a bg error at the dscontnuty, the ntal densty dstrbuton was smoothened usng the fve pont runnng average, and, n the second example below, an small artfcal numercal vscosty μ = n case of N = 4 was added at every tme step: + dx (8a) (8b) ρ ρ + µ ρ ρ + ρ, (8a) u u + u u + u. (8b) µ + dx In the thrd examples below, a bg dfference has occurred between SPH and PS solutons. ) Exponental Dstrbuton of the Intal Densty The ntal condtons are gven by x = x0 = ( ) LN, (83a) The boundary condtons are specfed as x 0.5L ρ = exp, 0.L (83b) u = 0. (83c) ρ = ρ, ρ = ρ ; (84a) 0 N N 436

24 H. Isshk ρ n x 0 ρ n x FD SPH0 PS Fgure 4. Rectangular dstrbuton of the ntal densty wth G = The computatonal condtons are as follows: u = u, u = u. (84b) 0 N N L = 4, N = 4, ν = 0., dt = , t = 0000dt, γ = dx for SPH0, h = dx for PS, G = 0.005, N _ = 8 for SPH0 and for PS. The numercal results are shown n Fgure 5. The FD uses the central dfferences for the frst and second space dervatves, and the precson of FD s consdered hgh. In ths example, the FD, SPH and PS solutons becomes smlar. ) The Trapezodal Dstrbuton of the Intal Densty The ntal condtons are gven by x = x0 = ( + 0.5) LN, (85a) end end cor 437

25 H. Isshk ρ n x 0 ρ n x FD SPH0 PS Fgure 5. Exponental dstrbuton of the ntal densty. 0 + ( x 0.35L) when 0.5L x < 0.35L L when 0.35L x < 0.65L ρ = 0 ( x 0.65L) when 0.65L x < 0.75L L otherwse, (85b) u = 0, (85c) where the ntal densty dstrbuton was smoothened usng the fve pont runnng average: ( ρ + ρ + ρ + ρ+ + ρ+ ) ρ. (86) 5 438

26 H. Isshk The boundary condtons are specfed as The computatonal condtons are as follows: ρ = ρ, ρ = ρ ; (87a) 0 N N u = u, u = u. (87b) 0 N N L = 4, N = 4, ν = 0.0, dt = , t = 30000dt, G = 0.005, γ = dx for SPH0, h = dx for PS, N _ = 4 for SPH0 and for PS. The numercal results are shown n Fgure 6. The FD uses the central dfferences for the frst and second space dervatves, and the precson of FD s consdered hgh. In ths example, small dfference s observed n the FD, SPH and PS solutons. 3) The Flat-Slope-Flat Dstrbuton of the Intal Velocty end cor end ρ n x 0 ρ n x FD SPH0 PS Fgure 6. Trapezodal dstrbuton of the ntal densty. 439

27 H. Isshk The ntal condtons are gven by x = x0 = LN, (88a) ρ =, (88b) 0.5 when x < 0.5L u = ( x 0.5L) when 0.5L x 0.75L (88c) L 0.5 when 0.75L x The boundary condtons are specfed as ρ =, = ; (89a) The computatonal condtons are as follows: 0 ρn u = 0.5, = 0.5. (89b) 0 un L = 4, N = 4, ν = 0.0, dt = , t = 0000dt, G = 0, γ = dx for SPH0, h = d x for PS, N _ = 8 for SPH0 and for PS. end cor Snce we assume G = 0 n ths example, the equaton becomes Burgers equaton. a) Comparson of FD Soluton wth the Analytcal One The central dfferences were used for the frst and second space dervatves. As shown n Fgure 7. The FD soluton s a very good approxmaton of the analytcal soluton []. b) Comparson of FD, SPH0 and PS Solutons Fgure 8 and Fgure 9 show the comparsons of ρ and u among FD, SPH0 and PS solutons. SPH0 and PS solutons do not gve good approxmatons. Wth respect to the velocty u, the dfference between SPH0 and PS s bg. 7. odfed Gaussan Weght Gaussan-type weghts of fnte support wth C contnuty are gven as follows: end 4 x x x x x αexp exp = α + 4 γ h h h γ w( x) = x πx α exp cos, γ h (90a), (90b) Fgure 7. Comparson of FD soluton wth the analytcal one ((a) FD soluton; (b) Exact soluton). 440

28 H. Isshk ρ n x 0 ρ n x FD SPH0 PS Fgure 8. Comparson of densty ρ. where α satsfes h x x hx α exp 0 exp d d x αh 0 x x h h γ h γ = w( x) dx w( x) dx = h = = 0 h x πx hx πx α exp cos dx α h exp cos d x. 0 0 γ h γ The frst and the second dervatves of w( x ) gven by Equaton (90a) are gven by x x exp 4 4 w x = α + x+ + x, (9a) h γ h γ h γ h γ (9) 44

29 H. Isshk u n x 0 u n x FD SPH0 PS Fgure 9. Comparson of velocty u x exp w x = α x + x + x. (9b) h γ h γ h γ γ h γ h γ h γ Numercal examples are shown n Fgure 0. When h s small, the accuracy becomes low. As has already ponted out n Secton 5.., there exst large errors at the boundares. The numercal results agree well wth the exact ones on overall. In the examples, γ s equal to dx = L/N = 0. and the accuracy seems suffcent when h s bgger than or equal to 4dx. 8. Conclusons The author has once shown how to obtan the partal dervatve n an rregular mesh usng the movng least 44

30 H. Isshk h dφ dx d φ dx Fgure 0. Effect of support h ( φ = sn ( πx L) ; L = 4, N = 40, γ = 0., h = 0., 0.4, 0.8). square method []. A smlar problem s dscussed from a dfferent vewpont n SPH. Gngold & onaghan [] and Lucy [3] have developed Smooth Partcle Hydrodynamcs method (SPH). We have extended SPH theoretcally n the present paper. In Vortex Blob ethod (VB), the abducton s approxmated reasonably. However, the dffuson of vortces can t be approxmated precsely. We have shown n the present paper that ths problem can be solved, f we apply the deas developed by SPH. ovng Partcle Sem-mplct method (PS) uses a smlar dscretzaton of the ntal and boundary value problem as SPH and VB. However, the mathematcal background of PS s not suffcent. We have shown n the present paper that the mathematcal background of the dscrete gradent operator s strengthened by applyng the deas developed by SPH. However, that of the dscrete Laplacan operator could not gven. 443

31 H. Isshk The dscrete gradent and Laplacan operators of SPH nclude the frst, frst and second dervatves of the weght functon wth respect to the space coordnates, respectvely. On the other hand, those of PS nclude only the weght functon tself. Ths may be the bggest dfference between the dscrete dfferental operators n SPH and PH. In the present paper, solutons by FD (Fnte Dfference ethod), SPH and PS are compared numercally n detal. ) Effects of mesh on the dscrete gradent and Laplacan operators were studed. PS showed good results wth respect to the random mesh. ) The FD, SPH and PS were appled to the ntal value problems, and the effects of the dfference of the soluton method were studed. In some cases, the solutons of ntal value problem showed a dfference between SPH and PS. 3) The dscrete Laplacan operator of SPH s senstve to the spacal dscontnutes of the soluton functon. Hence, n the case of the ntal value problem, the dscontnuty n the ntal condton should be smoothened beforehand, and the small amount of the artfcal vscosty should be ntroduced. 4) The author has once studed a mathematcal background of PS theoretcally [] and dd some numercal calculatons. In very lmted cases, the dscrete Laplacan operator of PS can be obtaned mathematcally. However, the generalzaton to the general mesh was not obtaned. Hence, we are oblged to apply the Laplacan operator as a bold approxmaton n case of the general mesh. 5) Recently, Ng, Hwang and Sheu [6] dscussed the accuracy of the dscrete Laplacan operator of PS theoretcally and numercally. They clarfed an mportant aspect of the propertes of the operator. As one of the propertes, they ponted out n n concluson of Ref. [6] that PS gave generally a favorable result n case of the rregular mesh. We also had a smlar mpresson as expressed n () above. The support of Gaussan weght n SPH s nfnte. In the present paper, weghts of a Gaussan-type of fnte support wth C contnuty were also gven. Acknowledgements The author thanks sncerely Prof. G. Yagawa, Toyo Unv., Japan for hs support and encouragement. References [] Isshk, H. (0) Dscrete Dfferental Operators on Irregular Nodes (DDIN). Internatonal Journal for Numercal ethods n Engneerng, 88, [] Gngold, R.A. and onaghan, J.J. (977) Smoothed Partcle Hydrodynamcs: Theory and Applcaton to Non-Sphercal Stars. onthly Notces of the Royal Astronomcal Socety, 8, mp;whole_paper=yes&type=printer&fletype=.pdf [3] Lucy, L.B. (977) A Numercal Approach to the Testng of the Fsson Hypothess. The Astronomcal Journal, 8, [4] Chorn, A.J. (973) Numercal Study of Slghtly Vscous Flow. Journal of Flud echancs, 57, [5] Yokoyama,., Kubota, Y., Kkuch, K., Yagawa, G. and ochzuk, O. (04) Some Remarks on Surface Condtons of Sold Body Plungng nto Water wth Partcle ethod. Advanced odelng and Smulaton n Engneerng Scences,, [6] Ng, K.C., Hwang, Y.H. and Sheu, T.W.H. (04) On the Accuracy Assessment of Laplacan odels n PS. Computer Physcs Communcatons, 85, [7] Isshk, H., Nagata, S. and Ima, Y. (04) Soluton of Vscous Flow around a Crcular Cylnder by a New Integral Representaton ethod (NIR). The Assocaton for Japan Exchange and Teachng,, fle:///c:/users/l/downloads/ pb%0().pdf [8] Koshzuka, S. and Oka, Y. (996) ovng Partcle Sem-Implct ethod for Fragmentaton of Incompressble Flud. Nuclear Scence and Engneerng, 3, [9] oscardn, L. and Dolag, K. (0) Chapter 4: Cosmology wth Numercal Smulatons. In: atarrese, S., Colp,., 444

32 H. Isshk Gorn, V. and oschella, U., Eds., Dark atter and Dark Energy, Sprnger, Berln, [0] onaghan, J.J. (99) Smoothed Partcle Hydrodynamcs. Annual Revew of Astronomy and Astrophyscs, 30, [] Wkpeda, Burgers Equaton

33