A New SPH Equations Including Variable Smoothing Lengths Aspects and Its Implementation

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1 COMPUTATIOAL MECHAICS ISCM007, July 30-August 1, 007, Beng,Chna 007 Tsnghua Unversty Press & Sprnger A ew SPH Equatons Includng Varable Smoothng Lengths Aspects and Its Implementaton Hongfu Qang*, Weran Gao Faculty of Mechancal & Propulson Engneerng, X an H-Tech Insttute, X an, Shaanx, Chna Emal: qang@63.net Abstract ew SPH equatons ncludng fully varable smoothng length aspects and ts mplementaton are proposed n ths paper. Unlke the exstng adaptve kernel SPH method, the fully varable smoothng lengths effects have been consdered essentally n the scheme based on the adaptve symmetrcal kernel estmaton. Among the new equatons, the evoluton equaton of densty s derved n essence, t s mplctly coupled wth varable smoothng length equaton; the momentum equaton and energy equaton are derved from Sprngel s fully conservatve formulaton SPH usng the symmetrcal kernel estmaton nstead of the scatter kernel estmaton algorthm. Because the new SPH densty evoluton equaton s mplctly coupled wth varable smoothng length equaton, an addtonal teraton process s employed necessary to solve the evoluton equatons of densty and the varable smoothng lengths equaton, and the SPH momentum equaton and the energy equaton s solved n nature, only the lttle cost s used n the teraton algorthm. The new equatons and ts algorthm are tested va two 1D shock-tube problems and a D Sedov problem, t s showed that the new algorthm corrects the varable smoothng lengths effect farly well, especally n the D Sedov problem, the pressure peak s captured by the presented method more accurately than that of Sprngel s scheme, and the accuracy of pressure at the center n Sedov problem s mproved also. The new method can deal wth the large densty gradent and large smoothng length gradent problems well, such as large deformaton and serous dstorton problems n hgh velocty mpact and blastng phenomenon. Key words: SPH, meshless, algorthm, shock-tube, Sedov problem ITRODUCTIO Usually the local nterpolaton resoluton s determned by the smoothng length h n SPH method. Early SPH smulatons used a fxed h for all partcles [1] or h s taken as a globally tme-dependent functon of the mean number densty of partcles n the system []. However, SPH method usng the constant smoothng length n space would yeld more accurately estmate n hgh partcle neghborhood regons than n low regons, and would furthermore not take full advantages of the partcle local dstrbuton. Consderng both the consstency and effcency, an SPH method wth local varable smoothng length algorthms was frstly presented by Gngold & Monaghan [3], whch adapts h accordng to the local number densty of partcles. Evard [4] also proposed the local varable smoothng length algorthms n SPH, he ponted out the h effects, whch refers to the effects of the varable spatal h on the estmaton of gradents, and ths knds of effects can be gnored safely. But accordng to research by Hernqust et al [5], the ntroducton of varyng smoothng length can lead to serous problems wth badly energy conservaton n certan stuatons. elson [6] conducted the research n detal on how the h affects the result, he ndcated that the addtonal h terms should appear n the partcle equatons of moton. Serna et al [7] also dd the smlar analyss. However, the above approach to correct the h effects leads to somewhat complex forms for the dynamcal equatons and 817

2 ntroduces nose nto smoothed estmates, and such verson of SPH method has not found wdespread usage fnally. Another dea to treat the nconsstent problems of varyng smoothng length s proposed by Sprngel et al [8], they derved a new formulaton of SPH whch employs varable smoothng lengths and conserves energy and entropy by usng the Lagrangan formalsm, and the method s equvalent to Thomas [9] equatons when constant smoothng length algorthm s adopted. Then Monaghan [10] also presented new SPH equatons va smlar way, whch mproves the relatonshp between the smoothng length and densty, and the conservaton propertes of the new equatons was analyzed n hs paper. The Sprngel SPH equatons [8] used the scatter kernel estmaton, but the scatter kernel estmaton produces more errors than the symmetrcal kernel estmaton method [11]. We found that the Sprngel s method corrects the h effects well, but worses the precson n the smulaton both at the front of expand waves and the contact dscontnues, where the smoothng length are not dstrbuted smoothly. In ths paper, new SPH equatons ncludng fully varable smoothng length aspects s presented, and the adaptve symmetrcal kernel estmaton algorthm s adopted. Among the new equatons, the evoluton equaton of densty s derved n essence, t s mplctly coupled wth varable smoothng length equaton; the momentum equaton and energy equaton are derved from Sprngel s fully conservatve formulaton SPH usng the symmetrcal kernel estmaton nstead of the scatter kernel estmaton algorthm. The mplementaton of new SPH equatons s also dscussed n ths paper, an addtonal teraton process s employed to solve the coupled densty equaton and varable smoothng length equaton, and the momentum equaton and energy equaton are solved n nature, only lttle cost s used n the teraton calculaton. Three test problems are presented to verfy the new equatons, there are the 1D Blast-Wave shock-tube problem, 1D Sögreen shock-tube problem and D Sedov problem. The numerc results show that the new equaton corrects the varable smoothng length effect farly well, especally n the D Sedov problem, the pressure peak s captured by the presented method more accurately than that of Sprngel s scheme, and the accuracy of pressure at the center n Sedov problem s mproved also. The new method can deal wth the large densty gradent and large smoothng length gradent problems well, such as large deformaton and serous dstorton problems n hgh velocty mpact and blastng phenomenon. SPH EQUATIOS WITH VARIABLE SMOOTHIG LEGTH 1. Evoluton Equaton of Densty As n SPH calculatons, the contnuous densty feld at the locaton r s evaluated as Eq. (1), ( ) = mw (, h) ρ r r r (1) = 1 r r s a smooth functon, usually referred to as the nterpolatng kernel, h s the smoothng length, or named as bandwdth of the kernel, whch determnes the kernel estmatng precson, and s the number of neghbors of partcle. W, h d = 1 lm W r r, h = δ r r. Where ρ s the densty of flud partcle, m s the mass of partcle, W(, h) The kernel functon satsfes the condtons ( r r ) r, and ( ) ( ) Ω Usually the smoothng lengths should vared to keep the number of neghbors for each partcle mantaned at a nearly fxed value, here, we select smoothng lengths by requrng that a fxed mass s contaned wthn the volume n h radus[8], vz. Eq. () σh ρ = M () d sph Where the d s the number of dmensons, and σ s constant number wth the values, π, 4 3 π correspond to dfferent dmensons. The use of Eq. () leads to the nconsstence spatal and temporal dstrbutng of h h r, t. smoothng length,.e., the smoothng length s both the functon of poston and tme, ( ) 818 h 0 = For the kernel estmaton wth varable smoothng length, the symmetrcal kernel estmaton can get hgher

3 order precson compared to the scatter kernel estmaton [11], we adopt the symmetrcal kernel estmaton, and the symmetrcal kernel estmaton form of Eq. () s ρ ( r ) = mw (3) = 1 1 Where W = W( r, h ), r = r r, h = ( h + h ) By dfferentatng Eq. (3) wth respect to tme, we get dρ d 1 dh dh W = mw = m W dt dt v + + dt dt h (4) Where v = v v, v s the velocty of partcle, ths s the new SPH evoluton equaton of densty ncludng varable smoothng length, and t equals to normal SPH evoluton equaton of densty when the smoothng length keep constant.. Equaton of Moton ow, we derved the new SPH equaton of moton based on Sprngel s fully conservatve formulaton of SPH [8], and replace the scatter kernel estmaton algorthm by the symmetrcal kernel estmaton. The Lagrangan L for the non-dsspatve moton of a flud s [10] 1 = ρ ( ρ, ) (5) L v u s dq Where v s the velocty, u s the thermal energy per unt mass, ρ s the densty and s s the entropy. For the strctly adabatc flow, from the frst law of thermodynamcs we know that u P = ρ ρ s Where P s pressure.. The SPH form of Eq. (5) s 1 L ( qq&, ) = mb vb u( ρb, sb) (7) b Where q = ( r1, Kr, h1, K h),ncludng the poston and smoothng length of partcles. Such process s consdered as dscretely approxmatng of contnues flud phase space to dmensonal. The second half of q,.e., smoothng length, s provded constrans Eq. (), n order to mantan the typcal number of smoothng neghbors for partcles. d ( ) = = 0, = 1 φ q σh ρ M K (8) sph We now obtan the equatons of moton from the Lagrangan equaton wth constrants Eq. (9) d L L φ =, = 1K dt q& q q λ = 1 Where λ s Lagrange multples. The second half of these equatons gves P ρ m d h h ρ d 1 d = σ ρλ + σ λ ρ h = 1 h (10) Where (6) (9) 819

4 ρ = mk Wk = m W (11) h k h h Among Eq. (11), the symmetrcal kernel estmaton s ntroduced, and W W(, h ) From the frst equatons of Eq. (6), we get 80 = r. dv P 1 ρ d m = m 1 σλ h ρ (1) dt = 1 ρ m P Usng ρ m W δ mk Wk k = 1 = + (13) The fnal equatons of moton are obtaned as dv P P = m f + f W dt = 1 ρ ρ (14) Where 1 ρ d f = 1 σλ h m P (15) ρ The λ n Eq. (15) s calculated from Eq. (10). Because of h s zero except the neghbor partcles, and the value of W ρ s lttle outsde one smoothng length from the center of partcle, we gnore the h h and smplfy the Eq. (15) nto Eq. (16). The result of numercal smulaton also support ths dea. 1 h ρ f = 1 + (16) dρ h The corrected SPH equaton of moton (14) s equal to the standard SPH equaton of moton f the effect of λ are gnored 3. Equaton of Thermal Energy From the frst law of thermodynamcs, we know that P Tds = du + Pdv = du d ρ (17) ρ Where T s the temperature, s s the entropy per unt mass, u s the thermal energy per unt mass, P s the pressure and v s the volume per unt mass. If the entropy mantans constant, the tme rate of change of thermal energy s du P d ρ = (18) dt ρ dt The SPH form of Eq. (18) s du P d ρ = (19) dt ρ dt 4. Varable Smoothng Length Equaton and Implementaton of Equatons The smoothng length h s the functon both of space and tme, by dfferentatng Eq. () wth respect to tme, we obtaned dh 1 h dρ = (0) dt d ρ dt

5 Because the dh dt s related to dρ dt, and the dρ dt s related to all ts neghbors dh dt and dh dt, t s dffcult to solve evoluton equaton of densty Eq. (4) and varable smoothng length equaton Eq. (0) drectly, here we employ an teraton process to solve these two equatons. Frstly let dh dt, and calculate the dρ dt through Eq. (4), then usng the ntermedate values of dρ dt to get dh dt from Eq. (0), and update the new dρ dt usng the value of dh dt va Eq. (4). Repeat the above process untl dρ dt has no sgnfcant change n one loop. The equaton of moton and equaton of energy are solved n nature. Usually only a lttle cost s spent n the teraton calculaton. UMERICAL TESTS In order to verfy the capabltes of the present method, two 1D shock problems and one D Sedov problems are tested, whch the smoothng length has changes sgnfcantly n the smulatons. Amongst them, the gas equaton of state as P = ( γ 1) ρu s used, where γ = 1.4. The cubc splne kernel s used [1] also. The correcton term of equaton of moton Eq. (14) n ths paper s calculated from Eq. (16), that means the effect of ρ h are gnored. The approprate artfcal vscosty [13] and artfcal heat [14] are ntroduced to stable the smulaton of shock, and the smulaton s updated usng a standard leapfrog ntegrator. 1. Blast-Wave Problem The Blast-Wave problem s a dffcult test whch nvolves extremely supersonc 5 flows, and the ntal pressure of the left sde to the dscontnuty s 10 tmes to the rght sde. The ntal condton s the deal gas wth Eq. (1). There are 800 partcles of equal mass mplementng n ths smulaton, and the ntal contact dscontnues s set at x = 0. The dscontnuous of physcal parameters are smoothed ntally [15]. ( ρ, u, p) = ( 1,0,1000 ) L (1) ( ρ, u, p) = ( 1,0,0.01) R The densty vs poston, velocty vs poston and pressure vs poston are showed n Fg 1. In ths test, the densty peak value s overestmated by about 10% and the energy conserves badly when standard SPH method s adopt. Compared wth the former, both the Sprngel s method and the present method n the paper performs well, and the smulaton error n densty peak value less than 1%. Furthermore, f the correcton on equaton of moton s gnored when f = 1, the new method also can get the correct results, unforturnately, the Sprngel s method can not obtan. Fgure 1: Results for the one-dmensonal blast-wave problem at t = s 800 partcles of equal mass are arranged on each sde of the ntal nterface. Results for dfferent SPH formulatons are compared wth densty profle (left), velocty profle (mddle) and pressure profle (rght). The Sprngel s method (trangles) new method presented n ths paper (dots) compares very well wth exact soluton (sold lne), however the standard SPH (squares) overestmated the densty peakvalue by about 10%.. Sögreen Problem The exact soluton of Sögreen problem contans two expand wave and a week constant contact dscontnues, the pressure value at contact s very lttle. Ths problem can check the ablty 81

6 of numercal method when smulate the low densty and low energy problem. The ntal condton s descrbe as Eq. (), and the setup of smulaton s same as above problem. ( ρ, u, p) = ( 1,,0.4 ) L () ρ, u, p = 1,,0.4 ( ) ( ) R The densty vs poston, velocty vs poston and pressure vs poston are showed n Fg 1. The standard SPH, Sprngel s method and the new method n ths paper all perform well. Among the result obtaned by Sprngel s method, there are small umps at the front of expand waves of densty and nternal profle, whereas the new method n ths paper and the standard SPH do not have ths problem. The new method get the lowest densty 0.04 at x = 0 (exact value s 0.015), yet the Standard SPH and Sprngel s method get the lowest densty great than Fgure : Results for the one-dmensonal Sögreen problem at t = 0.3s 800 partcles of equal mass are arraged on each sde of the ntal nterface. Results for dfferent SPH formulatons are compared wth densty profle (left), velocty profle (mddle) and pressure profle (rght), and all methods perform well. There are small umps at the front of expand waves of densty and pressure profle obtaned by Sprngel SPH (trangles), whereas the new method (dots) n ths paper do not have ths problem. And the new method get the lowest densty 0.04 at x = 0 (exact value s 0.015), yet the Standard SPH (squares) and Sprngel s method get the lowest densty great than D Sedov Problem The D Sedov Problem nvolves a cylndrcal blast wave from a δ functon ntal pressure perturbaton n a homogeneous medum at the rest. The analytcal soluton of ths problem was frstly derved by Sedov[16]. In ths test, there are = partcles arranged n a rectangle area, the ntal condton s Eq. (3), 5 T ρ ( 1, 0,1 10 ), x 0 T v = ( γ 1) e (3) P 1, 0,, x = 0 V0 Where e = 1 s the quantty of energy deposted at r = 0, V 0 s the volume of the center partcle. The standard SPH solved wth the evoluton equaton of densty cannot smulate ths problem successfully. The partcle dstrbuton at the tme t=0.05s s showed n Fg. 3, by comparson we fnd that the present method n ths paper gets more smoothng dstrbuton of partcles than Sprngel s method. Then the comparson of radal profles of densty and pressure s llustrated n Fg. 4, that the poston of the densty and pressure peak s captured by the presented method at r 0.3, t s very close to the exact soluton, however t s mproperly captured by Sprngel s method ( r 0.3 ). The accuracy of pressure at the center n Sedov problem s mproved also by usng the new method, but the center pressure get by Sprngel s method s much hgher than exact soluton. 8

7 Fgure 3: Partcle dstrbuton plot for D cylndrcal Sedov blast wave wth partcles at tme t=0.05s The sze of partcle crcle represents the smoothng length scale. Left: the results from Sprngel method. Rght: the results from the new method n ths paper. It shows that the smulated results by the new method get more smoothng partcle dstrbuton than that by Sprngel s method. Fgure 4: Radal profles of mass densty (left) and pressure (rght) for the D cylndrcal Sedov blast wave smulaton at t=0.05s The numercal results by the Sprngel s method (trangles) and the new method presented n ths paper (dots) are compared wth the exact soluton (sold lne). It shows that the new method obtaned more accurate than Sprngel s method both at poston of densty peak and the center pressure. COCLUSIOS We have derved a corrected SPH equaton ncludng varable smoothng length essentally, among the new equatons, the symmetrcal kernel estmaton s employed, that lead n mplctly couplng relaton of the evoluton equaton of densty and varable smoothng length equaton, the equaton of moton and equaton of energy are derved n the smlar way as Sprngel s fully conservatve formulaton of SPH. An teraton process s used to solve the coupled evoluton equaton of densty and the varable smoothng length equaton, and other equatons are solved drectly, only a lttle calculaton s needed n the teraton calculatons. 83

8 Two 1D shock problems and the D Sedov problems are presented to test the behavor of new formulaton and compared wth standard SPH method and Sprngel s method. The smulaton result demonstrates the new algorthms corrects the varable smoothng lengths effect farly well, especally n the D Sedov problem, the pressure peak s captured by the presented method more accurately than that of Sprngel s scheme, and the accuracy of pressure at the center n Sedov problem s mproved also. Ths new method can deal wth the large densty gradent and large smoothng length gradent problems well, such as large deformaton and serous dstorton problems n hgh velocty mpact and blastng phenomenon. Acknowledgements The support of ew Century Excellent Talents n Unversty (CET) and o for atonal 973 Program n Chna s gratefully acknowledged. REFERECES 1. Lucy LB. A numercal approach to the testng of the fsson hypothess. The Astronomcal Journal, 1977; 8(1): Gngold RA, Monaghan JJ. Smoothed partcle hydrodynamcs: theory and applcaton to non-sphercal stars. Mon. ot. R. astr. Soc., 1977; 181: Gngold RA, Monaghan JJ. Kernel estmates as a bass for general partcle methods n hydrodynamcs. Comput. Phys., 198; 46: Evrard AE. Beyond n-body: 3D cosmologcal gas dynamcs. Mon. ot. R. astr. Soc., 1988; 35: Hernqust L. Some cautonary remarks about smoothed partcle hydrodynamcs. The Astronomcal Journal, 1993; 404: elson RP, Papalozou JC. Varable smoothng lengths and energy conservaton n smoothed partcle hydrodynamcs. Mon. ot. R. astr. Soc., 1994; 70: Serna A, Alm JM, Cheze JP. Adaptve smooth partcle hydrodynamcs and partcle-partcle coupled codes: energy and entropy conservaton. The Astronomcal Journal, 1996; 461: Sprngel V, Hernqust L. Cosmologcal smoothed partcle hydrodynamcs smulatons: the entropy equaton. Mon. ot. R. Astr. Soc., 00; 333: Thomas PA, Couchman HMP. Smulatng the formaton of a cluster of galaxes. Mon. ot. R. Astr. Soc., 199; 57: Monaghan JJ. SPH compressble turbulence. Mon. ot. R. astr. Soc., 00; 335(5): Hernqust L, Katz. TreeSPH: a unfcaton of SPH wth the herarchcal tree method. The Astronomcal Journal Supplement Seres, 1989; 70: Monaghan JJ. Smoothed partcle hydrodynamcs. Annu. Rev. Astron. Astrophys., 199; 30: Monaghan JJ. Gngold RA. Shock smulaton by the partcle method SPH. J. Comput. Phys., 1983; 5: Monaghan JJ. SPH meets the shocks of noh. 1988, (unpublshed preprnt). 15. Monaghan JJ. SPH and Remann solver. J. Comput. Phys., 1997; 136: Sedov LI. Smlarty and Dmensonal Methods n Mechancs. Academc Press Inc., ew York, USA,