Tensor Smooth Length for SPH Modelling of High Speed Impact

Transcription

1 Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty , Lenna av. 36, Tomsk, Russa Abstract. SPH method wth tensor form of smoothng parameter s proposed for hgh speed mpact modellng. Calculaton of tensor smoothng parameter s based on deformaton of local coordnate system and can be obtaned from stran tensor evoluton. Stran ansotropy n such an approach does not cause mxng of the partcles whle mantanng the unform dstrbuton of partcles and the resultng soluton s more smooth. Weak varatonal formulaton s used to construct numercal ntegraton of moton equatons scheme and procedure of restorng of partcle consstency s used for calculaton of spatal dervatves. Boundary condtons for free surface and contact surface are realzed. Keywords: smoothed partcles, varable smoothng length, hgh speed mpact, SPH 1 Introducton Smoothed partcle hydrodynamcs was ntroduced as a numercal method for study of the moton of compressble gas wth self-gravty of arbtrary geometry n three dmensons n astro-physcs [1], [2]. Partcles are used to represent a sub-set of the flud elements n the Lagrangan descrpton of a flud, and spatal dervatves are calculated from analytcal nterpolaton formulae, SPH s well suted to problems n whch large deformatons can occur, such as the fragmentaton of self-gravtatng gas clouds. The spatal resoluton of SPH s determned by partcle densty and the smoothng length, h. Orgnally h was taken to be constant, and was a globally defned functon of the average densty of partcles n the system. More recently, however, t has become common for each partcle to have ts own tme dependent smoothng length whch corresponds to the local neghbors count. Full advantage s then taken of the Lagrangan nature of SPH, and the dynamc range n spatal resoluton of the method s ncreased. Tme-dependent smoothng lengths can, however, lead to problems wth energy conservaton n certan stuatons [4]. The problem arses from the fact that the use of varable smoothng lengths means that addtonal terms should appear n the partcle equatons of moton. Other mportant problem s lack of accuracy when varable smoothng length s used. Ths problem arses from the fact, 271

2 that SPH approcsmaton orgnally has form of analytcal nterpolaton, based on ntegraton, and there s some dfference between analytcal ntegraton and ts approxmaton va summaton over number of partcles. Nonunform partcle dstrbuton lead to partcle nconsstency problem, as a presence of boundary and varable smoothng length lead to one too. In should be noted, that hgh speed mpact s accompaned by large deformatons. When smoothng length s a scalar, smoothng kernel should have sphercal symmetry, and large ansotropc deformatons n ths case can cause non-physcal numercal fracture and partcle mxng. In ths paper we propose conservatve SPH wth varable tme-dependent tensor smooth parameter, and free surface boundary condton algorthm s proposed too. 1.1 SPH bascs Bass of SPH approxmaton s equaton f = f(x)w (x x, h)dx (1) where h s smoothng parameter, whch defne a radus of fluency for ponts, x s a space coordnate, W - smoothng functon. W (r, h) = C φ ( r /h) (2) h3 where C - ntegraton constant, φ (u) - cubcal b-splne Integraton constant defned as C = φ(u)dv (3) V where V s 1D, 2D or 3D volume. Functon φ(u) usally s cubcal B-splne: 0, u 2; φ (u) = 0.25 (2 u) 3, u (1, 2) ; 0.75 ( 1.0 u 2) (4) (2 u), u [0, 1] ; Spatal dervatves are defned va: f,α = f = x α f(x)w,α (x x, h)dx (5) Correspondng to (5) partcle approxmaton s wrtten as: f,α f = = f k W x,α (x k x, h) mk α ρ k (6) k where x k, f k, m k, ρ k radus-vector, approxmated functon value, mass and densty at k-th pont. 272

3 Equaton (6) have C 0 -consstency [3] near surfaces and boundares or at non-unform partcle dstrbuton, and for restorng partcle consstency specal approaches s needed. Accordng [3], a test vector-functon s ntroduced n form: Δ (x) = (1, x 0, x 1, x 2 ) (7) Approxmaton of test functon (1) and ts dervatves (5) can be found wth C 0 -consstency, but due to the fact that ths values are known, ths approxmaton can be used to construct correcton procedure and restore partcle consstency. Let s defne : { xα ; α > 1; Δ(x) α = (8) 1; α = 1; Matrx of approxmatons for ths functon at n-th pont, found va uncorrected SPH approxmaton (6): T βα (x n ) = m Δ α (x m x n ) W,β (x m x n, h) mk ; (9) ρk Lets defne correcton matrx as B n αβ = B αβ (x n ) = [ T αβ (x n ) ] 1 ; α, β = 1, 0, 1, 2; (10) Now, corrected approxmaton of functon f(x) value or ts dervatves can be found, f we buld full matrx of uncorrected approxmatons (6) and apply correcton matrx (10): f,α = f x α = T αβ { k m k ρ k f k ( W,β x k x, h ) } (11) Smoothng length h defnes maxmum nterpartcle dstance and therefore defnes partcle count. From the other hand, partcle count s related to nterpartcle dstance and smoothng length- greater partcle count allow to use smaller smoothng length. There s some analogy between smoothng length n SPH and space step n FEM/FDM. For calculaton of elastc plastc flow, approxmaton (11) s appled to Lagrangan of system: L = k m k ( v k v k 2 u k ) (12) where v k - s partcle speed, u k - s nternal energy (per mass). Internal energy change can be wrtten n form: du k dt = σ k : ε k (13) where σ k s stress tensor and ε k s stran rate tensor of partcle k respectvely. 273

4 Ths mples: Mathematcal and Informaton Technologes, MIT-2016 Mathematcal modelng dl dt = k m k ( v k a k σ k : ε k) (14) Snce the relaton between a small deformatons rate tensor and nodal veloctes s lnear: ε p j = 1 ( v + v ) j = m n ( ) np W 2 x j x ρn,k v n T p jk + vn j T p k (15) n Because of energy conservaton (14) and (15) leads to the relaton between the acceleratons of nodes and stress feld: ( ) a n m k γ = ρ k σk γjtjβw k,β kn /ρ n (16) k For mpact problem we use a condton of zero normal stress at free surface and equal normal stress and equal speeds on contact surface. In descrbed method ths boundary condtons does not need addtonal operatons, such as ghost partcles or specal ntegraton procedures for (5) to treat a boundary. Smoothng length h was selected to provde suffcent count of neghbors for each partcle for correcton matrx to be well defned. Usually neghbor count was from 11 to 16. Tme step s defned va Courant Fredrchs Lewy condton: Δt < mn [h/c], where C s the sound speed. At practce stablty condton for heat transfer problem s more weak whle droplet radus exceed 1 mm. When smoothng length h s scalar parameter ntensve deformatons lead to partcle mxng and artfcal fragmentaton. To avod ths effects we ntroduced a tensor smoothng parameter h j : W (r, h j ) = ( C det h j φ ( r j [h] 1 j Intal value h j = hi, equaton of evoluton of h j s: ) 2 ) (17) h j = h k v j,k (18) For scalar smoothng parameter spatal dervatves of smoothng functon have the form: W (r, h) = C r h 3 φ ( r/h ) / r = C r h 1 φ h 3 r 2 (19) u for tensor smoothng ones have smlar form: W (r, h j ) r = C r [h j ] 1 det h j r 2 φ u u=( r h 1 ) u=( r h 1 ) (20) 274

5 Fg. 1. Deformaton of local coordnate system and ts relaton to h j 3 Results A 38 μm radus stanless steel cylnder wth length of 190 μm s mpactng wth a velocty of 200 m/s on stanless steel substrate 64 μm thn. Parameters of steel: densty 7.87 g/cm3, E=200GPa, G=70 GPa, Ypl=1.2MPa. As t shown on Fg. 2, when mpact speed s relatvely small, results are smlar. Fg. 2. Scalar smoothng (I) and tensor smoothng (II) comparson at t=0.55 μs, v=200 m/s. Fg. 2, when mpact speed s relatvely small, results are smlar. Smoothng tensor remans almost sphercal n ths condton and smoothng kernel wth tensor parameter dffers slghtly from standard kernel wth scalar smoothng length. More dfferent results are obtaned for mpact speed 800 m/s (Fg. 3). Tensle nstablty and partcle mxng are not observed when tensor smoothng s used, and shape of materal ejecton s tracked more accurately. At the same tme, usage of constant scalar smoothng length lead to partcle mxng, numercal fracture (as t seen on bottom part of Fg. 3-I) and ntensve partcle clusterng. Also, symmetry of results obtaned wth tensor smoothng s much better. 1.2 Concluson Introducton of tensor smoothng length n varatonal SPH-code s qute smple and does not need sgnfcant calculaton cost, and provde more accurate results for hgh speed mpact modellng. 275

6 Fg. 3. Scalar smoothng (I) and tensor smoothng (II) comparson at t=0.03 μs, v=800 m/s. Acknowledgments. Ths work was supported by the Russan Scence Foundaton (RSF) No References 1. Lucy, L.B.: A numercal approach to the testng of fuson hypothess. Astronomcal Journal 82, (1977) 2. Gngold, R.A. and Monaghan, J.J.: Smoothed partcle hydro-dynamcs: theory and applcatons to non-sphercal stars. Monthly Notces of the Royal Astronomcal Socety 181, (1977) 3. Lu, M.B. and Lu, G.R.: Restorng partcle consstency n smoothed partcle hydrodynamcs. Appled Numercal Mathematcs 56(1), (2006) 4. Rchard P. Nelson, John C.B. Papalozou:Varable Smoothng Lengths and Energy Conservaton n Smoothed Partcle Hydrodynamcs. Mon.Not.Roy.Astron.Soc. 270 (1994) 1 276