Simulation of Dendritic Growth with Different Orientation by Using the Point Automata Method

Transcription

1 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 Simuatin f Dendritic Grwth with Different Orientatin by Using the Pint Autmata Methd A.Z. Lrbiecka 1 and B. Šarer 1,2 Abstract: The aim f this paper is simuatin f thermay induced iquid-sid dendritic grwth in tw dimensins by a cuped deterministic cntinuum mechanics heat transfer mde and a stchastic caized phase change kinetics mde that takes int accunt the undercing, curvature, kinetic and thermdynamic anistrpy. The stchastic mde receives temperature infrmatin frm the deterministic mde and the deterministic mde receives the sid fractin infrmatin frm the stchastic mde. The heat transfer mde is sved n a reguar grid by the standard expicit Finite Difference Methd (FDM). The phase-change kinetics mde is sved by the cassica Ceuar Autmata (CA) apprach and a nve Pint Autmata (PA) apprach. The PA methd was deveped and intrduced in this paper t circumvent the mesh anistrpy prbem, assciated with the cassica CA methd. Dendritic structures are in the CA apprach sensitive n the reative ange between the ce structure and the preferentia crysta grwth directin which is nt physica. The CA apprach is estabished n quadratic ces and Neumann neighbrhd. The PA apprach is estabished n randmy distributed pints and neighburhd cnfiguratin, simiar as appears in meshess methds. Bth methds prvide same resuts in case f reguar PA nde arrangements and neighbrhd cnfiguratin with five pints. A cmprehensive cmparisn between bth stchastic appraches has been made with respect t curvature cacuatins, dendrites with different rientatins f crystagraphic anges and types f the nde arrangements randmness. It has been shwn that the new methd can be used fr cacuatin f the dendrites in any directin. Keywrds: micrstructure mdeing, sidificatin, dendritic grwth, ceuar autmata methd (CA), pint autmata methd (PA), randm nde arrangement. 1 Labratry fr Mutiphase Prcesses, University f Nva Grica, Svenia 2 Crrespnding authr (B. Šarer) E-mai: bzidar.sarer@ung.si

2 70 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , Intrductin Sidificatin micrstructure is very imprtant since it infuences the prperties f the fina casting. Because f that has understanding and mdeing f micrstructures arge industria reevance. Hwever, the understanding f sidificatin prcess and reated micrstructures is very cmpicated. This is because it is affected by many interacting phenmena n different scaes, such as heat and sute transfer, fuid fw, thermdynamics f interfaces and s n [Rettenmayr and Buchmann (2006)]. Experiments that aw direct visuaizatin f micrstructure frmatin are difficut t perfrm. In the ast decade, severa numerica mdes, which can sve cmpicated transprt phenmena and phase transfrmatin under different bundary and initia cnditins, were deveped t cacuate varius micrstructure features f sidifying materias such as grain grwth with detais f sidificatin interface mrphgy. Amng f a numerica appraches Ceuar Autmata (CA) mdeing [Wfram (2002)] and phase fied mdeing [Qin and Waach (2003)] are the mst ppuar and widey used. We fcus n the CA based apprach in this paper. A cnsiderabe prgress n sidificatin micrstructure simuatin [Bettinger, Crie, Greer, Karma, Kurz, Rappaz and Trivedi (2000); Lrbiecka, Vertnik, Gjerkeš, Manjvić, Senčič, Cesar and Šarer (2009); Lrbiecka and Šarer (2010); Midwnik (2002)] has been made by the CA apprach. Rappaz and Gandin [Rappaz and Gandin (1993)] were the pineering researchers wh deveped the CA mde fr mdeing micrstructure in which nuceatin and grwth kinetics cud be cnsidered and grain structure with certain shapes and size were predicted. Gandin and Rappaz [Gandin and Rappaz (1994); Gandin and Rappaz (1997)] simuated the grain structure by cuping the CA technique fr the grain grwth with the finite eement methd (FEM) sver fr the heat fw (CA- FEM). Later Spitte and Brwn [Spitte and Brwn (1995)] cuped the CA with a finite difference sver (CA-FDM) fr sute diffusin during the sidificatin f casting t predict micrstructure. Unfrtunatey, the simpe CA mdes fr dendritic grwth suffer frm the strng impact f the anistrpy f the numerica grid. Cnsequences are that they tend t grw ny in the grid directin [Zhan, Wei and Dng (2008)]. It des nt matter which crystagraphic rientatin wi be chsen it wi aways shift the dendrite with respect t the grid axis. During the grwth prcesses f grains the crystagraphic rientatin axes f different grains have different divergence anges with respect t the crdinate system. In these cases is the grwth stage difficut t simuate by the CA methd. It is because the cnfiguratin f the CA mesh has a direct infuence n simuated structure and shape. Andersn [Andersn, Srvitz and Grest (1984)] and ater Spitte and Brwn [Spitte and Brwn (1989)] used a hexagna, rather than the standard square 2-D attice in rder t better represent the

3 Simuatin f Dendritic Grwth 71 grain anistrpy. But in genera even nw it is sti difficut t prpery mde the preferred crystagraphic rientatin. Rappaz and Gandin deveped a decentered square methd [Rappaz and Gandin (1993)] t try t sve this prbem, which turns ut t be very cmpicated. We present a nve Pint Autmata (PA) methd in this paper which fws the CA cncept and is abe t sve the mentined crystagraphic rientatin prbem. A basic feature f this methd is t distribute ndes randmy in the dmain instead f using reguar ces, which eads t different distances between the ndes and different neighbrhd cnfiguratins fr each f them. This new apprach was first prpsed by Janssens fr mdeing the recrystaisatin [Janssens (2000); Janssens (2003); Janssens (2003); Janssens (2010); Raabe, Kzeschnik, Midwnik and Nester (2007); Janssens]. [Lrbiecka, Vertnik, Gjerkeš, Manjvić, Senčič, Cesar and Šarer (2009)] were the first t cupe the cassica CA methd with a meshess methd instead f the FEM r FDM. They succesfuy predicted the grain structure in cntinuus casting f stee. Subsequenty, they repaced the CA methd by the PA methd in the same physica system [Lrbiecka and Šarer (2009)] and demnstrated the suitabiity f the PA methd fr ceuar t equiaxed and equiaxed t ceuar transitin simuatin in stee biets. The preiminary resuts f the dendritic grwth based n the PA apprach have been presented in [Lrbiecka and Šarer (2009)]. This apprach is expained and evauated in detais in the present paper where we numericay discuss a simpe physica mde which can simuate the dendritic frms during the sidificatin f pure metas frm its underced met. The deveped agrithm is abe t btain the dendritic mrphgy by sving the heat transfer equatin cuped with the sid fractin fied evutin thrugh the cacuatins f crysta grwth vecity, interface curvature, thermdynamic and kinetic anistrpy, respectivey. The present paper is structured in the fwing way: the CA and the PA methds are ined first, fwed by the descriptin f the gverning equatins f the heat transfer mde and the stchastic mde. The sutin f temperature fied and sid fractin is expained afterwards. The differences in numerica impementatin f the cassica CA and the new PA sutin prcedure are cmpared and discussed. The dendritic grwth is simuated fr ten different rientatins with the same randm nde arrangement with the PA methd. Afterwards, the infuence f fur different randm nde arrangements as we as different nde randmness was tested n tw different crystagraphic rientatins. Finay, the numerica resuts are shwn fr the cassica CA methd with and withut fuctuatins and cmpared t the resuts btained by the PA methd. Cncusins with systematicay isted characteristics f the PA methd and future devepments f the PA methd cmpete the present paper.

4 72 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , CA and PA initins Numerica mdes fr sving the micrstructure equatins can briefy be divided int tw categries: deterministic and stchastic [Stefanescu (2009)]. Stchastic mdeing represents a system where the physica phenmena are described by the randm numbers. As a cnsequence the utput data can vary frm ne simuatin t anther. The mst ppuar stchastic methds used t simuate the micrstructure frmatins are: Mnte Car methds, Randm Waker and CA apprach. CA stchastic methd [Wfram (2002)] represents ne f the numerica techniques, widey appied in mdeing sidificatin and recrystaizatin prcesses. This agrithm was first estabished by Neumann [Neumann (1987)] and is nwadays cmmny used in materia science. What fws are the basic eements f the CA methd n-d (n=1, 2, 3) space is divided int a discrete number f n-dimensina eements which are named ces (pygns and pyhedrns). a state is assigned t each CA ce, the neighbrhd cnfiguratin is ined deterministic r stchastic fr each CA ce, transitin rues are ined which create a new state f the ce as a functin f the states(s) f the ce(s) cnsisting f the previusy ined ca neighbrhd f the ce. The abve presented basic features f the CA system are cmmny impemented in the iterature. In the present wrk an aternative frmuatin t a cmmn CA methd is intrduced. What fws are the basic eements f this nve PA methd the starting pint is t distribute PA ndes (nt ces) randmy n the n-d cmputatina dmain, a state is assigned t each PA nde, the neighbrhd cnfiguratin is ined fr each nde separatey with respect t the chsen neighbrhd cnfiguratin, the neighbrhd f the nde incudes a randm ndes whse psitins are cated in the dmain f a circe in 2D r sphere in 3D. The number f the neighbrs can vary cay. The transitin rues are ined and they create a new state f the pint as a functin f the states(s) f the pints(s) cnsisting the ca neighbrhd cnfiguratin.

5 Simuatin f Dendritic Grwth 73 The irreguar (as named randm) PA ceuar transitins rues can be used in exacty the same way as fr the reguar apprach. In this sense the PA apprach is nt much different frm the cnventina ne, despite bringing many advantages isted in the cncusins. 3 Gverning equatins Thermay induced dendrite grwth is cnsidered in this paper. It is physicay described by the heat cnductin and phase change kinetics. The temperature fied is sved by the cassica deterministic methd and the phase change kinetics by the stchastic methd. 3.1 Temperature fied Cnsider a tw dimensina dmain Ω with bundary Γ fied with a phase change materia which cnsists f at east tw phases, sid and iquid, separated by an interfacia regin, which is usuay very thin in pure substances. The therma fied in such a system is gverned by the fwing equatin [Xu, Li, Liu and Liu (2008)] (ρh) = (λ T ) t (1) where ρ, h, λ, T represent materia density, specific enthapy, therma cnductivity and temperature, respectivey. The specific enthapy is cnstituted as h = c p T + f L (2) where c p, L, f represent the specific heat, the atent heat and iquid fractin, respectivey. A materia prperties are assumed cnstant fr simuatin simpicity. The sid and iquid fractins fw the rues f s + f = 1 (3) 1 fr T T s T f s (T ) = T T T s fr T s < T < T (4) 0 fr T T where T s, T, f s represent the sidus temperature, iquidus temperature and the sid fractin, respectivey. In case f pure substance are the sidus and the iquidus temperatures equa t the meting temperature T m. Hwever, fr the cmputatina purpses a narrw meting interva is aways present T > T m > T s. The meting temperature T m is ined as T m = 1 2 (T s + T ).

6 74 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 We search fr the temperature at time t 0 + t by assuming the initia cnditins T (p,t 0 ) = T 0 (p); p Ω (5) f s (p,t 0 ) = f s0 (p); p Ω (6) (where p represents the psitin vectr) and Neumann bundary cnditins T n (p,t) = F (p,t); p Γ, t 0 < t t 0 + t (7) where n represents the nrma n Γ and T 0, f s0,f represent knwn functin. 3.2 Phase change kinetics Interface undercing The phase change situatin can be achieved by undercing a iquid bew its meting temperature. When a sid seed is paced in such an underced met, sidificatin wi be initiated. Due t crysta anistrpy and perturbatins in the system, the grwth f the sid frm the seed wi nt be unifrm and an equiaxed dendritic crysta wi frm. Sid iquid interface is underced t the temperature T f ined as [Sait, Gdbeck-Wd and Muer-Krumbhaar (1988); Nakagawa, Narsume and Ohsasa (2006)] T f = T m ΓK (8) where Γ and K are the Gibbs-Thmsn cefficient and the interface curvature, respectivey Dendrite grwth kinetics The grwth prcess is driven by the ca undercing. The interface grwth vecity is given by the cassica sharp mde [Shin and Hng (2002)] V g (p,t) = µ K (T f T (p,t)); p Γ s, (9) where V g, µ K, Γ s, is the grwth vecity, interface kinetics cefficient and the sid iquid interface, respectivey. Dendrites aways grw in the specific crystagraphic rientatins. Therefre it is necessary t cnsider anistrpy in either the interfacia kinetics r surface energy (r bth). The present mde accunts fr the anistrpy in bth kinetics.

7 Simuatin f Dendritic Grwth Thermdynamic anistrpy The Gibbs-Thmsn cefficient can be evauated [Krane, Jhnsn and Raghavan (2009)] by taking int accunt the thermdynamic anistrpy reated t the crysta rientatin and type as fws Γ = Γ [ 1 δ t cs [ S ( θ θ de f )]] where S, θ, θ de f,δ t, Γ represent factrs which cntr the number f preferentia directins f the materia s anistrpy (S = 0 fr the istrpic case, S = 4 fr fur fd anistrpy and s n), grwth ange (ange between theycrdinate and the ine that cnnects the center f the mass f the dendrite and pint at Γ s,, see Fig. 1), the preferentia crystagraphic rientatin, thermdynamic anistrpy cefficient and the average Gibbs - Thmsn cefficient, respectivey Kinetic anistrpy The crysta grwth vecity is cacuated accrding t the crysta rientatin by taking int the cnsideratin the crysta grwth directin θ and the preferred rientatin θ de f. The crysta grwth vecity fws the equatin [Shin and Hng (2002)] V = V g (p,t) [ 1 + δ k cs ( S ( θ θ de f ))] where δ k represents the degree f the kinetic anistrpy, respectivey. 3.3 Cuping The mvement f the sid-iquid interface is gverned by the evutin f the temperature fied in the cmputatina dmain (Fig. 1). The dendritic structures are mdeed by the stchastic methd t track the interface mtin cuped t the determinate heat transfer cacuatins. We first describe the sutin f the temperature fied based n the FDM methd and subsequenty the transitin rues fr the CA (PA) methds fr cacuatin f sid fractin fied. The fwchart f the cacuatins is given in Fig. 18. (10) (11) 4 Sutin f the temperature fied A square dmain is cnsidered with ength. The number f pints in FDM mesh in xandydirectins is N. The tta number f FDM grid pints is N 2-4, since the fur crner ndes are nt cnsidered.

8 76 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 y p θ θ Γ x Γ s, Ω iquid Figure 1: Scheme f the dendrite grwth A unifrm FDM discretizatin is made with mesh distance x = y = a = /(N 1)as seen in Fig. 5 (tp). The sutin fr the temperature fied is perfrmed by the simpe expicit FDM. Sutin f the temperature fied in the dmain ndes is thus T i, j = T 0 i, j + tλ ρc p ( [ (T 0 i 1, j 2T 0 i, j + T 0 i+1, j )/ ( x 2)] + [ (T 0 i, j 1 2T 0 i, j + T 0 i, j+1 )/ ( y 2)] ) + + L c p ( f s i, j f 0s i, j ) (12) fr i = 2,3,...,N 1 and j = 2,3,...,N 1 The bundary ndes are cacuated (the Neumann bundary cnditins are set t F = 0W/m 2 ) as West T 1, j = T 2, j (13) fr j = 2,...,N 1 East T N, j = T N 1, j (14) fr j = 2,...,N 1 Nrth T i,n = T i,n 1 (15)

9 Simuatin f Dendritic Grwth 77 fr i = 2,...,N 1 Suth T i,1 = T i,2 (16) fr i = 2,...,N 1 where t, f 0s i, j, T 0 i, j, T 0 i+1, j, T 0 i 1, j,t 0 i, j+1, T 0 i, j 1 are the time step, initia sid fractin, initia temperature in the FDM centra, east, west, nrth and suth ndes, respectivey. 5 Sutin f the sid fractin fied We nw ine and discuss the eements f the cassica CA and the nve PA methds in detais. 5.1 Definitin f mesh and neighbrhd cnfiguratin Square ces with ength δx = y = a = /n where n = N 1, represents the number f ces in x and y directins are cnsidering in the CA apprach. In the PA apprach the square is divided in unifrm r nnunifrmy distributed ndes. Ces are nt ined Mesh and neighbrhd in the CA methd A basic initin f neighbrhd riginates frm the cassica CA apprach which perates n the grid divided int the square ces [Neumann (1987); Nastac (2004)]. The ce structure is depicted in Fig. 2. In ur cacuatins the Neumann cnfiguratin which takes int accunt ny the csest neighbr s ces during the cmputatin is appied. The cnventina square mesh structure is cmmny appied in CA cacuatins. It represents a square dmain cvered by the CA ces x CA i, j, y CA i, j cated exacty in the midde f fur FDM ndes, as it is depicted in Fig. 5 (midde). x CA i, j = 1 2 [x FDM i, j + x FDM i+1, j ] (17) y CA i, j = 1 2 [y FDM i, j + y FDM i, j+1 ] (18) Mesh and neighbrhd in the PA methd The PA nde grws with respect t the heat fw and with respect t the neighburhd cnfiguratin which is nw assciated with the psitin f the neighburing

10 78 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 c i, j { c = c, c, c, c i, j i+ 1, j i, j+ 1 i 1, j i, j 1 Figure 2: Graphica representatin f the Neumann neighbrhd cnfiguratin fr the cnventina CA methd } R H { i j } c c r r R i, j= i, j; i j < H Figure 3: Graphica representatin f the neighbrhd cnfiguratin prpsed fr the new PA methd PA ndes which fa int a circe [Janssens (2000); Janssens (2003)] with radius R H in 2-D r a sphere in 3-D. It means that each PA nde can in case f the randm mesh cntain different number and psitin f the neighbrs, which give varius pssibiities f neighbrhd cnfiguratins fr each nde. Fr the nve PA methd the randm nde arrangement is in the present paper generated frm the reguar CA mesh. T cnstruct the randm nde arrangements the CA ce centers are dispaced t the randmy chsen psitins and becme randm PA ndes x PA i, j, y PA i, j n the

11 Simuatin f Dendritic Grwth 79 cmputatina dmain (see Fig.5 bttm). x, y x + FDM i+1, j+1, y FDM i + 1, j + 1 FDM i, j+1 FDM i, j 1 x CA i, j, y CA i, j x PAi,j, y PAi, j x FDM i, j, y FDM i, j xfdmi,j+1, y FDMi, j+ 1 Figure 4: Schematic representatin f the reatinship between FDM ndes (4 crners), CA ce (center) and the randm PA nde The dispacement f each CA center is assumed ny in the square area appinted by the fur FDM ndes. The fwing prcedure is appied x PA i, j = x CA i, j + ε [2rand 1] (19) y PA i, j = y CA i, j + ε [2rand 1] (20) where x PAi, j, y PAi, j, ε represent crdinates f PA ndes and the scaing vaue 0 ε 0.49, respectivey.it must be emphasized that the PA prcedure is estabished n the randm ndes in genera. The heat transfer cacuatins are perfrmed n the reguar FDM ndes, which is briefy expained in Sectin Curvature cacuatins The interface curvature is apprximated by the cunting ce prcedure deveped by Sasikumar and Sreenivasan [Sasikumar and Sreenivasan (1994)] Cacuatin f curvature in the CA methd The expressin fr curvature K is given by the frmua [Krane, Jhnsn and Raghavan (2009)] K = 1 ( 1 2N ) s CA (21) a N t CA where N s CA and N t CA are the number f sid CA ces whse centers fa inside the circe f assumed radius R c and the tta number f CA ces whse centers fa inside the circe, respectivey (see Fig. 6).

12 80 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , Cacuatin f curvature in the PA methd The expressin fr PA is derived frm the expressin at the CA methd Eq.21 by assuming the average nde distance a instead f a. K = 1 ( 1 2N ) s PA (22) a N t PA where N s PA, N s PA are the number f sid PA ndes inside the circe f assumed radius R c and the tta number f PA ndes inside the circe, respectivey (see Fig. 7). The curvature f bth methds has been cacuated and cmpared n a circuar sid fractin arrangement with radius R = 10 µm, depicted in Fig. 8. Tw different types f R c have been chsen (R c = 1µm and R c = 5µm). It can be cncuded that with the higher radius R c the vaue f K becmes amst the same as in the cnventina CA apprach. This was depicted in Fig. 9 and Fig. 10, respectivey. 5.3 Phase change The crysta grwth vecity is cacuated accrding t the crysta rientatin. The envepe f the grain can be expressed by the Eq.11 which is depicted in Fig.11. Once a CA ce (r PA nde) becmes sid it starts t grw with respect t the neighbrhd cnfiguratin (see Fig. 2 and Fig. 3). Each f the CA ce (r the randm nde) can have tw pssibe states: iquid r sid. The CA ce (r PA nde) becmes sid thrugh the grwth prcess. The change f sid fractin f the CA ce r PA nde is cacuated frm the crysta grwth vecity. Fr a neighbrs f the treated sid CA ce (r sid PA nde), genera criterin d is checked which is represented by the fwing frmua d = (t) a i t = t 0 V i, j dt (23) (24) where a i represent engths frm the anayzed CA ce r PA nde t the nearest ne. If neighbr is ne f the fur nearest east, nrth, west, suth neighbrs then in the CA methd this distance becmes a i = a. In the PA methd a i (a i < R H ) represents the different distances t the neighbring PA ndes which fa int the circe with radius R H (see Fig. 14 and Fig. 15). When d a r d a i (Fig. 13 and Fig. 15) the grwing sid tuches the centre f the neighbring CA ce r PA nde and this ce/nde transfrms its state frm iquid t sid f spa = 1.

13 6 Simuatin f Dendritic Grwth 81 (19) (20) y nt crdinates f PA ndes and 9, respectivey.it must be emure is estabished n the randm ansfer cacuatins are perfrmed hich is briefy expained in Secs prximated by the cunting ce sikumar and Sreenivasan [Sasi- 4)]. x re in the CA methd re K is given by the frmua an (2009)] (21) y the number f sid CA ces circe f assumed radius R c and s whse centers fa inside the 6). re in the PA methd rived frm the expressin at the ming the average nde distance x (22) umber f sid PA ndes inside R c and the tta number f PA ctivey (see Fig. 7). ds has been cacuated and cmractin arrangement with radius ig. 8. Tw different types f y 1μ m and Rc = 5μm ). It can be her radius R c the vaue f K ben the cnventina CA apprach. Figure 5: Scheme f space discretizatin: (tp) FDM ndes x nd Fig. 10, respectivey. Figure 5: Scheme with fn space = 21, discretizatin: (midde), CA ces (tp) withfdm n = 20, ndes (bttm) with PA N = 21, (midde), CA ces with n = 20, (bttm) PA ndes ndes with with n = 20n = 20

14 82 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 Figure 6: Scheme shwing a circe sampe with R c = 2a fr cacuating the curvature f the cnventina CA methd (exampe: N s CA = 8 and N t CA = 12) Figure 7: Scheme shwing a circe sampe with R c = 2a fr cacuating the curvature fr the randm PA ndes (exampe: N s PA = 7 and N t PA = 11) crss sectin R Figure 8: Scheme f the area used t cmpare the curvature cacuatins by the CA and PA methds. R = 10µm. Fied area represents sid 6 FDM-PA-FDM transfer f temperature and sid fractin 6.1 FDM-PA transfer f temperature The btained vaues f temperature n reguar FDM grid (see Sectin 4) are in each time step transferred t randm PA grid accrding t the described scheme (Fig. 18).

15 Simuatin f Dendritic Grwth 83 The crss sectin f the curvature fr R c =1 1,5 1 0, ,5-1 -1,5 CA methd 1 K μm PA methd Figure 9: Cacuated curvature K with the CA and PA methd (ε = 0.49) fr R c = 1µm and a = a = 1µm with respect t the data depicted in Fig. 8 [ μm] The crss sectin f the curvature fr R c =5 1,5 1 K μm 1 0, ,5-1 -1,5 [ μm] CA methd PA methd Figure 10: Cacuated curvature K with the CA and PA-(A) methd fr R c = 5µm and a = a = 1µm with respect t the data depicted in Fig. 8 The fwing simpe interpatin frmua [Xu and Liu (2001)] is used in the present paper T PA i, j = (T i, j T i+1, j T i+1, j 3 + T i, j 4 )/ In case f FDM-CA the Eq.25 reduces t 4 i=1 i (25) T CA i, j = (T i, j+1 + T i+1, j+1 + T i+1, j + T i, j )/4 (26)

16 84 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 V i, j R H Figure 11: Schematic representatin f the shape functin (parameters see Tab.1) a i d a i d Figure 12: Grwth frnt wi nt reach the csest neighbr d < a i. The CA ce wi nt be cnverted t sid (exampe fr the Neumann neighbrhd cnfiguratin) Figure 13: Grwth frnt wi reach the csest neighbrd a i. The CA ce wi be cnverted t sid (exampe fr the Neumann neighbrhd cnfiguratin) where T PA i, j, T i, j, T CA i, j and i represent the temperature f the PA nde, the temperatures f the fur csest FDM ndes, the temperature fr the center CA ce and the distances t the nearest fur FDM ndes, respectivey. The cacuatin is repeated in each time step (see Fig. 16).

17 Simuatin f Dendritic Grwth 85 a i d a i d R H R H Figure 14: Grwth frnt wi nt reach the csest neighbr d < a i. The PA nde wi nt be cnverted t sid Figure 15: Grwth frnt wi reach the csest neighbr d a i. The PA nde wi be cnverted t sid T i, j + 1 T i+ 1, j+ 1 f s PA i, j+ 1 f spa i+ 1, j f s i, j T PAi, j f s PA i+ 1, j T i, j T i + 1, j Figure 16: Reatinship between fur FDM ndes and PA nde fr the cacuatin f the temperature vaues f s PA i, j Figure 17: Reatinship between the FDM nde and fur neighbring PA ndes fr the transfer f sid fractin 6.2 PA-FDM transfer f sid fractin The temperature fied at time t 0 + t can be cacuated frm the Eq.12 fr a FDM ndes. Then these vaues are recacuated t a PA ndes accrding t the Eq.25. Afterwards the PA prcedure takes pace (see Sectin 3). The utput infrmatin frm this eve f cacuatin is the vaue f sid fractin fr a randm PA ndes

18 86 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 f s PAi, j which have t be transferred t the FDM ndes t be abe t cacuate the new vaues f temperature (Fig. 17). The fwing equatin is appied f s i, j = ( f s PAi, j f s PAi+1, j f s PAi+1, j 3 + f s PAi, j 4 )/ In case f FDM-CA the Eq.27 reduces t 4 i=1 i (27) f s i, j = ( f s CAi, j+1 + f s CAi+1, j+1 + f s CAi+1, j + f s CAi, j )/4 (28) where f s i, j and f s PA represent the sid fractin fr the FDM ndes and fr the PA ndes, respectivey. 7 Numerica exampe 7.1 Numerica impementatin The mde was cded in Frtran. Fr the dendritic grwth in Fig.20 the CPU time varies frm 10 t 15 minutes depending n the input data. The sid PA nde r CA ce are depicted by cred pixe which can be bserved n the screen during the simuatin. 7.2 Prbem initin and discretizatin Initia cnditins. Simpified materia prperties presented in Tab. 1 fr pure auminum [Kammer (1999)] are used in a prepared numerica exampes. The prcess starts frm the predetermined sid seed psitin in ne singe PA r CA nde in the midde f the cmputatina dmain with the fwing initia cnditins f temperature K 1.5K and sid fractin f s = 1. A ther PA ndes are assumed t be iquid f s = 0 and FDM ndes with the temperature K. This data is cnstant with the prbem ined in the artice. The numerica exampes in the present paper are sved by the FDM based temperature cacuatins and CA r PA based sid fractin cacuatins. The cmputatina dmain f the square with = 350 µm in unifrm discretizatin N = 701. Mesh generatin. FDM and CA methds are aways cnstructed n a reguar nde arrangement in the present paper. In the PA apprach the randm nde arrangement needs t be cnstructed. The PA apprach was tested first with the predetermined nde arrangement PA-(A), see Fig. 20 and then with different types f randm nde arrangements: PA-(B), PA-(C), PA-(D), see Figs , respectivey (Tab.2). Time step. The time step used in FDM cacuatin f the temperature fied is imited by the frmua [Zhu and Hng (2001)] t FDM = a2 4.5D ; D = λ ρc p (29)

19 Simuatin f Dendritic Grwth 87 Set T0 and f 0 s PA frm the initia cnditins in FDM ndes Cacuatin f the new temperature fied in FDM ndes Transfer f temperatures frm FDM ndes t PA r CA ndes Time step. The time step used temperature fied is imited by (2001)] 2 a λ Δ tfdm = ; D = 4.5D ρc where D represents the therma tins f the sid fractin fied b fwing reatin is used [Dam fr assuming stabiity 2 a a Δ tca = η min, V D max p PA r CA cacuatin f f Transfer f f S frm the PA r CA ndes t the FDM ndes T0 S Set = T, f0 s = fs where η and V max represent the and the maximum grwth ve spectivey. Fr the stabiity f the cupe minimum f Δt CA and ΔtFDM s resuts f simuatins are shw graphic anges after Δ t = 6.82x10 s. use then in the vecity cacuati Figure 18: Fwchart f the therma fied and sid fractin cacuatins Figure 18: Fwchart f the therma fied and sid fractin Neighbrhd cnfiguratin. cacuatins csest neighbrhd cnfigura ger the vaue f R H is chsen where D represents A ther PA therma ndes diffusivity. are assumed Fr t be the iquid cacuatins f s = 0 and f FDM the sid dritic fractin and irreguar structures c tended area f neighbrs needs t fied by the CAndes andwith PA methd the temperature the fwing K reatin. This data is used is cnstant [Daming, tin Ru in the and PA methd. The radi Zhang (2004)] with fr assuming the prbem stabiity ined in the artice. The numerica exampes in the present paper are sved by the FDM based tem- kept at a minimum f 1.5 μm ( ) smaer vaues the dendritic sha perature a cacuatins and CA r PA based sid fractin cacuatins. The cmputatina dmain f the square with preferred rientatin is st as w t CA = η min V = 350, a2 (30) max D μm in unifrm discretizatin N = Simuated resuts where η and VMesh generatin. FDM and CA methds are aways cnstructed n a reguar nde arrangement in the present paper. max represent the psitive cnstant ess then 1 and thethe maximum dendritic mrphgies we grwth vecity FDM-CA and the nve FDM-P In f the a PA interface apprach ces, the randm respectivey. nde arrangement needs t be numerica exampes were prepar cnstructed. The PA apprach was tested first with the predetermined nde arrangement PA-(A), see Fig. 20 and then Frm CASE 1 t CASE 10 th with different types f randm nde arrangements: PA-(B), simuated by the PA methd wi PA-(C), PA-(D), see Figs , respectivey (Tab.2). rangement dented (PA-(A)) f graphic rientatins FDM Therma fuctuatins. In rder f the dendrite in the cnventi tuatins need t be intrduced i wing equatin is cmmn Therma nises are usuay pres fuctuatins F int the cacuati ing temperature r vecity [V

20 13 88 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 drite branches in x and y direcwith respect t the randm 165 nde ngement h h ] rati f primary dendrite arms x / y [-] x / y average ength rati [-] enght [μm] average ength f x and y [μm] 160 σ standard 155 deviatin f 150x / y [-] σ 145 standard deviatin f 140 enght [μm] preferred rientatin [ ] x branch y branch Figure 19: The engths f dendrite branches in x and y directins fr ten different crystagraphic rientatins, randm nde arrangement PA-(A), (see Fig.20) Fr the stabiity f the cuped FDM-CA-PA prcedure a minimum f t CA and t FDM shud be used. A depicted resuts f simuatins are shwn fr the different crystagraphic anges after 1500 time steps f the ength t FDM = s θ = CASE 1 at the nve PA cncept can be rwth prcess in any preferentia icatin prcess.

21 Simuatin f Dendritic Grwth θ = 5 θ = 20 CASE 2 CASE 5 θ = 10 θ = 25 CASE 3 CASE 6 θ = 15 θ = 30 CASE 4 CASE 7

22 90 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 θ = 35 θ = 5 CASE 8 CASE 11 θ = 40 θ = 5 CASE 9 CASE 12 θ = 45 θ = 5 CASE 10 CASE 13 Figure 20: Simuated dendritic grwth fr a singe dendrite fr different rientatins by the PA methd fr the same PA- (A) randm nde arrangement θ = 0,5,...,40,45 Figure 20: Simuated dendritic grwth fr a singe dendrite fr different rienta- tins by the PA methd fr the same PA-(A) randm nde arrangement θ de f = 0, 5,...,40,45 Figure 21: Simuated dendritic grwth withθ = 5 fr different randm nde tures: PA-(B), PA-(C), PA-(D),

23 Simuatin f Dendritic Grwth 91 Therma fuctuatins. In rder t avid the symmetric shape f the dendrite in the cnventina CA apprach sme fuctuatins need t be intrduced int the cacuatins. The fwing equatin is cmmny appied P = 1 + λ rand. Therma nises are usuay presented by putting the randm fuctuatins F int the cacuatins f atent heat, undercing temperature r vecity [Ver (2008)]. It this paper we use then in the vecity cacuatins V = V P. Neighbrhd cnfiguratin. In the CA apprach ny the csest neighbrhd cnfiguratin has been anayzed. Larger the vaue f R H is chsen in the PA methd mre dendritic and irreguar structures can be bserved. A mre extended area f neighbrs needs t be taken int the cnsideratin in the PA methd. The radius f neighbrhd shud be kept at a minimum f 1.5 µm in case f a = 0.5µm. Fr smaer vaues the dendritic shapes becme distrted and the preferred rientatin is st as we. 7.3 Simuated resuts The dendritic mrphgies were cacuated by the cassica FDM-CA and the nve FDM-PA appraches. The fwing numerica exampes were prepared Frm CASE 1 t CASE 10 the dendritic grwth prcess is simuated by the PA methd with the same randm nde arrangement dented (PA-(A)) fr the fwing ten crystagraphic rientatins θ de f = 0, θ de f = 5, θ de f = 10, θ de f = 15, θ de f = 20, θ de f = 25, θ de f = 30, θ de f = 35, θ de f = 40, θ de f = 45. Frm CASE 11 t CASE 19 the dendritic grwth prcess is simuated by the PA methd with different randm nde arrangements (PA-(B), PA-(C), PA-(D)) fr the fwing crystagraphic rientatins θ de f = 5, θ de f = 15, θ de f = 30. Frm CASE 20 t CASE 25 the dendritic grwth prcess is simuated by the PA methd with different randmness f the nde arrangement ε = 0.10, ε = 0.25 and ε = 0.49, fr the fwing θ de f = 5 and θ de f = 30 crystagraphic rientatins. Frm CASE 25 t CASE 28 the dendritic grwth prcess is simuated by the cnventina CA methd withut and with randm fuctuatins.

24 92 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 Frm CASE 29 t CASE 31 the dendritic grwth prcess is simuated by the PA methd incuding the factr respnsibe fr the crrectin in the engths f the x and y branches fr different randm nde arrangements (PA-(B)-F, PA-(C)-F, PA-(D)-F). The resuts have been arranged and represented in the fwing way. The FDM- PA cacuatins with different rientatins f the crystagraphic axis are depicted in Fig. 20 based n the same nde arrangements. The engths f the dendritic axes f these cacuatins are depicted in Fig. 19. Then Figs. 21,..,23 shw the FDM-PA resuts with the varied randm mesh structure fr a singe dendrite with θ de f = 5, θ de f = 15 and θ de f = 30, respectivey. The ength f the dendritie axes f theses cacuatins are depicted in Fig. 24. Fig. 25. Fig. 26 represents dendritic grwth fr a singe dendrite with θ de f = 5 and θ de f = 30 fr a different nde arrangement randmness. The simuatins are shwn fr the cnventina CA apprach with and withut randm fuctuatins in Fig. 27. Finay in Fig. 28 the resuts fr the PA methd, where the randmness crrectin factr is appied, are represented (see discussin in the next paragraph) Discussin f the resuts The rientatins f crystagraphic axes f different dendrites have different rientatins in genera. It is cmmny recgnized that this prcess is difficut t simuate by the cassica CA methd since the dendrite wi aways switch t 0 r 45 directin during the grwth. Our testing is thus primariy fcused n the grwth f the dendrite at different rientatins by the nve PA methd. Simuated exampes are fr the randm nde arrangements PA-(A),..., PA-(F) presented in Fig. 20, and Figs , respectivey. They shw that when empying the PA methd any f the crystagraphic rientatins can easiy be achieved. Resuts shw that the prper grwth directin is aways bserved with increasingy randm (ε 0.49) nde arrangement. Finay, fr the same input parameters the dendritic grwth prcess was simuated by the CA and PA methd fr the θ de f = 0 preferentia crystagraphic rientatin (see CASE 1 and CASE 27, respectivey). The engths f x and y branches were different in bth methds. This is due t the infuence f the randm nde arrangement and subsequent variabe distances between the ndes. In the CA methd the same vaue f a is taken whie fr the PA methd this distances are different and might vary between maximum x = y = 2εa and minimum x = y = 2(1 ε)a. It can be cncuded that the differences in the ength between x and y directins with respect t the randm nde arrangement are amst cnstant and kept bew 5%. The standard deviatin was cacuated fr the x and y engths f the dendritic arms and fr the rati between them (see Fig. 19 and Fig. 24). The fwing features

25 Simuatin f Dendritic Grwth 93 can be summarized frm Tab.3. The average ength f the dendrite at ten different rientatins and sme randm nde arrangement with ε = 0.49 is ± 5.18 µm. The average ength f the dendrite cacuated with fur different randm nde arrangement fr the fixed anges 5, 15 and 30 is ± 5.39 µm, ± 6.44 µm and ± 5.36 µm, respectivey. Frm this ne can cncude that the errrs caused by the rtatin f the dendrite are at the same rder as the errrs cussed by different randm nde arrangements. Fig.25 and Fig.26 demnstrate that when reducing ε frm 0.49 t 0.1 the PA starts t behave ike the CA and the prper simuatin f the dendrite is nt pssibe. We are t cse t the cassica nde structure in such case and CA imitatins appear. T achieve the same dendrite ength in PA methd as in the CA methd, an empirica factr, which mutipies the cacuated vecity in the PA methd, was added in the cde. It can be shwn that putting factr 1.25, (fr the randm nde arrangement ε= 0.49) in the PA cacuatins, the branches wi have the same ength in bth methds (see Fig. 28). In the present study it is nt necessary t put any therma fuctuatins in the PA methd. The randm nde arrangements in the PA methd repace the therma fuctuatins f the CA methd. Fig. 20 and Tab.3 shw that the nve PA cncept can be used t depict the dendritic grwth prcess in any preferentia rientatin during the sidificatin prcess. 8 Cncusins In this paper a nve PA methd is deveped and appied t mdeing the dendritic grwth prcess. Advantages f the deveped PA methd are N need fr mesh generatin r pygnisatin. Ony the nde arrangement has t be generated, but withut any gemetrica cnnectin between ndes. In the new PA methd the gverning equatins are sved with respect t the catin f pints (nt pygns) n the cmputatina dmain. The randm grid PA methd aws t rtate dendrites in any directin since it has a imited anistrpy f the nde arrangements. PA methd ffers a simpe and pwerfu apprach f CA type simuatins. It was shwn that bth methds are abe t quaitativey and quantitativey mde a diverse range f sidificatin phenmena in amst the same cacuatin time.

26 94 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 θ = 5 θ = 15 CASE 11 CASE 14 θ = 5 θ = 15 CASE 12 CASE 15 θ = 5 θ = 15 CASE 13 CASE 16 Figure 21: Simuated dendritic grwth fr a singe dendrite with θ de f = 5 fr different randm nde arrangement structures: PA-(B), PA-(C), PA-(D), respectivey Figure 22: Simuated dendritic grwth fr a singe dendrite with θ de f = 15 fr different randm nde arrangement structures: PA-(B), PA-(C), PA-(D), respectivey

27 Simuatin f Dendritic Grwth 95 θ = enght [μm] CASE PA-(A) PA-(B) PA-(C) PA-(D) 4 PA mesh x branch y branch θ = enght [μm] CASE PA-(A) PA-(B) PA-(C) PA-(D) 4 PA mesh x branch y branch θ = enght [μm] CASE 19 Figure 23: Simuated dendritic grwth fr a singe dendrite with θ de f = 30 fr different randm nde arrangement structures: PA-(B), PA-(C), PA-(D), respectivey PA-(A) PA-(B) PA-(C) PA-(D) 4 x branch PA mesh y branch Figure 24: The engths f the dendrite branches in x and y directins fr the θ de f = 5,θ de f = 15 and θ de f = 30 (frm tp t bttm) different rientatins, fr the randm nde arrangement (see Fig. 21, Fig. 22 and Fig. 23)

28 96 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , θ = 5 θ = 30 PA-(D) 4 CASE 20 CASE 23 θ = 5 θ = 30 PA-(D) 4 CASE 21 CASE 24 θ = 5 θ = 30 PA-(D) 4 in x and y = 30 (frm ndm nde g. 23) CASE 22 Figure 25: Simuated dendritic grwth fr a singe dendrite with θ Figure = 5 fr 25: different Simuated nde arrangement dendritic randmness ε = 0.1(PA-(E)), grwthε fr = 0.25 a(pa-(f)), singeε dendrite = 0.49 PA-(A) with frm the tp t bttm θ de f = 5 fr different nde arrangement randmness ε = 0.1(PA-(E)), ε = 0.25(PA-(F)), ε = 0.49PA-(A) frm the tp t bttm CASE 25 Figure 26: Simuated dendritic grwth fr a singe dendrite with θ de f = 30 fr different nde arrangement randmness ε = 0.1(PA-(E)), ε = 0.25(PA-(F)), ε = 0.49PA-(A) frm the tp t bttm

29 Simuatin f Dendritic Grwth 97 θ = 0 θ = 0 with factr CASE 26 CASE 29 θ = 0 θ = 0 with factr CASE 27 CASE 30 θ = 0 θ = 0 with factr CASE 28 Figure 27: Simuated dendritic grwth fr a singe dendrite fr θ de f = 0 by the CA methd withut λ = 0 (CASE 26) and with randm fuctuatins λ = 0.05 (CASE 27) and λ = 0.3(CASE 28), respectivey CASE 31 Figure 28: Simuated dendritic grwth fr a singe dendrite fr θ de f = 0 by the PA methd with factr 1.25 fr the PA-(B)-F, PA- (C)-F, PA-(D)-F randm nde arrangement, respectivey

30 98 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 Tabe 1: Nmina parameters used in the cacuatins Symb Vaue Unit ρ 2700 kg/m 3 T m K T s K T K λ 210 W/mK c p J/kgK L J/kg η Γ 1.6 x 10 7 Km δ t δ k S 4 1 R c 1.5 µm R H 2 µm µ K 2 m/sk 350 µm n 700 PA ndes/ CA ces N 701 FDM ndes The dimensin f the neighbrhd radius and generatin f the randm nde arrangement has t be chsen carefuy in rder t be abe t rtate the dendrite. Straightfrward nde refinement pssibiity. Straightfrward extensin t 3-D. The use f FDM-PA methd instead f FDM-CA methd impies transfer f the resuts frm the reguar FDM mesh t the irreguar PA nde arrangements and vice versa. This is nt the case in the cassica FDM-CA methd. A repacement f the FDM methd with a meshess [Aturi (2004); Liu and Gu (2005); Šarer, Vertnik and Perk (2005); Šarer and Vertnik (2006)] methd that is abe t directy cpe with irreguar nde arrangement is underway. Acknwedgement: The first authr wud ike t thank the Eurpean Marie Curie Research Training Netwrk INSPIRE fr psitin t study and research at the University f Nva Grica, Svenia. The secnd authr wud ike t thank

31 Simuatin f Dendritic Grwth 99 Tabe 2: Parameters used in the cacuatins CASE θ de f λ ε nde arangement PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(B) PA-(C) PA-(D) PA-(B) PA-(C) PA-(D) PA-(B) PA-(C) PA-(D) PA-(E) PA-(F) PA-(A) PA-(E) PA-(F) PA-(A) CA CA CA PA-(B)-F PA-(C)-F PA-(D)-F

32 100 Cpyright 2010 Tech Science Press CMC, v.18, n.1, pp , 2010 Tabe 3: The engths f dendrite branches in x and y directins and reated quantities with respect t the randm nde arrangement resuts methd x branch ength [µm] y branch ength [µm] rati f primary dendrite arms x/y [-] 5 PA-(A) PA-(B) PA-(C) PA-(D) PA-(A) PA-(B) PA-(C) PA-(D) PA-(A) PA-(B) PA-(C) PA-(D) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) PA-(A) x/y average ength rati [-] average ength f xandy [µm] σ standard deviatin f x/y [-] σ standard deviatin f enght [µm]

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