DELFT UNIVERSITY OF TECHNOLOGY

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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT A comparson of numercal models for one-dmensonal Stefan problems E. Javerre 1, C. Vuk, F.J. Vermolen, S. van der Zwaag ISSN Reports of the Delft Insttute of Appled Mathematcs Delft Supported by the Dutch Technology Foundaton (STW.

2 Copyrght 2005 by Delft Insttute of Appled Mathematcs, Delft, The Netherlands. No part of the Journal may be reproduced, stored n a retreval system, or transmtted, n any form or by any means, electronc, mechancal, photocopyng, recordng, or otherwse, wthout the pror wrtten permsson from Delft Insttute of Appled Mathematcs, Delft Unversty of Technology, The Netherlands.

3 A comparson of numercal models for one-dmensonal Stefan problems E. Javerre, C. Vuk, F.J. Vermolen and S. van der Zwaag Abstract In ths paper we present a crtcal comparson of the sutablty of several numercal methods, level set, movng grd and phase feld model, to address two well known Stefan problems n phase transformaton studes: meltng of a pure phase and dffusonal sold state phase transformatons n a bnary system. Smlarty solutons are appled to verfy the numercal results. The comparson shows that the type of phase transformaton consdered determnes the convenence of the numercal technques. Fnally, t s shown both numercally and analytcally that the sold-sold phase transformaton s a lmtng case of the sold-lqud transformaton. AMS classfcaton: 35R35; 65M06; 80A22 Keywords: Stefan problem; Phase transformatons; Smlarty solutons; Movng grd method; Level set method; Phase feld method 1 Introducton In Stefan problems, the boundary of the doman has to be found as part of the soluton. These problems descrbe several phenomena n nature, scence and socety, among others the meltng of the polar ce caps, orgnally studed by J. Stefan, the dendrtc soldfcaton problem [4, 12, 21], the decrease of oxygen n a muscle n the vcnty of a clotted bloodvessel [5], the etchng problem [26], the Amercan opton przng problem [10], or the phase transformatons n metallc alloys [22]. Ths paper deals wth a survey of exstng numercal technques for solvng one-dmensonal problems. In partcular, we consder the meltng problem and a sold state phase transformaton n parallel due to the resemblance n ther governng equatons, that we wll show afterwards. Exstence of a soluton was proved by Evans [6], whle the unqueness was proved by Douglas [7]. Moreover, the soluton of the Stefan problems we consder here satsfes the maxmum prncple n each phase. Further, t s possble to derve analytcal expressons for the soluton of these problems n an nfnte or sem-nfnte one dmensonal-space. Under these hypotheses the soluton s a functon of x s 0 t as proved by Hll [11], and t s often called the smlarty soluton. Several numercal methods have been developed to solve varous Stefan problems. Crank [5] provdes a good ntroducton to the Stefan problems and presents an elaborate collecton Delft Unversty of Technology, Delft, The Netherlands, e-mal: e.javerre@ew.tudelft.nl The authors want to thank the Dutch Technology Foundaton (STW for ts fnancal support. 3

4 of numercal methods used for these problems. Front trackng methods use an explct representaton of the nterface. Jurc and Tryggvason [12] used a fxed grd n space where some varables of the problem,.e. temperature, were calculated, and a movng grd on the nterface where the nterface heat sources were computed. Informaton from the nterface to the fxed grd was transferred va the mmersed boundary method. Segal et. al. [22] used an adaptve grd method n whch the movement of the grd was ntroduced nto the governng equatons by the use of the total tme dervatve (also called Arbtraran Lagrangan Euleran -ALEapproach. Murray and Lands [14] compared an adapted grd procedure wth a fxed grd, and showed that the adaptve grd method captures more accurately the nterface poston whereas the fxed grd algorthm gves a more precse heat dstrbuton n the whole doman. On the other hand, mplct methods are the natural alternatve to the front trackng methods. Wthn these mplct methods the most used are the enthalpy method, the level set method and the phase feld method. In the enthalpy method (see [5] and Chapter 9 of [15] the enthalpy functon s ntroduced. Ths functon measures the total heat of the system, and t has a jump dscontnuty at the nterface gven by the heat released (or absorbed durng the phase change. Ths dscontnuty s helpful to determne the nterface poston. Although ths method has been successfully appled to phase change problems by Voller et. al. [16, 17], and by Nedjar [18], t has only recently been generalzed to sold state phase transformatons wth a smple condton on the movng boundary, see Lam et. al. [19] for further detals. The level set method has ganed much popularty for solvng movng boundary problems. Frstly ntroduced by Osher and Sethan [20], t has already been generalzed to many problems [23, 24]. The level set functon captures the nterface poston as ts zero level set, and t s advected by the ntroducton of a hyperbolc equaton nto the governng set of equatons. The velocty feld, used to advect the level set functon, s qute dfferent wthn the applcatons of the level set method. Sussman et. al. [25] use the flud velocty to smulate ncompressble two-phase flows. Chen et. al. [4] use advecton equatons to extend the nterface velocty onto the whole doman n a soldfcaton problem and Adalstensson and Sethan [1] use a procedure based on the fast marchng methods to extend the front velocty n such a way that t does not destroy the dstance functon attrbute of the level set functon. In these last references the velocty feld s used for a numercal purpose. Fnally, the phase feld method s a wdely used method for phase transformaton problems. The doman s parameterzed by the phase feld functon whch equals a fxed constant n each phase, and vares rapdly, but smoothly, wthn these two values n the nterface regon. The phase transformaton occurs nsde ths nterface regon, whose thckness s an artfcal parameter of the model. There are several phase feld models n the lterature, but the most used are based on the Kobayash potental (see Wheeler et. al. [27] for phase transtons n bnary alloys or based on the Cagnalp potental (see Mackenze and Robertson [13] for the classcal meltng Stefan problem. The Kobayash potental s based on a fourth order polynomal wth fxed mnma at x = ±1 coupled wth a monotoncally ncreasng functon of the temperature to defne the free energy functonal, whereas the Cagnalp potental uses a double-well potental measured by a parameter and a lnear couplng wth the temperature. Fabbr and Voller [8] compare these models n detal. In both studes a dscretzaton wth an adaptve grd s used where the nterface poston concdes wth one grd node. In ther numercal experments they compare the solutons wth a fxed grd soluton, and the Kobayash potental shows a better agreement wth ths last soluton. In addton, an asymptotc analyss for the phase feld models s requred to check whether the phase feld soluton converges to the sharp nterface problem and to determne the parameters that appear n the formulaton 4

5 of the model. Ths asymptotc analyss s already done n [2], where Cagnalp proves the convergence to the Stefan and Hele-Shaw problems by takng the lmt n the parameters n a convenent way. In order to solve the governng equatons numercally Cagnalp and Ln [3] use a coarse grd where the temperature of the system was calculated and a fne grd for the phase feld functon. Mackenze and Robertson [13] propose to use an adaptve mesh wth a hgh resoluton n the nterface regon. Schmdt [21] uses fnte elements wth local refnement n the vcnty of the nterface to smulate dendrtc growth. An outlne of the paper s as follows. The governng equatons of both the meltng problem and the sold state phase transformaton problem wll be descrbed n Secton 2, together wth the analytcal expressons of the smlarty solutons for nfnte domans. The movng grd, level set and phase feld methods wll be presented n Secton 3. Some numercal results wll be gven n Secton 4 and the conclusons n Secton 5. 2 The physcal problems In the present paper we consder two classcal Stefan problems: the meltng problem and the sold state phase transformaton problem n bnary metallc alloys. In the meltng problem we have a lqud phase n contact wth a sold phase separated by the nterface, where the temperature s the meltng temperature. Ths problem s also called sold-lqud transformaton. The heat transport through the nterface causes ts dsplacement. In the second problem a volume of constant composton s surrounded by a dffusve phase. In the nterface between the partcle and the dffusve phase a constant concentraton s assumed, and the gradent of the concentraton causes the movement of the nterface. Ths problem s also called sold-sold transformaton. Moreover, we can thnk that the sold-sold transformaton s a partcular case of a sold-lqud transformaton wth zero thermal dffusvty n the lqud phase. We restrct ourselves to the one dmensonal problem. Hence, some physcal features of these problems lke the surface tenson (.e., the Gbbs-Thomson effect are not ncorporated here. The doman wll be denoted by Ω = [0, l] where l denotes the length. Ths doman wll be splt nto two phases, and the nterface separatng these phases wll consst of only one pont. Therefore, a functon s : IR + [0, l] wll assgn each tme t the poston of the nterface at ths tme s(t. 2.1 The meltng problem: a sold-lqud transformaton We consder the doman Ω = [0, l] where we have a materal that s n ts lqud state n a certan regon of Ω we call Ω lq (t = [0, s(t and t s n ts sold state n the rest of the doman Ω sol (t = Ω \ Ω lq (t = (s(t, l]. The pont separatng the lqud and the sold phases determnes the poston of the nterface s(t. We denote the temperature n the pont x at tme t by u(x, t. The governng equatons for ths problem are the heat equaton n both the lqud and the sold phases u ( u (x, t = Klq, x Ωlq (t, (1 t x x u ( (x, t = t x K sol u x, x Ω sol (t, (2 5

6 where K sol and K lq denote the thermal dffusvtes n the sold and the lqud phase respectvely, whch nvolve the heat capacty, densty and the heat conducton coeffcent of the materals. In ths study we assume them to be constant n tme and poston. The velocty v of the nterface s gven by the jump condton Lv = K sol u x (x, t x s(t K lq u x (x, t x s(t, (3 where L denotes the latent heat of soldfcaton. Equaton (3 s frequently called the Stefan condton. At the nterface we have the meltng temperature, that we choose here to be zero wthout loss of generalty: u(s(t, t = 0. (4 In ths problem thermally nsulated domans are consdered. Hence, no heat fluxes through the boundares of the doman Ω are allowed, whch leads to the homogeneous Neumann boundary condtons u (x, t = 0, x Ω. (5 x We consder a pecewse constant ntal heat dstrbuton u lq f x Ω lq = [0, s 0, u(x, 0 = 0 f x = s 0, u sol f x Ω sol = (s 0, l], where u lq and u sol are constants, generally postve and negatve respectvely, and s 0 denotes the ntal poston of the nterface,.e., s 0 = s(0. For ths problem the poston of the nterface s s a dfferentable functon. Therefore, the velocty of the nterface (v n Eq. (3 can be replaced by ds dt (t L ds dt (t = K u sol x (x, t u x s(t K lq x (x, t x s(t. (6 2.2 Phase transformatons n bnary alloys: sold-sold transformatons We consder the doman Ω = [0, l] that s composed by a partcle whose doman s denoted by Ω part (t = [0, s(t and a dffusve phase Ω dp (t = Ω \ Ω part (t = (s(t, l]. The pont separatng the partcle and the dffusve phase represents the nterface s(t. We consder the concentraton c of a certan materal wthn Ω, and we assume ths concentraton to be constant wthn the partcle. Therefore, we have the followng governng equatons c(x, t = c part, x Ω part (t, (7 c ( (x, t = D c t x x (x, t, x Ω dp (t, (8 where D denotes the dffusvty nsde the dffusve phase Ω dp. The velocty of the nterface v s derved from a mass balance through the nterface whch leads to 6

7 (c part c sol v = D c x (x, t x s(t, (9 where c sol s the nterface concentraton, that s, c(s(t, t = c sol. Further c part c sol, to avod an undefned velocty. We assume that the doman Ω s solated, and therefore there s no concentraton transported out of the doman Ω c (x, t = 0, x Ω. (10 x We assume a pecewse ntal concentraton as follows c part f x Ω part = [0, s 0, c(x, 0 = c sol f x = s 0, c 0 f x Ω dp = (s 0, l], where s 0 s the ntal poston of the nterface, and c part, c sol, c 0 > 0. Under these hypotheses s s a monotonous and dfferentable functon. Hence, the Stefan condton for the velocty of the nterface Eq. (9 can be expressed by (c part c sol ds c (t = D dt x (x, t x s(t. (11 If we take the thermal dffusvty of the lqud phase zero n the meltng problem and assgn L = c part c sol, then equatons (1, (2 and (3 are equvalent to (7, (8 and (9, beng the only dfference between the two models n the boundary value at the nterface, that smply realzes a translaton. 2.3 Smlarty solutons In our numercal experments we wll compare the solutons obtaned from the dfferent numercal methods wth the analytcal solutons that exst for the problems presented above. These solutons are expressed as functons of x s 0 t as proved n [11], and the doman Ω has to be nfnte or sem-nfnte. Hence, for Ω = IR the nterface poston s gven by s(t = s 0 + 2α t, where the constant α s obtaned by solvng the followng equaton α = Ksol πl erfc( u sol α Ksol α2 Klq exp( + K sol πl u lq α2 α exp(, (12 2 erfc( K Klq lq for the meltng problem and α = c0 c sol D c part c sol π exp( α2 D erfc( α D, (13 for the sold-sold transformaton problem. As we noted above, f we let K lq = 0, L = c part c sol and dentfy u sol wth the concentraton dfference c 0 c sol then equaton (12 and (13 are the same. When α s known, the temperature s gven by 7

8 u lq erfc ( α u Klq lq erfc ( x s 0 2 erfc ( 2 K lq t α + 2 erfc ( α, f x < s(t, u(x, t = Klq Klq u sol u sol erfc ( x s 0 2 K sol t erfc ( α, f x s(t, Ksol for the meltng problem, whereas the concentraton s gven by c part, f x < s(t, c(x, t = c 0 + (csol c 0 erfc ( x s 0 2 Dt erfc ( D α, f x s(t, (14 (15 for the sold-sold transformaton problem. 3 The numercal soluton methods In ths secton we present three numercal methods to solve the sold-lqud and the sold-sold transformatons. These are the movng grd, the level set and the phase feld methods. The movng grd method, as presented here, s easy to mplement and to ncrease the order of accuracy n the dscretzatons. Moreover, t leads to symmetrc matrces, whch are desrable to solve the large systems of equatons numercally. For hgher dmensonal methods t s convenent to ntroduce the dsplacement of the grd nto the governng equatons (ALE approach. However, an mplct dscretzaton of the convecton term wll lead to a nonsymmetrc matrx. The level set method captures the nterface poston mplctly and moves t accordng to a new artfcal equaton n the governng mathematcal model. It s known from lterature that mergng nterfaces are easy to handle wth the level set method. In addton, for ths method a fxed grd can be used, whch avods the mesh generaton at every tme step requred wth the movng grd. Fnally, the phase feld method allows a better agreement between the numercal model and the physcal problem, snce most of the drvng forces actng on the nterface (such as surface tenson are related to the phase feld parameters. However, an adaptve mesh wth a local refnement n the nterfacal regon seems to be necessary. For the sake of smplcty we wll restrct the presentaton of the numercal methods consdered n ths paper to the sold-lqud transformatons. Generalzaton of these procedures to the sold-sold transformatons s straghtforward, except for the phase feld model. 3.1 The movng grd method Here we present an nterpolatve movng grd method, n whch the grd s computed for each tme step and the soluton s nterpolated from the old grd to the new grd. The nterpolaton mght be abolshed by the ntroducton of the grd dsplacement nto the governng equatons wth the ALE approach. Ths technque s used by Murray and Lands [14] and Segal et. al. [22]. However, we prefer to use nterpolaton for the ease of mplementaton and snce the dfference s not decsve for 1D problems. Let N be the total number of grd ntervals, r of those le nsde the lqud phase and N r le nsde the sold phase. The grd s unform n each phase and the nterface s always located n the r th node. Due to the movement of the nterface, the grd should be adapted at 8

9 each tme step. To obtan the temperature at the next tme step, a backward Euler scheme s used for the tme dscretzaton of the heat equaton and central dfferences are used for the dscretzaton n space. Wth the temperature profle, the dsplacement of the nterface s calculated usng frst order accurate drectonal dervatves n each phase, and the grd s adapted to the new poston of the nterface. Fnally, the soluton s nterpolated from the prevous grd to the new one. The nterested reader s referred to [5] for further detals. 3.2 The level set method The level set method captures the poston of the nterface as the zero level set of a contnuous functon φ ntalzed as a dstance functon, x s 0 f x < s 0, φ(x, 0 = 0 f x = s 0, x s 0 f x > s 0, whch has been arbtrarly selected postve n the lqud phase. Ths functon s called the level set functon, and the nterface s mplctely represented by ts zero level set x = s(t φ(x, t = 0, t 0. The evoluton of the level set functon s derved from the above equaton takng nto account a contnuous extenson v of the nterface velocty (6 (or (11 respectvely as follows φ (x, t + v(x, t φ(x, t = 0, x Ω. (16 t x In our specfc applcaton the nterface velocty s only defned at the front poston tself. Hence, a contnuous extenson v of the nterface velocty s unavodable, and t s taken as the steady soluton of the next advecton equaton ṽ τ (x, τ + S( φ(x, t φ x (x, t ṽ (x, τ = 0 (17 x that propagates the front velocty n the correct upwnd drecton (see Chen et. al. [4]. Here τ denotes a fcttous tme step not related to the man tme step, and S denotes the sgn functon. For general dmensons, the normal vector to the nterface s gven by n = φ φ and the curvature of the nterface by κ = n. Furthermore, several smplfcatons can be done when φ s sgned dstance functon (see [24] for further detals. However, after solvng (16 the level set functon φ possbly s no longer a dstance functon. Then, a rentalzaton procedure s requred. Ths rentalzed level set s the steady-state soluton of φ ( τ (x, τ = S(φ(x, t 1 φ x (x, τ, (18 where τ s agan a fcttous tme and φ(x, t s the level set we want to set as a dstance functon. Note that the zero level set of φ s not changed, and φ = 1, whch characterzes 9 x

10 the dstance functons, n the steady soluton. The numercal dscretzaton of the set of governng equatons s done on a unform fxed grd wth grd spacng x. The front velocty s calculated by forward dfferences n the sold phase and backward dfferences n the lqud phase. The extenson of the front velocty and the advecton of the level set functon are done by a forward Euler scheme for the tme dscretzaton and an upwnd dscretzaton n space. Ths leads to stablty condtons for the tme steppngs: τ < x n the artfcal advecton of the front velocty, and max x Ω v(x, t t x < 1 n the advecton of the level set functon. The equaton for the rentalzaton s dscretzed by the frst order accurate Godunov s scheme presented n [25]. The heat equaton s dscretzed usng a backward Euler scheme n tme and central dfferences n space. For the neghbourng nodes to the nterface we use the second order dervatves of the quadratc Lagrangan nterpolaton polynomals that approxmate the soluton n the vcnty of the nterface from the approprate sde of the nterface. When a grd node changes phase (.e. the nterface crosses t the dscretzaton of the heat equaton should be adapted. When ths happens, the node n queston s not ncluded n the dscretzaton, and the soluton at ths node s obtaned by nterpolaton from the neghborng nodes wthn the same phase. Ths procedure s slghtly dfferent from the method presented recently by Gbou et. al. n [9], where the temperature n the conflctve node s adapted to the nterface poston before the heat equaton s solved. 3.3 The phase feld method The phase feld method uses a functon φ(x, t whch characterzes the phase of the system at each pont x and tme t. Ths functon, that s called the phase feld functon, assumes an nterface regon of thckness ε where the phase transtons occur. Ths s clearly dfferent from the movng grd and level set methods where a sharp nterface s consdered. The phase feld functon φ s gven by φ(x, t = { 1 f x s n the lqud phase at tme t, 1 f x s n the sold phase at tme t, and the nterface regon s characterzed by 1 < φ(x, t < 1. Then the evoluton of the system s descrbed by the followng system of two coupled partal dfferental equatons νξ 2 φ t = δf δφ, u t + L φ 2 t = ( u K, x x where K s the approprate dffusvty constant n each phase,.e., K = K lq where φ > 0 and K = K sol where φ < 0. Further, L denotes the latent heat and F denotes a free energy functonal whch s a functon of φ as well as other varables of the problem, and δf δφ denotes the varatonal dervatve of F wth respect to φ. The parameter ν s a relaxaton tme and ξ s related wth the thckness of the nterface regon. We use the Cagnalp model, for whch the asymptotc analyss s done by Cagnalp [2]. The free energy functonal F s expressed by 10

11 F(φ, u = Ω [ 1 2 ξ2( φ ] 2 + f(φ, u dx, x where f s the so-called free energy densty whch conssts of a double-well potental measured by a parameter a and a term couplng u wth φ f(φ, u = 1 8a (φ uφ. The two mnma of f establsh the stable states of the problem (.e. the lqud and sold states, whch are slghtly dsplaced from ts physcal values φ = ±1 due to the nfluence of the parameter a. Hence to mnmze ths nfluence the parameter a should be chosen small. Further, the nterface thckness s gven by the relaton ε = ξ a. The phase feld vares rapdly from 1 to 1 wthn the nterface regon, whch motvates the use of an adaptve mesh procedure. Here the approach presented by Mackenze and Robertson n [13] s used. The adaptve mesh s constructed by an equdstrbuton prncple x+1 (t x (t M( x, td x = 1 N 1 0 M( x, td x, for = 0, 1,..., N 1, (19 where N s the number of space ntervals we consder n our spatal doman Ω and M s a montor functon related wth the thckness of the nterfacal regon. In ths case M(x, t = γβ(t + sech ( x s(t 2ε 1, β(t = 0 sech ( x s(t 2ε dx, (20 where γ > 0 s a parameter chosen by the user. The parameter γ must be chosen postve to ensure that the montor functon M s postve and not zero to avod the clusterng of all the grd nodes nsde the nterface regon, snce the number of grd nodes placed wthn the N 1+γ. nterface regon s approxmately Fnally, the use of the Cagnalp potental and the adaptve mesh procedure leads to the followng system of dfferental equatons νξ 2( Dφ Dt dx dt Du Dt dx dt φ = ξ 2 2 φ x u x + L 2 x 2 1 2a (φ3 φ + 2u, φ ( u = K x x x (Dφ Dt dx dt (21a, (21b where the total-tme dervatve (ALE approach, see also Secton 3.1 has been used to ncorporate the mesh movement nto the governng equatons. These equatons are solved separately. It has to be remarked that the mesh at the new tme step s requred to solve system (21. We use the followng algorthm. Frst we estmate the nterface poston at the new tme step by s n+1 = 2s n s n 1. Note that a fxed tme step s used. Then, the mesh at the new tme step s calculated wth the equdstrbuton prncple Eqs. (19-(20, whch s determned by solvng the non-lnear equaton 11

12 [ γβ(t n+1 x n+1 + 2ε sn 1 ( tanh( xn+1 s n+1 2ε sn 1 ( tanh( sn+1 2ε ] = β(tn+1 (γ + 1, N for x n+1, = 0,..., N. Thereafter, the phase feld at the new tme step s calculated from Eq. (21a, where the temperature at the prevous tme step s used. For the computaton of the phase feld functon, we use the followng dscretzaton νξ 2( φ n+1 φ n t xn+1 x n t φ n +1 φn 1 2ξ 2 (φ n+1 = h n +1 + hn h n hn+1 +1 φn+1 h n+1 +1 φ n+1 1 φn+1 h n+1 1 [ (φ n+1 3 φ n+1 ] + 2u n 2a, for = 1,..., N 1. The value at the boundary nodes s gven by the boundary condtons. We use the Newton method to solve ths non-lnear system of equatons up to a gven tolerance. Subsequently, the temperature dstrbuton s obtaned by substtuton of the phase feld at the new tme step nto ts tme dervatve n Eq. (21b. We use the followng dscretzaton u n+1 u n t xn+1 x n t u n +1 un 1 h n +1 + hn + L 2 (φ n+1 φ n t xn+1 x n t φ n +1 φn 1 = h n +1 + hn 2 h n hn+1 ( K n+1 u n+1 +1 un+1 +1/2 h n+1 K n+1 u n+1 1/2 u n+1 1, for = 1,..., N 1. The value at the boundary nodes s obtaned from the boundary condtons. Note that central dfferences have been used to dscretze the convectve terms snce the phase transformaton s dffuson controlled. We refer the nterested reader to [13] for further detals. In the post-processng of the results, the nterface poston s calculated as the zero of the phase feld functon by lnear nterpolaton. In addton, Mackenze and Robertson [13] study the exstence of a soluton for the nonlnear system comng from Eq. (21a. They found a suffcent condton for the exstence of a numercal soluton based on the tme step: t < 2νaξ 2 = 2νε 2, whch reveals the numercal dffcultes that arse when we try to recover the sharp nterface problem wth ε small. However, n ther and our numercal experments t s possble to use a larger tme step, although ts selecton s very senstve to the other parameters n the phase feld model. 4 Numercal results In ths secton we present some numercal results for both the sngle phase (sold-sold transformaton and the two phases (sold-lqud transformaton problems. Our am s to mmc the frst Stefan problem by solvng the second Stefan problem, for whch the phase feld method can be appled. We also examne the convergence of the movng grd and the level set methods for the sngle phase problem. The data used through all the computatons are: the concentraton nsde the partcle c part = 0.53, the concentraton on the nterface c sol = 0 h n+1 12

13 , the ntal concentraton n the dffusve phase c 0 = 0.1, the dffusvty constant D = 1, the doman length l = 1 and the ntal poston of the nterface s 0 = Sold-sold transformaton In ths secton we use the movng grd and the level set methods to smulate the sngle phase problem presented above Movement of the nterface Concentraton hstory at x= poston concentraton Smlarty soluton MGM, N = 100 LSM, N = tme 0.1 Smlarty solton MGM, N = 100 LSM, N = tme (a Interface poston vs. tme (b Concentraton hstory at x = 0.25 Fgure 1: The sold-sold transformaton problem wth the movng grd (MGM and the level set (LSM methods. In Fgure 1(a the evoluton of the nterface postons for the movng grd, the level set methods and the smlarty soluton (obtaned from Eq. (13 s shown. It appears that there s a close agreement wth the smlarty soluton n the begnnng of the smulaton. However, n the numercal smulatons a fnte doman Ω s consdered whereas for the smlarty soluton the doman Ω s nfnte. Ths causes the dvergence of the numercal solutons wth respect to the smlarty soluton f tme evolves. In Fgure 1(b the concentraton at x = 0.25 versus the tme s compared agan wth the smlarty soluton gven by Eq. (15. For both the movng grd and the level set methods we take N = 100. For the movng grd method the tme step s t = , whereas n the level set method the tme step s determned by the CFL condton for stablty, whch prescrbes t n = CF L x v n wth CF L < 1. In our calculatons we have used CF L = 0.1. Ths choce made the tme step to vary from to , whch was used as an upper bound to avod excessvely large tmes steps due to small nterface veloctes. In Table 1 the convergence to the smlarty soluton of both the movng grd and the level set methods s examned, wth fnal tme for the numercal ntegraton t end = 0.1. The tme step was refned when N was ncreased n both methods. Frst order convergence n the nterface poston s observed for both methods, although a slghtly hgher accuracy s observed wth the movng grd method. Ths s lkely due to the dfferences n the grd spacng and tme steppngs for both methods. 13

14 Movng Grd Method Level Set Method N s h (0.1 s s h s h (0.1 s s h Table 1: Convergence analyss for the Movng Grd Method (left and the Level Set Method (rght. Smlarty soluton s(0.1 = Sold-lqud transformaton In ths secton we mmc the problem presented above wth a sold-lqud transformaton problem. Hence, we consder the next ntal temperature dstrbuton 0.53, n the lqud phase (0 x < s 0 u(x, 0 = 0, on the nterface x = s 0 0.1, n the sold phase (s 0 < x 1 where the ntal poston of the nterface s s 0 = 0.2 agan. The latent heat L = c part c sol = 0.53 and the thermal dffusvty n the sold phase s K sol = D = 1 and n the lqud phase s ether K lq = 0.05, 0.01 or As mentoned n Secton 2.3 the nterface poston of the smlarty soluton s determned by a parameter α whch s calculated from the ntal data n Eq. (12. In Table 2 values of α are gven for dfferent values of K lq, compared wth the value of α n the sold-sold transformaton problem Eq. (13. Note that the sold phase has been artfcally super-heated to mmc the sold-sold phase transformaton problem presented above, whch s obtaned for K lq = 0. The three methods were also tested on classcal Stefan problems, and the same knd of behavor for the numercal solutons as presented here has been observed. K lq α Table 2: Values of α for dfferent values of K lq. In Fgure 2(a the nterface poston wth the Movng Grd, the Level Set and the Phase Feld methods s presented. Comparson wth the smlarty soluton s only correct for small tmes, due to the bounded doman. Hence t end = 0.25 s used as the fnal tme for the smulaton. In Fgure 2(b the temperature hstory n x = 0.25 s presented, and the numercal solutons are consstent wth the smlarty soluton gven by Eq. (14. For the movng grd method N = 200 grd ntervals were used for the entre doman Ω, of whch r = 100 were n the lqud phase. The tme step used was t = For the level set method we took N = 200, 14

15 whle the CFL parameter was 0.1. The tme step vared from to , whch was used as a upper bound. Fnally, for the phase feld method a = and ξ = , whch led to an nterface thckness of ε = and relaxaton tme ν = 1. Further we use N = 200 and γ = 1, whch mples that about 100 nodes were placed wthn the nterface regon. As mentoned n Secton 3.3 the crteron on the tme steppng found by Mackenze and Robertson [13] leads to t < whch s useless for practcal purposes. However, as they also ponted n [13] t has been possble to obtan satsfactory results wth larger tme steps. In ths case we used t = The ntal temperature dstrbuton was obtaned from the smlarty soluton Eq. (14 usng an ntal tme t 0 = 0.01, snce t was seen that a dscontnuous ntal temperature dstrbuton causes nstabltes n the phase feld method. The same experment has been done for K lq = 0.05 and K lq = 0.01, for whch ξ = and ξ = were used respectvely. The followng observatons can be done: decreasng the nterface thckness mght requre to decrease the tme step (although n our calculatons t = was used and satsfactory results were obtaned, decreasng the dffusvty K lq wll requre to ncrease the ntal tme t 0 to obtan an ntal temperature dstrbuton that s suffcently smooth nsde the nterface regon. Further decreasng the thermal dffusvty K lq requres a decrease of the nterface thckness K lq = 0.005, Movement of the nterface 0.1 K lq = 0.005, Temperature hstory at x= poston temperature tme Smlarty soluton MGM, N=200, r=100 LSM, N=200 PFM, N= Smlarty soluton MGM, N=200, r=100 LSM, N=200 PFM, N= tme (a Interface poston vs. tme (b Temperature hstory at x = 0.25 Fgure 2: The sold-lqud transformaton problem wth the Movng Grd, the Level Set and the Phase Feld methods. 5 Conclusons In ths paper we consder two Stefan problems resultng from phase transformatons: the meltng problem and a dffusonal phase transformaton n bnary alloys where only n one of the two phases the soluton of the dffuson equaton s determned, whereas the soluton s constant n the other phase. Numercal solutons of those problems have been obtaned wth the movng grd, the level set and the phase feld methods. The formulaton of the problems and the exstng smlarty solutons show the resemblance between the two problems. In fact, the dffusonal one phase transformaton s a specal case of 15

16 the meltng problem. From the numercal computatons several conclusons can be obtaned: Both the movng grd and the level set methods are sutable numercal models for the sold state phase transformaton, and ther accuracy s comparable. The same can be concluded for the meltng problem. Furthermore, for ths problem the phase feld method s also sutable. From the reported results (Fgure 2 one mght conclude that the phase feld method gves a better approxmaton of the nterface poston and temperature profle. However, the grd resoluton wthn the nterfacal regon s much hgher wth the phase feld method than wth the movng grd and the level set methods, whch also leads to larger computatonal cost. The phase feld method has shown to be applcable for the meltng problem, even when the dfference of the thermal propertes n each phase s remarkable. Unfortunately t s hard to derve approprate phase feld parameter values from the physcal parameters of the phases nvolved, whch lmts the true predctve power of ths technque. The movng grd and the level set methods do not suffer from ths hgh dependence of the physcal parameters of the problem. Ther nput values can be obtaned rather easly from tables and phase-dagrams. The nterpolatve approach presented for the movng grd method can be replaced by the ALE approach, whch ntroduces the dsplacement of the grd nto the governng equatons, leadng to solvng a convecton-dffuson equaton, whch s more convenent for hgher dmensonal problems. Topologcal changes, whch nvolve mergng or breakng of the nterface, are dffcult to model wth the movng grd method. However, ths s easly handled n the level set method. The dffcultes n the level set method are the extenson of the front velocty and the rentalzaton, although they can be overcome wthout much effort. Ths motvates us to nvestgate the level set method for hgher dmensonal problems. References [1] D. Adalstensson and J.A. Sethan, The fast Constructon of Extenson Veloctes n Level Set Methods, J. Comput. Phys. 148 ( [2] G. Cagnalp, Stefan and Hele-Shaw type models as asymptotc of the phase-feld equatons, Phys. Rev. A 39 ( [3] G. Cagnalp and J.T. Ln, A numercal analyss of an Ansotropc Phase Feld Model, IMA J. Appl. Math. 39 ( [4] S. Chen, B. Merrman, S. Osher and P. Smereka, A Smple Level Set Method for Solvng Stefan Problems, J. Comp. Phys. 135 ( [5] J. Crank, Free and movng boundary problems, Clarendon Press, Oxford, 1984 [6] G.W. Evans, A note on the exstence of a soluton to a problem of Stefan, Q. Appl. Math. 9 ( [7] J. Douglas, A unqueness theorem for the soluton of a Stefan problem, Proc. Amer. Math. Soc. 8 (

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