Anisotropic, adaptive finite elements for the computation of a solutal dendrite

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1 Interfaces and Free Boundaries 5 (23), Anisotropic, adaptive finite elements for te computation of a solutal dendrite E. BURMAN AND M. PICASSO Institut de Matématiques, École Polytecnique Fédérale de Lausanne, 115 Lausanne, Switzerland [Received 2 June 22 and in revised form 31 October 22] We compute solutions of solutal pase-field models for dendritic growt of an isotermal binary alloy using anisotropic mes refinement tecniques. Te adaptive strategy is based on anisotropic a posteriori estimators using a superconvergent recovery tecnique in te form of te Zienkiewicz Zu error estimator. Te pase-field model contains an anisotropic strongly nonlinear second order operator modelling te dendritic brances; tis strong nonlinearity is included in te a posteriori error estimators by using a monotonicity result. Te monotonicity olds for pase-field anisotropy below a certain tresold value beyond wic tere are no known well-posedness results. We present computational results for bot regimes sowing te performance of te proposed metod. 1. Introduction In tis paper we consider te problem of computing solutions to a pase-field model for an isotermal binary alloy. Te model consists of a parabolic system of nonlinear equations modelling te pase-field, φ, wic is an order parameter taking te value 1 in te solid pase and in te liquid pase, making te transition in a continuous fasion in te tin solid-liquid interface, known as te musy zone (see Fig. 1.1) and te concentration of te binary alloy c, also taking values between zero and one depending on te mass fraction of te alloy constituants. φ = 1 Solid Tickness of te solid-liquid interface φ = Liquid FIG Te pase-field φ. A general matematical formulation of tis solidification problem, including te models of [23] and [22], is te following. Given a bounded domain of R 2 wit outer unit normal n, c, φ Corresponding autor, supported by te Swiss National Foundation. erik.burman@epfl.c marco.picasso@epfl.c c European Matematical Society 23

2 14 E. BURMAN & M. PICASSO H 1 () and a time interval (, T ) we consider te problem of finding φ, c : (, T ) R suc tat α φ div(a( φ) φ)) S(c, φ) = t in (, T ), (1.1) c t div(d 1(φ) c + D 2 (c, φ) φ) = in (, T ), (1.2) A( φ) φ n = on (, T ), (1.3) (D 1 (φ) c + D 2 (c, φ) φ) n = on (, T ), (1.4) φ() = φ, c() = c in. (1.5) Here α R is a positive parameter and A( ) is te matrix defined for ξ R 2 \ by ( a A(ξ) = 2 (θ(ξ)) a(θ(ξ))a ) (θ(ξ)) a(θ(ξ))a (θ(ξ)) a 2, (1.6) (θ(ξ)) were θ(ξ) denotes te angle between te vector ξ and te first component of te ortonormal cartesian basis (e 1, e 2 ), tat is, cos θ(ξ) = ξ e 1 ξ, and a is te real-valued function defined by a(θ) = 1 + ā cos(κθ), (1.7) wit ā (te anisotropy parameter) and κ a positive integer corresponding to te number of dendritic brancing directions. In [2] it is sown tat (1.1) (1.5) as a unique weak solution, in te isotropic case ā = (tus A( ) = I). Furtermore in tis case a priori error estimates, a posteriori and adaptive finite elements are available in [16, 17]. Existence of a weak solution in te anisotropic case ā = as been proved in [8]. More precisely, if S, D 1 and D 2 are bounded Lipscitz functions satisfying < D s D 1 (φ) D l φ R, S(c, ) = S(c, 1) = c R, D 2 (, φ) = D 2 (1, φ) = φ R, and if ā 1/(κ 2 1), ten problem (1.1) (1.5) as a weak solution φ, c L 2 (, T ; H 1 ()) H 1 (, T ; (H 1 ()) ). Moreover, if te functions S and D 2 are extended by zero outside te interval (, 1), ten a maximum principle olds for φ and c. In [7] an a priori error estimate in te L 2 (, T ; H 1 ()) norm is proved under te same condition on ā as above and assuming te existence of a sufficiently regular solution. Note tat te Lipscitz constant for S is proportional to te square of one over te interface tickness; for a typical example of te form of S we refer to te numerical section. Our goal is to perform anisotropic, adaptive finite element computations for solving efficiently (1.1) (1.5) in te general case wen ā =. Anisotropic error estimates based on te anisotropic interpolation estimates of [14, 13] are derived, using tecniques similar to tose presented in [18,

3 COMPUTATION OF A SOLUTAL DENDRITE 15 19]. Since (1.1) is strongly nonlinear, te difficulty is to relate te true H 1 error to te equation residual. An outline of te paper is as follows: in Section 2 we recall some results on anisotropic interpolation, in Section 3 we discretize te problem in space and derive anisotropic a posteriori error estimates. In Sections 4 and 5 we propose a time discretization and an adaptive algoritm based on a simplified anisotropic error indicator. Te algoritm is ten tested on some problems wit varying degree of pysical anisotropy ā in Section Anisotropic interpolation estimates For any < < 1, let T be a conforming triangulation of into triangles K wit diameter K less tan. Let V be te usual finite element space of continuous, piecewise linear functions on te triangles of T. For any triangle K of te mes, let T K : ˆK K be te affine transformation wic maps te reference triangle ˆK into K. Let M K be te Jacobian of T K, tat is, x = T K (ˆx) = M K ˆx + t K. Since M K is invertible, it admits a singular value decomposition M K = RK T Λ KP K, were R K and P K are ortogonal and were Λ K is diagonal wit positive entries. In te following we set ( ) ( ) λ1,k r T 1,K Λ K = and R λ K =, 2,K wit te coice λ 1,K λ 2,K. A simple example of suc a transformation is x 1 = H ˆx 1, x 2 = ˆx 2, wit H, tus ( ) ( ( H 1 M K =, λ 1,K = H, λ 2,K =, r 1,K =, r ) 2,K = 1) (see Fig. 2.1). 1 ˆx 2 T K x 2 r 2,K r T 2,K 1 ˆx 1 r 1,K H x 1 FIG A simple example of transformation from element ˆK to K. In te framework of anisotropic meses, te classical minimum angle condition must be avoided. However, it is required tat, for eac vertex, te number of neigbouring vertices is bounded above, uniformly wit respect to te mes size. Also, for any patc K (te set of triangles aving a vertex common wit K), te diameter of te corresponding reference patc K, tat is, K = ( K), must be uniformly bounded independently of. Tis latter ypotesis excludes some distorted patces (see Fig. 2.2). Let I : H 1 () V be a Clément or Scott Zang like interpolation operator. We now recall some interpolation results due to [14, 13]. TK 1

4 16 E. BURMAN & M. PICASSO ˆx 2 x 2 T K ˆK 1 ˆx 1 K x 1 1 H 1 H 1 H H 1 H FIG Example of an acceptable patc (top): te size of te reference patc K does not depend on te aspect ratio H/. Example of a nonacceptable patc (bottom): te size of te reference patc K now depends on te aspect ratio H/. PROPOSITION 2.1 (Lemma 2.3 and Proposition 2.2 of [13]) Tere is a constant C not depending on te mes size nor on te mes aspect ratio suc tat, for all v H 1 (), for all K T, for all edges e of K, we ave v I v L 2 (K) C(λ2 1,K (rt 1,K G K(v)r 1,K ) + λ 2 2,K (rt 2,K G K(v)r 2,K )) 1/2, ( λ1,k v I v L 2 (e) C1/2 K (r T 1,K λ G K(v)r 1,K ) + λ ) 1/2 2,K (r T 2,K 2,K λ G K(v)r 2,K ). 1,K Here G K (v) denotes te 2 2 matrix defined by ( ) v 2 G K (v) = dx T x 1 T K v v dx x 1 x 2 T v v dx T x 1 x 2 ( ) v 2. dx x 2 T

5 COMPUTATION OF A SOLUTAL DENDRITE Space discretization. Anisotropic a posteriori error estimates Te semi-discrete finite element problem corresponding to discretization in space of (1.1) (1.5) is te following. Assuming φ and c to be continuous, we set φ () = r φ, c () = r c. For eac t [, T ] we find φ (t) and c (t) in V suc tat α φ t v + A( φ ) φ v S(c, φ )v =, (3.1) c t w + D 1 (φ ) c w + D 2 (c, φ ) φ w =, (3.2) for all v, w V. Te well-posedness of te semi-discretized problem, for eac fixed, and for eac ā < 1, follows by using estimates similar to tose in [8] and te fact tat A( φ ) φ A( ψ ) ψ L φ ψ for all φ, ψ in V, wic is proved in [7]. In order to derive sarp a posteriori error estimates, we need te following result, proved in [7], corresponding to strong ellipticity of A( ). LEMMA 3.1 Let A( ) be te operator defined by (1.6) and let te convexity condition ā <1/(κ 2 1) old. Ten tere exists µ (depending on ā) suc tat, for all φ, ψ H 1 (), we ave µ (φ ψ) 2 L 2 (A( φ) φ A( ψ) ψ, (φ ψ)). (3.3) () We also need to assume tat te error in te L 2 (, T ; L 2 ()) norm converges faster tan te error in te L 2 (, T ; H 1 ()) norm, tat is, tere are two constants C > and s ], 1] suc tat for every mes T we ave ( φ φ 2 L 2 () + c c 2 L 2 () ) C( max K T λ 2,K ) 2s ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 ). (3.4) () Let us comment on tis assumption in te frame of isotropic meses, tat is to say, wen λ 1,K and λ 2,K are of order for all K T. In [7], an Euler sceme wit continuous, piecewise linear finite elements is proposed for solving (3.1) (3.2). A priori error estimates are obtained for isotropic meses wenever ā < 1/(κ 2 1). More precisely, assuming sufficient regularity of te solution, te error in te L 2 (, T ; H 1 ()) norm is sown to be O( + τ), τ being te time step. On te oter and, O( 2 + τ) convergence in te L (, T ; L 2 ()) norm is proved for isotropic meses in [16], but only in te case wen ā =. Terefore, we expect te following assumption: ( φ φ 2 L 2 () + c c 2 L 2 () ) C2s ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 () ) to old for isotropic meses wit s = 1 wen optimal convergence results old in bot L 2 (, T ; H 1 ()) and L (, T ; L 2 ()) norms. Note tat tis assumption as been used in [17] to obtain a posteriori error estimates for isotropic finite elements and in te case wen ā =. Wen considering anisotropic meses, in te above estimate sould be replaced by max K T λ 2,K, wic yields (3.4). Tis assumption is cecked numerically in Section 6.

6 18 E. BURMAN & M. PICASSO Te first step in deriving a posteriori error estimates consists in bounding te error by te equations residuals R φ (φ, c ) and R c (φ, c ), defined by R φ (φ, c ), v = R c (φ, c ), v = R φ (φ, c ), R c (φ, c ) (H 1 ()) a.e. in (, T ), ( α φ ) t v + A( φ ) φ v S(c, φ )v, (3.5) ( ) c t v + D 1(φ ) c v + D 2 (c, φ ) φ v, (3.6) for all v H 1 (), a.e. in (, T ). Here, stands for te duality pairing between H 1 () and its dual. Proceeding as in [17], we ave te following result. PROPOSITION 3.2 Let φ, c L 2 (, T ; H 1 ()) H 1 (, T ; (H 1 ()) ) be te weak solution of (1.1) (1.5) wit ā < (k 2 1) 1, let φ, c be te solution of (3.1) (3.2) and let R φ (φ, c ), R c (φ, c ) be defined by (3.5) (3.6). Assume tat φ, c L (, T ; H 2 ()) and define M 2 = D 2 L (R 2 ). Ten tere exists (depending on te constant C in (3.4) and on te Lipscitz constants of S, D 1 and D 2 ) suc tat, for every mes T satisfying max K T λ 2,K, we ave α 2 (φ φ )(T ) 2 L 2 () + µ (φ φ ) 2 2 L 2 () + µd s 8M2 2 (c c )(T ) 2 L 2 () + µd2 T s 8M2 2 (c c ) 2 L 2 () α 2 (φ φ )() 2 L 2 () + µd s 8M2 2 (c c )() 2 L 2 () + R φ (φ, c ), φ φ + µd s T R c (φ, c ), c c. (3.7) Proof. Using te fact tat φ is a weak solution of (1.1), we ave α t ) (φ φ ), φ φ + (A( φ) φ A( φ ) φ (φ φ ) ( = S(c, φ) α φ ) (φ φ ) A( φ ) φ (φ φ ). t Integrating between t = and t = T, using te definition (3.5) and te strong ellipticity (3.3), we obtain α 2 (φ φ )(T ) 2 L 2 () + µ 4M 2 2 (φ φ ) 2 L 2 () α T 2 (φ φ )() 2 L 2 () + R φ (φ, c ), φ φ + (S(c, φ) S(c, φ ))(φ φ ). (3.8)

7 COMPUTATION OF A SOLUTAL DENDRITE 19 On te oter and, using te fact tat c is a weak solution of (1.2), we ave t (c c ), (c c ) + D 1 (φ ) (c c ) 2 = ( D 2 (c, φ) φ (c c ) c t (c c )) D 1 (φ ) c (c c ) + (D 1 (φ ) D 1 (φ)) c (c c ). Tus, a time integration between t = and t = T, te fact tat D s D 1 (φ ) and te use of definition (3.6) lead to 1 2 (c c )(T ) 2 L 2 () + D s (c c ) 2 L 2 () 1 T 2 (c c )() 2 L 2 () + R c (φ, c ), c c + (D 2 (c, φ ) φ D 2 (c, φ) φ) (c c ) + (D 1 (φ ) D 1 (φ)) c (c c ). (3.9) Let β be a positive number to be cosen later. Adding (3.8) and β times (3.9) yields α 2 (φ φ )(T ) 2 L 2 () + µ (φ φ ) 2 L 2 () + β 2 (c c )(T ) 2 L 2 () + βd s (c c ) 2 L 2 () α 2 (φ φ )() 2 L 2 () + β 2 (c c )() 2 L 2 () T + R φ (φ, c ), φ φ + β R c (φ, c ), c c + (S(c, φ) S(c, φ ))(φ φ ) + β (D 2 (c, φ ) φ D 2 (c, φ) φ) (c c ) + β (D 1 (φ ) D 1 (φ)) c (c c ). (3.1) We now focus on te last tree terms of (3.1). Since S is Lipscitz tere is a constant C S independent of suc tat (S(c, φ) S(c, φ ))(φ φ ) C S ( c c L 2 () + φ φ L 2 () ) φ φ L 2 ().

8 11 E. BURMAN & M. PICASSO Terefore, by (3.4), tere is a constant C S independent of suc tat (S(c, φ) S(c, φ ))(φ φ ) C S ( max K T λ 2,K ) 2s Let us now turn to te second of te last tree terms of (3.1). We ave (D 2 (c, φ ) φ D 2 (c, φ) φ) (c c ) ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 ). (3.11) () = (D 2 (c, φ) D 2 (c, φ )) φ (c c ) D 2 (c, φ ) (φ φ ) (c c ). Since D 2 is bounded by M 2, using Young s inequality, we obtain D 2 (c, φ ) (φ φ ) (c c ) ( 1 M 2 2η (φ φ ) 2 L 2 () + η ) 2 (c c ) 2 L 2, (3.12) () were η > will be cosen in wat follows. On te oter and, using te Hölder inequality, we obtain (D 2 (c, φ) D 2 (c, φ )) φ (c c ) φ L 6 () D 2(c, φ) D 2 (c, φ ) L 3 () (c c ). (3.13) Since te imbedding of H 1 () in L 6 () is continuous and since by assumption φ L (, T ; H 2 ()), φ L 6 () is bounded a.e. in [, T ]. Moreover, since D 2 is Lipscitz we ave D 2 (c, φ) D 2 (c, φ ) L 3 () C( φ φ L 3 () + c c L 3 () ). Applying te above estimate in (3.13) yields (D 2 (c, φ) D 2 (c, φ )) φ (c c ) C We now use a Gagliardo Nirenberg inequality [6] to obtain (D 2 (c, φ) D 2 (c, φ )) φ (c c ) C ( φ φ L 3 () + c c L 3 () ) (c c ) L 2 (). ( φ φ 1/2 H 1 () φ φ 1/2 L 2 () + c c 1/2 H 1 () c c 1/2 L 2 () ) (c c ) L 2 ().

9 COMPUTATION OF A SOLUTAL DENDRITE 111 Finally, by repeated application of te Caucy Scwarz inequality and Young inequality, togeter wit (3.4), we obtain (D 2 (c, φ) D 2 (c, φ )) φ (c c ) C( max K T λ 2,K ) s/2 ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 ). (3.14) () Let us now turn to te last term of (3.1). Since D 1 is Lipscitz we ave, using te Hölder inequality, (D 1 (φ) D 1 (φ )) c (c c ) C c L 6 () φ φ L 3 () (c c) L 2 (). Proceeding as for te previous term we obtain, using te fact tat c L (, T ; H 2 ()), a Gagliardo Nirenberg inequality and (3.4), (D 1 (φ) D 1 (φ )) c (c c ) C( max K T λ 2,K ) s/2 Estimates (3.11), (3.14) and (3.15) applied in (3.1) yield α T 2 (φ φ )(T ) 2 L 2 () + µ (φ φ ) 2 L 2 () + β 2 (c c )(T ) 2 L 2 () + βd s ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 ). (3.15) () (c c ) 2 L 2 () α 2 (φ φ )() 2 L 2 () + β 2 (c c )() 2 L 2 () T + R φ (φ, c ), φ φ + β R c (φ, c ), c c ( 1 + βm 2 2η (φ φ ) 2 L 2 () + η ) 2 (c c ) 2 L 2 () + C ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 () ) (3.16) wit C = (C S (max K T λ 2,K ) 2s + C(max K T λ 2,K ) s/2 ), β, η > and C, C S independent of. We now coose β and η so tat µ βm 2 2η > and βd s βm 2η >, 2

10 112 E. BURMAN & M. PICASSO for instance we can coose β and η suc tat βm 2 2η = µ 4 and βm 2 η 2 = βd s 4, tat is, Wit tis coice, (3.16) becomes η = D s and β = µd s. 2M 2 4M 2 2 α 2 (φ φ )(T ) 2 L 2 () + 3µ 4 Let be suc tat (φ φ ) 2 L 2 () + µd s 8M2 2 (c c )(T ) 2 L 2 () + 3µD2 s 16M2 2 (c c ) 2 L 2 () α 2 (φ φ )() 2 L 2 () + µd s 8M2 2 (c c )() 2 L 2 () + R φ (φ, c ), φ φ + µd s T R c (φ, c ), c c + C 4M 2 2 ( (φ φ ) 2 L 2 () + (c c ) 2 L 2 ). (3.17) () C S 2s + Cs/2 < 3µ 4 and C S 2s + Cs/2 < 3µD2 s 16M2 2. For instance we can coose suc tat wit {( ) K 1/(2s) ( ) K 2/s } = min, 2C S 2C ( ) µ K = min 4, µd2 s 16M2 2 and we obtain (3.7) for every mes T suc tat max K T λ 2,K. Keeping in mind tat C S is inversely proportional to te interpase tickness squared we note tat we can expect te dependence of on te interpase tickness to be linear wen s = 1. Our goal now is to provide an upper bound for te equations residuals R φ (φ, c ) and R c (φ, c ) defined in (3.5) (3.6). For tis purpose, we proceed as in [18, 19] and introduce an anisotropic, explicit error estimator. For eac interior edge of T, let us coose an arbitrary normal direction n, and let [ξ] denote te jump of ξ across te edge. For eac edge of T lying on te boundary, we set [ξ] to twice te inner side value of ξ.

11 COMPUTATION OF A SOLUTAL DENDRITE 113 PROPOSITION 3.3 Let φ, c be te solution of (3.1) (3.2) and let R φ (φ, c ), R c (φ, c ) be defined by (3.5) (3.6). For eac triangle K of te mes, let λ 1,K, λ 2,K, r 1,K, r 2,K and G K ( ) be defined as in Section 2. Ten tere is a constant C depending only on te interpolation constants of Proposition 2.1 suc tat for all v L 2 (, T ; H 1 ()) we ave, using for simplicity te notation S = S(c, φ ), D 1 = D 1 (φ ), D 2 = D 2 (c, φ ), R φ (φ, c ), v C ( α φ div(a( φ ) φ ) S t K T + 1 ) 2λ 1/2 [A( φ ) φ n] L 2 ( K) 2,K L 2 (K) (λ 2 1,K (rt 1,K G K(v)r 1,K ) + λ 2 2,K (rt 2,K G K(v)r 2,K )) 1/2, R c (φ, c ), v C ( c div(d 1 c + D 2 φ ) t K T L 2 (K) + 1 [ ] ) c 2λ 1/2 D 1 n + D φ 2 n 2,K L 2 ( K) (λ 2 1,K (rt 1,K G K(v)r 1,K ) + λ 2 2,K (rt 2,K G K(v)r 2,K )) 1/2. (3.18) Proof. We will sketc te proof for te first estimate only. Te second estimate follows in te same manner. Let v be an element of L 2 (, T ; H 1 ()). Using (3.1) and (3.5) we ave R φ (φ, c ), v = R φ (φ, c ), v v v V a.e. in (, T ). From te definition of R φ (φ, c ) we ave R φ (φ, c ), v v = K T { K ( φ Integrating by parts in te last term we obtain R φ (φ, c ), v v = K T { t K ) } S(c, φ ) (v v ) + A( φ ) φ (v v ). K ( φ t Tus, te Caucy Scwarz inequality yields R φ (φ, c ), v v K T { φ t K ) div(a( φ ) φ ) S(c, φ ) (v v ) } [A( φ ) φ n] (v v ). div(a( φ ) φ ) S(c, φ ) v v L 2 (K) L 2 (K) } [A( φ ) φ n] L 2 ( K) v v L 2 ( K). (3.19)

12 114 E. BURMAN & M. PICASSO It ten suffices to coose v = I v, were I is Clément s interpolant, to use te interpolation estimates of Proposition 2.1 and te fact tat to conclude. λ 2,K ˆK K λ 1,K ˆK Putting togeter te results of Propositions 3.2 and 3.3, we obtain te main teoretical result of tis paper. THEOREM 3.4 Let φ, c be te weak solution of (1.1) (1.5), and let φ, c be te solution of (3.1) (3.2). Assume tat φ, c L (, T ; H 2 ()) and define M 2 = D 2 L (R 2 ). For eac triangle K of te mes, let λ 1,K, λ 2,K, r 1,K, r 2,K and G K ( ) be defined as in Section 2. Ten tere is a constant C depending only on te interpolation constants of Proposition 2.1 suc tat for every mes T wit max K T λ 2,K sufficiently small, we ave α 2 (φ φ )(T ) 2 L 2 () + µ 2 (φ φ ) 2 L 2 () + µd s 8M2 2 (c c )(T ) 2 L 2 () + µd2 s 8M2 2 (c c ) 2 L 2 () α 2 (φ φ )() 2 L 2 () + µd s 8M2 2 (c c )() 2 L 2 () ( + C α φ div(a( φ ) φ ) S(c, φ ) t K T L 2 (K) + 1 ) 2λ 1/2 [A( φ ) φ n] L 2 ( K) 2,K (λ 2 1,K (rt 1,K G K(φ φ )r 1,K ) + λ 2 2,K (rt 2,K G K(φ φ )r 2,K )) 1/2 + C µd2 T ( s c 4M2 2 div(d 1 (φ ) c + D 2 (c, φ ) φ ) t K T + 1 [ 2λ 1/2 D 1 (φ ) c n + D 2(c, φ ) φ ] ) n 2,K L 2 ( K) L 2 (K) (λ 2 1,K (rt 1,K G K(c c )r 1,K ) + λ 2 2,K (rt 2,K G K(c c )r 2,K )) 1/2. (3.2) Estimate (3.2) is not a usual a posteriori error estimate since φ and c are still involved in te rigt and side. We ten proceed as in [18, 19] and introduce an estimator based on superconvergent recovery, namely a Zienkiewicz Zu (Z-Z) like estimator [24, 3, 25]. More precisely, we consider te simplest Z-Z error estimator as defined in [21, 1]. For instance, te Z-Z error estimator corresponding to (φ φ ) is defined by te difference between φ and an approximate L 2 projection of φ onto V 2, namely: ( ) ( ) φ η ZZ (I Π η ZZ 1 (φ ) ) (φ ) = η2 ZZ = x 1 (φ ) ( ) φ. (3.21) (I Π ) x 2

13 Here Π : g L 2 () Π g V is defined by r ((Π g) v ) = COMPUTATION OF A SOLUTAL DENDRITE 115 g v v V, were r denotes te usual Lagrange interpolant. In oter words, from constant values of φ on triangles, we build values at vertices P using te formula ( ) ( ) φ φ K Π (P ) x P K 1 K x 1 ( ) φ = 1 K T ( ) Π (P ) φ. x 2 x 2 K K P K K T K P K K T From [2, 4, 21] we know tat for a certain class of meses (namely parallel meses) and for smoot solutions, Z-Z like error estimators are asymptotically exact. Our anisotropic error indicator corresponding to φ φ is obtained by replacing te matrix G K (φ φ ) in (3.2) by te matrix G K (φ ) defined by G K (φ ) = K K (η ZZ 1 (φ )) 2 dx η ZZ 1 (φ )η ZZ 2 (φ ) dx K η1 ZZ (φ )η2 ZZ (φ ) dx. (3.22) (η2 ZZ (φ )) 2 dx Finally, our anisotropic error indicator corresponding to φ φ is defined on eac triangle K by ( α φ div(a( φ ) φ ) S(c, φ ) t K T L 2 (K) + 1 ) 2λ 1/2 [A( φ ) φ n] L 2 ( K) 2,K K (λ 2 1,K (rt 1,K G K (φ )r 1,K ) + λ 2 2,K (rt 2,K G K (φ )r 2,K )) 1/2. (3.23) Accordingly, our anisotropic error indicator corresponding to c c is defined on eac triangle K by ( c div(d 1 (φ ) c + D 2 (c, φ ) φ ) t K T L 2 (K) + 1 [ 2λ 1/2 D 1 (φ ) c n + D 2(c, φ ) φ ] ) n 2,K L 2 ( K) 4. Time discretization (λ 2 1,K (rt 1,K G K (c )r 1,K ) + λ 2 2,K (rt 2,K G K (c )r 2,K )) 1/2. (3.24) Given an integer N, we set τ = T /N te time step, t n = nτ, n =,..., N. Assuming te initial data φ and c to be continuous, we set φ = r φ, c = r c, were r is te classical Lagrange interpolant. Ten, for eac n = 1,..., N, we find φ n in V suc tat

14 116 E. BURMAN & M. PICASSO α φn φn 1 v + τ A( φ n 1 ( S(c n 1 ) φ n v for all v V. Ten we find c n in V suc tat, φ n 1 ) + S φ (cn 1, φ n 1 )(φ n φn 1 ) ) v = (4.1) c n cn 1 w + D 1 (φ n τ ) cn w + D 2 (c n 1, φ n ) φn w = (4.2) for all v V. Equation (4.1) corresponds to an implicit Euler discretization of (3.1), wit only one Newton step at eac time step. Let C be suc tat C φ S (c, φ) for all c, φ R2. Ten problem (4.1) is well-posed for all ā < 1 wen τ α/c (by te Lax Milgram lemma). Tis limitation prevents te use of large time steps and is well known in te framework of mean curvature flows [11, 12]. 5. An adaptive algoritm We now propose an adaptive algoritm, te time step τ being fixed. Te goal is to build anisotropic triangulations T n, n = 1,..., N, suc tat te relative estimated error in te L2 (, T ; H 1 ()) norm is close to a preset tolerance TOL. Proceeding as in [19] we introduce c τ, te continuous, piecewise linear approximation in time defined by c τ (x, t) = t tn 1 τ c n (x) + tn t c n 1 (x), t n 1 t t n, x. (5.1) τ Ten our adaptive algoritm aims at building anisotropic triangulations T n, n = 1,..., N, suc tat Nn=1 K T.75 TOL (η n,k (c τ )) 2 c 1.25 TOL. (5.2) τ 2 Here η n,k (c τ ) is te simplified error indicator corresponding to (3.24), defined on [t n 1, t n ] K by t n [ ] (η n,k (c τ )) 2 1 = cτ t n 1 2λ 1/2 n 2,K L 2 ( K) (λ 2 1,K (rt 1,K G K (c τ )r 1,K ) + λ 2 2,K (rt 2,K G K (c τ )r 2,K )) 1/2, (5.3) were G K (c τ ) is defined as in (3.22). A sufficient condition to satisfy (5.2) is to build, for eac n = 1,..., N, an anisotropic triangulation T n suc tat.75 2 TOL 2 t n c τ 2 (η n,k (c τ )) TOL 2 t n c τ 2 NV n t n 1 for all triangles K T n, were NVn is te number of vertices of te mes T n. We ten proceed as in [18, 19] to build suc an anisotropic mes, using te BL2D mes generator [5]. NV n t n 1

15 COMPUTATION OF A SOLUTAL DENDRITE 117 FIG Typical profiles of te pase field (left) and concentration field (rigt). Te pase field as values zero or one, except in te pase cange region. Te concentration field canges rapidly across te pase cange region, but may also vary outside te pase cange region. Te reasons for using te simplified error indicator (5.3) instead of (3.23) and (3.24) are te following. First, we ave disregarded te error estimator related to te pase field φ because, wen computing solutal dendrites, altoug bot φ and c vary strongly in te small region corresponding to te solid-liquid interface, te function c may also vary in oter regions, wereas φ does not (see Fig. 5.1). Secondly, in order to simplify te implementation of te error indicator, we ave kept only te jump of te gradients across te edge, wic is te generic term in explicit error estimators. Teoretical arguments for suc a coice in te frame of elliptic problems can be found in [9]. 6. Numerical results We consider te model of [22], te notations being tose of [15]. Te source term S in (1.1) is defined by S(c, φ) = 1 λ 2 φ(1 φ)(1 2φ) + 5m ( ) l λγ φ2 (1 φ) 2 c 1 φ + kφ c l if φ 1, wereas S(c, φ) = if φ < or φ > 1. Here λ is te tickness of te solid-liquid interface, m l is te liquidus slope in te pase diagram, Γ is te Gibbs Tomson isotropic coefficient, c l is te liquid concentration in te pase diagram, and k is te pase diagram partition coefficient (tus c s = kc l, were c s is te solid concentration in te pase diagram). Te coefficient α in (1.1) equals 1/(µ k Γ ), were µ k is te interface kinetic coefficient. Te first term in te definition of S(, ) is noting but te derivative of a double-well potential tat forces φ to ave values zero or one, except in te pase cange region. In te limit wen λ goes to zero, te second term in te definition of S(, ) links te normal velocity of te solid-liquid interface, some anisotropic measure of te interface curvature, and te concentration field. Te function D 1 in (1.2) is given by D 1 (φ) = D s + 1 φ 1 φ + kφ (D l D s ) if φ 1, wereas D 1 (φ) = D l if φ < and D 1 (φ) = D s if 1 < φ. Here D s and D l are te solid and liquid diffusion coefficients. Finally, te function D 2 in (1.2) is given by (1 k)c(1 c) D 2 (c, φ) = D 1 (φ) 1 φ + kφ if c 1, wereas D 2 (c, φ) = if c < or c > 1. All te pysical parameters are given below in te international MKSA unit system.

16 118 E. BURMAN & M. PICASSO 6.1 A simple test case wit exact solution Our first goal is to validate numerically assumption (3.4) in te frame of anisotropic meses. For tis purpose, we set te computational domain to = [.2,.2] 2 and we add source terms in (1.1) (1.2) so tat φ and c are given by φ(x 1, x 2, t) = c(x 1, x 2, t) = 1 tan((x 1 vt)/δ), 2 were v = and δ = 1 5. Diriclet boundary conditions are prescribed on te vertical sides of, and omogeneous Neumann boundary conditions on te orizontal sides. Te isolines of φ and c at time t = and t =.5 are sown in Fig Te pysical parameters D 1 and D 2 involved in te definition of S are given in Table 6.1, te time step is τ = and is small enoug so tat te error due to time discretization is negligible. Meses wit strong anisotropy are used to validate assumption (3.4), an example of mes being sown in Fig TABLE 6.1 Test case wit exact solution: parameters used for te computations λ m l Γ c s c l D s D l µ k TABLE 6.2 Various convergence results for te travelling wave solution Anisotropic meses refined in bot orizontal and vertical directions 1 2 e L 2 e H 1 ei ZZ ei A Anisotropic meses refined in orizontal direction only 1 2 e L 2 e H 1 ei ZZ ei A Anisotropic meses refined in vertical direction only 1 2 e L 2 e H 1 ei ZZ ei A In Table 6.2, errors in te L 2 (, T ; L 2 ()) and L 2 (, T ; H 1 ()) norms (resp. e L 2 and e H 1) are reported wen using anisotropic meses (1 2 denotes te mes size in orizontal and vertical directions). Also, te effectivity indices ei ZZ and ei A corresponding to te Zienkiewicz Zu error estimator (3.21) and our simplified error indicator (5.3) are sown. Clearly, wen te mes is refined

17 COMPUTATION OF A SOLUTAL DENDRITE 119 FIG Test case wit exact solution: isolines (from.1 to.9) of φ and c at time t = (left) and t =.5 (rigt). FIG Test case wit exact solution: te mes wit size 1 2 =.5.1. in te orizontal direction, assumption (3.4) olds wit s = 1 since te L 2 (, T ; L 2 ()) error converges at order two and te L 2 (, T ; H 1 ()) error at rate one. However, wen te mes is refined in te wrong (vertical) direction, ten te error does not decrease and (3.4) does not old. 6.2 Computations wit small anisotropy We now consider te following pysical situation. At initial time, te computational domain is liquid, wit omogeneous concentration.2. Ten a circular solid seed of diameter and concentration.15 is placed at te center of. Te pysical parameters are now given in Table 6.3 and are taken from [15, Table 1, column B], except c s and c l. TABLE 6.3 Parameters used for te computations λ m l Γ c s c l D s D l µ k

18 12 E. BURMAN & M. PICASSO FIG Computations wit small anisotropy, ā =.4. Adapted meses (left column), concentration isovalues (middle column) and pase isovalues (rigt column), from t = to t = 1 s, wit TOL =.625 (6.25% estimated relative error). Row 1: t =.5 s, 6874 vertices. Row 2: t =.25 s, 1449 vertices. Row 3: t =.5 s, 1717 vertices. Row 4: t =.75 s, vertices. Row 5: t = 1 s, vertices.

19 COMPUTATION OF A SOLUTAL DENDRITE FIG Computations wit small anisotropy, ā =.4. Concentration profile along te diagonal of te computational domain at time.5 (solid line) and 1 s (dotted line). FIG Computations wit small anisotropy, ā =.4. Zooms of te results at final time. Top row: isolines of φ from.1 to.9. Bottom row: mes.

20 122 E. BURMAN & M. PICASSO FIG Computations wit small anisotropy, ā =.4. Concentration isovalues at final time t = 1 s wit various values of te tolerance TOL. From left to rigt: TOL =.25, final adapted mes 1987 vertices; TOL =.125, 673 vertices; TOL =.625, vertices; TOL =.3125, vertices. We first present computations in te case wen te anisotropy parameter ā is small. We set te number of dendrite arms κ = 4 and coose ā =.4 so tat ā < 1/(κ 2 1).667. Te time step is τ = and te final time is t = 1, making te total number of time steps 2. In Fig. 6.3, te adapted meses, concentration and pase fields corresponding to an adaptive computation wit tolerance TOL =.625 (6.25% estimated relative error) are reported. Te concentration profile along te diagonal of te computational domain is reported in Fig Te concentration c and pase φ appear to be smoot, but exibit strong gradients across te solid-toliquid transition zone, terefore te mes is strongly refined in te neigbourood of te solid-liquid interface. Zooms of te results are sown in Fig Te adaptive algoritm generates 4 meses from initial to final time. Te computation takes about 4 ours on a Pentium III 1.2 Gz PC, wit a required memory of less tan 3 Mb. Convergence of te finite element solution wit respect to te mes size is reported in Fig Te maximal anisotropy ratio of te mes during a simulation was approximately 3, witout any a priori upper bound imposed by te adaptive metod. 6.3 Computations wit large anisotropy We now coose te anisotropy parameter ā > 1/(κ 2 1).667, namely ā =.1. In tis case tere are no known existence results for te system in L 2 (, T ; H 1 ()); in fact, only solutions in te Young measure sense are known to exist [8, 1]. However, due to te Lax Milgram Lemma, te linearized problem (4.1) remains well-posed. FIG Computations wit large anisotropy, ā =.1. Concentration isovalues from t = to t = 1 s, wit TOL =.625 (6.25% estimated relative error). Column 1: t =.5 s, 5138 vertices. Column 2: t =.25 s, 1253 vertices. Column 3: t =.5 s, 258 vertices. Column 4: t =.75 s, 2865 vertices. Column 5: t = 1 s, vertices. In Fig. 6.7, te concentration fields corresponding to an adaptive computation wit tolerance TOL =.625 are reported. A zoom of te results at final time, Fig. 6.8, sows tat te gradient is discontinuous in some regions, for instance close to te dendrite tip. We explain tis penomenon in te following manner.

21 COMPUTATION OF A SOLUTAL DENDRITE 123 FIG Computations wit large anisotropy, ā =.1. Zooms of te results at final time. Top row: isolines of φ from.1 to.9. Bottom row: mes. Let us consider te functional g( φ) dx, were g(ξ) = a 2 (θ(ξ)) ξ 2 /2 and wose Frécet derivative is div(a( φ) φ). In Fig. 6.9, we ave plotted isolines of te anisotropic function ξ a 2 (θ(ξ)) ξ 2 wit small and large anisotropy. As proved in [8], tis function is convex if ā < 1/(κ 2 1), tis being te case wen ā =.4, but not wen ā =.1. However, wen ā > 1/(κ 2 1), te function ξ a 2 (θ(ξ)) ξ 2 is locally convex wenever ξ satisfies were θ m solves cos θ(ξ) cos θ m and sin θ(ξ) sin θ m, a(θ m ) + a (θ m ) cos θ m sin θ m =. Wen ā =.1, tis yields an angle θ m degrees as reported in Fig In tis zone te functional g(ξ) coincides wit its convex envelope Qg(ξ) given by

22 124 E. BURMAN & M. PICASSO 1 1 θ m FIG Isolines.1,.2,.3,.4,.5 of te function ξ a 2 (θ(ξ)) ξ 2 wen ā =.4 (left) and ā =.1 (rigt). FIG Scematic representation of te anisotropic function ξ g(ξ) = a 2 (θ(ξ)) ξ 2 (solid line) and te corresponding convex envelope ξ Qg(ξ) (dotted line). Te angle reported in te figure corresponds to θ m suc tat a(θ m ) + a (θ m ) cos θ m sin θ m =. Qg(ξ) = a 2 (θ(ξ)) ξ 2 2 if cos θ(ξ) cos θ m and sin θ(ξ) sin θ m, a 2 (θ(ξ))ξ1 2 2 cos 2 θ m if cos θ(ξ) > cos θ m, a 2 (θ(ξ))ξ2 2 2 cos 2 θ m if sin θ(ξ) > sin θ m ; (6.1) see Fig. 6.1 for a scematic representation of Qg( ). Turning once again to Fig. 6.8, we observe tat te gradient jump corresponds to an angle approximately 2θ m in te regions were te concentration gradient is discontinuous and tat te oscillations do not seem to take place on te scale of te mes but are distributed wit different distances. Tus, as expected, φ avoids te ig energy directions were cos θ( φ) > cos θ m or sin θ( φ) > sin θ m, tat is to say, te region were te function ξ a 2 (θ(ξ)) 2 ξ 2 is nonconvex. It is owever very difficult to draw any conclusions weter or not te dendritic brances exibit microstructure. In order to examine tese effects closer we propose to make computations using te convexified problem wit div(a( φ) φ) replaced by te Frécet derivative of

23 COMPUTATION OF A SOLUTAL DENDRITE 125 Qg( φ) dx + λ 2 2,K φ 2 dx, K T K were λ 2,K is defined as in Section 2. Te last term in te above convexified functional corresponds to a mes dependent artificial viscosity added to counter te loss of coercivity in te zones were g(ξ) > Qg(ξ). In Fig we ave reported results corresponding to te convexified functional wit ā =.1 and TOL =.625. Comparing wit Fig. 6.8, we observe tat te general form of te dendrite brances remains te same, wit sarp dendrite tips and sarp corners at te base of te brances. However te regions on te sides of te brances were φ is nonsmoot ave disappeared. Here te gradient of te solution of te convexified problem φ c points in te ig energy direction, so tat (g( φc ) Qg( φc )) dx >. Furtermore tis last quantity does not decrease wit decreasing tolerance. Tis numerical result suggests tat after a certain time T tere is no H 1 () solution to problem (1.1) (1.5) wen ā > 1/(κ 2 1), but only Young measure solutions, wit microstructure appearing on te sides of te dendrite brances. In oter words, te approximating sequence converges to a solution of te problem only in te sense of Young measures, were an admissible Young measure (a priori nonunique) can be derived from te corresponding convexification (6.1). FIG Computations wit large anisotropy, ā =.1. Convexified functional. Zooms of te results at final time. Top row: isolines of φ from.1 to.9. Bottom row: mes.

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