APPROXIMATION OF CRYSTALLINE DENDRITE GROWTH IN TWO SPACE DIMENSIONS. Introduction

Transcription

1 Acta Mat. Univ. Comenianae Vol. LXVII, 1(1998), pp APPROXIMATION OF CRYSTALLINE DENDRITE GROWTH IN TWO SPACE DIMENSIONS A. SCHMIDT Abstract. Te pase transition between solid and liquid in an undercooled liquid leads to dendritic growt of te solid pase. Te problem is modelled by te Stefan problem wit a modified Gibbs-Tomson law, wic includes te anisotropic mean curvature corresponding to a surface energy tat depends on te direction of te interface normal. A finite element metod for discretization of te Stefan problem is described wic is based on a weak formulation of te anisotropic mean curvature flow. Numerical experiments wit a nearly crystalline anisotropy are presented. Introduction Pase transition between liquid and solid pase wit dendritic growt of te solid pase inside an undercooled liquid can be modeled by te Stefan problem wit a modified Gibbs-Tomson law wic accounts for anisotropic surface tension and kinetic undercooling [8]. We denote by θ(x, t) te temperature in a domain Ω R n and by Γ(t) te moving free boundary wit normal velocity V and anisotropic mean curvature C α. Te sign of tese two scalar quantities is cosen suc tat for a growing convex solid, bot V and C α are positive. Let ν denote te outer normal to te solid region. Te system of equations consists of te eat equation (1) in bot solid and liquid regions, te Stefan condition (2), and te Gibbs-Tomson relation (3). Assuming for simplicity unit termal conductivities in bot pases and unit latent eat, te system reads (1) (2) (3) θ t θ = 0 in Ω \ Γ(t), [ ] θ + V = 0 on Γ(t), ν θ + β(ν)v + C α = 0 on Γ(t), togeter wit initial conditions for θ and Γ as well as boundary conditions for te temperature. We assume tat Γ does not meet te domain s boundary and Received November 11, Matematics Subject Classification (1991 Revision). Primary 65C20, 65N30, 65N50, 80A22, 80-08, 35K55, 35R35.

2 58 A. SCHMIDT will not cange its topology but stays equivalent to te initial interface, like a spere S n 1. Te modified Gibbs-Tomson relation (3) enables te growt of dendritic structures, as it incorporates te influence of an anisotropic mobility and an anisotropic interface energy, wic can be motivated by te underlying crystalline structure of te solid. For tis reason one assumes tat te kinetic coefficient β and te interface energy depend only on te direction of te surface normal ν. Tis is appropriate for te case of single solid crystals. Besides te normal velocity of te interface, te Gibbs-Tomson relation includes an anisotropic mean curvature (or weigted mean curvature) C α, wic reduces to te mean curvature of te interface (scaled by some material constant) in te case of constant isotropic interface energy. Taylor [12] compares several different formulations of (weigted) mean curvature for smoot and crystalline anisotropy, owever, a weak formulation wic is well suited for manifolds of weak regularity (only Lipscitz or H 1 ) and a finite element approximation is not directly included. In [13], geometric models for crystal growt based on tese curvature formulations are presented. Bellettini and Paolini [3] treat mean curvature flow in te context of Finsler geometries, were not only te interface but also te surrounding space exibits some anisotropic structure. Tis leads to somewat different equations. Rybka considers te Stefan problem wit purely crystalline interface motion and studies existence and uniqueness of solutions as well as asymptotic beaviour for vanising solid pase. Dziuk [4] uses a weak formulation of te mean curvature for a finite element approximation of motion by mean curvature. Convergence of te semidiscrete metod for curves (discretized in space) was sown in [5]. Recently, tis convergence result was extended to te case of smoot anisotropies [6]. In [2], [11] we presented a finite element metod for te approximation of dendritic growt in two and tree space dimensions, were we use a Gibbs-Tomson law of te form θ + β(ν)v + α(ν)c Γ = 0 on Γ(t), were C Γ is te usual mean curvature. Tis includes te general case of weigted mean curvature for smoot anisotropies in two dimensions, but not in tree dimensions. Te metod is based on Dziuk s discretization of mean curvature evolution [4]. Recently, Veeser [14] proved convergence and error estimates for te corresponding semidiscretization in space in two dimensions. Here we consider te twodimensional situation and describe te extension of te above mentioned finite element metod to te case of true anisotropic interface energy wic enables te finite element approximation of te general weigted mean curvature case also in tree dimensions as well as te consideration of non smoot anisotropies. First numerical results for nearly crystalline anisotropy in two space dimensions are presented.

3 APPROXIMATION OF 2d CRYSTALLINE DENDRITE GROWTH Motion by Anisotropic Curvature In tis section we study te first variation of te anisotropic surface energy, wic is connected to te motion by anisotropic mean curvature via te corresponding gradient flow. Let Γ be a smoot closed curve embedded in R 2 and ν te outer unit normal vector field on Γ. Te isotropic surface energy E (or measure) of Γ is given by E(Γ) := 1 do. Te first variation of E involves te mean curvature C Γ of Γ, as we will see below, and te mean curvature flow is te gradient flow for E. Motion by anisotropic curvature can be derived in a similar way, as we describe now. Given an anisotropic surface energy weigt function α : S 1 R, α > 0, wic depends only on te normal direction, te anisotropic surface energy E α (Γ) is defined as E α (Γ) := α(ν) do. Let α be extended 1-omogeneously to te wole R 2 suc tat α(λν) = λα(ν) for all λ 0. We want to derive weak formulas of te first variation of E α and te corresponding gradient flow. Let Γ be given via a periodic parametrization x : I R R 2 wit do(x s ) = > 0 te surface element at x and ν(x s ) = xs / x s te unit normal to Γ (were denotes rotation by π/2). For a periodic perturbation ϕ : I R 2 of x, te first variation of E α (x) := E α (Γ) in direction of ϕ is E α (x), ϕ = d dε E α (x + εϕ) ε=0 = d dε α(ν(x s + εϕ s )) do(x s + εϕ s ) ε=0 I = Dα(ν(x s )) Dν(x s ) ϕ s do(x s ) + α(ν(x s )) D(do)(x s ) ϕ s, I were we only used te cain rule. Te gradient Dα(ν) is sometimes called Can- Hoffman ξ-vector in te literature. Using te explicit formulas for ν and do, te variation can be expressed in te following way (were, denotes te euclidian scalar product of vectors in R 2 ): E α (x), ϕ = = I I ( x Dα s ( Dα ( x s ), Γ Γ 2 ϕ s, x s ), ϕ s x s x s xs, ϕ s + α ( x + α s ( x s ) x s, ϕ s ) x s, ϕ s ),

4 60 A. SCHMIDT since ϕ s x s x ϕ s s, xs x = s ϕ s x s x s ϕ s, x s x = xs s x s ϕ s, xs x s. Tis leads to te following non parametric formulation for a perturbation ϕ : Γ R 2 : (4) E α (Γ), ϕ = Γ ( ) Dα(ν), id id, ϕ + α(ν) id, ϕ do, were is te tangential gradient (or covariant derivative) on Γ and id: Γ R 2 is te identity vector field on R 2 (te embedding of Γ). Tis formulation is well suited for an isoparametric finite element discretization, see Section 2. In te isotropic case, α = const, integration by parts gives on closed curves or for perturbations wic vanis on Γ E α(γ), ϕ = α id, ϕ do, Γ were is te Laplace-Beltrami operator on Γ. Now, id is just te mean curvature vector C Γ of Γ, wic gives E α(γ) = α C Γ. Tus, te gradient flow for E α is in te isotropic case just mean curvature flow V = α C Γ, were te velocity of a moving manifold is equal to α times mean curvature. Wit te given sign convention for te scalar quantities V and C Γ, it olds νv = V = α C Γ = ανc Γ. Similarly, te gradient flow for anisotropic surface energy is te anisotropic curvature flow (or weigted mean curvature flow) for all smoot perturbations ϕ, and wit Γ V, ϕ do = E α(γ), ϕ C α := E α(γ), we ave te anisotropic curvature flow equation V = C α. Depending on te smootness of α and Γ, tis equation olds only in a weak sense. For smoot α it is easy to sow tat tis leads (for curves) to V = ( α + α ) C Γ, if α(ν) = α(γ), were γ = (ν, e 1 ) is te angle between te normal and te e 1 coordinate axis. Te connection between anisotropic curvature flow V = C α and te Gibbs- Tomson relation βv = C α θ is given after conversion to scalar quantities (by multiplying wit ±ν, suc tat te scalar mean curvature for a convex solid pase is positive, and te scalar velocity for an expanding solid is positive), and by using te temperature as an additional forcing term.

5 APPROXIMATION OF 2d CRYSTALLINE DENDRITE GROWTH 61 Using te 1-omogeneity of α, we can derive a simpler expression for te first variation, wic is not so well suited for a finite element formulation of anisotropic mean curvature flow like (4), but can be used to evaluate te anisotropic curvature. α(xs + εϕ s ) = Dα(xs ), ϕ s ε=0 E α (x), ϕ = d dε = I Dα I ( x s Γ ), ϕ s, wic leads to te intrinsic equation (5) E α(γ), ϕ = Dα(ν), ϕ do. Wulff Sapes For a given anisotropy α, te Wulff sape W α is defined as W α = { x R n x, ν α(ν) ν S n 1}. It is known tat anisotropic curvature flow of an initially convex curve converges to a self-similarly srinking Wulff sape [7]. If te polar grap of 1/α (te Frank diagram) is strictly convex (i.e. α + α > 0), ten W α is strictly convex, too. For crystalline anisotropies of k-fold symmetry, (6) α k (γ) = ε max { cos(γ 2πj/k); j = 0,..., k 1 }, α k + α k is concentrated at discrete angles and vanises almost everywere, and te corresponding Wulff sape is a regular k-polygon. For crystalline curvature flow witout driving forces, te Wulff sape faces are te only directions wic appear in an evolving manifold. We will see in te numerical results tat Wulff sape faces are preferred in te free boundary problem, too. 2. A Finite Element Metod for Dendrite Growt Let V H 1 (Ω) and V,0 = V H0 1 (Ω) be finite element spaces of continuous piecewise polynomial functions on a regular triangulation of Ω. After time discretization wit time step size τ m = t m t m 1, we denote by θ m V te discrete temperature and by Γ m a continuous, piecewise polynomial discrete interface at time t m. Let W m H1 (Γ m ) be te isoparametric finite element space of piecewise polynomial functions on Γ m. Te discretization of te interface is totally independent from te temperature mes, and coupling between bot is only done via integrals over te free boundary. For discretization of te interface evolution, we proceed according to [4] by parametrizing Γ m over Γm 1 via x m : Γm 1 R 2, x m m 1 (W ) 2. After multiplying te Gibbs-Tomson relation (3) by te normal ν in order to get a vector I

6 62 A. SCHMIDT valued equation, we use equation (4) for a variational formulation and derivation of a finite element metod. Tis leads to a igly nonlinear equation for x m, as te normal ν m (wic is a parameter to α and Dα) and te tangential derivative on Γ m all depend on te parametrization. To avoid tis nonlinear problem, we use a linearization wic uses te normal vector field ν m 1 and tangential derivatives corresponding to te old interface Γ m 1. Tis leads to te following semi-implicit, linearized problem for x m m 1 (W ) 2 : β xm id ϕ + Dα(ν m 1 ), id x m, ϕ + α(ν m 1 ) x m, ϕ Γ m 1 τ m = θ m 1 ν m 1, ϕ, ϕ (W m 1 ) 2. Γ m 1 As we are using isoparametric finite elements, we can define te new discrete interface by Γ m := xm (Γm 1 ). After integration by parts in te solid and liquid subdomains wit a test function ψ V, te eat equation (1) leads to a parabolic problem for θ m V wit a rigt and side given by a line integral: Ω θ m θm 1 ψ + θ m, ψ = τ m Γ m V ψ, ψ V. We use te Gibbs-Tomson relation (3) to replace te velocity V and get Ω θ m θm 1 ψ + θ m, ψ + τ m Γ m 1 β θm ψ = Γ m 1 β C αψ, ψ V, wic gives a more stable discretization by including te temperature at te interface implicitely. Te rigt and side is evaluated by calculating C α, 1 β ψ ν = E α(γ m ), 1 β ψ ν. Instead of using ere (4), we can take advantage of te simpler expression (5). Te two finite element metods described above are coupled to a metod for simulation of te free boundary problem. Te simplest coupling is te following: In eac time step, te coupled metod first computes te new discrete interface Γ m using te old temperature θ m 1. Te new temperature θ m is ten computed by using tis new interface. More elaborate couplings are possible, using extrapolation tecniques or iger order time discretizations. Adaptive Metods. Instead of using one single finite element space V for all time steps, te triangulation is adapted to te actual solution based on a posteriori

7 APPROXIMATION OF 2d CRYSTALLINE DENDRITE GROWTH 63 error indicators and local refinement and coarsening of te mes, see [11]. Tis leads to finite element spaces V m for te discrete temperature θm. An additional mes transfer operator I m is needed in order to project te old temperature θm 1 from V m 1 to te new space. For adaptive modifications of te temperature mes, we use local refinement and coarsening of triangles based on bisection metods [1]. Tis leads to compatible triangulations after mes canges suc tat an interpolation error θ m 1 I mθm 1 > 0 is introduced only during coarsening, not by local refinement operations. Te interface discretization as to be adapted, too. During te evolution, large parts of te interface can grow out of initially small interface regions. For curves, tis adaption can be done by bisecting polygon pieces, wic grow longer tan a specified tresold lengt. Tis simple criterion generates usually good numerical results. More elaborate error indicators for local adaption of te polygon can be derived. 3. Numerical Simulations wit Nearly Crystalline Anisotropy Simulations wit smoot anisotropy were presented in [2], [11]. Our aim ere is to approximate dendritic growt wit te crystalline anisotropy α k (ν) given in terms of α k (γ), compare (6). Te equation for anisotropic mean curvature flow (4) makes sense if Γ is Lipscitz, because ten te Sobolev space H 1 (Γ) is well defined [15] and id Γ H 1 (Γ) olds. Te Lipscitz continuity of our discrete interfaces Γ m is guaranteed by te definition via piecewise polynomial continuous parametrizations. For te anisotropy, additional regularity is needed in order to be able to evaluate α(ν) and Dα(ν). As Γ is only Lipscitz, te normal vector field ν is only L ; te discrete normal ν m is piecewise smoot but jumps at discrete points. For integration of Dα(ν), we now need α C 1 -regularity. Tis means tat te crystalline anisotropy α k, wic is only Lipscitz, can not be used directly. We use a periodic regularization α k,δ (γ) C 1 (R) wic is defined in te following way. Set α k (γ) α k,δ (γ) = δ + p(γ) (2j + 1)π if γ > δ, j = 0,..., k 1, k oterwise, were δ > 0 is te regularization parameter and p(γ) are pieces of quadratic polynomials suc tat α k,δ C 1 olds, see Figure 1. Corresponding Wulff sapes are sown in Figure 2. Te simulations presented below were done wit δ = 0.1. Numerical results. We present results from two simulations. Bot are done wit piecewise quadratic finite element discretizations of te temperature and te

8 64 A. SCHMIDT delta=0.0 delta= gamma Figure 1. Crystalline anisotropy α 6 (solid) and regularization α 6,δ (dotted) for δ = 0.1. Figure 2. Wulff sapes for crystalline anisotropy α 6 (left) and regularization α 6,0.1 (rigt). interface curve. Te domain is Ω = ( 4, +4) 2, te initial curve is a circle of radius R 0 = 0.1. Te anisotropic coefficients in te Gibbs-Tomson relation are α = α 6,0.1 and β(ν) = 0.01 (isotropic kinetic coefficient). Boundary value for te temperature is θ = 0.5 on Ω. In te first simulation, te parameter for temperature mes adaption is tol = We use a fixed time step size of τ = Te simulation runs over a time interval [0, 2.25] of 9000 time steps. Te curve pieces are allowed to ave a lengt of 0.005, wit bisection refinement if tey grow longer. Figure 3 sows te evolving interface at time t = 0.0, 0.25, 0.5,..., 2.25 (after 0, 1000,..., 9000 time steps). Te outer square depicts te domain boundary Ω. In Figure 4, a zoom to te upper rigt part of te interfaces is sown. It sows clearly tat te Wulff sape faces are preferred directions for te interface, too. But in regions of ig velocity, especially near te dendrite tips, were te temperature is more negative and tus te forcing term in te mean curvature equation is larger, intermediate directions appear, too. Zooms of te locally adapted temperature meses at t = 0.25, 2.0 are sown in Figures 5 and 6.

9 APPROXIMATION OF 2d CRYSTALLINE DENDRITE GROWTH 65 Figure 3. Example 1 Interface curves at t = 0.0, 0.25, 0.5,..., Figure 4. Example 1 zoom of interface curves at t = 0.0, 0.25, 0.5,..., 2.25.

10 66 A. SCHMIDT Figure 5. Example 1 zoom of temperature mes at t = Figure 6. Example 1 zoom of temperature mes at t = 2.0.

11 APPROXIMATION OF 2d CRYSTALLINE DENDRITE GROWTH 67 Te complete meses consist of 3316 and triangles. Te meses are igly refined were te temperature gradient exibits a large variation. Tis gradient varies most rapidly near fast moving parts of te interface, compare te Stefan condition (2). Tis results in te local refinement near te interface. Te second simulation is included to demonstrate some numerical effects, wic may appen if time step size or mes size are cosen too coarse. Te simulation is done wit a fixed time step size τ = and te parameter for temperature mes adaption is tol = All oter parameters are te same as for te first simulation. Te interface curve at time t = 1.25 is sown in Figure 7. Figure 7. Example 2 too large time steps may result in interface topology canges. Due to te larger noise introduced by te coarser numerical approximation, side brances of te interface develop more easily. Additionally, te interface curve intersects itself, wic is equivalent to a cange of topology of te solid pase (te code did not ceck for suc singularities). In our numerical experiments, suc self intersections occur only wen te time step size is too large; time step size reduction always eliminates suc singularities.

12 68 A. SCHMIDT References 1. Bänsc E., Local mes refinement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg. 3 (1991), Bänsc E. and Scmidt A., A finite element metod for dendritic growt, Computational crystal growers worksop, AMS Selected Lectures in Matematics, (J. E. Taylor, ed.), 1992, pp Bellettini G. and Paolini M., Anisotropic motion by mean curvature in te context of Finsler geometry, Hokkaido Mat. J. 25 (1996), Dziuk G., An algoritm for evolutionary surfaces, Numer. Mat. 58 (1991), , Convergence of a semi-discrete sceme for te curve sortening flow, Mat. Models Metods Appl. Sci. 4 (1994), , Discrete anisotropic curve sortening flow, preprint, Freiburg, Gage M. E., Evolving plane curves by curvature in relative geometries, Duke Mat. J. 72 (1993), Gurtin M. E., Toward a nonequilibrium termodynamics of two-pase materials, Arc. Ration. Mec. Anal. 100 (1988), Rybka P., A crystalline motion: Uniqueness and geometric properties, SIAM J. Appl. Mat. 57 (1997), , Crystalline version of te Stefan problem wit Gibbs-Tomson law and kinetic undercooling, to appear in Advances in Diff. Equ Scmidt A., Computation of tree dimensional dendrites wit finite elements, J. Comput. Pys. 125 (1996), Taylor J. E., Mean curvature and weigted mean curvature, Acta metall. mater. 40 (1992), Taylor J. E., Can J. W. and Handwerker C. A., Geometric models of crystal growt, Acta metall. mater. 40 (1992), Veeser A., Error estimates for semidiscrete dendritic growt, in preparation. 15. Wloka J., Partial differential equations, Cambridge University Press. A. Scmidt, Institut für Angewandte Matematik, Universität Freiburg, Germany