International Series of Numerical Mathematics, Vol. 154, 327–337 c 2006 Birkh¨
auser Verlag Basel/Switzerland

Adaptive Finite Elements with High Aspect Ratio for Dendritic Growth of a Binary Alloy Including Fluid Flow Induced by Shrinkage Jacek Narski and Marco Picasso Abstract. An adaptive phase field model for the solidification of binary alloys in two space dimensions is presented. The fluid flow in the liquid due to different liquid/solid densities is taken into account. The unknowns are the phase field, the alloy concentration and the velocity/pressure in the liquid. Continuous, piecewise linear finite elements are used for the space discretization, a semi-implicit scheme is used for time discretization. An adaptive method allows the number of degrees of freedom to be reduced, the mesh triangles having high aspect ratio whenever needed. Numerical results are presented for dendritic growth of four dendrites.

1. Introduction In recent years, considerable progress has been made in numerical simulation of solidification processes at microscopic scale [1]. Although sharp interface [2, 3] and level-set models [4, 5] have proved to be efficient, the phase field method emerged as a method of choice in order to simulate dendritic growth in binary alloys [6, 7, 8, 9, 10, 11]. In phase field models, the location of the solid and liquid phases in the computational domain is described by introducing an order parameter, the phase field, which varies smoothly from one in the solid to zero in the liquid through a slightly diffused interface. The main difficulty when solving numerically phase field models is due to the very rapid change of the phase field (and also of the concentration field in alloys) across the diffuse interface, whose thickness has to be taken very small (between 1 and 10 nm) to correctly capture the physics of the phase transformation. A high spatial resolution is therefore needed to describe the smooth transition. In order to reduce the computational time and the number of grid points adaptive isotropic finite elements [12, 13] have been used. Further reduction of the number of nodes has been achieved using adaptive finite elements with high aspect ratio [14, 15]. Jacek Narski is supported by the Swiss National Science Foundation.

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The influence of inter-dendritic liquid flow has already been taken into account in dendritic simulations [16, 17, 18, 19, 20]. Also, the inter-dendritic liquid flow induced by shrinkage – that is to say the flow induced by the fact that solid and liquid densities are different – has been considered [21, 22, 23, 24, 25]. The goal of this paper is to take into account the inter-dendritic liquid flow due to shrinkage, using adaptive finite elements with high aspect ratio as in [14, 15]. Numerical results show that the method is capable of predicting micro-porosity. The outline of the paper is the following. In the following section, we present the model and briefly discuss the numerical method. Numerical results are presented in section 3, where the pressure drop due to shrinkage is observed during solidification of four dendrites.

2. The model The equations governing the solidification process are derived using a volume averaging technique in a similar way as in [16, 17]. The key idea is to develop two sets of equations (for the solid and liquid phases) and transform them into one set using averaging over small volume and introducing average quantities. In the following we present and discuss the averaged mass, momentum and species conservation equation for binary alloy undergoing a solid/liquid phase transition. As in [21, 22, 23, 24, 25], we take into account the fact that the solid and liquid densities are different. 2.1. Mass conservation The solidification of a binary alloy in a bounded domain Ω of R2 between time 0 and tend is considered. Let φ : Ω × (0, tend ) → R be the phase field describing presence of solid (φ = 1) or liquid (φ = 0). The phase field φ varies smoothly but rapidly from zero to one in a thin region of width δ, the so-called solid/liquid diffused interface. Let ρs and ρl be the constant solid and liquid densities (for most alloys ρl < ρs ). Then the average density ρ : Ω × (0, tend ) → R is defined by ρ = ρs φ + ρl (1 − φ).

(1)

Let vs , vl be the solid and liquid velocities, respectively. In this model, the solid velocity vs is a known constant (in most cases vs = 0), whereas vl is unknown. Then, the average velocity v : Ω × (0, tend ) → R2 is defined by ρv = ρs φvs + ρl (1 − φ)vl .

(2)

Averaging the mass conservation equation in the solid and liquid regions yields    ∂ ρs φ + ρl (1 − φ) + div ρs φvs + ρl (1 − φ)vl = 0, ∂t which can be rewritten, using (1) and (2): ∂ρ + div (ρv) = 0. ∂t

(3)

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It should be noted that, in the sharp interface limit (that is to say when the width of the solid-liquid interface δ tends to zero), then the phase field φ is the characteristic function of the solid so that the density ρ becomes a step function and (3) has to be understood in the sense of distributions. Then, the following relation holds on the solid/liquid interface: [ρ]V + [ρv · n] = 0

(4)

where [·] denotes the jump of the inside quantity across the interface, V is the normal velocity of the solid/liquid interface and the vector n denotes the normal to the interface. For instance, when solid is not moving (vs = 0), this condition reduces to ρs − ρl V, (5) vl · n = ρl see Figure 1.

Solid V vl Liquid Figure 1. Solidification of a solid seed with density ρs larger than the liquid density ρl : as the solid/liquid interface moves with normal velocity V toward the boundary of the calculation domain Ω, liquid flows with velocity vl toward the solid. At this point it should be noted that certain geometrical configurations are incompatible with the mass conservation equation (3). This is the case when a liquid region is surrounded by a solid region, see Figure 2 where examples of compatible and not compatible configurations are shown. 2.2. Momentum conservation Averaging the momentum conservation equation in the solid and liquid regions yields    ∂ ρs φvs + ρl (1 − φ)vl + div ρs φvs ⊗ vs + ρl (1 − φ)vl ⊗ vl ∂t   − div φσs + (1 − φ)σl = 0,

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Figure 2. Examples of solidification configurations that are compatible and not compatible with equation (3). The first two configurations are compatible with (3) since some external liquid can enter the computational domain in order to feed all liquid regions. The last two configurations are not compatible with (3) since external liquid cannot feed the liquid region located at the center of the computational domain. where σs , σl are the solid and liquid stress tensors, respectively. Our goal is to obtain a momentum equation for the unknown average velocity v. First, the definitions (1) (2) together with (3) are used to eliminate the liquid velocity vl in the momentum equation:

 φρs ∂v + (ρv · ∇)v + div ρ ρ (v − vs ) ⊗ (v − vs ) ∂t (1 − φ)ρl   − div φσs + (1 − φ)σl = 0. Moreover, the solid mechanical deformation is neglected, thus σs = 0. Also, given a penalty parameter ε $ 1, the penalty term 1 2 φ (v − vs ) ε is added to the momentum equation in order to force the average velocity field v to equal the solid velocity vs in the solid region (φ = 1). Finally, the liquid stress tensor (1 − φ)σl is replaced by −pI + 2µl (v) where p is the average pressure, µl the liquid viscosity and (v) = 1/2(∇v + ∇v T ) the rate of deformation tensor of the average velocity v. Therefore, the momentum equation can be rewritten in the whole computational domain Ω as:

 φρs ∂v + (ρv · ∇)v + div ρ (v − vs ) ⊗ (v − vs ) ρ ∂t (1 − φ)ρl   1 − 2div µl (v) + ∇p + φ2 (v − vs ) = 0. (6) ε

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It should be stressed that in the liquid far from the solid/liquid interface (φ = 0), the mass and momentum equations (3) (6) reduce to the incompressible NavierStokes equations whereas in the solid region (φ = 1), due to the penalty term, the velocity equals the solid velocity vs as ε becomes small. 2.3. Species conservation We proceed as in [20], eq. (33) and (34). Let C be the average massic concentration of the alloy defined by ρC = ρs φCs + ρl (1 − φ)Cl , where Cs , Cl are the solid and liquid concentrations, respectively. Introducing the constant partition coefficient k k=

Cs Cl

yields ρC kρC and Cs = . (7) kρs φ + ρl (1 − φ) kρs φ + ρl (1 − φ) Averaging the species conservation gives    ∂ ρs φCs + ρl (1 − φ)Cl + ∇ · ρs φCs vs + ρl (1 − φ)Cl vl ∂t   − ∇ · Ds ρs φ∇Cs + Dl ρl (1 − φ)∇Cl = 0, Cl =

where Ds , Dl are the constant solid and liquid diffusion coefficients. Using (7) and (2), the quantities Cs , Cl and vl can be eliminated to obtain

  ρC ∂(ρC) + div ρv + (k − 1)ρs φvs ∂t kρs φ + ρl (1 − φ) 

ρC(ρl − kρs ) ∇φ = 0, − div D(φ)∇(ρC) + D(φ) kρs φ + ρl (1 − φ) where we have set D(φ) =

kρs φDs + ρl (1 − φ)Dl . kρs φ + ρl (1 − φ)

Introducing the volumic concentration c = ρC, the above equation writes

  c ∂c + div ρv + (k − 1)ρs φvs ∂t kρs φ + ρl (1 − φ)   ˜ φ)∇φ = 0, (8) − div D(φ)∇c + D(c, where we have set ˜ φ) = D(φ) D(c,

c(ρl − kρs ) . kρs φ + ρl (1 − φ)

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2.4. Phase-field As in [17], we consider a standard phase-field equation for the solid phase moving with constant velocity vs : 



   φ(1 − φ)(1 − 2φ) 1 ∂φ + vs · ∇φ = Γ div A(∇φ)∇φ − µk ∂t δ2 

φ(1 − φ) c −T . (9) + T m + ml kρs φ + ρl (1 − φ) δ Here µk denotes the kinetic mobility, Γ is the Gibbs-Thomson coefficient. The term div(A(∇φ)∇φ) is the functional derivative (that is to say the Fr´echet derivative) of the surface energy
  2 1 a θ ∇φ(x) |∇φ(x)|2 dx, 2 Ω where a is the real-valued function defined by a(θ) = 1 + a ¯ cos(4θ), with a ¯ the anisotropy parameter and where θ(ξ) denotes the angle between a vector ξ ∈ R2 \ {0} and the first component of the orthonormal Cartesian basis (e1 , e2 ), that is cos θ(ξ) =

ξ · e1 . ξ

Therefore the matrix A(·) in (9) is defined for ξ ∈ R2 \ {0} by

 a2 (θ(ξ)) −a(θ(ξ))a (θ(ξ)) A(ξ) = , a(θ(ξ))a (θ(ξ)) a2 (θ(ξ)) The term φ(1 − φ)(1 − 2φ) in (9) is the derivative of the double well which forces the phase field to values close to zero or one. Finally, the last term in (9) is a source term accounting for the energy due to the solid-liquid phase transformation, where Tm is the melting temperature of the pure substance, ml is the slope of the liquid line in the equilibrium phase diagram. The temperature field T is a known linear function of space and time T (x, t) = T0 + tT˙ + Gd · x

(10)

where T˙ and G are given constants and d is a selected solidification direction in space. It should be noted that the velocity field v is not present in equation (9) so that, in the sharp interface limit, the classical Gibbs-Thomson relation between the interface velocity V , the curvature and the concentration c is recovered. Therefore, the effect of the fluid motion on Gibbs-Thomson relation is neglected. We refer to [24, 25] for more general models including such effects.

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2.5. Summary of the model The goal of the present model is to find the phase field φ : Ω × (0, tend ) → R, the volumic concentration c : Ω × (0, tend ) → R, the velocity v : Ω × (0, tend ) → R2 and the pressure p : Ω × (0, tend ) → R, satisfying equations (9), (8), (6) and (3). Natural boundary conditions apply on the boundary of the calculation domain Ω for φ, c and v. Moreover, initial conditions at time t = 0 must be prescribed for φ, c and v. Existence and uniqueness of solutions for this model in the absence of liquid flow and for sufficiently small a ¯ (small anisotropy) are proved in [26]. A posteriori error estimation and adaptive finite elements are presented in [14]. Existence and convergence of solutions in presence of liquid flow is an open problem. 2.6. Numerical method Equations (9), (8), (6) and (3) are discretized in time using an order one semiimplicit scheme. Space discretization is based on continuous, piecewise linear finite elements on triangular adapted meshes. In order to reduce the number of degrees of freedom, the triangles may have large aspect ratio whenever needed. The refinement/coarsening criterion is based on a posteriori error estimates and the adaptive algorithm has already been presented for elliptic problems [27], parabolic problems [28], Stokes problem [29], dendritic growth [14] and coalescence [15].

3. Numerical experiments The liquid flow due to shrinkage is now computed around four dendrites. We place 2 × 2 = 4 dendritic seeds in a square Ω of size 0.001 m and let the system evolve observing the liquid flow and the pressure drop during the process. The distance between the seeds is 0.0002 m. The temperature is given by (10) with T0 = 993.8 K, T˙ = 10 K/s and G = 0. The values of the physical parameters are given in Table 1. Tm 1000 K

k Ds 0.5 5 10−10 m2 /s

a ¯ µk ml 0.04 0.0015 m/(Ks) −260 K

Dl Γ 5 10−9 m2 /s 5 10−7 Km

ρs 1000 kg/m3

ρl 950 kg/m3

µl 0.014 kg/(ms)

Table 1. Values of the physical parameters. The solidification shrinkage causes the liquid to flow toward the center of the square. When the dendrites are sufficiently big, one can observe negative pressure appearing in the almost closed central region of the computational domain Ω. The velocity field is changing significantly during the experiment, the maximum observed velocity being 0.0001 m/s. In the beginning, where distance between dendrites is big, the maximum velocity is small, namely 0.00001 m/s. However, when neighboring dendrite tips are sufficiently close to each other, the velocity changes

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rapidly in order to supply the central region of the domain with new material. When dendrites grow further, the spacing between dendrite tips of neighboring dendrites gets smaller and that limits the amount of liquid that can pass toward the center, which causes the pressure drop. Furthermore, the concentration in the center gets much higher than in the rest of the domain. Figures 3 and 5 show the evolution of the system in the case of 4 dendrites.

Figure 3. Dendritic growth of 4 dendrites at times t = 0.15 and 0.225s. Left: adapted meshes (39442 vertexes at final time). Middle: concentration. Right: pressure.

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Figure 4. Dendritic growth of 4 dendrites at time t = 0.225 s. Velocity field.

Figure 5. Dendritic growth of 4 dendrites at time t = 0.225 s. Zoom of the concentration and adapted mesh. Same scale as in Figure 3.

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[20] W.J. Boettinger, J.A. Warren, C. Beckermann, and A. Karma. Phase-field simulation of solidification. Annu. Rev. Mater. Res., 32:163–194, 2002. [21] D.M. Anderson, G.B. McFadden, and A.A. Wheeler. A phase-field model of solidification with convection. Phys. D, 135(1-2):175–194, 2000. [22] D.M. Anderson, G.B. McFadden, and A.A. Wheeler. A phase-field model with convection: sharp-interface asymptotics. Phys. D, 151(2-4):305–331, 2001. [23] J.C. Heinrich and D.R. Poirier. Convection modeling in directional solidification. C. R. Mecanique, 332:429–445, 2004. [24] M. Griebel, W. Merz, and T. Neunhoeffer. Mathematical modeling and numerical simulation of freezing processes of a supercooled melt under consideration of density changes. Comp. Vis. Sci., 1(4):201–219, 1999. [25] Y. Sun and C. Beckermann. Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations. Phys. D, 198(3-4):281–308, 2004. [26] E. Burman and J. Rappaz. Existence of solutions to an anisotropic phase-field model. Math. Methods Appl. Sci., 26(13):1137–1160, 2003. [27] M. Picasso. Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator. Comm. Numer. Methods Engrg., 19(1):13– 23, 2003. [28] M. Picasso. An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems. SIAM J. Sci. Comput., 24(4):1328– 1355 (electronic), 2003. [29] M. Picasso. An adaptive algorithm for the stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Appl. Numer. Math., 54(3-4):470–490, 2005. Jacek Narski and Marco Picasso Institut d’Analyse et Calcul Scientifique Ecole Polytechnique F´ed´erale de Lausanne CH-1015 Lausanne, Switzerland e-mail: {jacek.narski,marco.picasso}@epfl.ch