Fully Implicit, Adaptive Grid Methods for Phase-Field Simulation of Solidification in Pure Metals and Alloys

A fully-implicit numerical method based upon adaptively refined meshes for the simulation of pure materials and binary alloy solidification in 2D is presented. In addition we combine a second-order fully-implicit time discretisation scheme with variable steps size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, is shown. Due to the high non-linearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour. Introduction In order to model and simulate crystal growth in pure materials and alloys the phase-field method is one of the most popular and powerful techniques (e.g.[1, 2, 3, 4, 5] ). However, the nature of the phase-field models leads to coupled systems of highly nonlinear and unsteady partial differential equations (PDEs). Typically, this complexity has led modellers to rely primarily on relatively simple numerical methods, however in this work we aim to demonstrate that it is possible, and indeed advantageous, to make use of advanced numerical methods, such as adaptivity, implicit schemes and multigrid. For phase-field models, in which the phase variable, φ, is constant in the two phases and only varies in the thin interface region, the use of mesh adaptivity is a natural choice. Adaptive mesh refinement was applied to phase-field models for pure materials solidification, e.g. [6, 7, 8], and has subsequently also been used for model of binary alloy solidification, e.g. [9, 10]. This method leads to very fine mesh resolution only in the interface region and therefore allows the use of large domains to prevent boundary effects. Another important, and related, factor is the choice of a suitable time integration method. Widely used methods are explicit methods such as Euler’s method. However, when using explicit methods a major constraint in the computation is the time-step restriction in order to assure the stability of the scheme. Implicit methods are more expensive per step than explicit ones because intermediate approximations have to be solved from a system of nonlinear algebraic equations. However, implicit methods (e.g. [8, 9]) are important because of their superior stability properties, which allow larger time steps. In this work we use the A-stable implicit second-order Backward Differentiation Formula (BDF2) [11], which is combined with variable step size selection, to obtain an adaptive time and space method. Especially for the simulation of dendritic growth, variable time stepping is valuable because of the variation in the interface velocity over time. Explicit schemes are not generally able to exploit this since the step size selected is typically the maximum stable time step: and when mesh adaptivity is used this can be very small. Here we demonstrate the advantages of the implicit method by considering the case of binary alloy solidification where the system of PDEs is more complex than in the thermal case. Nevertheless, the advantage of the proposed method could also been demonstrated for pure materials solidification where the method has been applied to the phase-field model of Karma and Rappel [12]. The considered phase-field model here is the isothermal case of the coupled heat and solute phase-field model of Ramirez, Beckermann, Karma and Diepers [4]. These authors propose that the results of this model are independent of the interface width, thus making this model especially attractive. This model is an extension of the phase-field model for pure materials [12] and binary alloys [3, 13]. The model is described briefly in the next section before we describe the proposed discretisation methods in detail. In order for the implicit time-stepping scheme to be viable it is essential that the large systems of nonlinear algebraic equations, that occur at each time step, are solved as efficiently as possible. In order to achieve this a nonlinear multgrid solver, based upon [14], has been implemented. This is demonstrated to behave almost optimally on both uniformly and locally refined grids. Finally we compare our proposed method to other discretisation methods with respect to CPU time and accuracy by comparing total time and interface positions as well as tip velocities. Some typical simulation results are also included. A detailed description of the discretisation method and the full set of results can be found in [15]. Phase-Field model The Phase-Field model used here is a variation of the coupled thermal-solute model for the simulation of microstructure formation in dilute binary alloys, given in [4]. In this paper we only consider the isothermal case in which the model reduces to a pure solute model by fixing the thermal undercooling and choosing an infinitely large Lewis number. The authors in [4] showed that the simulation results for the isothermal case agree exactly with those results found by using the model given in [3]. The microstructure is represented by the phase variable φ which divides the liquid and the solid phase by a diffuse interface. The solid and liquid phases correspond to φ = 1 and φ = −1 respectively, and in the interface region φ varies smoothly between the bulk values. The governing equations, in dimensionless forms for vanishing kinetic effects [4], are A(ψ) ∂φ ∂t = A(ψ)∇φ+ 2A(ψ)A′(ψ) [ ∂ψ ∂x ∂φ ∂x + ∂ψ ∂y ∂φ ∂y ]