Numerical solution of a phase field model for polycrystallization processes in binary mixtures

Abstract

We consider the numerical solution of a phase field model for polycrystallization in the solidification of binary mixtures in a domain Ω ⊂ R 2 . The model is based on a free energy in terms of three order parameters: the local orientation Θ of the crystals, the local crystallinity φ, and the concentration c of one of the components of the binary mixture. The equations of motion are given by an initial-boundary value problem for a coupled system of partial differential equations consisting of a regularized second order total variation flow in Θ, an L 2 gradient flow in φ, and a W 1,2 (Ω) * gradient flow in c. Based on an implicit discretization in time by the backward Euler scheme, we suggest a splitting method such that the three semidiscretized equations can be solved separately and prove existence of a solution. As far as the discretization in space is concerned, the fourth order Cahn-Hilliard type equation in c is taken care of by a C 0 Interior Penalty Discontinuous Galerkin approximation which has the advantage that the same finite element space can be used as well for the spatial discretization of the equations in Θ and φ. The fully discretized equations represent parameter dependent nonlinear algebraic systems with the discrete time as a parameter. They are solved by a predictor corrector continuation strategy featuring an adaptive choice of the time-step. Numerical results illustrate the performance of the suggested numerical method.

Numerical simulation of the formation of spherulites in polycrystalline binary mixtures

“…In fact, a uniform choice τ m = T/M only works, if M is chosen sufficiently large which would require an unnecessary huge amount of time steps. An appropriate way to overcome this difficulty is to consider (5.2a), (5.2b) as parameter dependent nonlinear systems with the time as a parameter and to apply a predictor-corrector continuation strategy with an adaptive choice of the time steps (cf., e.g., [5,12,13,14]). Given the pair (Θ m−1 , Φ m−1 ), the time step size τ m−1,0 = τ m−1 , and setting k = 0, where k is a counter for the predictor-corrector steps, the predictor step for (5.2a) consists of constant continuation leading to the initial guesses…”

“…In [8], the authors suggested a phase field model featuring an orientational free energy density f ori in the phase field variables (local degree of crystallinity ϕ and orientation angle Θ) which takes care of misorientation. As a function of Θ, the orientational free energy density is related to the total variation of Θ. Consequently, from a mathematical point of view one has to consider Θ in the Banach space BV of functions of bounded variation as has been done in [13] for the two-field Kobayashi-Warren-Carter model [15] and in [14] for a three-field phase field model for polycrystalline growth in binary mixtures. As far as the minimization of functionals in terms of the total variation is concerned, there is a fundamental result stated in [16].…”

“…Specifying mobilities M ϕ and M Θ , the dynamics of the polycrystallization process can be described by a coupled system of evolutionary initial-boundary value problems consisting of an evolution inclusion in Θ and an evolution equation in ϕ. Following the approach in [13] and [14], in section 4 we suggest a splitting scheme based on an implicit discretization in time which decouples the evolutionary problems such that at each time step the evolution inclusion in Θ is solved first followed by the evolution equation in ϕ. We show that the split system is related to two minimization problems for functionals in BV and the Sobolev space W 1,2 that both admit a solution as can be shown by tools from the calculus of variations.…”