Focusing the latent heat release in 3D phase field simulations of dendritic crystal growth

Abstract

We investigate a family of phase field models for simulating dendritic growth of a pure supercooled substance. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for the spatial distribution of latent heat release during solidification. In this context, several existing forms of the reaction term are analyzed. Inspired by the known conclusions of matched asymptotic analysis, we propose a new variant (the 'ΣP1-P' model) that is simple enough to allow mathematical and numerical analysis and robust enough to be applicable to solidification under very large supercooling. The important component of the model (the Σ limiter) can also be incorporated into the original models to extend the range of their applicability. The individual models are tested in a number of numerical simulations focusing on mesh-dependence and model parameter settings. When the phase interface thickness is kept large with respect to the microscopic capillary length to make numerical computations feasible, the parameters of the Σ limiter can be tuned to improve agreement with previous models. The results obtained using the ΣP1-P model exhibit a good quantitative agreement with experimental data from rapid solidification of nickel melts.

Existence of Optimal Control for Dirichlet Boundary Optimization in a Phase Field Problem

“…For this particular model, we get 1 6 (see, e.g. [28]). The initial conditions for the temperature and phase field are given by ini 1 , ini 1 .…”

“…It is composed of the heat equation and the phase field (Allen-Cahn) equation. The model is introduced in dimensionless form in accordance with the related works [6,27,28]. The goal of the optimization, which is described by the cost functional , is to achieve a certain shape of the solid body at final time .…”

“…If necessary, the magnitude of the control can be reduced by increasing the regularization parameter 0 in the cost functional (1). Derivation of the dimensionless model and the meaning of the original dimensional quantities can be found in [28].…”

“…Phase field models (PFM) find utility in various areas, where interface tracking is of interest [7,9]. Applications include, but are not limited to, the modeling of phase transitions in materials science [8,9,12,21,22,25,28,30,33], crack propagation [11] or tumor growth [14]. The first of the three mentioned applications is addressed in this article.…”

“…A particular PFM that governs the solidification of a supercooled melt [4,27,28] is considered as the state equation of the system. It consists of the heat equation with a latent heat term and the phase field equation.…”