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. Our aim is to remove limitations inherent to some existing models both in terms of the applicability to physically realistic situations and the feasibility of mathematical and numerical analysis. The central object of interest is the reaction term in the Allen-Cahn equation, which is responsible for spatial distribution of latent heat release during solidification. Several existing forms of the reaction term are analyzed and new variants are proposed, with consistent asymptotic behavior as the interface thickness tends to zero. The resulting models are tested in a number of numerical simulations focusing on mesh-dependence, model parameter settings, and the applicability to solidification under very large supercooling.

Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification

“…This paper provides a detailed proof of existence of the weak solution and convergence of the finite volume scheme to this solution for the isotropic phase field model suitable for solidification modeling in polyhedral domains covered by admissible polyhedral meshes. We consider a general form of the reaction term in the phase field equation which allows to apply the presented results to existing models [30] as well as several new variants of the phase field model presented in our work [42]. We show that introducing an artificial limiter of the reaction term makes it possible to perform the analysis while not affecting the simulation results [42].…”

“…We consider a general form of the reaction term in the phase field equation which allows to apply the presented results to existing models [30] as well as several new variants of the phase field model presented in our work [42]. We show that introducing an artificial limiter of the reaction term makes it possible to perform the analysis while not affecting the simulation results [42]. A semi-discrete form of the scheme is used, leaving temporal discretization up to the reader's choice.…”

“…For the purposes of the analysis, we consider the isotropic PFM with a generically formulated reaction term, which is well suited for practical applications: In our related work [42], we propose a novel form of the PFM which is compatible with the presented numerical analysis. In addition, we also introduce its anisotropic variant.…”