A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling

A method for incorporating multi-resolution capabilities within pre-existing global 3-D spherical mantle convection codes is presented. The method, which we term "geometric multigrid refinement", is based upon the application of a multigrid solver on non

Multiphase field modelling of alloy solidification

We present an approach to alloy solidification modelling that incorporates binary interface energies in a manner that correctly reproduces the associated theoretical angles at triple junctions in eutectic solidification. We find that simply applying the principle that the correct binary junction behaviour is recovered when only two phases are present is insufficient. Previous research (Toth \cite{Toth2016}) recommends a modification of the surface energy by adding an energy barrier at the triple junction, and we explore alternative models that would benefit from this approach. The main approach we recommend here, though, is to extend the minimal model of Folch and Plapp \cite{folch2003,FolchPlapp2005}, which, without modification, is limited to $120^\circ$ junction angles. This is achieved by a linear transformation of this formulation, and facilitated by an analytical multiphase solution presented here for the first time

Bracket formalism applied to phase field models of alloy solidification

We present a method for coupling current phase field models of alloy solidification into general continuum modelling. The advantages of this approach are to provide a generic framework for phase field modelling, give a natural and thermodynamically consistent extension to non-isothermal modelling, and to see phase field models in a wider context. The bracket approach, introduced by Beris and Edwards, is an extension of the Poisson bracket of Hamiltonian mechanics to include dissipative phenomena. This paper demonstrates the working of this formalism for a variety of alloy solidification models including multi phase, multi species with thermal and density dependency. We present new models by deriving temperature equations for single and more general phase field models, and postulate a density dependent formulation which couples phase field to flow

Three dimensional thermal-solute phase field simulation of binary alloy solidification

We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) are controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources

A new approach to multi-phase formulation for the solidification of alloys

This paper demonstrates that the standard approach to the modelling of multi-phase field dynamics for the solidification of alloys has three major defects and offers an alternative approach. The phase field formulation of solidification for alloys with multiple solid phases is formed by relating time derivatives of each variable of the system (e.g. phases and alloy concentration), to the variational derivative of free energy with respect to that variable, in such a way as to ensure positive local entropy production. Contributions to the free energy include the free energy density, which drives the system, and a penalty term for the phase field gradients, which ensures continuity in the variables. The phase field equations are supplemented by a constraint guaranteeing that at any point in space and time the phases sum to unity. How this constraint enters the formulation is the subject of this paper, which postulates and justifies an alternative to current methods