P.C. Bollada

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.

Simulations of three-dimensional dendritic growth using a coupled thermo-solutal phase-field model

Using a phase field model, which fully couples the thermal and solute concentration field, we present simulation results in three dimensions of the rapid dendritic solidification of a class of dilute alloys at the meso scale. The key results are the prediction of steady state tip velocity and radius at varying undercooling and thermal diffusivities. Less computationally demanding 2-dimensional results are directly compared with the corresponding 3-dimensional results, where significant quantitative differences emerge. The simulations provide quantitative predictions for the range of thermal and solutal diffusivities considered and show the effectiveness and potential of the computational techniques employed. These results thus provide benchmark 3-dimensional computations, allow direct comparison with underlying analytical theory, and pave the way for further quantitative results.

An adaptive mesh method for phase-field simulation of alloy solidification in three dimensions

We present our computational method for binary alloy solidification which takes advantage of high performance computing where up to 1024 cores are employed. Much of the simulation at a sufficiently fine resolution is possible on a modern 12 core PC and the 1024 core simulation is only necessary for very mature dendrite and convergence testing where high resolution puts extreme demands on memory. In outline, the method uses implicit time stepping in conjunction with an iterative solver, adaptive meshing and a scheme for dividing the work load across processors. We include three dimensional results for a Lewis number of 100 and a snapshot for a mature dendrite for a Lewis number of 40.

Towards a 3-dimensional phase-field model of non-isothermal alloy solidification

We review the application of advanced numerical techniques such as adaptive mesh refinement, implicit time-stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the 3-dimensional phase field problem for coupled heat and solute transport during non-isothermal alloy solidification. Using such techniques it is shown that such models are tractable for modest values of the Lewis number (ratio of thermal to solutal diffusivities). Solutions to the 3-dimensional problem are compared with existing solutions to the equivalent 2-dimensional problem.

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

We show that the standard approach to the modeling of multi-phase field dynamics for the solidification of alloys has three major defects and offer an alternative more successful formulation. The model contains an action made up of a free energy functional of the temperature, concentration and the phase variables. In addition there is a penalty functional for the gradients of the phase variables, which keeps all field variables continuous – the phase field method. A variation of this action is related to time derivatives of the variables in the field. At any physical point there exists up to N phases, i, each of which represent the proportion of each phase at that point, thus implying the constraint, ΣNi = 1 i = 1. The standard Lagrange multiplier treatment for imposing this constraint has three major defects: non-reduction to standard single phase formulation; a dependence on the value of N; generation of unphysical additional phases. We demonstrate a multi-phase formulation that avoids these two defects and, partly as a consequence, does not generate spurious additional phases. Moreover, this aim is achieved without losing the active part that three non-zero phases should play if present at any point.

Phase-Field Modelling of Intermetallic Solidification

Many important intermetallic compounds display a faceted morphology during solidification close to equilibrium but adopt a more continuous, dendritic like morphology with increasing departure from equilibrium. We present a phase-field model of solidification that is able to both reproduce the Wulff shape at low driving force and to simulate a continuous transition from faceted to dendritic growth as the driving force is increased. A scaled ratio of the (perimeter)2 to the area is used to quantify the extent of departure from the equilibrium shape.

Thermo-Solutal Modelling of Microstructure Formation during Multiphase Alloy Solidification - a New Approach

This paper shows how to move from a specification of free energy for the solidification of a binary alloy to the dynamical equations using the elegance of a dissipative bracket analogous to the Poisson bracket of Hamiltonian mechanics. A key new result is the derivation of the temperature equation for single-phase thermal-solutal models, which contains generalisations and extra terms which challenge standard models. We also present, for the first time, the temperature equation for thermal multi-phase field models. There are two main ingredients: one, the specification of the free energy in terms of the time and space dependent field variables:

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

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Faceted and dendritic morphology change in alloy solidification

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A numerical approach to compensate for phase field interface effects in alloy solidification

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