A recorded presentation by Professor Peter Jimack, Dean, Faculty of Engineering, Professor of Scientific Computing, University of Leeds.
A widely used class of mathematical models for the description of phase-change problems is based around the phase-field formulation. In this approach the mathematically sharp interface between the solid and liquid phases is assumed diffuse, allowing the definition of a continuous (differentiable) order parameter which represents the phase of the material (typically −1 in the liquid and +1 in the solid regions). The evolution of this phase variable is governed by a free energy functional which can be solved using standard techniques for partial differential equations (PDEs) without explicitly tracking the solid-liquid interface, thus allowing the simulation of arbitrarily complex morphologies. In this talk we will consider one such class of model, based upon a system of highly nonlinear parabolic PDEs, for the simulation of the solidification a non-isothermal binary alloy. The challenges of this model include the need to resolve a moving feature (the solid-liquid interface) at very small length scales, the existence of vastly different time scales (leading to severe stiffness) and the desire for simulations to be in three space dimensions. In order to develop efficient and reliable simulation software it has been necessary to incorporate mesh adaptivity (for locally enhanced spatial resolution), an implicit stiff integrator in time, and the use of multigrid methods to solve the resulting nonlinear algebraic system of equations at each time step. Furthermore, extensions to three space dimensions require the use of parallel implementations of the above on the high performance computing (HPC) facility at Leeds.