Phase-Field Software for the Quantitative Prediction of Coupled Thermal-Chemical Alloy Solidification

The modelling of solidification microstructures has become an area of intense interest in recent years. The properties of large-scale cast products are strongly influenced by the physics of processes occurring on the microscopic and mesoscopic lengths scales. One of the most fundamental and all pervasive microstructures produced duringsolidification is the dendrite. Remnants of these dendritic microstructures often survive subsequent processing operations such as rolling and forging and the length scales established by the dendrite can influence not only the final grain size but also the micro- and hence the macro-segregation patterns.Theoretical and experimental studies of the development of dendrites have been numerous in recent years, and one of the most powerful techniques to emerge for modelling dendritic microstructures is the phase-field method. The novelty of the phase-field method is that 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. The evolution of this phase parameter is governed by a free energy functional which can be solved using standard techniques for partial differential equations without explicitly tracking the solid-liquid interface, thus allowing the simulation of arbitrarily complex morphologies. However, in order obtain realistic solidification structures the width of the diffuse interface must be much smaller than the smallest structural feature to be resolved. Consequently, phase-field simulations usually require very fine meshing, but equally they are ideally suited to adaptive mesh refinement.Here, our aim is to build upon our prior work, also funded by the EPSRC, which has allowed us to develop the first fully implicit, fully adaptive (in space and time) numerical tool for the solution to the coupled set of equations governing both the evolution of the phase-field and the transport of heat and mass. The adaptivity allows very fine numerical resolution of the phase interface, whilst the fully implict time integration allows us to tackle problems with physically realistic diffusion parameters (that are not attainable with semi-implicit or explicit schemes due to their stiffness). The high nonlinearity of these problems has previously prevented researchers from applying fully implicit schemes, and so our main contribution has been to overcome this problem by solving the resulting systems of nonlinear algebraic equations using a robust and efficient nonlinear multigrid method.In this proposal we seek to take the final steps towards developing a software tool that will permit simulations to be undertaken that can yield quantitative predictions of physical solidification behaviour in realistic materials for the first time. This will require two additional innovative steps: the generalisation of our current methodology to three space dimensions, and the application of high performance computing techniques. Three space dimensions are required in order to provide physically realistic interfaces and HPC is required to make such simulations feasible in terms of both memory and cpu requirements. We will be able to undertake these steps successfully by building upon our existing 2-d adaptive, implicit phase-field work, combined with our prior experience with the development of both 3-d adaptive and 3-d HPC software.