Multi-Scale Modelling Using 3-Dimensional Adaptive Meshing with an Implicit, Multigrid Solver: A Crystallization Example

"We review the application of advanced numerical techniques such as adaptive mesh refmement, implicit time-stepping, multigrid solvers and massively parallel implementations as a route to obtaining solutions to the 3-dimensional phase-field problem with a domain size and interface resolution previously possible only in 2-dimensions. Using such techniques it is shown that such models are tractable even as the interface width approaches the solute capillary length.IntroductionMulti-scale modelling impinges upon many areas of science, engineering and technology where the interplay between processes occurring on disparate length and/or timescales can affect the properties of systems in a complex, and sometimes unforeseen, manner. One such is the processing of materials where interactions at the atomic scale, through the nano- and microstructural scale can have a profound affect on the macro-scale response of engineering components. Here we focus on one such problem, the crystallisation of alloy melts, to illustrate how a range of advanced numerical techniques make the modelling of such systems tractable.The modelling of solidification structures, in particular the growth of dendritic crystals, is a subject of intense and enduring interest within the scientific community, both because dendrites are a prime example of spontaneous pattern formation and they have a pervasive influence on the engineering properties of metals. For alloy systems, which form the vast majority of engineering materials, solidification involves the rejection of both solute and latent heat from the growing solid, forming a coupled diffusion problem in which both the chemical and thermal diffusion equations should be solved. The multi-scale nature of this problem arises from the disparity between the thermal and solutal diffusivities for most liquid metals, which are typically 10-5 m2 s-1 and 10-9 m2 s-1 respectively. The ratio of these two, referred to as the Lewis number (Le =Dia), is therefore typically of the order 10,000 and gives a measure of the multi-scale nature of the problem. In fact, in many cases the thermal diffusivity is so much larger than its solutal counterpart that thermal diffusion is considered instantaneous, wherein the system can be approximated as isothermal such that the multi-scale calculation does not need to be performed. This is certainly a reasonable approach in most conventional castings. However, there is a class of solidification problem in which this approach is far from satisfactory, namely Rapid Solidification (RS). The application of isothermal phase-field models in such situations are highly likely to produce spurious results or to fail totally, particularly for temperatures below To (the temperature at which the free energies of the solid and liquid phases are equal at the concentration of the bulk alloy), wherein partitionless solidification may occur."