Abstract
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.
Highlights
We here present our computational approach to simulating, at the meso-scale, three dimensional, non-isothermal alloy solidification from an initial small, spherical seed into a mature, dendritic crystal
A feature of a mature dendrite is the geometric complexity of its evolving two-dimensional surface
Taking the thickness of the interface to be ≈ 1, we find the size of a mature dendrite grows to ∼ 300, requiring the domain size to be significantly greater still (depending on the thermal field this may need to be O (1000), or even more), we see that the interface region of interest is very much smaller than the overall domain
Summary
We here present our computational approach to simulating, at the meso-scale, three dimensional, non-isothermal alloy solidification from an initial small, spherical seed into a mature, dendritic crystal. A feature of a mature dendrite is the geometric complexity of its evolving two-dimensional surface (see Fig. 1 for a typical snapshot in time) This makes tracking of the surface a difficult task in sharp interface models. The numerical solution to this phase field model (described in detail ) requires methods to solve a time-dependent, highly nonlinear system of PDEs, of parabolic type, and capable of resolving varying length and time scales. The computational problem is: non-linear, three-dimensional, stiff, involves multiple length scales to capture small phase and large temperature fields, multi-time scale associated with the Lewis number and, to establish a mature dendrite, requires a long simulation time.
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