Abstract
Dendritic growth is computed with automatic adaptation of an anisotropic and unstructured finite element mesh. The energy conservation equation is formulated for solid and liquid phases considering an interface balance that includes the Gibbs-Thomson effect. An equation for a diffuse interface is also developed by considering a phase field function with constant negative value in the liquid and constant positive value in the solid. Unknowns are the phase field function and a dimensionless temperature, as proposed by [1]. Linear finite element interpolation is used for both variables, and discretization stabilization techniques ensure convergence towards a correct non-oscillating solution. In order to perform quantitative computations of dendritic growth on a large domain, two additional numerical ingredients are necessary: automatic anisotropic unstructured adaptive meshing [2,[3] and parallel implementations [4], both made available with the numerical platform used (CimLib) based on C++ developments. Mesh adaptation is found to greatly reduce the number of degrees of freedom. Results of phase field simulations for dendritic solidification of a pure material in two and three dimensions are shown and compared with reference work [1]. Discussion on algorithm details and the CPU time will be outlined.
Highlights
The phase field approach is a method of choice for simulating interfacial pattern formation phenomena in solidification
Tracking is avoided by introducing an order parameter, or phase field, which varies smoothly from one value in the liquid to another value in the solid across a spatially diffuse interface region related with a thickness W
This field naturally distinguishes the solid and liquid phases and converts the problem of simulating the advance of a sharp boundary to that of solving a stiff system of partial differential equations that govern the evolution of the phase and diffusion fields
Summary
The phase field approach is a method of choice for simulating interfacial pattern formation phenomena in solidification. Tracking is avoided by introducing an order parameter, or phase field , which varies smoothly from one value in the liquid to another value in the solid across a spatially diffuse interface region related with a thickness W. This field naturally distinguishes the solid and liquid phases and converts the problem of simulating the advance of a sharp boundary to that of solving a stiff system of partial differential equations that govern the evolution of the phase and diffusion fields. In a diffuse interface context, instead of solving the equations on each solid/liquid domain with the given interface conditions, we may obtain a set of equations valid in the whole domain by using a free energy approach
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