Application of the level-set method to a mixed-mode driven Stefan problem in 2$D$ and 3$D$

Abstract

This study focusses on the growth of small precipitates within a matrix phase (see also den Ouden et al., Comput Mater Sci 50:2397–2410, 2011). The growth of a precipitate is assumed to be affected by the concentration gradients of a single chemical element within the matrix phase at the precipitate/matrix boundary and by an interface reaction, resulting into a mixed-mode formulation of the boundary condition on the precipitate/matrix interface. Within the matrix phase we assume that the standard diffusion equation applies to the concentration of the considered chemical element. The formulated Stefan problem is solved using a level-set method (J Comput Phys 79:12–49, 1988) by introducing a time-dependent signed-distance function for which the zero level-set describes the precipitate/matrix interface. All appearing hyperbolic partial differential equations are discretised by the use of Streamline-Upwind Petrov–Galerkin finite-element techniques (Comput Vis Sci 3:93–101, 2000). All level-set related equations are solved on a background mesh, which is enriched with interface nodes located on the zero-level of the signed-distance function. The diffusion equation is solved in the diffusive phase. Simulations with the implemented methods for the growth of various precipitate shapes show that the methods employed in this study correctly capture the evolution of the precipitate/matrix interface including topological changes. At the final stage of growth/dissolution physical equilibrium is attained. We also observe that our solutions show mass conservation as the time-step and element-size tend to zero.