  The grain envelope model (GEM) describes the growth of envelopes of dendritic
crystal grains during solidification. Numerically the growing envelopes are
usually tracked using an interface capturing method employing a phase field
equation on a fixed grid. Such an approach describes the envelope as a diffuse
interface, which can lead to numerical artefacts that are possibly detrimental.
In this work, we present a sharp-interface formulation of the GEM that
eliminates such artefacts and can thus track the envelope with high accuracy.
The new formulation uses an adaptive meshless discretization method to solve
the diffusion in the liquid around the grains. We use the ability of the
meshless method to operate on scattered nodes to accurately describe the
interface, i.e., the envelope. The proposed algorithm combines parametric
surface reconstruction, meshless discretization of parametric surfaces, global
solution construction procedure and partial differential operator approximation
using monomials as basis functions. The approach is demonstrated on a
two-dimensional h-adaptive solution of diffusive growth of dendrites and
assessed by comparison to a conventional diffuse-interface fixed-grid GEM.
