  The growth of dendritic grains during solidification is often modelled using
the Grain Envelope Model (GEM), in which the envelope of the dendrite is an
interface tracked by the Phase Field Interface Capturing (PFIC) method. In the
PFIC method, an phase-field equation is solved on a fixed mesh to track the
position of the envelope. While being versatile and robust, PFIC introduces
certain numerical artefacts. In this work, we present an alternative approach
for the solution of the GEM that employs a Meshless (sharp) Interface Tracking
(MIT) formulation, which uses direct, artefact-free interface tracking. In the
MIT, the envelope (interface) is defined as a moving domain boundary and the
interface-tracking nodes are boundary nodes for the diffusion problem solved in
the domain. To increase the accuracy of the method for the diffusion-controlled
moving-boundary problem, an \h-adaptive spatial discretization is used, thus,
the node spacing is refined in the vicinity of the envelope. MIT combines a
parametric surface reconstruction, a mesh-free discretization of the parametric
surfaces and the space enclosed by them, and a high-order approximation of the
partial differential operators and of the solute concentration field using
radial basis functions augmented with monomials. The proposed method is
demonstrated on a two-dimensional \h-adaptive solution of the diffusive growth
of dendrite and evaluated by comparing the results to the PFIC approach. It is
shown that MIT can reproduce the results calculated with PFIC, that it is
convergent and that it can capture more details in the envelope shape than PFIC
with a similar spatial discretization.
