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.

翻译：在凝固过程中，枝晶晶粒的生长通常采用晶粒包络模型（GEM）进行模拟，其中枝晶的包络是通过相场界面捕捉（PFIC）方法追踪的界面。在PFIC方法中，固定网格上求解相场方程以追踪包络位置。尽管该方法通用且稳健，但会引入某些数值伪影。本研究提出一种替代方案，采用无网格（尖锐）界面追踪（MIT）方法求解GEM，该方法使用直接、无伪影的界面追踪。在MIT中，包络（界面）被定义为移动域边界，界面追踪节点即为域内扩散问题求解的边界节点。为增强扩散控制移动边界问题的求解精度，采用h-自适应空间离散化，即在包络附近细化节点间距。MIT结合了参数曲面重建、参数曲面及其包围空间的免网格离散化，以及使用径向基函数（增广单项式）对偏微分算子和溶质浓度场的高阶逼近。通过二维h-自适应求解枝晶扩散生长的算例验证了所提方法，并将其结果与PFIC方法进行对比评估。研究表明，MIT能够复现PFIC的计算结果，具有良好的收敛性，且在相同空间离散化条件下可捕捉到比PFIC更丰富的包络形状细节。