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：凝固过程中枝晶 grains 的生长通常采用晶粒包络模型（GEM）进行建模，其中枝晶包络是通过相场界面捕捉（PFIC）方法追踪的界面。在 PFIC 方法中，在固定网格上求解相场方程以追踪包络的位置。PFIC 虽然通用且鲁棒，但会引入某些数值伪影。本文提出了一种求解 GEM 的替代方法：采用无网格（锐利）界面追踪（MIT）公式，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，界面追踪节点是域内扩散问题的边界节点。为提高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，从而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网格自由离散化，以及使用径向基函数（经单项式增强）对偏微分算子及溶质浓度场的高阶近似。通过在二维 h 自适应离散化下求解枝晶扩散生长问题展示了所提方法，并通过与 PFIC 方法的结果对比进行了评估。结果表明，MIT 能够复现 PFIC 的计算结果，具有收敛性，并且在相似空间离散化下能够捕捉到比 PFIC 更丰富的包络形状细节。