A novel adaptive numerical approach is developed for an accurate and computationally efficient phase-field modelling of dendritic solidification. The adaptivity is based on the dynamic quadtree domain decomposition. A quadtree decomposes the computational domain into rectangular sub-domains of different sizes. Each sub-domain is extended to ensure overlap communication between neighbouring sub-domains. In each sub-domain, uniform distribution of computational nodes is generated. The product between the node density and the sub-domain area is fixed to ensure the h-adaptivity. The adaptive approach provides the highest density of computational nodes at the solid-liquid interface and the lowest density in the bulk of the phases. The meshless radial basis function generated finite difference (RBF-FD) method is applied for the spatial discretisation of the partial differential equations which arise from the phase-field model. The RBF-FD method is especially appealing since it allows straightforward spatial discretisation of partial differential equations on scattered node distributions. The use of scattered node distribution reduces the discretisation-induced anisotropy in the phase-field modelling of dendritic growth. The forward Euler scheme is used for temporal discretisation. The adaptive time-stepping is employed to speed up the calculations further. The performance of the novel numerical approach is tested for dendritic solidification of supercooled pure melts and supersaturated dilute binary alloys at arbitrary preferential growth directions. The impact of the numerical parameters on the accuracy and computational efficiency is thoroughly analysed. It is shown that the RBF-FD method, defined on scattered node distribution, together with the space-time adaptive approach, represents an accurate and efficient technique for solving the phase-field models of dendritic solidification.

开发了一种新的自适应数值方法，用于对枝晶凝固进行精确且计算有效的相场建模。自适应性基于动态四叉树域分解。四叉树将计算域分解为不同大小的矩形子域。扩展每个子域以确保相邻子域之间的重叠通信。在每个子域中，生成计算节点的均匀分布。节点密度与子域面积的乘积是固定的，以保证h-自适应性。自适应方法在固-液界面处提供最高密度的计算节点，而在大部分相中提供最低密度。无网格径向基函数生成有限差分 (RBF-FD) 方法应用于由相场模型产生的偏微分方程的空间离散化。RBF-FD 方法特别吸引人，因为它允许对分散节点分布上的偏微分方程进行直接的空间离散化。分散节点分布的使用减少了在树枝状生长的相场建模中离散化引起的各向异性。前向欧拉方案用于时间离散化。采用自适应时间步长进一步加快计算速度。针对过冷纯熔体和过饱和稀二元合金在任意优先生长方向上的枝晶凝固测试了新数值方法的性能。深入分析了数值参数对精度和计算效率的影响。结果表明，基于离散节点分布定义的 RBF-FD 方法与时空自适应方法一起，代表了求解枝晶凝固相场模型的一种准确有效的技术。