AI 中文总结
针对多尺度界面演化问题，提出基于梯度平滑方法与分层自适应移动结构化网格结合的强形式相场求解器，可自动定位界面分辨率，减少自由度总数，降低计算成本，在大规模问题中效率显著提高。
AI 中文摘要
具有演化界面的多尺度问题在科学和工程中普遍存在。相场模型是模拟计算力学和材料建模中界面主导现象的有力工具，但将其应用于大规模问题时，常受限于在整个域上解析薄扩散界面的高计算成本。本文提出一种高效的强形式相场求解器，它将梯度平滑方法（GSM）与分层自适应移动结构化网格相结合，能在窄界面区域自动定位分辨率，同时在主体域保持粗离散化。引入分层细化设计以在界面上保持局部均匀分辨率，使GSM离散化在远离界面处网格非均匀性很强时仍保持整体二阶精度。尽管GSM的每个自由度成本高于标准有限差分格式，但自适应框架大幅减少了自由度总数，与均匀网格方法的二次缩放相比实现了近线性计算缩放。基于Allen-Cahn和Cahn-Hilliard方程的数值例子表明，所提出的自适应GSM求解器在界面演化中提供了所需精度，同时比现有弱形式和强形式求解器具有更有利的计算复杂度O(N)，对于具有薄界面或相对于整个域较小界面面积分数的大规模问题效率显著提高。
英文摘要
Multiscale problems with evolving interfaces are ubiquitous in science and engineering. Phase-field models are a powerful tool for simulating interface-dominated phenomena in computational mechanics and materials modeling, but their application to large-scale problems is often constrained by the high computational cost of resolving thin diffuse interfaces over the entire domain. This paper presents an efficient strong-form phase-field solver that couples the Gradient Smoothing Method (GSM) with a hierarchical adaptive and moving structured mesh, enabling automatic localization of resolution within a narrow interfacial region while retaining coarse discretization in bulk domains. A layered refinement design is introduced to preserve locally uniform resolution across the interface, allowing the GSM discretization to maintain overall second-order accuracy despite strong mesh non-uniformity away from the interface. Although GSM incurs a higher per-degree-of-freedom cost than standard finite-difference schemes, the adaptive framework substantially reduces the total number of degrees of freedom, resulting in near-linear computational scaling compared with the quadratic scaling of uniform-grid approaches. Numerical examples based on the Allen-Cahn and Cahn-Hilliard equations demonstrate that the proposed adaptive GSM solver delivers desired accuracy for interface evolution while attaining more favorable computational complexity, O(N), than existing weak-form and strong-form solvers, becoming significantly more efficient for large-scale problems with thin interfaces or a small interfacial area fraction relative to the whole domain.
