We present an adaptive scheme for isogeometric phase-field modeling, to perform suitably graded hierarchical refinement and coarsening on both single- and multi-patch geometries by considering truncated hierarchical spline constructions which ensures $C^1$ continuity between patches. We apply the proposed algorithms to the Cahn-Hilliard equation, describing the time-evolving phase separation processes of immiscible fluids. We first verify the accuracy of the hierarchical spline scheme by comparing two classical indicators usually considered in phase-field modeling, for then demonstrating the effectiveness of the grading strategy in terms of accuracy per degree of freedom. A selection of numerical examples confirms the performance of the proposed scheme to simulate standard modes of phase separation using adaptive isogeometric analysis with smooth THB-spline constructions.

翻译：我们提出了一种适用于等几何相场建模的自适应方案，通过采用确保块间$C^1$连续性的截断层次样条构造，在单块及多块几何上实现适宜分级层次细化与粗化。将该算法应用于描述不混溶流体时变相分离过程的Cahn-Hilliard方程。首先通过对比相场建模中通常考虑的两类经典指标验证层次样条方案的精度，进而展示分级策略在每自由度精度方面的有效性。一系列数值算例证明，该方案利用光滑THB样条构造进行自适应等几何分析时，能够有效模拟相分离的标准模式。