In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries. The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint inclusions. The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information. We use the Hanzawa transformation to transform the problem onto a fixed reference domain. Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically. We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters. Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem. We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations.

翻译：本研究采用解析与数值技术研究具有移动边界的相变模型。该模型呈现两个相关的空间尺度，分别对应宏观相与微观相，二者通过不相交夹杂物相互作用。夹杂物的收缩或增长遵循修正的吉布斯-汤姆逊定律，该定律依赖于宏观温度但无需曲率信息。我们采用Hanzawa变换将问题转换至固定参考域，随后运用不动点论证证明了系统在有限时间区间内的适定性。由于模型的非线性特性以及宏观参数（由依赖于夹杂物尺寸的微分方程给出），该问题的数值求解计算成本高昂。本文提出一种预计算法：在离线阶段求解多个单元问题，随后采用插值方案确定所需参数。此外，我们提出半隐式时间步进法以处理问题的非线性特性。我们系统分析了预计算与时间步进过程的误差，并通过数值模拟验证了理论结果。