I-MCHM：用于多尺度椭圆问题的界面多连续体 homogenization 方法
I-MCHM: Interface Multicontinuum Homogenization for Multiscale Elliptic Problems
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中文总结 AI 辅助
针对高对比度多尺度椭圆问题，提出I-MCHM方法，通过构造双边约束局部基与界面修正的粗双线性形式，实现无逐点传输律的精准升尺度，经多几何数值实验验证有效性。
中文摘要 AI 辅助
我们提出了一种用于高对比度椭圆问题的界面多连续体 homogenization 方法（I-MCHM），该问题的子域可具有不同的微观模式和不等数量的连续体。多连续体模型将微观场升尺度为宏观连续体量；此处的连续体为高渗透率和低渗透率区域，宏观变量是在尺寸为 $H_\epsilon$ 的观测单元上精细尺度解的相应局部平均值。尽管精细尺度解在材料界面处连续，但这些宏观连续体场在界面处不一定重合。此外，当相邻子域保留不同数量的连续体时，无法定义一一对应的粗传输条件。因此，标准的子域级多连续体 homogenization 可确定体方程，但未明确这些粗变量之间的相互作用。I-MCHM 通过在界面邻域构造双边约束局部基，并在子域邻接图上组装包含体切割单元积分和界面段修正的粗双线性形式，从而填补了这一空白，且无需指定逐点传输律。对直线、曲线和三结几何（包括连续体耦合消融和不等连续体数量）的数值实验，证明了所得升尺度模型的准确性。
英文摘要
We introduce an interface multicontinuum homogenization method (I-MCHM) for high-contrast elliptic problems whose subdomains may have distinct microscopic patterns and unequal numbers of continua. Multicontinuum models upscale microscopic fields to macroscopic continuum quantities; here the continua are the high- and low-permeability regions, and the macroscopic variables are the corresponding local averages of the fine-scale solution on observing cells of size $H_ε$. Although the fine-scale solution is continuous across a material interface, these macroscopic continuum fields need not coincide on the interface. Moreover, when adjacent subdomains retain different numbers of continua, a one-to-one coarse transmission condition cannot be defined. Standard subdomain-wise multicontinuum homogenization therefore determines the bulk equations but leaves the interaction among these coarse variables unspecified. I-MCHM closes this gap by constructing two-sided constrained local bases on interface neighborhoods and assembling a coarse bilinear form with bulk cut-cell integrals and interface-segment corrections over the subdomain adjacency graph, without prescribing a pointwise transmission law. Numerical experiments on straight, curved, and triple-junction geometries, including continuum-coupling ablations and unequal continuum counts, demonstrate the accuracy of the resulting upscaled model.
