This work presents a novel unfitted finite element framework to simulate coupled surface-bulk problems in time-dependent domains, focusing on fluid-fluid interactions in animal cells between the actomyosin cortex and the cytoplasm. The cortex, a thin layer beneath the plasma membrane, provides structural integrity and drives shape changes by generating surface contractile forces akin to tension. Cortical contractions generate Marangoni-like surface flows and induce intracellular cytoplasmic flows that are essential for processes such as cell division, migration, and polarization, particularly in large animal cells. Despite its importance, the spatiotemporal regulation of cortex-cytoplasm interactions remains poorly understood and computational modelling can be very challenging because surface-bulk dynamics often lead to large cell deformations. To address these challenges, we propose a sharp-interface framework that uniquely combines the trace finite element method for surface flows with the aggregated finite element method for bulk flows. This approach enables accurate and stable simulations on fixed Cartesian grids without remeshing. The model also incorporates mechanochemical feedback through the surface transport of a molecular regulator of active tension. We solve the resulting mixed-dimensional system on a fixed Cartesian grid using a level-set-based method to track the evolving surface. Numerical experiments validate the accuracy and stability of the method, capturing phenomena such as self-organised pattern formation, curvature-driven relaxation, and cell cleavage. This novel framework offers a powerful and extendable tool for investigating increasingly complex morphogenetic processes in animal cells.

翻译：本文提出了一种新颖的非拟合有限元框架，用于模拟时变域中的耦合表面-体相问题，重点关注动物细胞中肌动球蛋白皮层与细胞质之间的流-流相互作用。皮层是位于质膜下方的薄层，通过产生类似于张力的表面收缩力来维持结构完整性和驱动形状变化。皮层收缩产生类似马兰戈尼效应的表面流，并诱导胞内细胞质流动，这对细胞分裂、迁移和极化等过程至关重要，尤其在大型动物细胞中。尽管其重要性显著，但皮层-细胞质相互作用的时空调控机制仍不明确，且由于表面-体相动力学常导致细胞大幅变形，计算建模极具挑战性。为解决这些问题，我们提出了一种锐界面框架，该框架独特地结合了用于表面流的迹有限元方法与用于体相流的聚合有限元方法。该方法无需重新网格化即可在固定笛卡尔网格上实现精确稳定的模拟。模型还通过活性张力分子调控因子的表面输运纳入了机械化学反馈。我们采用基于水平集的方法在固定笛卡尔网格上求解所得混合维度系统，以跟踪演化中的表面。数值实验验证了该方法的精度与稳定性，能够捕捉自组织模式形成、曲率驱动弛豫以及细胞分裂等现象。这一新颖框架为研究动物细胞中日益复杂的形态发生过程提供了强大且可扩展的工具。