Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations struggle with topology changes, and per-cell representations are computationally too demanding for large-scale simulations. To address these challenges, we propose a novel cell-centered approach based on differentiable Voronoi diagrams. Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly. In this way, we substantially reduce the number of problem variables, eliminate the need for explicit contact handling, and ensure continuous geometry changes during topological transitions. Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies. Finally, we extend our differentiable Voronoi diagrams to enable coupling with arbitrary rigid and deformable boundaries. We apply our approach to a diverse set of examples, highlighting splitting and merging of cells as well as neighborhood changes. We illustrate applications to inverse problems by matching soap foam simulations to real-world images. Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times.

翻译：在细胞机械系统中导航拓扑转变是现有模拟方法面临的重大挑战。虽然抽象模型缺乏细胞层面的预测能力，但显式网络表示难以处理拓扑变化，而逐细胞表示对于大规模模拟而言计算成本过高。为应对这些挑战，我们提出一种基于可微Voronoi图的新型细胞中心方法。该方法通过将每个细胞表示为Voronoi站点，隐式定义界面网络的形状与拓扑。通过这种方式，我们大幅减少了问题变量数量，消除了显式接触处理的必要性，并确保拓扑转变过程中几何变化的连续性。网络位置的闭式导数使得采用牛顿类方法对各类逐细胞能量进行模拟成为可能。最后，我们扩展了可微Voronoi图以支持与任意刚性和可变形边界的耦合。我们将方法应用于包含细胞分裂、融合以及邻域变化等多种实例，并通过将肥皂泡模拟结果匹配到真实图像来展示其在逆问题中的应用。与显式细胞模型的对比分析表明，我们的方法在显著提升计算速度的同时实现了定性可比的结果。