Lubricated Immersed Boundary Method: Numerical Analysis, Benchmarking, and Applications

流固耦合在生物问题和工程应用中很常见。在许多感兴趣的情况下，例如红细胞通过微循环的运动和心脏瓣膜的打开和关闭，近接触结构之间的小规模流体流动对整体行为很重要。这些问题中多尺度的存在给建模和仿真都带来了挑战。而现有的方法，如自适应网格细化使用算法技术来桥接这些尺度并产生显著的计算成本，在这里的研究开发了一种替代方法，其中控制方程在小尺度上被更简单的渐近极限取代。本项目的目标是开发一种改进的浸入边界方法，该方法使用润滑理论来模拟近接触结构之间的流体流动。浸入边界法是一种广泛应用的流固耦合建模方法。这种方法长期存在的一个局限性是，近距离接触的结构往往会粘在一起，因此它们需要非物理的大力才能分离。润滑浸入边界法利用润滑理论作为子网格模型，克服了经典方法的这一困难。由于润滑理论在两个结构之间的间距趋于零的极限下是精确的，因此随着结构之间的距离越来越近，子网格模型的精度也会提高。本文对润滑浸入边界法进行了分析和基准测试。具体目标包括进行多尺度数值分析，开发三维有效实现，以及研究基于浸入界面方法的高阶精确扩展。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluid-structure interaction is common in biological problems and engineering applications. In many cases of interest, such as red blood cell motion through the microcirculation and the opening and closing of the heart valves, small-scale fluid flows between structures in near-contact are important for the overall behavior. The presence of multiple scales in these problems poses challenges for both modeling and simulation. Whereas existing methods such as adaptive mesh refinement use algorithmic techniques to bridge these scales and incur significant computational costs, here the research develops an alternative method in which the governing equations are replaced at small scales by a much simpler asymptotic limit.The goal of this project is to develop an improved version of the immersed boundary method that uses lubrication theory to model fluid flows between structures in near contact. The immersed boundary method is a widely used approach for modeling fluid-structure interaction. A longstanding limitation of this method is that structures that are in near contact tend to stick together, so that they require unphysically large forces to separate. The lubricated immersed boundary method uses lubrication theory as a subgrid model to overcome this difficulty of the classical method. Because lubrication theory is exact in the limit that the spacing between two structures goes to zero, the accuracy of the subgrid model improves as structures come closer together. This research analyzes and benchmarks the lubricated immersed boundary method. Specific aims include performing a multiscale numerical analysis, developing efficient implementations in three dimensions, and investigating higher-order accurate extensions based on the immersed interface method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.