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

流固耦合是生物问题和工程应用中的常见问题。在许多令人感兴趣的情况下，例如红细胞在微循环中的运动和心脏瓣膜的打开和关闭，近接触结构之间的小范围流体流动对整体行为是重要的。这些问题中多个尺度的存在给建模和仿真带来了挑战。虽然现有的方法，如自适应网格加密使用算法技术来弥合这些尺度，并导致大量的计算成本，但在这里，研究开发了另一种方法，在小尺度上用一个简单得多的渐近极限来代替控制方程。本项目的目标是开发一种改进的浸没边界方法，该方法使用润滑理论来模拟近接触结构之间的流体流动。浸没边界法是一种广泛应用的流固耦合数值模拟方法。这种方法的一个长期局限性是，接近接触的结构往往会粘在一起，因此它们需要超大的力才能分开。浸没润滑边界法采用润滑理论作为亚网格模型，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接近于零的极限情况下是准确的，所以随着结构之间的距离越来越近，亚网格模型的精度也会提高。本研究对润滑浸没边界法进行了分析和基准。具体目标包括执行多尺度数值分析，开发三维高效实现，以及基于沉浸式界面方法调查更高阶的准确扩展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.