Fast Algorithms for Simulating the Collective Swimming of Microorganisms

流体动力学相互作用在微生物的集体动力学中起着至关重要的作用。该项目的主要目标是探索和开发有效的算法来模拟一大群微生物在三维粘性流体中的集体游泳。该项目产生的计算方法将为理解精子和纤毛等微游泳者如何在人体内共同执行各种生理功能提供新的工具。它们还将有助于揭示细菌和藻类等微生物聚集并形成菌落的方式和原因。微生物通常必须在粘液和聚合物等弹性结构中穿行；它们在非牛顿流体中的运动性近年来引起了极大的兴趣。所提出的方法可以推广到研究粘弹性网络内微生物的集体游动。除了水动力相互作用外，空间相互作用和化学相互作用也对微生物的集体行为产生了深远的影响。该项目为开发包含各种相互作用的集体动力学的更全面的数学模型奠定了基础。在足够高的密度下，微游泳者，如细菌、纤毛、精子和藻类，表现出显著的集体运动，具有重要的生物学意义。例如，精子在游向卵子的过程中既有竞争，也有合作；气道中的纤毛共同跳动，将黏液和外来颗粒赶出肺部。这些现象可以用几种方法来模拟，所有这些方法都需要求解流固耦合方程。其中，正则化Stokeslets方法和Rotne-Prager-Yamakawa张量方法的优点是不需要三维欧拉网格，而是使用底层方程的基本解。然而，这两种方法所需的计算都需要使用密集矩阵，并且对于微型游泳者数量很大的实际模型来说，它们往往很大，而且成本很高。此外，在游动的微生物中，模式可能需要很长时间才能出现和发展，这使得模拟更具挑战性。该项目有以下三个目标：1。开发快速计算矩阵-向量乘积的算法，以及用上述大而密集的矩阵求解线性系统的算法；利用这些算法来研究(a)被一个表面限制的一大群精子的集体游动，(b)稠密的跳动的纤毛所引起的流场，(c)大量自由的微型游泳者所引起的流场；采用并行时间方法加速集体游泳的时变仿真并在使用数百个计算机内核时比较空间和时间并行化的效率。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。

Hydrodynamic interactions play a crucial role in the collective dynamics of microorganisms. The main goal of this project is to explore and develop efficient algorithms for simulating the collective swimming of a large group of microorganisms in a three-dimensional viscous fluid. The computational methods resulting from this project will provide new tools to understand how micro-swimmers such as sperm and cilia collectively perform various physiological functions inside the human body. They will also help shed new light on how and why microorganisms such as bacteria and algae aggregate and form colonies. Microorganisms often have to navigate through elastic structures such as mucus and polymers; their motility in a non-Newtonian fluid has attracted significant interest in recent years. The proposed methods can be extended to study the collective swimming of microorganisms inside a viscoelastic network. Besides hydrodynamic interactions, steric and chemical interactions also have profound impacts on the collective behaviors of microorganisms. This project lays a foundation for the development of more comprehensive mathematical models for collective dynamics that incorporate a variety of interactions.At high enough density, micro-swimmers, such as bacteria, cilia, sperm and algae, exhibit remarkable collective motions which bear significant biological implications. For example, sperm swim both competitively and collaboratively to reach the egg, and cilia in the airways beat collectively to propel mucus and foreign particles out of the lung. These phenomena can be modeled by several methods, all of which entail solving equations of fluid-structure interaction. Among them, the method of regularized Stokeslets and the Rotne-Prager-Yamakawa tensor have the advantage of not requiring a 3D Eulerian grid and using the fundamental solutions to the underlying equations instead. However, the computations required by both methods entail the use of dense matrices, and they tend to be large and very costly to work with for practical models in which the number of micro-swimmers is large. In addition, patterns can take a long time to emerge and develop in swimming microorganisms, making the simulation even more challenging. The project has the following three objectives:1.Develop fast algorithms for computing matrix-vector products and for solving linear systems with the aforementioned large, dense matrices;2.Employ these algorithms to investigate (a) the collective swimming of a large group of sperm confined by a surface, (b) the flow field induced by a dense mat of beating cilia, and (c) the flow field induced by a large number of free micro-swimmers;3.Accelerate the time-dependent simulation of collective swimming by implementing parallel-in-time methods; and compare the efficiency of spatial and temporal parallelization when hundreds of computer cores are used.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.