Fast Algorithms for Simulating the Collective Swimming of Microorganisms

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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.