Collaborative Research: Algorithm and Theory for Interface Computations

流体与浸入水中的弹性结构相互作用的现象在自然界和日常生活中比比皆是。这类流体-结构相互作用(FSI)问题包括微生物的游泳、鸟类的飞行、心脏中的血液流动以及树叶在风中的变形。了解此类FSI问题的一个有效方法是通过计算机模拟。许多FSI问题导致具有挑战性的计算问题，需要在现有算法的基础上进行重大改进。浸没边界(IB)方法是FSI问题的一种常用计算方法，本项目的主要目的之一是了解IB方法的数学性质，以帮助开发更快、更健壮的数值算法。另一个目标是调整IB方法，使其能够处理流体(水)可以通过弹性结构的问题。这些问题对于理解生物细胞的运动和形状变化尤为重要。我们工作的这一细胞生物学方面将与实验生物物理学家合作进行。该项目将在这些问题以及更广泛的数学和计算科学方面介绍和培训本科生和研究生。该项目包括两个主要目标，即理论和算法开发，以解决具有移动膜界面的计算问题。在理论方面，PI将建立浸没边界(IB)方法的收敛理论。IB方法是一种广泛应用于流体结构相互作用问题的数值方法，但它的收敛性质却鲜为人知。IB方法的收敛分析将是为流体结构相互作用算法建立的第一批方法之一，在该算法中使用了双重网格；一个用于流体，另一个用于弹性结构。这样的分析将阐明网格和时间离散化参数对IB方法稳定性的影响。在算法方面，PI将开发一种数值方案来处理离子的电扩散和在存在可变形弹性膜的情况下的跨膜水流动。与传统的流体结构相互作用问题不同，渗透水流动问题的一个新特征是界面膜不相对于局部流体速度运动，并且这种滑移速度由跨膜界面扩散的化学物质浓度的跳跃控制。流体结构相互作用用IB方法处理，而化学扩散用笛卡尔嵌入边界方法处理。这一算法将被应用于研究电生理/渗透水流和细胞力学之间的相互作用，这是一个理论上探索很少但其重要性日益明显的领域。

Phenomena in which a fluid interacts with an immersed elastic structure abound in nature and in every day life. Such fluid-structure interaction (FSI) problems include the swimming of microorganisms, the flying of birds, blood flow in the heart and the deformation of leaves to the wind. One powerful way to understand such FSI problems is through computer simulation. Many FSI problems lead to challenging computational problems that require significant improvements over currently available algorithms. The immersed boundary (IB) method is a popular computational method for FSI problems, and one major goal of this project is to understand the mathematical properties of the IB method to aid in the development of faster and more robust numerical algorithms. Another goal is to adapt the IB method so that it can handle problems in which fluid (water) can flow through an elastic structure. Such problems are particularly important for the understanding of movement and shape changes of biological cells. This cell-biological aspect of our work will be performed in collaboration with experimental biophysicists. The project will introduce and train undergraduate and graduate students in these problems and more broadly in the mathematical and computational sciences. This project consists of two major aims in theory and algorithmic development for computational problems with moving membrane interfaces. On the theoretical side, the PIs will establish a convergence theory for the immersed boundary (IB) method. The IB method is a widely used numerical method for fluid structure interaction problems, but despite its popularity, its convergence properties are poorly understood. Convergence analysis for the IB method will be one of the first to be established for a fluid structure interaction algorithm in which a dual grid is used; one for the fluid and another for the elastic structure. Such an analysis will clarify the effect of grid and time discretization parameters on the stability properties of the IB method. On the algorithmic side, the PIs will develop a numerical scheme to handle electrodiffusion of ions and transmembrane water flow in the presence of deformable elastic membranes. A novel feature of the osmotic water flow problem in contrast to conventional fluid structure interaction problems is that the interfacial membrane does not move with respect to the local fluid velocity and that this slip velocity is controlled by the jump in concentration of a diffusing chemical across the membrane interface. The fluid structure interaction will be treated with the IB method whereas chemical diffusion will be treated using a cartesian embedded boundary method. This algorithm will be applied to study the interplay between electrophysiology/osmotic water flow and cell mechanics, an area that is poorly explored theoretically but whose importance is becoming increasingly clear.