Collaborative Research: Algorithm and Theory for Interface Computations

流体与浸入式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动、鸟类的飞翔、心脏中的血液流动以及树叶在风中的变形。理解此类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，这些问题要求对当前可用的算法进行重大改进。浸入边界（IB）方法是FSI问题的一种流行的计算方法，本项目的一个主要目标是了解IB方法的数学性质，以帮助开发更快、更鲁棒的数值算法。另一个目标是调整IB方法，使其能够处理流体（水）流过弹性结构的问题。这些问题对于理解生物细胞的运动和形状变化尤为重要。这项工作的细胞生物学方面将与实验生物物理学家合作进行。该项目将通过对这些问题的研究，培养数学和计算科学领域的本科生和研究生。本项目包括两个主要目标，在理论和算法开发的计算问题与移动膜界面。在理论方面，pi将建立浸入边界（IB）方法的收敛理论。IB方法是一种广泛应用于流固耦合问题的数值方法，但尽管它很受欢迎，但人们对其收敛性知之甚少。IB方法的收敛分析将是首先为使用双网格的流固相互作用算法建立的收敛分析之一；一个是流体结构，另一个是弹性结构。这样的分析将澄清网格和时间离散参数对IB方法稳定性的影响。在算法方面，pi将开发一个数值方案来处理在可变形弹性膜存在下离子的电扩散和跨膜水流。与传统的流体结构相互作用问题相比，渗透水流问题的一个新特征是，界面膜不随局部流体速度移动，而这种滑动速度是由膜界面上扩散的化学物质浓度的跳跃控制的。流体结构相互作用将用IB方法处理，而化学扩散将使用笛卡尔嵌入边界方法处理。该算法将应用于研究电生理/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较少但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 everyday 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 in the wind. One powerful way to understand such FSI problems is through computer simulation. Many FSI problems lead to challenging computational problems that call for 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 the work will be performed in collaboration with experimental biophysicists. The project will train undergraduate and graduate students in the mathematical and computational sciences through research on these problems.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.