A Theoretical and Algorithmic Study of the Immersed Boundary Method

该提案是使用2009年美国复苏和再投资法案（公法111-5）提供的资金授予的。浸入边界法是一种用于流固耦合问题数值模拟的常用方法。尽管浸入边界法很受欢迎，但其收敛性却知之甚少。本文的目标是建立浸入边界法的收敛理论，并发展有效的隐式方法来克服应用中经常遇到的数值刚度。浸入边界法的一个显著特点是使用正则化的狄拉克δ函数来处理计算域内尖锐界面的存在。这个想法现在被广泛使用的浸没边界方法以外的多相和复杂的流动模拟的背景下，前跟踪和水平集方法。这里发展的理论也应该有助于我们理解这种方法。浸入边界法应用中的主要困难之一是数值刚度，需要从业者使用非常小的时间步长，以避免数值不稳定。建立在以前的工作，有效的隐式方法将被开发，以克服这个困难。现象，流体和弹性材料相互作用在自然界和工程丰富。重要的例子包括心脏中的血液流动，鸟类和昆虫的游泳和飞行，以及工业过程中出现的流动。计算研究对于理解这类现象是非常重要的，而浸入边界法是模拟这类问题最流行的计算方法之一。在这个项目中，我们寻求开发更准确和有效的方法来执行浸入式边界计算。其中一个重要的组成部分是开发一个理论框架，以了解浸入边界计算的收敛特性。这样的收敛理论应作为一个坚实的基础上，开发更好的算法来模拟流动与浸没的弹性结构。

This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The immersed boundary method is a popular numerical scheme to simulate problems problems of fluid-structure interaction. Despite the popularity of the immersed boundary method, its convergence properties are poorly understood. The goal of this proposal is to build a convergence theory of the immersed boundary method and to develop efficient implicit methods to overcome numerical stiffness often encountered in application. A salient feature of the immersed boundary method is the use of regularized Dirac delta functions to handle the presence of sharp interfaces within the computational domain. This idea is now widely used beyond the immersed boundary method in multiphase and complex flow simulations in the context of front-tracking and level-set methods. The theory developed here should also contribute toour understanding of such methods. One of the major difficulties in applications of the immersed boundary method is numerical stiffness, necessitating the practitioner to use very small time steps to avoid numerical instability. Building upon previous work, efficient implicit methods will be developed to overcome this difficulty.Phenomena in which fluid and elastic materials interact abound in nature and engineering. Important examples include blood flow in the heart, swimming and flying of birds and insects, and flows that arise in industrial processes. Computational investigation is very important in understanding such phenomena, and the immersed boundary method is one of the most popular computational methods to simulate such problems. In this project, we seek to develop more accurate and efficient ways to perform immersed boundary computations. One important ingredient for this is to develop a theoretical framework to understand convergence properties of immersed boundary computations. Such a convergence theory should serve as a firm foundation upon which to develop better algorithms to simulate flows with immersed elastic structures.