Mathematical Sciences: Immersed Interface Methods

数学科学：沉浸式接口方法

• 批准号: 9303404
• 负责人: Randall LeVeque
• 金额: $ 22万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-08-15 至 1997-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303404&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Immersed; Interface; Methods

9303404 LeVeque Many important practical problems lead to partial differential equations whose solutions have discontinuities or nonsmoothness across some interface. In solving these problems numerically, it is very convenient to use a uniform Cartesian grid in spite of the fact that the interface may cut between grid points. The investigator and his colleagues develop finite difference methods that aim to give highly accurate solutions at all grid points while maintaining the efficiency and ease of implementation of uniform grid methods. Several specific problems are studied in depth. One goal is to develop improved versions of Peskin's "immersed boundary method" for incompressible fluid dynamics in regions with complicated moving boundaries, such as blood flow in a beating heart. Hyperbolic wave equations with discontinuous coefficients arise in modeling the structure of the earth in seismic oil exploration. Other applications include porous media equations arising in oil reservoir simulation and groundwater transport, and multi-phase solidification problems. In modeling fluid motion near rigid boundaries, an additional problem arises in solving an ill-conditioned system of equations for the strength of discontinuities at the boundary. Iterative methods for solving such problems are studied. Simulating the behavior of complicated structures in the real world typically involves solving large systems of equations that cannot be solved exactly. Instead the solution must be approximated by numerical methods on high performance computers. The investigators study problems in which there is a boundary or interface that has a complicated shape and may be moving in time. Examples include the surface of a body of water or a bubble surrounded by fluid, the surface of a beating heart, or the boundary between oil and some fluid that is injected into the earth to force oil out of an oil field. In these examples the equations being solved model the flow o f some fluid. Another problem is to study the motion of the boundary between melting ice and the surrounding water, or between different phases of a substance more generally. In this case the equations model the conduction of heat. In oil exploration it is necessary to model the motion of seismic waves in the earth and the manner in which they reflect off interfaces between different kinds of rock deep within the earth. The goal of the project is to develop relatively simple methods that can be used to solve a wide variety of such problems with complicated boundaries or interfaces. ***

9303404 LeVeque 许多重要的实际问题都会导致偏微分方程的解在某些界面上具有不连续性或不光滑性。 在数值求解这些问题时，尽管网格点之间的界面可能会切割，但使用统一的笛卡尔网格是非常方便的。 研究人员和他的同事开发了有限差分方法，旨在在所有网格点上提供高精度的解决方案，同时保持统一网格方法的效率和易于实施。 对几个具体问题进行了深入研究。 一个目标是开发 Peskin 的“浸没边界法”的改进版本，用于具有复杂移动边界的区域（例如跳动的心脏中的血流）中的不可压缩流体动力学。 具有不连续系数的双曲波动方程出现在地震石油勘探中的地球结构建模中。 其他应用包括油藏模拟和地下水输送中出现的多孔介质方程以及多相凝固问题。 在模拟刚性边界附近的流体运动时，在求解边界处不连续强度的病态方程组时会出现另一个问题。 研究了解决此类问题的迭代方法。 模拟现实世界中复杂结构的行为通常涉及求解无法精确求解的大型方程组。 相反，必须在高性能计算机上通过数值方法来近似解决方案。研究人员研究的问题中存在形状复杂且可能随时间移动的边界或界面。例如，水体的表面或被流体包围的气泡、跳动的心脏的表面或石油与注入地球以迫使石油流出油田的某些流体之间的边界。 在这些示例中，所求解的方程模拟了某些流体的流动。 另一个问题是研究融化的冰与周围的水之间的边界的运动，或者更广泛地研究物质的不同相之间的边界的运动。 在这种情况下，方程模拟了热传导。 在石油勘探中，有必要对地震波在地球中的运动以及地震波在地球深处不同类型岩石之间的界面反射的方式进行建模。 该项目的目标是开发相对简单的方法，可用于解决具有复杂边界或界面的各种此类问题。 ***