Mathematical Sciences: Immersed Interface Methods

数学科学：沉浸式接口方法

• 批准号: 9626645
• 负责人: Randall LeVeque
• 金额: $ 24万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626645&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Immersed; Interface; Methods

LeVeque 9626645 The investigator and his colleagues develop immersed interface methods and apply them to a variety of problems. Immersed interface methods are a class of methods for solving partial differential equations whose solutions have discontinuities or nonsmoothness across some interface(s). The idea is to use a uniform Cartesian grid in spite of the fact that the interface may cut between grid points. The goal of this work is to develop finite difference methods in two and three space dimensions that give highly accurate solutions at all grid points while maintaining the efficiency and ease of implementation of uniform grid methods. Pointwise second order accurate methods have already been developed for several classes of problems, including certain elliptic, parabolic, and hyperbolic equations with discontinuous coefficients across fixed interfaces. Stokes flow and solidification problems with moving interfaces have also been solved. Extensions to a number of other problems are currently underway. Many important practical problems lead to differential equations in regions of 2- or 3-dimensional space that are geometrically complicated, and that contain interfaces across which the nature of the solution changes. These equations can rarely be solved exactly, and large-scale computation is required to obtain well-resolved solutions over multi-dimensional regions. The goal of this work is to develop efficient computational methods to approximate solutions of such problems. Several specific problems are studied in depth: (1) One goal is to develop improved methods for incompressible fluid dynamics in regions with complicated moving elastic boundaries, based on a method of Peskin's that is widely used in biophysical and physiological modelling, e.g., of the heart, blood vesssels, the inner ear, etc. (2) Wave equations in heterogeneous materials arise in numerous applications, including ultrasound imaging and therapy, ocean acoustics, and seismic p ropagation in the earth. The "forward problem" consists of solving the equations forward in time given the material properties and locations of interfaces. The harder "inverse problem" requires determining the material properties and locations of interfaces from measurements made at the surface. Ultrasound imaging requires solving the inverse problem for acoustics equations. Seismic exploration of the earth, heavily used in the search for oil, for example, requires solving inverse problems for elastic wave equations or for simplified travel-time equations. The methods developed in this work are applied to both forward and inverse problems. (3) In electrical impedance tomography, an inverse problem is solved for the electrical conductivity of an object. This technique is used in medical imaging and is being studied for the purpose of locating unexploded land mines, a major health risk in many countries today. Inverse problems of this type are being studied in this work. (4) Porous media equations arising in oil reservoir simulation and groundwater transport requiremodeling discontinuities in permeability and porosity at geological interfaces, as well as moving front representing interfaces between oil and water, for example. Accurate modeling of groundwater flow is important in studying contamination by toxic wastes and remediation of contaminated sites. (5) Multi-phase solidification problems involve moving interfaces between the two phases (e.g. ice and water) and may be coupled with fluid dynamics effects as well as heat transfer. Practical applications include studying the effect of ocean currents on ice shelves, injection molding in manufacturing processes, and chemical etching.

LeVeque 9626645 研究人员和他的同事开发了浸入式界面方法并将其应用于各种问题。 浸入界面法是一类求解偏微分方程的方法，其解在某些界面上具有不连续性或不光滑性。 这个想法是使用统一的笛卡尔网格，尽管事实上界面可能在网格点之间切割。 这项工作的目标是开发两个和三个空间维度的有限差分方法，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。 已经针对几类问题开发了逐点二阶精确方法，包括某些在固定界面上具有不连续系数的椭圆、抛物线和双曲方程。 移动界面的斯托克斯流动和凝固问题也已得到解决。 目前正在解决许多其他问题。 许多重要的实际问题都会导致几何复杂的 2 维或 3 维空间区域中的微分方程，并且包含解性质发生变化的界面。 这些方程很少能够精确求解，并且需要大规模计算才能在多维区域上获得良好解析的解。 这项工作的目标是开发有效的计算方法来近似解决此类问题。 深入研究了几个具体问题：（1）一个目标是在具有复杂移动弹性边界的区域中开发不可压缩流体动力学的改进方法，该方法基于广泛应用于生物物理和生理建模的 Peskin 方法，例如， (2) 异质材料中的波动方程出现在许多应用中，包括超声成像和治疗、海洋声学以及地球中的地震传播。 “正向问题”包括在给定材料属性和界面位置的情况下及时求解方程。 更难的“反问题”需要根据表面测量来确定材料特性和界面位置。 超声成像需要解决声学方程的反问题。 例如，广泛用于寻找石油的地球地震勘探需要求解弹性波方程或简化的走时方程的反演问题。 这项工作中开发的方法适用于正向和逆向问题。 (3)在电阻抗断层扫描中，求解物体的电导率的反问题。 该技术用于医学成像，并且正在研究用于定位未爆炸的地雷，这是当今许多国家的主要健康风险。 这项工作正在研究这种类型的反问题。 (4) 例如，油藏模拟和地下水输送中出现的多孔介质方程需要对地质界面处渗透率和孔隙度的不连续性以及代表油和水之间界面的移动前沿进行建模。 地下水流的准确建模对于研究有毒废物污染和污染场地修复非常重要。 (5) 多相凝固问题涉及两相（例如冰和水）之间的移动界面，并且可能与流体动力学效应以及传热耦合。 实际应用包括研究洋流对冰架的影响、制造过程中的注塑以及化学蚀刻。