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调查员及其同事开发了浸泡的界面方法，并将其应用于各种问题。 沉浸式界面方法是解决偏微分方程的一类方法，其解决方案在某些界面上具有不连续性或非平滑度。 这个想法是使用统一的笛卡尔网格，尽管界面可能会在网格点之间切开。 这项工作的目的是在两个和三个空间维度上开发有限的差异方法，这些方法在所有网格点上提供了高度准确的解决方案，同时保持统一网格方法的效率和易于实施。 已经开发了针对多种问题的二阶准确方法，包括某些椭圆，抛物线和双曲线方程，并在固定接口之间具有不连续系数。 也已经解决了移动界面的Stokes流量和凝固问题。 目前正在进行许多其他问题的扩展。 许多重要的实际问题导致在几何复杂的2-或3维空间区域的微分方程中，并且包含溶液性质在变化的接口。 这些方程式很少可以准确地解决，并且需要进行大规模计算以在多维区域获得良好的解决方案。 这项工作的目的是开发有效的计算方法来近似此类问题的解决方案。 深入研究了几个特定问题：（1）一个目标是根据一种基于佩斯金的方法，开发具有复杂移动弹性界限的区域中不可压缩的流体动力学方法，该方法广泛用于生物物理和生理模型中海洋声学和地球上的震荡。 “正向问题”包括在界面的材料属性和位置，在时间内求解方程。 更难的“反问题”需要从表面进行的测量结果确定接口的材料特性和位置。 超声成像需要解决声学方程的逆问题。 例如，在寻找石油的大量使用地球的地震探索需要解决弹性波方程或简化的旅行时间方程的反问题。 这项工作中开发的方法都应用于前进和反问题。 （3）在电阻抗断层扫描中，解决了物体的电导率的逆问题。 该技术用于医学成像，并正在研究目的是为了找到未爆炸的土地矿山，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）在石油储层模拟和地下水传输中产生的多孔介质方程要求在地质界面的渗透性和孔隙率中不连续，以及移动前面代表油与水之间的界面。 地下水流的准确建模对于研究有毒废物污染和修复受污染部位的污染很重要。 （5）多相凝固问题涉及两个阶段之间的界面（例如冰和水），并且可以与流体动力学效应以及热传递相结合。 实际应用包括研究洋流对冰架的影响，制造过程中的注射成型以及化学蚀刻。