Moving Mesh Methods for Numerical Solution of Time Dependent Partial Differential Equations in Two and Three Spatial Dimensions

二维和三维时变偏微分方程数值解的移动网格法

• 批准号: 0074240
• 负责人: Weizhang Huang
• 金额: $ 9万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074240&HistoricalAwards=false
• 关键词: Moving; Mesh; Methods; Numerical; Solution

The investigator continues to develop adaptive moving mesh methodsfor the numerical solution of time dependent, multi-dimensionalpartial differential equations. The research work will be focusedon a new moving mesh method, the moving mesh partial differentialequation approach proposed by the investigator and his collaborators.The approach has been implemented in one and two dimensions forgenerating non-singular structured and unstructured adaptive meshesand successfully applied to a number of problems. Moreover,the approach has led to a unifying framework describing previousmethods, providing a new theoretical underpinning, and buildingreliable new methods. The objectives of the proposal are tofurther improve the efficiency and robustness of the two dimensionalmethod, to apply it to practical problems, and to implementthe three dimensional method.This project is concerned with the development of new computationalmethods which are essential to enhance the ability of scientists andengineers to solve large scale computational problems that are crucialto our economy, environment, and security. The research is focused ondevelopment of adaptive numerical techniques or mesh adaptation methods,where the special moving features of the particular problem being solvedare adapted to. Mesh adaptation has recently played an indispensable rolein the numerical simulation of many large-scale problems arising fromscience, engineering, and industry, such as those involving shockwaves, boundary layers, ignition propagation fronts, and multi-materialinterface. These problems have a distinct common feature, that is, theirsolution changes significantly only in a small portion of the physicaldomain and the resolution of the solution in this portion dominatesthe quality of the whole simulation. Standard (non-adaptive) techniquesoften fail to solve these problems because they spend effort evenly onthe entire domain and thus require formidable resources of computerCPU time and memory to obtain a reasonable degree of resolution.On the other hand, adaptive mesh methods gain significant economiesby paying most attention to the small portion of the physical domainwhere the solution changes most. The moving mesh method under studyis a natural type of adaptive mesh methods which are designed tocapture the moving features of the physical solution. The methodis suitable for parallel computing and has proven to be an indispensabletool for use in the simulation of many industrial manufacturing problems.As an important part of the proposed research project, two specificapplications will be focused on. The first will be the numericalsimulation of chemical transport in groundwater aquifers.Groundwater supplies much of the water use in the UnitedStates. The wide spread degradation of groundwater quality fromchemical contamination has recently prompted extensive research forsimulating and predicting chemical behaviors in the subsurface.The application of the moving mesh methods will provide accurate,efficient, and robust numerical algorithms for simulating chemicaltransport in groundwater and therefore for effectively protectingand managing the groundwater resources. The other application will beon the analysis of dynamic stall of airfoil for better understandingthe physical mechanisms which cause the unsteady flowbehavior in the high-angle-of-attack flight condition found commonwith modern fighter and civil transport aircrafts.

研究人员继续发展自适应移动网格方法的数值解的时间依赖性，多维偏微分方程。本文的研究工作将集中在一种新的移动网格方法上，即由研究者及其合作者提出的移动网格偏微分方程方法，该方法已在一维和二维中实现，用于生成非奇异的结构和非结构自适应网格，并成功地应用于一些问题。此外，该方法已经导致了一个统一的框架，描述以前的方法，提供了一个新的理论基础，并建立可靠的新方法。该项目的目标是进一步提高二维方法的效率和鲁棒性，将其应用于实际问题，并实现三维方法。该项目涉及新的计算方法的发展，这些方法对于提高科学家和工程师解决对我们的经济，环境和安全至关重要的大规模计算问题的能力至关重要。研究的重点是自适应数值技术或网格自适应方法的发展，其中所解决的特定问题的特殊运动特征是适应的。网格自适应技术在科学、工程和工业领域的许多大规模问题的数值模拟中起着不可或缺的作用，如激波、边界层、点火传播前沿和多材料界面等。这些问题有一个共同的特点，即它们的解只在物理域的一小部分发生显著变化，而这一部分的解的分辨率决定了整个模拟的质量。标准（非自适应）技术往往不能解决这些问题，因为它们在整个域上均匀地花费精力，因此需要强大的计算机CPU时间和内存资源来获得合理程度的分辨率。另一方面，自适应网格方法通过将大多数注意力集中在解决方案变化最大的物理域的一小部分而获得显着的经济效益。移动网格法是一种自然的自适应网格法，它的目的是捕捉物理解的移动特征. 该方法适用于并行计算，并已被证明是一个不可缺少的工具，用于模拟许多工业制造问题。作为一个重要的一部分，拟议的研究项目，两个具体的应用程序将集中在。第一个将是数值模拟的化学品输运在地下水含水层。地下水供应了大量的水在美国的使用。地下水化学污染引起的水质恶化引起了人们对地下水化学行为模拟和预测的广泛研究，移动网格方法的应用将为地下水化学迁移模拟提供准确、高效和鲁棒的数值算法，从而有效地保护和管理地下水资源。另一个应用是分析翼型的动态失速，以便更好地理解现代战斗机和民用运输机常见的大迎角飞行条件下引起非定常流动行为的物理机制。