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.

研究者继续开发自适应移动网格方法，用于时间依赖性的多二维差分方程的数值解决方案。研究工作将重点介绍一种新的移动网格方法，该方法是研究人员及其合作者提出的移动网格部分差异化方法。该方法已在一个和二维中实施，却忘记了忘记了非单独的结构化和非结构化的自适应网格，并成功地应用了许多问题。此外，该方法导致了一个统一的框架，描述了先前的方法，提供了新的理论基础和可靠的新方法。该提案的目标是提高二维方法的效率和鲁棒性，将其应用于实际问题，并实施三维方法。该项目与新的计算方法的发展有关，这些计算方法对于增强了科学家和开发人员解决大规模计算问题的能力至关重要，这些计算问题是我们的经济至关重要的，环境，环境和环境，环境，和环境，环境，和环境，环境，和环境，和环境，和环境，和环境，和环境，和环境。这项研究集中于自适应数值技术或网状适应方法的开发，其中要解决的特定问题的特殊运动特征适应了。网格的适应性最近扮演着必不可少的角色，对许多大规模问题的数值模拟，这些大规模问题是由于涉及冲击波，边界层，点火传播界和多材料界面而引起的许多大规模问题。这些问题具有独特的共同特征，即它们的解决方案仅在物理域的一小部分和解决方案的分辨率中发生显着变化，而在整个模拟的这一部分中，解决方案的分辨率。标准（非自适应）技术通常无法解决这些问题，因为它们在整个域上均匀地花费了努力，因此需要计算时间和记忆的大量资源才能获得合理的分辨率。另一方面，自适应网格方法，通过向解决方案的物理领域的一小部分变化，从而使大量的经济性获得了大量的经济性。研究下的移动网格方法是一种天然类型的自适应网格方法，该方法是设计了物理溶液的移动特征。 适用于并行计算的方法，已被证明是用于模拟许多工业制造问题的IndispensableTool。作为拟议研究项目的重要组成部分，将重点关注两个特定应用程序。首先是地下水含水层化学运输的数值模拟。地下水提供美国的大部分用水。地下污染的地下水质量质量的广泛降解最近引发了广泛的研究，可以实现和预测地下中的化学行为。移动网格方法的应用将提供准确，高效且可靠的数值算法，以在地下水中模拟地下水中的化学交通，从而有效地保护了地下资源，从而有效地保护了地下资源。另一个应用将对机翼动态摊位进行分析，以更好地理解在高攻击飞行条件下导致不稳定的流动行为的物理机制，发现了与现代战斗机和民用运输飞机。