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: 9626107
• 负责人: Weizhang Huang
• 金额: $ 5.78万
• 依托单位: University of Kansas Main Campus
• 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=9626107&HistoricalAwards=false
• 关键词: Moving; Mesh; Methods; Numerical; Solution

9626107 Huang The investigator develops a moving mesh method for the numerical solution of two- and three-dimensional time-dependent PDEs (partial differential equations) that extends the MMPDE (Moving Mesh PDE) approach introduced by the investigator and his coworkers for PDEs in one spatial dimension. With this approach, an MMPDE is formulated explicitly based upon the heat flow equation and the theory of harmonic maps. The MMPDE is employed to move the mesh, to concentrate nodes in regions where the physical solution has large variations, and to control certain mesh qualities, in particular the skewness. Techniques are investigated to discretize and solve efficiently the extended system consisting of the underlying physical PDE and the MMPDE. A number of related issues are considered - e.g., the computation of a reference mesh for an arbitrary connected domain, the use of flow directional control, the study of preconditioning techniques, the examination of the space-time finite element method, and the investigation of robustness and reliability of the method. Moreover, the method is applied to problems in airfoil analysis and wing design, moving boundary problems, and problems involving blow-up solutions. The new algorithms and computational techniques are simultaneously implemented in mathematical software that could find widespread usage among scientists and engineers. This project is concerned with the development of new computational methods that are essential to enhance the ability of scientists and engineers to solve large scale computational problems that are crucial to our economy, environment, and security. The research is focused on development of so-called adaptive numerical techniques, where the special features of the particular problem being solved are adapted to. More specifically, the places (mesh points) where the solution to the scientific problem is being approximated are moved with time to adapt to the changes in the solu tion. Mesh adaptation has recently played an indispensable role in the numerical solution of the problems, as, supercomputers notwithstanding, such problems can generally not otherwise be solved satisfactorily. This is because in many problems of science and engineering, there is a small portion of the physical domain where large changes in the solution occur over very small separations in the mesh. Numerical solution of these problems using fixed uniform meshes is formidable because millions of mesh points are required to resolve the physical phenomena. On the other hand, use of adaptive mesh methods can significantly reduce the number of mesh points and thus economies can be gained. Unlike other adaptive mesh methods, the moving mesh method under study changes the mesh smoothly in time to adapt to the key solution features. It is suitable for parallel computing. The moving mesh method should be very useful in the numerical simulation of many industrial manufacturing problems. A particular area is airfoil analysis and wing design. For example, in the airfoil and wing design the airfoil or wing shape is computed when a pressure distribution is specified. It will save computing resources tremendously if the flow solution and the computational mesh, and hence the airfoil shape, can be updated simultaneously, as these methods do. This advantage holds for a variety of other applications as well, such as the problem of studying how and why industrial and house fires start. The investigator believes that a `moving mesh' approach is the best way to reduce computing time and improve accuracy, and that the actual technology transfer for these methods is close at hand.

9626107 Huang研究者开发了一种移动的网格方法，用于用于二维时间和三维时间依赖性PDE（部分微分方程）的数值解，该方法扩展了研究人员及其同事在一个空间上的PDES引入的MMPDE（移动网格PDE）方法。 使用这种方法，MMPDE根据热流程方程和谐波图理论明确配制。 MMPDE用于移动网格，将浓缩节点集中在物理溶液具有较大变化并控制某些网格质量的区域，尤其是偏度。 研究了由基础物理PDE和MMPDE组成的扩展系统离散和解决的技术。 考虑了许多相关问题 - 例如，对任意连接域的参考网格计算，流动方向控制的使用，预处理技术的研究，对时空有限元方法的检查以及该方法的鲁棒性和可靠性的研究。 此外，该方法应用于机翼分析和机翼设计，移动边界问题以及涉及爆炸解决方案的问题。 新算法和计算技术是在数学软件中同时实施的，这些软件可以在科学家和工程师中发现广泛使用。 该项目涉及开发新的计算方法，这些方法对于增强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。 这项研究的重点是开发所谓的自适应数值技术，在该技术中，要解决的特定问题的特殊特征是适应的。 更具体地说，在近似科学问题的解决方案的位置（网格点）随着时间的流逝而移动以适应解决方案的变化。 网格适应最近在问题的数值解决方案中起着必不可少的作用，因为尽管如此，超级计算机通常不能令人满意地解决这些问题。 这是因为在科学和工程的许多问题中，物理领域的一小部分，在网格中的小分离中，解决方案发生了很大的变化。 使用固定均匀网格对这些问题的数值解非常强大，因为需要数百万个网格点才能解决物理现象。 另一方面，使用自适应网格方法可以显着减少网格点的数量，从而获得经济体。 与其他自适应网格方法不同，研究中的移动网格方法会及时平稳地更改网格以适应关键解决方案特征。 它适用于并行计算。 移动网格方法对于许多工业制造问题的数值模拟应该非常有用。 特定区域是机翼分析和机翼设计。 例如，在指定压力分布时，在机翼和机翼设计中计算机翼或机翼形状。 如果可以像这些方法一样，可以同时更新流量解决方案和计算网格以及翼型的形状，从而极大地节省计算资源。 该优势也适用于其他各种应用程序，例如研究工业和房屋火灾的开始方式以及为什么开始的问题。 研究人员认为，“移动网格”方法是减少计算时间并提高准确性的最佳方法，并且这些方法的实际技术传输即将到来。