Fast, High Order Vortex Methods Based on Deforming Basis Functions

基于变形基函数的快速高阶涡旋方法

• 批准号: 9971800
• 负责人: Louis Rossi
• 金额: $ 7.1万
• 依托单位: University of Massachusetts Lowell
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971800&HistoricalAwards=false
• 关键词: Fast; Order; Vortex; Methods; Based

9971800This project expands and enhances existing knowledge of a new category of vortex method based on deforming basis functions. Vortex methods approximate the vorticity field of a flow as a linear combination of localized basis functions. These basis functions move with the flow velocity at the basis function centroid, and the velocity field is calculated from the vorticity field through a Biot-Savart integral. For flows dominated by isolated regions of vorticity, vortex methods offer considerable advantages because they are naturally adaptive in the sense that computational elements are dedicated exclusively to regions that have vorticity. Most vortex methods use rigid, axisymmetric basis functions though there are some exceptions. This project studies vortex methods that use basis functions which deform with local flow deviations, and builds upon the recent development of a high spatial order vortex method based on deforming elliptical Gaussians. Elliptical Gaussians are special in the sense that they represent self-similar, exact solutions to the relevant advection-diffusion equation. To implement such a scheme, one must develop methods for evaluating the Biot-Savart integral for an elliptical Gaussian and identify the relevant convergence parameters for the method as a whole. Though a two-dimensional scheme has already been developed, the Principle Investigator will enhance this method by developing a fast summation algorithm. The Principle Investigator will extend this method to three dimensions, providing a crucial link between vortex stretching and the geometry of vortex filaments. The new method will be used to perform careful calculations of a variety of problems including vortex dipole collisions and jet transients. Finally, the Principle Investigator will extend this approach to moisture transport through unsaturated porous media.This work provides a means of quickly and accurately calculating fluid flow properties on a wide range of problems including but not limited to aerospace applications, industrial processes, ocean currents and atmospheric flows of all sorts. The Principal Investigator will also extend these concepts along new lines to develop methods that can accurately calculate the motion and diffusion of moisture and contaminants in unsaturated soils. This type of method, called a "vortex method", is unusual in that these schemes are naturally adaptive. Naturally adaptive methods dedicate computational resources exclusively to the dominant regions of the flow. These methods simulate flows by calculating the evolution of the local angular momentum in the fluid. Often, the angular momentum is restricted to a small fraction of the total volume of the fluid. For instance, weather systems are driven by a collection of storm systems that represent concentrated regions of angular momentum. Using such a method, one can reconstruct the entire flow field based solely on the evolution of the angular momentum which means that the computer only need perform calculations over a small area to capture the entire flow field. Vortex methods have the added advantage of being easily parallelized on multiple processor computers so that one can take full advantage of supercomputing facilities and networks of connected computers. Also, the Principal Investigator will apply these techniques to an entirely new type of problem involving flow through unsaturated porous media such as dry soils, clays or concrete. Similar to vortex methods, this new technique takes advantage of the fact that the movement of moisture is dominated by narrow "preferred paths" occupying a relatively small fraction of the total domain of interest. These paths are created by moisture-media interactions, and the computer need only dedicate its resources to those regions containing moisture. Finally, many aspects of these activities make excellent undergraduate research projects. These research activities will enhance students' interests and knowledge in mathematics, environmental science and high performance computing. Thus, in addition to its scientific merits, this project will give students at the University of Massachusetts Lowell meaningful research experiences and enhance collaboration across several disciplines.

9971800该项目扩展并增强了基于变形基函数的新型涡旋方法的现有知识。 涡旋方法将流动的涡度场近似为局部基函数的线性组合。 这些基函数随着基函数质心处的流速移动，并且速度场是通过比奥-萨伐尔积分从涡度场计算出来的。 对于由涡度的孤立区域主导的流动，涡旋方法提供了相当大的优势，因为它们自然地具有自适应性，因为计算元素专门用于具有涡度的区域。 大多数涡旋方法使用刚性的轴对称基函数，但也有一些例外。 该项目研究使用随局部流动偏差变形的基函数的涡旋方法，并建立在最近开发的基于变形椭圆高斯的高空间阶涡旋方法的基础上。 椭圆高斯函数的特殊之处在于它们代表相关平流扩散方程的自相似、精确解。 为了实现这一方案，必须开发评估椭圆高斯的毕奥-萨伐尔积分的方法，并确定整个方法的相关收敛参数。 尽管二维方案已经开发出来，但原理研究者将通过开发快速求和算法来增强该方法。 原理研究员将这种方法扩展到三个维度，提供涡旋拉伸和涡旋丝几何形状之间的关键联系。 新方法将用于对各种问题进行仔细计算，包括涡旋偶极子碰撞和射流瞬变。 最后，原理研究员将这种方法扩展到通过不饱和多孔介质的水分传输。这项工作提供了一种快速、准确地计算各种问题的流体流动特性的方法，包括但不限于航空航天应用、工业过程、洋流和各种大气流动。 首席研究员还将沿着新的思路扩展这些概念，以开发能够准确计算非饱和土壤中水分和污染物的运动和扩散的方法。 这种类型的方法称为“涡流方法”，其不同寻常之处在于这些方案具有天然的适应性。自然自适应方法将计算资源专门用于流的主要区域。 这些方法通过计算流体中局部角动量的演变来模拟流动。 通常，角动量仅限于流体总体积的一小部分。 例如，天气系统是由代表角动量集中区域的一系列风暴系统驱动的。 使用这种方法，人们可以仅根据角动量的演化来重建整个流场，这意味着计算机只需要在一个小区域上进行计算即可捕获整个流场。 涡旋方法还有一个额外的优点，即可以在多处理器计算机上轻松并行化，从而可以充分利用超级计算设施和连接计算机的网络。 此外，首席研究员还将把这些技术应用于一种全新类型的问题，涉及流过不饱和多孔介质（如干燥土壤、粘土或混凝土）的问题。 与涡流方法类似，这种新技术利用了这样一个事实，即水分的运动由狭窄的“首选路径”主导，占据整个感兴趣区域的相对较小部分。 这些路径是由水分与介质相互作用创建的，计算机只需将其资源专用于那些含有水分的区域。 最后，这些活动的许多方面都成为优秀的本科生研究项目。 这些研究活动将增强学生在数学、环境科学和高性能计算方面的兴趣和知识。 因此，除了其科学优点外，该项目还将为马萨诸塞大学洛厄尔分校的学生提供有意义的研究经验，并加强跨多个学科的合作。