Mathematical Sciences: Applied and Computational Mathematics and Experimental Analysis

数学科学：应用与计算数学和实验分析

• 批准号: 9403915
• 负责人: Jerry Bona
• 金额: $ 33.5万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403915&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applied; Computational; Mathematics

9403915 Bona This project comprises a broadly-based investigation of some fundamental aspects of mechanics, including the development of some of the relevant mathematical techniques and tools. Included within its auspices are theoretical and experimental studies of various fluid flows, studies of water waves and formation of sand structures in the near- shore zone of large bodies of water, and theoretical and experimental studies of liquid crystals. One of the larger investigations will be concerned with the modelling of water waves with an eye toward a better understanding of the interaction of waves travelling in opposite directions. Another interesting study is concerned with flows in which there is a balance between dissipative effects and surface tension. Still another sub-project will reconsider some classical flows such as flow past a sphere in a pipe, Taylor-Couette flow and the related issues of stability and vortex breakdown. The complex mathematical modelling needed to describe exotic materials like liquid crystals will also be high on the agenda. Allied to all of these investigations will be the development of specific theoretical results in partial differential equations and in numerical analysis. Especially important in the latter area will be work on highly efficient methods for the solution of large linear systems arising in the algorithms for the solution of complicated nonlinear differential equations. The resulting computer codes will be used in conjunction with experimental work on all the forementioned topics and the outcomes interpreted within the relevant theoretical framework. This work is exciting for several reasons. First, many of the issues being addressed are not only of fundamental theoretical importance, but they also have potential to provide new insights into engineering and technological design and processing. Secondly, because of the presence of the fluid mechanics laboratory sited within the mathematics department at Pe nn State, the group has the ability to combine theoretical, numerical and experimental work in a rarely realized close collaboration. In addition, some of the work on development of optimal numerical techniques will have impact in a much broader arena of science and engineering activity than just the confines of the present study.

[403915]博纳：这个项目包括对力学一些基本方面的广泛研究，包括开发一些相关的数学技术和工具。在它的主持下，包括对各种流体流动的理论和实验研究，对大型水体近岸地带的水波和沙结构形成的研究，以及对液晶的理论和实验研究。其中一项较大的研究将涉及水波的建模，以期更好地理解反方向传播的水波之间的相互作用。另一个有趣的研究是关于在耗散效应和表面张力之间存在平衡的流动。还有一个子项目将重新考虑一些经典的流动，如在管道中流过球体，Taylor-Couette流动以及稳定性和漩涡破裂的相关问题。描述像液晶这样的奇异材料所需的复杂数学模型也将被提上日程。与所有这些研究相关的将是偏微分方程和数值分析中具体理论结果的发展。在后一个领域中，特别重要的是在求解复杂非线性微分方程的算法中出现的求解大型线性系统的高效方法的研究。由此产生的计算机代码将与所有上述主题的实验工作结合使用，并在相关理论框架内解释结果。这项工作令人兴奋的原因有几个。首先，许多正在解决的问题不仅具有基本的理论重要性，而且它们也有可能为工程和技术设计和处理提供新的见解。其次，由于位于宾州州立大学数学系的流体力学实验室的存在，该小组有能力将理论、数值和实验工作结合起来，以一种很少实现的密切合作。此外，一些关于发展最佳数值技术的工作将对科学和工程活动的更广泛领域产生影响，而不仅仅是目前研究的范围。