DynSyst_Special_Topics: Collaborative Research: Reduced Dynamical Descriptions of Infinite-Dimensional Nonlinear systems via a-Priori Basis Functions from Upper Bound Theories

DynSyst_Special_Topics：协作研究：通过上界理论的先验基函数简化无限维非线性系统的动态描述

• 批准号: 0927587
• 负责人: Charles Doering
• 金额: $ 24万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0927587&HistoricalAwards=false
• 关键词: DynSyst_Special_Topics; Collaborative; Research; Reduced; Dynamical

The aim of this interdisciplinary collaborative research project is to develop a novel model reduction technique for forced dissipative infinite-dimensional dynamical systems by employing basis functions computed using upper bound theories. Like popular Proper Orthogonal Decomposition (POD) based methods, this approach associates the condensed variables needed for model reduction with coherent structures and captures nonlinear interactions between these linear modes via Galerkin projection and finite-dimensional truncation. Unlike empirical POD methods, however, this new method does not require extensive data sets from experiments or direct numerical simulations of the governing partial differential equations (PDEs) and thus yields truly predictive reduced models. The theoretical and computational methodology will be developed in the context of a particular physical system, thermal convection in fluid saturated porous media, that is of considerable environmental and technological importance and an ideal testbed for new ideas.This research will contribute to the development of a general methodology for deriving simplified mathematical models of highly complex dynamical systems arising in diverse areas of science and engineering. In many applications of interest (e.g., control of various fluid flows to achieve drag reduction for oil pumped in pipelines or for air flowing past commercial jets, or for estimation of carbon dioxide sequestration by porous rock material for reducing global warming), direct numerical simulations based on the complete governing mathematical equations are infeasible using even the world's fastest high-performance supercomputers. This project will address these challenges using novel mathematical techniques to derive simplified equations directly from the governing physical laws that are amenable to practical computation and analysis.

这个跨学科合作研究项目的目的是通过采用上限理论计算的基函数，开发一种用于强制耗散无限维动力系统的新型模型简化技术。 与流行的基于本征正交分解 (POD) 的方法一样，该方法将模型简化所需的压缩变量与相干结构相关联，并通过伽辽金投影和有限维截断捕获这些线性模式之间的非线性相互作用。 然而，与经验 POD 方法不同，这种新方法不需要来自实验的大量数据集，也不需要对控制偏微分方程 (PDE) 进行直接数值模拟，从而产生真正的预测简化模型。理论和计算方法将在特定的物理系统（流体饱和多孔介质中的热对流）的背景下开发，该系统具有相当大的环境和技术重要性，并且是新想法的理想测试平台。这项研究将有助于开发用于推导科学和工程不同领域中出现的高度复杂动力系统的简化数学模型的通用方法。 在许多令人感兴趣的应用中（例如，控制各种流体流动以减少管道中泵送的油或流过商用喷气机的空气的阻力，或者估计多孔岩石材料封存的二氧化碳以减少全球变暖），直接数值模拟即使使用世界上最快的高性能超级计算机，基于完整的控制数学方程的计算也是不可行的。该项目将使用新颖的数学技术来解决这些挑战，直接从适用于实际计算和分析的物理定律导出简化方程。