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.

该跨学科协作研究项目的目的是通过采用使用上限理论计算的基础功能来开发一种新型的模型还原技术，以强迫耗散无限二维动力学系统。 与流行的基于正交的正交分解（POD）方法一样，这种方法将模型还原模型所需的凝结变量与相干结构相关联，并通过Galerkin投影和有限维度截断捕获这些线性模式之间的非线性相互作用。 但是，与经验POD方法不同，这种新方法不需要从实验或管理部分微分方程（PDE）的直接数值模拟中进行大量数据集，因此产生了真正的预测性降低模型。理论和计算方法将在特定的物理系统（流体饱和多孔培养基中的热对流）中开发，这具有相当大的环境和技术重要性，并且是对新思想的理想测试床。这项研究将有助于开发一种普遍的方法，用于开发一种在科学和发动机多样性领域中引起高度复杂动态系统的简化数学模型的通用方法。 在许多感兴趣的应用中（例如，控制各种流体流量以减少管道中的油或流过经过商业飞机的空气，或通过多孔岩石材料估算二氧化碳序列的碳，以减少全球变暖），基于完整的数学方程式，使用全球范围的高度良好的高度高高的高度超高的高度可降低数值的直接数值。该项目将使用新颖的数学技术来解决这些挑战，以直接从适合实用计算和分析的人身定律中得出简化的方程。