INSPIRE: Nonlinear Data Reduction applied to Dense Granular Media

INSPIRE：应用于密集颗粒介质的非线性数据缩减

• 批准号: 1248071
• 负责人: Konstantin Mischaikow
• 金额: $ 45万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1248071&HistoricalAwards=false
• 关键词: INSPIRE; Nonlinear; Data; Reduction; applied

This INSPIRE award is partially funded by the Applied Mathematics and the Topology program in the Division of Mathematical Sciences in the Directorate for Mathematical and Physical Sciences and the Granular Materials Program in the Division of Chemical, Bioengineering, Environmental, and Transport Systems in the Engineering Directorate. It will support work that uniquely combines state of the art experimental techniques, large scale molecular dynamics simulations, and cutting edge theoretical developments in topological data analysis for the purpose of understanding and predicting the behavior of dense granular materials (DGM). In particular, it is based on the novel proposition of using persistence diagrams, a relatively new concept in applied algebraic topology, as the fundamental modeling tool for DGM. The core tasks include the following. (1) Develop an efficient computational framework for working with persistence diagram data arising from spatiotemporal systems. (2) Using discrete element simulations (DEM) and experiments, carry out precise, well-defined, and complete characterizations of forces and stresses in DGM, including spatial and temporal variability of a given system and quantification of the differences between simulations and experiments. (3) Formulation of an optimum combination of experiments and simulations to be used to analyze a given system, leading to reliable predictions for the nature and evolution of macroscopic quantities describing a system including stresses and bulk and shear moduli. (4) Use of persistence diagrams to characterize spatial and temporal fluctuations, including the dependence on system size, on the physical dimensions (e.g. 2D versus 3D), and particle properties such as shape or friction. (5) Study the topology of the space of persistence diagrams. Characterize the dynamics of DGM by using purely topological techniques to study time series reconstructed dynamics.Granular materials in the dense fluid-like and solid states present one of the most significant modeling challenges of our times. These materials appear in a broad spectrum of practical settings from heavy industry to pharmaceuticals. As such, they are of intense interest to the engineering world. The way in which grains pack and in which forces are carried in DGM presents a significant puzzle for the soft condensed matter and statistical physics communities. Associated with force and packing complexity are challenging mathematical issues related to the analysis of complex high dimensional multiscale spatiotemporal data. DGM are inherently high dimensional systems in which geometry plays a fundamental role. However because of their granular nature they cannot be usefully approximated by analytic continuum models and thus a clear method of reduction to a tractable problem is lacking. Our goal is to demonstrate that new ideas associated with computational topology provide an efficient, faithful and coherent approach to providing tractable models for the temporal evolution of spatial structures in DGM. The abstract nature of the topological methods being developed imply that the tools developed in this context will be applicable to a wide range of systems demonstrating complex spatiotemporal structures.

该INSPIRE奖的部分资金来自数学和物理科学局数学科学部的应用数学和拓扑学计划，以及工程局化学、生物工程、环境和运输系统部的颗粒材料计划。它将支持将最先进的实验技术、大规模分子动力学模拟和拓扑数据分析中的前沿理论发展独特地结合在一起，以了解和预测致密颗粒材料(DGM)的行为的工作。特别是，它基于应用代数拓扑学中一个相对较新的概念--持久图作为DGM的基本建模工具这一新颖的命题。核心任务包括以下内容。(1)开发一个高效的计算框架来处理来自时空系统的持久性图数据。(2)利用离散元模拟(DEM)和实验，对DGM中的力和应力进行精确、明确和完整的表征，包括给定系统的空间和时间变化以及模拟和实验之间的差异的量化。(3)建立实验和模拟的最佳组合，用于分析给定的体系，从而对描述体系的宏观变量的性质和演变进行可靠的预测，包括应力、体积和剪切模数。(4)使用持续性图来描述空间和时间波动，包括对系统大小的依赖、对物理维度的依赖(例如，2D与3D)以及诸如形状或摩擦的粒子属性。(5)研究持久图空间的拓扑。通过使用纯粹的拓扑技术来研究时间序列重建的动力学来表征DGM的动力学。致密的流体和固体状态下的颗粒材料是我们这个时代最重要的建模挑战之一。这些材料出现在从重工业到制药的广泛实用环境中。因此，它们引起了工程界的强烈兴趣。对于软凝聚态物质和统计物理学界来说，DGM中颗粒的堆积和力的携带方式是一个重大的谜题。与力和堆积复杂性相关的是与分析复杂的高维多尺度时空数据相关的具有挑战性的数学问题。DGM本质上是高维系统，几何在其中起着基本的作用。然而，由于它们的颗粒性，它们不能有效地用解析连续模型来近似，因此缺乏一种清晰的方法来归结为一个可处理的问题。我们的目标是证明与计算拓扑学相关的新思想提供了一种有效、可靠和连贯的方法来为DGM中空间结构的时间演化提供易于处理的模型。正在开发的拓扑方法的抽象性质意味着，在此背景下开发的工具将适用于展示复杂时空结构的广泛系统。