INSPIRE: Nonlinear Data Reduction applied to Dense Granular Media

该INSPIRE奖部分由数学和物理科学理事会数学科学部的应用数学和拓扑学计划以及工程理事会化学，生物工程，环境和运输系统部的颗粒材料计划资助。它将支持独特地结合最先进的实验技术，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的工作，以理解和预测致密颗粒材料（DGM）的行为。 特别是，它是基于使用持久性图，一个相对较新的概念，在应用代数拓扑，作为DGM的基本建模工具的新命题。核心任务包括以下内容。(1)开发一个有效的计算框架，用于处理来自时空系统的持久性图数据。(2)使用离散元模拟（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差异的量化。 (3)用于分析给定系统的实验和模拟的最佳组合的公式化，导致描述系统的宏观量的性质和演变的可靠预测，包括应力和体积和剪切模量。(4)使用持久性图来表征空间和时间波动，包括对系统大小、物理尺寸（例如2D与3D）以及颗粒特性（例如形状或摩擦）的依赖性。(5)研究持久性图空间的拓扑结构。通过使用纯拓扑技术研究时间序列重构动力学来表征DGM的动力学。处于致密流体和固体状态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的实际环境中。因此，它们引起了工程界的浓厚兴趣。在DGM中颗粒的包装方式和力的携带方式为软凝聚态和统计物理学界提出了一个重要的难题。 与力和包装的复杂性是具有挑战性的数学问题，涉及到复杂的高维多尺度时空数据的分析。 DGM本质上是高维系统，其中几何形状起着重要作用。然而，由于它们的颗粒性质，它们不能有效地近似分析连续模型，因此缺乏一个明确的方法减少到一个易于处理的问题。 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，忠实和连贯的方法，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意味着在这方面开发的工具将适用于广泛的系统，展示复杂的时空结构。

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.