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.
