A significant part of the research in applied fields of science and engineering focuses on the systems built up from discrete objects. This includes the systems relevant to materials science, such as dry and wet granular systems, but also many other soft-matter systems such as foams, colloids, and liquid crystals. There is also an increasingly relevant and active field of active matter where the systems of interest are built out of particles governed by some type of internal forces, such as bacteria and the similar. Going back to the systems relevant to materials science, one could note important applications, involving trillions of dollars per year in the US alone. Despite wide-ranging appearance of systems built out of granular particles, our ability to predict their behavior lags far behind that for more conventional materials such as Newtonian fluids. Similar, even stronger, conclusions could be reached for the other particulate-based systems that are just becoming to be considered. Lack of continuum-based models for many of the listed systems requires carrying out discrete element simulations that focus on modeling particle-particle interactions. Due to increased computational power, current simulations are able to provide realistic description of the experimental systems and can often be used with predictive power. However, the separation of spatial and temporal scales describing particles and their interactions, and of those describing meso- or macro-scales that are of interest when considering properties of a system as a whole, leads to increasingly large and essentially unmanageable amount of data. This proposal focuses on development of a technique, based on computational homology, which allows us to reach deeper understanding of the dynamical properties of the considered complex systems by extracting required information from these large data sets.
The proposed work is based on topological data analysis, and in particular it focuses on development of techniques based on persistent homology, algebraic topological techniques from nonlinear dynamics, and algorithms and software to compute homological invariants that are capable of identifying and characterizing the nonlinear dynamics of complex spatio-temporal systems. These techniques will be applied to the results of discrete element simulations of particulate-based systems that will be developed in parallel. These simulations will consider large number of particles interacting by both attractive and repulsive forces, both in two and three spatial dimensions. We will consider circular/spherical particles, as well as the particles of polygonal/polyhedral shapes. As an outcome of the proposed project, we expect to develop much better understanding of the dynamical properties of the considered systems, which will be then passed to scientists and engineers working on their applications. The implications of success in this project are far reaching. Developing new computationally efficient mathematical tools for understanding and predicting the dynamics of complex patterns on large scale data sets provides the foundations for the analysis of a wide range of problems involving complex nonequilibrium systems. In the context of particulate-based systems, this includes (i) dry granular matter built out of particles interacting by repulsive force, with the examples coming from nature - including avalanches, debris flows, and earthquakes, technology - processing of coal, ores, and pharmaceuticals; (ii) `wet' systems built out of particles interacting by a combination of repulsion and attraction, in particular relevant to porous media applications, and (iii) a number of multiphase systems including suspensions and active matter. In all of these systems, understanding and prediction of complex behavior of special structures is desired.
|Effective start/end date||9/15/15 → 8/31/19|
- National Science Foundation: $130,996.00