A novel interdisciplinary approach, combining the modern theory of dynamical systems, advanced simulation programs, and sophisticated visualization techniques shall be used to investigate complex granular flows having important engineering and industrial applications, with a focus on density relaxation and related phenomena such as jamming and force chains. Approximate continuum infinite-dimensional dynamical models shall be constructed using a method developed by the investigators as well as by taking more conventional long-wave limits. Once these models are constructed, they will be analyzed to determine such behaviors as stability, bifurcations and transitions to chaos. In addition, aspects of these granular flows such as jamming and force chains shall be rigorously characterized and analyzed in the context of nonlinear dynamics. The dynamical systems component will be extensively fine-tuned, tested and verified using a highly developed discrete simulation code and novel computer visualization techniques tailored especially for nonlinear dynamics. Both the code and the visualization techniques have already proven to be effective in predicting macroscopic properties in several granular flow regimes such as vibrating beds, and also in illuminating complex behavior in dynamical systems. The envisaged interplay among the dynamical systems, simulation code, and visualization components will be intimate and encompassing, so as to optimize the outcomes of the proposed project. Granular flow research is a data intensive activity that benefits from multiscale, cross-institutional collaborations, which necessitate extensive cyberinfrastructure (CI) development and reuse. A goal is the creation of algorithms and software for open source CI systems; facilitated via Purdues involvement in the TeraGrid Partnership and NJITs commitment to the development of widely accessible computational science resources. The dynamical systems/simulation/visualization (DSSV) approach will be applied as follows: a) Developing an effective paradigm for combining continuum approximations, simulations and visualization towards the goals of analyzing and predicting density relaxation related phenomena, and tailoring the outcomes for CI integration and reuse. b)Rigorously characterizing, analyzing and predicting such behaviors and artifacts as jamming and force chains for a range of granular flows in one, two and three dimensions, and developing novel methods for their detection. c) Proving results about the integrability of the continuum limits of integrable systems as well as new KAM type theorems for these systems; quantifying the accuracy of the approximations; and devising novel simulation and visualization schemes for studying the associated granular flows.
This involves the blending of rigorous dynamical systems analysis, simulations and computer visualization to produce new insights and predictive tools for density relaxation, as well as the development of rigorous analysis strategies for jamming and force chain detection. A team comprised of a mathematician and a mechanical engineer at NJIT, and a computer scientist from Purdue, with extensive experience in mathematical modeling, dynamical systems analysis, granular flow simulations, and dynamically oriented visualization, shall conduct the research. This team has a successful track record of combining their expertise to solve outstanding problems related to the proposed project.
The project will serve as a paradigm for obtaining effective approximate models for a wide range of granular flows of industrial importance. In addition, dissemination through publications, presentations at conferences, industrial sites and government laboratories, Web posting, CI reuse and the inclusion of outcomes in graduate courses will be done. Moreover, a substantial effort will be made to recruit highly qualified graduate and undergraduate students - especially from underrepresented groups - as research assistants on the project, thereby providing them with a unique opportunity to participate in leading-edge interdisciplinary research. In addition, a special issue of Mechanics Research Communications devoted to the research focus of this project is planned, and several outcomes from this project are slated to be included in a book on integrability analysis of infinite-dimensional dynamical systems now being completed.
1. Introduction Nearly every product or commodity in use today is constituted and/or derived from granular materials through mining, agriculture, and/or chemical processing. The processing and handling of particulates (whether in the form of dry mixtures or multiphase materials) constitutes the world’s largest industrial complex, contributing trillions of dollars towards the global economy. Recent major advances have significantly increased our understanding of flowing particulates. Nevertheless, there are few methods available for accurately predicting granular flow phenomena in complex systems. The aim of the project was to address some fundamental deficiencies in granular flow research through an interactive approach combining theory and advanced particle-level simulations at the NJIT with sophisticated visual analysis tools rooted in nonlinear dynamics to be developed at Purdue. 2. Research Strategy and Goals A dynamical systems-simulation-visualization (DSSV) approach was used to investigate granular flows via a combination of approximate dynamical systems models, discrete element simulations and computer visualization techniques. The DSSV strategy is summarized in Fig. 1. We chose density relaxation of particles as our focus as in the vertical tapping of particles confined to a container (see Fig. 2). The primary approximate dynamical system used in the project to predict the evolution of granular flows was the BSR model devised by D. Blackmore, R. Samulyak and A. Rosato, is shown in Fig. 3. where u is the velocity, ρ is the density, e is the external force per unit mass and Φ is the interaction force. Various reductions of this model were also employed, including a 2D difference equation to analyze the motion. The goals were as follows: Development of procedures for analyzing granular flows based on novel continuum approximations and visualization strategies coupled with advanced simulations. Demonstration of how the DSSV approach can identify key features of the dynamics that govern granular flows.. Rigorous methods for analyzing phenomena such as density waves so they can be readily detected using the DSSV. New mathematics concerning integrable continuum limits, novel methods for employing simulations and visualization for granular flows. The two investigators at New Jersey Institute of Technology (NJIT) have collaborated on granular flow research for the past several years, while the investigator at Purdue has established a strong record of using visualization techniques for nonlinear dynamics. 3. Project Outcomes Most of the envisaged outcomes for this project were obtained, and we obtained some results, which were not anticipated. 3a. Outcome summary The outcomes of the project can be summarized as: One book published. One edited volume published,. One book chapter published,. Eleven conference papers written and presented. Seventeen papers published in leading journals, and eight papers in press, under review, or in preparation for submission to journals. Twenty-three talks, mostly invited, given at prestigious conferences and unversities. Four minisymposia organized - two at SIAM Dynamical Systems conferences and one in a conference in Cetara. Two technologies developed: (i) A Voronoi-Gui code for 1D tapping; and (ii) A 1D code for solving the BSR model. The Granular Science Laboratory (GSL) at NJIT website (http://web.njit.edu/~rosato/) was enhanced and LIGGGHTS (http://cfdem.dcs-computing.com/?q=node/8) was launched. Four graduate students served as research assistants on the project, as did four undergraduate (two REU) students. 3b. Some outcome details A more detailed account of the project outcomes is as follows: (1) Prof. Rosato made numerous improvements of his simulation code (see Fig. 4). (2) Hao Wu, Prof. Blackmore’s PhD student, devised a proof of the well- posedness of the BSR model. (3) We established by comparison with Rosato's simulations that the BSR model is an efficient and accurate predictor of granular flow dynamics, at least for 1D flows. (4) Blackmore and his collaborators have proved new integrability results for systems of the BSR type. Integrability of these dynamical systems implies the existence of solitons. (5) We have also obtained some significant results on the existence and stability of traveling wave type solutions of BSR systems. (6) Hao Wu has devised accurate, efficient semidiscrete numerical solution methods for BSR models. Density (wave) evolution and dynamics are shown in Fig. 5. This work is part of our CI reuse efforts (7) A paper published in Physica D demonstrated the power of the BSR and other reduced models in studying granular flows (8) Significant progress has been made in comparing simulation and visualization results. (9) Blackmore, Rosato and Tricoche organized a successful two-part minisymposium at the SIAM DS-13. Blackmore and Rosato have also organized a related minisymposium at SIAM DS-15 (10) Prof. Rosato co-organized a conference with a strong granular dynamics theme in Cetara, Italy, and edited a related special issue of Mechanics Research Communications. (11) Blackmore, Rosato and others published several papers on novel integrabilty criteria and their applications. (12) Blackmore and his collaborators employed insights from this project to obtain several new ways of analyzing strange attractors in dynamical systems.
