## Granular Materials

My research has focused primarily on the simulation of slow, steady, dense granular flows through converging hoppers with varied geometries. Depending on the formulation, these problems are often systems of hyperbolic conservation laws or elliptic systems. In either case, the differential equations are often coupled with an algebraic constraint (the yield condition, describing the relationship among the stress tensor components when a granular material deforms), giving rise to a differential-algebraic (DAE) system.

Here are some exampes of the types of problems I've worked on:

• Granular Flows Through Nonaxisymmetric Hoppers

Much work has been done with flows through axisymmetric hoppers (like right circular cones) but in practice there are a wide variety of hoppers shapes, many of which are not axisymmetric. An interesting extension of the usual similarity flows in axisymmetric hoppers is to look for similarity flows in nonaxisymmetric but pyramidal hoppers. (By "nonaxisymmetric but pyramidal" we mean that the cross section at any one height is similar to the cross-section at another height.) The resulting problems are elliptic and numerical simulation of the flows reveals circulation cells. Such circulation is easily observed in physical experiments, but the inherent symmetry assumptions in simpler models precluded the observation of this circulation in simulations.

This image shows different types of symmetry. Upper left: axisymmetric, no transition vertically (Jenike). Upper right: Axisymmetric, vertical transition in wall angle (Previous work, see below). Lower left: Nonaxisymmetric, no transition vertically (this work). Lower right: Nonaxisymmetric, transition vertically (open problem).

There is much work yet to be done in this area, including comparison of different constitutive models, delineation of the mass flow limit for various hopper geometries, and cooperation with physicists and engineers to compare simulations with laboratory experiments.

Below are some sample images from these simulations in a hopper with a nearly square cross-section. Only one quarter of the domain is represented. (The other three quadrants are the same, by symmetry.) The code is written in Matlab.

• Well-posed Hyperbolic Free Boundary Value Problem with Dirichlet Boundary Conditions

Given the problems encountered in the earlier research in hopper transitions (see below), one proposal was that the incorporation of a free boundary, like at the exit at the bottom of a hopper, might lead to more physical results. To investigate this possibility, my colleague David Schaeffer and I showed that in a smaller model problem that solutions exist and are unique when a free boundary is present.

The model problem is a variation of Schaeffer's antiplane shear model proposed. In this model, as above, the problem is hyperbolic with an algebraic constraint (which mimics the yield condition).

• Abrupt changes in wall properties for axisymmetric hoppers

In this work, Pierre Gremaud and I examined the effect of a change in the wall material or the angle of wall inclination on an established, steady flow. For example, in a transition from a smooth to a rough wall, the stress field would develop a shock that would then propagate through the lower portion of the hopper. The reverse transition, from rough to smooth, would result in a rarefaction that eventually interacts with itself to develop a shock.

This work revealed an important difficulty in this class of problems: a general way to describe physically meaningful boundary value problems. To be more precise, this work computes solutions to systems of hyperbolic conservation laws with algebraic constraints. Even if one starts with physically meaningful data long the initial manifold, the solution will propagate through the hopper from that manifold and is not guaranteed to remain physically meaningful. The particular issue we observed was that the sign of work done by friction would flip from positive to negative, meaning that the force of friction was creating, not dissipating, energy.

These fields were computed with a high order Runge-Kutta Discontinuous Galerkin method, implemented from scratch in C. Here are some visualizations from the simulations:

## Colleagues

In the course of my work, the following persons have been important and should be duly credited.

• Pierre Gremaud, North Carolina State University. My PhD advisor while I was at NCSU. The best colleague I have and a good friend, I could not be where I am today without his mentoring and encouragement.
• David G. Schaeffer, Duke University. My mentor during my post-doctorate work at Duke. The sharpest mathematician I've ever known, and one of the biggest influences in my life as a researcher and teacher.

## Publications

1. On the Computation of Steady Hopper Flows II: von Mises Materials in Various Geometries, with P.A. Gremaud and M. O'Malley, NCSU-CRSC Tech Report CRSC-TR03-46, to be published in J. Comput. Phys. [ PDF, 6.5M ]
2. A Well-Posed Free Boundary Value Problem for a Hyperbolic Equation With Dirichlet Boundary Conditions, with D.G. Schaeffer, SIAM J. Math. Anal., (36) 2004, p.256-271. [ PDF, 181K ]
3. Secondary Circulation in Granular Flow Through Nonaxisymmetric Hoppers, with P.A. Gremaud and D.G. Schaeffer, SIAM J. Appl. Math., (64) 2003, p.583-600. [ PDF, 800K ]
4. On the Computation of Steady Hopper Flows: I, Stress Determination for Coulomb Materials, with P.A. Gremaud, NCSU-CRSC Tech Report CRSC-TR99-35, J. Comput. Phys., 166 (2001), p.63-83. [ PDF, 11M ]
5. Simulation of Gravity Flows of Granular Materials in Hoppers, with P.A. Gremaud, in Discontinuous Galerkin Methods, Theory, Computation and Applications, B. Cockburn, C.W. Shu, G. Karniadakis Eds., Lecture Notes in Computational Science and Engineering, #11 (2000), Springer Verlag, p.125-134. [ PDF, 150K ]
6. Similarity solution for hopper flows, with P.A. Gremaud and M. Shearer, Proceedings of the SIAM/AMS Conference on Nonlinear PDEs, Dynamics and Continuum Physics, J. Bona, K. Saxton, R. Saxton, Eds., AMS Contemporary Mathematics Series, #255 (2000), p.79-95. [ PDF, 300K ]

## Resources

In the course of my work, various bits of software get produced which might be of interest to others. Any that I can release I will publish here.

• Visualization of flow in nonaxisymmetric hoppers: foogui-0.5 for Windows 2000 or Windows XP.
• See Windows instructions
• Shows stress, velocity components for nonaxisymmetric Von Mises flow in a converging hopper.
• Simple interface to adjust internal angle of friction and coefficient of wall friction.
• Binary distribution with source. Runs "out of the box", no extra libraries required.
• Building from source requires GTK+ 2.4 and PLPlot. Configuration of Makefile probably required.
• To do: Suggestions from engineers. Matsuoka/Nakai and Tresca flows. Visualization improvements. Save images and data to file.
• Visualization of flow in nonaxisymmetric hoppers: foogui-0.5 for GNU/Linux.
• Shows stress, velocity components for nonaxisymmetric Von Mises flow in a converging hopper.
• Simple interface to adjust internal angle of friction and coefficient of wall friction.
• Source distribution only.
• Building from source requires GTK+ 2.4 and PLPlot. Configuration of Makefile probably required.
• To do: Suggestions from engineers. Matsuoka/Nakai and Tresca flows. Visualization improvements. Save images and data to file.