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G. William Baxter - Cellular Automata Models of Flowing Granular Media 

I am studying cellular automata models of gravity driven flows of asymmetric grains through a hopper. I am particularly interested in orientation and density effects in these flows.
Since the correct continuum equations describing the flow of a granular material are the subject of considerable debate, it is not possible to directly model them on a computer. Simulations using molecular dynamics are possible and very successful, but MD simulations are best for small numbers of grains with simple shapes and simple interactions. [Although computer speed and algorithmic sophistication has increased dramatically in recent years, it is still true that computational complexity can quickly become a limiting factor.]
Cellular automata are a natural choice for this problem. The characteristics of an automata are as follows.
  • Automata operate on a lattice (here, hexagonal.) 
  • Automata operate in discrete time steps. 
  • Automata define a discrete value on each lattive site (here, a site may be a wall or grain or a hole. Grains are described by their orientation and motion direction.) 
  • Automata use a rule to determine how the values at each lattice site change at each time step. 
In the present case, grains are described by their orientation and direction of motion. They may have one of three orientations, along the basis vectors of the lattive, and one of seven velocity or motion directions, toward one of the six nearest neighbor sites or at rest. The most challenging component is the rule. This model uses an energy minimization rule. We define an energy function which is a function of the orientation and motion of both a grain and its nearest neighbors. The new orientation and motion of each grain is chosen to most lower its energy.
In this model, typical lattices have 1/4 million sites. We begin with an empty lattice with wall sites forming a hopper. With the outlet plugged, grains are allowed to fall in. After filling, the output is unplugged and the grains are allowed to flow out. The resulting flows have much in common with real flows in a similar geometry.


Top 2000 iterations after the flow begins, the flow looks like this. Colors denote grain orientations. White is walls; black is holes. 
Bottom A density picture of the above-left image. Here the gray level gives the local grain density.


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