Finding shapes that optimize packing density
Finding shapes that optimize packing density
Wednesday, November 18, 2015
How can we systematically investigate the relationship between particle shape and granular properties in disordered aggregates of granular particles? Shape is infinitely variable, rendering
an exhaustive search infeasible. Furthermore, though it is straightforward to generate an aggregate given a particle shape, it is much more difficult to take desired packing properties as a starting point in identifying the appropriate particle shape.
To tackle this problem we use an approach recently developed by our group (see here or here), in which we pair an evolutionary algorithm with molecular dynamics simulations to navigate this enormous search space in a systematic way. In our simulations we compose each particle from spheres of varying diameters, gaining the ability to approximate both convex and non-convex shapes while maintaining feasibility of computational time and resources. Because granular media is sensitive to processing and boundary conditions, our simulations recreate experimental protocols as closely as possible, and we experimentally verify the results of our optimizations in the lab with 3d-printed copies of the shapes identified by the evolutionary algorithm.
Consider the following problem, which seems simple but so far as no known optimal solution: What is the particle shape that produces the most densely packed random aggregate when poured under gravity into a (infinitely) large box? Allow for arbitrary shape (convex or non-convex) but assume for simplicity that only one single kind of particle is poured and that friction is turned off (corresponding to experiments where the box is vibrated gently to allow particles to move around a bit and thus explore configurations that maximize the packing density). Mono-disperse spheres give a packing density of ~0.64, so which shapes do better?
In a first project we considered particle shapes formed by bonding together spheres of different radii at different angles, but without allowing for sphere-sphere overlap. This limited the search space to shapes that had some residual corrugation and it lead to the discovery of a “Mouse” particle as the densest packer under these conditions (~0.67), comprised of a central sphere with 2 “ears” of each 1/3 its diameter, placed apart 70 degrees:
•Marc Z. Miskin and Heinrich M. Jaeger, “Evolving Design Rules for the Inverse Granular Packing Problem”, Soft Matter 10, 3708–3715 (2014). link
The movie on the left shows snapshots from the search the evolutionary algorithm performed in order to find the optimal shape. Note that the optimizers “decides” to drop the radii of several spheres to zero, asymptotically settling on the 3-sphere “Mouse”. The image on the right shows a simulated packing.
The second project extended this approach to overlapping spheres, thereby vastly increasing the search space by also enabling smooth, continuous surfaces with a precision proportional to the number of constituent spheres. The surprising result was the discovery of a class of planar, triangular particles that proved to be the densest packers (~0.73):
•Leah K. Roth and Heinrich M. Jaeger, Optimizing Packing Fraction in Granular Media Composed of Overlapping Spheres, Soft Matter 12, 1107-1115 (2016). link
The graph below shows the evolution of the particle shape, starting from a randomly picked shape on the left. Each particle’s shape is formed from up to 10 spheres whose position, radius and amount of overlap are variables the optimizer can change. With each generation of the evolutionary algorithm the shape gets mutated in order to improve the packing fraction (plotted on the vertical axis) of an aggregate of roughly 950 particles poured into a box (fixed bottom but periodic boundary conditions on the sides of the box). An example of such aggregate is shown in the inset. The procedure runs 40 DEM simulations (using LAMMPS) of such aggregates in parallel, one for each shape tested. From this the optimizer picks the best performer, which is then used as the starting point for further mutations to produce the next generation. The data points show the median performance of the population of 40 members per generation, while the shaded band gives the performance range for the 25th to 75th percentile. The final shape the optimizer settled on is the planar, roughly equilateral triangle shown also in the figure at the very top (panel c). Smoothing out the dimples (see panel d) does not change the packing fraction significantly.