EUREKA AUGUST 2002 COVER FEATURE STORY

Packed shapes key multiple designs

New software speeds design tasks as varied as modelling the flow of real particles or counting the number of beans to be packed in a jar. Tom Shelley reports

Reducing particles to patterns of pixels or voxels allows realistic modelling of many real world problems, from designing packaging for loose groceries to complex segregation effects, which occur when particles of different shapes and densities are shaken.
The name of the software is DigiPac, developed by Dr Xiaodong Jia and colleagues in the Institute for Particle and Engineering at Leeds University. Its approach is derived from image processing and is based on reducing each shape to a pattern of whole pixels, or picture elements, or in 3D, voxels, or volume elements, moving on a square lattice. The shapes are allowed to move randomly, one lattice grid at a time. Diagonal moves are composed of two/three orthogonal moves. The consequent use of integers or whole numbers avoids the need for floating point calculations and greatly reduces computation time. Rounding off to integers introduces initial errors, but such errors can be greatly reduced by using finer resolutions. The area ratio error incurred by representing a circle by 76 pixels on a 10 x 10 grid is 3.2% or 0.6% by representing a circle by 316 pixels on a 20 x 20 grid. During the packing process, translational movements do not incur rounding errors and special measures are taken to minimise errors arising from rotations.

The shapes can be simple or complex, derived from scanning in real world particles. In 2D, they may be scanned in using a simple image scanner. But in 3D they require the use of either 3D x-ray microtomography, if they are inside something else, or a 3D optical scanner. This contrasts with previously available software only able to cope with spheres or other simple geometrical shapes such as ellipses, spheroids and cylinders.
Dr Jia says that the key to modelling their behaviour is to apply "very simple rules for each pixel or voxel." For example, if studying coating behaviour, there can be a rule that causes small particles to move towards big particles. Stickiness and cohesiveness can be simulated using probabilities, which may not represent a strict reproduction of physical behaviour but result in particle behaving at least qualitatively in a realistic way.
Particles in a coating can take any form
2D simulations take minutes to accomplish on a PC, sufficient to demonstrate effects and behaviour. But more realistic 3D simulations take hours. However, the 2D simulations are often good enough to permit study of real world effects, even if 3D simulation is necessary to verify them more quantitatively.


The simplest task in 2D or 3D is to model the packing of particles in a container. Particles can be of many different shapes, and quite large. It is also possible to model their behaviour as they fall through a hopper, and/or build a heap. In the simplest model of dog bone shaped particles, which does not take rebounding or particle rotation into account, one might find a packing density of, say, 0.39. This approach is known as a 'rain' model. But with a 50% rebounding probability with rotation, this rises to 0.6. Boundary conditions can either be periodical or solid walls. In theory, with solid walls, the software could be used to study how more commuters could be packed into trains, by digitising a range of humans and train interiors, and applying whatever rules the train companies care to apply.
With periodical boundary conditions, a particle moves out of one side and comes in from the opposite side at the same time. This allows bulk properties to be predicted by a small sample. The width of the simulation box should be at least ten times bigger than that of the largest particle. With solid walls, in containers of particles or on trains, the packing is usually less dense near the wall than in the central part.
As well as studying assemblages of loose particles, it is also possible to study the effects of reinforcing particles and chopped fibres of different sizes and shapes, in chopped fibre reinforced plastics and metal matrix composites. It may well be that a combination of sizes and shapes could achieve greater strengths than single sized, chopped fibre lengths, but hitherto, there has been no computer tool available to model this. Compaction, force distribution and strength of both loose assemblages and composites can all be deduced.
If a bed of particles is vibrated, it is possible to study size segregation effects, and how size distribution and shape affect these. This is very important in studying the behaviour of building materials, such as concrete and asphalt, which initially consist of heterogeneous mixtures of particles of different densities, size and shape. Segregation can, of course, lead to fairly disastrous results in the end product, be it building or road.

By imposing a rule that reduces surface area in order to minimise surface energy, it is possible to model sintering. By studying the particle pixels or voxels in contact with each other, it is possible to model thermal conductivity. Simulations then show how some areas of particles tend to get heated more than others. And by studying pixels or voxels representing the gaps in between, it is possible to model gas and liquid permeability, simulating both pressure distributions and flow rates.
The initial research was supported by EPSRC, the Keyworth Institute and the White Rose Faraday Packaging Partnership. Now the team is looking for consultancy and support for research projects. Licensing of software for use by clients for their own investigations may follow later. And work is in hand to extend modelling capabilities to include deformable shapes, and to verify predictions.
(More information from www.particles.leeds.ac.uk and X.Jia@leeds.ac.uk )
Design Pointers
* New modelling method allows the more realistic simulation of the behaviour of particles of different sizes and shapes.
* Packing, flow, heat transfer, permeability and compaction can all be simulated
* It may now be possible to use to tool to discover new types of composite reinforcement which could not previously be investigated other than by an excessively long trial and error process
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