Lionel Pournin - Research

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Pouring 64000 spherosimplices into a cylindrical container.

My research interests include discrete geometry and numerical simulation of granular media. How do these two (seemingly distant) subjects relate to one another?

Numerical simulation of granular media consists in reproducing the evolution with time of an assembly of particles (typically sand, pebbles, rice...) on a computer. This requires, among other things, a physical model for the contacts and an algorithm to detect the occurrence of these contacts. The contact detection algorithm is the reason why discrete geometry comes into play.

The contact detection method implemented in the simulation code I contributed to uses weighted Delaunay triangulations, also sometimes called regular triangulations. This method was designed by Didier Müller in the two-dimensional case [1] and Jean-Albert Ferrez for three-dimensional spherical grains [2,3], both under the supervision of Prof. Tom Liebling. During my PhD, I extended the Distinct Element simulation Method and Jean-Albert's C++ code to handle non-spherical particle shapes which I called spheropolyhedra [4].

The convergence of the triangulations-based contact detection method is conditioned to the connectedness of flip-graphs associated to three dimensional point configurations, which is an open problem in discrete geometry. See my page on flip-graphs for some details on this problem.

Along these directions, I proved that the flip-graph of the 4-dimensional cube (or tesseract), is connected. This was another long-standing open problem in discrete geometry.

The granular media simulation code I contributed to can be used to investigate many phenomena occuring for assemblies of particles. These phenomena have a fundamental importance to both engineers and physicists. If you are interested in computer simulation and physics of granular media, you may visit my page on numerical simulation.

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[2]

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[4]