Lionel Pournin - Research

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[Flip-graphs]

[Tesseract]

[Simulation]

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Pouring 64000 spherosimplices into a cylindrical container.

My main research interests are in geometry, but I also worked on the numerical simulation of granular media. How do these 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 geometry comes into play.

The contact detection method implemented in the simulation code I contributed to develop 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]. I extended the method and Jean-Albert Ferrez's code to handle non-spherical particle shapes: spheropolyhedra [4].

The convergence of the triangulations-based contact detection method is conditioned to the connectedness of the flip-graph associated to a finite subset of the 3-dimensional Euclidean space, a long-standing open problem in discrete geometry.

Along these directions, I proved that the flip-graph of the (set of vertices of the) 4-dimensional cube (also sometimes called tesseract), is connected.

[1]

[2]

[3]

[4]