Sphere Packing and Applications to Granular Structure Modeling

This paper presents a new sphere packing algorithm for generating granular structures in either two or three dimensions. Such a structure is often modeled by a parallelepiped containing spherical balls in three dimensions or by a rectangle filled with disks in two dimensions. The grains (spherical balls or disks) are separated by interfaces with specific thicknesses, called grain boundaries, and their size correspond to a size distribution experimentally obtained. The geometrical modeling of such a structure, which consists in determining the repartition of the set of disjoint grains according to the above specifications, can then be considered as the classic sphere packing problem. The proposed method is a constructive algorithm based on an advancing-front approach, which is well known in a different context, namely mesh generation. Since the use of the advancing-front approach leads to empty areas near front collisions, a point relocation algorithm, using weighted Delaunay triangulation, is then introduced to balance the local density on the whole structure. Moreover, we propose a method to transform spherical balls (disks) into polyhedral (polygonal) cells similar to the real grain shape. Numerical 2D and 3D examples are provided to illustrate the capability and the efficiency of our approach. The algorithms and techniques presented here can find applications to generate aggregates in all fields concerned by the granular structures such as metallurgy, ceramics, soil science, cements, biomechanics, etc.