Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary value problems. We demonstrate an intimate connection between pattern formations and optimal random packing on the plane. The main study is based on the following two points. First, the diffusive flux in reaction-diffusion systems is approximated by piecewise linear functions in the framework of structural approximations. This leads to a discrete network approximation of the considered continuous problem. Second, the discrete energy minimization yields optimal random packing of the domains (disks) in the representative cell. Therefore, the general problem of pattern formations based on the reaction-diffusion equations is reduced to the geometric problem of random packing. It is demonstrated that all random packings can be divided onto classes associated with classes of isomorphic graphs obtained form the Delaunay triangulation. The unique optimal solution is constructed in each class of the random packings. If the number of disks per representative cell is finite, the number of classes of isomorphic graphs, hence, the number of optimal packings is also finite.
From suspensions to porous multilayers: microstructure formation and particle packing in drying colloidal films
Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IET-2), Forschungszentrum Jülich（于利希研究中心埃尔朗根-纽伦堡亥姆霍兹可再生能源研究所（IET-2））；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
Formation of cylindrical shells via sphere packing from fluidized beds
Faculdade de Engenharia Mecânica, UNICAMP-Universidade Estadual de Campinas（坎皮纳斯州立大学机械工程学院）；Laboratoire PIMM, CNRS, Arts et Métiers Institute of Technology, Cnam（国立工艺学院PIMM实验室）；Faculty of Physics, University of Duisburg-Essen（杜伊斯堡-埃森大学物理系）
