Pattern formations and optimal packing
- Speaker(s)
- Vladimir Mityushev
- Affiliation
- Department of Computer Sciences and Computer Methods, Pedagogical University Krakow
- Date
- Oct. 28, 2015, 2:15 p.m.
- Room
- room 4050
- Seminar
- Seminar of Biomathematics and Game Theory Group
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.
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.