Time evolution of three-dimensional cellular systems: Computer modeling based on vertex-type models

An effective computer modeling of time evolution of three-dimensional cellular systems like soap froths and crystalline grain aggregates has been devised, which captures the essence of difficult correlation effects of neighboring cells. This can be achieved by eliminating the continuous degrees of freedom besides the immediate vicinity of the center of a singular region of space, that is, an intersection of interfaces from the original full-curvature drien equation of motion of interfaces, thus leaving a set of equations of motion for such intersections, i.e. vertices. To actually carry out this projection operation each interface is divided intoa set of two-dimensional simplexes. A derivation of the model equations is given in the most general possible form. Various results including topological characteristics of three-dimensional cellular patterns were obtained using the simpler version of these vertex equations, among which the result for the average growth rate of f-sided cells is presented. An application to some specific cellular systems is also discussed.