Method of Multilevel Homogenization of Hierarchical Periodic Structures

The mathematical description of hierarchical multilevel structures is presented based on the implementation of the idea of introduction of differently scaled coordinates and multi-parametric asymptotic resolution. A method of multilevel homogenization for hierarchical periodic structures is suggested, that allows determining effective characteristics of the structures and setting their microscopic properties at various levels as well. The calculation of effective elastic characteristics of composite materials as an example of the method application is considered. The method is shown to allow obtaining analytical expressions for elastic characteristics in a number of cases.