An approach to solving mechanics problems for materials with multiscale self-similar microstructure

This article proposes an efficient method for solving mechanics boundary value problems formulated for domains with multiscale self-similar microstructure. In particular, composite materials for which one of the phases has a fractal-like structure with scale cut-offs are considered. The boundary value problems are solved using a finite element procedure with enriched shape functions that incorporate information about the geometric complexity. The use of these shape functions makes possible the definition of a unique, parametrically defined model from which the solution for configurations with an arbitrary number of scales can be derived. The proposed method is primarily useful for structures with a large number of self-similar scales for which using the usual finite element method would be too expensive. In order to exemplify the method, a 2D composite with fractal microstructure is considered and several boundary value problems are solved.