On generalized boundary conditions for mesoscopic volumes in computational homogenization

Abstract

This paper investigates generalized boundary conditions for mesoscopic statistical volume elements (SVEs) in computational homogenization methods. Lagrange multipliers are used to enforce the average values of the macroscopic quantity over the SVE. Spring-like constraints are simultaneously applied at the boundary in order to take into account microscopically perturbed interaction between the mesoscopic volume and its surroundings. This approach provides apparent homogenized parameters, which are located between the results generated by classical uniform kinematic constraining and uniform static loading of the SVE. Since the method combines both these solutions, the term generalized boundary conditions is used. A single parameter with the dimension of length is recognized and adjusted to obtain good homogenization results, even for small SVEs. The method was implemented within the finite element framework and the formula for calculating apparent stiffness tensor, in the convenient form of a single matrix equation, was given. 2D and 3D reference homogenization examples are also provided. The presented approach is especially suitable for random composites, as no assumptions concerning the SVE geometry, such as periodicity, are necessary.