A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models

Recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements (RVEs) with greater numbers of inclusions than have been previously studied. Imposing periodic boundary conditions on very large RVEs can mean enforcing thousands of constraint equations. In addition, a periodic mesh is essential for enforcing the constraints. The present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed. The present study demonstrates the method for two-dimensional triangular and square RVEs of periodically-spaced regular hexagonal and square arrays of composite material containing fibers of equal radii. To impose the boundary conditions along the edges, this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges. It is shown that the method eliminates the need for the conventional node-coupling scheme for imposing periodic boundary conditions, consequently reducing the number of unknowns to the interior degrees of freedom of the RVE along with a small number of global parameters. The method is demonstrated for periodic and non-periodic mesh designs.