Virtual element method for computational homogenization of composite and heterogeneous materials

In this study, a two-dimensional multi-region framework, based on the use of the Virtual Element Method (VEM), is developed for computational materials homogenization and applied to different classes of widely employed heterogeneous materials. The VEM has recently emerged as a powerful generalisation of the Finite Element Method capable of dealing with very general polygonal mesh elements, including non-convex or highly distorted elements. Such features are appealing for the treatment of problems whose analysis domains present complex or statistical morphological features, which would generally require careful and time-consuming mesh/data preparation and regularization. In this work, the lowest-order VEM for two-dimensional elastostatics is employed for the homogenization of polycrystalline materials and unidirectional fibre-reinforced composites. In both cases, artificial micro-morphologies are usually generated resorting to automatic algorithms aimed at approximating/reproducing the statistical microscopic features of real materials. In such a context, the likely presence of morphological irregularities, and subsequent mesh distortions, usually requires caution and the employment of sophisticated mesh regularization procedures. The study demonstrates how the inherent features of the VEM can be conveniently exploited for such classes of problems, as the method allows the relaxation of the requirements on the mesh quality, yet providing accurate numerical results.