  In this paper, we study applications of the virtual element method (VEM) for
simulating the deformation of multiphase composites. The VEM is a Galerkin
approach that is applicable to meshes that consist of arbitrarily-shaped
polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis
functions are defined as the solution of a local elliptic partial differential
equation, and are never explicitly computed in the implementation of the
method. The stifness matrix of each element is built by using the elliptic
projection operator of the internal virtual work (bilinear form) and it
consists of two terms: a consistency term that is exactly computed (linear
patch test is satisfied) and a correction term (ensures stability) that is
orthogonal to affine displacement fields and has the right scaling. The VEM
simplifies mesh generation for a multiphase composite: a stiff inclusion can be
modeled using a single polygonal or polyhedral element. Attributes of the
virtual element approach are highlighted through comparisons with Voronoi-cell
lattice models, which provide discrete representations of material structure.
The comparisons involve a suite of two-dimensional linear elastic problems:
patch test, axisymmetric circular inclusion problem, and the deformation of a
three-phase composite. The simulations demonstrate the accuracy and flexibility
of the virtual element method.
