The virtual element method for arbitrarily distorted 8-node bricks

Standard isoparametric 8-node hexahedral Finite Elements (FEs) are among the most widely used elements for the simulation of 3D problems. While these elements provide excellent results in the case of regular meshes, their accuracy may deteriorate with increasing distortion, finally failing in the case of degenerate elements, where the Jacobian determinant of the geometry mapping vanishes inside the element. In the case of simulation domains of complex geometries, the generation of a mesh of regular or only mildly distorted brick elements can be extremely difficult and time consuming, and in some cases it is even impossible to obtain a mesh free of degenerate elements. Three-dimensional Virtual Elements (VEs) are well known to be highly robust with respect to extreme element distortion; however, at least in the conforming setting, they are restricted to polyhedral elements with planar faces. To enhance the usability of VEs and to make possible their integration in meshes of FE isoparametric bricks, thus significantly reducing the meshing time, in this work we formulate a new 8-node, order 1, hexahedral VE with curved faces, that we name virtual brick. With reference to a simple reaction-diffusion problem, it is shown that the proposed element is convergent and fully compatible with regular 8-node isoparametric FEs. Furthermore, the integrals on the element faces and volume can be computed using the standard Gauss quadrature rules of the isoparametric FE, a property of great importance in the case of nonlinear problems. All these properties hold also in the degenerate case, as long as the virtual brick faces do not intersect with each other.