Deltahedral self-stabilized virtual elements for 3D linear elastostatics problems

The Virtual Element Method (VEM), a new generation of the traditional finite element method (FEM), allows for arbitrary polyhedral meshes, with elements not necessarily convex. However, the stiffness matrix of the Virtual Element (VE) in most cases requires stabilization, which is one of the main limitations of the VEM. This paper presents a new type of 3D self-stabilized VE, based on a Hu-Washizu variational approach for 3D linear elastostatics. The surface of the new element is composed of triangular faces, resembling the Greek letter Delta (Delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}). Due to its unique geometric features, the new VE is named Deltahedron element. The advantage of triangles over faces of arbitrary polygonal shapes is that the displacement model is polynomial on the faces and therefore is not virtual. 8-node Deltahedra with 12 triangular faces are of particular interest as they can be smoothly coupled to a flat face of an 8-node 3D finite element (brick element). Numerical tests have been conducted on highly distorted, self-stabilized, deltahedral 8-node elements, including non-convex shapes, and they have shown good accuracy and expected convergence rates. The issue of integrals computation is also discussed in detail.