Abstract
SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of ‐continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four‐node quadrilateral and eight‐node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.
Highlights
For linear and nonlinear transient analysis, underintegrated finite elements (FE) with hourglass stabilization is widely adopted
SUMMARY In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods
The rank-deficiency of K allows spurious singular modes to be present that can pollute the numerical solution. This deficiency is cured by adding a correction term to (10), and the various remedies for underintegrated finite elements fall under the umbrella of what has come to be known as hourglass control methods
Summary
For linear and nonlinear transient analysis, underintegrated finite elements (FE) with hourglass stabilization is widely adopted. Use of the one-point quadrature, combined with some form of stabilization, has been a topical area of research since the early 1980s. This feature is available in many of the finite element software codes [1,2,3,4,5]. Use of the one-point quadrature leads to a rank-deficient (singular) stiffness matrix and the presence of hourglass (solid continua) or spurious singular (Poisson equation) modes. The derivation of the VEM for the bilinear isoparametric four-node quadrilateral element is presented, and direct comparisons to the approach of Belytschko and coworkers [8,9,10] are made.
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