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Hourglass stabilization and the virtual element method

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journal contribution
posted on 06.12.2016, 15:38 by A. Cangiani, G. Manzini, A. Russo, N. Sukumar
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. 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 inline image-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.


A.C. gratefully acknowledges support from the College of Science and Engineering of the University of Leicester and the support of the EPSRC (grant EP/L022745/1). G.M. gratefully acknowledges the support of the LDRD-ER project #20140270 ‘From the finite element method to the virtual element method’ at the Los Alamos National Laboratory and of the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research. A.R. gratefully acknowledges the research support from the University of Milano–Bicocca. N.S. gratefully acknowledges the research support of the National Science Foundation through contract grant CMMI-1334783 to the University of California at Davis.



International Journal for Numerical Methods in Engineering, 2015, 102 (3-4), pp. 404-436 (33)

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/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics


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International Journal for Numerical Methods in Engineering







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