TY - JOUR
T1 - On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes
AU - Gain, Arun L.
AU - Talischi, Cameron
AU - Paulino, Glaucio H.
N1 - Funding Information:
We are thankful to the support from the US National Science Foundation under grant numbers 1321661 and 1437535 , and from the Donald B. and Elizabeth M. Willett endowment at the University of Illinois at Urbana-Champaign . We also acknowledge partial support provided by Tecgraf/PUC-Rio (Group of Technology in Computer Graphics) , Rio de Janeiro, Brazil. Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors or sponsoring agencies. We acknowledge Prof. Anton Evgrafov for suggesting, during the World Congress of Computational Mechanics (WCCM 2012, Brazil), that we investigate the class of mimetic numerical methods in order to develop our polygonal discretization scheme in three dimensions—his suggestion was the inception of the work in the present paper. We would also like to acknowledge the anonymous reviewers for their insightful comments which helped to improve the present work.
Publisher Copyright:
© 2014 Elsevier B.V.
PY - 2014/12/1
Y1 - 2014/12/1
N2 - We explore the recently-proposed Virtual Element Method (VEM) for the numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the linear elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
AB - We explore the recently-proposed Virtual Element Method (VEM) for the numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the linear elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
KW - Mimetic Finite Difference
KW - Polyhedral meshes
KW - Polytopes
KW - Virtual Element Method
KW - Voronoi tessellations
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U2 - 10.1016/j.cma.2014.05.005
DO - 10.1016/j.cma.2014.05.005
M3 - Article
AN - SCOPUS:84908330879
SN - 0374-2830
VL - 282
SP - 132
EP - 160
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -