On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes

Arun L. Gain, Cameron Talischi, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)132-160
Number of pages29
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Dec 1 2014
Externally publishedYes


  • Mimetic Finite Difference
  • Polyhedral meshes
  • Polytopes
  • Virtual Element Method
  • Voronoi tessellations

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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