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

By Arun L. Gain, Cameron Talischi, Glaucio H. Paulino. 2013

424 citations recorded by the source corpus.

Abstract 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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/63188d3477293f13229e7ffd093f4c91f15f14cc

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Citations in the corpus

A multigrid algorithm for the $p$-version of the Virtual Element Method (2017, background): “VEMs for linear and nonlinear elasticity have been developed in [18, 21, 50], whereas VEMs for plate bending, Cahn-Hilliard, Stokes, and Helmholtz problems have been addressed in [6, 8, 34, 55].”
Conforming and nonconforming virtual element methods for elliptic problems (2015, positive): “Concurrently, the VEM framework has been extended to the solution of plate-bending problems [14], linear elasticity problems in two and three spatial dimensions [6, 21], the Steklov eigenvalue problem [27], the simulation of discrete fracture networks [11], and the twodimensional streamline formulat”
Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods (2015, methodology): “To evaluate the integral in Equation (12), two schemes are adopted [53]: a nodal quadrature and a conforming interpolant quadrature.”
Hourglass stabilization and the virtual element method (2015, background): “Recent studies on the VEM have examined elasticity [28, 29] and plate bending [26].”
Virtual Element Methods for general second order elliptic problems on polygonal meshes (2016, neutral): “See in particular, for Mimetic Finite Diﬀerences 15 , and mostly 10 , 27 (and the references therein), and 1 , 2 , 11 , 6 , 7 , 16 , 17 , 26 , 28 , 30 for Virtual Elements.”
Divergence free Virtual Elements for the Stokes problem on polygonal meshes (2015, methodology): “The Virtual Element Method has been developed successfully for a large range of problems, see for instance [1–4, 7, 8, 12, 14, 19, 22, 34, 40, 49].”
A Plane Wave Virtual Element Method for the Helmholtz Problem (2015, background): “The basic VEM introduced on the Poisson problem has then been extended to highly regular [11], non conforming [2] and discontinuous approximating spaces [16], to general elliptic [8] and mixed problems [6,7,14], as well as to Stokes [1], plate problems [15], linear and non linear elasticity [5, 10, ”
A low order 3D virtual element formulation for finite elasto–plastic deformations (2017, background): “Many application of virtual elements can be related to the linear elastic range, see [1] and [2].”
Virtual Element Methods for general second order elliptic problems on polygonal meshes (2014, neutral): “See in particular, for Mimetic Finite Diﬀerences, [3], [13], [14], [17], [24], [26], [27], [25], and mostly [16], [49] (and the references therein), and [1], [2], [19], [10], [11], [18], [28], [29], [45], [46], [50], [53], [64] for Virtual Elements.”
The Virtual Element Method with curved edges (2017, background): “Since its first appearence, VEM has shared a very good success both in the mathematical and engineering literature: a very small sample of works is given by [25, 32, 12, 18, 40, 42, 11, 9, 31, 26, 3, 20].”
Stability Analysis for the Virtual Element Method (2016, neutral): “In this paper, we do not investigate on how to select τ E , but we address the reader to [4, 19, 14] for some study on such an issue.”