Virtual Elements for Linear Elasticity Problems

By L. Beirão da Veiga, Franco Brezzi, L. Donatella Marini. SIAM Journal on Numerical Analysis, 2013

596 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/5446f2a487b370726d5009706025fe2e50bbfc13

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

A virtual element method with arbitrary regularity (2014, methodology): “We refer the interested reader to Beirão da Veiga et al. (2013) for a thorough discussion concerning this issue.”
The Hitchhiker's Guide to the Virtual Element Method (2014, methodology): “It has been shown in [1, 5, 6] how to compute from (5.”
Virtual element for the buckling problem of Kirchhoff–Love plates (2019, background): “in a large range of problems in fluid and solid mechanics; see for instance [2, 3, 4, 7, 8, 9, 11, 12, 15, 16, 21, 22, 23, 24, 33, 40, 43, 44].”
A displacement‐based finite element formulation for general polyhedra using harmonic shape functions (2014, methodology): “More analysis is needed to put the present research in context with the Virtual Element Method.”
The Virtual Element Method with curved edges (2017, background): “t concerns the L2-error, the behaviour is again the expected one O(hk+1). Although wedidnotprovethe L2-errorestimateinthepresentpaper, thiscouldbederivedcombining thetoolsherepresentedwithSection2.7in[5]. Test 5.1. WeconsiderthecurveddomainΩ describedby Ω := {(x,y) s.t 0 ≤x≤1, and g 1(x) ≤y≤g 2(x)},”
Topology optimization using polytopes (2013, unknown): “We use the Virtual Element Method to numerically solve the governing elasticity problem.”
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes (2013, background): “…solution, namely that the Galerkin error is bounded by a constant multiple of the error in the best approximation of u in Vgh : ‖u− uh‖1,Ω ≤ C inf wh∈Vgh ‖u−wh‖1,Ω (12) By the approximation property of the finite element space, the right hand side is O(h), and the convergence rate is linear.”
Divergence free Virtual Elements for the Stokes problem on polygonal meshes (2015, methodology): “As a consequence, the scheme in [5] can be immediately extended to the Stokes problem, thus yielding a stable VEM family (that would be comparable to the Crousiex-Raviart finite element family).”
Virtual Element Methods for general second order elliptic problems on polygonal meshes (2014, unknown): “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.”
Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method (2017, unknown): “The Virtual Element Method was recently developed as a generalization of Mimetic Finite Differences,[29, 30], and has been applied to a wide number of problems, such as plate bending problems [31], elasticity problems [32, 33], Stokes problems [34] and the Steklov eigenvalue problem [35].”
Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods (2015, background): “[57, 58]: K2 D ̨P (37) where ̨ D ̨ trace.”
New perspectives on polygonal and polyhedral finite element methods (2014, methodology): “The Galerkin formulation and implementation of polygonal and polyhedral finite elements are discussed in Section 3, with emphasis on connecting the virtual element method [64, 65, 66] to barycentric finite element methods.”
Hourglass stabilization and the virtual element method (2015, background): “Recent studies on the VEM have examined elasticity [28, 29] and plate bending [26].”