The Hitchhiker's Guide to the Virtual Element Method

Veiga, L. Beirão da; Brezzi, Franco; Marini, L. D.; Russo, A.

doi:10.1142/s021820251440003x

Cited by 639 publications

(540 citation statements)

References 37 publications

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“…First introduced in [4] and extended in [5,6,3,21,2,26], the Virtual Element Method allows the use of quite general non-degenerate and star-shaped polygons to mesh the spatial domain, even including the possibility of straight angles. In the present framework, we take advantage from this flexibility to easily build a mesh which, on each fracture, is locally or globally conforming to the traces.…”

Section: The Discrete Dfn Problemmentioning

confidence: 99%

“…where the pseudo-projectionΠ 0 k : V δi → P k is defined, as in [3], by local projections, using Π ∇ E,k v δi in place of v δi to compute the moments of order k − 1 and k:…”

Section: The Vem Settingmentioning

confidence: 99%

“…In this work we recall some results concerning the application of the Virtual Element Method [4,3,2] to the steady state simulation of the flow in DFNs [1,22,27,30,24,32,23,15,16,17,18,19,9,10,8,13]. In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity.…”

Section: Introductionmentioning

confidence: 99%

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The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

Benedetto¹,

Berrone²,

Borio³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We discuss several issues concerning the application of the Virtual Element Method (VEM) to the flow in fractured media modeled by the Discrete Fracture Network (DFN) model. Due to the stochastic nature of the computational domains, several geometrical complexities make the computations very challenging. The geometrical flexibility provided by the Virtual Element Method can be exploited to mutually couple local problems, either by resorting to a Mortar approach, or by allowing for the global conformity of the local meshes, while keeping the computational cost under control. We describe these two approaches in detail and we test them on a realistic test case, showing the viability of the two approaches.

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Section: The Discrete Dfn Problemmentioning

confidence: 99%

“…where the pseudo-projectionΠ 0 k : V δi → P k is defined, as in [3], by local projections, using Π ∇ E,k v δi in place of v δi to compute the moments of order k − 1 and k:…”

Section: The Vem Settingmentioning

confidence: 99%

See 1 more Smart Citation

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

Benedetto¹,

Berrone²,

Borio³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…In this section we introduce the Virtual Element discretization of problem (4). From now on, we will assume that Ω is a polygonal domain in R 2 .…”

Section: Virtual Element Discretizationmentioning

confidence: 99%

“…The Virtual Element Method (see, e.g., [3] for an introduction to the method and [4] for the details of its practical implementation) is characterized by the capability of dealing with very general polygonal/polyedral meshes and by the possibility of easily implementing highly regular discrete spaces. Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element.…”

Section: Introductionmentioning

confidence: 99%

Vem and Topology Optimization on Polygonal Meshes

Antonietti¹,

Bruggi²,

Scacchi³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Topology optimization is a fertile area of research that is mainly concerned with the automatic generation of optimal layouts to solve design problems in Engineering. The classical formulation addresses the problem of finding the best distribution of an isotropic material that minimizes the work of the external loads at equilibrium, while respecting a constraint on the assigned amount of volume. This is the so-called minimum compliance formulation that can be conveniently employed to achieve stiff truss-like layout within a two-dimensional domain. A classical implementation resorts to the adoption of four node displacement-based finite elements that are coupled with an elementwise discretization of the (unknown) density field. When regular meshes made of square elements are used, well-known numerical instabilities arise, see in particular the so-called checkerboard patterns. On the other hand, when unstructured meshes are needed to cope with geometry of any shape, additional instabilities can steer the optimizer towards local minima instead of the expected global one. Unstructured meshes approximate the strain energy of truss-like members with an accuracy that is strictly related to the geometrical features of the discretization, thus remarkably affecting the achieved layouts. In this paper we will consider several benchmarks of truss design and explore the performance of the recently proposed technique known as the Virtual Element Method (VEM) in driving the topology optimization procedure. In particular, we will show how the capability of VEM of efficiently approximating elasticity equations on very general polygonal meshes can contribute to overcome the aforementioned mesh-dependent instabilities exhibited by classical finite element based discretization techniques.

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Interpolation and Quasi‐Interpolation inh‐ andhp‐Version Finite Element Spaces

Apel

Melenk

2017

Encyclopedia of Computational Mechanics Second Edition

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Interpolation operators map a function to an element of a finite element space. Unlike more general approximation operators, interpolants are defined locally. Estimates of the interpolation error , that is, the difference , are of utmost importance in numerical analysis. These estimates depend on the size of the finite elements, the polynomial degree employed, and the regularity of . In contrast to interpolation the term quasi‐interpolation is used when the regularity is so low that interpolation has to be combined with regularization. This chapter gives an overview of different interpolation operators and their error estimates. The discussion includes the ‐version and the ‐version of the finite element method, interpolation on the basis of triangular/tetrahedral and quadrilateral/hexahedral meshes, affine and nonaffine elements, isotropic and anisotropic elements, and Lagrangian and other elements.

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The Hitchhiker's Guide to the Virtual Element Method

Cited by 639 publications

References 37 publications

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

Vem and Topology Optimization on Polygonal Meshes

Interpolation and Quasi‐Interpolation inh‐ andhp‐Version Finite Element Spaces

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