A simple and effective gradient recovery scheme and a posteriori error estimator for the Virtual Element Method (VEM)

Chi, Heng; Veiga, L. Beirão da; Paulino, Gláucio H.

doi:10.1016/j.cma.2018.08.014

Cited by 48 publications

(42 citation statements)

References 63 publications

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“…In the framework of Structural Mechanics, after the initial papers [26,30,15,43], many applications and developments were pursued by the community. Among the many publications, we here refer to a long but non exhaustive list [20,29,33,6,5,21,3,42,44,4,8,7,16,41,34,27,22]. Finally, it must be noted that other polytopal methods exist in the Structural Mechanics literature; we here limit ourselves in citing [39,18,23,28] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Curvilinear Virtual Elements for 2D solid mechanics applications

Artioli

Veiga

Dassi

2020

Computer Methods in Applied Mechanics and Engineering

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In the present work we generalize the curvilinear Virtual Element technology, introduced for a simple linear scalar problem in a previous work, to generic 2D solid mechanic problems in small deformations. Such generalization also includes the development of a novel Virtual Element space for displacements that contains rigid body motions. Our approach can accept a generic black-box (elastic or inelastic) constitutive algorithm and, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. Rigorous theoretical interpolation properties for the new space on curvilinear elements are derived. We undergo an extensive numerical test campaign, both on elastic and inelastic problems, to assess the behavior of the scheme. The results are very promising and underline the advantages of the curved VEM approach over the standard one in the presence of curved geometries. * artioli@ing.uniroma2.it †

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Section: Introductionmentioning

confidence: 99%

Curvilinear Virtual Elements for 2D solid mechanics applications

Artioli

Veiga

Dassi

2020

Computer Methods in Applied Mechanics and Engineering

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“…The integration on the element boundary is associated with the line and face integrals in 2D and 3D, respectively. One notes that the face integral in 3D can be evaluated by triangulation of faces . Finally, we can show that, for any

v_{h} \in scriptV ((), E)

, the following relation holds between the two projections:

\nabla ((), Π^{0} v_{h}) = Π^{0} ((), italic\nabla v_{h}) .

…”

Section: Basic Elastodynamic Vem Formulationmentioning

confidence: 91%

“…One notes that the face integral in 3D can be evaluated by triangulation of faces. 7,18,20 Finally, we can show that, for any v h ∈  (E), the following relation holds between the two projections:…”

Section: -Projection Operatorsmentioning

confidence: 94%

“…Alternatively, a scalar‐based stabilization scheme can be used, and

{\overset{true‾}{boldK}}_{E, s}

is expressed as

{\overset{true‾}{boldK}}_{E, s} = α_{s} {({boldI}_{n} - {boldP}^{0})}^{T} ((), I_{n} - P^{0}),

where α s is a stiffness scaling parameter. Previously, tr( C )/ n C was used as the scaling parameter . Additionally, the present study proposes to use the maximum diagonal component of the consistent part for the scaling parameter, ie, α s = max ([ K E , c ] ii ), in the scalar‐based stabilization scheme.…”

Section: Basic Elastodynamic Vem Formulationmentioning

confidence: 99%

“…Previously, tr(C)/n C was used as the scaling parameter. 9,20 Additionally, the present study proposes to use the maximum diagonal component of the consistent part for the scaling parameter, ie,…”

Section: Stiffness Matrixmentioning

confidence: 99%

See 2 more Smart Citations

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Park

Chi

Paulino

2019

Numerical Meth Engineering

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Summary We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.

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Error estimation and mesh adaptivity for the virtual element method based on recovery by compatibility in patches

D’Altri

Patruno

Artioli

et al. 2020

Numerical Meth Engineering

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Summary In this article, a recovery by compatibility in patches (RCP)‐based a posteriori error estimator is proposed for the virtual element method (VEM), and it is utilized to drive adaptive mesh refinement processes in two‐dimensional elasticity problems. In RCP, recovered stresses are obtained by minimizing the complementary energy of patches of elements over a set of assumed equilibrated stress modes. To this aim, the explicit knowledge of displacements is only needed along the patch boundaries and no knowledge of superconvergent points is required, so making the RCP naturally suitable for the VEM. The a posteriori error estimation is conducted by comparing the stress field of a standard displacement‐based VEM solution and the stress field obtained through RCP. The procedure is simple, and it does not require ad hoc modifications for small patches. The capability of this RCP‐based error estimator to drive adaptive mesh refinements is successfully demonstrated through various numerical examples.

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A simple and effective gradient recovery scheme and a posteriori error estimator for the Virtual Element Method (VEM)

Cited by 48 publications

References 63 publications

Curvilinear Virtual Elements for 2D solid mechanics applications

Curvilinear Virtual Elements for 2D solid mechanics applications

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Error estimation and mesh adaptivity for the virtual element method based on recovery by compatibility in patches

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