The Virtual Element Method with curved edges

Veiga, L. Beirão da; Russo, A.; Vacca, Giuseppe

doi:10.1051/m2an/2018052

Cited by 108 publications

(125 citation statements)

References 55 publications

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“…In the present section we briefly review the space proposed in [12] and introduce also an alternative space that has the additional advantage of including rigid body motions. Indeed, the space proposed for the diffusion problem in [12] has good approximation properties and is therefore suitable also as a displacement space for problems in structural mechanics. On the other hand, such space does contain constant polynomials (that is translations of the body) but not (linearized) rigid body rotations.…”

Section: Description Of the Virtual Spacementioning

confidence: 99%

“…We here quickly review the space of [12], here extended to the vector-valued version. As usual, we define the space element by element.…”

Section: The Original Spacementioning

confidence: 99%

“…Among the many possibilities, we here cite two among the most known, that are isoparametric Finite Elements (see for instance [24]) and Isogeometric Analysis (see for instance [32,25]). As shown in [12], it turns out that for Virtual Elements in 2D the extension to cases with curved edges (and thus able to approximate exactly the geometry of interest) is quite natural. By changing the definition of the edge-wise space, the method is able to accommodate for general boundaries described by a given parametrization (for instance obtained by CAD).…”

Section: Introductionmentioning

confidence: 99%

“…In the present work we start from the basic construction introduced in [12], that was on a simple (scalar) linear model problem, and generalize it to the more complex situation of small deformation problems in structural mechanics. Our approach, in the spirit of [15,6,5], can accept a generic black-box (elastic or inelastic) constitutive algorithm but, in addition, can make use of curved edges thus leading to an exact approximation of the geometry.…”

Section: Introductionmentioning

confidence: 99%

“…Our approach, in the spirit of [15,6,5], can accept a generic black-box (elastic or inelastic) constitutive algorithm but, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. When the Virtual Spaces of [12] are applied in the present context of structural mechanics, a potential drawback appears. Indeed, such spaces (although holding optimal approximation properties) contain constant functions but do not contain linear ones.…”

Section: Introductionmentioning

confidence: 99%

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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: Description Of the Virtual Spacementioning

confidence: 99%

“…We here quickly review the space of [12], here extended to the vector-valued version. As usual, we define the space element by element.…”

Section: The Original Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Curvilinear Virtual Elements for 2D solid mechanics applications

Artioli

Veiga

Dassi

2020

Computer Methods in Applied Mechanics and Engineering

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Modeling curved interfaces without element‐partitioning in the extended finite element method

Chin

Sukumar

2019

Numerical Meth Engineering

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In this paper, we model holes and material interfaces (weak discontinuities) in two-dimensional linear elastic continua using the extended finite element method on higher-order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme-a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one-dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used.For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h-refinement (p = 1, 2, … , 5) and exponential rates of convergence with p-refinement (p = 1, 2, … , 7) for elastostatic problems with holes and material inclusions on Cartesian pth-order spectral finite element meshes. KEYWORDSBézier curves, homogeneous numerical integration method, interface stabilization, level set methods, material interface enrichment, spectral X-FEM Int J Numer Methods Eng. 2019;120:607-649.wileyonlinelibrary.com/journal/nme

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Numerical integration in the virtual element method with the scaled boundary cubature scheme

Chin,

Dassi,

Manzini

et al. 2024

Numerical Meth Engineering

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The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

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The Virtual Element Method with curved edges

Cited by 108 publications

References 55 publications

Curvilinear Virtual Elements for 2D solid mechanics applications

Curvilinear Virtual Elements for 2D solid mechanics applications

Modeling curved interfaces without element‐partitioning in the extended finite element method

Numerical integration in the virtual element method with the scaled boundary cubature scheme

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