On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials

AbstractThe virtual element method allows to revisit the construction of Kirchhoff-Love elements because the $$C^1$$ C 1 -continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of $$C^1$$ C 1 -continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for $$C^0$$ C 0 -continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these $$C^1$$ C 1 -continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

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First-order VEM for Reissner–Mindlin plates

Computational Mechanics ◽

10.1007/s00466-021-02095-1 ◽

2021 ◽

Author(s):

A. M. D’Altri ◽

L. Patruno ◽

S. de Miranda ◽

E. Sacco

Keyword(s):

Stiffness Matrix ◽

Piecewise Linear ◽

Piecewise Linear Approximation ◽

Element Formulation ◽

Transverse Displacement ◽

Virtual Element Method ◽

First Order ◽

Independent Variables ◽

Virtual Element ◽

Displacement Based

AbstractIn this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

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The nonconforming virtual element method for plate bending problems

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820251650041x ◽

2016 ◽

Vol 26 (09) ◽

pp. 1671-1687 ◽

Cited By ~ 29

Author(s):

Jikun Zhao ◽

Shaochun Chen ◽

Bei Zhang

Keyword(s):

Error Analysis ◽

Function Space ◽

General Framework ◽

Plate Bending ◽

Virtual Element Method ◽

Order Of Accuracy ◽

Plate Elements ◽

Bending Problems ◽

Virtual Element ◽

Element Method

We develop the nonconforming virtual element method for linear plate bending problems. A class of nonconforming virtual elements is constructed, which is [Formula: see text]-continuous. Like the classical nonconforming plate elements, it relaxes the continuity requirement for the function space to some extent. Further, the virtual element is constructed for any order of accuracy and adapts to complicate element geometries. We present a general framework on the error analysis for the nonconforming virtual element method, highlighting the main difference with the conforming one.

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A virtual element method for the vibration problem of Kirchhoff plates

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2017041 ◽

2018 ◽

Vol 52 (4) ◽

pp. 1437-1456 ◽

Cited By ~ 16

Author(s):

David Mora ◽

Gonzalo Rivera ◽

Iván Velásquez

Keyword(s):

Thin Plates ◽

Optimal Order ◽

Transverse Displacement ◽

Computational Domain ◽

Virtual Element Method ◽

Order Error ◽

Vibration Problem ◽

Kirchhoff Plates ◽

Virtual Element ◽

Element Method

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose anH2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

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FINITE ELEMENTS SIMULATION OF HEAT TRANSFER IN THIN PLATES WITH TEMPERATURE DEPENDENT CONDUCTIVITY

10.26678/abcm.encit2018.cit18-0431 ◽

2018 ◽

Author(s):

Gleydson Yuri Ramos Silva ◽

Maria Laura Martins-Costa ◽

ROGERIO GAMA

Keyword(s):

Heat Transfer ◽

Finite Elements ◽

Thin Plates ◽

Temperature Dependent ◽

Temperature Dependent Conductivity ◽

Finite Elements Simulation

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Physical Ultrasonics of Composites

10.1093/oso/9780195079609.001.0001 ◽

2011 ◽

Cited By ~ 3

Author(s):

Dale Chimenti ◽

Stanislav Rokhlin ◽

Peter Nagy

Keyword(s):

Composite Laminates ◽

Stiffness Matrix ◽

Composite Plates ◽

Anisotropic Media ◽

Anisotropic Materials ◽

Ultrasonic Waves ◽

Laminated Plates ◽

Structure Characteristic ◽

Major Advance

Physical Ultrasonics of Composites is a rigorous introduction to the characterization of composite materials by means of ultrasonic waves. Composites are treated here not simply as uniform media, but as inhomogeneous layered anisotropic media with internal structure characteristic of composite laminates. The objective here is to concentrate on exposing the singular behavior of ultrasonic waves as they interact with layered, anisotropic materials, materials which incorporate those structural elements typical of composite laminates. This book provides a synergistic description of both modeling and experimental methods in addressing wave propagation phenomena and composite property measurements. After a brief review of basic composite mechanics, a thorough treatment of ultrasonics in anisotropic media is presented, along with composite characterization methods. The interaction of ultrasonic waves at interfaces of anisotropic materials is discussed, as are guided waves in composite plates and rods. Waves in layered media are developed from the standpoint of the "Stiffness Matrix", a major advance over the conventional, potentially unstable Transfer Matrix approach. Laminated plates are treated both with the stiffness matrix and using Floquet analysis. The important influence on the received electronic signals in ultrasonic materials characterization from transducer geometry and placement are carefully exposed in a dedicated chapter. Ultrasonic wave interactions are especially susceptible to such influences because ultrasonic transducers are seldom more than a dozen or so wavelengths in diameter. The book ends with a chapter devoted to the emerging field of air-coupled ultrasonics. This new technology has come of age with the development of purpose-built transducers and electronics and is finding ever wider applications, particularly in the characterization of composite laminates.

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Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽

10.1007/s10092-021-00418-5 ◽

2021 ◽

Vol 58 (3) ◽

Author(s):

Elena Bachini ◽

Gianmarco Manzini ◽

Mario Putti

Keyword(s):

Arbitrary Order ◽

Surface Geometry ◽

Virtual Element Method ◽

Anisotropic Character ◽

Local Reference System ◽

Local Reference ◽

Local Parametrization ◽

Manufactured Solution ◽

Virtual Element ◽

Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

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