scholarly journals First-order VEM for Reissner–Mindlin plates

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.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

scholarly journals A virtual element method for the von Kármán equations

2021 ◽  
Vol 55 (2) ◽  
pp. 533-560
Author(s):  
Carlo Lovadina ◽  
David Mora ◽  
Iván Velásquez
Keyword(s):  
Optimal Error Estimates ◽  
Transverse Displacement ◽  
Discrete Problem ◽  
Elastic Plates ◽  
Virtual Element Method ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Virtual Element ◽  
Von Kármán ◽  
Element Method

In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming inH2for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed forhsmall enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

scholarly journals A dual hybrid virtual element method for plane elasticity problems

2020 ◽  
Vol 54 (5) ◽  
pp. 1725-1750
Author(s):  
Edoardo Artioli ◽  
Stefano de Miranda ◽  
Carlo Lovadina ◽  
Luca Patruno
Keyword(s):  
Closed Form Solution ◽  
Plane Elasticity ◽  
Form Solution ◽  
Stability And Convergence ◽  
Virtual Element Method ◽  
Convergence Results ◽  
Elasticity Problems ◽  
Virtual Element ◽  
Displacement Based ◽  
Element Method

A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically tested on two benchmarks with closed form solution, and on a typical microelectromechanical system. The numerical outcomes have proved that the dual hybrid scheme represents a valid alternative to the more classical low-order displacement-based Virtual Element Method.

scholarly journals A virtual element formulation for general element shapes

2020 ◽  
Vol 66 (4) ◽  
pp. 963-977 ◽  
Author(s):  
P. Wriggers ◽  
B. Hudobivnik ◽  
F. Aldakheel
Keyword(s):  
Considerable Progress ◽  
Element Formulation ◽  
Virtual Element Method ◽  
Polynomial Degree ◽  
General Element ◽  
Arbitrary Polynomial ◽  
Arbitrary Geometries ◽  
Virtual Element ◽  
Element Method

Abstract The virtual element method is a lively field of research, in which considerable progress has been made during the last decade and applied to many problems in physics and engineering. The method allows ansatz function of arbitrary polynomial degree. However, one of the prerequisite of the formulation is that the element edges have to be straight. In the literature there are several new formulations that introduce curved element edges. These virtual elements allow for specific geometrical forms of the course of the curve at the edges. In this contribution a new methodology is proposed that allows to use general mappings for virtual elements which can model arbitrary geometries.

scholarly journals Computational homogenization of transient chemo-mechanical processes based on a variational minimization principle

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Elten Polukhov ◽  
Marc-André Keip
Keyword(s):  
Material Point ◽  
Computational Homogenization ◽  
Element Formulation ◽  
Macroscopic Scale ◽  
Minimization Principle ◽  
First Order ◽  
Independent Variables ◽  
Variational Framework ◽  
Variational Minimization ◽  
Conforming Finite Element

Abstract We present a variational framework for the computational homogenization of chemo-mechanical processes of soft porous materials. The multiscale variational framework is based on a minimization principle with deformation map and solvent flux acting as independent variables. At the microscopic scale we assume the existence of periodic representative volume elements (RVEs) that are linked to the macroscopic scale via first-order scale transition. In this context, the macroscopic problem is considered to be homogeneous in nature and is thus solved at a single macroscopic material point. The microscopic problem is however assumed to be heterogeneous in nature and thus calls for spatial discretization of the underlying RVE. Here, we employ Raviart–Thomas finite elements and thus arrive at a conforming finite-element formulation of the problem. We present a sequence of numerical examples to demonstrate the capabilities of the multiscale formulation and to discuss a number of fundamental effects.

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

2021 ◽  
Author(s):  
P. Wriggers ◽  
B. Hudobivnik ◽  
O. Allix
Keyword(s):  
Finite Elements ◽  
Stiffness Matrix ◽  
Anisotropic Materials ◽  
Continuity Condition ◽  
Thin Plates ◽  
Virtual Element Method ◽  
Quadrilateral Elements ◽  
Plate Elements ◽  
New Ideas ◽  
Virtual Element

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.

scholarly journals A virtual element method for the vibration problem of Kirchhoff plates

2018 ◽  
Vol 52 (4) ◽  
pp. 1437-1456 ◽  
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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