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.

On Von Kármán Equations and the Buckling of a Thin Circular Elastic Plate

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Joanna Janczewska ◽  
Anita Zgorzelska
Keyword(s):  
Elastic Plate ◽  
Compressive Load ◽  
Load Parameter ◽  
Elastic Plates ◽  
Plate Structures ◽  
Von Karman ◽  
Fredholm Map ◽  
Von Kármán Equations ◽  
Von Kármán ◽  
Nonlinear Deformations

AbstractWe shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero. We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A.Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu.I. Sapronov we will investigate the shape of bifurcation branches.

scholarly journals Virtual element method for quasilinear elliptic problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2450-2472 ◽  
Author(s):  
A Cangiani ◽  
P Chatzipantelidis ◽  
G Diwan ◽  
E H Georgoulis
Keyword(s):  
A Priori ◽  
Quasilinear Equation ◽  
Nonlinear Coefficient ◽  
Optimal Order ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Fixed Point Iterations ◽  
Quasilinear Elliptic ◽  
Element Method

Abstract A virtual element method for the quasilinear equation $-\textrm{div} ({\boldsymbol \kappa }(u)\operatorname{grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

scholarly journals The nonconforming Virtual Element Method for eigenvalue problems

2019 ◽  
Vol 53 (3) ◽  
pp. 749-774 ◽  
Author(s):  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca
Keyword(s):  
Eigenvalue Problems ◽  
Inner Product ◽  
Optimal Order ◽  
Large Set ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Discrete Eigenvalue ◽  
Order Error ◽  
Virtual Element ◽  
Element Method

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of theL2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

scholarly journals The fully nonconforming virtual element method for biharmonic problems

2017 ◽  
Vol 28 (02) ◽  
pp. 387-407 ◽  
Author(s):  
P. F. Antonietti ◽  
G. Manzini ◽  
M. Verani
Keyword(s):  
Numerical Approximation ◽  
Optimal Error Estimates ◽  
Approximation Space ◽  
Virtual Element Method ◽  
Optimal Error ◽  
Order Of Accuracy ◽  
Element Discretization ◽  
Virtual Element ◽  
Biharmonic Problems ◽  
Element Method

In this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

scholarly journals Equilibrium analysis of an immersed rigid leaflet by the virtual element method

2021 ◽  
pp. 1-50
Author(s):  
L. Beirão da Veiga ◽  
C. Canuto ◽  
R. H. Nochetto ◽  
G. Vacca
Keyword(s):  
Numerical Technique ◽  
Angular Position ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Bisection Algorithm ◽  
Computational Mesh ◽  
Virtual Element ◽  
Element Method

We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology.

scholarly journals Some estimates of virtual element methods for fourth order problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingguang Guan
Keyword(s):  
Error Estimates ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Regularity Estimates ◽  
Interpolation Operators ◽  
Virtual Element ◽  
The Right ◽  
Element Method

<p style='text-indent:20px;'>In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.</p>

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.

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

CALCOLO ◽  
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.

scholarly journals A Review on Analytical Modeling for Collapse Mode Capacitive Micromachined Ultrasonic Transducer of the Collapse Voltage and the Static Membrane Deflections

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 714
Author(s):  
Jiujiang Wang ◽  
Xin Liu ◽  
Yuanyu Yu ◽  
Yao Li ◽  
Ching-Hsiang Cheng ◽  
...  
Keyword(s):  
Analytical Modeling ◽  
Ultrasonic Transducer ◽  
Governing Equations ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Collapse Mode ◽  
Capacitive Micromachined Ultrasonic Transducer ◽  
Gap Height ◽  
Von Kármán ◽  
Analytical Expressions

Analytical modeling of capacitive micromachined ultrasonic transducer (CMUT) is one of the commonly used modeling methods and has the advantages of intuitive understanding of the physics of CMUTs and convergent when modeling of collapse mode CMUT. This review article summarizes analytical modeling of the collapse voltage and shows that the collapse voltage of a CMUT correlates with the effective gap height and the electrode area. There are analytical expressions for the collapse voltage. Modeling of the membrane deflections are characterized by governing equations from Timoshenko, von Kármán equations and the 2D plate equation, and solved by various methods such as Galerkin’s method and perturbation method. Analytical expressions from Timoshenko’s equation can be used for small deflections, while analytical expression from von Kármán equations can be used for both small and large deflections.

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