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 an H2(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.
Accepté le :
DOI : 10.1051/m2an/2017041
Mots clés : Virtual element method, Kirchhoff plates, spectral problem, error estimates
@article{M2AN_2018__52_4_1437_0,
author = {Mora, David and Rivera, Gonzalo and Vel\'asquez, Iv\'an},
title = {A virtual element method for the vibration problem of {Kirchhoff} plates},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1437--1456},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017041},
mrnumber = {3875292},
zbl = {1407.65274},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017041/}
}
TY - JOUR AU - Mora, David AU - Rivera, Gonzalo AU - Velásquez, Iván TI - A virtual element method for the vibration problem of Kirchhoff plates JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1437 EP - 1456 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017041/ DO - 10.1051/m2an/2017041 LA - en ID - M2AN_2018__52_4_1437_0 ER -
%0 Journal Article %A Mora, David %A Rivera, Gonzalo %A Velásquez, Iván %T A virtual element method for the vibration problem of Kirchhoff plates %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1437-1456 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017041/ %R 10.1051/m2an/2017041 %G en %F M2AN_2018__52_4_1437_0
Mora, David; Rivera, Gonzalo; Velásquez, Iván. A virtual element method for the vibration problem of Kirchhoff plates. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1437-1456. doi : 10.1051/m2an/2017041. http://www.numdam.org/articles/10.1051/m2an/2017041/
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