The interior penalty virtual element method for the two-dimensional biharmonic eigenvalue problem

authored by: Jian Meng, Bing Bing Xu, Fang Su, Xu Qian
Abstract: The biharmonic eigenvalue problem is a fourth order eigenmodel appearing in many applications of the mechanics, fluid and inverse scattering theory. In this paper, we introduce the interior penalty virtual element method for the biharmonic eigenvalue problem in two dimensions. It preserves the symmetric positive-definiteness of the continuous problem and reduces the total number of required degrees of freedom. Considering standard assumptions on polygonal meshes, we prove the correct approximation of spectrum for the proposed virtual element scheme. Necessitated by supporting the convergence analysis, representative numerical examples are reported, including the optimal convergence on different meshes, the associate vibration and buckling problems with clamped, simply supported and Cahn–Hilliard boundary conditions, together with developing Serendipity version dropping the internal-to-element degrees of freedom.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): National University of Defense Technology
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 436
No. of pages: 20
ISSN: 0045-7825
Publication date: 01.03.2025
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.cma.2024.117685 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{d6b07bec68b443188b92703d1ae91ae2,
title = "The interior penalty virtual element method for the two-dimensional biharmonic eigenvalue problem",
abstract = "The biharmonic eigenvalue problem is a fourth order eigenmodel appearing in many applications of the mechanics, fluid and inverse scattering theory. In this paper, we introduce the interior penalty virtual element method for the biharmonic eigenvalue problem in two dimensions. It preserves the symmetric positive-definiteness of the continuous problem and reduces the total number of required degrees of freedom. Considering standard assumptions on polygonal meshes, we prove the correct approximation of spectrum for the proposed virtual element scheme. Necessitated by supporting the convergence analysis, representative numerical examples are reported, including the optimal convergence on different meshes, the associate vibration and buckling problems with clamped, simply supported and Cahn–Hilliard boundary conditions, together with developing Serendipity version dropping the internal-to-element degrees of freedom.",
keywords = "Biharmonic eigenvalue problem, Error estimates, Interior penalty VEM, Spectral approximation, The vibration and buckling problems of Kirchhoff–Love plate",
author = "Jian Meng and Xu, {Bing Bing} and Fang Su and Xu Qian",
note = "Publisher Copyright: {\textcopyright} 2024 Elsevier B.V.",
year = "2025",
month = mar,
day = "1",
doi = "10.1016/j.cma.2024.117685",
language = "English",
volume = "436",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}