A Virtual Element Numerical Approximation for the Vibration Problem of a Thin Plate
DOI:
https://doi.org/10.5540/03.2026.012.01.0241Palabras clave:
Vibration Plate, Virtual Element Method, Clamped Plate, Simply Supported, Numerical ResultsResumen
We focus our attention in the development of a virtual element method for the approximation of the vibration problem of a thin plate modeled by Kirchhoff-Love equations. We introduce a weak variational formulation based on the Sobolev space H2. In addition, we propose a discretization by means of the lowest-order non conforming elements. We show that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates. Finally, we report some numerical tests supporting our theoretical results.
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D. Adak, M. Mora, and I. Velásquez. “A C0-nonconforming virtual element method for the vibration and buckling problems of thin plates”. In: Computer Methods in Applied Mechanics and Engineering 403 (2023), pp. 1–24. doi: 10.1016/j.cma.2022.115763.
P. F. Antonietti, G. Manzini, and M. Verani. “The fully nonconforming virtual element method for biharmonic problems”. In: Mathematical Models and Methods in Applied Sciences 28.2 (2018), pp. 387–407. doi: 10.1142/S0218202518500100.
L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. “Basic principles of virtual element methods”. In: Mathematical Models and Methods in Applied Sciences 23.1 (2013), pp. 199–214. doi: 10.1142/S0218202512500492.
P. G. Ciarlet. The Finite Element Method for Elliptic Problems. 2nd rev. ed. SIAM, 2002. isbn: 978-0-898715-14-9.
F. Dassi and I. Velásquez. “Virtual element method on polyhedral meshes for bi-harmonic eigenvalues problems”. In: Computers & Mathematics with Applications 121 (2022), pp. 85–101. doi: 10.1016/j.camwa.2022.07.001.
P. Grisvard. Elliptic Problems in Non-Smooth Domains. Vol. 24. Monographs and Studies in Mathematics. Boston: Pitman, 1985. isbn: 978-0273086569.
F. Lepe, D. Mora, G. Rivera, and I. Velásquez. “A virtual element method for the Steklov eigenvalue problem allowing small edges”. In: Journal of Scientific Computing 88 (2021), pp. 1–21. doi: 10.1007/s10915-021-01555-3.
D. Mora, G. Rivera, and I. Velásquez. “A virtual element method for the vibration problem of Kirchhoff plates”. In: ESAIM. Mathematical Modelling and Numerical Analysis 52.4 (2018), pp. 1437–1456. doi: 10.1051/m2an/2017041.
J. Zhao, B. Zhang, S. Chen, and S. Mao. “The Morley–type virtual element for plate bending problems”. In: Journal of Scientific Computing 76.1 (2018), pp. 610–629. doi: 10.1007/s10915-017-0632-3.
