Numerical simulations of a mixed finite element method for damped plate vibration problems

Ruxin Zhang; Zhe Yin; Ailing Zhu; Ruxin Zhang; Zhe Yin; Ailing Zhu

doi:10.3934/mmc.2023002

Mathematical Modelling and Control

2023, Volume 3, Issue 1: 7-22. doi: 10.3934/mmc.2023002

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Research article

Numerical simulations of a mixed finite element method for damped plate vibration problems

School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250358, China

Received: 03 September 2022 Revised: 26 December 2022 Accepted: 06 January 2023 Published: 14 February 2023

The mixed finite element method can reduce the requirement for the smoothness of the finite element space and simplify the interpolation space for finite elements, and hence is especially effective in solving high order differential equations. In this work, we establish a mixed finite element scheme for the initial boundary conditions of damped plate vibrations and prove the existence and uniqueness of the solution of the semi-discrete and backward Euler fully discrete schemes. We use linear element approximation for the introduced intermediate variables, conduct the error analysis, and obtain the optimal order error estimate. We verify the efficiency and the accuracy of the mixed finite element scheme via numerical case studies and quantify the influence of the damping coefficient on the frequency and amplitude of the vibration.
- mixed finite element,
- semi-discrete schemes,
- backward Euler fully discrete schemes,
- optimal order error estimate,
- numerical simulation
Citation: Ruxin Zhang, Zhe Yin, Ailing Zhu. Numerical simulations of a mixed finite element method for damped plate vibration problems[J]. Mathematical Modelling and Control, 2023, 3(1): 7-22. doi: 10.3934/mmc.2023002

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Abstract

The mixed finite element method can reduce the requirement for the smoothness of the finite element space and simplify the interpolation space for finite elements, and hence is especially effective in solving high order differential equations. In this work, we establish a mixed finite element scheme for the initial boundary conditions of damped plate vibrations and prove the existence and uniqueness of the solution of the semi-discrete and backward Euler fully discrete schemes. We use linear element approximation for the introduced intermediate variables, conduct the error analysis, and obtain the optimal order error estimate. We verify the efficiency and the accuracy of the mixed finite element scheme via numerical case studies and quantify the influence of the damping coefficient on the frequency and amplitude of the vibration.

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