Stable and accurate numerical methods for generalized Kirchhoff-Love plates

TL;DR

This paper introduces stable, second-order accurate numerical methods for solving the generalized Kirchhoff-Love plate model under various boundary conditions, validated through stability analysis, numerical tests, and applications like natural frequency identification and resonance simulation.

Contribution

The paper develops and analyzes stable finite-difference schemes with explicit and implicit time-stepping for generalized Kirchhoff-Love plates, including practical stability criteria and applications.

Findings

Numerical schemes are stable and second-order accurate.

Results are consistent with experimental data for thin plates.

Methods successfully identify natural frequencies and simulate resonance phenomena.

Abstract

Efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff-Love plate model subject to three common physical boundary conditions: (i) clamped; (ii) simply supported; and (iii) free. We solve the model equation by discretizing the spatial derivatives using second-order finite-difference schemes, and then advancing the semi-discrete problem in time with either an explicit predictor-corrector or an implicit Newmark-Beta time-stepping algorithm. Stability analysis is conducted for the schemes and the results are used to determine stable time steps in practice. A series of carefully chosen test problems are solved to demonstrate the properties and applications of our numerical approaches. The numerical results confirm the stability and 2nd-order accuracy of the algorithms, and are also comparable with experiments for similar thin plates. As an application, we illustrate a strategy to identify the natural frequencies of a plate using our numerical methods in conjunction with a fast Fourier transformation (FFT) power spectrum analysis of the computed data. Then we take advantage of one of the computed natural frequencies to simulate the interesting physical phenomena known as resonance and beat for a generalized Kirchhoff-Love plate.