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Stable and accurate numerical methods for generalized Kirchhoff–Love plates

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Abstract

Kirchhoff–Love plate theory is widely used in structural engineering. In this paper, 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. The generalization stems from the inclusion of additional physics to the classical Kirchhoff–Love model that accounts for bending only. 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 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.

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Acknowledgements

This research was supported by the Louisiana Board of Regents Support Fund under contract No. LEQSF(2018-21)-RD-A-23. L. Li is grateful to Professor W.D. Henshaw of Rensselaer Polytechnic Institute (RPI) for helpful conversations. Portions of this research were conducted with high performance computational resources provided by the Louisiana Optical Network Infrastructure (http://www.loni.org).

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Authors and Affiliations

  1. Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504, USA

    Duong T. A. Nguyen & Longfei Li

  2. Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095, USA

    Hangjie Ji

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Appendix A: Nodal line patterns for the eigenvalue problem

Appendix A: Nodal line patterns for the eigenvalue problem

We show the results of the eigenvalue problem (31) here. Nodal lines of the first 25 eigenmodes (with multiplicity) for the square plate with clamped edges and the annular plate with simply supported boundaries are shown in Figs. 18 and 19, respectively. The eigenmodes plotted for each degenerated pair are arbitrary so they can be asymmetric.

Fig. 18
figure 18

Nodal lines of the first 25 eigenmodes (with multiplicity) of the clamped square plate. There are 6 degenerate pairs of eigenmodes, so that only 19 distinct normalized eigenvalues are represented. Values reported in the plots are the natural frequencies in increasing order

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Fig. 19
figure 19

Nodal lines of the first 25 eigenmodes (with multiplicity) of the simply supported annular plate. There are 11 degenerate pairs of eigenmodes, so that only 14 distinct eigenvalues are represented. Values reported in the plots are the natural frequencies in increasing order

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Nguyen, D.T.A., Li, L. & Ji, H. Stable and accurate numerical methods for generalized Kirchhoff–Love plates. J Eng Math 130, 6 (2021). https://doi.org/10.1007/s10665-021-10163-x

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