Conservative Numerical Methods for the Full von Karman Plate Equations

Stefan Bilbao, Olivier Thomas, Cyril Touze, Michele Ducceschi

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

This article is concerned with the numerical solution of the full dynamical von K{\'a}rm{\'a}n plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von K{\'a}rm{\'a}n system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semi-discrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of
plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretisations.
Original languageEnglish
Number of pages24
JournalNumerical Methods for Partial Differential Equations
DOIs
Publication statusPublished - 2015

Keywords / Materials (for Non-textual outputs)

  • conservative numerical methods
  • Hamiltonian methods
  • nonlinear plate vibration

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