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Conservative numerical methods for the Full von Kármán plate equations
Authors:Stefan Bilbao  Olivier Thomas  Cyril Touzé  Michele Ducceschi
Affiliation:1. Acoustics and Audio Group, James Clerk Maxwell Building, University of Edinburgh, United Kingdom;2. Arts et Métiers ParisTech, Lille, France;3. IMSIA (Institute of Mechanical Sciences and Industrial Applications), UMR 8193 CNRS‐EDF‐CEA‐ENSTA, Université Paris‐Saclay, Palaiseau Cedex, France
Abstract:This article is concerned with the numerical solution of the full dynamical von Kármán 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 Kármán 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 semidiscrete (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 discretizations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1948–1970, 2015
Keywords:conservative numerical methods  Hamiltonian methods  nonlinear plate vibration
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