Conservative numerical methods for the Full von Kármán plate equations

Stefan Bilbao (1) , Olivier Thomas (2) , Cyril Touzé (3, 4) , Michele Ducceschi (3, 4, 1)

1 Acoustics and Audio Group
2 LSIS- INSM - Laboratoire des Sciences de l'Information et des Systèmes : Ingénierie Numérique des Systèmes Mécaniques
3 UME - Unité de Mécanique
4 IMSIA - UMR 9219 - Institut des Sciences de la mécanique et Applications industrielles

This article is concerned with the numerical solution of the full dynamical von Karman plate equations for geometrically nonlinear (large-amplitude) vibration. 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 Karman 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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Mécanique [physics.med-ph]; Vibrations [physics.class-ph]

Stefan Bilbao, Olivier Thomas, Cyril Touzé, Michele Ducceschi. Conservative numerical methods for the Full von Kármán plate equations. Numerical Methods for Partial Differential Equations, 2015, 31 (6), ⟨10.1002/num.21974⟩. ⟨hal-01206323⟩