Conservative numerical methods for the Full von Kármán plate equations

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. 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.

Wave turbulence theory of elastic plates

“…Fig. 1 shows this geometric construction and we can see that for any wavenumber p 1 and k 1 , an intersection exists between the revolution paraboloid of origin O with the one that has B for origin, satisfying therefore precisely the conditions imposed in (52). As expected, this problem has 5 degrees of freedom, but one of this variable p 1 is not integrated, and so, the domain of integration becomes a three dimensional manifold.…”

“…In fact, periodic boundary conditions provide the natural framework to investigate wave turbulence. On the other hand, methods using finite difference scheme would be more appropriate to account for general boundary conditions although it might be less accurate to capture the wave dynamics [32,52]. As described below the pseudo-spectral method uses the real space for computing the non-linear term while the integration is performed in the Fourier space thanks to an Adams-Bashford scheme.…”

Cascades and transitions in turbulent flows

“…Elastic wave turbulence thus provides a unique opportunity where theoretical work based on weak wave turbulence can be tested [658]. Relations similar to von Kármán-Howarth-Monin relations have been derived in [659] and compared with experimental data in [660][661][662][663][664][665][666] and with numerical simulations in [658,664,[667][668][669][670][671]. This crossing of theory, experiments and numerical simulations has brought up number of issues such as the effect of finite dissipation coefficients and the role of coherent structures in the formation of the spectrum that are often neglected in theoretical work.…”

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

“…The transverse and in-plane displacements are written w(r, t) and u(r, t) at time t and position r. The vectors u and r are two-dimensional vectors of the mid-plane of the plate. The equations of motions can be written [57]:…”