https://hal-mines-paristech.archives-ouvertes.fr/hal-00504869Marmorat, Jean-PaulJean-PaulMarmoratCMA - Centre de Mathématiques Appliquées - MINES ParisTech - École nationale supérieure des mines de Paris - PSL - Université Paris sciences et lettresPayre, G.G.PayreDépartement de Génie Mécanique - EPM - École Polytechnique de MontréalZolésio, Jean-PaulJean-PaulZolésioOPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - JAD - Laboratoire Jean Alexandre Dieudonné - UNS - Université Nice Sophia Antipolis (1965 - 2019) - COMUE UCA - COMUE Université Côte d'Azur (2015-2019) - CNRS - Centre National de la Recherche Scientifique - UCA - Université Côte d'AzurINLN - Institut Non Linéaire de Nice Sophia-Antipolis - UNS - Université Nice Sophia Antipolis (1965 - 2019) - COMUE UCA - COMUE Université Côte d'Azur (2015-2019) - CNRS - Centre National de la Recherche ScientifiqueA convergent finite element scheme for a wave equation with a moving boundaryHAL CCSD1990Finite element schemeWave equation[INFO.INFO-AU] Computer Science [cs]/Automatic Control EngineeringPrudon, MagalieJean-Paul Zolésio2010-07-21 16:29:512022-08-04 17:05:422010-07-21 16:29:51enBook sections10.1007/BFb00067031We wish to consider in this paper the numerical approximation of the solution of a wave equation when the boundaries of the spatial domain are moving. This problem has many practical applications in engineering science. One encounters wave systems in evolving domains in widely disseminated situations, such that rolling or unrolling antennas of space satellites, decoding the sound waves emitted by moving underwater objects or simulating the displacement of crane cables. In order to obtain computer simulations of this situations, one may try to make use of the following idea: a first discretization of the partial differential equation with respect to the space variable leads to a second order ordinary differential system M(h)q(t) + K(h)q(t) = F(t). The discretization parameter h gives typically the size of a cell, the number of such cells being held constant during the simulation. When the domain evolves with the time, the parameter h is allowed to vary, and one has to solve M(h(t))q(t) + K(h(t))q = F(t). We shall give evidence in this paper that the results given by such methods are false, as opposed to those obtained by using the concept of convected dense family defined in [2]. One may find in this reference a new proof of the existence of solution for the continuous problem which generalizes the Galerkin method on basis convected from OMEGA(t) to OMEGA0. This approach gives a practical way to generate the convergent numerical solutions we are looking for.