Conservative coupling between finite elements and retarded potentials. Application to vibroacoustics

Pascal Grob 1 Patrick Joly 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : The numerical simulation of vibroacoustics is concerned with the radiation of sound emitted by thin vibrating mechanical structures. We present a numerical method that combines boundary elements (integral equation) for the acoustic wave equation with standard finite elements for the mechanics. The originality of our work is that we consider the time domain problem and use retarded potentials for writing the integral equation. We establish a nonstandard variational formulation of this new problem, in space-time for the acoustic equation, and in space only for the mechanic equation. The basic ideas for the discretization are the following: (i) Space finite elements and finite differences with time step $\Delta t$ (a $\theta$-scheme, $ 0 \leq \theta \leq 1/2$) are used for the discretization of the mechanic equation. (ii) Space-time finite elements are used for the discretization of the acoustic equation, which is "projected" two times on two staggered time grids of time step $2 \Delta t$. The use of staggered twice larger time grids for the discretization of the acoustic equation (see (ii) above) plays a key role in the cancellation of the "coupling terms" (between the two equations), which is crucial in the energy analysis. Copyright © 2007 Society for Industrial and Applied Mathematics
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Pascal Grob, Patrick Joly. Conservative coupling between finite elements and retarded potentials. Application to vibroacoustics. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2007, 29 (3), pp.1127-1159. ⟨10.1137/050647141⟩. ⟨hal-00976796⟩

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