# Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem

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m2an0322.dvi ESAIM: M2AN ESAIM: Mathematical Modelling and Numerical Analysis Vol. 38, No 4, 2004, pp. 653–672 DOI: 10.1051/m2an:2004028 ANALYSIS OF A COUPLED BEM/FEM EIGENSOLVER FOR THE HYDROELASTIC VIBRATIONS PROBLEM ∗ Mauricio A. Barrientos1, Gabriel N. Gatica2 ,†, Rodolfo Rodr´ıguez2, † and Marcela E. Torrejo´n2 Abstract. A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported. Mathematics Subject Classification. 65N25, 65N30, 65N38, 70J30, 74F10, 76Q05. Received: May 15, 2003. Revised: April 8, 2004. 1. Introduction In this paper we analyze a coupled boundary/finite element method to numerically solve a spectral problem arising in fluid-solid interactions: the computation of free hydroelastic vibrations. In particular, we consider the problem of determining the harmonic vibrations of a coupled system consisting of an elastic vessel containing an incompressible fluid. The most direct approach to solve this problem is to discretize a coupled formulation in terms of solid dis- placements and fluid pressure (see for instance [23]). However, such strategy leads to non symmetric eigenvalue problems, which are usually hard to solve. Another procedure has been considered in [5] (see also [3] and [4]). It is based on using displacement variables to describe the fluid, discretized by lowest-degree Raviart-Thomas finite elements on a

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