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

Published online by Cambridge University Press:  15 August 2004

Mauricio A. Barrientos ,
Gabriel N. Gatica ,
Rodolfo Rodríguez  and
Marcela E. Torrejón
Mauricio A. Barrientos
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Chile.
Gabriel N. Gatica
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Rodolfo Rodríguez
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Marcela E. Torrejón
Affiliation:
GIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
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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.

Keywords

Fluid-structure interactionhydroelasticityadded massBEM/FEM.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 4 , July 2004 , pp. 653 - 672
Copyright
© EDP Sciences, SMAI, 2004

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References

Alonso, A., Dello Russo, A., Otero-Souto, C., Padra, C. and Rodríguez, R., An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids. Comput. Visual. Sci. 4 (2001) 67–78. CrossRef
I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, North-Holland, Amsterdam (1991) 641–787.
Bermúdez, A., Durán, R. and Rodríguez, R., Finite element solution of incompressible fluid-structure vibration problems. Internat. J. Numer. Methods Eng. 40 (1997) 1435–1448. 3.0.CO;2-P>CrossRef
A. Bermúdez, R. Durán and R. Rodríguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp. 67 (1998) 111–136.
Bermúdez, A. and Rodríguez, R., Finite element analysis of sloshing and hydroelastic vibrations under gravity. ESAIM: M2AN 33 (1999) 305–327. CrossRef
Bermúdez, A., Rodríguez, R. and Santamarina, D., A finite element solution of an added mass formulation for coupled fluid-solid vibrations. Numer. Math. 87 (2000) 201–227. CrossRef
P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, North-Holland, Amsterdam (1991) 17–351.
Costabel, M., Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19 (1988) 613–621. CrossRef
Ervin, V.J., Heuer, N. and Stephan, E.P., On the h-p version of the boundary element method for Symm's integral equation on polygons. Comput. Methods Appl. Mech. Eng. 110 (1993) 25–38. CrossRef
G.N. Gatica and G.C. Hsiao, Boundary-Field Equation Methods for a Class of Nonlinear Problems. Longman, Harlow, Pitman Res. Notes Math. Ser. 331 (1995).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA, Monogr. Stud. Math. 24 (1985).
Hamdi, M., Ousset, Y. and Verchery, G., A displacement method for the analysis of vibrations of coupled fluid-structure systems. Internat. J. Numer. Methods Eng. 13 (1978) 139–150. CrossRef
G.C. Hsiao, On the boundary-field equation methods for fluid-structure interactions, in Problems and Methods in Mathematical Physics (Chemnitz, 1993), L. Jentsch and F. Tröltzsch, Eds. Teubner, Stuttgart, Teubner-Texte Math. 134 (1994) 79–88.
Hsiao, G.C. and Wendland, W.L., A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58 (1977) 449–481. CrossRef
Hsiao, G.C., Kleinman, R.E. and Roach, G.F., Weak solutions of fluid-solid interaction problems. Math. Nachr. 218 (2000) 139–163. 3.0.CO;2-S>CrossRef
G.C. Hsiao, R.E. Kleinman and L.S. Schuetz, On variational formulations of boundary value problems for fluid-solid interactions. Elastic Wave Propagation (Galway, 1988). North-Holland, Amsterdam, North-Holland Ser. Appl. Math. Mech. 35 (1989) 321–326. CrossRef
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).
D. Mercier, Some systems of PDE on polygonal networks, in Partial Differential Equations on Multistructures (Luminy, 1999), F.A. Mehmeti, J. von Below and S. Nicaise Eds., Dekker, New York, Lect. Notes Pure Appl. Math. 219 (2001) 163–182.
Mercier, D., Problèmes de transmission sur des réseaux polygonaux pour des systèmes d'EDP. Ann. Fac. Sci. Toulouse Math. 10 (2001) 107–162. CrossRef
H.J.-P. Morand and R. Ohayon, Fluid-Structure Interaction. J. Wiley & Sons, Chichester (1995).
Ryan, P., Eigenvalue and eigenfunction error estimates for finite element formulations of linear hydroelasticity. Math. Comp. 70 (2001) 471–487. CrossRef
M.E. Torrejón, Solución Numérica de Problemas de Vibraciones Hidroelásticas. Degree Thesis in Mathematical Engineering, Universidad de Concepción, Chile (2003).
O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method. Mc Graw Hill, London (1989).