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The order of convergence of eigenfrequencies
in finite element approximations
of fluid-structure interaction problems

Authors: Rodolfo Rodríguez and Jorge E. Solomin
Journal: Math. Comp. 65 (1996), 1463-1475
MSC (1991): Primary 65N25, 65N30; Secondary 70J30, 73K70, 76Q05
MathSciNet review: 1344621
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure vibration problems. We improve estimates given recently for compressible and incompressible fluids. To do this, we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators.

References [Enhancements On Off] (What's this?)

  • 1. I. Babu\v{s}ka and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol. II (ed. P. G. Ciarlet and J. L. Lions) North Holland, Amsterdam, 1991. CMP 91:14
  • 2. A. Bermúdez, R. Durán, M.A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes, SIAM J. Numer. Anal., 32 (1995), 1280--1295. CMP 95:15
  • 3. A. Bermúdez, R. Durán and R. Rodríguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp. (submitted)
  • 4. A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg. (submitted)
  • 5. Alfredo Bermúdez and Rodolfo Rodríguez, Finite element computation of the vibration modes of a fluid-solid system, Comput. Methods Appl. Mech. Engrg. 119 (1994), no. 3-4, 355–370. MR 1316022, 10.1016/0045-7825(94)90095-7
  • 6. Jacqueline Boujot, Mathematical formulation of fluid-structure interaction problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 239–260 (English, with French summary). MR 896242
  • 7. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
  • 8. H. C. Chen and R. L. Taylor, Vibration analysis of fluid-solid systems using a finite element displacement formulation, Internat. J. Numer. Meth. Eng., 29 (1990) 683-698.
  • 9. Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. I. The problem of convergence, RAIRO Anal. Numér. 12 (1978), no. 2, 97–112, iii (English, with French summary). MR 0483400
  • 10. J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 2: Error estimates for the Galerkin methods, R.A.I.R.O. Anal. Numér., 12 (1978) 113-119. MR 58:34046
  • 11. M. Hamdi, Y. Ouset and G. Verchery, A displacement method for the analysis of vibrations of coupled fluid-structure systems, Internat. J. Numer. Methods Eng., 13 (1978) 139-150.
  • 12. Henri J.-P. Morand and Roger Ohayon, Interactions fluides-structures, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 23, Masson, Paris, 1992 (French). MR 1180076
  • 13. B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427–453. MR 606505, 10.1090/S0025-5718-1981-0606505-9
  • 14. L. Olson and K. Bathe, Analysis of fluid-structure interactions. A direct symmetric coupled formulation based on the fluid velocity potential, Comp. Struct., 21 (1985) 21-32.
  • 15. P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
  • 16. J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965.

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Additional Information

Rodolfo Rodríguez
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 4009, Concepción, Chile

Jorge E. Solomin
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 4009, Concepción, Chile

Keywords: Fluid-structure, eigenvalue problems
Received by editor(s): January 30, 1995
Additional Notes: Partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
Article copyright: © Copyright 1996 American Mathematical Society