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

Rodríguez, Rodolfo; Solomin, Jorge

doi:10.1090/S0025-5718-96-00739-9

The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems
HTML articles powered by AMS MathViewer

by Rodolfo Rodríguez and Jorge E. Solomin PDF

Math. Comp. 65 (1996), 1463-1475 Request permission

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

I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol. II (ed. P. G. Ciarlet and J. L. Lions) North Holland, Amsterdam, 1991.
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.
A. Bermúdez, R. Durán and R. Rodríguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp. (submitted)
A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg. (submitted)
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, DOI 10.1016/0045-7825(94)90095-7
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, DOI 10.1051/m2an/1987210202391
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, DOI 10.1007/978-1-4612-3172-1
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.
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 483400, DOI 10.1051/m2an/1978120200971
Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
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.
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
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, DOI 10.1090/S0025-5718-1981-0606505-9
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.
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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965.