Finite element analysis of compressible and incompressible fluid-solid systems

Bermúdez, Alfredo; Durán, Ricardo; Rodríguez, Rodolfo

doi:10.1090/S0025-5718-98-00901-6

Finite element analysis of compressible and incompressible fluid-solid systems
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by Alfredo Bermúdez, Ricardo Durán and Rodolfo Rodríguez PDF

Math. Comp. 67 (1998), 111-136 Request permission

Abstract:

This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem. Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

References

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), no. 4, 1280–1295. MR 1342293, DOI 10.1137/0732059
A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems. Int. J. Numer. Methods Eng. 40 (1997) 1435–1448.
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
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
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
Tunc Geveci, B. Daya Reddy, and Howard T. Pearce, On the approximation of the spectrum of the Stokes operator, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 129–136 (English, with French summary). MR 1015922, DOI 10.1051/m2an/1989230101291
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
M. Hamdi, Y. Ouset and G. Verchery, A displacement method for the analysis of vibrations of coupled fluid-structure systems, Int. 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
R. Ohayon and E. Sánchez-Palencia, On the vibration problem for an elastic body surrounded by a slightly compressible fluid, RAIRO Anal. Numér. 17 (1983), no. 3, 311–326 (English, with French summary). MR 702140, DOI 10.1051/m2an/1983170303111
Rodolfo Rodríguez and Jorge E. Solomin, The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems, Math. Comp. 65 (1996), no. 216, 1463–1475. MR 1344621, DOI 10.1090/S0025-5718-96-00739-9
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. Sanchez Hubert and E. Sánchez-Palencia, Vibration and coupling of continuous systems, Springer-Verlag, Berlin, 1989. Asymptotic methods. MR 996423, DOI 10.1007/978-3-642-73782-4
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7