Article
MSC:
35P05,
35Q35,
47A75,
49R05,
65N25,
74F10,
74H45,
76Q05 | MR 3164572 | Zbl 06346368 | DOI: 10.1007/s10492-014-0037-7
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
eigenvalue problem; fluid-solid vibration; variational characterization; minmax principle; maxmin principle
eigenvalue problem; fluid-solid vibration; variational characterization; minmax principle; maxmin principle
Summary:
Small amplitude vibrations of an elastic structure completely filled by a fluid are considered. Describing the structure by displacements and the fluid by its pressure field one arrives at a non-selfadjoint eigenvalue problem. Taking advantage of a Rayleigh functional we prove that its eigenvalues can be characterized by variational principles of Rayleigh, minmax and maxmin type.
References:
[1] Alonso, A., Russo, A. D., Padra, C., Rodríguez, R.: A posteriori error estimates and a local refinement strategy for a finite element method to solve structural-acoustic vibration problems. Adv. Comput. Math. 15 (2001), 25-59. DOI 10.1023/A:1014243118190 | MR 1887728 | Zbl 1043.74041
[2] Babuška, I., Osborn, J.: Eigenvalue problems. Handbook of Numerical Analysis. Volume II: Finite Element Methods (Part 1) P. Ciarlet et al. North-Holland Amsterdam (1991), 641-787. MR 1115240
[3] Belytschko, T.: Fluid-structure interaction. Comput. Struct. 12 (1980), 459-469. DOI 10.1016/0045-7949(80)90121-2 | Zbl 0457.73076
[4] Bennighof, J. K.: Vibroacoustic frequency sweep analysis using automated multi-level substructuring. Proceedings of the AIAA 40$^ th$ SDM Conference, St. Louis, Missouri, 1999 Department of Aerospace Engineering & Engineering Mechanics, The University of Texas Austin (1999).
[5] Bermúdez, A., Gamallo, P., Noguieras, M. R., Rodríguez, R.: Approximation of a structural acoustic vibration problem by hexahedral finite elements. IMA J. Numer. Anal. 26 (2006), 391-421. DOI 10.1093/imanum/dri032 | MR 2218639
[6] Bermúdez, A., Rodríguez, R.: Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems. Math. Comput. 71 (2002), 537-552. DOI 10.1090/S0025-5718-01-01335-7 | MR 1885614 | Zbl 0992.74066
[7] Craggs, A.: The transient response of a coupled plate-acoustic system using plate and acoustic finite elements. Journal of Sound and Vibration 15 (1971), 509-528. DOI 10.1016/0022-460X(71)90408-1
[8] Deü, J.-F., Larbi, W., Ohayon, R.: Variational formulation of interior structural-acoustic vibration problem. Computational Aspects of Structural Acoustics and Vibrations G. Sandberg et al. CISM International Centre for Mechanical Sciences 505 Springer, Wien (2009), 1-21.
[9] Everstine, G. C.: A symmetric potential formulation for fluid-structure interaction. Journal of Sound and Vibration 79 (1981), 157-160. DOI 10.1016/0022-460X(81)90335-7
[10] Morand, H., Ohayon, R.: Substructure variational analysis of the vibrations of coupled fluid-structure systems. Finite element results. Int. J. Numer. Methods Eng. 14 (1979), 741-755. DOI 10.1002/nme.1620140508 | Zbl 0402.73052
[11] Olson, L. G., Bathe, K.-J.: Analysis of fluid-structure interactions. A direct symmetric coupled formulation based on the fluid velocity potential. Comput. Struct. 21 (1985), 21-32. DOI 10.1016/0045-7949(85)90226-3 | Zbl 0568.73088
[12] Petyt, M., Lea, J., Koopmann, G. H.: A finite element method for determining the acoustic modes of irregular shaped cavities. Journal of Sound and Vibration 45 (1976), 495-502. DOI 10.1016/0022-460X(76)90730-6
[13] Rodríguez, R., Solomin, J. E.: The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems. Math. Comput. 65 (1996), 1463-1475. DOI 10.1090/S0025-5718-96-00739-9 | MR 1344621 | Zbl 0853.65111
[14] Sandberg, G., Göransson, P.: A symmetric finite element formulation for acoustic fluid-structure interaction analysis. Journal of Sound and Vibration 123 (1988), 507-515. DOI 10.1016/S0022-460X(88)80166-4
[15] Stammberger, M.: On an unsymmetric eigenvalue problem governing free vibrations of fluid-solid structures. PhD thesis. Institute of Numerical Simulation, Hamburg University of Technology Hamburg (2010).
[16] Stammberger, M., Voss, H.: Automated multi-level sub-structuring for fluid-solid interaction problems. Numer. Linear Algebra Appl. 18 (2011), 411-427. DOI 10.1002/nla.734 | MR 2760061 | Zbl 1249.65271
[17] Stammberger, M., Voss, H.: On an unsymmetric eigenvalue problem governing free vibrations of fluid-solid structures. ETNA, Electron. Trans. Numer. Anal. (electronic only) 36 (2009-2010), 113-125. MR 2780001 | Zbl 1237.74028
[18] Voss, H., Stammberger, M.: Structural-acoustic vibration problems in the presence of strong coupling. J. Pressure Vessel Technol. 135 (2013), paper 011303. DOI 10.1115/1.4007251
Search
Partner of
