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February  2016, 9(1): 269-287. doi: 10.3934/dcdss.2016.9.269

Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem

Salim Meddahi 1, and  David Mora 2,

1. 

Departamento de Matemáticas, Universidad de Oviedo, Facultad de Ciencias, Calvo Sotelo s/n, 33007 Oviedo, Spain
2. 

Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

Received  September 2014 Revised  February 2015 Published  December 2015

We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments.
Keywords: Fluid-structure interaction, error estimates., sloshing, mixed elasticity equations, eigenvalue problem, elastoacoustic vibrations, finite elements.
Mathematics Subject Classification: Primary: 65N15, 65N25, 65N30; Secondary: 74F1.
Citation: Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269
References:
[1]

D. N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity, Japan J. Appl. Math., 1 (1984), 347-367. doi: 10.1007/BF03167064.  Google Scholar

[2]

D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), 1699-1723. doi: 10.1090/S0025-5718-07-01998-9.  Google Scholar

[3]

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. doi: 10.1137/0732059.  Google Scholar

[4]

A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg., 40 (1997), 1435-1448. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1435::AID-NME119>3.0.CO;2-P.  Google Scholar

[5]

A. Bermúdez, P. Gamallo, L. Hervella-Nieto, R. Rodríguez and D. Santamarina, Fluid-structure Acoustic Interaction, Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods (eds. S. Marburg and B. Nolte), Springer, 2008. Google Scholar

[6]

A. Bermúdez and R. Rodríguez, Finite element analysis of sloshing and hydroelastic vibrations under gravity, RAIRO - Math. Model. Numer. Anal. ($M^2AN$), 33 (1999), 305-327. doi: 10.1051/m2an:1999117.  Google Scholar

[7]

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120. doi: 10.1017/S0962492910000012.  Google Scholar

[8]

D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Comm. Pure Appl. Anal., 8 (2009), 95-121. doi: 10.3934/cpaa.2009.8.95.  Google Scholar

[9]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[10]

M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer, Berlin, 1988.  Google Scholar

[11]

J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence, RAIRO Anal. Numér., 12 (1978), 97-112.  Google Scholar

[12]

G. N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D, SIAM J. Numer. Anal., 50 (2012), 1648-1674. doi: 10.1137/110836705.  Google Scholar

[13]

P. Grisvard, Problèmes aux limites dans les polygones. Mode d'emploi, EDF, Bulletin de la Direction des Études et Recherches (Serie C), 1 (1986), 21-59.  Google Scholar

[14]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/S0962492902000041.  Google Scholar

[15]

L. Kiefling and G. C. Feng, Fluid-structure finite element vibration analysis, AIAA J., 14 (1976), 199-203. doi: 10.2514/3.61357.  Google Scholar

[16]

S. Meddahi, D. Mora and R. Rodríguez, Finite element spectral analysis for the mixed formulation of the elasticity equations, SIAM J. Numer. Anal., 51 (2013), 1041-1063. doi: 10.1137/120863010.  Google Scholar

[17]

S. Meddahi, D. Mora and R. Rodríguez, Finite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem, Comput. Math. Appl., 68 (2014), 1733-1750. doi: 10.1016/j.camwa.2014.10.016.  Google Scholar

[18]

H. J.-P. Morand and R. Ohayon, Fluid Structure Interaction, J. Wiley & Sons, Chichester, 1995. Google Scholar

show all references

References:
[1]

D. N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity, Japan J. Appl. Math., 1 (1984), 347-367. doi: 10.1007/BF03167064.  Google Scholar

[2]

D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), 1699-1723. doi: 10.1090/S0025-5718-07-01998-9.  Google Scholar

[3]

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. doi: 10.1137/0732059.  Google Scholar

[4]

A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg., 40 (1997), 1435-1448. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1435::AID-NME119>3.0.CO;2-P.  Google Scholar

[5]

A. Bermúdez, P. Gamallo, L. Hervella-Nieto, R. Rodríguez and D. Santamarina, Fluid-structure Acoustic Interaction, Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods (eds. S. Marburg and B. Nolte), Springer, 2008. Google Scholar

[6]

A. Bermúdez and R. Rodríguez, Finite element analysis of sloshing and hydroelastic vibrations under gravity, RAIRO - Math. Model. Numer. Anal. ($M^2AN$), 33 (1999), 305-327. doi: 10.1051/m2an:1999117.  Google Scholar

[7]

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120. doi: 10.1017/S0962492910000012.  Google Scholar

[8]

D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Comm. Pure Appl. Anal., 8 (2009), 95-121. doi: 10.3934/cpaa.2009.8.95.  Google Scholar

[9]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[10]

M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer, Berlin, 1988.  Google Scholar

[11]

J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence, RAIRO Anal. Numér., 12 (1978), 97-112.  Google Scholar

[12]

G. N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D, SIAM J. Numer. Anal., 50 (2012), 1648-1674. doi: 10.1137/110836705.  Google Scholar

[13]

P. Grisvard, Problèmes aux limites dans les polygones. Mode d'emploi, EDF, Bulletin de la Direction des Études et Recherches (Serie C), 1 (1986), 21-59.  Google Scholar

[14]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/S0962492902000041.  Google Scholar

[15]

L. Kiefling and G. C. Feng, Fluid-structure finite element vibration analysis, AIAA J., 14 (1976), 199-203. doi: 10.2514/3.61357.  Google Scholar

[16]

S. Meddahi, D. Mora and R. Rodríguez, Finite element spectral analysis for the mixed formulation of the elasticity equations, SIAM J. Numer. Anal., 51 (2013), 1041-1063. doi: 10.1137/120863010.  Google Scholar

[17]

S. Meddahi, D. Mora and R. Rodríguez, Finite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem, Comput. Math. Appl., 68 (2014), 1733-1750. doi: 10.1016/j.camwa.2014.10.016.  Google Scholar

[18]

H. J.-P. Morand and R. Ohayon, Fluid Structure Interaction, J. Wiley & Sons, Chichester, 1995. Google Scholar

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