Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem

Mauricio A. Barrientos; Gabriel N. Gatica; Rodolfo Rodríguez; Marcela E. Torrejón

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 4, page 653-672
  • ISSN: 0764-583X

Abstract

top
A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported.

How to cite

top

Barrientos, Mauricio A., et al. "Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem." ESAIM: Mathematical Modelling and Numerical Analysis 38.4 (2010): 653-672. <http://eudml.org/doc/194232>.

@article{Barrientos2010,
abstract = { A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported. },
author = {Barrientos, Mauricio A., Gatica, Gabriel N., Rodríguez, Rodolfo, Torrejón, Marcela E.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Fluid-structure interaction; hydroelasticity; added mass; BEM/FEM.; added mass formulation; convergence; error estimates},
language = {eng},
month = {3},
number = {4},
pages = {653-672},
publisher = {EDP Sciences},
title = {Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem},
url = {http://eudml.org/doc/194232},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Barrientos, Mauricio A.
AU - Gatica, Gabriel N.
AU - Rodríguez, Rodolfo
AU - Torrejón, Marcela E.
TI - Analysis of a coupled BEM/FEM eigensolver for the hydroelastic vibrations problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 4
SP - 653
EP - 672
AB - A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported.
LA - eng
KW - Fluid-structure interaction; hydroelasticity; added mass; BEM/FEM.; added mass formulation; convergence; error estimates
UR - http://eudml.org/doc/194232
ER -

References

top
  1. A. Alonso, A. Dello Russo, C. Otero-Souto, C. Padra and R. Rodríguez, An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids. Comput. Visual. Sci.4 (2001) 67–78.  Zbl1017.74068
  2. I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, North-Holland, Amsterdam (1991) 641–787.  Zbl0875.65087
  3. A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems. Internat. J. Numer. Methods Eng.40 (1997) 1435–1448.  
  4. A. Bermúdez, R. Durán and R. Rodríguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp.67 (1998) 111–136.  Zbl0898.73060
  5. A. Bermúdez and R. Rodríguez, Finite element analysis of sloshing and hydroelastic vibrations under gravity. ESAIM: M2AN33 (1999) 305–327.  Zbl0961.74057
  6. A. Bermúdez, R. Rodríguez and D. Santamarina, A finite element solution of an added mass formulation for coupled fluid-solid vibrations. Numer. Math.87 (2000) 201–227.  Zbl0998.76046
  7. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, P.G. Ciarlet and J.L. Lions Eds., Vol. II, North-Holland, Amsterdam (1991) 17–351.  Zbl0875.65086
  8. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal.19 (1988) 613–621.  
  9. V.J. Ervin, N. Heuer and E.P. Stephan, On the h-p version of the boundary element method for Symm's integral equation on polygons. Comput. Methods Appl. Mech. Eng.110 (1993) 25–38.  Zbl0842.65076
  10. G.N. Gatica and G.C. Hsiao, Boundary-Field Equation Methods for a Class of Nonlinear Problems. Longman, Harlow, Pitman Res. Notes Math. Ser.331 (1995).  Zbl0832.65126
  11. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, MA, Monogr. Stud. Math.24 (1985).  Zbl0695.35060
  12. M. Hamdi, Y. Ousset 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.  Zbl0384.76060
  13. G.C. Hsiao, On the boundary-field equation methods for fluid-structure interactions, in Problems and Methods in Mathematical Physics (Chemnitz, 1993), L. Jentsch and F. Tröltzsch, Eds. Teubner, Stuttgart, Teubner-Texte Math.134 (1994) 79–88.  
  14. G.C. Hsiao and W.L. Wendland, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl.58 (1977) 449–481.  Zbl0352.45016
  15. G.C. Hsiao, R.E. Kleinman and G.F. Roach, Weak solutions of fluid-solid interaction problems. Math. Nachr.218 (2000) 139–163.  Zbl0963.35043
  16. G.C. Hsiao, R.E. Kleinman and L.S. Schuetz, On variational formulations of boundary value problems for fluid-solid interactions. Elastic Wave Propagation (Galway, 1988). North-Holland, Amsterdam, North-Holland Ser. Appl. Math. Mech.35 (1989) 321–326.  
  17. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).  Zbl0948.35001
  18. D. Mercier, Some systems of PDE on polygonal networks, in Partial Differential Equations on Multistructures (Luminy, 1999), F.A. Mehmeti, J. von Below and S. Nicaise Eds., Dekker, New York, Lect. Notes Pure Appl. Math.219 (2001) 163–182.  Zbl0973.35071
  19. D. Mercier, Problèmes de transmission sur des réseaux polygonaux pour des systèmes d'EDP. Ann. Fac. Sci. Toulouse Math.10 (2001) 107–162.  
  20. H.J.-P. Morand and R. Ohayon, Fluid-Structure Interaction. J. Wiley & Sons, Chichester (1995).  Zbl0834.73002
  21. P. Ryan, Eigenvalue and eigenfunction error estimates for finite element formulations of linear hydroelasticity. Math. Comp.70 (2001) 471–487.  Zbl0969.65103
  22. M.E. Torrejón, Solución Numérica de Problemas de Vibraciones Hidroelásticas. Degree Thesis in Mathematical Engineering, Universidad de Concepción, Chile (2003).  
  23. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method. Mc Graw Hill, London (1989).  Zbl0991.74002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.