Finite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem

The aim of this paper is to analyze an elastoacoustic vibration problem employing a dual-mixed formulation in the solid domain. The Cauchy stress tensor and the rotation are the primary variables in the elastic structure while the standard pressure formulation is considered in the acoustic fluid. The resulting mixed eigenvalue problem is approximated by a conforming Galerkin scheme based on the lowest order Lagrange and Arnold-Falk-Winther finite element subspaces in the fluid and solid domains, respectively. We show that the scheme provides a correct approximation of the spectrum and prove quasi-optimal error estimates. Finally, we report some numerical experiments.

[1]  Douglas N. Arnold,et al.  Mixed finite element methods for linear elasticity with weakly imposed symmetry , 2007, Math. Comput..

[2]  I. Babuska,et al.  Locking effects in the finite element approximation of elasticity problems , 1992 .

[3]  J. Rappaz,et al.  On spectral approximation. Part 1. The problem of convergence , 1978 .

[4]  Alfredo Bermúdez,et al.  Analysis of a finite element method for pressure/potential formulation of elastoacoustic spectral problems , 2002, Math. Comput..

[5]  J. Z. Zhu,et al.  The finite element method , 1977 .

[6]  G. C. Feng,et al.  Fluid-Structure Finite Element Vibrational Analysis , 1974 .

[7]  Alfredo Bermúdez,et al.  Finite element computation of the vibration modes of a fluid—solid system , 1994 .

[8]  JEAN DESCLOUX,et al.  On spectral approximation. Part 2. Error estimates for the Galerkin method , 1978 .

[9]  Daniele Boffi,et al.  Finite element approximation of eigenvalue problems , 2010, Acta Numerica.

[10]  Salim Meddahi,et al.  Analysis of the Coupling of Lagrange and Arnold-Falk-Winther Finite Elements for a Fluid-Solid Interaction Problem in Three Dimensions , 2012, SIAM J. Numer. Anal..

[11]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

[12]  G. Gatica.  Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations , 2006 .

[13]  A. Bermúdez,et al.  FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS , 1997 .

[14]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[15]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[16]  Pablo Gamallo,et al.  Finite element analysis of pressure formulation of the elastoacoustic problem , 2003, Numerische Mathematik.

[17]  Salim Meddahi,et al.  Finite Element Spectral Analysis for the Mixed Formulation of the Elasticity Equations , 2013, SIAM J. Numer. Anal..

[18]  M. Fortin,et al.  Reduced symmetry elements in linear elasticity , 2008 .

[19]  R. Ohayon,et al.  Fluid-Structure Interaction: Applied Numerical Methods , 1995 .

[20]  C. P. Gupta,et al.  A family of higher order mixed finite element methods for plane elasticity , 1984 .

[21]  Salim Meddahi,et al.  Analysis of the Coupling of Primal and Dual-Mixed Finite Element Methods for a Two-Dimensional Fluid-Solid Interaction Problem , 2007, SIAM J. Numer. Anal..

[22]  Roderick Wong,et al.  A tribute to Professor Philippe G. Ciarlet on his 70th birthday , 2008 .

[23]  A. Bermúdez,et al.  Fluid–Structure Acoustic Interaction , 2008 .

[24]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[25]  A. Bermúdez,et al.  Finite element vibration analysis of fluid-solid systems without spurious modes , 1995 .

[26]  K. Bathe,et al.  A mixed displacement-based finite element formulation for acoustic fluid-structure interaction , 1995 .

[27]  A. Bermúdez,et al.  Finite element analysis of sloshing and hydroelastic vibrations under gravity , 1999 .