Finite element analysis of compressible and incompressible fluid-solid systems

Bermúdez, Alfredo; Durán, Ricardo G.; Rodrı́guez, Rodolfo

doi:10.1090/s0025-5718-98-00901-6

Cited by 38 publications

(73 citation statements)

References 16 publications

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“…The following characterization (see [12]), Proof. It is a simple variation of that of Theorem 3.1 in [4]. In fact, for all (u, 0) ∈ K, clearly T(u, 0) = (u, 0).…”

Section: Characterization Of the Spectrum And A Priori Estimatesmentioning

confidence: 79%

“…A similar problem was considered in [4], but for a closed vessel completely filled with fluid and neglecting the gravity effects. In this section we extend the results in that reference to cover our problem.…”

Section: Variational Formulationmentioning

confidence: 99%

“…An extension to incompressible fluids has been made in [4], where error estimates independent of the acoustic speed have been obtained. The same discretization can be used in our case.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

“…Such discretization yields a sparse linear symmetric eigenvalue problem. In [4] it is shown that this method can be adapted to deal with incompressible fluids too.…”

Section: Introductionmentioning

confidence: 99%

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Finite element analysis of sloshing and hydroelastic vibrations under gravity

Bermúdez

Rodrı́guez

1999

ESAIM: M2AN

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Abstract. This paper deals with a finite element method to solve fluid-structure interaction problems.More precisely it concerns the numerical computation of harmonic hydroelastic vibrations under gravity. It is based on a displacement formulation for both the fluid and the solid. Gravity effects are included on the free surface of the fluid as well as on the liquid-solid interface. The pressure of the fluid is used as a variable for the theoretical analysis leading to 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 non conforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and optimal error estimates are given. Finally, numerical results are shown.Résumé. Cet article concerne une méthode d'éléments finis pour la résolution de problèmes d'intérac-tion d'un fluide avec une structure. Plus précisement il s'agit de calculer les vibrations hydroélastiques harmoniques sous gravité. La méthode est basée sur une formulation en déplacementsà la fois pour le solide et le fluide. Les effets de gravité sont inclus sur la surface libre du fluide et sur l'interphase entre le fluide et le solide. La pression dans le fluide est utilisée comme variable pour l'analyse théorique de la méthode ce qui conduità un problème mixte aux valeurs propres bien posé. L'élément triangulaire de Raviart-Thomas du plus bas degré est utilisé pour discrétiser le fluide ; pour le solide on utilise deséléments finis linéaires par morceaux classiques. La condition de transmission cinématiquè a l'interphase est prise en compte de façon faible ce qui donne une discrétisation non conforme. La méthode ne produit pas des modes parasites rotationnels pour des fréquences non nulles. On démontre aussi la convergence et des estimations d'erreur qui sont optimales. Finalement, quelques résultats numériques sont présentés.

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“…The following characterization (see [12]), Proof. It is a simple variation of that of Theorem 3.1 in [4]. In fact, for all (u, 0) ∈ K, clearly T(u, 0) = (u, 0).…”

Section: Characterization Of the Spectrum And A Priori Estimatesmentioning

confidence: 79%

Section: Variational Formulationmentioning

confidence: 99%

“…An extension to incompressible fluids has been made in [4], where error estimates independent of the acoustic speed have been obtained. The same discretization can be used in our case.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

“…Such discretization yields a sparse linear symmetric eigenvalue problem. In [4] it is shown that this method can be adapted to deal with incompressible fluids too.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finite element analysis of sloshing and hydroelastic vibrations under gravity

Bermúdez

Rodrı́guez

1999

ESAIM: M2AN

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“…In [27], the authors investigate various a posteriori estimators for stabilized mixed approximations of the Stokes problem. A posteriori error estimators for some quasi-newtonian fluids are considered in [33] and for combined fluidsolid systems in [10]. Error indicators based on superconvergence of finite element approximations for Stokes and Navier-Stokes equations are studied in [50].…”

Section: Introductionmentioning

confidence: 99%

Two-sided a posteriori estimates for a generalized Stokes problem

Repin

Stenberg²

2009

J Math Sci

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The paper is concerned with deriving computable majorants and minorants of the difference between the exact solution of for the so-called three-field formulation of the generalized Stokes problem and any functions from the admissible (energy) spaces that contain velocity, pressure and stress fields. Physical motivation of this problem is related to models of viscous fluids with polymeric chains. For the the case of uniform Dirichlét boundary conditions this model and respective numerical approximation methods were analyzed in [14]. In the present paper, we consider the generalized Stokes problem with mixed Dirichlét/Neumann boundary conditions and variable viscosity in the context of a posteriori error analysis. For the velocity, pressure, and stress fields we derive two-sided functional a posteriori error estimates. The estimates are practically computable, sharp (i.e., have no gap between the left-and right-hand sides), and are valid for arbitrary functions from the respective functional classes. The estimates are derived by transformations of the integral identity that defines the solution (this method was suggested and used in [39,40] for certain classes of elliptic type problems). Error majorants are given by weighted sums of the terms that present penalties for violations of all the relations of the problem considered with the weights defined by the constants in the Friederichs-Poincáre and Ladyzhenskaja-BabuskaBrezzi inequalities, respectively.

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Finite element computation of sloshing modes in containers with elastic baffle plates

Bermúdez

Rodrı́guez

Santamarina

2002

Numerical Meth Engineering

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SUMMARYA ÿnite element method to approximate the vibration modes of a plate in contact with an incompressible uid is analysed in this paper. The e ect of the uid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral ÿnite elements. Gravity waves on the free surface of the liquid are considered in the model. The plate is modelled by Reissner-Mindlin equations discretized by MITC3 locking-free elements.Implementation issues are discussed and numerical experiments are presented. In particular, the method is compared with analytical approximations and with an experimental study which has been recently reported.

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Finite element analysis of compressible and incompressible fluid-solid systems

Cited by 38 publications

References 16 publications

Finite element analysis of sloshing and hydroelastic vibrations under gravity

Finite element analysis of sloshing and hydroelastic vibrations under gravity

Two-sided a posteriori estimates for a generalized Stokes problem

Finite element computation of sloshing modes in containers with elastic baffle plates

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