Luis Hervella-Nieto

Error Estimates for Low-Order Isoparametric Quadrilateral Finite Elements for Plates

This paper deals with the numerical approximation of the bending of a plate modeled by Reissner--Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is based on the family of elements called MITC (mixed interpolation of tensorial components). We consider two lowest-order methods of this family on quadrilateral meshes. Under mild assumptions we obtain optimal H1 and L2 error estimates for both methods. These estimates are valid with constants independent of the plate thickness. We also obtain error estimates for the approximation of the plate vibration problem. Finally, we report some numerical experiments showing the very good behavior of the methods, even in some cases not covered by our theory.

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family. Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding L 2 -estimate for a load problem which is proven here. This optimal order L 2 -estimate is of interest in itself. Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.

An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems

The aim of this paper is to introduce an “exact” bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a nonintegrable absorbing function. “Exactness” must be understood in the sense that this technique allows exact recovering of the solution to time-harmonic scattering problems in unbounded domains. In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space. The resulting weak formulation can be numerically solved by using standard finite elements. The high accuracy of this approach is numerically demonstrated as compared with a classical PML technique.

Finite element computation of the vibrations of a plate-fluid system with interface damping

Abstract This paper deals with a finite element method to compute the vibrations of a coupled fluid–solid system subject to an external harmonic excitation. The system consists of an acoustic fluid and a plate, with a thin layer of a noise damping viscoelastic material separating both media. The fluid is described by displacement variables whereas the plate is modeled by Reissner–Mindlin equations. Face elements are used for the fluid and MITC3 elements for the bending of the plate. The effect of the damping material is taken into account by adequately relaxing the kinematic constraint on the fluid–solid interface. The nonlinear eigenvalue problem arising from the free vibrations of the damped coupled system is also considered. The dispersion equation is deduced for the simpler case of a fluid in a hexahedral rigid cavity with an absorbing wall. This allows computing analytically its eigenvalues and eigenmodes and comparing them with the finite element solution. The numerical results show that the coupled finite element method neither produces spurious modes nor locks when the thickness of the plate becomes small. Finally the computed resonance frequencies are compared with those of the undamped problem and with the complex eigenvalues of the above nonlinear spectral problem.

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness

Computation of the vibration modes of plates and shells by low-order MITC quadrilateral finite elements

t>0

An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems

, and introduce appropriate scalings for the physical parameters so that these problems attain a limit when

Review in Sound Absorbing Materials

t\to 0

Finite element computation of three-dimensional elastoacoustic vibrations

. We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method.

Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

This paper deals with the approximation of the vibration modes of plates and shells using the MITC4 finite element method. We use the classical Naghdi model over a reference domain. We assess the performance of this approach for both structures by means of numerical experiments.