Rodolfo Rodriguez | Universidad Centroamericana (UCA). El Salvador (original) (raw)

Papers by Rodolfo Rodriguez

Research paper thumbnail of The order of convergence of Eigenfrequencies in finite element approximations of fluid-structure interaction problems

Mathematics of Computation, 1996

In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure ... more In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure vibration problems. We improve estimates given recently for compressible and incompressible fluids. To do this, we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators.

Research paper thumbnail of Finite Element Vibration Analysis of Fluid–Solid Systems without Spurious Modes

Siam Journal on Numerical Analysis, 1995

... {Ha'1(div,Q,) x [HL+/3(Qs)]} n Gv GV-[1 Page 7. 1286 BERMUDEZ, DURAN, MU... more ... {Ha'1(div,Q,) x [HL+/3(Qs)]} n Gv GV-[1 Page 7. 1286 BERMUDEZ, DURAN, MUSCHIETTI, RODRIGUEZ, AND SOLOMIN We conclude this section by giving an analogue of Theorem 2.5 for (f, g) ? Go This result will be used for the proof of Lemma 5.7 below. THEOREM 2.8. ...

Research paper thumbnail of On the asymptotic exactness of error estimators for linear triangular finite elements

Numerische Mathematik, 1991

This paper deals with a-posteriori error estimates for piecewise linear finite element approximat... more This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution. One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

Research paper thumbnail of The problem of the selection of ana posteriori error indicator based on smoothening techniques

International Journal for Numerical Methods in Engineering, 1993

This paper addresses the problem of assessing the quality of an a posteriori error estimate of a ... more This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2-projections is analysed in the case of translation-invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one-dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.

Research paper thumbnail of Finite element computation of three-dimensional elastoacoustic vibrations: J

In this paper we solve the interior elastoacoustic problem in a 3D domain. Displacement variables... more In this paper we solve the interior elastoacoustic problem in a 3D domain. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation we use a non standard discretization consisting of classical linear tetrahedral finite elements for the solid and Raviart-Thomas elements of lowest order for the fluid. A new unknown is introduced on the interface between solid and fluid to impose the trasmission conditions. * This work was partially supported by FONDECYT (Chile) through its grant No. 1.960.615 and Programa de Cooperación Científica con Iberoamérica (MEC, Spain).

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

Numerische Mathematik, 2000

We consider the approximation of the vibration modes of an elastic plate in contact with a compre... more 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 t>0t>0t>0 , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when tto0t\to 0tto0 . 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.

Research paper thumbnail of FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID-STRUCTURE VIBRATION PROBLEMS

International Journal for Numerical Methods in Engineering, 1997

... will be disrupted as follows: New York 0630 EDT to 0830 EDT; London 1130 GMT to 1330 GMT; Sin... more ... will be disrupted as follows: New York 0630 EDT to 0830 EDT; London 1130 GMT to 1330 GMT; Singapore 1730 SGT to 1930 SGT. ... Research Article. FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS. ALFREDO BERMÚDEZ ...

Research paper thumbnail of A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Numerische Mathematik, 2000

A finite element method to approximate the vibration modes of a structure in contact with an inco... more A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

Research paper thumbnail of Finite element analysis of compressible and incompressible fluid-solid systems

Mathematics of Computation, 1998

This paper deals with a finite element method to solve interior fluid-structure vibration problem... more This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding 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 nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

Research paper thumbnail of Finite Element Methods in Local Active Control of Sound

Siam Journal on Control and Optimization, 2004

The active control of sound is analyzed in the framework of the mathematical theory of optimal co... more The active control of sound is analyzed in the framework of the mathematical theory of optimal control. After setting the problem in the frequency domain, we deal with the state equation, which is a Helmholtz partial differential equation. We show existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure. Two optimization problems are successively considered. The first one concerns the choice of phases and amplitudes of the actuators to minimize the noise at the sensors location. The second one consists in determining the optimal actuators placement. Both problems are then numerically solved. Error estimates are settled and numerical results for some tests are reported.

Research paper thumbnail of A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations

Siam Journal on Numerical Analysis, 2002

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic M... more The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain.

Research paper thumbnail of Quality assessment of the a-posteriori error estimation in finite elements

Finite Elements in Analysis and Design, 1992

The properties of an a-posteriori error estimator are examined with reference to the results of n... more The properties of an a-posteriori error estimator are examined with reference to the results of numerical experiments. In particular, attention is given to various aspects of the relation between the theoretical understanding of the estimator and benchmark selection. The estimator is shown to behave as expected from its properties defined in two theorems.

Research paper thumbnail of On the asymptotic exactness of Bank-Weiser's estimator

Numerische Mathematik, 1992

In this paper we analyze an error estimator introduced by Bank and Weiser. We prove that this est... more In this paper we analyze an error estimator introduced by Bank and Weiser. We prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes. By considering a simple example we show that this is not true for general meshes.

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

Mathematics of Computation, 1999

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissne... more 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.

Research paper thumbnail of Finite element computation of the vibration modes of a fluid---solid system

Computer Methods in Applied Mechanics and Engineering, 1994

ABSTRACT

Research paper thumbnail of Analysis of the Efficiency of an a Posteriori Error Estimator for Linear Triangular Finite Elements

Siam Journal on Numerical Analysis, 1992

Research paper thumbnail of Finite Element Analysis of a Quadratic Eigenvalue Problem Arising in Dissipative Acoustics

Siam Journal on Numerical Analysis, 2000

A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic... more A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered. The problem is shown to be equivalent to the spectral problem for a non-compact operator and a thorough spectral characterization is given.

Research paper thumbnail of The order of convergence of Eigenfrequencies in finite element approximations of fluid-structure interaction problems

Mathematics of Computation, 1996

In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure ... more In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure vibration problems. We improve estimates given recently for compressible and incompressible fluids. To do this, we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators.

Research paper thumbnail of Finite Element Vibration Analysis of Fluid–Solid Systems without Spurious Modes

Siam Journal on Numerical Analysis, 1995

... {Ha'1(div,Q,) x [HL+/3(Qs)]} n Gv GV-[1 Page 7. 1286 BERMUDEZ, DURAN, MU... more ... {Ha'1(div,Q,) x [HL+/3(Qs)]} n Gv GV-[1 Page 7. 1286 BERMUDEZ, DURAN, MUSCHIETTI, RODRIGUEZ, AND SOLOMIN We conclude this section by giving an analogue of Theorem 2.5 for (f, g) ? Go This result will be used for the proof of Lemma 5.7 below. THEOREM 2.8. ...

Research paper thumbnail of On the asymptotic exactness of error estimators for linear triangular finite elements

Numerische Mathematik, 1991

This paper deals with a-posteriori error estimates for piecewise linear finite element approximat... more This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution. One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.

Research paper thumbnail of The problem of the selection of ana posteriori error indicator based on smoothening techniques

International Journal for Numerical Methods in Engineering, 1993

This paper addresses the problem of assessing the quality of an a posteriori error estimate of a ... more This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2-projections is analysed in the case of translation-invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one-dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.

Research paper thumbnail of Finite element computation of three-dimensional elastoacoustic vibrations: J

In this paper we solve the interior elastoacoustic problem in a 3D domain. Displacement variables... more In this paper we solve the interior elastoacoustic problem in a 3D domain. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation we use a non standard discretization consisting of classical linear tetrahedral finite elements for the solid and Raviart-Thomas elements of lowest order for the fluid. A new unknown is introduced on the interface between solid and fluid to impose the trasmission conditions. * This work was partially supported by FONDECYT (Chile) through its grant No. 1.960.615 and Programa de Cooperación Científica con Iberoamérica (MEC, Spain).

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

Numerische Mathematik, 2000

We consider the approximation of the vibration modes of an elastic plate in contact with a compre... more 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 t>0t>0t>0 , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when tto0t\to 0tto0 . 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.

Research paper thumbnail of FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID-STRUCTURE VIBRATION PROBLEMS

International Journal for Numerical Methods in Engineering, 1997

... will be disrupted as follows: New York 0630 EDT to 0830 EDT; London 1130 GMT to 1330 GMT; Sin... more ... will be disrupted as follows: New York 0630 EDT to 0830 EDT; London 1130 GMT to 1330 GMT; Singapore 1730 SGT to 1930 SGT. ... Research Article. FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS. ALFREDO BERMÚDEZ ...

Research paper thumbnail of A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Numerische Mathematik, 2000

A finite element method to approximate the vibration modes of a structure in contact with an inco... more A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

Research paper thumbnail of Finite element analysis of compressible and incompressible fluid-solid systems

Mathematics of Computation, 1998

This paper deals with a finite element method to solve interior fluid-structure vibration problem... more This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding 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 nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

Research paper thumbnail of Finite Element Methods in Local Active Control of Sound

Siam Journal on Control and Optimization, 2004

The active control of sound is analyzed in the framework of the mathematical theory of optimal co... more The active control of sound is analyzed in the framework of the mathematical theory of optimal control. After setting the problem in the frequency domain, we deal with the state equation, which is a Helmholtz partial differential equation. We show existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure. Two optimization problems are successively considered. The first one concerns the choice of phases and amplitudes of the actuators to minimize the noise at the sensors location. The second one consists in determining the optimal actuators placement. Both problems are then numerically solved. Error estimates are settled and numerical results for some tests are reported.

Research paper thumbnail of A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations

Siam Journal on Numerical Analysis, 2002

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic M... more The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain.

Research paper thumbnail of Quality assessment of the a-posteriori error estimation in finite elements

Finite Elements in Analysis and Design, 1992

The properties of an a-posteriori error estimator are examined with reference to the results of n... more The properties of an a-posteriori error estimator are examined with reference to the results of numerical experiments. In particular, attention is given to various aspects of the relation between the theoretical understanding of the estimator and benchmark selection. The estimator is shown to behave as expected from its properties defined in two theorems.

Research paper thumbnail of On the asymptotic exactness of Bank-Weiser's estimator

Numerische Mathematik, 1992

In this paper we analyze an error estimator introduced by Bank and Weiser. We prove that this est... more In this paper we analyze an error estimator introduced by Bank and Weiser. We prove that this estimator is asymptotically exact in the energy norm for regular solutions and parallel meshes. By considering a simple example we show that this is not true for general meshes.

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

Mathematics of Computation, 1999

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissne... more 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.

Research paper thumbnail of Finite element computation of the vibration modes of a fluid---solid system

Computer Methods in Applied Mechanics and Engineering, 1994

ABSTRACT

Research paper thumbnail of Analysis of the Efficiency of an a Posteriori Error Estimator for Linear Triangular Finite Elements

Siam Journal on Numerical Analysis, 1992

Research paper thumbnail of Finite Element Analysis of a Quadratic Eigenvalue Problem Arising in Dissipative Acoustics

Siam Journal on Numerical Analysis, 2000

A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic... more A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered. The problem is shown to be equivalent to the spectral problem for a non-compact operator and a thorough spectral characterization is given.