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• Comparison of some finite element methods for solving the diffusion-convection-reaction equation

  R. Codina

  (1996). Research Report, Nº PI101

  
  

  Abstract

  In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard in an effort to emphasize the similarities and differences between them. In particular, it is shown that the classical SUPG method is very similar to an explicit version of the Characteristic-Galerkin method, whereas the Taylor-Galerkin method has a stabilization effect similar to a sub-grid scale model, which is un turn related to the introduction of bubbles functions.

  Abstract
  In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard [...]

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• A fractional step method for the solution of the compressible Navier-Stokes equations

  R. Codina, M. Vázquez, O. Zienkiewicz

  (1997). Research Report, Nº PI118

  
  

  Abstract

  In this paper we present a summary of the splitting technique for both compressible and incompressible flows previously proposed in [22, 23, 7]. Also, we extend it to the case of a fully implicit treatment of the viscous and convective terms of the momentum equations. For incompressible flows, this scheme for a non-standard treatment of the boundary conditions. For compressible flows the continuity equation involves two variables which must be related through the equation of state. Convective terms of the conservation equations to be solved are stabilized by means of a Characteristic – Galerkin scheme. Also, in the presence of shocks some additional dissipation is needed. Both numerical techniques are explained here taking the transport of a scalar quantity as a model problem.

  Abstract
  In this paper we present a summary of the splitting technique for both compressible and incompressible flows previously proposed in [22, 23, 7]. Also, we extend it to the [...]

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• On stabilized finite element methods for linear systems of convection-diffusion-reaction equations

  R. Codina

  (1997). Research Report, Nº PI126

  
  

  Abstract

  A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a 1D model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner-Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.

  Abstract
  A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach [...]

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• Error estimation including pollution assesment for nonlinear finite element analysis

  A. Huerta, P. Díez

  (1998). Research Report, Nº PI132

  
  

  Abstract

  The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).

  Abstract
  The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of [...]

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• High-order accuarte time-stepping schemes for convection-diffusion problems

  J. Donea, B. Roig, A. Huerta

  (1998). Research Report, Nº PI135

  
  

  Abstract

  A major objective of the study described in the present paper is to achieve a high accuracy in the time integration of transient convection-diffusion problems and to eventually combine this with new methods for space discretization, such as meshless methods. In this way, a uniformly high-order accurate methodology could be made available for the numerical solution of convection-diffusion problems. Both Padé approximations of the exponential function and Runge-Kutta methods are considered for deriving multi-stage schemes involving first time derivatives only, thus easier to implement than standard Taylor-Galerkin schemes which incorporate second and third time derivatives. After a brief discussion of the stability and accuracy properties of the stability and accuracy properties of the multi-stage schemes and the presentation of illustrative examples, the paper closes with some considerations on the coupling of high-order accurate temporal schemes and meshless methods for the spatial representation.  

  Abstract
  A major objective of the study described in the present paper is to achieve a high accuracy in the time integration of transient convection-diffusion problems and to eventually [...]

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• Adaptive finite element strategies based on error assessment

  A. Huerta, A. Rodríguez-Ferran, P. Diez

  (1998). Research Report, Nº PI136

  
  

  Abstract

  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or errors indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are elearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.  

  Abstract
  Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or errors [...]

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• Arbitrary Lagrangian-Eulerian formulation for fluid-rigid body interaction

  J. Sarrate, A. Huerta, J. Donea

  (1998). Research Report, Nº PI137

  
  

  Abstract

  A new formulation for two-dimensional fluid-rigid body interaction problems is developed. In particular, vortex-induced oscillations of a rigid boy in viscous incompressible flow are studied. The incompressible Naiver-Stokes equations are used to describe the motion of the fluid, while it is assumed that the rigid body is mounted on a system consisting of a spring and a dashpot. An arbitrary Lagrangian-Eulerian formulation (ALE) is used in order to account for large boundary motion. A general formulation for coupled problem is obtained by uncoupling the translation motion of the body from its rotational motion and developing a specific algorithm to efficiently handle the nonlinear dependence of the rotations. This general formulation ca be easily applied to multi-body problems. Two numerical examples involving either translations and rotations are presented as an illustration of the proposed methodologies for fluid-rigid body interaction.

  Abstract
  A new formulation for two-dimensional fluid-rigid body interaction problems is developed. In particular, vortex-induced oscillations of a rigid boy in viscous incompressible [...]

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• Modelo constitutivo elastoplástico anisótropo en deformaciones finitas para análisis de materiales compuestos reforzados con fibras

  E. Car, S. Oller, E. Oñate

  (1998). Research Report, Nº PI139

  
  

  Abstract

  En este trabajo se presenta un modelo constitutivo elastoplástico anisótropo generalizado en deformaciones finitas para el análisis de materiales compuestos multifase utilizando el método de los elementos finitos. En la modelización del material compuesto se utiliza la teoría de mezclas en la cual se inserta el modelo constitutivo elastoplástico propuesto. El comportamiento elástico de un sólido anisótropo se simula con la teoría de la elasticidad clásica, mientras que el comportamiento no proporcional de un sólido anisótropo se simula mediante un modelo elastroplástico anisótropo en régimen de grandes deformaciones basado en una formulación isótropa equivalente. Este modelo asume la existencia de un espacio real anisótropo y de un espacio ficticio isótropo, en el cual se resuelve un problema ficticio. Ambos espacios están relacionados a través de una transformación lineal utilizando un tensor de cuarto orden que contiene la información del material real. Por último se presenta los detalles de la implementación numérica del modelo propuesto en un código de elementos finitos y ejemplos de aplicación del modelo al análisis del comportamiento lineal y no-lineal de materiales compuestos.  

  Abstract
  En este trabajo se presenta un modelo constitutivo elastoplástico anisótropo generalizado en deformaciones finitas para el análisis de materiales compuestos [...]

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• The "BAJEL" project. A tool for hydrodynamic design of passenger ships

  J. García-Espinosa, L. Rojas, J. Cabezas, J. Alonso

  (1998). Research Report, Nº PI140

  
  

  Abstract

  The BAJEL project consists on the validation and final development of a comprehensive computer system to support the hydrodynamic design of ships. It aims at contributing to the optimal design of the new generations of ships, specially passenger vessels, improving their navigation conditions.   This project is developed as a result of the joint work of a center specialized in Computational Analysis (CIMNE) and two Hydrodynamic Research Centers (ETSIN and Canal de El Pardo) within the framework of the industrial reality represented by the company BAZAN.   This work presents the objectives of the project, describing the computer tools that have served as the basis and starting point for it. It also presents the achievements that in this field have achieved in recent years the member of the research teams.   Finally, different results of studies carried out on real geometries of ships with special dedication to high-speed passenger ships are included.

  Abstract
  The BAJEL project consists on the validation and final development of a comprehensive computer system to support the hydrodynamic design of ships. It aims at contributing to [...]

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• Estudio comparativo de métodos numéricos para resolver ecuaciones en derivadas parciales lineales

  D. Vives

  (1998). Research Report, Nº PI141

  
  

  Abstract

  La principal ventaja del método de los elementos finitos y del método de los volúmenes finitos es la capacidad que tienen de tratar con dominios ciertamente complejos de manera simple dándole un carácter local a la aproximación. Estos métodos dividen el dominio en un número finito de subdominios, que cumplen unas condiciones geométricas regulares. Para el caso 2-dimensional, ambos métodos no presentan grandes dificultades para la generación de una malla, mientras que en el caso 3-dimensional ello resulta bastante complicado y es uno de los problemas que actualmente se están estudiando con mayor profundidad, ya que, en la mayoría de los casos, la creación de una malla resulta computacionalmente más caro que la resolución numérica del problema. Principalmente por este motivo, se empezaron a estudiar los denominados métodos sin malla. Los primeros métodos se basaron es un intento de generalizar el método de las diferencias finitas a mallas irregulares. Otro de los primeros métodos estudiados son los denominados Smooth Particle Hydrodynamics (SPH). Estos métodos trabajan la ausencia de contornos, aunque no son tan precisos, como los métodos de elementos finitos regulares. Esta clase de métodos suele utilizarse frecuentemente para modelar fenómenos astrofísicos que no posean contornos. Recientemente, se han estudiado una clase de métodos, donde no es necesaria la creación de una malla. Los principios métodos son el método de los elementos difusos (DE), cuyo principal precursor fue Nayroles y el método Element-Free Galerkin (EFG). Ambos se basan en que las funciones interpolatorias son polinomios que tratan de aproximar la función en los nodos por un método de aproximación por mínimos cuadrados. En los dos casos no basta para formular las ecuaciones de Galerkin con una colección de nodos y una descripción del contorno. La diferencia entre ambos métodos radica es que en los métodos de los elementos difusos no se consideran necesarios. Los dos métodos son consistentes y bastante estables, aunque sustancialmente más caros que los métodos SPH. Gracias a un estudio de Duarte y Oden y a otro de Babuska y Melenk se han comprendido con mayor profundidad, y dicen que son métodos basados en la partición de la unidad. Estos personajes junto con Liu fueron los primeros en probar la convergencia de estos dos métodos. Finalmente, para concluir, hay que destacar la presencia de otra clase de métodos denominados Reproducing Kernel Particle (RKP), cuyo principal precursor fue Liu. En la sección 2, veremos una introducción teórica a algunos métodos como son elementos finitos, el método de los elementos difusos, el método EFG, los métodos SPH y finalmente el método de las diferencias finitas para el caso bidimensional. En la sección 3, abordaremos algunos aspectos sobre la programación de ellos, que ha sido efectuada, en el caso 1-dimensional, donde se han programada todos los anteriormente citados, salvo los SPH. La sección 4, contendrá breves remarcar sobre la programación de los métodos en el caso 2-dimensional, donde se han implementado los mismos métodos que en el caso 1-dimensional, con la excepción única de reemplazar el método de los elementos finitos por el de las diferencias finitas. La sección 5 y 6 se comentarán los resultados obtenidos para el caso 1-dimensional y 2-dimensional respectivamente. La seccón 7 contendrá algunas conclusiones y comentarios sobre el estudio.

  Abstract
  La principal ventaja del método de los elementos finitos y del método de los volúmenes finitos es la capacidad que tienen de tratar con dominios ciertamente [...]

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