COLLECTIONS

Documents published in Scipedia

• A stabilized finite element approximation of low speed thermally coupled flows

  J. Principe, R. Codina

  Int. J. Num. Meths. for Heat and Fluid Flows. (2008). Vol. 18 (7-8), pp. 835-867

  
  

  Abstract

  Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed. Design/methodology/approach – The formulation, that allows us to consider convection dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity-pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed. Findings – A full Newtown linearization turns out to be very efficient for steady-state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity-pressure and temperature happens to be more appropriate for transient problems. Research limitations/implications – A fractional step scheme, splitting also momentum and continuity equations, could be further analysed. Practical implications – The results presented in the paper are useful to decide the solution strategy for a given problem. Originality/value – The numerical implementation of a stabilized finite element approximation of thermally coupled flows is described. The implementation algorithm is developed considering several possibilities for the solution of the discrete nonlinear problem.

  Abstract
  Purpose – The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number [...]

  • 9
  •  read

• Algebraic pressure segregation methods for the incompressible Navier-Stokes equations

  S. Badia, R. Codina

  Archives of Comp. Meths. Engng. (2008). Vol. 15 (3), pp. 1-52

  
  

  Abstract

  This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled.

  Abstract
  This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexactLU block [...]

  • 11
  •  read

• Finite element approximation of the modified Boussinesq equations using a stabilized formulation

  R. Codina, J. González-Ondina, G. Díaz-Hernández, J. Principe

  Int. J. Numer. Meth. Fluids (2008). Vol. 57 (9), pp. 1305-1322

  
  

  Abstract

  In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system of equations, namely, the need to use stable velocity‐depth interpolations, which can be overcome by the use of a stabilization technique. The one described in this paper is based on the splitting of the unknowns into their finite element component and the remainder, which we call the subgrid scale. We also discuss the treatment of high‐order derivatives of the mathematical model and describe the time integration scheme

  Abstract
  In this work, we present a finite element model to approximate the modified Boussinesq equations. The objective is to deal with the major problem associated with this system [...]

  • 6
  •  read

• Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscale

  R. Codina

  Applied Numerical Mathematics (2008). Vol. 58 (3), pp. 264-283

  
  

  Abstract

  In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf–sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. It is shown that this formulation is stable and optimally convergent for an adequate choice of the algorithmic parameters on which the method depends.

  Abstract
  In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility [...]

  • 9
  •  read

• Finite element approximation of the hyperbolic wave equation in mixed form

  R. Codina

  Comput. Methods Appl. Mech. Engrg., (2008). Vol. 197 (13-16), pp. 1305-1322

  
  

  Abstract

  The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector field to transform the problem into a first-order problem in space and time. We explain why the standard Galerkin method is inappropriate to solve this problem, and propose as alternative a stabilized finite element method that can be cast in the variational multiscale framework. The unknown is split into its finite element component and a remainder, referred to as subscale. As original features of our approach, we consider the possibility of letting the subscales to be time dependent and orthogonal to the finite element space. The formulation depends on algorithmic parameters whose expression is proposed from a heuristic Fourier analysis.

  Abstract
  The purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form, that is, introducing an auxiliary vector [...]

  • 6
  •  read

• Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach

  S. Badia, R. Codina

  Int. J. Numer. Meth. Fluids (2008). Vol. 56 (4), pp. 351-382

  
  

  Abstract

  In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been replaced by a pressure Poisson equation obtained at the discrete level. In these methods it is the velocity instead of the pressure the extrapolated unknown. Moreover, predictor–corrector schemes are suggested, again motivated by the new monolithic system. Key implementation aspects are discussed, and a complete stability analysis is performed. We end with a set of numerical examples in order to compare these methods with classical pressure‐correction schemes.

  Abstract
  In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been [...]

  • 11
  •  read

• On some fluid-structure iterative algorithms using pressure segregation methods. Applications to aeroelasticity

  S. Badia, R. Codina

  Int. J. Numer. Meth. Engng. (2007). Vol. 72 (1), pp. 46-71

  
  

  Abstract

  In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure segregation methods for the solution of the fluid problem and fixed point iterative algorithms for the fluid-structure coupling. These coupling algorithms are applied to the aeroelastic simulation of suspension bridges. We assess flexural and torsional frequencies for a given inflow velocity. Increasing this velocity we reach the value for which the flutter phenomenon appears.

  Abstract
  In this paper we suggest some algorithms for the fluid-structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure [...]

  • 6
  •  read

• Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

  S. Badia, R. Codina

  Numerische Mathematik (2007). Vol. 107 (4), pp. 533-557

  
  

  Abstract

  In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.

  Abstract
  In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using [...]

  • 5
  •  read

• Shockwaves in spillways with the particle finite element method

  F. Salazar, J. San-Mauro, M. Celigueta, E. Oñate

  Comp. Part. Mech. (2020). Vol. 7 (1), pp. 87-99

  
  

  Abstract

  Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. These disturbances are propagated downstream and need to be considered in the design of the chute walls. In dam spillways, where flow rates are often high, this phenomenon can have significant implications for the cost and complexity of the solution. It has been traditionally analysed by means of reduced-scale experimental tests, as it has a clear three-dimensional character and therefore cannot be approached with two-dimensional numerical models. In this work, the ability of the particle finite element method (PFEM) to reproduce this phenomenon is analysed. PFEM has been successfully applied in previous works to problems involving high irregularities in free surface. First, simple test cases available in the technical bibliography were selected to be reproduced with PFEM. Subsequently, the method was applied in two spillways of real dams. The results show that PFEM is capable of capturing the shockwave fronts generated both in the contractions and in the expansions that occur behind the spillway piers. This suggests that the method may be useful as a complement to laboratory test campaigns for the design and hydraulic analysis of dam spillways with complex geometries.

  Abstract
  Changes in direction and cross section in supercritical hydraulic channels generate shockwaves which result in an increase in flow depth with regard to that for uniform regime. [...]

  • 12
  •  read

• An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics

  O. Guasch, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (45-48), pp. 4672-4689

  
  

  Abstract

  An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization.

  Abstract
  An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element [...]

  • 17
  •  read