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• On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems

  R. Codina, S. Badia

  Comput. Methods Appl. Mech. Engrg., (2006). Vol. 195 (23-24), pp. 2900-2918

  
  

  Abstract

  In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities and the pressure. The first family of schemes consists of classical fractional step methods, of which we discuss several possibilities for the pressure extrapolation and the time integration of first and second order. The second family consists of schemes based on an explicit treatment of the pressure in the momentum equation followed by a Poisson equation for the pressure. It turns out that this “staggered” treatment of the velocity and the pressure is stable. Finally, we present predictor-corrector methods based on the above schemes that aim to converge to the solution of the monolithic time integration method. Apart from presenting these schemes and check its numerical performance, we also present a complete set of stability results for the fractional step methods that are independent of the space stability of the velocity-pressure interpolation, that is, of the classical inf-sup condition.

  Abstract
  In this paper we treat several aspects related to time integration methods for the incompressible Navier-Stokes equations that allow to uncouple the calculation of the velocities [...]

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• Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions

  A. Coppola‐Owen, R. Codina

  Int. J. Numer. Meth. Fluids (2005). Vol. 49 (12), pp. 1287-1304

  
  

  Abstract

  In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. We propose a solution based on enriching the pressure shape functions on the elements cut by the interface. The enrichment is used to enable the pressure gradient to be discontinuous at the interface, thus improving the ability to simulate the behaviour of fluids with different density under a gravitational force. The additional shape function used is local to each element and the corresponding degree of freedom can therefore be condensed prior to assembly, making the implementation quite simple on any existing finite element code.

  Abstract
  In this paper we present a problem we have encountered using a stabilized finite element method on fixed grids for flows with interfaces modelled with the level set approach. [...]

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• Error estimates for an operator-splitting method for incompressible flows

  J. Blasco, R. Codina

  Applied Numerical Mathematics (2004). Vol. 51 (1), pp. 1-17

  
  

  Abstract

  In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain first order error estimates in the time step size, both for the intermediate and the end-of-step velocities of the method; we also give some error estimates for the pressure solution.

  Abstract
  In this paper we provide an error analysis of a fractional-step method for the numerical solution of the incompressible Navier–Stokes equations. Under mild regularity [...]

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• A finite element model for the simulation of lost foam casting

  G. Houzeaux, R. Codina

  Int. J. Numer. Meth. Fluids (2004). Vol. 46 (2), pp. 203-226

  
  

  Abstract

  In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes in play during a casting. We also briefly comment on the possible physical and numerical models to envisage the numerical simulation. Next we present a model which aims at solving “part of” the complexities of the casting, together with a simple energy budget that enables to obtain an equation for the velocity of the metal front advance. Once the physical model is established we develop a finite element method to solve the governing equations. The numerical and physical methodologies are then validated through the solution of a two and a three-dimensional example. Finally, we briefly discuss some possible improvements of the numerical model in order to catch more physical phenomena.

  Abstract
  In this paper, we present a numerical model to simulate lost foam casting processes. We first introduce this particular casting in order to catch the different physical processes [...]

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• Approximation of the incompressible Navier–Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes

  R. Codina, O. Soto

  Comput. Methods Appl. Mech. Engrg., (2004). Vol. 193 (15-16), pp. 1403-1419

  
  

  Abstract

  This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. The former consists of adding a least-square form of the component orthogonal to the finite element space of the convective and pressure gradient terms; this allows to deal with convection-dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Likewise, we put special emphasis on the use of anisotropic grids. In particular, we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples. The preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described.

  Abstract
  This paper describes a finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales and a pressure segregation. [...]

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• A stabilized finite element predictor–corrector scheme for the incompressible Navier–Stokes equations using a nodal‐based implementation

  R. Codina, A. Folch

  Int. J. Numer. Meth. Fluids (2004). Vol. 44 (5), pp. 483-503

  
  

  Abstract

  A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to segregate the pressure and a nodal based implementation is presented in this paper. The stabilization consists of adding a least‐squares form of the component orthogonal to the finite element space of the convective and pressure gradient terms, which allows to deal with convection‐dominated flows and to use equal velocity–pressure interpolation. The pressure segregation is inspired in fractional step schemes, although the converged solution corresponds to that of a monolithic time integration. Finally, the nodal‐based implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix and right‐hand side vector of the final algebraic system to be solved. After appropriate approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over the elements and thus making the calculations much faster. Some issues related to this implementation for fractional step and our predictor–corrector scheme, which is the main contribution of this paper, are discussed

  Abstract
  A finite element model to solve the incompressible Navier–Stokes equations based on the stabilization with orthogonal subscales, a predictor–corrector scheme to [...]

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• A Dirichlet/Neumann domain decomposition method for incompressible turbulent flows on overlapping subdomains

  G. Houzeaux, R. Codina

  Computers and Fluids (2004). Vol. 33 (5-6), pp. 771-782

  
  

  Abstract

  When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility is the use of domain decomposition methods. The domain decomposition method we present in this work aims at coupling overlapping subdomains in relative motion using a Dirichlet/Neumann coupling. The method is applied to the solution of incompressible and turbulent flows.

  Abstract
  When one wants to simulate flows with moving bodies and when there is no possible way of prescribing simple boundary conditions in any frame of reference, one possibility [...]

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• An iteration-by-subdomain overlapping Dirichlet/Robin domain decomposition method for advection–diffusion problems

  G. Houzeaux, R. Codina

  Journal of Computational and Applied Mathematics (2003). Vol. 158 (2), pp. 243-276

  
  

  Abstract

  We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning of the computational domain. We derive interface equations to study the convergence of the method and show its properties through four numerical examples. The mathematical framework is general and can be applied to derive overlapping versions of all the classical nonoverlapping methods.

  Abstract
  We present a new overlapping Dirichlet/Robin Domain Decomposition method. The method uses Dirichlet and Robin transmission conditions on the interfaces of an overlapping partitioning [...]

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• A Chimera method based on a Dirichlet/Neumann(Robin) coupling for the Navier–Stokes equations

  G. Houzeaux, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2003). Vol. 192 (31-32), pp. 3343-3377

  
  

  Abstract

  We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain method based on Dirichlet/Neumann(Robin) coupling. The DD method we propose is not only geometric but also algorithmic, for the solution on each subdomain is obtained on separate processes and the exchange of information between the subdomains is carried out by a master code. This strategy is very flexible as it requires almost no modification to the original numerical code. Therefore, only the master code has to be adapted to the numerical codes and the strategies used on each subdomain. As a basic flow solver, we a use stabilized finite element method.

  Abstract
  We present a Chimera method for the numerical solution of incompressible flows past objects in relative motion. The Chimera method is implemented as an iteration-by-subdomain [...]

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• Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

  R. Codina

  Comput. Methods Appl. Mech. Engrg., (2002). Vol. 191 (39-40), pp. 4295-4321

  
  

  Abstract

  In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.

  Abstract
  In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable [...]

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