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• A free surface finite element model for low Froude number mould filling problems on fixed meshes

  H. Coppola-Owen, R. Codina

  Int. J. Numer. Meth. Fluids (2011). Vol. 66 (7), pp. 833-851

  
  

  Abstract

  The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position of the interface and the resulting flow field increases. The usual two‐phase flow model provides poor results for such flow. In order to overcome the difficulties, a free surface model that applies boundary conditions at the interface accurately is used. Moreover, the use of wall laws on curved boundaries also fails in the case of low Froude number flows. To solve this second problem, we combine wall laws with ‘do nothing’ boundary conditions. A special feature of our approach is that ‘do nothing’ boundary conditions are only applied in the normal direction. These two key ingredients together with the Level Set method allow us to simulate three‐dimensional mould filling problems borrowed directly from the foundry.

  Abstract
  The simulation of low Froude number mould filling problems on fixed meshes presents significant difficulties. As the Froude number decreases, the coupling between the position [...]

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• Approximation of the inductionless MHD problem using a stabilized finite element method

  R. Planas, S. Badia, R. Codina

  Journal of Computational Physics (2011). Vol. 230 (8), pp. 2977-2996

  
  

  Abstract

  In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD problem couples the Navier–Stokes equations and a Darcy-type system for the electric potential via Lorentz’s force in the momentum equation of the Navier–Stokes equations and the currents generated by the moving fluid in Ohm’s law. The key feature of the FE formulation resides in the design of the stabilization terms, which serve several purposes. First, the formulation is suitable for convection dominated flows. Second, there is no need to use interpolation spaces constrained to a compatibility condition in both sub-problems and therefore, equal-order interpolation spaces can be used for all the unknowns. Finally, this formulation leads to a coupled linear system; this monolithic approach is effective, since the coupling can be dealt by effective preconditioning and iterative solvers that allows to deal with high Hartmann numbers.

  Abstract
  In this work, we present a stabilized formulation to solve the inductionless magnetohydrodynamic (MHD) problem using the finite element (FE) method. The MHD [...]

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• Approximation of the thermally coupled MHD problem using a stabilized finite element method

  R. Codina, N. Hernández-Silva

  Journal of Computational Physics (2011). Vol. 230 (4), pp. 1281-1303

  
  

  Abstract

  A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization terms, which serve several purposes. First, convective dominated flows in the Navier–Stokes and the heat equation can be dealt with. Second, there is no restriction in the choice of the interpolation spaces of all the variables and, finally, flows highly coupled with the magnetic field can be accounted for. Different aspects related to the design of the final fully discrete and linearized algorithm are also discussed.

  Abstract
  A numerical formulation to solve the MHD problem with thermal coupling is presented in full detail. The distinctive feature of the method is the design of the stabilization [...]

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• Spatial approximation of the radiation transport equation using a subgrid-scale finite element method

  M. Ávila, R. Codina, J. Principe

  Comput. Methods Appl. Mech. Engrg., (2011). Vol. 200 (5-8), pp. 425-438

  
  

  Abstract

  In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition of the unknowns into resolvable and subgrid scales, with an approximation for the latter that yields a problem to be solved for the former. This approach allows us to design the algorithmic parameters on which the method depends, which we do here when the discrete ordinates method is used for the directional approximation. We concentrate on two stabilized methods, namely, the classical SUPG technique and the orthogonal subscale stabilization. A numerical analysis of the spatial approximation for both formulations is performed, which shows that they have a similar behavior: they are both stable and optimally convergent in the same mesh-dependent norm. A comparison with the behavior of the Galerkin method, for which a non-standard numerical analysis is done, is also presented.

  Abstract
  In this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation. These methods are based on the decomposition [...]

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• Long-term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical sub-grid scale modeling

  S. Badia, R. Codina, J. Gutiérrez-Santacreu

  SIAM J. Numer. Anal. (2010). Vol. 48 (3), pp. 1013-1037

  
  

  Abstract

  Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure stability) and the velocity stability loss for high Reynolds numbers. These methods enrich the Galerkin formulation with a sub-grid component that is modelled. In fact, the effect of the sub-grid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum. Thus, they also act as effective large-eddy simulation turbulence models and allow to compute flows without the need to capture all the scales in the system. In this article, we consider a dynamic sub-grid model that enforces the sub-grid component to be orthogonal to the finite element space in L 2 sense. We analyze the long-term behavior of the algorithm, proving the existence of appropriate absorbing sets and a compact global attractor. The improvements with respect to a finite element Galerkin approximation are the long-term estimates for the sub-grid component, that are translated to effective pressure and velocity stability. Thus, the stabilization introduced by the sub-grid model into the finite element problem is not deteriorated for infinite time intervals of computation. 

  Abstract
  Variational multiscale methods lead to stable finite element approximations of the NavierStokes equations, both dealing with the indefinite nature of the system (pressure [...]

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• Finite element approximation of turbulent thermally coupled incompressible flows with numerical sub-grid scale modeling

  R. Codina, J. Principe, M. Ávila

  Int. J. Numer. Meth. Heat and Fluid Flow (2010). Vol. 20 (5), pp. 492-516

  
  

  Abstract

  The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient and orthogonal to the finite element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. This strategy allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view and discuss important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation

  Abstract
  The objective of this work is to describe a variational multiscale finite element approximation for the incompressible Navier-Stokes equations using the Boussinesq approximation [...]

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• Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

  S. Badia, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2010). vol. 199 (25-28), pp. 1654-1667

  
  

  Abstract

  We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2-norm.

  Abstract
  We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. [...]

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• On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes

  J. Principe, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (21-22), pp. 1386-1402

  
  

  Abstract

  In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated. In order to incorporate the distortion of the mesh into this approximation, we transform the equation for the subgrid scale within each element to the shape-regular reference domain. The expression for the subgrid scale arises from an approximate Fourier analysis and the identification of the wave number direction where instabilities are most likely to occur. The final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation.

  Abstract
  In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting [...]

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• The fixed‐mesh ALE approach applied to solid mechanics and fluid–structure interaction problems

  J. Baiges, R. Codina

  Int. J. Numer. Meth. Engng. (2009). Vol. 81 (12), pp. 1529-1557

  
  

  Abstract

  In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. The main feature of the method is that it properly accounts for the advection of information as the domain boundary evolves. To achieve this, we use an Arbitrary Lagrangian–Eulerian (ALE) framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. For solid mechanics problems subject to large strains, the fixed‐mesh (FM)‐ALE method avoids the element stretching found in fully Lagrangian approaches. For FSI problems, FM‐ALE allows for the use of a single background mesh to solve both the fluid and the structure.

  Abstract
  In this paper we propose a method to solve Solid Mechanics and fluid–structure interaction problems using always a fixed background mesh for the spatial discretization. [...]

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• The dissipative structure of variational multiscale methods for incompressible flows

  J. Principe, R. Codina, F. Henke

  Comput. Methods Appl. Mech. Engrg., (2010). Vol. 199 (13-16), pp. 791-801

  
  

  Abstract

  In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.

  Abstract
  In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. [...]

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