COLLECTIONS

Documents published in Scipedia

• On unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics

  S. Badia, R. Codina, R. Planas

  Journal of Computational Physics (2013). Vol. 234, pp. 399-416

  
  

  Abstract

  In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation is the fact that it always converges to the physical solution, even for singular ones. A detailed set of numerical experiments have been performed in order to validate our approach.

  Abstract
  In this work, we propose a new stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]

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• A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes

  J. Baiges, R. Codina, F. Henke, S. Shahmiri, W. Wall

  Int. J. Numer. Meth. Engng. (2012). Vol. 90 (5), pp. 636-658

  
  

  Abstract

  In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element‐discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations.

  Abstract
  In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic [...]

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• A third‐order velocity correction scheme obtained at the discrete level

  H. Owen, R. Codina

  Int. J. Numer. Meth. Fluids (2012). Vol. 69 (1), pp. 52-72

  
  

  Abstract

  In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed into a discrete pressure Poisson equation and a velocity extrapolation is used. In Badia et al. (IJNMF, 2008, p. 351), where the method was introduced, the discrete Laplacian that appears in the pressure Poisson equation is approximated by a continuous one using an extrapolation for the pressure. In this work we explore the possibility of actually solving the discrete Laplacian. This introduces significant differences because the pressure extrapolation is avoided and only a velocity extrapolation is needed. Our numerical results indicate that it is the second‐order pressure extrapolation which makes third‐order methods unstable. Instead, second‐order velocity extrapolations do not lead to instabilities. Avoiding the pressure extrapolation allows to obtain stable solutions in problems that become unstable when the Laplacian is approximated. A comparison with a pressure correction scheme is also presented to verify the well‐known fact that the use of a second order pressure extrapolation leads to instabilities. Therefore we conclude that it is the combination of a velocity correction scheme with a discrete Laplacian that allows to obtain a stable third‐order scheme by avoiding the pressure extrapolation.

  Abstract
  In this work we explore a velocity correction method that introduces the splitting at the discrete level. In order to do so, the algebraic continuity equation is transformed [...]

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• A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions

  S. Badia, R. Codina

  SIAM J. Numer. Anal. (2012). Vol. 50 (2), pp. 398-417

  
  

  Abstract

   A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural H(curl 0; Ω) norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal-based interpolations are known to suffer from spurious convergence upon mesh refinement.

  Abstract
   A new mixed finite element approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous [...]

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• Stokes, Maxwell and Darcy: A single finite element approximation for three model problems

  S. Badia, R. Codina

  Applied Numerical Mathematics (2012). Vol. 62 (4) pp. 246-263

  
  

  Abstract

  In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We briefly review the formulations we have proposed for these three problems independently in a unified manner, stressing the advantages of our approach. In particular, for Darcyʼs problem we are able to design stabilized methods that yield optimal convergence both for the primal and the dual problems. In the case of Maxwellʼs problem, the formulation we propose allows one to use continuous finite element interpolations that converge optimally to the continuous solution even if it is non-smooth. Once the formulation is presented for the three model problems independently, we also show how it can be used for a problem that combines all the operators of the independent problems. Stability and convergence is achieved regardless of the fact that any of these operators dominates the others, a feature not possible for the methods of which we are aware.

  Abstract
  In this work we propose stabilized finite element methods for Stokesʼ, Maxwellʼs and Darcyʼs problems that accommodate any interpolation of velocities and pressures. We [...]

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• A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem

  S. Badia, R. Codina

  Applied Mathematics and Computation (2011). Vol. 218 (8), pp. 4276-4294

  
  

  Abstract

  Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG–dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functins continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.

  Abstract
  Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are [...]

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• The Fixed‐Mesh ALE approach for the numerical simulation of floating solids

  J. Baiges, R. Codina, H. Coppola-Owen

  Int. J. Numer. Meth. Fluids (2011). Vol. 67 (8), pp. 1004-1023

  
  

  Abstract

  In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. 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 framework, the distinctive characteristic being that at each time step results are projected onto a fixed, background mesh. We pay special attention to the tracking of the various interfaces and their intersections, and to the approximate imposition of coupling conditions between the solid and the fluid. 

  Abstract
  In this paper, we propose a method to solve the problem of floating solids using always a background mesh for the spatial discretization of the fluid domain. The main feature [...]

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• Finite element dynamical subgrid-scale approximation of low Mach number flow equations

  M. Ávila, J. Principe, R. Codina

  Journal of Computational Physics (2011). Vol. 230, pp. 7988-8009

  
  

  Abstract

  In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number equations, which are obtained as a limit of the compressible Navier Stokes equations in the small Mach number. In contrast to the commonly used Boussinesq approximation, this model permits to take volumetric deformation into account. Although the former is more general than the later, both systems have similar mathematical structure and their numerical approximation can suffer the same type of instabilities. We propose a stabilized finite element approximation based on the the variational multiscale method, in which a decomposition of the approximating space into a coarse scale resolvable part and a fine scale subgrid part is performed. Modeling the subscale and taking its effect on the coarse scale problem into account, results in a stable formulation. The quality of the final approximation (accuracy, efficiency) depends on the particular model. The distinctive features of our approach are to consider the subscales as transient and to keep the scale splitting in all the nonlinear terms. The first ingredient permits to obtain an improved time discretization scheme (higher accuracy, better stability, no restrictions on the time step size). The second ingredient permits to prove global conservation properties. It also allows us to approach the problem of dealing with thermal turbulence from a strictly numerical point of view. Numerical tests show that nonlinear and dynamic subscales give more accurate solutions than classical stabilized methods. 

  Abstract
  In this work we propose a variational multiscale finite element approximation of thermally coupled low speed flows. The physical model is described by the low Mach number [...]

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• Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales

  R. Codina, J. Baiges

  Int. J. Numer. Meth. Engng. (2011). Vol. 87 (1-5), pp. 386-411

  
  

  Abstract

  Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized finite element approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non‐conforming part, a hybrid formulation for the latter and a simple approximation for the unknowns involved in the hybrid problem.

  Abstract
  Terms involving jumps of stresses on boundaries are proposed for the finite element approximation of the Stokes problem and the linear elasticity equations. These terms are [...]

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• Thermal coupling of fluid and structural response of a tunnel induced by fire

  B. Schrefler, R. Codina, F. Pesavento, J. Principe

  Int. J. Numer. Meth. Engng. (2010). Vol. 87 (1-5), pp. 361-385

  
  

  Abstract

    In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling strategy is based on a Dirichlet/Neumann non‐overlapping domain decomposition of the problem, which is carried out by developing a master code that controls solvers dedicated to the fluid mechanics and to the solid mechanics simulation. The computational fluid dynamics formulation consists of a stabilized finite element approximation of the low Mach number equations based on the subgrid scale concept, that allows us to deal with convection‐dominated problems and to use equal order interpolation of velocity and pressure. The thermo‐structural model of the tunnel vault, that considers a multiphase porous material where pores are partly filled with liquid and partly by gas, is specially devised for the simulation of concrete at high temperatures and consists of balance equations for mass conservation of dry air, mass conservation of water species (both in the liquid and gaseous state), enthalpy conservation and linear momentum conservation taking phase changes into account. The developed algorithm is applied to the problem of the response of a tunnel to a fire. We consider the combustion process as a heat release which can vary usually from 1thinspaceMW for small car fires to 100 MW for catastrophic fires. The released heat is transferred to the concrete walls of the tunnel which could cause extensive and heavy damage of the structure.

  Abstract
    In this work we present the progress in the development of an algorithm for the simulation of thermal fluid–structural coupling in a tunnel fire. The coupling [...]

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