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• Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem

  E. Castillo, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 579-605

  
  

  Abstract

  In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to use equal interpolation for the problem unknowns σ–u–p (elastic deviatoric stress–velocity–pressure) and to stabilize dominant convective terms. Starting from residual-based stabilized formulations, the proposed method introduces a term-by-term stabilization which is shown to have a superior behavior when there are stress singularities. A general discontinuity-capturing technique for the elastic stress component is also proposed, which allows one to eliminate the local oscillations that can appear when the Weissenberg number We is high and the fluid flow finds an abrupt change in the geometry. The formulations are tested in the classical 4:1 planar contraction benchmark up to We=5 in the inertial case, with Reynolds number Re=1, and up to We=6.5 in the quasi non-inertial case, with Re=0.01. The standard Oldroyd-B constitutive model is used for the rheological behavior and linear and quadratic elements for the spatial approximation.

  Abstract
  In this paper, three-field finite element stabilized formulations are proposed for the numerical solution of incompressible viscoelastic flows. These methods allow one to [...]

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• Stabilized stress–velocity–pressure finite element formulations of the Navier–Stokes problem for fluids with non-linear viscosity

  E. Castillo, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2014). Vol. 279, pp. 554-578

  
  

  Abstract

  The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. The first is associated with the incompressibility and loss of stability in the calculation of the stress field, and the second with the dominant convection. The first type of instabilities can be overcome by choosing an interpolation for the unknowns that satisfies the appropriate inf–sup conditions, whereas the dominant convection requires a stabilized formulation in any case. This paper proposes two stabilized schemes of Sub-Grid Scale (SGS) type, differing in the definition of the space of the sub-grid scales, and both allowing to use the same interpolation for the variables σ–u–p (deviatoric stress, velocity and pressure), even in problems where the convection component is dominant and the velocity–stress gradients are high. Another aspect considered in this work is the non-linearity of the viscosity, modeled with constitutive models of quasi-Newtonian type. This paper includes a description of the proposed methods, some of their implementation issues and a discussion about benefits and drawbacks of a three-field formulation. Several numerical examples serve to justify our claims.

  Abstract
  The three-field (stress–velocity–pressure) mixed formulation of the incompressible Navier–Stokes problem can lead to two different types of numerical instabilities. [...]

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• Stability, convergence and accuracy of stabilized finite element methods for the wave equation in mixed form

  S. Badia, R. Codina, H. Espinoza

  SIAM J. Numer. Anal. (2014). Vol. 52 (4), pp. 1729-1752

  
  

  Abstract

  In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns. The variational forms corresponding to different functional settings are treated in a unified manner through the introduction of length scales related to the unknowns. Stability and convergence analysis is performed together with numerical experiments. It is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods.  

  Abstract
  In this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form. These stabilized finite element [...]

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• Large eddy simulation of low Mach number flows using a dynamical and nonlinear finite element subgrid scale model

  M. Ávila, R. Codina, J. Principe

  Computers and Fluids (2014). Vol. 99, pp. 44-66

  
  

  Abstract

  Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is to analyze the behavior of our numerical method in the large eddy simulation (LES) of thermally coupled turbulent flows at low Mach number. Methods: Our numerical method is a stabilized finite element approximation based on 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 as LES model) depends on the particular subscale 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. Another important contribution of this work is the extension of the orthogonal subgrid scale method widely tested for incompressible flows to variable density flows, using a density-weighted L 2 product to define the orthogonality of the subscales and the finite element spaces. Results: Referring to numerical testing, we present numerical results for a laminar testcase validation that shows the dissipative behavior of the different stabilized methods. Then, we present results of the numerical simulation of two turbulent flow problems, the turbulent channel flow with large temperature differences in the wall normal direction at Res ¼ 180, and the turbulent thermally driven cavity with aspect ratio 4. The behavior of the method is evaluated by comparison against results available in the literature obtained using LES and direct numerical simulation (DNS). They are explained based on a careful analysis of the dissipative structure of the method, showing the physical interpretation of the subgrid scale method presented. Conclusion: The material presented here is a clear indication of the potential of the method to model all kinds of turbulent thermally coupled flows. The formulation is the same in laminar and turbulent regimes

  Abstract
  Objective: In this article we study the approximation to thermal turbulence from a strictly numerical point of view, without the use of any physical model. The main goal is [...]

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• A Sommerfeld non-reflecting boundary condition for the wave equation in mixed form

  H. Espinoza, R. Codina, S. Badia

  Comput. Methods Appl. Mech. Engrg., (2014). Vol. 276, pp. 122-148

  
  

  Abstract

  In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D.

  Abstract
  In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial [...]

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• Error analysis of discontinuous Galerkin methods for the Stokes problem under minimum regularity

  S. Badia, R. Codina, T. Gudi, J. Guzmán

  IMA Journal of Numerical Analysis (2014). Vol. 34 (2), pp. 800-819

  
  

  Abstract

  In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs to[H10(Ω)]d  and the pressurep∈L20(Ω). First, we analyse standard DG methods assuming that the right-hand side f belongs to[H−1(Ω)∩L1(Ω)]d.A DG method that is well defined for f belonging to [H−1(Ω)]d  is then investigated. The methods under study include stabilized DG methods using equal-order spaces and inf–sup stable ones where the pressure space is one polynomial degree less than the velocity space.

  Abstract
  In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs [...]

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• On the design of discontinuous Galerkin methods for elliptic problems based on hybrid formulations

  R. Codina, S. Badia

  Comput. Methods Appl. Mech. Engrg., (2013). Vol. 263, pp. 158-168

  
  

  Abstract

  The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation of the problem involving as unknowns the main field in the interior of the element domains and its fluxes and traces on the element boundaries. Rather than working with this three-field formulation, fluxes are modeled using finite difference expressions and then the traces are determined by imposing continuity of fluxes, although other strategies could be devised. This procedure is applied to four elliptic problems, namely, the convection–diffusion equation (in the diffusion dominated regime), the Stokes problem, the Darcy problem and the Maxwell problem. We justify some well known dG methods with some modifications that in fact allow to improve the performance of the original methods, particularly when the physical properties are discontinuous.

  Abstract
  The objective of this paper is to present a new framework for the design of discontinuous Galerkin (dG) methods for elliptic problems. The idea is to start from a hybrid formulation [...]

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• Immersed stress method for fluid structure interaction using anisotropic mesh adaption

  E. Hachem, S. Feghali, R. Codina, T. Coupez

  Int. J. Numer. Meth. Engng. (2013). Vol. 94 (9), pp. 805-825

  
  

  Abstract

  This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with complex geometry. First, a new stabilized three‐field stress, velocity, and pressure finite element formulation is presented for modeling the interaction between the fluid (laminar or turbulent) and the rigid body. The presence of the structure will be taken into account by means of an extra stress in the Navier–Stokes equations. The system is solved using a finite element variational multiscale method. We combine this method with anisotropic mesh adaptation to ensure an accurate capturing of the discontinuities at the fluid–solid interface. We assess the behavior and accuracy of the proposed formulation in the simulation of 2D and 3D time‐dependent numerical examples such as the flow past a circular cylinder and turbulent flows behind an immersed helicopter in a forward flight.

  Abstract
  This paper presents advancements toward a monolithic solution procedure and anisotropic mesh adaptation for the numerical solution of fluid–structure interaction with [...]

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• Anisotropic adaptive meshing and monolithic variational multiscale method for fluid-structure interaction

  E. Hachem, S. Feghali, R. Codina, T. Coupez

  Computers and Structures (2013). Vol. 122, pp. 88-100

  
  

  Abstract

  This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. The fluid–solid interfaces are captured using a level-set method. A new a posteriori error estimate, based on the length distribution tensor approach and the associated edge based error analysis, is then used to ensure an accurate capturing of the discontinuities at the fluid–solid interface. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The presence of the structure will be taken into account by means of an extra stress tensor in the Navier–Stokes equations. The system is solved using a stabilized three-field, stress, velocity and pressure finite element (FE) formulation. It consists in the decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales and also in the efficient enrichment of the extra constraint. We assess the accuracy of the proposed formulation by simulating 2D and 3D time-dependent numerical examples such as: falling disk in a channel, turbulent flows behind an airfoil profile and flow behind an immersed vehicle.

  Abstract
  This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. [...]

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• A variational multiscale method with subscales on the element boundaries for the Helmholtz equation

  J. Baiges, R. Codina

  Int. J. Numer. Meth. Engng. (2013). Vol. 93 (6), pp. 664-684

  
  

  Abstract

  In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite elements. The expression for the subscales is obtained by imposing the continuity of fluxes across the interelement boundaries. The stabilization parameter is determined by performing a dispersion analysis, yielding the optimal values for the different discretizations and finite element mesh configurations. The performance of the method is compared with that of the standard Galerkin method and the classical Galerkin least‐squares method with very satisfactory results. Some numerical examples illustrate the behavior of the method. 

  Abstract
  In this paper, we apply the variational multiscale method with subgrid scales on the element boundaries to the problem of solving the Helmholtz equation with low‐order finite [...]

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