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• A numerical strategy to compute optical parameters in turbulent flow. Application to telescopes

  R. Codina, J. Baiges, D. Pérez-Sánchez, M. Collados

  Computers and Fluids (2010). Vol. 39 (1), pp. 87-98

  
  

  Abstract

  We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier-Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.

  Abstract
  We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible [...]

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• Approximate imposition of boundary conditions in immersed boundary methods

  R. Codina, J. Baiges

  Int. J. Numer. Meth. Engng. (2009). Vol. 80 (11), pp. 1379-1405

  
  

  Abstract

  We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition

  Abstract
  We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element [...]

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• Computational aeroacoustics of viscous low speed flows using subgrid scale finite element methods

  O. Guasch, R. Codina

  Journal of Computational Acoustics (2009). Vol. 17 (3), pp. 309-330

  
  

  Abstract

  A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure is followed that makes use of Lighthill's acoustic analogy in the frequency domain. The procedure has been conceptually divided into three steps. In the first one, the incompressible Navier–Stokes equations are solved to obtain the flow velocity field. In the second step, Lighthill's acoustic source term is computed from this velocity field and then Fourier transformed to the frequency domain. Finally, the acoustic pressure field is obtained by solving the corresponding inhomogeneous Helmholtz equation. All equations in the formulation are solved using subgrid scale stabilized finite element methods. The main ideas of the subgrid scale numerical strategy are outlined and its benefits when compared to the Galerkin approach are described. As numerical examples, the aerodynamic noise generated by flow past a two-dimensional cylinder and by flow past two cylinders in parallel arrangement are addressed.

  Abstract
  A methodology to perform computational aeroacoustics (CAA) of viscous low speed flows in the framework of stabilized finite element methods is presented. A hybrid CAA procedure [...]

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• Mathematical models for thermally coupled low speed flows

  J. Principe, R. Codina

  Adv. Theor. Appl. Mech. (2009). Vol. 2 (2), pp. 93-112

  
  

  Abstract

  In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat exchange (the isentropic regime) this limit is well understood and rigorous results are available. When heat exchange is considered different simplified models can be obtained, the most famous being the Boussinesq approximation. Here a unified formal justification of these models is presented, paying special attention to the relation between the low Mach number and the Boussinesq approximations. Precise conditions for their validity are given for classical problems in bounded domains.

  Abstract
  In this paper we review and clarify some aspects of the asymptotic analysis of the compressible Navier Stokes equations in the low Mach number limit. In the absence of heat [...]

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• Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems

  S. Badia, R. Codina

  SIAM J. Numer. Anal. (2009). Vol. 47 (3), pp. 1971-2000

  
  

  Abstract

  In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart from the interest of this fact, the important issue is that we are able to deal with both problems at the same time, in a completely unified manner, in spite of the fact that the functional setting is different. Concerning the stabilization formulation, we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters, both in what refers to stability and to accuracy. This choice is shown to be crucial in the case of Darcy's problem. As an additional feature of this work, we treat two types of stabilized formulations, showing that they have a very similar behavior  

  Abstract
  In this paper we propose stabilized finite element methods for both Stokes' and Darcy's problems that accommodate any interpolation of velocities and pressures. Apart [...]

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• A numerical approximation of the thermal coupling of fluids and solids

  J. Principe, R. Codina

  Int. J. Numer. Meth. Fluids (2009). Vol. 59 (11), pp. 1181-1201

  
  

  Abstract

  In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context.

  Abstract
  In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including [...]

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• On a multiscale approach to the transient Stokes problem: Dynamic subscales and anisotropic space–time discretization

  S. Badia, R. Codina

  Applied Mathematics and Computation (2009). Vol. 207 (2), pp. 415-433

  
  

  Abstract

  In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. We define an anisotropic time–space discretization as a family of time–space partitions that does not satisfy the condition h2⩽Cδt with C uniform with respect to h and δt. Standard residual-based stabilization techniques are motivated by a multiscale approach, approximating the effect of the subscales onto the large scales. One of the approximations is to consider the subscales quasi-static (neglecting their time derivative). It is well known that these techniques are unstable for anisotropic time–space discretizations. We show that the use of dynamic subscales (where the subscales time derivatives are not neglected) solves the problem, and prove optimal convergence and stability results that are valid for anisotropic time–space discretizations. Also the improvements related to the use of orthogonal subscales are addressed.

  Abstract
  In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. [...]

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• The fixed-mesh ALE approach for the numerical approximation of flows in moving domains

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

  Journal of Computational Physics (2009). Vol. 228 (5), pp. 1591-1611

  
  

  Abstract

  In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation to be presented falls within the category of fixed-grid methods. Even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed-grid method, our approach can be applied together with any technique to impose approximately boundary conditions, although we also describe the one we actually favor. Our main concern is to properly account for the advection of information as the domain boundary evolves. To achieve this, we use an arbitrary Lagrangian–Eulerian framework, the distinctive feature being that at each time step results are projected onto a fixed, background mesh, that is where the problem is actually solved.

  Abstract
  In this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization, and therefore the formulation [...]

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• Finite element approximation of the three field formulation of the Stokes problem using arbitrary interpolations

  R. Codina

  SIAM J. Numer. Anal. (2009). Vol. 47 (1), pp. 699-718

  
  

  Abstract

  The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis.  

  Abstract
  The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of [...]

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• Subscales on the element boundaries in the variational two-scale finite element method

  R. Codina, J. Principe, J. Baiges

  Comput. Methods Appl. Mech. Engrg., (2009). Vol. 198 (5-8), pp. 838-852

  
  

  Abstract

  In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. The key idea is that the subscales on the element boundaries must be such that the transmission conditions for the unknown, split as its finite element contribution and the subscale, hold. In particular, we consider the scalar convection–diffusion–reaction equation, the Stokes problem and Darcy’s problem. For these problems the transmission conditions are the continuity of the unknown and its fluxes through element boundaries. The former is automatically achieved by introducing a single valued subscale on the boundaries (for the conforming approximations we consider), whereas the latter provides the effective condition for approximating these values. The final result is that the subscale on the interelement boundaries must be proportional to the jump of the flux of the finite element component and the average of the subscale calculated in the element interiors.

  Abstract
  In this paper, we introduce a way to approximate the subscales on the boundaries of the elements in a variational two-scale finite element approximation to flow problems. [...]

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