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• Interpolation with restrictions between finite element meshes for flow problems in an ALE setting

  A. Pont, R. Codina, J. Baiges

  Comput. Methods Appl. Mech. Engrg., (2017). Vol. 110 (13), pp. 1203-1226

  
  

  Abstract

  The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission of information between finite element meshes. Because the Lagrangian projection of results from one mesh to another is a dissipative method, a new conservative interpolation method has been developed. A series of constraints, such as the conservation of mass or energy, are applied to the interpolated arrays through Lagrange multipliers in an error minimization problem, so that the resulting array satisfies these physical properties while staying as close as possible to the original interpolated values in the L2 norm. Unlike other conservative interpolation methods that require a considerable effort in mesh generation and modification, the proposed formulation is mesh independent and is only based on the physical properties of the field being interpolated. Moreover, the performed corrections are neither coupled with the main calculation nor with the interpolation itself, for which reason the computational cost is very low.

  Abstract
  The need for remeshing when computing flow problems in domains suffering large deformations has motivated the implementation of a tool that allows the proper transmission [...]

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• Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations

  E. Castillo, R. Codina

  Mathematical Models and Methods in Applied Sciences (2017). Vol. 51 (4), pp. 1407-1427

  
  

  Abstract

  In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.

  Abstract
  In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field [...]

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• An adaptive Fixed-Mesh ALE method for free surface flows

  J. Baiges, R. Codina, A. Pont, E. Castillo

  Comput. Methods Appl. Mech. Engrg., (2017). Vol. 313, pp. 159-188

  
  

  Abstract

  In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background domain. This mesh is successively refined and unrefined at each time step in order to focus the computational effort on the spatial regions where it is required. Some of the main ingredients of the formulation are the use of an Arbitrary-Lagrangian–Eulerian formulation for computing temporal derivatives, the use of stabilization terms for stabilizing convection, stabilizing the lack of compatibility between velocity and pressure interpolation spaces, and stabilizing the ill-conditioning introduced by the cuts on the background finite element mesh, and the coupling of the algorithm with an adaptive mesh refinement procedure suitable for running on distributed memory environments. Algorithmic steps for the projection between meshes are presented together with the algebraic fractional step approach used for improving the condition number of the linear systems to be solved. The method is tested in several numerical examples. The expected convergence rates both in space and time are observed. Smooth solution fields for both velocity and pressure are obtained (as a result of the contribution of the stabilization terms). Finally, a good agreement between the numerical results and the reference experimental data is obtained.

  Abstract
  In this work we present a Fixed-Mesh ALE method for the numerical simulation of free surface flows capable of using an adaptive finite element mesh covering a background [...]

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• Finite element approximation of the viscoelastic flow problem: A non-residual based stabilized formulation

  E. Castillo, R. Codina

  Computers & Fluids (2017). Vol. 142, pp. 72-78

  
  

  Abstract

  In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns σ−u−p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.

  Abstract
  In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based [...]

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• A pseudo-compressible variational multiscale solver for turbulent incompressible flows

  L. Yang, S. Badia, R. Codina

  Computational Mechanics (2016). Vol. 58 (6), pp. 1051-1069

  
  

  Abstract

  In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with 5123cells on a 12 cores 64 GB ram machine.

  Abstract
  In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. [...]

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• Residual‐based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics

  O. Guasch, P. Sánchez-Martín, A. Pont, J. Baiges, R. Codina

  Int. J. Numer. Meth. Fluids (2016). Vol. 82, pp. 839-857

  
  

  Abstract

  The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf–sup compatibility condition can be built in the case of no mean flow, that is, for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual‐based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared‐solenoidal mean flow, and for the aeolian tone generated by flow past a two‐dimensional cylinder.

  Abstract
  The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non‐uniform mean flow. As for any hybrid computational aeroacoustics [...]

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• A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

  Ö. Türk, D. Boffi, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2016). Vol. 310, pp. 886-905

  
  

  Abstract

  In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.

  Abstract
  In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) [...]

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• Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

  C. Bayona, J. Baiges, R. Codina

  Int. J. Numer. Meth. Heat and Fluid Flows (2016). Vol. 26 (3-4), pp. 1240-1271

  
  

  Abstract

  Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Even though this formulation is relatively well known, some particular features that have been applied with great success in other flow problems are incorporated. Design/methodology/approach The orthogonal subgrid scales, the non-linear tracking of these subscales, and their time evolution are applied. Moreover, a systematic way to design the matrix of algorithmic parameters from the perspective of a Fourier analysis is given, and the adjoint of the non-linear operator including the volumetric part of the convective term is defined. Because the subgrid stabilization method works in the streamline direction, an anisotropic shock capturing method that keeps the diffusion unaltered in the direction of the streamlines, but modifies the crosswind diffusion is implemented. The artificial shock capturing diffusivity is calculated by using the orthogonal projection onto the finite element space of the gradient of the solution, instead of the common residual definition. Temporal derivatives are integrated in an explicit fashion. Findings Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. The non-linearity introduced by the anisotropic shock capturing method has less effect in the convergence behavior to the steady state. Originality/value A complete investigation of the stabilized formulation of the compressible problem is addressed.

  Abstract
  Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written [...]

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• Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics

  O. Guasch, A. Pont, J. Baiges, R. Codina

  Computers and Fluids (2016). Vol. 133, pp. 129-139

  
  

  Abstract

  The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.

  Abstract
  The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) [...]

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• A stabilized finite element method for the mixed wave equation in an ALE framework with application to diphthong production

  O. Guasch, M. Arnela, R. Codina, H. Espinoza

  Acta Acustica united with Acustica (2016). (preprint) Vol. 102, pp. 94-106

  
  

  Abstract

  Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity field. Indeed, this becomes the natural option in many problems, such as those involving waves propagating in moving domains, because the equations can easily be set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Yet, when attempting a standard Galerkin finite element solution (FEM) for them, it turns out that an inf-sup compatibility constraint has to be satisfied, which prevents from using equal interpolations for the approximated acoustic pressure and velocity fields. In this work it is proposed to resort to a subgrid scale stabilization strategy to circumvent this condition and thus facilitate code implementation. As a possible application, we address the generation of diphthongs in voice production.

  Abstract
  Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity [...]

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