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• Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations

  R. Codina, J. Blasco

  Numerische Mathematik (2000). Vol. 87 (1), pp. 59-81

  
  

  Abstract

  The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the L2 norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this.

  Abstract
  The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. [...]

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• Fourier analysis of an equal‐order incompressible flow solver stabilized by pressure gradient projection

  G. Buscaglia, F. Basombrío, R. Codina

  Int. J. Numer. Meth. Fluids (2000). Vol. 34 (1), pp. 65-92

  
  

  Abstract

  Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes equations, here denoted by pressure gradient projection (SPGP) method. The analysis is motivated by spurious waves that pollute the computed pressure in start‐up flow simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behaviour of stabilized methods with remarkable accuracy, even though the original Navier–Stokes setting must be significantly simplified to apply them. In the steady state case, good estimates for the stabilization parameters are obtained. In the transient case, spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though this implies additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons with Galerkin–least‐squares (GLS) method and Chorin's projection method are also addressed

  Abstract
  Fourier analysis techniques are applied to the stabilized finite element method (FEM) recently proposed by Codina and Blasco for the approximation of the incompressible Navier–Stokes [...]

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• A nodal-based implementation of a stabilized Finite Element Method for Incompressible Flow Problems

  R. Codina

  Int. J. Numer. Meth. Fluids (2000). Vol. 33 (5), pp. 737-766

  
  

  Abstract

  The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are conducted to check its performance. This method is able to deal with all the instabilities that the standard Galerkin method presents, namely, the pressure instability, the instability arising in convection dominated situations and also the less popular instabilities found when the Navier-Stokes equations have a dominant Coriolis force, or there is a dominant absorption term arising from the small permeability of the medium where the flow takes place. The second objective is to describe a nodal-based implementation of the finite element formulation introduced. This implementation is based on an a priori calculation of the integrals appearing in the formulation and then the construction of the matrix approximations, this matrix and this vector can be constructed directly for each nodal point, without the need to loop over element and thus making the calculations much faster. In order to be able to do this, all the variables have to be defined at the nodes of the finite element mesh, not on the elements. This is also so for the stabilization parameters of the formulation. However, doing this given rise to questions regarding the consistency and the conservation properties of the final scheme that are addressed in this paper.

  Abstract
  The objective of this paper is twofold. First, a stabilized finite element method for the incompressible Navier-Stokes is presented, and several numerical experiments are [...]

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• On stabilized finite element methods for linear systems of convection-diffusion-reaction equations

  R. Codina

  Comput. Methods Appl. Mech. Engrg., (2000). Vol. 188 (200), pp. 61-82

  
  

  Abstract

  A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a 1D model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner-Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.

  Abstract
  A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach [...]

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• Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection

  R. Codina, J. Blasco

  Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3-4), pp. 277-300

  
  

  Abstract

  In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the Laplacian of the pressure and the divergence of this new vector field. This leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures. In the case of the transient equations, we consider the possibility of treating the pressure gradient projection either implicitly or explicity. In the first case, the number of unknowns of the problem is substantially increased with respect to the standard Galerkin formulation. Nevertheless, iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns (velocity and pressure). When this vector field is treated explicitly, the increment of computational cost of the stabilized formulation with respect to the Galerkin method is very low. We provide a stability estimate for the case of the simple backward Euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection.

  Abstract
  In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown [...]

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• Finite element simulation of the filling of thin moulds

  R. Codina, O. Soto

  Int. J. Numer. Meth. Engng. (1999). Vol. 46 (9), pp. 1559-1573

  
  

  Abstract

  In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, such as the Q2/P1 element (tri‐quadratic velocities and piecewise linear, discontinuous pressures) or stabilized finite element formulations. The tracking of the free surface is based on the Volume‐Of‐Fluid (VOF) method. The velocity profile is assumed to be parabolic in the direction normal to the mid‐plane, so that one element along the width of the mould is enough to reproduce this profile if this element is quadratic. The velocity is prescribed to zero on the upper and lower surfaces and the normal to the mid‐plane is also prescribed to zero. In the case of div‐stable elements, the pressure profile is prescribed to be constant along the width of the mould. This can be achieved by using as interpolation degrees of freedom the pressure values at the element centroid and its derivatives in the directions tangent and normal to the mid‐plane, and prescribing the latter to zero. No modifications are needed when stabilized formulations are employed. To advance in time the function used in the VOF technique, we use a constant velocity across the width of the mould, which is taken as the projection on the tangent plane of each element of the nodal velocities. This is needed in order to have mass conservation.

  Abstract
  In this work we present a finite element formulation to simulate the filling of thin moulds. The model has as starting point a 3‐D approach using either div‐stable elements, [...]

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• The characteristic‐based‐split procedure: an efficient and accurate algorithm for fluid problems

  O. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vázquez, P. Ortiz

  Int. J. Numer. Meth. Fluids (1999). Vol. 31 (1), pp. 359-392

  
  

  Abstract

  In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self‐adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi‐implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic‐based‐split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised.

  Abstract
  In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them [...]

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• Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows

  R. Codina, O. Soto

  Int. J. Numer. Meth. Fluids (1999). Vol. 30 (3), 309-334

  
  

  Abstract

  The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution.

  Abstract
  The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε [...]

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• A fractional-step finite-element method for the Navier–Stokes equations applied to magma-chamber withdrawal

  R. Folch, M. Vázquez, R. Codina, J. Martí i Castell

  Computers and Geotechnics (1999). Vol. 25 (3), pp. 263-275

  
  

  Abstract

  We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows the use of equal interpolation spaces for the pressure and velocity fields and can be used to solve both compressible and incompressible flows. The standard Galerkin method is used to space-discretize the equations in which convective terms are dominant because the equations are reformulated in a characteristics co-moving frame providing thus the required artificial diffusion in a consistent way. The algorithm depends on four different parameters. Depending on their values, a fully implicit, a semi-implicit or a fully explicit solution can be obtained. The proposed algorithm may be useful in solving a wide spectrum of problems in geology, engineering and geosciences. Several flows frequently encountered in practical applications such as incompressible, slightly compressible or perfect gas are taken into account. As an example, we use it to model the withdrawal of magma from shallow chambers during explosive volcanic eruptions. Our model constitutes a first attempt to characterize the temporal evolution of the most revelant physical parameters during such a process.

  Abstract
  We develop an algorithm to solve numerically the Navier–Stokes equations using a finite element method. The algorithm uses a fractional step approach that allows [...]

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• Numerical solution of the incompressible Navier–Stokes equations with Coriolis forces based on the discretization of the total time derivative.

  R. Codina

  Journal of Computational Physics (1999). Vol. 148 (2), pp. 467-496

  
  

  Abstract

  In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approximation is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the characteristic Galerkin technique for convection-dominated flows.

  Abstract
  In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on [...]

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