Computational Fluid Mechanics : Scipedia

CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter
R. Codina, O. Zienkiewicz

Commun. Numer. Meth. Engng (2001). Vol 18 (2), pp. 99-112

Abstract
In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem.
Abstract
In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first [...]
- 18
- read
Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection
R. Codina, J. Blasco, G. Buscaglia, A. Huerta

Int. J. Numer. Meth. Fluids (2001). Vol. 37 (4), pp. 419-444

Abstract
We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure interpolations. The method consists in introducing the project of the pressure gradient and adding the difference between the pressure Laplacian and divergence of this new field to the incompressibility equations, both multiplied by suitable algorithmic parameters. The main purpose of this paper is to discuss how to deal with the new variable in the implementation of the algorithm. Obviously, it could be treated as one extra unknown, either explicitly or as a condensed variable. However, we take for granted that the only way or another. Here we discuss some iterative schemes to perform this uncoupling of the pressure gradient projection (PGP) from the calculation of the velocity and the pressure, both for stationary and the transient of the linearization loop and the iterative segregation of the PGP, whereas in the second the main dilemma concerns the explicit or implicit treatment of the PGP.
Abstract
We discuss in this paper some implementation aspects of a finite element formulation for the incompressible Navier-Stokes which allows the use of equal order velocity-pressure [...]
- 10
- read
Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations
J. Blasco, R. Codina

Applied Numerical Mathematics (2001). Vol. 38 (4), pp. 475-497

Abstract
In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the time discretization we focus on a fully implicit, monolithic scheme. We provide some error estimates for the fully discrete solution which show that the velocity is first order accurate in the time step and attains optimal order accuracy in the mesh size for the given spatial interpolation, both in the spaces L2(Ω) and H10(Ω); the pressure solution is shown to be order 12 accurate in the time step and also optimal in the mesh size. These estimates are proved assuming only a weak compatibility condition on the approximating spaces of velocity and pressure, which is satisfied by equal order interpolations.
Abstract
In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the [...]
- 5
- read
Pressure stability in fractional step finite element methods for incompressible flows
R. Codina

Journal of Computational Physics (2001). Vol. 170 (1), pp. 112-140

Abstract
The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second order scheme in which part of the extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.
Abstract
The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows. For the classical first order projection [...]
- 10
- read
Transmission conditions with constraints in finite element domain decomposition methods for flow problems
G. Houzeaux, R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol 17 (3), pp. 179-190

Abstract
This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD algorithm is based on the iterative update of the boundary conditions on the interfaces between the subregions, the so‐called transmission conditions. The transmission conditions involve the essential and natural boundary conditions of the weak form of the problem, and should ensure strong continuity of the velocity and weak continuity of the traction. As a first approach, the transmission conditions are interpolated using the classical Lagrange interpolation functions. Conservation problems might arise when two adjacent subdomains have a sensibly different mesh spacing. In order to conserve any desired quantity of interest, an interface constraining is introduced: continuity of the transmission conditions are constrained under a scalar conservation equation. An example of mass conservation illustrates the algorithm.
Abstract
This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD [...]
- 5
- read
A stabilized finite element method for generalized stationary incompressible flows
R. Codina

Comput. Methods Appl. Mech. Engrg., (2001). Vol. 190 (20-21), pp. 2681-2706

Abstract
In this paper we describe a finite element formulation for the numerical solution of the stationary Navier-Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection-diffusion-reaction equations and then we apply it to linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.
Abstract
In this paper we describe a finite element formulation for the numerical solution of the stationary Navier-Stokes equations including Coriolis forces and the permeability [...]
- 22
- read
A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation
E. Oñate

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 182 (3-4), pp. 355-370

Abstract
A stabilized finite element formulation for incompressible viscous flows is derived. The starting point are the modified Navier-Stokes equations incorporating naturally the necessary stabilization terms via a finite increment calculus (FIC) procedure. Application of the standard finite element Galerkin method to the modified differential equations leads to a stabilized discrete system of equations overcoming the numerical instabilities emanating from the advective terms and those due to the lack of compatibility between approximate velocity and pressure fields. The FIC method also provides a natural explanation for the stabilization terms appearing in all equations for both the Navier-Stokes and the simpler Stokes equations. Transient solution schemes with enhanced stabilization properties are also proposed. Finally a procedure for computing the stabilization parameters is presented.
Abstract
A stabilized finite element formulation for incompressible viscous flows is derived. The starting point are the modified Navier-Stokes equations incorporating naturally the [...]
- 20
- read
Stabilization of incompressibility and convection through orthogonal sub-scale in finite element methods
R. Codina

Comput. Methods Appl. Mech. Engrg., (2000). Vol. 190 (13-14), pp. 1579-1599

Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first of them is the stabilization thought the pressure gradient projection, which consists of adding a certain least-squares form of the difference between the pressure gradient and its L² projection onto the discrete velocity space in the variational equations of the problem. The second is a sub-grid scale method, whose stabilization effect is very similar to that of the Galerkin/least-squares method for the Stokes problem. It is shown here that the first method can also be recast in the framework of sub-grid scale method with a particular choice for the space of sub-scales. This leads to a new stabilization procedure, whose applicability to stabilize convection is also studied in this paper.
Abstract
Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper. The first [...]
- 7
- read
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. [...]
- 4
- read
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 [...]
- 3
- read
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 [...]
- 3
- read
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 [...]
- 17
- read
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 [...]
- 10
- read
Smoothed surface gradients for panel methods
J. D'Elía, M. Storti, S. Idelsohn

Advances in Engineering Software (2000). Vol. 31 (5), pp. 339-346

Abstract
A weak form to compute the dipolar and monopolar surface gradients, related to a low-order panel method, is shown. The flow problem is formulated by means of a three-dimensional potential model and the discretization is based on Morino's formulation for the perturbation velocity potential. On the body surface, this representation reduces to a boundary integral equation with the source (or monopolar) and the doublet (or dipolar) densities. The first of the two is found by application of the boundary flow condition, and the second one is the unknown over the body surface. A lower panel method is used for the analytic integrations of both the monopolar and dipolar influence coefficients. The surface velocity field is computed after solving the linear system, with a strong and a weak form of the Stokes theorem, which is oriented to fairly non-structured panel meshes. The proposed method is validated by comparing the numerical results with analytical ones for an isolated sphere and includes a prediction over a car-like configuration.
Abstract
A weak form to compute the dipolar and monopolar surface gradients, related to a low-order panel method, is shown. The flow problem is formulated by means of a three-dimensional [...]
- 6
- read
A closed form for low-order panel methods
J. D'Elía, M. Storti, S. Idelsohn

Advances in Engineering Software (2000). Vol. 31 (5), pp. 347-353

Abstract
A closed form for the computation of the dipolar and monopolar influence coefficients related to a low-order panel method is shown. The flow problem is formulated by means of a three-dimensional potential model; the method of discretization is based on the Morino formulation for the perturbation velocity potential. On the body surface this representation reduces to an integral equation with the source (or monopolar) and the doublet (or dipolar) densities. The former is found by application of the boundary condition, and the latter is the unknown over the surface of the body. The lower panel method is used for the analytical integrations of the monopolar and dipolar influence coefficients, with special attention to avoid a logarithmic singularity in the monopolar matrix when flat fairly structured meshes that are common in ship-wave calculations are used.
Abstract
A closed form for the computation of the dipolar and monopolar influence coefficients related to a low-order panel method is shown. The flow problem is formulated by means [...]
- 0
- read
A finite point method for incompressible flow problems
E. Oñate, C. Sacco, S. Idelsohn

Computing and Visualization in Science (2000). Vol. 3 (1), pp. 67-75

Abstract
A stabilized finite point method (FPM) for the meshless analysis of incompressible fluid flow problems is presented. The stabilization approach is based in the finite increment calculus (FIC) procedure developed by Oñate [14]. An enhanced fractional step procedure allowing the semi-implicit numerical solution of incompressible fluids using the FPM is described. Examples of application of the stabilized FPM to the solution of two incompressible flow problems are presented.
Abstract
A stabilized finite point method (FPM) for the meshless analysis of incompressible fluid flow problems is presented. The stabilization approach is based in the finite increment [...]
- 16
- read
Numerical aerodynamic analysis of large buildings using a finite element model with application to a telescope building
R. Codina, C. Morton, E. Oñate, O. Soto

Int. J. of Num. Meths. for Heat and Fluid Flow (2000). Vol. 10 (5-6), pp. 616-633

Abstract
Presents a numerical strategy for the aerodynamic analysis of large buildings, with an application to the simulation of the air flow within a telescope building. The finite element formulation is presented first, and then the methodology followed to obtain significant data from the calculations is described. The quality of the ventilation of the building is defined by the average residence times, and the feasibility of this ventilation by the actions created on the instruments and the general flow pattern.
Abstract
Presents a numerical strategy for the aerodynamic analysis of large buildings, with an application to the simulation of the air flow within a telescope building. The finite [...]
- 15
- read
Finite element solution of free‐surface ship‐wave problems
S. Idelsohn, E. Oñate, C. Sacco

Int. J. Numer. Meth. Engng. (1999). Vol. 45 (5), pp. 503-528

Abstract
An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface.
Abstract
An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes [...]
- 9
- read
A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces
M. Cruchaga, E. Oñate

Comput. Methods Appl. Mech. Engrg., (1999). Vol. 173, pp. 241-255

Abstract
In the present work a generalized streamline finite element formulation able to deal with incompressible flow problems is presented. In the finite element framework, this technique allows the use of equal order interpolation for the unknowns of the problem: velocity and pressure. In this context, stable and convergent solutions can be obtained without requiring tuning parameters defined outside this model. The tracking of moving surfaces is also included in the numerical model. This formulation has been checked in 21) and 3D tests.
Abstract
In the present work a generalized streamline finite element formulation able to deal with incompressible flow problems is presented. In the finite element framework, this [...]
- 15
- read
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 [...]
- 2
- read

show more

information

About this publication

How to submit

Open access

Contact

CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter

Implementation of a stabilized finite element formulation for the incompressible Navier-Stokes equations based on a pressure gradient projection

Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations

Pressure stability in fractional step finite element methods for incompressible flows

Transmission conditions with constraints in finite element domain decomposition methods for flow problems

A stabilized finite element method for generalized stationary incompressible flows

A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation

Stabilization of incompressibility and convection through orthogonal sub-scale in finite element methods

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

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

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

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

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

Smoothed surface gradients for panel methods

A closed form for low-order panel methods

A finite point method for incompressible flow problems

Numerical aerodynamic analysis of large buildings using a finite element model with application to a telescope building

Finite element solution of free‐surface ship‐wave problems

A generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces

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

information

Sign up at Scipedia and complete your profile

Click on "CREATE A DOCUMENT" in your profile page

Upload your manuscript in LaTeX or Word, or create it from scratch with our online editor

Click on "SUBMIT FOR PUBLICATION" at the manuscript page, and select your preferred journal or collection

Click on "SUBMIT FOR PUBLICATION" at the manuscript page, select your preferred collection and complete the submission form

The editorial board will ensure that the publishing requirements are adhered to, before your paper is published