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• A numerical model to track two‐fluid interfaces based on a stabilized finite element method and the level set technique

  R. Codina, O. Soto

  Int. J. Numer. Meth. Fluids (2002). Vol. 40 (1-2), pp. 293-301

  
  

  Abstract

  The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. The stabilization technique employed allows us to use equal velocity–pressure interpolations and to deal with convection–dominated flows. The tracking of the fluid interface is based on the level set technique. A novel smoothing technique of this surface based on ideas of signal processing is presented.

  Abstract
  The objective of this work is to present a stabilized finite element formulation for transient incompressible flows and to apply it to the tracking of two‐fluid interfaces. [...]

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• Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ale framework

  S. Badia, R. Codina

  Computing and Visualization in Science (2002). Vol. 4 (3), pp. 167-174

  
  

  Abstract

  In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semi-discrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.

  Abstract
  In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, [...]

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• 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 [...]

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• 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 [...]

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• Numerical modeling of magma withdrawal during explosive caldera-forming eruptions

  A. Folch, J. Martí i Castell, R. Codina

  Journal of Geophysical Research (2001). Vol. 106 (B8), pp. 16 163-16 175

  
  

  Abstract

  We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such eruptions as a piston-like system in which the host rock is assumed to subside as a coherent rigid solid. Magma behaves as a Newtonian incompressible fluid below the exsolution level and as a compressible gas-liquid mixture above this level. We consider caldera-forming eruptions within the frame of fluid-structure interaction problems, in which the flow-governing equations are written using an arbitrary Lagrangian-Eulerian (ALE) formulation. We propose a numerical procedure to solve the ALE governing equations in the context of a finite element method. The numerical methodology is based on a staggered algorithm in which the flow and the structural equations are alternatively integrated in time by using separate solvers. The procedure also involves the use of the quasi-Laplacian method to compute the ALE mesh of the fluid and a new conservative remeshing strategy. Despite the fact that we focus the application of the procedure toward modeling caldera-forming eruptions, the numerical procedure is of general applicability. The numerical results have important geological implications in terms of magma chamber dynamics during explosive caldera-forming eruptions. Simulations predict a nearly constant velocity of caldera subsidence that strongly depends on magma viscosity. They also reproduce the characteristic eruption rates of the different phases of an explosive calderaforming eruption. Numerical results indicate that the formation of vortices beneath the ring fault, which may allow mingling and mixing of parcels of magma initially located at different depths in the chamber, is likely to occur for low-viscosity magmas. Numerical results confirm that exsolution of volatiles is an efficient mechanism to sustain explosive caldera-forming eruptions and to explain the formation of large volumes of ignimbrites.

  Abstract
  We propose a simple physical model to characterize the dynamics of magma withdrawal during the course of caldera-forming eruptions. Simplification involves considering such [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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• 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 [...]

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