Saturation Equation

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Koffi B Fadimba - One of the best experts on this subject based on the ideXlab platform.

  • a linear backward euler scheme for a class of degenerate advection diffusion Equations a mathematical analysis of the convergence in l 0 t0 l2 ω and in l2 0 t0 h1 ω
    Analysis and Applications, 2014
    Co-Authors: Koffi B Fadimba
    Abstract:

    This paper concerns itself with establishing convergence estimates for a linearized scheme for solving numerically the Saturation Equation. In a previous paper, error estimates were obtained for the same scheme in L2(0, T0;L2(Ω)). In this work, we establish error estimates for the linear scheme in L∞(0, T0;L2(Ω)) and in L2(0, T0;H1(Ω)) (in the discrete norms). Under certain realistic conditions, we show that, if the regularization parameter β and the spatial discretization parameter h are carefully chosen in terms of the time-stepping parameter Δt, the convergence, in these spaces, is at least of order O((Δt)α) for some determined α > 0, function of a parameter μ > 0 defined in the problem. Examples of possible choices of β and h in terms of Δt are given.

  • convergence estimates for a linear backward euler scheme for the Saturation Equation
    The Open Applied Physics Journal, 2012
    Co-Authors: Koffi B Fadimba
    Abstract:

    In a previous work, stability and consistency results were established for a linearized Euler scheme for the Saturation Equation. In this paper we continue the mathematical analysis of the scheme, in preparation for its numerical treatment in a future work. We use the regularity results, obtained previously, to establish error estimates in L 2 () for the linear scheme. This work is done with the degenerate nature of the Saturation Equation in mind, but it is also valid for the non degenerate case like the concentration Equation. We show that, if the regularization parameter  and the spatial discretization parameter h are carefully chosen in terms of the time stepping parameter t , the convergence is at least of order O((t)  ) for some determined  >0 . Examples of choices of  and h are given. We also establish a new (at our knowledge) regularity result for the continuous Galerkin formulation of the Saturation Equation and a new regularity result for the linear scheme. 2000 Mathematics Subject Classification. 35Q35, 65M06, 65M15, 65M60. 0)

  • a linear backward euler scheme for the Saturation Equation regularity results and consistency
    Journal of Computational and Applied Mathematics, 2010
    Co-Authors: Koffi B Fadimba
    Abstract:

    We consider a linearization of a numerical scheme for the Saturation Equation (or porous medium Equation) @[email protected][email protected][email protected]?f(S)[email protected][email protected]?k(S)@?S=0, through first order expansions of the fractional function f and the inverse of the function K(s)[email protected]!"0^sk(@t)[email protected], after a regularization of the porous medium Equation. We establish a regularity result for the Continuous Galerkin Method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme. We then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work.

  • Error Analysis for a Galerkin Finite Element Method Applied to a Coupled Nonlinear Degenerate System of Advection-diffusion Equations
    Computational Methods in Applied Mathematics, 2006
    Co-Authors: Koffi B Fadimba
    Abstract:

    We consider a standard Galerkin Method applied to both the pressure Equation and the Saturation Equation of a coupled nonlinear system of degenerate advection-diffusion Equations modeling two-phase immiscible flow through porous media. After regularizing the problem and establishing some regularity results, we derive error estimates for a semi-discretized Galerkin Method. A decoupled nonlinear scheme is then proposed for a fully discretized (backward in time) Galerkin Method, and error estimates are derived for that method. We also prove existence and uniqueness for the nonlinear operator intervening in the backward time discretization.

  • Galerkin finite element method for a class of porous medium Equations
    Nonlinear Analysis: Real World Applications, 2004
    Co-Authors: Koffi B Fadimba, Robert Sharpley
    Abstract:

    We study the numerical approximation of the Saturation Equation which arises in the formulation of two phase fluid flow through porous media, idealized as either a convex bounded polyhedral domain or a domain with smooth boundary. This Equation is degenerate and the solutions are not guaranteed to be sufficiently smooth for direct numerical approximation. Through regularization, a family of approximate non-degenerate problems is considered along with their numerical approximations. Error estimates are established for appropriately transformed continuous Galerkin approximations, followed by corresponding error estimates for a fully discretized Galerkin method for this class of problems.

Francisco Chinesta - One of the best experts on this subject based on the ideXlab platform.

  • an efficient solver of the Saturation Equation in liquid composite molding processes
    International Journal of Material Forming, 2010
    Co-Authors: J A Garcia, Francisco Chinesta, Ll Gascon, Edu Ruiz, Ll Gascon, Edu Ruiz, François Trochu
    Abstract:

    A major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy’s law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of Saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of Saturation. The modified Saturation Equation is obtained as a result, which is a non-linear advection-diffusion Equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified Saturation Equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.

  • An efficient solver of the Saturation Equation in liquid composite molding processes
    International Journal of Material Forming, 2010
    Co-Authors: J A Garcia, Francisco Chinesta, Ll Gascon, E. Ruiz, F Trochu
    Abstract:

    International audienceA major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy's law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of Saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of Saturation. The modified Saturation Equation is obtained as a result, which is a nonlinear advection-diffusion Equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified Saturation Equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds

  • A flux limiter strategy for solving the Saturation Equation in resin transfer molding process simulation
    Composites Part A: Applied Science and Manufacturing, 2010
    Co-Authors: Juan Antonio Garcia, Ll Gascon, Ll Gascon, Francisco Chinesta
    Abstract:

    The main aim of this work is to propose a numerical procedure for solving the Saturation Equation in RTM process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model proposed here considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The model consists of an elliptic Equation governing the pressure distribution and a transport hyperbolic Equation describing the evolution of the Saturation in RTM. A global flux limiter fixed mesh strategy is proposed for solving the transport Equation with a source term. The flux limiter method has the ability to limit the extra numerical diffusion introduced by standard first-order schemes. This formulation can lead to improvements of existing RTM flow simulation codes and optimize the injection process. Preliminary numerical results are presented to validate this new approach.

  • Models for evaluating the Saturation in Liquid Composites Molding: a numerical journey
    2009
    Co-Authors: Francisco Chinesta, Ll Gascon, Ll Gascon, Juan Antonio Garcia
    Abstract:

    The main aim of this work is the proposal of a numerical procedure for solving the Saturation Equation in Resin Transfer Molding process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model here proposed considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The main numerical issues will be discussed.

  • A flux limiter strategy for solving the Saturation Equation in RTM process simulation
    2008
    Co-Authors: Juan Antonio Garcia, Francisco Chinesta, Ll Gascon, Ll Gascon, François Trochu
    Abstract:

    The main aim of this work is the proposal of a numerical procedure for solving the Saturation Equation in RTM process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model here proposed considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The model consists of an elliptic Equation governing the pressure distribution and a transport hyperbolic Equation governing the evolution of the Saturation. A global flux limiter fixed mesh strategy is proposed for solving the transport Equation with source term which describes the Saturation evolution in RTM process. The flux limiter method has the ability to limit the extra numerical diffusion introduced by standard first-order schemes. This formulation could be an appealing choice for improving RTM flow simulations and optimize injection processes. Some preliminary numerical results are presented in order to validate the proposed numerical strategy.

Ll Gascon - One of the best experts on this subject based on the ideXlab platform.

  • A two-phase flow model to simulate mold filling and Saturation in Resin Transfer Molding
    International Journal of Material Forming, 2016
    Co-Authors: Ll Gascon, Edu Ruiz, Juan Antonio Marin Garcia, Francois Lebel, F Trochu
    Abstract:

    This paper addresses the numerical simulation of void formation and transport during mold filling in Resin Transfer Molding (RTM). The Saturation Equation, based on a two-phase flow model resin/air, is coupled with Darcy's law and mass conservation to simulate the unsaturated filling flow that takes place in a RTM mold when resin is injected through the fiber bed. These Equations lead to a system composed of an advection–diffusion Equation for Saturation including cap-illary effects and an elliptic Equation for pressure taking into account the effect of air residual Saturation. The model intro-duces the relative permeability as a function of resin satura-tion. When capillary effects are omitted, the hyperbolic nature of the Saturation Equation and its strong coupling with Darcy Equation through relative permeability represent a challenging numerical issue. The combination of the constitutive physical laws relating permeability to Saturation with the coupled sys-tem of the pressure and Saturation Equations allows predicting the Saturation profiles. The model was validated by compari-son with experimental data obtained for a fiberglass reinforce-ment injected in a RTM mold at constant flow rate. The satu-ration measured as a function of time during the resin impreg-nation of the fiber bed compared very well with numerical predictions.

  • an efficient solver of the Saturation Equation in liquid composite molding processes
    International Journal of Material Forming, 2010
    Co-Authors: J A Garcia, Francisco Chinesta, Ll Gascon, Edu Ruiz, Ll Gascon, Edu Ruiz, François Trochu
    Abstract:

    A major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy’s law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of Saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of Saturation. The modified Saturation Equation is obtained as a result, which is a non-linear advection-diffusion Equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified Saturation Equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.

  • An efficient solver of the Saturation Equation in liquid composite molding processes
    International Journal of Material Forming, 2010
    Co-Authors: J A Garcia, Francisco Chinesta, Ll Gascon, E. Ruiz, F Trochu
    Abstract:

    International audienceA major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy's law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of Saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of Saturation. The modified Saturation Equation is obtained as a result, which is a nonlinear advection-diffusion Equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified Saturation Equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds

  • A flux limiter strategy for solving the Saturation Equation in resin transfer molding process simulation
    Composites Part A: Applied Science and Manufacturing, 2010
    Co-Authors: Juan Antonio Garcia, Ll Gascon, Ll Gascon, Francisco Chinesta
    Abstract:

    The main aim of this work is to propose a numerical procedure for solving the Saturation Equation in RTM process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model proposed here considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The model consists of an elliptic Equation governing the pressure distribution and a transport hyperbolic Equation describing the evolution of the Saturation in RTM. A global flux limiter fixed mesh strategy is proposed for solving the transport Equation with a source term. The flux limiter method has the ability to limit the extra numerical diffusion introduced by standard first-order schemes. This formulation can lead to improvements of existing RTM flow simulation codes and optimize the injection process. Preliminary numerical results are presented to validate this new approach.

  • Models for evaluating the Saturation in Liquid Composites Molding: a numerical journey
    2009
    Co-Authors: Francisco Chinesta, Ll Gascon, Ll Gascon, Juan Antonio Garcia
    Abstract:

    The main aim of this work is the proposal of a numerical procedure for solving the Saturation Equation in Resin Transfer Molding process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model here proposed considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The main numerical issues will be discussed.

Juan Antonio Garcia - One of the best experts on this subject based on the ideXlab platform.

  • A flux limiter strategy for solving the Saturation Equation in resin transfer molding process simulation
    Composites Part A: Applied Science and Manufacturing, 2010
    Co-Authors: Juan Antonio Garcia, Ll Gascon, Ll Gascon, Francisco Chinesta
    Abstract:

    The main aim of this work is to propose a numerical procedure for solving the Saturation Equation in RTM process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model proposed here considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The model consists of an elliptic Equation governing the pressure distribution and a transport hyperbolic Equation describing the evolution of the Saturation in RTM. A global flux limiter fixed mesh strategy is proposed for solving the transport Equation with a source term. The flux limiter method has the ability to limit the extra numerical diffusion introduced by standard first-order schemes. This formulation can lead to improvements of existing RTM flow simulation codes and optimize the injection process. Preliminary numerical results are presented to validate this new approach.

  • Models for evaluating the Saturation in Liquid Composites Molding: a numerical journey
    2009
    Co-Authors: Francisco Chinesta, Ll Gascon, Ll Gascon, Juan Antonio Garcia
    Abstract:

    The main aim of this work is the proposal of a numerical procedure for solving the Saturation Equation in Resin Transfer Molding process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model here proposed considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The main numerical issues will be discussed.

  • A flux limiter strategy for solving the Saturation Equation in RTM process simulation
    2008
    Co-Authors: Juan Antonio Garcia, Francisco Chinesta, Ll Gascon, Ll Gascon, François Trochu
    Abstract:

    The main aim of this work is the proposal of a numerical procedure for solving the Saturation Equation in RTM process simulation. In order to analyze in more detail the progressive impregnation of a fibrous preform by a fluid resin, the numerical model here proposed considers the flow through a partially saturated medium, including the dependence of permeability on the Saturation degree. The model consists of an elliptic Equation governing the pressure distribution and a transport hyperbolic Equation governing the evolution of the Saturation. A global flux limiter fixed mesh strategy is proposed for solving the transport Equation with source term which describes the Saturation evolution in RTM process. The flux limiter method has the ability to limit the extra numerical diffusion introduced by standard first-order schemes. This formulation could be an appealing choice for improving RTM flow simulations and optimize injection processes. Some preliminary numerical results are presented in order to validate the proposed numerical strategy.

C. Marquet - One of the best experts on this subject based on the ideXlab platform.

  • QCD traveling waves at non-asymptotic energies
    Physics Letters B, 2005
    Co-Authors: C. Marquet, R. Peschanski, Gregory Soyez
    Abstract:

    Abstract Using consistent truncations of the BFKL kernel, we derive analytical traveling-wave solutions of the Balitsky–Kovchegov Saturation Equation for both fixed and running coupling. A universal parametrization of the “interior” of the wave front is obtained and compares well with numerical simulations of the original Balitsky–Kovchegov Equation, even at non-asymptotic energies. Using this universal parametrization, we find evidence for a traveling-wave pattern of the dipole amplitude determined from the gluon distribution extracted from deep inelastic scattering data.

  • The Balitsky–Kovchegov Equation in full momentum space
    Nuclear Physics, 2005
    Co-Authors: C. Marquet, Gregory Soyez
    Abstract:

    Abstract We analyse the Balitsky–Kovchegov (BK) Saturation Equation in momentum space and solve it numerically. We confirm that, in the limit where the transverse momentum of the incident particle k is much bigger than the momentum transfer q, the Equation admits traveling-wave solutions. We extract the q dependence of the Saturation scale Q s ( Y ) and verify that Q s ( Y = c s t e ) scales as max ( q , Q T ) , where Q T is the scale caracterizing the target.

  • The Balitsky–Kovchegov Equation in full momentum space
    Nuclear Physics A, 2005
    Co-Authors: C. Marquet, G. Soyez
    Abstract:

    We analyse the Balitsky-Kovchegov (BK) Saturation Equation in momentum space and solve it numerically. We confirm that, in the limit where the transverse momentum of the incident particle k is much bigger than the momentum transfer q, the Equation admits travelling-wave solutions. We extract the q dependence of the Saturation scale Q_s(Y) and verify that Q_s(Y=cste) scales as max(q,Q_T), where Q_T is the scale caracterizing the target.Comment: 11 pages, 7 figures, minor corrections, version appeared in Nucl. Phys.

  • QCD traveling waves at non-asymptotic energies
    Physics Letters B, 2005
    Co-Authors: C. Marquet, R. Peschanski, G. Soyez
    Abstract:

    Using consistent truncations of the BFKL kernel, we derive analytical traveling-wave solutions of the Balitsky-Kovchegov Saturation Equation for both fixed and running coupling. A universal parametrization of the ``interior'' of the wave front is obtained and compares well with numerical simulations of the original Balitsky-Kovchegov Equation, even at non-asymptotic energies. Using this universal parametrization, we find evidence for a traveling-wave pattern of the dipole amplitude determined from the gluon distribution extracted from deep inelastic scattering data.Comment: 10 pages, 5 figures, minor revision, version to appear in PL

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