Stabilized methods and post-processing techniques for miscible displacements

Coutinho, Álvaro L. G. A.; Dias, Claudia Mazza; Alves, José L. D.; Landau, Luiz; Loula, Abimael F. D.; Malta, Sandra M. C.; Castro, Rigoberto Gregorio Sanabria; Garcia, E.L.M.

doi:10.1016/j.cma.2003.12.031

Cited by 22 publications

(14 citation statements)

References 19 publications

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“…In our case, where time-dependent behaviour is of interest and diffusion is very small, formulations employing (17) and (16) are, for all practical purposes, equivalent. Numerical examples presented in the next section indicate that omitting the time-derivative term from the definition of Z for time-dependent problems leads to discontinuities that are overly diffuse.…”

Section: Discrete Formulation and Discontinuity Capturingmentioning

confidence: 99%

“…In the absence of the source term, and because one typically employs discontinuity capturing for very small or zero physical diffusion, definition (17) reduces to (15) for the steady problem, and to (16) for the time-dependent case. In our case, where time-dependent behaviour is of interest and diffusion is very small, formulations employing (17) and (16) are, for all practical purposes, equivalent.…”

Section: Discrete Formulation and Discontinuity Capturingmentioning

confidence: 99%

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YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery

Bazilevs

Calo

Tezduyar

et al. 2007

Numerical Methods in Fluids

142

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SUMMARYThe YZ discontinuity-capturing operator, recently introduced in (Encyclopedia of Computational Mechanics, Vol. 3, Fluids. Wiley: New York, 2004) in the context of compressible flows, is applied to a time-dependent, scalar advection-diffusion equation with the purpose of modelling drug delivery processes in blood vessels. The formulation is recast in a residual-based form, which reduces to the previously proposed formulation in the limit of zero diffusion and source term. The NURBS-based isogeometric analysis method, proposed by Hughes et al. (Comput. Methods Appl. Mech. Eng. 2005; 194:4135-4195), was used for the numerical tests. Effects of various parameters in the definition of the YZ operator are examined on a model problem and the better performer is singled out. While for low-order B-spline functions discontinuity capturing is necessary to improve solution quality, we find that high-order, highcontinuity B-spline discretizations produce sharp, nearly monotone layers without the aid of discontinuity capturing. Finally, we successfully apply the YZ approach to the simulation of drug delivery in patientspecific coronary arteries.

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Section: Discrete Formulation and Discontinuity Capturingmentioning

confidence: 99%

Section: Discrete Formulation and Discontinuity Capturingmentioning

confidence: 99%

YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery

Bazilevs

Calo

Tezduyar

et al. 2007

Numerical Methods in Fluids

142

View full text Add to dashboard Cite

show abstract

“…For a summary of the early literature on stabilized methods see Brooks and Hughes [10]. Recent work on stabilized methods is presented in [1,8,9,11,14,[18][19][20][21]28,33,[41][42][43].…”

Section: Introductionmentioning

confidence: 99%

The role of continuity in residual-based variational multiscale modeling of turbulence

et al. 2007

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This paper examines the role of continuity of the basis in the computation of turbulent flows. We compare standard finite elements and non-uniform rational B-splines (NURBS) discretizations that are employed in Isogeometric Analysis (Hughes et al. in Comput Methods Appl Mech Eng, 194:4135-4195, 2005). We make use of quadratic discretizations that are C 0 -continuous across element boundaries in standard finite elements, and We also find that the effect of continuity is greater for higher Reynolds number flows.

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“…Improved post-processing techniques that reduce these problems can be found e.g. in [15,17]. This approach has been restricted to continuous (H 1 -conforming) pressure FE spaces.…”

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confidence: 99%

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

Badia

Codina

2010

Computer Methods in Applied Mechanics and Engineering

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Abstract. We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic the different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L 2 -norm.Key words. Darcy's problem, stabilized finite element methods, characteristic length scale, orthogonal subgrid scales AMS subject classifications. 65N30, 35Q301. Introduction. Darcy's problem governs the flow of an incompressible fluid through a porous medium. It is composed by the Darcy law that relates the fluid velocity (the flux) and the pressure gradient and the mass conservation equation. In flow in porous media, a proper functional setting for this problem is to consider the flux in H(div, Ω) and the pressure in L 2 (Ω). This yields a saddle-point problem that is well posed due to inf-sup conditions known to hold at the continuous level, and that allow one to obtain stability estimates for the pressure and the velocity divergence.The Galerkin approximation of this indefinite system is a difficult task, because the continuous inf-sup conditions are not naturally inherited by most finite element (FE) velocitypressure spaces. We can avoid these problems by invoking the Darcy law in the mass conservation equation, getting a pressure Poisson problem; this is an elliptic problem that can be easily approximated by the Galerkin technique and Lagrangian elements. The fluxes can be obtained as a postprocess by using a L 2 -projection. This approach is computationally appealing because pressure and velocity computations are decoupled and the implementation is easy. Unfortunately, this approach has two drawbacks: the loss of accuracy for the velocity and the very weak enforcement of the mass conservation equation. Improved post-processing techniques that reduce these problems can be found e.g. in [15,17]. This approach has been restricted to continuous (H 1 -conforming) pressure FE spaces. However, the continuous pressure admits discontinuities, e.g. in regions with jumps of the physical properties (conductivity), and this approach leads to poor accuracy in the vicinity of these regions.Th...

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Stabilized methods and post-processing techniques for miscible displacements

Cited by 22 publications

References 19 publications

YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery

YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery

The role of continuity in residual-based variational multiscale modeling of turbulence

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

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