A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations

Shakib, Farzin; Hughes, Thomas J.R.; Johan, Zdeněk

doi:10.1016/0045-7825(91)90041-4

Cited by 548 publications

(436 citation statements)

References 23 publications

Supporting

Mentioning

423

Contrasting

Unclassified

Order By: Relevance

“…The dual estimatesq = nnp i=1q i andr = nnp i=1r i , whereq i andr i verify the local problems given in (22), verify the hypothesis of theorem 2 and therefore…”

Section: Theoremmentioning

confidence: 99%

“…The appropiate selection of this parameter is not discussed here since the primary goal of this work is to show the performance of the error estimation strategy. Thus, the stabilization parameter is chosen following [22]. However, the error estimation procedure is valid for any choice of the definition of the stabilization parameter, see for instance [23,24,25,26].…”

Section: Numerical Examplesmentioning

confidence: 99%

See 1 more Smart Citation

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

show abstract

“…The dual estimatesq = nnp i=1q i andr = nnp i=1r i , whereq i andr i verify the local problems given in (22), verify the hypothesis of theorem 2 and therefore…”

Section: Theoremmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Mariné

Dı́ez

Huerta

2012

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…Considering that the fluid (water body) is incompressible, of the Newtonian type and in hydrostatic equilibrium, so the variations of density ρ are not significant (SHAKIB et al, 1991).…”

Section: Hydrodynamic Modelmentioning

confidence: 99%

A transport modeling of the carbon-nitrogen cycle at Igapó I Lake - Londrina, Paraná State, Brazil

et al. 2012

View full text Add to dashboard Cite

ABSTRACT. This work is a contribution to a better understanding of the effect that domestic sewage discharges may cause in a water body, specifically at Igapó I Lake, in Londrina, Paraná State, Brazil. The simulation of the dynamics of pollutant concentrations throughout the water body was conducted by means of structured discretization of the geometry of Igapó I Lake, together with the finite differences and the finite elements methods. Firstly, the hydrodynamic flow (without the pollutants), modeled by NavierStokes and pressure equations, was numerically resolved by the finite differences method, and associated with the fourth order Runge-Kutta procedure. After that, by using the hydrodynamic field velocity, the flow of the reactive species (pollutants) was described through a reaction transport model, restricted to the carbon-nitrogen cycle. The reaction transport model was numerically resolved by the stabilized finite elements method, by means of a semi-discrete formulation. A qualitative analysis of the numerical simulations provided a better understanding of the dynamics of the processes involved in the flow of the reactive species, such as the dynamics of the nitrification process, of the biochemical demand of oxygen and of the level of oxygen dissolved in the water body at Igapó I Lake.Keywords: water quality model, pollutants, numerical simulation, finite differences method, finite elements method.Uma modelagem do transporte do ciclo carbono-nitrogênio no Lago Igapó I -Londrina, Estado do Paraná, Brasil RESUMO. Este artigo é uma contribuição para um melhor entendimento do efeito que uma descarga de esgoto doméstico pode causar num corpo d'água, em particular, no lago Igapó I, Londrina, Paraná, Brasil. A simulação da dinâmica das concentrações dos poluentes em todo o corpo d'água é realizada por meio de uma discretização estruturada da geometria do lago Igapó I, juntamente com os métodos de diferenças finitas e de elementos finitos. Primeiramente, o escoamento hidrodinâmico (sem os poluentes), modelado pelas equações de Navier-Stokes e de pressão, é resolvido numericamente pelo método de diferenças finitas, associado ao procedimento de Runge-Kutta de quarta ordem. Em seguida, utilizando o campo de velocidades fornecido pelo modelo hidrodinâmico, descreve-se o escoamento de espécies reativas (restrito ao ciclo carbono-nitrogênio), por meio de um modelo de transporte de poluentes com reações. O modelo de transporte de poluentes com reações é resolvido numericamente pelo método de elementos finitos estabilizados, através de uma formulação semi-discreta. Uma análise qualitativa das simulações numéricas proporcionou uma melhor compreensão da dinâmica dos processos envolvidos no escoamento de espécies reativas, tais como a dinâmica do processo de nitrificação, da demanda bioquímica de oxigênio e do nível de oxigênio dissolvido no corpo d'água do lago Igapó I.Palavras-chave: modelo de qualidade de água, poluentes, simulação numérica, método de diferenças finitas, método de elementos finitos.

show abstract

“…The SUPG method has been widely used by several authors for compressible flows [1,6,7,40,47]. Other popular stabilization techniques for compressible flows were based on the Galerkin least squares (GLS) method [8,15,46] which coincides with the original SUPG method under some specified conditions. Using of the fractional step concept [9,10], Zienkiewicz and co-workers introduced the characteristic-based split (CBS) method [11,[50][51][52], which benefited from the anisotropic shock capturing term presented by Codina [12].…”

Section: Introductionmentioning

confidence: 99%

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Kouhi

Oñate

2015

Comput Mech

View full text Add to dashboard Cite

A new implicit stabilized formulation for the numerical solution of the compressible NavierStokes equations is presented. The method is based on the Finite Calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors [24]. In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.

show abstract

A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations

Cited by 548 publications

References 23 publications

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

A transport modeling of the carbon-nitrogen cycle at Igapó I Lake - Londrina, Paraná State, Brazil

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Contact Info

Product

Resources

About