An alpha-adaptive approach for stabilized finite element solution of advective-diffusive problems with sharp gradients

Oñate, Eugenio; García, Julio Martínez; Idelsohn, Sergio

doi:10.1016/s0922-5382(98)80017-2

Cited by 20 publications

(38 citation statements)

References 19 publications

(30 reference statements)

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“…The underlined term in Equation (3) introduces a non-locality effect in the standard equilibrium equations (dN/dx = 0). Distance h is the characteristic length of the discrete problem and its value depends on the material properties and the parameters of the discretization method chosen (such as the grid size) [19][20][21][22][23][24][25][26]. Note that for h → 0 the standard infinitesimal form of the balance equation (dN/dx = 0) is recovered.…”

Section: Basic Concepts Of the Finite Calculus (Fic) Methodsmentioning

confidence: 99%

“…These terms are essential in order to introduce the necessary stabilization for the discrete solution of some problems using whatever numerical technique. Details of the derivation of Equation (4) for steady state and transient advective-diffusive and fluid flow problems can be found in References [19][20][21][22].…”

Section: Basic Concepts Of the Finite Calculus (Fic) Methodsmentioning

confidence: 99%

“…The FIC approach has been successfully used to derive stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems [19][20][21][22][23][24][25][26]. The same ideas were applied in Reference [27] to derive a stabilized formulation for quasi-incompressible and incompressible solids allowing the use of linear triangles and tetrahedra.…”

Section: Introductionmentioning

confidence: 99%

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Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Oñate

Rojek

Taylor

et al. 2004

Numerical Meth Engineering

112

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SUMMARYMany finite elements exhibit the so-called 'volumetric locking' in the analysis of incompressible or quasi-incompressible problems. In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi-implicit and explicit 2D and 3D non-linear transient dynamic analysis of an impact problem and a bulk forming process are presented.

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Section: Basic Concepts Of the Finite Calculus (Fic) Methodsmentioning

confidence: 99%

Section: Basic Concepts Of the Finite Calculus (Fic) Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Oñate

Rojek

Taylor

et al. 2004

Numerical Meth Engineering

112

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“…Other applications of the FIC scheme to incompressible flows and convection-diffusion problems are presented in [31][32][33][34][35].…”

Section: Fic Scheme For Navier-stokes Equationsmentioning

confidence: 99%

“…Within the family of stabilization techniques, the Finite Increment Calculus (FIC, or Finite Calculus in short) formulation has been successfully implemented for the stabilization of advectivediffusive transport and incompressible fluid flow problems by Oñate and co-workers [30][31][32][33][34][35][36][37]. In this paper, a FIC-based stabilized formulation for the numerical solution of the compressible Navier-Stokes equations is considered in the context of Galerkin FEM using an implicit scheme.…”

Section: Introductionmentioning

confidence: 99%

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

Kouhi

Oñate

2015

Comput Mech

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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.

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A general procedure for deriving stabilized space-time finite element methods for advective-diffusive problems

Oñate

Manzan

1999

Int. J. Numer. Meth. Fluids

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An alpha-adaptive approach for stabilized finite element solution of advective-diffusive problems with sharp gradients

Cited by 20 publications

References 19 publications

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

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

A general procedure for deriving stabilized space-time finite element methods for advective-diffusive problems

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