Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

By Ramon Codina. 2002

473 citations recorded by the source corpus.

Abstract In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2713cd69d61b300e170a9b0a19e29add49736e20

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, methodology): “, Galerkin least-square methods [5, 7, 17, 37], pressure projection methods [8,14,18], pressure gradient projection methods [13,30], and local projection stabilized methods [6, 39,51].”
The particle finite element method. An overview (2004, unknown): “Similar values for τi (usually τi = τ is taken) are used in other works from ad-hoc extensions of the 1D advective-diffusive problem [Codina et al. (1998); Codina and Blasco (2000); Codina (2002); Codina and Zienkiewicz (2002); Cruchaga and Oñate (1997; 1999); Donea and Huerta (2003); Franca and Fre”
Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations (2009, methodology): “Following [14], the stabilization parameter is designed through sK w h2Kðmþ jajK hKÞ (1).”
Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation (2009, methodology): “The variational multiscale method provides the theoretical foundation from which stabilization methods such as the algebraic subgrid scales [80] and the orthogonal subgrid scales [48] are derived.”
Numerical modeling of friction stir welding processes (2013, methodology): “The OSS technique, originally developed to fulfill the incompressibility condition in CFD problems (see [26,28,27]), has been exploited in the solid mechanics context to deal with elastic incompressibility and J2plasticity (isochoric) problems (see [5,17,21,22,54]).”
Finite element modeling of multi-pass welding and shaped metal deposition processes (2010, methodology): “The orthogonal sub-grid scale approach [20,21] is assumed as an attractive alternative to circumvent the Babuska-Brezzi stability condition [22].”
Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales (2008, unknown): “A heuristic justification of (9)–(12) based on a Fourier analysis of the problem of which Ũ is solution can be found in [17].”
Mixed linear/linear simplicial elements for incompressible elasticity and plasticity (2003, methodology): “In the case of higher order elements these derivatives can be neglected, leading to a method which is still consistent, but with a non-optimal rate of convergence [22].”
ON THE ORTHOGONAL SUB-GRID SCALE PRESSURE STABILIZATION IN FINITE DEFORMATION J2 PLASTICITY (2004, methodology): “In Codina [27], [28] the Orthogonal Subgrid Scales (OSGS) method was introduced, leading to a better sustained and better performing stabilization procedures.”
Modular vs. non-modular preconditioners for fluid-structure systems with large added-mass effect (2008, methodology): “where c1 and c2 are appropriate constants, justified in [10] through a Fourier analysis.”
Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows (2015, methodology): “The subscale as a time dependent variable of the problem was introduced in [16,22].”
An Object-oriented Environment for Developing Finite Element Codes for Multi-disciplinary Applications (2010, methodology): “[18] using equal order pressure-velocity elements stabilized by Orthogonal SubScales (OSS) [ 19 ], or Finite Increment Calculus (FIC) [63].”
A variational multiscale finite element method for multiphase flow in porous media (2005, methodology): “Our formulation is different from other multiscale approximations [12,14] in that: (1) the nonlinearity of the equations is retained at the time of invoking the multiscale split; and (2) the approximate solution is never reconstructed from pointwise values of the coarse- and subgrid-scale components”