Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection

By Ramon Codina, J. Blasco. 2000

179 citations recorded by the source corpus.

Abstract In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the Laplacian of the pressure and the divergence of this new vector field. This leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures. In the case of the transient equations, we consider the possibility of treating the pressure gradient projection either implicitly or explicity. In the first case, the number of unknowns of the problem is substantially increased with respect to the standard Galerkin formulation. Nevertheless, iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns (velocity and pressure). When this vector field is treated explicitly, the increment of computational cost of the stabilized formulation with respect to the Galerkin method is very low. We provide a stability estimate for the case of the simple backward Euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/76d0b6f712f43d598b36189df5adb1f8d5733c7c

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Citations in the corpus

A stabilized finite element method for the Stokes problem based on polynomial pressure projections (2004, methodology): “The pressure-Poisson [10], [5], the Douglas-Wang [7], and the pressure gradient projection method [13] are all examples of stabilized methods that are not associated with an optimization setting.”
Least-Squares Finite Element Methods (2009, background): “Notable among these efforts have been penalty and stabilized FEMs, e.g., for the Stokes problem, stabilized FEMs [4]–[5], [ 6 ], [12], [15], [29], [30], [36], [37], [44] recover advantages 2 and 3 but fail to recover advantage 4c and often lose advantage 4b.”
Mixed linear/linear simplicial elements for incompressible elasticity and plasticity (2003, background): “In [19,20], the concept of orthogonal sub-grid scales (OSGS) was introduced, which leads to well sustained and better performing stabilization procedures, see [21,22].”
A residual correction method based on finite calculus (2003, background): “…is introduced and shown to give alternative solution, including one which leads to the Brezzi-Pitkäranta (Brezzi and Pitkäranta, 1984) or Galerkin least squares (Hughes et al., 1989) result of adding a Laplacian term as well as an orthogonal sub-scale form (Codina, 2000; Codina and Blasco, 2000).”
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]).”
Stabilization of Low-order Mixed Finite Elements for the Stokes Equations (2004, methodology): “One example, motivated by fractional step algorithms for time-dependent problems, is the pressure gradient projection (PGP) method (see [6, 7, 13]) and the related local pressure gradient stabilization (LPS) method [4].”
Finite calculus formulation for incompressible solids using linear triangles and tetrahedra (2004, methodology): “This method was first proposed by Codina [15, 16] for the stabilization of convection and incompressibility effects in fluid flow problems.”
Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows (2001, methodology): “In general, it can be taken close to the critical time step of the explicit time integration of (5) and (6) (see [22, 24 ] for further discussion).”