Abstract

In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf-sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. The motivation of the method is fully described. It is also shown that this formulation is table and optimally convergent for an adequate choice of the algorithmic on which the method depends.

Abstract

In this paper we present a stabilized finite element formulation to solve the Oseen equations as a model problem involving both convection effects and the incompressibility restriction. The need for stabilization techniques to solve this problem arises because of the restriction in the possible choices for the velocity and pressure spaces dictated by the inf–sup condition, as well as the instabilities encountered when convection is dominant. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the assumption that unresolved subscales are orthogonal to the finite element space. It is shown that this formulation is stable and optimally convergent for an adequate choice of the algorithmic parameters on which the method depends.

Abstract

In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.

Abstract

The treatment of (near-)incompressibility is a major concern for the simulation of rubber-like parts, or forming processes. The use of mixed finite element methods is known to prevent the locking of the F.E. approximation in the incompressible limit. However, the stability of these formulations is conditionned by the fullfilment of the inf-sup condition. Recently, finite elements method has evolved with the introduction of the partition of unity. The X-FEM uses it to remove the need to mesh (and remesh) physical surfaces. In this paper, a strategy is proposed for the treatment of holes within X-FEM in the incompressible setting. Numerical examples show that F.E. convergence rate is preserved and that the inf-sup condition is passed.