Logarithmic conformation reformulation in viscoelastic flow problems approximated by a VMS-type stabilized finite element formulation

Abstract

The log-conformation reformulation, originally proposed by Fattal and Kupferman [1], allows computing incompressible viscoelastic problems with high Weissenberg numbers which are impossible to solve with the typical three-field formulation. By following this approach, in this work we develop a new stabilized finite element formulation based on the logarithmic reformulation using the Variational Multiscale (VMS) method as stabilization technique, together with a modified log-conformation formulation. Our approach follows the term-by-term stabilization proposed by Castillo and Codina [2] for the standard formulation, which is more effective when there are stress singularities. The formulation can be used when the relaxation parameter is set to zero, and permits a direct steady numerical resolution. The formulation is validated in the classical benchmark flow past a cylinder and in the well-known planar contraction 4:1, achieving very accurate, stable and mesh independent results for highly elastic fluids.

Analysis of a stabilized finite element approximation for a linearized logarithmic reformulation of the viscoelastic flow problem

“…However, in problems where the solution has strong gradients, we have found (3.7) more robust, similarly to what it is explained in [7]. For a detailed motivation and numerical experimentation using this method, see [28].…”

“…So, if = 1 and 0,min = 0, the original change of variables proposed in [14] is recovered. It is worth to remark that in the numerical experiments we have found useful to take small, so that 0 < ; this has allowed us to obtain converged solutions that we have not been able to get for = 1; see [28].…”

“…In this paper we will use a slightly different scaling of the logarithmic reformulation. In contrast with the original one, our change of variables will be non-singular when the Weissenberg number is close to zero and the flow is Newtonian [28]; a similar idea was presented in [33].…”

“…These ideas were applied and extended in [8][9][10][11] for the Navier-Stokes problem and the three-field Stokes problem considering the space of the sub-grid scales orthogonal to the FE space. The viscoelastic fluid flow problem was stabilized following a VMS framework in [6,7] and in the logarithmic formulation the method was tested in [28] for some numerical examples in which the Weissenberg number is relevant. This type of stabilization is also applied in [22,23].…”

An efficient stabilized finite element scheme for simulating viscoelastic flows

“…He 20 implemented a cell‐based smoothed FEM to spatially discretize the governing equations, with the characteristic‐based split scheme for temporal discretization. More recently, Moreno et al 21 proposed a new stabilized finite element formulation based on the variational multiscale method together with a modified LCR, which is effective in handling the problems with stress singularities and permits a direct steady numerical computation.…”