Higher order finite element techniques for viscoelastic flow problems with change of type and singularities

Abstract

The stability, accuracy and computational efficiency of higher order Galerkin ( hp-type) methods with and without separation of the extra stress tensor into purely elastic and viscous components for steady inertial flow of viscoelastic fluids under change of type conditions have been investigated. It has been shown that the hp-Galerkin technique without splitting the stress gives rise to a stable discretization with an exponential convergence rate toward the exact solution, but with an upper viscoelastic Mach number limitation for a given elasticity number. The hp-Galerkin method with the separation of the stress (EVSS/ hp-Galerkin) is also stable with an exponential convergence rate toward the exact solution and in the parameter span of this study no Mach or elasticity number limitations have been detected. In contrast, lower order Galerkin discretization based on separation of the stress (EVSS/Galerkin) does not provide a stable discretization to problems with change of type while the streamline upwind version of this technique (EVSS/SU) does. However, the EVSS/SU method has an extremely slow convergence rate toward the exact solution which makes its use for this class of problems prohibitively expensive. Based on a comparison of higher and lower methods, we have shown that higher order methods are highly efficient and effective for inertial flows with change of type and are superior to their lower order counterparts. In addition, the stability of this class of techniques for steady viscoelastic flows with singularities has been studied. These studies are based on a one-dimensional problem in which a discontinuity is imposed by a Dirac delta function. Various discretization techniques are considered and it is shown that when the extra stress tensor is fractioned into viscous and elastic components, a stable and accurate discretization is obtained. For this particular discretization the optimum performance is obtained when the velocity field is approximated with a polynomial degree of one order higher than the viscoelastic extra stress tensor. Moreover, a special discretization that utilizes a polynomial space for the extra stress tensor that is higher by one than the velocity vector produces a stable discretization when the extra stress tensor is not partitioned. However, the performance of this discretization is not as good as the one based on splitting the stresses.