A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems

Abstract

In this work, we discuss special numerical techniques for viscoelastic flow problems given in log-conformation reformulation (LCR). In particular, we consider Oldroyd-B and Giesekus-type fluids. We utilize a fully coupled monolithic finite element approach that treats all the numerical variables simultaneously. Thus, it is possible to do a direct steady approach and to avoid pseudo-time stepping with correspondingly small time step sizes in the case of a nonsteady approach. The Newton method handles the discrete nonlinear system, which results from the FEM discretization with consistent edge-oriented FEM stabilization techniques. In each nonlinear step, a direct sparse solver or a geometrical multigrid solver with special Vanka smoother deals with the resulting linear subproblems. Moreover, local grid refinement helps to reduce the computational efforts and to increase the accuracy of functional values. The merit of the presented methodology, for the well-known ‘flow around cylinder’ benchmark problem, is that we can obtain the discrete approximations by using a direct steady approach. Thus, the numerical effort can be rather independent of the examined We numbers. Furthermore, the ‘black box’ techniques can deal with any given viscoelastic models easily, hereby showing the same advantageous numerical convergence behaviour of the above mentioned fluids.