Efficient FEM Solvers for Incompressible Nonlinear Flow Models

We present special numerical techniques for non-Newtonian flows including non-isothermal, shear/pressure dependent and particularly viscoelastic effects which are based on the logconformation reformulation (LCR), including Oldroyd-B and Giesekus-type fluids as prototypical models. We utilize a fully coupled monolithic finite element approach which treats the velocity, pressure, temperature and the logarithm of the conformation stress tensor simultaneously. As a consequence, fully implicit time stepping and even direct steady approaches are possible such that pseudo-time stepping with correspondingly small time step sizes, which typically depend by stability reasons on the spatial mesh size, can be avoided. Based on an accurate FEM discretization with consistent edge-oriented stabilization techniques for the convective operators, the corresponding nonlinear systems for velocity, pressure, temperature and conformation stress tensor are treated by a discrete Newton method, and local grid refinement is applied to reduce the computational efforts and to increase the accuracy. Then, a special geometrical multigrid solver with modified Vanka smoother is used for the resulting linear subproblems to maintain high accuracy and robustness, particularly w.r.t. different rheological behaviour but also regarding different problem sizes and type of nonlinearity. The presented methodology is analyzed for the well-known `flow around cylinder' problem and other prototypical flow configurations of benchmarking character. Similar to other authors, simulation results for Oldroyd-B fluids can be obtained with the LCR approach for a wide range of Weissenberg (We) numbers. The merit of our approach is that we can obtain the discrete approximations by a direct steady approach with a numerical effort which is rather independent of the examined We numbers. Moreover, the same `black box' techniques can be applied to Giesekus flow models which seem to lead to acceptable approximations for a much higher range of We numbers, hereby showing the same advantageous numerical convergence behaviour.