In this talk we shall discuss the monolithic Newton multigrid FEM approach
for the simulation of general nonlinear incompressible flow problems with
complex rheology. The governing equations arise from model problems with nonisothermal
behavior, pressure and shear dependent viscosity in viscoelastic fluids.
The well-known HWNP is overcome with LCR formulation wich guarantees the
positive definiteness of the discrete conformation tensor. The coupling between the
velocity gradient and the elastic stress, which leads to the restriction for the choice
of FEM approximation spaces, and the hyperbolic nature of the problem are handled
with edge-oriented stabilization. The nonlinearity is treated with Newton-type
solver for nonregular problems taking into account the special properties of the partial
operators which arise due to the differentiation of the corresponding nonlinear
viscosity function. The resulting linearized system inside of the outer Newton
solver is a typical nonsymmetric saddle point problem is solved using the geometrical
multigrid with a Vanka-like smoother. Based on the existing software packages
for the numerical simulation of complex fluid flows (FeatFlow), the new numerical
methods and algorithmic tools have been used to deal with the challenging models,
as for instance, granular material, particulate flow, or viscoelastic fluid models.
We present some realistic examples of nonlinear fluids which are modeled by
non-Newtonian models in complex geometry to illustrate some of these numerical
techniques
Key words: Monolitic, Viscoelastic flow, LCR reformulation, Edge-Oriented
stabilization, Finite Element Method, Newton method, multigrid solver, Vanka
smoother