Computation of Die Swell behind a Complex Profile Extrusion Die Using a Stabilized Finite Element Method for Various Thermoplastic Polymers
Final Report Abstract
The design of plastics-extrusion dies continues to rely on both experimental studies and the experience of the die designer. Trial-and-error procedures are an integral part of the process. This is mostly due to the lack of intuition with respect to the viscoelastic flow behavior of plastic melts. In this respect, numerical simulations can assist the engineer in the design process. At the current state of the art, it is still a challenge to include the effect of die swell into such simulations. Aspects like instability and high computational effort prevent an easy usability. The finite element method being one of the most important numerical methods, it is worthwhile to be able to solve the Navier-Stokes equations together with a viscoelastic material model on deforming domains in such a framework, with the aim of predicting die swell. Ever since the first simulation attempts in the seventies, the simulations suffered from the so-called the high-Weissenberg number problem: The simulations only converged to a solution if the viscoelasticity was under a certain bound. This bound was much smaller than what experiments suggested. Even worse, these bounds became most often smaller with increased numerical accuracy (e.g. finer meshes). As a result of the project, a novel formulation for all Oldroyd-type viscoelastic models was developed. This new model, formulated in log-conformation form, is immediately discretizable with a variety of numerical methods. In particular, it allows for fully-implicit coupling with the Navier-Stokes equations, meaning that the new constitutive equation can be combined into one single non-linear system of equations together with the Navier-Stokes equations. This means that it is now amenable to a non-linear solver of the users choice, e.g., the Newton-Raphson algorithm. Detailed analysis of the new model was provided. In particular, the existance of a solution for the Giesekus model, discretized with space-time finite elements could be proven for fixed grids. This is a significant first step with respect to overcoming the high-Weissenberg number problem. Furthermore, results regarding the inf-sup stability condition with respect to the chosen finite element interpolation functions were obtained. The numerical methods were applied to the case of an air bubble rising in a viscoelastic drop. The respective simulation results are available to the public under https://zenodo.org/record/1324421#.W-LVei2ZN-U .
Publications
- The fully-implicit log-conformation formulation and its application to three-dimensional flows. Journal of Non-Newtonian Fluid Mechanics 223, 209–220 (2015)
Knechtges, P.
(See online at https://doi.org/10.1016/j.jnnfm.2015.07.004) - Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE. Computers & Mathematics with Applications, 72(11), 2808–2822 (2016)
Zwicke, F., Knechtges, P., Behr, M., Elgeti, S.
(See online at https://doi.org/10.1016/j.camwa.2016.10.010) - Space-time NURBS-enhanced finite elements for free-surface flows in 2D. International Journal for Numerical Methods in Fluids 81, 426–450 (2016)
Stavrev, A., Knechtges, P., Elgeti, S. Huerta, A.
(See online at https://doi.org/10.1002/fld.4189) - An ultraweak DPG method for viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics, 247, 107–122 (2017)
Keith, B., Knechtges, P., Roberts, N. V., Elgeti, S., Behr, M., Demkowicz, L.
(See online at https://doi.org/10.1016/j.jnnfm.2017.06.006) - Simulation of Viscoelastic Free-Surface Flows PhD thesis (RWTH Aachen, 2018)
Knechtges, P.
(See online at https://doi.org/10.18154/RWTH-2018-229719)
