Monolithic FEM multigrid techniques for the simulation of viscoelastic flow

Transcription

1 Monolithic FEM multigrid techniques for the simulation of viscoelastic flow A. Ouazzi, H. Damanik, S. Turek, J. Hron Institute of Applied Mathematics, LS III, TU Dortmund European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2009 June 29 July Uppsala University, Sweden Page 1

2 Governing equations Generalized Navier-Stokes equations Viscous stress Elastic stress Viscoelastic flow models Page 2

3 Quasi-Newtonian models Viscous stress Power law model Carreau model Schaeffer model (granular flow) Non-isothermal model Page 3

4 Constitutive models Elastic stress (Oldroyd/Maxwell/Jeffreys) Upper/Lower convective derivative Johnson Segelman terms Problems Blow up phenomena for time dependent problem High Weissenberg Number Problem (HWNP!) Page 4

5 HWNP Different highly developed models Oldroyd A/B, Maxwell A/B, Jeffreys Phan-Thien Tanner, Phan-Thien, Giesekus Different numerical methods FEM, FVM, FDM, DEVSS,DG, SUPG HWNP remains Kinetic energy for two different We numbers Zoom shows oscillation..!! Reformulation Problem reformulation Page 5

6 Conformation tensor reformulation Conformation Tensor (Oldroyd-B) (Lee & Xu) Using the identity Change of variable Conformation tensor reformulation Rate type expression Integral expression positive definite of exponential type Positivity preserving discretizations Page 6

7 LCR formulation Conformation reformulation (Fattal & Kupferman) The diagonalizing transformation Transformation and decomposition of velocity gradient The symmetric part The anti-symmetric part Page 7

8 LCR formulation Conformation reformulation (Fattal & Kupferman) New conformation tensor reformulation Log Conformation Reformulation (LCR) Change of variable Positivity preserving via LCR Page 8

9 LCR equations Page 9

10 Variational formulations Standard Navier-Stokes New non-symmetric bilinear forms due to LCR Page 10

11 Variational formulations New nonlinear tensor variational form due to LCR Energy equation with friction Source terms Page 11

12 Problem formulation Set Find such that Typical saddle point problem! Page 12

13 Compatibility conditions Compatibility condition for existance and uniqueness What about LCR! Page 13

14 Compatibility conditions for LCR The `NEW non-symmetric bilinear forms due to LCR Page 14

15 FEM Discretization FEM discretization approximations for velocity-stresstemperature-pressure Edge-oriented stabilization for (in preparation) Same finite element interpolation velocity and stress convective dominated problem Practically the convective terms are the main source of the instability! Page 15

16 Nonlinear Solver Newton with damping results in the solution of the form Inexact Newton The Jacobian matrix is approximated using finite differences Typical saddle point problem! Page 16

17 Linear solver Monolithic multgrid solver Standard geometric multigrid aproach Full restrictions and prolongations Local MPSC via Vanka-like smoother Coupled Monolithic Multigrid Solver! Page 17

18 Solvers M-FEM Multgrid solver for 4:1 contraction Stable nonlinear solver w.r.t. adaptivity Stable multigrid solver w.r.t. adaptivity More investigation w.r.t discrete jacobian evaluation and stabilization Q: Optimal way to apply Newton! Page 18

19 Viscoelastic benchmark Driven cavity (Oldroyd-B) Streameline for, and With increasing of We Shift of streameline to the right Increases of bottom right vortex Decreases of bottom left vortex Direct steady simulation for non-steady derived formulation! Page 19

20 Viscoelastic benchmark 4:1 contraction (Oldroyd-B) Lip vortex growth Page 20

21 Viscoelastic benchmark Planar flow around cylinder (Oldroyd-B) Half Domain Vs. Full Domain Only with LCR & next EO-FEM we hope that HWNP will be history! Page 21

22 Viscoelastic benchmark Axial stress w.r.t. X-curved for half domain vs. full Domain (HD/ FD) (Oldroyd-B) H.D. vs. F.D. H.D. vs. F.D. Q: is it appropriate to impose the symmetric condition for higher We number! Page 22

23 Summary New numerical and algorithmic tools are available using Monolithic Finite Element Method (M-FEM) Log Conformation Reformulation (LCR) Edge Oriented stabilization (EO-FEM) Fast Multigrid Solver with local MPSC smoother for the simulation of viscoelastic flow Advantages No CFL-condition restriction due to the fully coupling Positivity preserving Large order and local adaptivity Page 23