Efficient FEM-multigrid solver for granular material

Transcription

1 Efficient FEM-multigrid solver for granular material S. Mandal, A. Ouazzi, S. Turek Chair for Applied Mathematics and Numerics (LSIII), TU Dortmund STW user committee meeting Enschede, 25th September, 2014

2 Contents 1 Introduction 2 Mathematical formulation 3 Numerical Results 4 Conclusion and outlook

3 Contents 1 Introduction 2 Mathematical formulation 3 Numerical Results 4 Conclusion and outlook

4 Generalized Navier-Stokes equation Conservation of mass u = 0 in Conservation of momentum Dt = p + σ + f in along with some initial and boundary conditions ρ Du Flow Rheology σ = 2η(γ II, p) γ; where γ = 1 2 ( u + ut ), and 2γ II = γ : γ = γ 2 = tr[( γ) 2 ] Nonlinear and coupled equations

5 Miscellaneous models Fluid models η(γ II, p) = η 0, for Newtonian fluid = η 0 ( γ 2 + ɛ) m 2 1, for Power law fluid Granular materials η(γ II, p) = 2p sin φ 1, Schaeffer model γ = 2p(sin φ + b cos φ γ n ) 1, Schaeffer-Tardos model γ 2 = αp 2 γ + βdp, Poliquen model δ p ρ + γ d

6 Contents 1 Introduction 2 Mathematical formulation 3 Numerical Results 4 Conclusion and outlook

7 Bilinear Form Find (u, p) X M such that 2η(γ II, p)d(u) : D(v)dx+ (u. u)vdx + p div vdx = fvdx, v X; with q div udx = 0, q M Compact form Find (u, p) X M such that L(u, p)u, v + N(u)u, v + Bp, v = fvdx, v X; }{{} Au }{{} g q, B T u = 0, q M

8 Nonlinear discrete system FEM discretization We take X h X and M h M Find (ũ, p) X h M h such that ( ) (ũ ) ( ) A B g = B T 0 p 0 Nonlinear saddle point problem!

9 Newton solver Motivation Strongly coupled problem Solution x n+1 = (ũ, p), Residual equation R(x n ) x n+1 = x n + w n[ R(x n )] 1R(x n x ) Automatic damping control w n for each nonlinear step Quadratic convergence when iterative solutions are close to the actual one

10 Newton solver Jacobian with respect to the Diffusive term 2η(γ II (u l ), p l )D(u) : D(v)dx η(γ II (u l ), p l )[D(u l ) : D(u)][D(u l ) : D(v)]dx η(γ II (u l ), p l )[D(u l ) : D(v)]pdx = fvdx 2η(γ II (u l ), p l )D(u l ) : D(v)dx, v V Jacobian with respect to the Convective term (u l. u)vdx + (u. u l )vdx v X,

11 Newton solver Diffusive term 2η(γ II (u l ), p l )D(u) : D(v)dx L η(γ II (u l ), p l )[D(u l ) : D(u)][D(u l ) : D(v)]dx L η(γ II (u l ), p l )[D(u l ) : D(v)]pdx B = fvdx 2η(γ II (u l ), p l )D(u l ) : D(v)dx, v V Convective term (u l. u)vdx N + (u. u l )vdx N v X,

12 Linear system Final discrete problem Compute ũ and p by solving ( ) (ũ ) ( ) A B Resũ = B T 0 p Res p where Aũ = [(L + δ d L )(ũ l, p l ) + (N + δ c N )(ũ l )]ũ, B p = [B + δ p B (ũ l, p l )] p Unusual saddle point problem!

13 Linear solver Multigrid techniques Direct Gauss elimination as coarse-grid solver General VANKA smoother with block-diagonal preconditioner F-cycle multigrid Intergrid transfer and coarse grid correction based on the underlying mesh hierarchy and the finite elements

14 Contents 1 Introduction 2 Mathematical formulation 3 Numerical Results 4 Conclusion and outlook

15 Benchmark problem Figure : Geometry for the flow around cylinder configuration Figure : Computational mesh for the flow around cylinder configuration

16 Featflow Why Featflow Basic flow solver for incompressible fluids Supports higher order (space/time) FEM Use of unstructured meshes Dynamic adaptive grid formulation FEM based tools FEM characteristics Stable FE spaces for velocity/pressure and velocity/stress interpolation; e.g. Q2/P1 Special treatments of the convective term - EO FEM, TVD/FCT

17 Inner solver Solver Nonlinearity handled by Newton method Monolithic Multigrid techniques for the auxiliary linearized problem Vanishing shear rate taken care by the regularization parameter Appropriate module to ensure unique and positive pressure Advantages Inf-sup stable (LBB condition) for velocity and pressure Higher order is good for accuracy Discontinuous pressure is good for the solver

18 Results Shear-thinning fluid η(γ II, p) = η 0 ( γ 2 + ɛ) m 2 1, η 0 = 10 3, m = 1.5, ɛ = 0.1 MG Drag Lift Solver Statistics Level Ref 1 Feat2 Ref 1 Feat2 Ref 1 Feat /3 8/ /2 4/ /2 3/ /2 3/2 Efficient nonlinear and linear solver Ref 1 : Damanik et al. Monolithic Newton-multigrid solution techniques for incompressible nonlinear flow models, International Journal for Numerical Methods in Fluids, Volume 71, Issues 2, pages

19 Results Shear-thickening fluid η(γ II, p) = η 0 ( γ 2 + ɛ) m 2 1, η 0 = 10 3, m = 3.0, ɛ = 0.1 MG Drag Lift Solver Statistics Level Ref 1 Feat2 Ref 1 Feat2 Ref 1 Feat /2 6/ /2 3/ /2 3/ /1 3/1 Ref 1 : Damanik et al. Monolithic Newton-multigrid solution techniques for incompressible nonlinear flow models, International Journal for Numerical Methods in Fluids, Volume 71, Issues 2, pages

20 Results Test case η(γ II, p) = η 0 exp(p), η 0 = 10 3 Fluid Level Drag Lift Solver Statistics E E-02 5/3 Test case E E-02 3/ E E-02 3/ E E-02 2/1.5

21 Contents 1 Introduction 2 Mathematical formulation 3 Numerical Results 4 Conclusion and outlook

22 Conclusion Summary Newton solver for nonlinearity Multigrid techniques for the linearized problem Numerical examples 1 non-newtonian fluids including shear thinning and shear thickening fluid 2 Test case with pressure dependant viscosity Continuous Newton solver Nonlinear Continuous Linear Discrete Linear FEM discr MG Solution

23 Outlook Current work Benchmarking of Poliquen model Test with split-bottom geometry To incorporate Tardos model [ ρ Du = p + Dt Questions q(p, ρ) D 1 n ui ρ t + (ρu) = 0 q(p, ρ) u = D 1 p n ui How to fix the pressure range? 2D version of split-bottom geometry? ( D 1 n ui ) ] + ρg; n = 2, 3

24 THANK YOU!