Efficient Numerical Methods and Simulationtechniques for Granular flow

Abderrahim Ouazzi, Stefan Turek

Institut fur Angewandte Mathematik und Numerik, LS3,

Universitat Dortmund,

Germany

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Examples of Applications

Paddles in a mixer or granulator

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Examples of Applications

Flow around inserts during emptying of bins

and hoppers

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Schaeffer Law

Incompressible Stokes problem

− ∇ · [2ν(DII(u), p)D(u)] + ∇ p = f, ∇ · u = 0 (1)

the nonlinear viscosity ν(·, ·) is a function of DII(u) and p,

DII(u) = 12D(u) : D(u) = 1

2

∑i,j Di,j(u)Di,j(u).

Power law defined for

ν(z, p) = ν0zr2−1

Bingham law defined for

ν(z, p) = ν0z−12

Schaeffer law (including the pressure) defined for

ν(z, p) = pz−12

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Nonlinear Solver

Let ul being the initial state, the (continuous) Newton method consists offinding u such that

∫

Ω2ν(DII(u

l), pl)D(u) : D(v)dx

+

∫

Ω2∂1ν(DII(u

l), pl)[D(ul) : D(u)][D(ul) : D(v)]dx

+

∫

Ω2∂2ν(DII(u

l), pl)[D(ul) : D(v)]pdx

=

∫

Ωfv −

∫

Ω2ν(DII(u

l), pl)D(ul) : D(v)dx, ∀v ∈ V , (2)

where ∂iν(·, ·); i = 1, 2 is the partial derivative of ν related to the first and

second variable respectively.

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New Linear Algebraic Problem

The algorithm consists of finding (u, p) as solution of the linear system

A(ul, pl)u+ δdA

∗(ul, pl)u+Bp+ δpB∗(ul, pl)p = Ru(ul, pl),

BTu = Rp(ul, pl),

(3)

where Ru(·, ·) and Rp(·, ·) denote the corresponding nonlinear residual termsfor the momentum and continuity equations, and the matrix A∗(ul, pl) andB∗(ul, pl) are defined as follows respectively

〈A∗(ul, pl)u,v〉 =

∫

Ω2∂1ν(DII(u

l), pl)[D(ul) : D(u)][D(ul) : D(v)]dx. (4)

〈B∗(ul, pl)p,v〉 =

∫

Ω2∂2ν(DII(u

l), pl)[D(ul) : D(v)]pdx. (5)

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Spatial discretization

Quadrilateral Rannacher-Turek Stokes Element

P1

P2

P3P4

M1

M2

M3

M4M

E2E1

Advantage:

Stable and efficient for incompressible flow.

Compact data structures.

Disadvantage:“Not” satisfying discrete Korn’s inequality

∑

τ∈τh‖v‖H1(τ) ≤ c(‖v‖20,τ + ‖D(v)‖20,τ )

12 (6)

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Stabilized Rannacher-Turek Stokes Element

Remedy: Stabilized R-T FEM

The stabilization consists of adding the following bilinear form

∑

E∈EI∪ED

1

|E|

∫

E[φi][φj ]ds (7)

for all basis function φi and φj with a weighted parameter s = s(ν) which willact as ‘free’ stabilization parameter. Then the corresponding matrix S isdefined as:

< Su, v >=∑

E∈EI∪ED

1

|E|

∫

E[u][v]ds (8)

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Linear Multigrid solver

Vanka smoother as defect correction

[ul+1

pl+1

]=

[ul

pl

]+ ωl

∑i

(F + S∗|Ωi B + δpB

∗|Ωi

BT|Ωi 0

)−1 [Ru(ul, pl)

Rp(ul, pl)

]

(9)

with matrix F = A + δdA∗. For the preconditioning step only a part of the

matrix, i.e. F + S∗, is taken

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Comparison with high order scheme

Coarse mesh and geometrical details for ‘Flow around a cylinder’

Mesh information Q1/Q0 Q2/P1

Level Elements Vertices Total unknowns Total unknowns1 156 130 702 1533

2 572 520 2686 5927

3 2184 2080 10608 23295

4 8528 8320 42016 92351

5 33696 33280 167232 367743

6 133952 133120 667264 −

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Comparison with high order scheme

ν(DII, p) = (ε+DII(u))−α/2

Level Elements Drag Lift M p NNL/AVL

α = 0.9

2 Q1/Q0 819.49 2.5201 14.22 13/2

Q2/P1 920.59 2.1805 16.76 14/27

3 Q1/Q0 917.89 3.1958 15.54 14/2

Q2/P1 941.51 3.3310 16.02 15/70

4 Q1/Q0 916.02 3.7381 15.74 12/2

Q2/P1 953.94 3.9217 15.82 16/208

5 Q1/Q0 935.13 3.9954 15.82 15/3

Q2/P1 957.64 4.0587 15.87 33/368

6 Q1/Q0 946.22 4.0592 15.85 13/5

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Efficiency of the nonlinear solver

Shear dependent viscosity flow

ν(DII, p) = (ε+DII(u))−α/2

Q1/Q0

α 0.9

ε = 10−2 Newton FixpointLevel NNL/AVL CPU NNL/AVL CPU

2 13/2 60 74/2 228

3 14/2 215 114/2 1359

4 12/2 740 125/2 5871

5 15/3 3957 119/2 21735

6 13/5 25926 109/2 80438

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Efficiency of the nonlinear solver

Shear dependent viscosity flow

ν(DII, p) = (ε+DII(u))−α/2

ε = 10−2 Newton Fixpointα NNL/AVL CPU NNL/AVL CPU

0.1 7/2 7 10/2 9

0.5 7/2 8 27/2 32

0.9 9/3 16 113/3 190

1.0 12/4 52 394/2 889

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Efficiency of the nonlinear solver

Pressure dependent viscosity flow

ν(DII, p) = exp(βp) Newton Fixpointβ NNL/AVL CPU NNL/AVL CPU

5× 10−2 5/2 9 10/2 21

1× 10−1 6/2 13 15/2 33

2× 10−1 7/4 33 30/3 86

ν(DII, p) = βp Newton Fixpointβ NNL/AVL CPU NNL/AVL CPU

5× 10−2 9/6 63 94/4 296

1× 10−1 8/7 65 103/4 426

2× 10−1 7/16 120 124/8 1042

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Steady state flow in Silos

Schaeffer flow under gravity forceν(DII, p) = βp

ν(DII, p) = (DII(u))−1/2

ν(DII, p) = βp(DII(u))−1/2

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Steady state flow in Couette device

ν(DII, p) = exp(βp)

ν(DII, p) = (DII(u))−1/2

ν(DII, p) = exp(βp)(DII(u))−1/2

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