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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