Numerical schemes for viscoplastic
avalanches
E. Fernandez-Nieto, J.M. Gallardo,
Paul Vigneaux•
• Unité
de Mathématiques
Pures et Appliquées
Ecole Normale Supérieure de Lyon – France
GdR CNRS “EGRIN” – 2nd Meeting
Domaine de Chalès – July 1st, 2014.
Introduction
The model
Outline
1
The model
2
The schemes
3
Numerical results
The schemes
Numerical results
Conclusion
Introduction : Wet snow avalanche, Oisans, 2013
Courtesy P. Etard.
Introduction
The model
The schemes
Numerical results
Conclusion
Introduction - Thin layers of viscoplastic fluids
Present objectives
1
Simulate viscoplastic flows : with Bingham constitutive
law
2
Thin layers on inclined planes : lubrication or
shallow-water models to reduce computational cost
3
Schemes able to catch stationary states
a blend of variational inequalities, finite-volumes schemes
and well-balanced philosophy
Introduction
The model
The schemes
Plasticity - The origin
1916 ; 1922 →
Numerical results
Conclusion
Introduction
The model
The schemes
Numerical results
Plasticity
an
flu
id
Bingham fluid
N
ew
to
ni
Shear stress
Modern formalism →
Yield stress
Shear rate
Conclusion
Introduction
The model
Outline
1
The model
2
The schemes
3
Numerical results
The schemes
Numerical results
Conclusion
Introduction
The model
The schemes
Numerical results
Asymptotic model - Domain description
Conclusion
Introduction
The model
The schemes
Numerical results
Conclusion
Asymptotic model - Domain description
We consider a general bottom for the solid boundary.
For this, let Ω ⊂ R2 be a fixed bounded domain and
D(t) = {(x, z) ∈ Ω × R / b(x) < z < b(x) + h(t, x)},
where h(t, x) is the thickness of the fluid and x = (x1 , x2 ).
Γs (t) := {(x, z) ; x ∈ Ω, z = h(t, x)},
Γb (t) := ∂D(t) \ Γs (t)
the free and bottom surfaces. v := (v1 , v2 ), the horizontal
component of the velocity field and w, the vertical one, i.e.
u = (v, w).
Asymptotic model - Derivation at a glance
Z
Hρ0 St∂t V 0 · (Ψ − V 0 ) + V 0 · ∇x V 0 (Ψ − V 0 ) dX
∀Ψ,
Z Ω
βV 0 · (Ψ − V 0 )dX
+
Ω
Z
2
+
HηD(V 0 ) : D(Ψ − V 0 )dX
Ω Re
Z
2
+
Hηdivx V 0 (divx Ψ − divx V 0 )dX
Re
ZΩ
q
q
2
2
+
τy BH
|D(Ψ)| + (divx Ψ) − |D(V 0 )|2 + (divx V 0 )2 dX
Ω
Z
Z
1
1
≥ 2
Hρ0 FΩ · (Ψ − V 0 )dX − 2 (H)2 Z ρ0 fz (divx Ψ − divx V 0 )dX
Fr Ω
Fr Ω
(1)
Rk1 : τy = 0 : classical 2D viscous SW, cf. Gerbeau-Perthame
Rk2 : for more details on model derivation → Bresch et al.
Advances in Math. Fluid Mech. pp 57-89. 2010
Introduction
The model
The schemes
Numerical results
Conclusion
Asymptotic model - 1D version
(x, t) ∈ [0, L] × [0, T ]. H = H(x, t), etc. + ρ0 = cte
External forces : fx = −g sin α, fz = −g cos α.
∂H
∂(HV )
+
= 0,
∂t
∂x
Z L 1
2
H ∂t V (Ψ − V ) + ∂x (V )(Ψ − V ) dx
2
0
Z L
Z L
+
βV (Ψ − V )dx +
4ηH∂x (V )∂x (Ψ − V )dx
0
0
L
Z
(2)
+
√
τy 2H |∂x (Ψ)| − |∂x (V )| dx
0
Z
≥
L
Z
H(fΩ + fz ∂x b)(Ψ − V )dx +
0
0
L
H2
fz (∂x Ψ − ∂x V )dx,
2
∀Ψ
(3)
Introduction
The model
Outline
1
The model
2
The schemes
3
Numerical results
The schemes
Numerical results
Conclusion
Introduction
The model
The schemes
Numerical results
Conclusion
Semi-discretization in time
H n+1 − H n ∂(H n V n )
+
= 0,
∆t
∂x
L
(4)
1
V n+1 − V n
n+1
n 2
n+1
H
(Ψ − V
) + ∂x ((V ) )(Ψ − V
) dx
∆t
2
0
Z L
Z L √
n+1
n+1
+
βV
(Ψ − V
)dx +
τy 2H n |∂x Ψ| − |∂x V n+1 | dx
Z
n
0
Z
0
L
4ηH n ∂x (V n+1 )∂x (Ψ − V n+1 )dx ≥
Z L Z L
(H n )2
H n fΩ + fz ∂x b (Ψ − V n+1 )dx −
fz (∂x Ψ − ∂x V n+1 )d
2
0
0
+
0
Observe : problems on H n+1 and V n+1 are decoupled
Introduction
The model
The schemes
Numerical results
Conclusion
Velocity problem - Augmented Lagrangian
Following Glowinski et al. (’83, ’07) ⇒ minimization problem
Augmented Lagrangian func. s.t. its saddle point is the solution
Uzawa like algorithm to find this saddle point :
Solve ∂V Lr (V , q, µ) = 0 for V (linear problem)
Lr (V , q, µ) non differentiable in q
but q can be solved explicitly
Update the Lagrange multiplier µ and loop
Convergence to the unique solution ⇒ V n+1
Introduction
The model
The schemes
Numerical results
Conclusion
Velocity problem - Augmented Lagrangian
Compute qik +1 locally (at {xi }i ) :

k
k
 0 if |µ + r ∂x (V )| < τy ,
√
q k +1 =
1
k
k
k
k
 r (µ + r ∂x (V )) − τy 2 sgn(µ + r ∂x (V )) otherwise
Solve for V k +1 the linear system :
V k +1 − V n
∆t
(6)
+ βV k +1 − ∂x 4ηH n ∂x (V k +1 ) − ∂x rH n ∂x (V k +1 )
(H n )2
Hn
= (fΩ + fz ∂x b) H n + ∂x fz
−
∂x ((V n )2 ) + ∂x (H n (µk − rq k +1(7
)
2
2
H
n
Update Lagrange multiplier :
µk +1 = µk + r ∂x V k +1 − q k +1
(8)
Introduction
The model
The schemes
Numerical results
Conclusion
Known facts I
For L.A. methods, the parameter r in
µk +1 = µk + r ∂x V k +1 − q k +1
is known to influence the speed of convergence of the algorithm
i.e. the number of iterations to reach V n+1
Introduction
The model
The schemes
Numerical results
Known facts II
There exists an optimal r in practice :
r → +∞ is prevented by the fact that
it also deteriorates condition number of the linear pb
For L.A. methods, except in very simple cases,
no way to derive the optimal r .
Conclusion
Introduction
The model
The schemes
Numerical results
Conclusion
An alternative
Bermudez - Moreno method : Comput. Math. Appl., 7(1) :43-58, 1981.
“Duality methods for solving variational inequalities.”
General structure close to L.A. :
(
A(V k ) + ωB ∗ (B(V k )) + B ∗ (θk ) = L,
θk +1 = Gλω (B(V k ) + λθk ).
where Gλω is the Yosida regularization of the operator
accounting for the non differentiable term "|∂x (V k )|".
In short, for a given problem, one can use a general way
to derive the optimal ω, via eigenvalue problems.
Ex : "exact" in 1D ; "numerical" in 2D for Bingham
(9)
Velocity problem - Bermudez-Moreno
Find V k +1 ∈ V solution of the following linear problem :
n
H
+ β V k +1 − ∂x ((4ηH n + ω) ∂x V k +1 ) − ∂x (ω∂x V k +1 )
∆t
Hn n Hn
1
=
V −
∂x ((V n )2 ) + ∂x ((H n )2 fz ) + H n (fΩ + fz ∂x b) + ∂x θk .
∆t
2
2
Update the so-called BM multiplier θk +1 via ξ k +1 = ∂x V k +1 + λθk
and

√
√
−ω ξ k +1 +τy 2H n (x)


if ξ k +1 > λτy 2H n (x),

1−λ
ω
 k +1
√
√
θk +1 = ξ λ
if ξ k +1 ∈ [−λτy 2H n (x), λ τy 2H n (x)],

√


 −ω ξk +1 −τy 2H n (x) if ξ k +1 < −λτ √2H n (x).
y
1−λ ω
(10)
Note that this computation is again local in space.
n
ωopt (Hmax
)
=
2
n
Hmax
L
n
+β
+ 4ηHmax
.
∆t
Nπ 2
(11)
Introduction
The model
The schemes
Numerical results
Conclusion
Height problem and Spatial discretization
Looking at the global problem ... (P)n,k :
 k +1 n
H
−H

+ ∂x (H n V n ) = 0,
 ∆t


 k +1
n
H n V ∆t−V + βV k +1 − ∂x 4ηH n ∂x (V k +1 ) − ∂x rH n ∂x (V k +1 )




= (f + f ∂ b) H n + ∂ (H n )2 fz − H n ∂ ((V n )2 ) + ∂ (H n (µk − rq k +1 )).
Ω
z x
x
x
2
2 x
(12)
... & invoking the “SWE structure”
with various source terms, including the duality terms
⇒ we choose finite-volume for spatial discretization of (P)n,k
Introduction
The model
The schemes
Numerical results
Conclusion
"Unified" viscoplast. Well-Balanced finite volume
• System form :
k +1
W
− Wn
n
n
+ ∂x F (W ) − ∂x ((4ηH n + δ n )I ∂x W k +1 )
D(W )
∆t
= −βI W k +1 + S(W n )∂x σ k ,
• Flux approximation :
n
k
φ(Win , Wi+1
, {ζj+1/2
}j=i+1
j=i−1 )
=
n
F (Win ) + F (Wi+1
)
2
1 n
j=i+1
n
k
))
− Qi+1/2 (Wi+1
− Win + G({ζj+1/2
}j=i−1
2
n
• Numerical viscosity matrix, Qi+1/2
, various possibilities :
(modified) Lax-Friedrichs, Rusanov [diagonal],
Roe, HLL, Lax-Wendroff, Force, Gforce [not diagonal].
← source terms treatment of Chacón et al. SIAM JSC 2007
Introduction
The model
The schemes
Numerical results
Conclusion
Distributing the system “on speed and height”
Linear (sub)problem on V
An V k +1 = b n,k ,
n,(1)
b n,k
= bi
i
n,k ,(2)
+ bi
(13)
n,k ,(3)
+ bi
,
where
n,(1)
bi
=
Hin
=
n,k ,(3)
= Hi
bi
bi+1 − bi−1
fΩ + fz
2 ∆x
,
k
k
ζi+1/2
− ζi−1/2
n,k ,(2)
bi
,
∆x
j=i+1
n
k
n
n
k
[φ(Wi−1
, Win , {ζj+1/2
}j=i+1
j=i−1 )]2 − [φ(Wi , Wi+1 , {ζj+1/2 }j=i−1 )]2
n
∆x
n,k ,(3)
n
Rk : If Qi+1/2
diagonal, b i
n,(3)
bi
. Reduce comput. cost
.
Introduction
The model
The schemes
Numerical results
Conclusion
Distributing the system “on speed and height”
Sub-Problem on H
D(Win )
−
j=i+1
j=i
n
n
k
n
n
k
Wik +1 − Win φ(Wi , Wi+1 , {ζj+1/2 }j=i−1 ) − φ(Wi−1 , Wi , {ζj+1/2 }j=i−2 )
+
∆t
∆x
1
k +1
k +1
n
n
k +1
k +1
n
n
(W
−W
)−(4ηH
+δ
)I
(W
−W
)
(4ηH
+δ
)I
i−1/2
i−1/2
i+1/2
i+1/2
i+1
i
i
i−1
∆x 2
= − βI Wik +1 + S(Win )
n,k
σ n,k
i+1/2 − σ i−1/2
∆x
(14)
.
where
n
Hi+1/2
n
H n + Hi+1
= i
,
2
j=i+1
k
G({ζj+1/2
}j=i−1
)
1
=
fz

σ n,k
i+1/2
fΩ xi+1/2 + fz
bi +bi+1
2
=

,
k
ζi+1/2
fΩ ∆x + fz (bi+1 − bi ) +
∆(ζ+δ n ∂x V )ki+1/2
Hi+1/2
!
,
0
Compute H n+1 = H k +1 with the first component of (14) and the
most recent "duality multiplier", i.e, ζ n+1
Introduction
The model
The schemes
Numerical results
Well-Balancing and Wet/Dry treatment
To design ∆(ζ + δ n ∂x V )ki+1/2 , use a flux limiter
in conjunction with a convex combination of a 2nd order
and a 1st order approximation of ∂x (ζ + δ n ∂x V )ki+1/2
Wet/Dry correction : check whether rigid or fluid
no numerical diffusion in H discretiz.
and local equilibrium of pressure
Properties :
recover standard w/d treatment [WD] when fluid
and natural resting state when rigid
[WD] Castro et al. Mathematical and Computer Modelling 42(3-4), 2005.
Conclusion
Well-Balanced property of the coupled scheme
Two stationary solutions : V = 0 and
√ 0
Pi
P[N/2] 0 0
H
−
H
If fΩ ∆x
j
j=1
j=1 j
≤ τy 2Hi+1/2 ∀ i, then ↑ is stat. sol. (25)
Theorem
Let (H = H(x); V ≡ 0) be a stationary solution of (3), and assume that the
proposed numerical scheme uses the following initialization for ζ :
1
ζi+1/2
= −∆x
X
i
j=1
[N/2]
Hj0 −
X
j=1
Hj0
fΩ + fz
0
Hi+1
+ bi+1 − (Hi0 + bi )
,
∆x
then, the scheme exactly preserves both stationary solutions : (i) horizontal
free surface and (ii) flat height, verifying (25), over a bottom b.
Introduction
The model
Outline
1
The model
2
The schemes
3
Numerical results
The schemes
Numerical results
Conclusion
Introduction
The model
The schemes
Numerical results
Conclusion
The duct flow case
An analytical solution for Poiseuille-Bingham - H = 1 and
Z
∀Ψ,
L
∂t V (Ψ − V ) + 4η∂x (V )∂x (Ψ − V )
Z
0
L
√
Z
τy 2 (|∂x Ψ| − |∂x V |) dx ≥
+
0
(15)
L
f (Ψ − V )dx.
0
where f is the pressure gradient in the direction of the flow.
Analytical stationary solution :

2
L
if 0 ≤ χ ≤ χy ,
f  2 − χy
(16)
VBP (χ) =
8η  L − χy 2 − (χ − χy )2 if χy < χ ≤ L .
2
where ξ = |x − 2L |, ξo =
2
√
2τy /f and the domain is x ∈ [0, L].
Introduction
The model
The schemes
Numerical results
The duct flow case
Convergence of order 2 in space for both AL and BM
Conclusion
Introduction
The model
The schemes
The duct flow case
Duality variables : AL and BM
Numerical results
Conclusion
Introduction
The model
The schemes
Numerical results
Conclusion
Optimal parameters
320
A.L.
B.M.
topt
300
k
280
260
240
220
200
0.4
0.6
0.8
1
1.2
1.4
{r,t}
1.6
1.8
2
2.2
Introduction
The model
The schemes
Numerical results
Well-Balanced : 2 stationary solutions
Conclusion
Introduction
The model
The schemes
Numerical results
Baby avalanche : test 3. Initial condition
Conclusion
Baby avalanche : test 3. Final time (a)
Baby avalanche : test 3. Final time (b)
Baby avalanche : test 3. Final time, mesh conv.
Baby avalanche : test 3. Comput. cost
Conclusions
Summary
derivation of a Shallow-Water Bingham model
Re = O(1) and 1st order slip BC
valid for null slope and up to moderate slopes
design of Well-Balanced schemes
which allow to catch stationary states by coupling
duality methods and Finite-Volume methods...
... taking into acc. wet/dry fronts on general slopes
BM : a way to determine optimal duality parameters
Perspectives
extension of such schemes to simulations in 2D space
More details : J. of Comput. Phys. (2014) Vol 64, pp 55-90