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Random front propagation infractional diffusive systems
Andrea MENTRELLIDepartment of Mathematics & CIRAM
University of Bologna, [email protected]
with Gianni PAGNINI (Ikerbasque/BCAM, Bilbao)
Didicated to Professor Francesco Mainardi
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
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Outline
Introduction & Scope of the work
The Level Set MethodOverview
Tracking of deterministic fronts
Tracking of random fronts: local/non-local cases
Random fronts driven by time-fractional diffusion
equationFundamental solutions
The Mainardi function
Numerical results
Conclusions
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Introduction & Scope of the Work
Many problems involve moving interfaces:
. . .
Reacting flows: irreversible single-step chemical reactions
(fresh mixture→products) instantaneously occurring at the
interface
Wildland fires: the fireline is an interface separating
burnt/unburnt regions
The Level Set Method (LSM) [Osher and Sethian, 1988] is a
successfultechnique for tracking moving interfaces with complex
evolving topology
Recently, LSM has been applied to the case of random fronts [GP
& Bonomi,2011; GP & Massidda, 2013; GP & AM, 2013]
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Overview (I)
Q. What are the main ingredients of the LSM?
The LSM relies on an implicit representation of the moving
interface Γ(t) as ahypersurface in Rd (d = 1,2,3)
An auxiliary function γ : Rd × [0,∞[ is defined such that (given
γ0 = const)the γ0-level set (γ = γ0) represents the evolving
interface Γ(t)
The motion of Γ(t) is thus determined by the evolution equation
for
γ(x(t), t) = γ0
i.e.DγDt
= 0
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Overview (II)
The evolution equation for the γ0-level set can be rewritten
as(
DDt =
∂∂t + V · ∇
)DγDt
=∂γ
∂t+ V (x, t) · ∇γ
=∂γ
∂t− V (x, t) · n̂ ‖∇γ‖
=∂γ
∂t− V(x, t) ‖∇γ‖ = 0
wheren̂ = − ∇γ
‖∇γ‖V(x, t) : Rate of Spread (ROS) of the front
The evolution equation reads:∂γ
∂t= V(x, t) ‖∇γ‖
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
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The Level Set Method, Tracking of Deterministic Fronts (I)
Let x0 be a point belonging to the interface Γ0 ≡ Γ(t = 0).
Its deterministic trajectory x(t |x0) is given by
dxdt
= V(x, t) x(0|x0) = x0
and the region Ω(t) enclosed by Γ(t) is identified by the
indicator ϕ
ϕ : S ⊆ Rd → {0, 1} ϕ (x, t) =
{1 if γ(x, t) ≥ γ00 elsewhere
The deterministic region enclosed by Γ(t) is: Ω(t) = {x : ϕ(x,
t) = 1}
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Deterministic Fronts (II)
[from Wikipedia]
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
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The Level Set Method – Tracking of Random Fronts (I)
Let x0 be the same point as before, belonging to the interface
Γ0 ≡ Γ(t = 0).
The ω-realization of its random trajectory is defined as
Xω(t ,x0) Xω(0,x0) = x0
The random trajectory is problem-dependent
The initial condition Xω(0,x0) = x0 is independent by the
realization andequals the deterministic position
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts (II)
In the ω-realization the γ0-level set is given by
γω0 (xΓω , t) =∫
Γ0
γ(xΓ0 ,0) δ(xΓω − Xω(t ,xΓ0)) dxΓ0
=
∫Γγ(xΓ, t) δ(xΓω − Xω(t ,xΓ)) dxΓ
The auxiliary γ function in the ω-realization is
γω(x, t) =∫Sγ(x, t) δ(x− Xω(t ,x)) dx
Here, it has been assumed the incompressibility-like condition J
=dxΓ0dxΓ
= 1
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts (III)Let’s focus
on γω(x, t) =
∫S γ(x, t) δ(x− X
ω(t ,x)) dx
〈ϕω(x, t)〉 = 〈∫Sϕ(x, t) δ(x− Xω(t ,x)) dx〉
=
∫Sϕ(x, t) 〈δ(x− Xω(t ,x))〉dx
=
∫Sϕ(x, t) f (x; t |x) dx = ϕe(x, t)
〈·〉 is the ensemble averagef (x; t |x) = 〈δ(x− Xω(t ,x))〉 is the
probability density function (PDF) of thedistribution of the
particles on Γ(t)
ϕe(x, t) is the effective indicator
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
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The Level Set Method, Tracking of Random Fronts (IV)
Q. A short recap?In the deterministic case the region Ω(t) is
“indicated” by
ϕ : S ⊆ Rd → {0, 1} ϕ (x, t) =
{1 if γ(x, t) ≥ γ00 elsewhere
(1)
In the random case the region Ω(t) is “indicated” by
ϕe : S ⊆ Rd → [0, 1] ϕe(x, t) =∫Sϕ(x, t) f (x; t |x) dx (2)
f (x; t |x) is the probability density function (PDF)
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Local Case
(I)Q. What about f (x; t |x)?
For local processes f (x; t |x) is governed by a general
evolution equation
∂f∂t
= Ex f (x; t |x) (Ex is some spatial operator)
Reynolds transport theorem + Divergence theorem:
∂ϕe∂t
=∂
∂t
∫Ω(t)
f (x; t |x) dx =∫
Ω(t)
∂f∂t
dx +∫∂Ω(t)
f V(x, t) · n̂ ds
=
∫Ω(t)
∂f∂t
dx +∫
Ω(t)∇x · (V(x, t) f ) dx
= Ex ϕe +∫
Ω(t)∇x · (V(x, t) f ) dx
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Local Case
(II)
If the rate of spread V(x, t) depends only on the front
curvature κ(x, t) = 12∇ · n̂,i.e. V(x, t) = V(κ, t)
∂ϕe∂t
= Ex ϕe +∫
Ω(t)
(V · ∇xf dx +
∂V∂κ
f ∇xκ · n̂ + 2 f Vκ)
dx
which is a reaction-diffusion equation coupled to the LSM
equation
The interface propagation is affected by
Type of random process, Ex and f (. . . an example? ordinary
diffusion)Rate of Spread, VMean front curvature, κ
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Local Case
(III)
Q. Is it always so complicated? ∂ϕe∂t = Ex ϕe +
∫Ω(t)
(V · ∇xf dx + ∂V∂κ f ∇xκ · n̂ + 2 f Vκ
)dx
When the normal to the front n̂ is constant:
the curvature is null: κ = 0
V(κ, t) = V(t)∇x f = −∇x f
the evolution equation for ϕe becomes:
∂ϕe∂t
= Ex ϕe + V(t) ‖∇ϕe‖
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Non-Local Case
(I)Q. . . . and where is theMainardi function?
When the random motion of the front particles is non-local and
governed by thetime-fractional diffusion equation, then f (x; t) is
the fundamental solution of thefractional PDE
D2β∗ f = K ∇2f 0 < β ≤ 1/2f (x, t = 0) = δ(x)
where D2β∗ is the fractional derivative in the Caputo sense
0 < β < 1/2 → anomalous diffusion
β = 1/2 → normal diffusion (local case): ∂f∂t
= K ∇2f
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Non-Local Case
(II)The fundamental solution of the time-fractional diffusion-wave
equation in the1D case is [Mainardi, 1996]
f1D(x , t) =12
1√K tβ
Mβ(| x |√K tβ
)whereMβ is theMainardi function
Mβ (z) =∞∑
n=0
(−z)n
n! Γ[−βn + (1− β)]
=1π
∞∑n=1
(−z)n−1
(n − 1)!Γ(βn) sin(πβn)
Note: For β = 1/2: M1/2 (z) = 1√π exp(−z2/4
)Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Non-Local Case
(III)
The fundamental solution of the time-fractional diffusion
equation in the 2D caseis [Hanyga, 2002]
f2D(x, t) =1
(2π)2 i K
∫ i∞−i∞
ests2β−1K0
(sβ
r√K
)
K0: Modified Bessel Function of the Second Kind
In the 2D case it is not possible to write f in terms of theM
function
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
The Level Set Method, Tracking of Random Fronts – Non-Local Case
(IV)
The fundamental solution of the time-fractional diffusion
equation in the 3D caseis [Hanyga, 2002]
f3D(r , t) = −1
2πr∂
∂rf1D(r , t)
= − 14π√
K tβ r∂
∂rMβ
(r√K tβ
)
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results Fundamental Solutions
Development of a Python/Fortran95 library (pyMlib) for the
numericalcomputation of the fundamental solutions in the 1D, 2D and
3D cases bymeans of available algorithms [Gorenflo et al., 2002;
Hanyga, 2002]
0 1 2 3 4 5x
0.0
0.2
0.4
0.6
0.8
1.0
f 1D(x
)
1D
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
0 2 4 6 8 10r
10-910-810-710-610-510-410-310-210-1100
f 2D(x
)
2D
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
0.0 0.5 1.0 1.5 2.0r
10-2
10-1
100
101
f 3D(x
)
3D
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
Coupling with standard Level Set Method library (lsmlib) [Chu
& al., 2009]
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (I)
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=5, t=0.01
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=5, t=5
Q. Where is the front?
Arbitrarily, one can assume, for example: Ωe(t) = {x : ϕe(x, t)
≥ 0.5}
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (II)
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=100, t=0.01
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=100, t=5
3000 2000 1000 0 1000 2000 3000 4000x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=100, t=200
Q. Do the fronts propagate with the same speed?
No. And the regions Ωe(t) enclosed by the interface vary as
well.
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (III)
0.0 0.5 1.0 1.5 2.0 2.5 3.0t
7
8
9
10
11
s
front velocity (K=100)
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0t
0.94
0.96
0.98
1.00
|Ω|/|Ω
0|
random/deterministic marked region (K=100)
Q. Can the “weakening” due to diffusion extinguish Ωe(t)?
Yes, and when ϕmaxe is below the threshold, Ωe vanishes
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (IV)
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=500, t=0.01
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
80 60 40 20 0 20 40 60 80 100x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=500, t=5
3000 2000 1000 0 1000 2000 3000 4000x
0.0
0.2
0.4
0.6
0.8
1.0
ϕe
K=500, t=200
Q. Can the diffusion be so strong as to extinguish Ωe(t)?
Yes. As K ↗ and/or β ↘ this can happen
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (V)
0.0 0.1 0.2 0.3 0.4 0.5t
120
100
80
60
40
20
0
20
s
front velocity (K=500)
β=0.2
β=0.3
β=0.4
β=0.5
0.0 0.1 0.2 0.3 0.4 0.5t
0.0
0.2
0.4
0.6
0.8
1.0
|Ω|/|Ω
0|
random/deterministic marked region (K=500)
Q. What is the effect of subdiffusive processes?
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Numerical Results, 1D Random Front Propagation (VI)
0 10 20 30 40 50 60 70 80t
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ϕm
ax
e
ϕmaxe
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
0 2 4 6 8 10t
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ϕm
ax
e
ϕmaxe (zoom)
Subdiffusion processes can extinguish the region Ω(t) more then
ordinarydiffusion processes
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Conclusions
Implementation of a library to study the propagation of random
interfacesin presence of anomalous diffusion in the framework of
the level setmethod (LSM) in 1D, 2D and 3D
The major effect of anomalous diffusion pointed out here (in 1D)
is anincreased weakening of the front with respect to ordinary
diffusion
Work in progress: investigations in 2D and 3D with various
profiles of Γ0
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
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References[Gorenflo et al., 2002)] R. Gorenflo, J. Loutchko, Y.
Luchko, Computation of the Mittag-Leffler function Eα,β(z)and its
derivative, Fractional Calculus and Applied Analysis 5, 491–518
(2002)
[Hanyga, 2002] A. Hanyga, Multidimensional solutions of
time-fractional diffusion-wave equations, Proc. R. Soc.Lond. A 458,
933–957 (2002)
[Mainardi, 1996] F. Mainardi, Fractional Relxation-Ocillation
and Fractional Diffusion-Wave Phenomena, Chaos,Solitons &
Fractals 9, 1461–1477 (1996)
[Osher and Sethian, 1988] S. J. Osher, J. A. Sethian, Fronts
propagating with curvature-dependent speed:algorithm based on
Hamilton-Jacobi formulations, J. Comput. Phys. 79, 12–49 (1988)
[Osher and Fedkiw, 2002] S. J. Osher, R. P. Fedkiw, Level Set
Methods and Dynamic Implicit Surfaces,Springer-Verlag (2002)
[Pagnini and Bonomi, 2011] G. Pagnini, E. Bonomi, Lagrangian
formulation of turbulent premixed combustion,Phys. Rev. Lett. 107,
044503 (2011)
[Pagnini and Massidda 2013] G. Pagnini, L. Massidda, Modelling
turbulence effects in wildland fire propagationby the randomized
level-set method, SIAM J. Appl. Math. (submitted)
[Pagnini and Mentrelli, 2013] G. Pagnini, A. Mentrelli, Random
front tracking for wildland fire propagation, Nat.Hazards Earth
Syst. Sci. (submitted)
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013
-
Acknowledgements
AM & GP are grateful to the Italian National Group for
Mathematical Physics(GNFM/INdAM) for the financial support (Young
Researchers Project 2013,Development of a hyperbolic model in the
framework of Extended Thermodynamics for theanalysis of
combustion-wave interaction in turbulent premixed combustion)
Thank you for your attention!
Andrea MENTRELLI – FCPNLO “Fractional Calculus, Probability and
Non-local Operators”, Bilbao (Basque Country, Spain), November 6–8
2013