Random front propagation in fractional diffusive systems€¦ · The Level Set Method, Tracking of...

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

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


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]


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


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‖


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}


The Level Set Method, Tracking of Deterministic Fronts (II)

[from Wikipedia]


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


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


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


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)


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


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


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‖


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


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


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

)


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]


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}


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.


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


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


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?


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


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


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)


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!


Random front propagation in fractional diffusive systems€¦ · The Level Set Method, Tracking of...

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