Frattali e calcolo frazionario
(Fractals and fractional calculus)
Università degli Studi di Palermo, ItalyDipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica (DISAG), Viale delle Scienze Ed.8, 90128 Palermo. email:[email protected]
Mario Di Paola
OUTLINES
♦ EUCLIDEAN SPACE AND APPLICATIONS TO VARIOUS BRANCHES OF ENGINEERING
♦ THE SCALE PROBLEMS AND PARADOXES OF EUCLIDEAN REALM
♦ FRACTAL REALM AND INFORMATION FROM MICROSCALES
♦ FRACTIONAL DERIVATIVES
♦ FRACTALS VS FRACTIONAL OPERATORS
♦ CONCLUSIONS
Engineer is an Euclidean Man
POINT LINE SURFACE VOLUME[L 0] [L 1] [L 2] [L 3]
DIMENSIONS
0 1 2 3 4VOID VOID VOID VOID
DERIVATIVES
0 1 2 3 4VOID VOID VOID VOID
))(( xfDdxfd jjj =
PRIMITIVES
0 1 2 3 4VOID VOID VOID VOID
))(()...( xfIxdxf j=∫ ∫ ∫
Spring Dashpot Mass
Ku F= Cu F=ɺ M u F=ɺɺ
( )M u C u Ku F t+ + =ɺɺ ɺ
FFF
K c
DYNAMIC MODEL
F
K
C
M
DUHAMELL INTEGRAL
( )20 0
( )
2 ( )
M u C u Ku F t
u u u F t M f tςω ω+ + =
+ + = =
ɺɺ ɺ
ɺɺ ɺ20 0 ; = 2K M C Mω ς ω=
If ( ) ( )f t tδ= ( )tδ = Dirac’s Delta
( ) [ ] [ ]01
exp sinh t t tςω ωω
= −
( ) ( ) ( )
( ) ( ) ( )0
0
0
1e sin
t
tt
u t h t f d
t f dςω τ
τ τ τ
ω τ τ τω
− −
= −
= −
∫
∫
20 1ω ω ς= −
Impulse Response
0 2 4 6 8 10-0.2
-0.1
0.0
0.1
0.2
0.3
0 2 4 6 8 10-0.2
-0.1
0.0
0.1
0.2
0.3
∞
∞
( )f t
KELVIN UNIT
VISCOELASTIC MODEL
( )M u Cu Ku F t+ + =ɺɺ ɺ
( )Cu Ku F t+ =ɺ
( ) ( ) ( )0
tt F
u t e dC
α τ ττ− −= ∫
K Cα =
CONVOLUTION INTEGRAL
( )C E tε ε σ+ =ɺ
( )F t
K
c( )u t
( )u u f tα+ =ɺ
( ) [ ]exph t tα= − ImpulseResponse
ELASTIC MODEL
( )M u C u Ku F t+ + =ɺɺ ɺ
( )( )
F tu t
K=
MEMORY-LESS
Eσ ε=
K( )F t
( )u t
Hooke’s Law
SPRING VS CAUCHY
F F
( )u t
CAUCHY SOLID DOES NOT DEPEND ON THE SCALE LENGTH
EA x∆
1) The only actions exerted on the volume are the stress σij coming from adjacentvolumes and external body forces
2) The contact forces (stresses) are local ones
CONSEQUENCES OF THE CAUCHY REALM
1) THE COLLAPSE OF THE SPECIMEN ONLY DEPENDS OF THE STRESS
2) THE DEFORMATION OF A TENSILE BAR IS UNIFORM
3) VISCOELASTIC MODEL IS RULED BY A LINEAR DIFFERENTIAL EQUATION
All the conclusions are erroneous
1) The collapse depends on the “scale” of the material, the bigger the specimen the smaller ultimate stress and vice-versa (liberty ships)
2) The deformation of a tensile bar is greater at the ends (NON LOCAL EFFECTS)
3) For viscoelastic models the Kelvin-Voigt model does not fit experiments
AN /=σ
THE APPARENT PARADOXES MAY BE EXPLAINED IN THE
CONTEXT OF THE FRACTAL GEOMETRY
B. B. Mandelbrot: “The fractal geometry of nature”, W. H. Freeman and Company, 1983.
K. J. Falconer: “The geometry of fractal sets”. Cambridge University Press, 1985.
WHAT IS THE LENGTH OF THE GREAT BRITAIN COAST?
♦ IT DEPENDS ON THE LENGTH OF SPECIMEN FOR THE MEASURE
♦ THE SMALLER THE SPECIMEN THE BIGGER THE COAST LENGTH
♦ AT THE LIMIT WHEN ∆→0 THE LENGTH OF THE GREAT BRITAIN
COAST IS ∞
THE NATURE IS FRACTAL: COASTS, MOUNTAINS, TREES, RIVERS, BIOLOGICAL TISSUES…
COMMON POINT: SELF-SIMILARITY
FRACTAL GEOMETRY
• Sets with non-integertopological dimensions and preserving some invarianceat any observation scale are dubbed Fractals. An Example: The Cantor set
Length of element
0 9ℓ
n iteration
0ℓ
0 3ℓ 0 3ℓ
0 9ℓ 0 9ℓ 0 9ℓ
n → ∞
01 1 0
22 ; ; =
3 3N r L= = ℓ ℓ
0 01 ; ; N r L= = =ℓ ℓ
2 02 2 02
42 ; ; =
3 9N r L= = ℓ ℓ
00
22 ; ; = 0
3 3
nnn
n n n nN r L →∞= = →ℓ
ℓ
N. of elements Topological lenght
Preserving invarianceof the lenght with a realtopological dimension
[ ][ ]2
3j
j
Log N Logd
LogLog r
= − =
( ) 0
d dj jN r = ℓ
Parent
Generator
Pre
frac
tals
SELF-SIMILAR FRACTAL SETS (INVASIVE)
01 1 1 0
44 ; ;
3 3N r= = =ℓ
ℓ ℓ
0 0 01 ; ; N r= = =ℓ ℓ ℓ
2 02 2 2 02
164 ; ;
3 9N r= = =ℓ
ℓ ℓ
00
44 ; ;
3 3
nn
n n nn nN r= = =ℓ
ℓ ℓ
Triadic Curve Von Koch
[ ] [ ]4 3 1.262d Log Log= = MANDELBROT1980
0ℓ
0 3ℓ
n iteration
0 3ℓ 0 3ℓ
0 3ℓ
0 9ℓ 0 9ℓ
lim nn→∞
= ∞ℓ
Parent
Pre
frac
tals
Fractaln → ∞
Length of elementN. of elements Topological length
( ) 0
d dj jN r = ℓ
• A solid body withFractaldistribution of themassshows more details at finerobservation scale: Topological density is strictly dependentof the observation scale(colloids, polymers, liquid crystals…).
The Fractal Model of a Solid Body• A solid body withEuclideandistribution of themass shows the same details at any
observation scale: Topological density is independentof the observation scale.
L 2L
M
3L 4L
( )d
FM Lρ=
M M M [ ] 3/ costM Lρ = =
• The scale relation of the mass :
L / 3L 9L / 27L
• Meters
[ ] / dF M Lρ =
M M M M dimensiond =
THE FRACTAL DIMENSION: THE HAUSDORFF MEASURE
( )sup : ,
j j
j j
diam G
G x y x y G
δ = =
= = − ∈
• Introducing a countable collections of diameters of subsets the Housdorff-δ measure is provided by :
0 j≤ δ ≤ δ
• Housdorff measure is provided by: ( ) ( )0
lims sH V H Vδδ →=
THE RIGOROUS MEASURE OF A FRACTAL SET
δ
Gj
diam( )Gj
V
( ) { }1
inf :n
ssi i i
i
H V G G Gδ δ=
= ∈ ∑
jG
THE HAUSDORFF DIMENSION
( ) ( )0
lims sH V H Vδδ →=
s0
∞
Hs = ∆THE PROPERTIES OF THE HAUSDORFF MEASURE
• The scaling property: ( ) ( )s s sH V H Vλ λ=
• The Hölder Mapping property :( ) ( )f x f y c x yα− < − ( )( ) ( )s s sH f V c H Vα α=
• Hausdorff measure is invariant under rotation and translation:
• Hausdorff measure is additive for nearly disjoint sets:
( )sH V
FRACTAL vs EUCLIDEAN DIMENSIONS (HAUSDORFF)
0ℓ
0 3ℓ
0 9ℓ 0 9ℓ 0 9ℓ
0 3ℓ
0 9ℓ
1/i nδ δ= =
( )0 0
1
1 1lim limn n
s sns
i
H V nδ δδ δ→∞ →∞
→ →=
= =
∑
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
( ) ( )0
lims sH V H Vδδ →=
Scaling of a factor 1/3, by scalingand additive property
1 20, ; ,1
3 3L RF F = =
1 1
3 3
s ss s s s sF L R V VH H H H H
= + = +
[ ] [ ]log 2 log 3Hs = ∆ =
( )( )( )
; 1
; 1
0 ; 1
s
s
s
H V s
H V finite s
H V s
= ∞ <
= =
= >
1H s∆ = =
( ) { }1
inf :n
ssi i i
i
H V G G Gδ δ=
= ∈ ∑
DIMENSIONS OF FRACTAL SETS
Von Koch
1.262∆ =
length ∞
INVASIVE
Cantor
0.671∆ =
length 0
LACUNAR
DIMENSIONS
LACUNAR LINE
INVASIVE LINE
LACUNARVOLUME
INVASIVEVOLUME
LACUNARSURFACE
INVASIVE SURFACE0 1 2 3
POINT LINE SURFACE VOLUME
LACUNARIPERVOLUME
IPERVOLUME
Menger Sponge
[ ] [ ]20 3
2.726
Log Log∆ ==
Volume 0
LACUNAR
EUCLIDEAN GEOMETRY
0ℓ 0 bℓ 20 bℓ 0
jbℓ
1b <
00
0
( , ) exp 0
( , ) exp 0
xI x C
i t U t
ξξ
τ α τ α
− = − >
= − − >
ℓℓ
• In the Euclidean geometry, objects are exactly the same at each observation scales:
• Ordinary-type differential equations that the Duhamel form correspond to a convolution integral with exponential-type kernel
FRACTAL GEOMETRY
0ℓ 0 bℓ
0ℓ 0bℓ 20b ℓ
1b >
0
( , ) 0 1
( , ) 0 1
I x c x
Ui t t
h
α
ν
ξ ξ α
τ τ ν
−
−
= − < <
= − < <
α: related to fractal dimension
CURIE 1889
20 bℓ
0jbℓ
0jb ℓ
• TOP-DOWN APPROACH
• BOTTOM-UP APPROACH
SUMMING UP
Euclidean Fractals
0 1 2 0 1 2
t
x
β
γ
τ
ξ
−
−
−
−
0
0
1( ) exp( ) ( )
1( ) exp( ) ( )
t
t
u t t F dc
u x x l F dc
α τ τ τ
ξ ξ ξ
= − −
= − −
∫
∫
( ) ( ) ( )u t u t u tɺ ɺɺ
FRACTIONAL INTEGRALS
exp( )tα τ− −
exp( )x lξ− −( ) ( ) ( )u x u x u xɺ ɺɺ
( )( )( )( )D u t
D u x
β
γ
0
0
1 ( )( )
1 ( )( )
t
t
Fu t d
c t
Fu x d
c x
ββ
γγ
τ ττξ ξξ
=−
=−
∫
∫
Fractional Calculus (Generalization of the differential calculus):
Riemann, Liouville, Abel, Grunwald, Leytnikov, Leibnitz, Marchaud, Weyl, Caputo, Riesz, Samko…
Applications in mechanics:
• Viscoelastic behaviour
De L’Hopital asked Leibnitz: “What about D1/2 f ? ” (1695)
• Fracture Mechanics
• Non-local continuum mechanics
• Stochastic dynamics
(Atanackovic et al., 2002)
(Carpinteri et al., 2001)
(Lazopoulos, 2006, Di Paola, Zingales, 2008)
(Einstein-Smolochowsky, 1930; Di Paola, Pirrotta, Zingales, 2007; Cottone, Di Paola, 2007) 23
Why so limited diffusion in the field of mechanics ?
• Many definitions: (Riemann-Liouville, Marchaud, Grunwald-Leytnikov, Riesz, Weyl, Caputo, …) with a common point: Allfractional integral or derivatives are convolution integrals with power-law decay kernel
• It is very hard to tackle by hand
• Lack of any geometrical meaning
Books: Samko et al., 1987; Miller, Ross, 1993; Podlubny, 1987
24
• Many different symbologies
The importance of fractional calculus
• Classical derivatives:
( )2
2, , ,..., ,...,
j
j
df d f d ff x j N
dx dx dx∈
• Generalized derivatives:
( )( ) D f xα α ∈ℜ
• Bridge between differential and integral equation
• Fractional derivatives exhist also for non-differentiable functionsand for fractals
25
Riemann-Liouville Fractional Integral and Derivativ e
Since
( )( )( )
( )( )10
1
1 !
xn
n
fI f x d n
n x
ξξ
ξ−= ∈
− −∫ ℕ
DE L’HOPITAL ASKED LEIBNIZ: “WHAT ABOUT ?”( )1/2 1/2/d f x dx
( )( ) ( ) ( )( ) ( )1 2
0 0 0; ; ...
x x t
I f x f d I f x f d dtξ ξ ξ ξ= =∫ ∫ ∫CAUCHY
α
( )( )( )
( )( )0 1
0
1 x fI f x d
x
α
α
ξξ
α ξ+ −=
Γ −∫ α ∈ ℝ
RIEMANN – LIOUVILLE: simply extend the Cauchy formula to the real exponent
( )2
2
0
/d
df dx f ddx
ξ ξ
∞
= ∫
( )( )( )
( )( )0 1
0
1 m x
m
fdf x d
dx x
α
α
ξξ
α ξ+ −=
Γ −∫D[ ] 1m α
α
= +
∈ ℝ
Unbounded Domain
( )( )( )
( )( )1
1 x fI f x d
x
α
α
ξξ
α ξ+ −
−∞=Γ −∫ LEFT RL Fractional integral
( )( )( )
( )( )1
1
x
fI f x d
x
α
α
ξξ
α ξ
∞
− −=Γ −∫ RIGHT RL Fractional integral
( )( )( )
( )( )1
1 m x
m
fdf x d
dx x
α
α
ξξ
α ξ+ −
−∞=Γ −∫D LEFT RL Fractional derivative
( )( )( )
( )( )1
1 m
mx
fdf x d
dx x
α
α
ξξ
α ξ
∞
− −=Γ −∫D RIGHT RL Fractional derivative
[ ] 1m α
α
= +
∈ ℝ
Rules of Fractional Operators
( )( ) ( )( )m n m nI I f x I f x+=
All the rules of classical integrals and derivatives still apply!
( )( ) ( )( )I I f x I f xα β α β+= ,α β ∈ ℝ
,m n ∈ ℕ
( )( ) ( )( )m n m nf x f x+=D D D
( )( ) ( )( )f x f xα β α β+=D D D ,α β ∈ ℝ
,m n ∈ ℕ
Leibnitz’s rule
Integration by parts
⋮
Fourier Transform of RL Fractional Integrals & Derivatives
Fourier transform of the Riemann-Liouville Fractional Integral and Derivative
n ∈ ℕn
nn
α ∈ ℝαα
α α
EXAMPLE: FRACTIONAL DERIVATIVE OF SIN(X)
( ) sin( )f x x=
d /df x( ) sin( )f x x=
EXAMPLE: FRACTIONAL DERIVATIVE OF SIN(X)
d /df x
2 2d /df x
( ) sin( )f x x=
EXAMPLE: FRACTIONAL DERIVATIVE OF SIN(X)
d /df x2 2d /df x
3 3d /df x
( ) sin( )f x x=
4 4d /df x
EXAMPLE: FRACTIONAL DERIVATIVE OF SIN(X)
( )( )sin xγ
+D
( ) sin( )f x x=
γ
x
EXAMPLE: FRACTIONAL DERIVATIVE OF SIN(X)
( )( )sinI xγ
+
EXAMPLE: FRACTIONAL INTEGRAL OF SIN(X)
x
Marchaud (M) fractional derivative:
( )( ) ( )( )
( )( ) ( )
( ) ( )( )
( )( )10 0
1 <1
1 1
ndeff x f x f xd
D f x d d f xdx
γ γγ γ
ξ ξγξ ξ γγ γξ ξ
∞ ∞
± ±+
± −= = =
Γ − Γ −∫ ∫ D∓ ∓
For bounded domain:
( )( ) ( )( ) ( ) ( )( )
( )( ) ( )( ) ( ) ( )( )
ˆ 1
ˆ 1
a a
b b
f xf x f x
x a
f xf x f x
b x
γ γγ
γ γγ
γ
γ
+ +
− −
= +Γ − −
= +Γ − −
D D
D D
( )( ) ( )( ) ( )
( )( )
( )( ) ( )( ) ( )
( )( )
1
1
ˆ1
ˆ1
x
aa
b
bx
f x ff x d
x
f x ff x d
x
γγ
γγ
ξγ ξγ ξ
ξγ ξγ ξ
+
−
+
+
−=
Γ − −
−=
Γ − −
∫
∫
D
D
36
[ ],x a b∈
( )( ) ( )( ) ( ) ( ) ( )( )ˆ ˆ ; f x f x f x f xγ γ γ γ+ + − −= =D D D D,a b→ −∞ → ∞
Fourier transform( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )
=
=
=
i xf x e f x dx F
df x i f x i F
dx
D f x i f x i F
θθ
θ θ
α ααθ θ
θ
θ θ θ
θ θ θ
∞
−∞
±
ℑ =
ℑ = ℑ
ℑ = ℑ
∫
Fractional differential equations( )
( ) ( ) ( ) ( ) ( )( )
cx kx f t
Fi X kX F X
i c k
ωω ω ω ω ωω
+ =
+ = → =+
ɺ
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )
C D x kx f t
Fi C X kX F X
i C k
αα
αα α
α
ωω ω ω ω ωω
+ + =
+ = → =+
Fractals vs Fractional Derivatives
• Fractional derivatives of non-differentiable functionsmay exhist
• Fractional derivatives of fractals exist forα < Fractal Dimension
Euclidean Objects Fractal Objects
( )( ) ( )( ) ; n nI f x D f x n N∈ ( )( ) ( )( ) ; I f x D f xα α± ±
( )cx kx f t+ =ɺ
( ) ( ) ( )0
1t
k c tx t e f dc
τ τ τ− −= ∫
Convolution integralConvolution integral
( )( ) ( )C D x t f tαα ± =
( )( )
( ) ( )1
0
1t
x t t f dC
α
ατ τ τ
α−= −
Γ ∫
( ) ( )( ) ( )1 0
F
Xi c k i
ωωω ω
=+
Fourier Transform Fourier Transform( ) ( )
( )
FX
i Cαα
ωωω
=
Conclusions
• Real objects are fractals, then all physical phenomenamay not be interpreted as embedded in EuclideanSpace
• Fractals are nowhere differentiable in classical sensebut they may be differentiable using fractionalderivatives
• By making inverse path, coming from nano to actualscale, the physical laws do not exhibite classicalderivatives but fractional ones. The problem becomesimportant dealing with microstructures or when wewant to focus attention to cracks or megastructures.
