Complexity of Material Explored with Nonlinear Tools: … · doppelpendel/dPendulum .html. 44 Serge...

Master Student course given within ERASMUS Program, Nov 5th 2007 , PragueSerge Dos Santos

ComplexityComplexity of of Material Explored with Nonlinear Material Explored with Nonlinear ToolsTools : : from Physicalfrom Physical Concepts to Image Concepts to Image

Processing Processing

Serge Dos Santos, Serge Dos Santos, PhDPhDAssistant Assistant ProfessorProfessor

ENI Val de LoireENI Val de LoireLUSSI Université FrançoisLUSSI Université François--Rabelais de Tours Rabelais de Tours -- FRE 2448 CNRS FRE 2448 CNRS -- GIP UltrasonsGIP Ultrasons

Rue de la Chocolaterie BP 3410, FRue de la Chocolaterie BP 3410, F--41034 BLOIS cedex, France41034 BLOIS cedex, France

serge.serge.dossantosdossantos@@univuniv--tours.tours.frfr

Master studentMaster student course course proposed withinproposed within ERASMUS ERASMUS Program Program

2Master Student course given within ERASMUS Program, Nov 5th 2007 , PragueSerge Dos Santos

OutlineOutline�� acoustic properties of homogeneous medium (liquids, gazes and soacoustic properties of homogeneous medium (liquids, gazes and solids)lids)

�� complexity of heterogeneous and damaged materialscomplexity of heterogeneous and damaged materials

�� General acoustics and wave propagationGeneral acoustics and wave propagation

�� Introduction to Nonlinear SystemsIntroduction to Nonlinear Systems

�� History and simple experimentsHistory and simple experiments

�� concepts and generic physical behaviorsconcepts and generic physical behaviors

�� nonlinear dynamics, analysis and signal processingnonlinear dynamics, analysis and signal processing

�� Engineering applications and future trends related to structuralEngineering applications and future trends related to structural health health

monitoring of aeronautical structuremonitoring of aeronautical structure


Basics on metrology … Basics on metrology …

�� Principle : Principle :

use a measurement device related to the system under analysis use a measurement device related to the system under analysis


celerity in water

celerity = 1500 m/s

wavelength = celerity / frequency

1 MHz : wavelength = 1.5 mm100 MHz: wavelength = 15 micrometers

Principle Principle of of echographyechography

�� Principle : Principle :

The greater the ultrasound frequency, The greater the ultrasound frequency,

the smaller the size of the analyzed systemthe smaller the size of the analyzed system


acoustic properties of homogeneous acoustic properties of homogeneous medium (liquids, gazes and solids)medium (liquids, gazes and solids)


Complexity of ultrasonic wavesComplexity of ultrasonic waves


General acoustics and wave propagationGeneral acoustics and wave propagation


Acoustics and ultrasoundAcoustics and ultrasound

�� Provide fundamental theory on acoustic wave Provide fundamental theory on acoustic wave propagation in semipropagation in semi--infinite isotropic mediainfinite isotropic media

�� Study the impact of material discontinuities on Study the impact of material discontinuities on ultrasonic wave propagationultrasonic wave propagation

�� Explore the physics underlying several Explore the physics underlying several applications that involve bulk acoustic wavesapplications that involve bulk acoustic waves


ContentContent�� Bulk Acoustic Waves in semiBulk Acoustic Waves in semi--infinite mediainfinite media

�� Elastic theoryElastic theory�� Ultrasonic wave modesUltrasonic wave modes�� Properties of propagating wavesProperties of propagating waves

�� Material discontinuities: impact on ultrasonic Material discontinuities: impact on ultrasonic signalsignal�� Reflection and transmission (acoustic impedance)Reflection and transmission (acoustic impedance)

�� Applications of Bulk Acoustic WavesApplications of Bulk Acoustic Waves�� Acoustic imagingAcoustic imaging�� Nondestructive evaluation of material integrityNondestructive evaluation of material integrity�� Material characterizationMaterial characterization


Physics of a solid mediumPhysics of a solid medium

�� Traction forces & stressesTraction forces & stresses


molecules

molecular interactionwith a spring

U=particle displacment

U’

u=particle velocity=∂∂∂∂U/∂∂∂∂t

force

Displacement Displacement of of the medium the medium (or (or particleparticle) ) and particle velocityand particle velocity

Medium = Particles interconnected together through internal elastic forces


3D modeling3D modeling


Linear acousticsLinear acoustics�� Conservation of massConservation of mass

Acoustic hypothesis Acoustic hypothesis ((linearlinear) :) :


Linear acousticsLinear acoustics

�� Conservation of Conservation of the momentumthe momentum ::

�� forces forces related related to pressure variationsto pressure variations

�� densitydensity

�� accelerationacceleration

Acoustic hypothesis Acoustic hypothesis : :


Linear acousticsLinear acoustics�� State State equation equation (case of gazes) (case of gazes)

Acoustic hypothesisAcoustic hypothesis ::


Linear acousticsLinear acoustics�� Conservation + state Conservation + state equation equation

�� After elimination After elimination of of twotwo variables : variables :


Vibration of a stringVibration of a string�� Wave equationWave equation

�� SolutionSolution 2

22

2

2

x

yc

t

y

∂∂=

∂∂

L

Tc

ρ=

)()( 21 xctfxctfy ++−=


Wave propagationWave propagation


Longitudinal and transverse wavesLongitudinal and transverse waves

longitudinal transverse longitudinal transverse


Acoustic impedance of a mediumAcoustic impedance of a medium

�� Acoustic impedanceAcoustic impedance

�� AirAir�� Z= 415Z= 415 RaylsRayls

�� WaterWater�� Z= 1.5Z= 1.5 MRaylsMRayls


Acoustic intensityAcoustic intensity�� IntensityIntensity

�� Stationary wavesStationary waves

�� Progressive plane wavesProgressive plane waves


Acoustic power of a sourceAcoustic power of a source�� Acoustic power of a source : integration of Acoustic power of a source : integration of intensity around the sourceintensity around the source

�� Intrinsic characteristic of the sourceIntrinsic characteristic of the source


Acoustic intensity Acoustic intensity -- DecibelDecibel�� Puissance Puissance

�� Tic Tic --TacTac : microwatt: microwatt�� lanceurslanceurs :: mégawattsmégawatts

�� earring : differential sensitivity S with respect to Iearring : differential sensitivity S with respect to I

�� 0 dB = 0 decibel :0 dB = 0 decibel :

�� in air in air �� PascalPascal


Acoustic intensity Acoustic intensity -- DecibelDecibel


�� Let us Let us consider this problemconsider this problem ::

�� Continuity Continuity of pressure of pressure and particle velocityand particle velocity

�� L’impédance acoustique est donnée par :L’impédance acoustique est donnée par :

�� In In this this casecase

1 1 2p p p+ − ++ = +−+ =− 211 uuu

211

1112

1

1

1

1

11

2

2

11

11 Zpp

ppZZ

Z

p

Z

p

pp

u

p

uu

pp=

−+

⇒=−

+

+⇒=

−+

−+

−+

−+

−+

+

+

−+

−+

u

pZ =

Wave Wave guideguide


Reflection and Reflection and transmission coefficienttransmission coefficient

�� On x=0 On x=0 thethe ratio ratio betweenbetween pressure pressure isis givengiven by by amplitude ratioamplitude ratio

�� Incident Incident and reflected and reflected pressure pressure is given thanks is given thanks to to the continuity the continuity of pressureof pressure

1 1 1 2 11 2

1 1 1 2 1

P P P Z ZZ Z R

P P P Z Z

+ − −

+ − +

+ −= ⇒ = =− +

12

2

12

12 2111

ZZ

Z

ZZ

ZZRTRT

+=

+−

+=+=⇒=−


Reflection Reflection coefficient versus coefficient versus geometrical parametersgeometrical parameters

�� AssumingAssuming

�� We obtainWe obtain

GeometricalGeometrical effets du to effets du to the symmetrythe symmetry of of the mediumthe medium

2c

cZ

d

ρ=

21

21

12

12

dd

dd

ZZ

ZZR

+−

=+−

=


Mismatch impedance samplesMismatch impedance samples

impedance

1

2Z

impedance

Z

�� Direct propagationDirect propagation


Mismatch impedance samplesMismatch impedance samples�� Direct propagationDirect propagation


complexity of heterogeneous and damaged complexity of heterogeneous and damaged materialsmaterials


Damaged structures : macroscopic aspectsDamaged structures : macroscopic aspects


Composite Plates

Imaging of Complex MaterialsImaging of Complex MaterialsCracked Glass

Quentched AlloyMulti-layered Composites


Damaged structures : microscopic aspectsDamaged structures : microscopic aspects


degradation degradation ------> cracks> cracks

��Linear regionLinear region ��Linear regionLinear region��Nonlinear regionNonlinear region


AdvantageAdvantage of of ultrasonicultrasonic nonlinearnonlinear waveswaves

�� How to How to detect smallerdetect smaller cracks : cracks : �� Increase the frequency Increase the frequency of of ultrasoundultrasound ……�� consequenceconsequence : : increase increase of of attenuation attenuation … …

�� Solution : Solution : �� … … increase the ultrasonic increase the ultrasonic power …power …�� consequenceconsequence : : nonnon linears linears effets are effets are createdcreated ((harmonicsharmonics))

�� AdvantageAdvantage ::�� «« NaturalNatural » » increaseincrease of of thethe frequencyfrequency thanksthanks to to harmonicsharmonics


MedicalMedical applications of applications of ultrasonicultrasonic nonlinear nonlinear waveswaves

• Harmonic Imaging

Fréquence

E - Fondamental R - HarmoniqueHarmonicFondamental


Introduction to Nonlinear SystemsIntroduction to Nonlinear Systems

History and simple experiments

concepts and generic physical behaviors

nonlinear dynamics, analysis and signal processing


Double pendulum : modelDouble pendulum : model

1 1 1sinx l θ=

1 1 1cosy l θ= −2 1 1 2 2sin sinx l lθ θ= +

2 1 1 2 2cos cosy l lθ θ= − −

1 1 2 2

1 2 1 1 2 2 2( ) cos cos

V m gy m gy

V m m gl m glθ θ= += − + −

( )

( )( )

2 21 1 2 2

2 2 2 2 2 21 1 1 2 1 1 2 2 1 2 1 2 1 2

1

21

2 cos( )2

T m v m v

T m l m l l l lθ θ θ θ θ θ θ

= +

= + + + −& & & & &


……etet équationséquations..


Double pendulum : modelDouble pendulum : model

�� http://http://wwwwww.maths..maths.tcdtcd..ieie/~/~plynchplynch//SwingingSprinSwingingSpringg//doublependulumdoublependulum..htmlhtml

�� http://www.http://www.myphysicslabmyphysicslab.com/dbl_pendulum.ht.com/dbl_pendulum.htmlml


...and chaos......and chaos...

http://http://scienceworldscienceworld.wolfram.com/physics.wolfram.com/physics//DoublePendulumDoublePendulum.html.html


Approximation of nonlinear systems : Approximation of nonlinear systems :

linear systems … and limitationslinear systems … and limitations


Limits of linear model…Limits of linear model…

�� linearisationlinearisation ……

�� http://www.http://www.zfmzfm..ethzethz..chch//mecameca/applets//applets/doppelpendeldoppelpendel//doppelpendeldoppelpendel.html.html

�� http://www.http://www.zfmzfm..ethzethz..chch//mecameca/applets//applets/doppelpendeldoppelpendel//dPendulumdPendulum.html.html


Linear ModelingLinear Modeling

�� Modeling withModeling with LaplaceLaplace transformstransforms�� Automation, control systems, …Automation, control systems, …

�� Modeling with Fourier TransformsModeling with Fourier Transforms�� electronics, optics and acoustics, …electronics, optics and acoustics, …

�� Linear systems are described with Linear systems are described with �� Linear partial derivative equationsLinear partial derivative equations�� With constant coefficients With constant coefficients

�� Resolution is done thanks to the general properties of linearResolution is done thanks to the general properties of linear albegraalbegra(matrix formalism) (matrix formalism)

�� Behavior of Linear system can be computed numerically thanks to Behavior of Linear system can be computed numerically thanks to the the superposition principle superposition principle �� The final state is a superposition of discrete intermediate statThe final state is a superposition of discrete intermediate states (in es (in space and in time) space and in time)


Nonlinear signatureNonlinear signature�� Harmonics generationHarmonics generation�� intermodulation intermodulation �� Modulation, autoModulation, auto--modulationmodulation�� Amplitude dependant of “classical linear signatures”Amplitude dependant of “classical linear signatures”

�� Resonance frequencyResonance frequency�� attenuationattenuation

�� subsub--harmonicsharmonics�� Low frequency effects <Low frequency effects <--> slow dynamics> slow dynamics�� chaos...chaos...

�� Generic signature in various physical systems Generic signature in various physical systems �� MecanicsMecanics, optics, electronics, acoustics, control, optics, electronics, acoustics, control

Nonlinearity level


Excitation of Nonlinear Systems : Excitation of Nonlinear Systems : experimentsexperiments

�� Linear systems (amplitude is not critical)Linear systems (amplitude is not critical)�� time domain : pulse time domain : pulse

�� frequency domain : sine waves arefrequency domain : sine waves are eigeneigen--functionsfunctions

�� Nonlinear systems (amplitude is critical)Nonlinear systems (amplitude is critical)�� time domain : pulse amplitude must be known (calibration)time domain : pulse amplitude must be known (calibration)

�� frequency domain : sine waves are notfrequency domain : sine waves are not eigeneigen--functions (modulation)functions (modulation)

�� Attenuation and frequency are timeAttenuation and frequency are time--dependant (Slow dynamic)dependant (Slow dynamic)

�� scaling effects : how to take into account them systematicallyscaling effects : how to take into account them systematically

�� It depends It depends on on thethe systemsystem�� how to how to find suchfind such excitations ?excitations ?


Excitation of Nonlinear Systems : conceptExcitation of Nonlinear Systems : concept�� Linear systemsLinear systems

�� output spectrum properties are «output spectrum properties are « invariantinvariant » with respect to excitation» with respect to excitation�� lots of invariants including scaling effects, reciprocity and tilots of invariants including scaling effects, reciprocity and time reversalme reversal

�� Nonlinear systemsNonlinear systems�� spectrum is modified : spectrum representation in not an «spectrum is modified : spectrum representation in not an « invariantinvariant »»�� is it still interesting to look at frequency components ?is it still interesting to look at frequency components ?�� what is the nextwhat is the next candidatcandidat instead of sine wave excitation?instead of sine wave excitation?

�� time evolution of frequency representation ?time evolution of frequency representation ?

)().()(

)()()(

fEfRfS

tEtRtS

=∗=

R(t) not dependant of amplitude A

)2cos()( tfAtE π= R(t) S(t)

�� It depends It depends on on thethe systemsystem�� how to how to find suchfind such invariant ?invariant ?


Evolution …Evolution …""Tractatus logicoTractatus logico--philosophicusphilosophicus" , Ludwig Wittgenstein (1889 1951)" , Ludwig Wittgenstein (1889 1951)

«« The world is all that is the caseThe world is all that is the case »»

Unification ofUnification of mecanicsmecanics and thermodynamics (1900) and thermodynamics (1900) «« The world is statisticalThe world is statistical »»

After Einstein (1910) …After Einstein (1910) …«« The world is relativeThe world is relative »»

After QuantumAfter Quantum MecanicsMecanics (1930) (1930) «« The world is quantaThe world is quanta »»

Nonlinearity now is recognized as being fundamental in almost anNonlinearity now is recognized as being fundamental in almost any area of physics, notably y area of physics, notably hydrodynamics, optics, acoustics, and extends to chemistry, biolhydrodynamics, optics, acoustics, and extends to chemistry, biology, ecology [1]ogy, ecology [1]

Thus, todayThus, today«« The world is nonlinearThe world is nonlinear » »

[1] :«[1] :« Vision of Nonlinear Science in the 21th CenturyVision of Nonlinear Science in the 21th Century »,», HuertasHuertas, Chen,, Chen, MadanMadan, 1999, 1999


Every Every particular particular

problem needs problem needs a specific toola specific tool


ModelingModeling


Classical nonlinearityClassical nonlinearity


Nonlinear equation : example of Burger’s equationNonlinear equation : example of Burger’s equation(Nonlinear plane waves with attenuation)(Nonlinear plane waves with attenuation)

�� From state equation and momentum conservation :From state equation and momentum conservation :

�� Introducing scaling Introducing scaling effets effets

�� Second order in Second order in εε lead to the Burgerlead to the Burger’’s equation s equation

AttenuationAttenuation nonlinearitynonlinearity

Non linear parameterNon linear parameter


Consequences : nonlinear signatureConsequences : nonlinear signature

�� If attenuation is negligibleIf attenuation is negligible

�� Assuming Assuming

�� One obtainOne obtain

Harmonics (2Harmonics (2ωω11 , 2, 2ωω22 ))

Intermodulation Intermodulation productsproducts

((ωω11++ωω22 ; ; ωω11--ωω22 ))


«« nonclassicalnonclassical » nonlinearity» nonlinearity


TitaneTitane (Guyer et al.,Phys. Rev. Lett. 1995)

Elementary element

L(o) δ(o)δ(o) << L(o)

Elastic grainHysteretic element

Microscopic stress / strain relation:

Strain

StressPo Pc

1rK −

rγ

Strain

StressPo Pc

1rK −

rγ(Scalerandi et al.,

JASA 2003)

Homogenization in PM Space : non classical nonlinearity

Nonlinear Wave Propagation with PM space Nonlinear Wave Propagation with PM space modeling of damaged materialsmodeling of damaged materials


3D Kelvin notation model

11 12 13

12 11 13

13 13 33

44

44

66

0 0 0

0 0 0

0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

KelvinIJ

C C C

C C C

C C CC

C

C

C

=

NumericalNumerical Model DescriptionModel Description

0

1 iji

j

v

t x

τρ

∂∂ =∂ ∂ ij ijkl klCτ ε=

z (x3 axis)

• Crack orthogonal to z:bond, contact

• Geometrical nonlinearity negligible ; source of nonlinearity : the defect

• Due to geometry of considered cracks (penny shape): effective medium supposed to have a transverse isotropic behavior

Kelvinnotation

• Basic equations:


NumericalNumerical Model DescriptionModel Description

(5)1 1

(5)1

(1) (2) (3) (4) (5) (6)

(5) 2 (6) 2

1 0 0 0

1 0 0 0

0 0 0 01 1 11, , , , ,

0 1 0 02 01 2( ) 1 2( )0 0 1 0 00 0 0 1 0

ε εε

ε ε ε ε ε εε ε

− = = = = = = + +

(6)

(6)1

1

0

0

0

ε

(5,6)1 11 12 33 13( ) 4C C C X Cε = + − ± 2 2

13 11 12 338 ( )X C C C C= + + −with

• 6 eigenvectors of the elastic constants tensor corre spond to 6 eigenstress/eigenstrain vectors:

applied stress and created strain are in the same direction

a scalar PM-space model is used for each of these 6 directions

Inverse transformation in the initial space


Nonlinear Propagation in Hysteretic MaterialNazarov et al. Model (JASA 2000, Acoust. Phys. 2000)

( ) ( )( ) ( )εεαρεεεεσ &&s

gfK ++−= 1, 00

( ) ( )

( )

11

2 1 2

31

4 3 4

0 0

0 01,

0 0

0 0

n

n nm

n

n nm

if

iff

ifn

if

ε εγ εε εγ ε γ γ ε ε

ε εε εγ εε εγ ε γ γ ε ε

−+

−−

> > > <− + += < <− < >− +

&

&&

&

&

x

u

∂∂=ε

x

v

∂∂=ε&with and

mε +

mε −

Stress

Strain


SimulationsSimulations


1 4 100γ γ= =

2 3 10γ γ= =

20 40 60 80 100 120-12

-10

-8

-6

-4

-2

0

x 10-4

Distance (µm)

Str

ain

1020 1040 1060 1080 1100 1120

-12

-10

-8

-6

-4

-2

0

x 10-4

Distance (µm)

Str

ain

0

Simulation of Nonlinear Propagation in Hysteretic Material


1 2 3 4 50γ γ γ γ= = = =

20 40 60 80 100 120-12

-10

-8

-6

-4

-2

0

x 10-4

Distance (µm)

Str

ain

1020 1040 1060 1080 1100 1120

-12

-10

-8

-6

-4

-2

0

x 10-4

Str

ain

Distance (µm)

0

Simulation of Nonlinear Propagation in Hysteretic Material

Soliton ?


John Scott Russell (1808-1882)

Solitary Waves

http://www.ma.hw.ac.uk/~chris/scott_russell.html

- Scottish engineer at Edinburgh- Committee on Waves: BAAC


Reconstitution of Soliton (1995)

Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995


Model of Long Shallow Water WavesD.J. Korteweg and G. de Vries (1895)

22

2

3 1 2 1

2 2 3 3

g

t l x x

η ηη αη σ ∂ ∂ ∂= + + ∂ ∂ ∂

- surface elevation above equilibrium

- depth of water

- surface tension

- density of water

- force due to gravity

- small arbitrary constant

lTρgα

31

3

Tll

gσ

ρ= −

η


6 0t x xxxu uu u− + =

Nonlinear term Dispersion term

6 0t xu uu− = 0t xxxu u+ =

Korteweg-de Vries (KdV) Equation

3 2, , 2

2 3

g xt t x u

lη α

σ σ→ → − → − −

Model:

KdV equation :

steppen flatten

t

x

uu

tu

ux

∂=∂∂=∂

22

2

3 1 2 1

2 2 3 3

g

t l x x

η ηη αη σ ∂ ∂ ∂= + + ∂ ∂ ∂

Rescaling:


Stable Solutions

Steepen + Flatten = Stable

- Unchanging in shape- Bounded- Localized

Profile of solution curve:

Do such solutions exist?


Solitary Wave Solutions1. Assume traveling wave of the form:

2. KdV reduces to an integrable equation:

3. Cnoidal waves (periodic):

( , ) ( ),u x t U z z x ct= = −

3

36 0

dU dU d Uc U

dz dz dz− − + =

( )2( ) cn ,U z a bz kδ= +


2 2 2 2( , ) 2 sech ( 4 ) ) , 4u x t k k x k t c kδ = − − + =

Solitary waves (one-soliton):

2( ) sech2 2

c cU z z δ

= − +

x

- u

x

- u

x

- u

x

- u

x

- u


Other Types of Soliton

Equation de Sine-Gordon Equation

sinxx ttu u u− =

- Superconductors (Josephson tunneling effect) - Relativistic field theories

Nonlinear Schrödinger (NLS) Equation :

20t xxiu u u u+ + =

- Fiber optic transmission systems- Lasers


"Anywhere you find waves you find solitons"-Randall Hulet, Rice University, on creating solitons in Bose-Einstein condensates, Dallas Morning News, May 20, 2002

Many Faces of Solitons -Future of SolitonsQuantum Field Theory General Relativity

- Quantum solitons- Monopoles- Instantons

- Bartnik-McKinnon solitons (black holes)

Biochemistry

- Davydov solitons (protein energy transport)

Solitons as a new NDT pulse-echo techniques ??? ….

Non Destructive Testing ?


Symmetry of Nonlinear SystemsSymmetry of Nonlinear Systems


We need a super mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups.

- Sir Arthur Stanley Eddington

A falling drop of milk has circular symmetry…

…but after impact a ”crown” rises that only has 24 possible

rotations.

The Natural Language of Symmetry The Natural Language of Symmetry --Group TheoryGroup Theory

Infinite number of Infinite number of rotations and reflectionsrotations and reflections

SO(2)SO(2)

24 rotations and 12 24 rotations and 12 reflectionsreflections

DD2424


Symmetry of Nonlinear EquationsSymmetry of Nonlinear Equations

�� Solutions :Solutions :�� quadraturequadrature integrationintegration

�� separation of variablesseparation of variables

�� Theoretical point of view: invariance with respect toTheoretical point of view: invariance with respect to

�� This is a Lie continuous group of translation of This is a Lie continuous group of translation of parameter parameter aa


Lie Groups, Symmetry and InvariantsLie Groups, Symmetry and Invariants

�� Properties :Properties :�� solutions reveals the equation symmetrysolutions reveals the equation symmetry�� it works on Nonlinear Systemsit works on Nonlinear Systems�� symmetry exhibit conserved quantities: invariantssymmetry exhibit conserved quantities: invariants

�� ExamplesExamples

Symmetry Symmetry of of the equationthe equation

Time translation invarianceTime translation invarianceSpaceSpace translation invariancetranslation invarianceRotationalRotational invarianceinvariance

Conserved quantity Conserved quantity (invariants)(invariants)

EnergyEnergyMomentumMomentumAngular MomentumAngular Momentum

PropertyProperty

homogeneity homogeneity of timeof timehomogeneity homogeneity of of spacespaceisotropy isotropy of of spacespace


Examples (part 1) Examples (part 1)


Solutions of Burgers EquationSolutions of Burgers Equation

�� Nonlinear Burgers equationNonlinear Burgers equation

�� HopfHopf--Cole Cole transformationtransformation

�� Linear Linear Diffusion Diffusion equationequation

Lie Groups Lie Groups and Symmetriesand Symmetries


Heat equationHeat equation�� PDE (Partial derivative Equation)PDE (Partial derivative Equation)

Symmetry andSymmetry and Lie groups Lie groups

Change of variable Change of variable

�� Ordinary DifferentialOrdinary Differential EquationEquation

If b=2aIf b=2a


Spherical wavesSpherical waves

Symmetries andSymmetries and Lie groupsLie groups

Change of variable/Change of variable/functionfunction

�� Plane Plane waveswaves

Spherical waves Spherical waves --------> Plane > Plane waves waves : : Scaling effectScaling effect


�� Transformation of variables : time, scaling and spatial coordinaTransformation of variables : time, scaling and spatial coordinates, etctes, etc

�� Transformation of functions : velocity, acoustic stress, strain,Transformation of functions : velocity, acoustic stress, strain, etc...etc...

Which transformation ?Which transformation ?

Theoretical framework : Symmetry AnalysisTheoretical framework : Symmetry Analysis


�� Group transformation of variables and functions Group transformation of variables and functions

�� Invariance condition of EquationInvariance condition of Equation

�� Absolute set of complete invariantsAbsolute set of complete invariants

Symmetry of Nonlinear EquationsSymmetry of Nonlinear Equations


Assuming equation (1)

Transformation of coordinates ...

… is a symmetry of equation (1) if formulation is conserved, i.e.

infinitesimal generators of associated Lie group are given by :

coming from

Symmetry of Nonlinear EquationsSymmetry of Nonlinear Equations(generalization)(generalization)


Lie Algorithm and Lie Algorithm and reduction of equationsreduction of equations

Computation of all infinitesimals allow extraction of invariants given by

the following caracteristic equation

References


KZ equation = Burger’s equation with diffractionKZ equation = Burger’s equation with diffraction


Solutions of KZSolutions of KZ

�� Lapidus and RudenkoLapidus and Rudenko


Pre and post Signal Processing :Pre and post Signal Processing :

Time Reversal invarianceTime Reversal invarianceReciprocity invarianceReciprocity invariance

SymmetrisationSymmetrisation of excitationof excitation


Time Reversal for NEWSTime Reversal for NEWS

�� NEWS : Nonlinear Elastic Wave SpectroscopyNEWS : Nonlinear Elastic Wave Spectroscopy

�� What is the nonlinear signature due to damaged area ?What is the nonlinear signature due to damaged area ?

�� A high level of ultrasound is needed A high level of ultrasound is needed

�� How to localize sources of nonlinearity ?How to localize sources of nonlinearity ?

�� Time ReversalTime Reversal

�� RetrofocusedRetrofocused signal with high level of ultrasound (for NL effects) signal with high level of ultrasound (for NL effects)

�� Temporal focusing : reconstruction of coherent toneTemporal focusing : reconstruction of coherent tone--burst signalsburst signals

�� Spatial focusing : analysis on localized point : the focused regSpatial focusing : analysis on localized point : the focused region ion

(practically measurements are done with laser interferometers )(practically measurements are done with laser interferometers )

M. Fink ,M. Fink ,IEEE Trans on UFFC (1992)IEEE Trans on UFFC (1992)

K. Van DenK. Van Den AbeeleAbeele, P.A. Johnson, and A., P.A. Johnson, and A. SutinSutin. . Res.Res. NondestrNondestr.. EvalEval. ( 2000). ( 2000)


Nonlinear Elastic Wave Spectroscopy (NEWS)Nonlinear Elastic Wave Spectroscopy (NEWS)and Time Reversal (TR)and Time Reversal (TR)

�� Time Reversal combined with nonlinear signal processing toolsTime Reversal combined with nonlinear signal processing tools

GoursolleGoursolle T , T , CalléCallé S , S , Bou Matar Bou Matar O, Dos Santos S, O, Dos Santos S, in proceedings of the 2006 Vancouver IEEE Ultrasonic in proceedings of the 2006 Vancouver IEEE Ultrasonic

Symposium and J.Symposium and J. AcoustAcoust. Soc. Am., 122 (6), (2007). Soc. Am., 122 (6), (2007)

local nonlinear imaginglocal nonlinear imaging

Experimental validation :Experimental validation :J. Ulrich, P. A. Johnson, J. Ulrich, P. A. Johnson,

and R.A. and R.A. GuyerGuyer, , Phys. Rev.Phys. Rev. LettLett. (2007). (2007)


NEWSNEWS--TR and TRTR and TR--NEWSNEWS(concepts and results)(concepts and results)


Source

Receivers

Receivers

Rec

eive

rsClassical TRClassical TR

Source of nonlinearity (damaged region)

M. Fink ,M. Fink ,IEEE Trans on UFFC (1992)IEEE Trans on UFFC (1992)


Source

Receivers

Receivers

Rec

eive

rs

+ Harmonic filtering



NEWSNEWS--TRTR

Source of nonlinearity(damaged region)

P.-Y. Le Bas, K.E-A. Van Den Abeele, S. Dos Santos, T. Goursolle, and O. Bou Matar. Experimental analysis for nonlinear time reversal imaging of damaged materials. In Proceedings of the 9th European Conference on Non-Destructive Testing, Berlin, 2006

O Bou Matar, S. Dos Santos, S. Callé, T. Goursolle, S. Vanaverbeke, and K.E-A. Van Den Abeele. Simulations of nonlinear time reversal imaging of damaged materials. In Proceedingsof the 9th European Conference on Non- Destructive Testing, Berlin, 2006


• Retrofocusing on the source

• Focal spot size (5mm) linkedto the source (size, shape) and emission frequency

3D 3D LinearLinear retrofocusing retrofocusing (simulations) (simulations) mapping of the max of amplitude during the back pro pagation

O. Bou Matar, S. Callé, T. Goursolle, S. Dos Santos, 3D Simulations of Nonlinearity based Time Reversal Imaging of Damaged Materials, in proc of the International Congress on Ultrasonics, Vienna, April 9 - 13, 2007


Linear mediumLinear medium

Nonlinear Signal Processing :Nonlinear Signal Processing :SymmetrisationSymmetrisation of Excitation with Pulse Inversion (PI) of Excitation with Pulse Inversion (PI)

x(t)x(t)

-- x(t)x(t)

Nonlinear mediumNonlinear mediumx(t)x(t) y(t)y(t)

Nonlinear responseNonlinear response


Pulse Inversion (PI) methodPulse Inversion (PI) methodInversion group (CInversion group (C22) interpretation) interpretation

Extension for the 3rd order nonlinearity


Higher order Pulse Inversion (PI) methodHigher order Pulse Inversion (PI) method

C3 character table and irreducible representation

New « symmetrized » excitations


New New symmetrized symmetrized excitationsexcitations

Multiply initial excitation x(t) by Multiply initial excitation x(t) by coscos( 2 pi/3)= ( 2 pi/3)= --0.5 and sin (2 pi/3)= 0.5 and sin (2 pi/3)= √√3/2 , 3/2 , for examplefor exampleA, 2A, 4A, 8A, 16 A ; B=A, 2A, 4A, 8A, 16 A ; B=√√3/2 A, 2B, 4B, 8B, 16B 3/2 A, 2B, 4B, 8B, 16B with pulse inversion A with pulse inversion A �� --A and B A and B �� --BB


x

y

y

z

DefectRetropropagation of nonlinear components

3D 3D Nonlinear retrofocusing Nonlinear retrofocusing of NEWSof NEWS--TRTR

O. Bou Matar, S. Callé, T. Goursolle, S. Dos Santos, 3D Simulations of Nonlinearity based Time Reversal Imaging of Damaged Materials, in proc of the International Congress on Ultrasonics, Vienna, April 9 - 13, 2007


TRTR--NEWSNEWS


zoom

Time reversal

Principle of TRPrinciple of TR--NEWS (from experiments)NEWS (from experiments)

January 2002 – Experiment at LUSSI BloisEmission : contact transducerReception : BMI heterodyne interferometric laser


TR signal

TRTR--NEWS application : NEWS application : AERONEWS wing panel (may 2006, Prague)AERONEWS wing panel (may 2006, Prague)

initial excitationTime reversed signal

Analysisdirect signal


TRTR--NEWS resultsNEWS results�� Frequency mixing : f1=185 kHz ; f2= 495 kHz + pulse inversion Frequency mixing : f1=185 kHz ; f2= 495 kHz + pulse inversion

f1

f2

0 5 10 15 200

2

4

6

8

10

Final scan


TRTR--NEWS : reciprocity validation (3D simulations)NEWS : reciprocity validation (3D simulations)T. T. GoursolleGoursolle et al , in Proc of the IEEE symposium, et al , in Proc of the IEEE symposium, New York, Oct. 2007New York, Oct. 2007

TRTR--NEWSNEWS Classical Classical TRTR

1 2 3 4 50

0.5

1

1.5

2

retrofocused position

norm

aliz

ed a

mpl

itude

1' 2' 3' 4' 5'0

0.5

1

1.5

2

retrofocused position

norm

aliz

ed a

mpl

itude

NL signature (PI)


Improving focusing and Improving focusing and nonlinearity extraction for nonlinearity extraction for NEWSNEWS--TR and TRTR and TR--NEWSNEWS


Quality of the focusingQuality of the focusing�� January 2002 : LUSSI January 2002 : LUSSI –– Single Channel Single Channel

TRA TRA –– one emitterone emitter--one receiverone receiver

�� January 2006 : AERONEWS 2nd year January 2006 : AERONEWS 2nd year meeting meeting –– Multiple single frequency Multiple single frequency emitters (16) for “Breaking the symmetry emitters (16) for “Breaking the symmetry of excitation” of excitation” –– One receiverOne receiver

�� May 2006 : AERONEWS 1stMay 2006 : AERONEWS 1st pragueprague week week ––Multiple dual frequency emitters Multiple dual frequency emitters –– One One receivedreceived

�� ……


Strategies for focusing improvementStrategies for focusing improvement

�� Increase emitters (number, position, properties)Increase emitters (number, position, properties)

�� Increase reverberant properties (increase complexity !!! )Increase reverberant properties (increase complexity !!! )

�� Decrease symmetry in the medium !!!!! ( cancel modes and “regulaDecrease symmetry in the medium !!!!! ( cancel modes and “regular” behavior )r” behavior )

�� Optimize pre and post signal processing Optimize pre and post signal processing �� increase symmetry properties during the excitation process (advaincrease symmetry properties during the excitation process (advanced pulse nced pulse

inversion)inversion)�� ChirpChirp--coded excitation for improvement of the focusing spotcoded excitation for improvement of the focusing spot


0 200 400 600 8000

0.02

0.04

0.06

0.08

0.1

Spectrum Spectrum of of the chirpthe chirpexcitation x(t)excitation x(t)

Central Central frequency frequency : 400 kHz: 400 kHz

Signal Signal processing processing

ChirpChirp--coded NEWScoded NEWS--TRTR

Acoustic response Acoustic response

M. H. Pedersen, T.X. Misaridis, and J.A. Jensen, “Clinical Evaluation of Chirp-Coded Excitation in Medical Ultrasound”, Ultras. in Med. & Biol., 29, pp. 895-905 (2003)

�� ChirpChirp--coded excitation in linear ultrasound:coded excitation in linear ultrasound:�� transmit more energy per time without transmit more energy per time without

increasing the peak intensity increasing the peak intensity �� Increase the SNR ratioIncrease the SNR ratio

**

Correlation Correlation ~ impulse ~ impulse responseresponse


0 200 400 600 8000

0.02

0.04

0.06

0.08

0.1

Spectrum Spectrum of of the chirpthe chirpexcitation x(t)excitation x(t)

Central Central frequency frequency : 400 kHz: 400 kHz

Signal Signal processing processing

ChirpChirp--coded NEWScoded NEWS--TRTR

Acoustic response Acoustic response

�� PSTD simulations (PSTD simulations (smallersmaller amplitude amplitude isis neededneeded) : ) :

T. Goursolle, S. Dos Santos, S. Callé, and O. Bou Matar, 3D PSTD Simulations of NEWS-TR and TR-NEWS Methods: Application to Nonclassical Nonlinearity Ultrasonic Imaging, in proc. of the IEEE Ultrasonic Symposium, New York (2007)

Time ReversalTime Reversal


ChirpChirp--coded TRcoded TR--NEWS experiments NEWS experiments (symmetry invariance with respect to PI excitation)(symmetry invariance with respect to PI excitation)

Nonlinearity breaks invariance of convolution respo nse

x(t) � - x(t)h(t) � -h(t)y+(t) � y-(t)=y+(t)

Direct and Inverse normalized Time Reversal signal

3 ms coda zoom

Bracket n 3 after

fatigue tests

with Panametrics

transducers


ChirpChirp--coded TRcoded TR--NEWS (nonlinear response)NEWS (nonlinear response)x(t) � - x(t)h(t) � -h(t) + nonlinear responsey+(t) � y-(t)=y+(t) + nonlinear effects

Nonlinearity extractedNonlinearity extracted versus Amplifier versus Amplifier ResearchResearch (AR) (AR) levellevel of of the sweep the sweep excitationexcitation

Bracket sample

fatigue tests

with Panametrics

transducers

Normalized spectrum of TR signal


Engineering applications and future trends Engineering applications and future trends related to structural health monitoring of related to structural health monitoring of

aeronautical structureaeronautical structure


ExampleExampleEuropean Sixth Framework Program AERONEWSEuropean Sixth Framework Program AERONEWS

Health Monitoring of Aircraft byHealth Monitoring of Aircraft byNonlinear Elastic Wave SpectroscopyNonlinear Elastic Wave Spectroscopy

Problematic :Problematic :

�� NonNon--destructive Testing of Aeronautic Structures destructive Testing of Aeronautic Structures

�� Localization of cracks (hysteretic behavior)Localization of cracks (hysteretic behavior)

Tool : Tool :

�� Nonlinear Acoustics in Complex Medium Nonlinear Acoustics in Complex Medium

�� ExperimentsExperiments

�� Theory and simulationsTheory and simulations

AERONEWS PARTNERS

Belgium: Catholic University Leuven, Campus Kortrijk (KULeuven)Free University Brussels (VUB)ASCO Industries (ASCO)

Czech Republic: DAKELAeronautical Research and Test Institute (VZLU)Institute of Thermomechanics (ITASCR)

France: GIP ULTRASONS – LUSSI FRE 2448 CNRSNDT-Expert

Germany: Fraunhofer Institute for NDT, Saarbrücken (IZFP)Italy: Politecnico di Torino (POLITO)

University of Naples, Dept. of Aeron. Eng. (UNI-NA)Spain: Instituto de Acustica, CSIC

Boeing Research & Technology Europe – Madrid Sweden: BodyCote Celsius Saab Materialteknik (CSM)UK: University of Exeter

University of BristolUniversity of NottinghamCranfield University, College of Aeronautics, Cranfi eld

PARTNERS:12 Universities

and Research Institutes

6 SME’s and Industries

8 EU Countries

BUDGET:4.9 M€

EU Contribution3.6 M€


Application of NEWS on simple AERONEWS components,Application of NEWS on simple AERONEWS components,List of NEWS procedures focused onList of NEWS procedures focused on

- Nonlinear ultrasonic transmission and reflection comprising harmonic generation of a narrowband excitation (IZFP, ITASCR, UNEXE)

- Ultrasonic wave mixing (ITASCR, UNIVBRIS) and bi-spectral analysis (UNIVBRIS)- Nonlinear resonance and reverberation spectroscopy (KULAK)- Nonlinear time reversal acoustics (KULAK, GIP-U)- High frequency multi-sine broadband excitation (VUB, ASCO)- Calibrated phase modulation in bi-layered solids (GIP-U)- Frequency response function technique (UNI-Na) using arrays of actuators and sensors to determine a damage index by recognition-based neural network learning

- Second and third order parametric excitation (CSIC) profiting by the strong dispersivecharacter of flexural waves

- Phase-coded pulse-sequence (PCPS) technique (CSM, KULAK)

The partners from industry mainly focus on - Sample preparation, especially complex real parts (ASCO, VZLU, BR&TE, CSM)- Conventional NDT techniques, application and evaluation as reference (NDTE, CSM)- Concept of Structural Health Monitoring, integration of the NEWS techniques proposed by the project in real maintenance programs (BR&TE)


Application of NEWS on simple AERONEWS components,Application of NEWS on simple AERONEWS components,List of NEWS procedures focused onList of NEWS procedures focused on

XXPhasePhase--coded pulsecoded pulse--sequence (PCPS)sequence (PCPS)

XXXXSecond and third order parametric excitationSecond and third order parametric excitation

XXFrequency response function techniqueFrequency response function technique

XXXXCalibrated phase modulationCalibrated phase modulation

XXHigh frequency multiHigh frequency multi--sine broadband excitationsine broadband excitation

XXNonlinear time reversal acousticsNonlinear time reversal acoustics

XXNonlinear resonance spectroscopyNonlinear resonance spectroscopy

XXXXUltrasonic wave mixingUltrasonic wave mixing

XXXXHarmonic generation of a narrowband excitationHarmonic generation of a narrowband excitation

GlobalGlobalLocalLocalTechniqueTechnique


More detailed infoMore detailed info

Health Monitoring of Aircraft byHealth Monitoring of Aircraft byNonlinear Elastic Wave SpectroscopyNonlinear Elastic Wave Spectroscopy

FP6FP6--502927502927http://www.http://www.kuleuvenkuleuven--kortrijkkortrijk.be/.be/aeronewsaeronews


ConclusionConclusion� Nonlinear Acoustics provides a natural tool for exploring media below the wavelength

� Modeling nonlinear system is completely different than linear systems

� Superposition principle is not applicable

� All classical approaches (Fourier description, … ) must be used carefully

� Instrumentation must include calibration for nonlinear measurements

� consequence of the amplitude dependant property


AcknowledgementsAcknowledgementsThis course is supported by the European Union Sixth This course is supported by the European Union Sixth

Framework Program AERONEWSFramework Program AERONEWS

and the ERASMUS Program and the ERASMUS Program between between

České vysoké učení technickéČeské vysoké učení technické v v PrazePrazeCzech Technical University in PragueCzech Technical University in Prague

andandl‘l‘Ecole Nationale d’Ingénieurs du Ecole Nationale d’Ingénieurs du Val de Loire, Val de Loire, Blois Blois , FRANCE, FRANCE

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