Diffraction Effects on Ultrasonic Waves

Diffraction Effects on Ultrasonic WavesDiffraction Effects on Ultrasonic Waves

Radiated by a Transducer Mounted on the Radiated by a Transducer Mounted on the

Section of a Guide of Arbitrary GeometrySection of a Guide of Arbitrary Geometry

byby Karim JezzineKarim Jezzine and and Alain LhémeryAlain Lhémery

French Atomic Energy CommissionCEA - Saclay

• Context of developmentContext of development

• TheoryTheory• brief reviewbrief review• adaptation of Semi-Analytic F.E. methodadaptation of Semi-Analytic F.E. method

• Validation in the axisymmetric caseValidation in the axisymmetric case

• Transducer diffraction effectsTransducer diffraction effects• influence influence of aof apertureperture // bandwith bandwith in the in the a axisymmetric casexisymmetric case• 2D case2D case: : rectangular cross sectionrectangular cross section

• Summary – Summary – WWorkork in progress in progress

Context of developmentContext of development• industrial needs for simulation tools dealing with guided waves in CIVA (software platform for NDE simulation developed at CEA)

- prediction of transducer diffraction effects to optimise testing configuration (mode selection etc.)

- simulation of radiation, propagation, scattering by a defect and reception• aims: guides of arbitrary section

computer efficiency => 3D computational methods hopeless• 1st application: unusual configuration of testing where the transducer (emitter/receiver) is mounted on the guide section

z

transducer

waveguide

guide axis

S

guide section

• Context of developmentContext of development

• TheoryTheory• brief reviewbrief review• adaptation of Semi-Analytic F.E. methodadaptation of Semi-Analytic F.E. method

• Validation in the axisymmetric caseValidation in the axisymmetric case

• Transducer diffraction effectsTransducer diffraction effects• influence influence of aof apertureperture // bandwith bandwith in the in the a axisymmetric casexisymmetric case• 2D case2D case: : rectangular cross sectionrectangular cross section

• Summary – Summary – WWorkork in progress in progress

Theory: Theory: brief reviewbrief review

• basic idea: benefit of the symmetry of translation to restrict computations to the guide section modal decomposition

n nnn

n nnn

tzkj

tzkj

n

n

eyxAzyxAzyx

eyxAzyxAzyx

)(

)(

),(~),,(),,(

),(~),,(),,(

nn

nn

σσσ

uuu propagation along z

mixed: either ux, uy,zz , or, xz,yz,uz at z=0

pure: either ux,uy,uz , or xz,yz,zz (e.g.: piezo transducer in direct contact) at z=0

modal decomposition : un , n

imposed end conditions: mixed or pure• find An knowing :

• direct projection of the source terms on the mode basis impossible for pure end conditions

nzz(n)n(source)zz

nyz(n)n

nxz(n)n

σAσ

σAσA

~

~0~0

minimisation of residual boundary values at z = 0

Gregory & Gladwell , Quart. J. Mech. Appl. Math. (1989) Puckett & Peterson, Ultrasonics (2005)

• requirement: initial computation of modes

- roots of analytical solutions (dispersion equation) for multi-layered plates or cylinders (expl. Disperse, NDT group at Imperial College)

Lowe, I.E.E.E. Trans. UFFC (1995);

+ : « exact » dispersion equations – multi-layered- : two geometries only (plate, cylinder)

- Semi-Analytical Finite Element (S.A.F.E.) method:

Dong & Nelson, J. Appl. Mech. (1972); Gavrić, J. Sound. Vib. (1995); Hayashi & Rose, Mat. Eval. (2003);Damljanović & Weaver, J. Acoust. Soc. Am. (2004);

+ : arbitrary section + easy to account for anisotropy, viscoelasticity

- : computer intensive at high frequencies

Theory: Theory: brief review brief review (contd.)(contd.)

computation of modesin complex cases

Theory: Theory: adaptation of SAFE to pb. of radiation from the sectionadaptation of SAFE to pb. of radiation from the section

• meshing of the section by finite elements:

elements: 1D (plate/axisym.) 2D (arbitrary section)

0)( 22 iiii3

i2

i1 dMdKKK kjk

• application of the principle of virtual work at the i-th element:quadratic eigensystem (size 9x9) in k

)(),(])[;,,,( tkzji eyxikzyx dNu

matrix of interpolation (quadratic) functions

nodal displacement

z-propagator

• displacement field at the i-th element (for 2D case):

Theory: Theory: adaptation of SAFE… adaptation of SAFE… (contd.)(contd.)

• assembly of a 3M x 3M quadratic eigensystem (M nodes)

0)( 22 MddKKK 321 kjk

• once solved: 6M eigenvalues (wavenumbers)

- real values: propagative modes- imaginary values: evanescent ‘’- complex values: inhomogeneous ‘’

6M eigenvectors (corresponding displacement)

1st elt.

2nd elt.

(system #1)

• account of source terms in SAFE:

- existing: on the guiding surface source modelled as an external force

Liu & Achenbach, J. Appl. Mech. (1995), Zhuang et al., J. Appl. Mech. (1999),Hayashi et al., J. Acoust. Soc. Am. (2003)

- here: on the section

problem closely related to that of the scattering from the free end of a semi-infinite guide:

SAFE: - Rattanawangcharoen et al., J. Appl. Mech. (1994), - Taweel et al., Int. J. Solids Struct. (2000), - Galan & Abascal, Int. J. Numer. Meth. Engng. (2002)

Le Clézio, PhD thesis Bordeaux 1 University (2001)

=> source modelled as a vertical boundary condition

Theory: Theory: adaptation of SAFE… adaptation of SAFE… (contd.)(contd.)

Theory: Theory: adaptation of SAFE… adaptation of SAFE… (contd.)(contd.)

• source mounted on the guide section at z = 0:

- one selects modes (obtained from System #1) that make sense for z > 0: 3M - stress tensor deduced from eigenvector displacement at the M nodes (xi , yi )

- piston-source modelled as source of normal stress

M

niiyz(n)n

M

niixz(n)n

M

niizz(n)nii(source)zz

yxσAyxσA

yxσAyxσ

3

1

3

1

3

1

),(~0),(~0

),(~)0,,(i = 1,…, M (system #2)

• time-domain solutions obtained by Fourier synthesisin system#1, the various matrices are frequency-independent, not the overall system

• Context of developmentContext of development

• TheoryTheory• brief reviewbrief review• adaptation of Semi-Analytic F.E. methodadaptation of Semi-Analytic F.E. method

• Validation in the axisymmetric caseValidation in the axisymmetric case

• Transducer diffraction effectsTransducer diffraction effects• influence influence of aof apertureperture // bandwith bandwith in the in the a axisymmetric casexisymmetric case• 2D case2D case: : rectangular cross sectionrectangular cross section

• Summary – Summary – WWorkork in progress in progress

Validation in the axisymmetric caseValidation in the axisymmetric case

d

transducer E cylindrical guide transducer R

receiver output

2/

0),()(

d

z drrtrvtO

• Simulation of a transducer in reception

d

• (System #1) + (System #2) + IFFT:

d

transducer cylindrical guide

zfield points

field prediction

Validation in the axisymmetric caseValidation in the axisymmetric case

2/

0),(

)(d

z drrtrv

tO

• (very recent) example from Puckett & Peterson (Ultrasonics 43(3), 2005)

configuration:- d = 25 mm- z = 250 mm- fused quartz- piston-like transd.- Gaussian pulses: 1107 kHz – 6.7 % of relative bandwidth

(12 prop. modes)

d

transducer E cylindrical guide

z

transducer R

measured (Ultrasonics 43(3), 2005)60 80 100 120 14040

simulated (present model)60 80 100 120 14040

• Context of developmentContext of development

• TheoryTheory• brief reviewbrief review• adaptation of Semi-Analytic F.E. methodadaptation of Semi-Analytic F.E. method

• Validation in the axisymmetric caseValidation in the axisymmetric case

• Transducer diffraction effectsTransducer diffraction effects• influence influence of aof apertureperture // bandwith bandwith in the in the a axisymmetric casexisymmetric case• 2D case2D case: : rectangular cross sectionrectangular cross section

• Summary – Summary – WWorkork in progress in progress

Transducer diffraction effects (axisym.)Transducer diffraction effects (axisym.)

0 d / 2 =12.5mm

t

B-scans at z = 250 mm

axial displacement uz (r,t)

d Øfield points

z

aperture:

Ø = 25 mm200 µs

0 d / 2 =12.5mm

t

aperture:

Ø = 12.5 mm400 µs

0 d / 2 =12.5mm

t

aperture:point-source

960 µs

Transducer diffraction effects (axisym.)Transducer diffraction effects (axisym.)6

.7 %

20

%4

0 %

excitation pulse T and R : 12.5-mm-Øbw

T and R : 25-mm-Ø

Transducer diffraction effects Transducer diffraction effects (2D case)(2D case)• Guide of rectangular cross-section 20x10mm (Steel)• 4 types of modes (2 axes of symmetry)• 11 propagative modes at f=200 kHz

mode shape (Uz):Flexural Y Flexural X Torsional Extensional

500 elements

Transducer diffraction effects (2D case)Transducer diffraction effects (2D case)

Excitation pulse (fc=200kHz, bp=10%) :

Frequency (MHz)

Gro

up v

eloc

ity

(mm

/s) extensional modes

z=1m

circular:Ø = 10 mm

Ø = 10 mm

transducer Etransducer R

S

z dxdytyxv

tO

),,(

)(

rectangular: 20 x 10 mm²

Transducer diffraction effects (2D case)Transducer diffraction effects (2D case)

Rectangular transducer

Circular transducer

• Context of developmentContext of development

• TheoryTheory• brief reviewbrief review• adaptation of Semi-Analytic F.E. methodadaptation of Semi-Analytic F.E. method

• Validation in the axisymmetric caseValidation in the axisymmetric case

• Transducer diffraction effectsTransducer diffraction effects• influence influence of aof apertureperture // bandwith bandwith in the in the a axisymmetric casexisymmetric case• 2D case2D case: : rectangular cross sectionrectangular cross section

• Summary – Summary – WWorkork in progress in progress

Summary – Work in progressSummary – Work in progress

• Summary:

- SAFE method extended to case of transducer mounted on guide section:- can deal with arbitrary guide (geometry, anisotropy) with symmetry of translation- very efficient numerically (computation in the sole section, i.e. 2D or even 1D)- validated by comparison with existing results for cylinder (theo. – exp.)

- importance of transducer diffraction effects:- requires a proper simulation tool to be predicted- easily studied using SAFE computations- as strong here as in the case of radiation from the guiding surface

• Work in progress: - implementation in CIVA

- scattering by a crack normal to the guide axis computed by SAFE- experimental validation of our own

more stuff…

visco-elastic absorbing layervisco-elastic absorbing layer

used to deal with sections of infinite extent:Hooke’s tensor in the absorbing layer has

an increasing (with r) imaginary part

Liu & Achenbach, J. Appl. Mech. (1994)

Transducer diffraction effects: Transducer diffraction effects: embeddeembedded guided guide

rsteel

cement

• axisymmetry: still a 1D computation in the present case• no more real-valued wavenumbers, imaginary parts standing for the leakage of energy in the cement of propagative modes in the steel core.

1

-2.50 a

0

1

-30 a

0

1

-60 a

01

-30 a

0

u z(r)

-2.5

0

-4.50 a

1.2

-0.20 a

0

1

0 a

0

1.5

-30 a

0

u r(r)

L(0,1) L(0,2) L(0,3) L(0,4)471.4 m-1562.1 m-1818.9 m-11049.8 m-1

471.7 – 6.2 i m-1561.7 – 14.6 i m-1819.0 – 8.5 i m-11045.6 – 34.6 i m-1

at z=0, 500 kHz – steel cylinder: free – embedded in cement

Transducer diffraction effects: Transducer diffraction effects: embeddeembedded guided guide

transducer E cylindrical guide transducer R= transducer E

Ø2a

Transducer diffraction effectsTransducer diffraction effects:: apertureaperture

otherwise,0)(

,1)(

rA

arrA

otherwise,0)(

2/,1)(

rA

arrA

araa

ar

rA

arrA

2/,2

)2(cos1

)(

2/,1)(

receiver output

a

z drrtrvrAtO0

),()()(

Transducer diffraction effectsTransducer diffraction effects:: apertureaperture

-7.5 dB-7.5 dB

-9.3 dB-9.3 dB

ur (r,t) receiver outputaperture uz (r,t)

-9.9 dB-9.9 dB

transducer E cylindrical guide transducer R= transducer E

Ød

Transducer diffraction effectsTransducer diffraction effects:: apertureaperture

otherwise,0)(

2/,1)(

rA

drrA

otherwise,0)(

4/,1)(

rA

drrA

receiver output

2/

0),()()(

d

z drrtrvrAtO

3 excitation pulses :- same center freq.: 1107kHz- 3 bandwidths: 6.7, 20, 40%

• Combination of effects of apperture in transmission and in reception and effects of relative bandwidth

0)( 22 iiii

3i2

i1 dMdKKK kjk

iE

rdr1HT1

i1 BcBK

NLNLB 0r,r1

TMzMrzr ddddp) (mode,,1,1, ...pd

00

00

0

00

10

00

00

011

0

rLLr

)(0)(0)(0

0)(0)(0)()(

321

321

rrr

rrrr

N

000

02

02

02

Hc

mixed end conditions and axisymmetry: direct projection of the modes Duncan Fama, Quart. J. Mech. Appl. Math. (1972)

Herczynski & Folk, Quart. J. Mech. Appl. Math. (1989)

n

zz(n)n(source)zzn

r(n)n(source)r σAσuAu ~and~

orthogonality relation Fraser, J. Sound Vib. (1975)

mnrdruuQ mzznz

a

nrzmrmn ifexcept 0)~~~~(~

)()(

0

)()(

rdruuQ

A rcesouzznz

a

nrzourcesr

nn

n )~~~(~1

)()(

0

)()(

zguide axis

pure end conditions: direct projection impossible

nzz(n)n(source)zz

nrz(n)n

σAσ

σA

~

~0

minimization of residual boundary values at z = 0

Gregory & Gladwell , Quart. J. Mech. Appl. Math. (1989) Puckett & Peterson, Ultrasonics (2005)

Theory: Theory: brief reviewbrief review (contd.)(contd.)

Transducer diffraction effectsTransducer diffraction effects:: apertureaperture

0 d / 2 =12.5mm

t

0 d / 2 =12.5mm

t

0 d / 2 =12.5mm

t

B-scans at z = 250 mm

radial displacement ur (r,t)

d Øfield points

z

aperture:

Ø = 25 mm

aperture:

Ø = 12.5 mm

aperture:point-source

200 µs

400 µs

960 µs

-9.3 dB-9.3 dB

-7.5 dB-7.5 dB

+1.8 dB+1.8 dB

Transducer diffraction effectsTransducer diffraction effects:: apertureaperture

aperture:

Ø = 12.5 mm

aperture:point-source

d Ø

z

received signal

at z = 250 mm

aperture:

Ø = 25 mm

40 60 80 100

µs120 140 160

Theory: Theory: validation validation vs. vs. exact resultsexact results

• Cylindrical case: Pochhammer’s solution for wavenumbers(roots of an exact equation)

here: 1MHz, steel cylinder of 20-mm-Ø – 9 propagative modes

623.5870.4991.51049.41253.61550.51757.61884.32101.930 elts

623.4870.2991.31049.41253.41550.51757.61884.22101.320 elts

619.8866.1988.11047.71247.61548.91757.31884.12094.110 elts

557.1784.1951.41035.81174.21525.91753.11883.12058.45 elts

S.A

.F.E

.

(wavenumbers in m-1)

L(0,9)L(0,8)L(0,7)L(0,6)L(0,5)L(0,4)L(0,3)L(0,2)L(0,1)mode:

623.4870.4991.41049.41253.61550.51757.61884.32102.0exact

0.06 ‰ error 10.6 % with 5 elements 0.1 ‰ error 0.6 % with 10 elements0 % error 0.3 ‰ with 20 elements0 % error 0.16 ‰ with 30 elements

Transducer diffraction effectsTransducer diffraction effects

-10

z/Ø

0

-20

-30

-40

-50

-60

0.2 0.4 0.6 0.8 10

ampl

itude

in d

B

• Amplitude variation (z) of propagative, evanescent and inhomogeneous modes with imaginary part < 1500 m-1 as radiated in the previous configuration

at z = 250mm, only thepropagative modes contribute to the receivedsignal

Transducer diffraction effects: Transducer diffraction effects: excitation spectrumexcitation spectrum

• Same center frequency, 3 different bandwidths

6.7

%2

0 %

40

%

excitation pulse simulated received signalbw