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SISOM 2007 and Homagial Session of the Commission of Acoustics, Bucharest 29-31 May

NUMERICAL SIMULATION OF ULRASONIC WAVE PROPAGATION

WITH WEDGED TRANSDUCERS

Cristian RUGINA

Institute of Solid Mechanics, Romanian Academy, e-mail: [email protected]

The numerical simulation of ultrasonic wave propagation are useful in nondestructive testing as a toolfor a better understanding the signals captured on defect scopes screens. The method used L.I.S.Asimilar to the classical finite difference method, in the 2D case. The model of the ultrasonic

transducer used in simulation is described, and its parameters determined. A comparison between

numerical results and experimental data is done on a real case of nondestructive testing.

1. INTRODUCTION

The practical applications of elastic waves propagation [1], [2], [3], can be found, mainly, in Ultrasonic

Nondestructive Testing (Ultrasonic NDT) and Ultrasonic Nondestructive Evaluation (Ultrasonic NDE). In

industry, with Ultrasonic NDT, flaws (void, inclusions, cracks, corrosion, etc) in elastic bodies can be

detected. The Ultrasonic NDT consists in the creation of short pulses of elastic wave (


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Numerical simulation of Ulrasonic wave Propagation with wedged transducers501

where - displacements, - density and - stiffness tensor,w S ( )

ax

is denoted ( )a or ( ),a when the

implicit summation convention is used, and2

2

aw

t

is denoted (aw 1, 2, 3a= ).

In the 2D case the wave equation becomes:

( ), , , 1, 2 2,1k k kk k hh h kh k w w w w k h+ + = = = , (2)

where , , , ,1 1111 S= 2 2222 S= 1122 S= 1212 S= = + , S- the stiffness tensor; - materials density.

The discretization steps are and on both directions in space and in time. The materials

physical properties in each grid cell are the elastic constants and the materials density ( , , , , , ).

These physical properties are supposed to be constant within each cell.

1 2

1 2

The material displacement are denoted in the direction and in the direction, at the t

time step in the grid point . Omitting the subscripts whenever equal to , or

, ,t i ju x , ,t i jv y

( , )i j t i j , and using the

following index conventional notation:1 - , 2 - ( 1( 1, 1)i j+ + , 1)i j + , 3 - ( 1, 1)i j , 4 - ( 1, 1)i j+ ,

5 - , 6 - ( , , 7 -( 1, )i j+ 1)i j+ ( 1, )i j , 8 - ( , 1)i j ,

the method gives, for the homogeneous case (all the cells around a grid point ( , have the same physical

properties), the recurrences formulas:

)i j

1 1 5 7 1 6 8 1 1 1 2 3 4 1

1 2 6 8 2 5 7 2 2 1 2 3 4 1

( ) ( ) 2( 1) ( )

( ) ( ) 2( 1) ( )t t

t t

u u u u u u v v v v u

v v v v v v u u u u v

+

+

= + + + + + +

= + + + + + +

,

. (3)

where

2 2 2

2 2

1 2

( 1,2), ( 1, 2; 3 ),

4 k k

k k

k h

k k h k = = = = = = . (4)

For the inhomogeneous case (at least one of the four cell cells around the grid point have

different physical properties) the recurrences formulas are:

( , )i j

( ) ( ) ( )

(1) (1)2 (

5 6 7 81 1 5 6 7 82 2 2 2 2 2

1 2 1 2 1 2

4 4' ' ' '

5 5 6 6 7 7 8 8

1 11 2 1 2 1 2

(2) (2)2

5 6 7 81 1 5 6 72 2 2

1 2 1

2 2

1 1 11

1) ,

1 ,4 4

2

t t

k k

k k k

i i

t t

u u u u u u u u

v v g v g v g v g

v v v v v v

+

= =

+

= + + + + +

+ + + + +

= + + + +

v

( ) ( ) ( )

(2 )

82 2 2

2 2 1

4 4' ' ' '

5 5 6 6 7 7 8 8

1 11 2 1 2 1 2

2 ,

1 1 11 1 .

4 4

k k

k k k

i i

v v

u u g u g u g u= =

+

+ + + +

g u

(5)

where

4

1

1

4k

i=

= , ( ) k k k= + 1,2,3,4k= ,

( ) ( )(1) 1 1111 k k kS= = , ( ) ( )(2 )

2 2222 k k kS= = 1,2,3,4k= ,

(6)


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Cristian RUGIN 502

( ) ( )( )1 1(1)5 4 11

2

= + , ( ) ( )( )2 2(2)6 11

2

= + 2 ,( ) ( )( )1 1(1)7 3

1 3

2= + , ( ) ( )( )2(2 )8 3 4 = + ,2

1

2

( )5 4 11

2

= + , ( )6 11

2

= + 2 , ( )7 21

2

= + ,3 ( )8 31

2

= + ,4

( ) k k kg = ( )1..k= .4 ,' ' ' '

5 4 1 5 1 2 7 2 3 8 3 4, , ,g g g g g g g g g g g= g= = =

3. COMPARISON BETWEEN EXPERIMENTAL AND COMPUTED RESULTS

The ex ultrasonicwave propagation, with experimental data is presented in fig. 3.1. The defect scope Krautkramer USD 10ge

perimental setup used to compare the results obtained by the numerical simulation of

nerates electric pulse and "listen" the echoes. But this specialized apparatus shows on his screen thecaptured signal, modified for an easier interpretation of results by authorized persons. For the recording andtransmission of the raw signals in digital form, the Combiscope Fluke 3380A was connected to thetransducer and also to a PC by a serial interface RS232, so that real experimental signals could be compared

with those obtained by numerical simulation.

Krautkramer USD 10 Fluke PM3380A

CombiscopeDefect scope

Analized body

TransducerComputer

with RS-232

serial port

Fig. 3.1. The experimental setup used to compare experimental results with computed results.

The used tran amer MPU-045/4 (fig. 3.2).

sducer is with normal incidence and is an equivalent of the transducer Krautkr

piezoelement Damping material Absorbing Boundary Conditions

PlexiglassSignal source

24 mm

h

l

Plexiglass

h

24 mm

Signal source

l

a b cFig. 3.2. Transducer Krautkramer MPU-045/4(a), compo parts of the transducer (b)

The model of the transducer, as a done; some effects as dispersionand high attenuation had to be neglected. The thin film of couplant is not taken into account explicitly in thetran

nent

and equivalent model used in simulations (c).

complex device, could not totally be

sducer model, but implicitly. The transducers produced by different firms are encapsulated and theelastic properties and geometric dimensions of the component parts are partially secrets and cannot bemeasured without destroying the transducer. But the firms give in the their documentation the spectrum andthe shape of the signal present at the active surface of the transducer, essential elements in the transducer's

modeling.Even if the waveform is indicated by the producers, the waveform of the signal introduced in simulations


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( )s tis to be determined, depending on the used couplant. I considered a signal that approximate very well

the measured signal. The signal ( )s t has been obtained from a sinusoidal signal of frequency f , but of

finite length, modulated with 2 functions, 11( ) a tf t e

= and 22( ) 1 a tf t e

= , definite

for 0 _ _t t emission transducer . The first function 1( )f t characterizes a low band pass filter and the

second function ( )2f t characterizes a high band pass filter. T ct produces a pass band filter, similar

to t er. So, the analytical expression of the signal ( )

heir produ

he one the transdproduced by uc s t is:

( )1 2( ) (1 ) sin 2a t a t s t A e e f t = , (7)

where A , 1a , 2a , f , _ _t emission transducerare the signals parameters.

e resonance frequIn our case the transducer has th ency of 4MHz ( 64 10f Hz= ). The parameter

ransducerhas been taken 10 s into account were

A

_ _t emis ion t . The other parameters of the signal takens

4.4= , 60.6 10a = , 62

1.2 10a1

= .elocities of longitudinal and transversal waves in steel has been taken

59 d 0

The propagation v

33 /lv m , an 313 /v m s= . The steel density has been taken kg m = .

holes (fig. 3.3.a) with geometrical dimensi tion (fig. 3.3.b).

( )steel ( )l steel ( )steel

The comparison between experiment and numerical simulation has been done with an steel body withons that have been used in computa

s

3

7700 /

38 mm

19 mm 44 mm

a b

Fig. 3.3. The experimental setup with the posit e transducer MPU-045/4 on a steel body with holes (a)

and the geome in s tak n imulation (b).

ion of th

trical shape ection, with the transducers positioning, e in s

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34-0.4

-0.2

0

0.2

.

s

V

Fig. 3.4. Graphical representation of the measured signal (red) and of the simulated signal (black)

Graphical r n in fig 3.4.

The sources of errors are geometrical imperfection of the cylinder (bevel cant of the cylinder, approximate

pa

epresentation of the measured and simulated signal, and their envelopes are show

rallelism of the cylinder surfaces, rugosity of the surfaces) imperfect position of the transducer (bad

concentricity of the transducer and cylinder, approximate parallelism of the transducer's and cylinder's

actives surfaces due to the coupling medium). These geometrical and positioning differences between the

simulated (ideal) case and the experimental (real) case, compared to the wavelength of the ultrasonic wave


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Cristian RUGIN 504

can lead to this kind of errors.

The duration between 0s 36s can be considered sufficient, because in ultrasonic NDT only thewav

in fig 3.4 that in the first part, 0s 10s, the measured and computed signals are verydiff

n in this case, in fig 3.5 we representedthe

e's firsts reflections give the majority of information. As it can be seen in figs 3.5 and 3.6, the scatteringof the ultrasonic wave after multiple reflections and mode conversions, make unusable the signal after thesefirsts reflections.

It can be seenerent. This corresponds to the generation of the ultrasonic wave, and the transducer's model differs very

much from the real transducer. But this part of the signal is not interesting in the NDT, the important thingsbeing the firsts reflections. The purpose of the modeled transducers is to generate into the material asimulated ultrasonic wave that approximate very well the real ultrasonic wave, so that the reflections andmode conversions phenomena are as close as possible to the reality.

For a visual appreciation of the ultrasonic elastic wave propagatiofield of the displacements modulus at different time steps, and in fig 3.6 the field of displacements by it's

two components u and v (transversal and longitudinal) at the same time steps. These snapshots at differenttime steps or movies of propagation can explain very well the propagation phenomena that take place in thetested elastic body and so the signal shape on defect scopes screens.

6.0 s

12.0 s

18.0 s

20.0 s

Fig. 3.5. The computed field of the displacements modulus of an ultrasonic wave propagation

in a steel body with hole having dimensions presented in fig 3.3.


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6.0 s

12.0 s

18.0 s

20.0 s

v uFig. 3.6. The computed field of the displacements, by its longitudinal v and transversal u components,

of ultrasonic wave propagation in an steel body with hole having dimensions presented in fig 3.3.

Acknowledgements:The author is grateful to NATIONAL UNIVERSITY RESEARCH COUNCILfor

financial support by the grants CNCSIS nr.120/2006 and nr.55/2007.

REFERENCES

1. ACHENBACH, J.D., Wave propagation in elastic Solids, North-Holland, Amsterdam, 1973.

2. BREKHOVSKIKH, L.M., Waves in Layered Media, Academic Press 1980.

3. HARKER, A.H.,Elastic Waves In Solids with applications to Nondestructive Testing of Pipelines, IOP Publishing Ltd, Bristol,1988.

4. STRIKWERDA, J., Finite Difference Schemes and Partial Differential Equations, Chapman&Hall Publishing, 1989.5. DELSANTO, P.P., WHITCOMBE, T., CHASKELIS, H., MIGNOGNA, R., Connection Machine Simulation of ultrasonic wave

propagation in materials. I : The one-dimensional case, Wave Motion, 16, pp. 65-80, 1992.6. DELSANTO, P.P., SCHECHTER, R., CHASKELIS, H., MIGNOGNA, R., KLINE, R., Connection Machine Simulation of

ultrasonic wave propagation in materials. II : The two-dimensional case,Wave Motion, 20, pp. 295-314, 1994.7. SCHECHTER, R.S., CHASKELIS, H.H., MIGNOGNA, R.B., DELSANTO, P.P., Visualization of 2-dimensional Acoustic

Wave in Solids Using Parallel Processing Techniques, Rev. of Progress in Quantitative Nondestructive Evaluation, 13, 1994.8. CHIROIU, V., CHIROIU, C., DELSANTO, P.P., RUGIN, C., SCALERANDI, M., Propagation of ultrasonic waves in

nonlinear multilayered media, Journal of Computational Acoustics, 2001.

9. RUGINA, C.,Bidimensional simulation of elastic wave propagation with applications in nondestructive testing, Proceedings ofthe annual Symposium of the Institute of Solid Mechanics SISOM2004, 20-21 May 2004, pp.231-239.

numerical simulation of ultrasonic wave propagationu_19

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