Analysis of the modeling of echo responses from circular concavity defects using impulse response and discrete representation methods

Physics Procedia 00 (2009) 000–000

www.elsevier.com/locate/procedia

International Congress on Ultrasonics, Universidad de Santiago de Chile, January 2009

Analysis of the modeling of echo responses from circular concavity

defects using impulse response and discrete representation methods

Paulo Orestes Formigoni, Julio C. Adamowski, Flávio Buiochi*

School of Engineering at University of São Paulo, Department of Mechatronics Engineering Av. Prof. Mello Moraes, 2231, CEP 05508-900, São Paulo, Brazil

Elsevier use only: Received date here; revised date here; accepted date here

Abstract

In this work the impulse response and the discrete representation method are applied to model the acoustic wave generated by ultrasonic transducers, its interaction with the defect, and the echoes received by the transducer. Several simulations are performed to determine the echo responses arising from a concave circular defect on a plane surface, using a planar circular piston emitting pulses in water. The simulations are calculated using the Matlab software. The effects of defect size and field position on both amplitude and shape of the echo responses are investigated. Moreover, experimental measurements of the acoustic field are compared with the simulations. The experimental results, based on the above-mentioned process, were obtained using 1.6 and 2.25MHz transducers.

PAC: 43:20 El

Keywords: acoustic field; ultrasound; simulation; impulsive response; pulse-echo

1. Introduction

The preventive maintenance of pipelines requires finding corrosion spots. This can be done by using a quality control nondestructive testing device, based on the pulse-echo mode. It tries to get as much information as possible from the existing defect. However, identifying the defect geometry is not an easy task in realistic situation. A solution for identifying different geometries would be simulating models of propagation in media containing defects and setting up a database of the obtained signals. In this paper two methods are described: impulse response and discrete representation.

The impulse response method [1] is based on the Rayleigh integral that calculates the echo response using the closed-form analytical solutions obtained by simple geometries [2]. The second method is a computational method

* Corresponding author. Tel.: 55-11-3091-9762; fax: 55-11-3091-5461. E-mail address: [email protected].

Physics Procedia 3 (2010) 847–853

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that predicts the echo response from arbitrary-geometry defects, using a model that calculates the longitudinal wave evolution caused by interfaces [3], which is based on the impulse response method [1] and on the discrete representation method [4], [5].

Both methods allow recognizing the variability of echoes from different defects and selecting the optimal broadband ultrasonic transducer, which improves the axial and lateral resolutions. The models are suitable for all field regions and can be performed for any excitation waveform. The validation of the wave propagation models implemented in the ongoing paper requires experimental corroboration, which is done by using broadband ultrasonic transducers operating in pulse-echo mode and a circular concavity defect embedded in planar reflector.

2. Impulse response method

The Rayleigh integral can be simplified because the circular piston is symmetrical, and the exact analytical solution of the velocity potential impulse response at a field point P is given by [1]:

) ( 2 ) , (r t c ct h _P :

S

G , (1)

where c is the acoustic propagation velocity, and ȍ(ct) is the angle of an arc included on the emitter surface and

centered on P’ (P projection in plane z = 0), as shown in Fig. 1. The arcs are drawn by using piston surface points, such that the impulse excitations arrive at point P at time t = r / c, where r is the distance between point P and any point on the arc. Arc angleȍ(ct) explicit expressions were given by different authors [1], [7], [8].

2.1. Pressure response for circular reflector

Fig. 1 presents the geometry used to determine the pulse-echo response. An emitter of radius a transmits an acoustic pulse into the medium that contains a circular reflector of radius b with its center located at Q(xoff, 0, z). The

faces of the reflector and the emitter are parallels.

a b r= ct z x z Tx : ct O reflector x xo ff E m itter/ rec eiv er P Q y P ’

Fig.1 Geometry used to determine the pulse-echo response for a circular piston interrogating a circular reflector axially unaligned.

Using the McLaren and Weight’s solution [2] for a uniform piston interrogating a reflector of acoustic impedance greater than that of the medium, the total acoustic pressure ¢p²(t) over the surface of the emitter/receiver is calculated by a temporal convolution between the excitation signal and the total pressure impulse response

: ) (t v ) (t pi² ¢ ) ( ) ( ) (t vt p t p² ¢ i² ¢ (2)

² ¢

The total pressure impulse response p_i (t)can be simplified, considering two cases. The first, when the reflector does not intercept the emitter axis (xoff!b):

³

  w w w w ² b x b x off off xdx x t t z x h t t z x h c t) ( , ,) ( , ,) ( ) ( U T ) (x , (3) ¢p_i where

T

is given by [8]: ¸ ¹ ¨ © 2xxoff ) , (rPt ¸ · ¨ §x xoff b x) 2arccos ( 2 2 2 T , (4)

and h G is the velocity potential impulse response at point P(x,0,z) given by (1).

d

The second condition, when the reflector does intercept the axis of the emitter (xoff b). The total pressure impulse

response is:

³

_w w ² b xoff xdx x t t z x h t t z x h c t 0 ) ( ) , , ( ) , , ( ) ( U T  w w _{, (5)} ¢pi where

T

( )

x

is given by [8]: off off off off x b x x b x x x b x x x  d  ! ¸ ¸ ¹ · ¨ ¨ © §   if 2 if 2 arccos 2 ) ( 2 2 2 S T ₍₆₎

2.2. Impulse response for concave circular defect

The impulse response calculated from a small circular reflector is subtracted from the one calculated from a large circular reflector, resulting in the impulse response from a ring. In order to obtain the impulse response from a circular concavity defect, the above technique can be repeated several times, until it reaches the deeper spot in the defect, represented by a tiny disk, as shown in Fig. 2. Each concentric ring can be displaced of dz along the z axis, thus approximating from the concave circular geometry. Moreover, the concentric rings can be shifted xoff from the z

axis (center of transducer) in order to simulate the transducer approaching the defect. In case xoff equals zero, the

centers of the defect and the transducer are aligned.

The planar reflector is approximated to a finite annular surface around the defect, large enough to intercept the major portion of the significant incident energy of the acoustic field.

'b xoff Transducer Reflector z Rings Defect z O x rigid baffle aperture (SA) y Pi soft baffle dSi interface (SI) dSa defect rib rai i rG b rG a r G dSb nG

Fig.2 Rings superposed to generate the geometry of a defect. Fig.3 Arbitrary geometry used to determine the pulse-echo mode response using discrete representation method.

3. Discrete representation method

The computational method solution proposed by Buiochi et al. [3], which calculates the acoustic field through interfaces, is easily used to calculate the pulse-echo responses using the same theoretical concepts. The proposed solution is an approximated method that operates by dividing the aperture and the interface with a defect into elementary areas, as shown in Fig. 3. The radiated and reflected acoustic fields result from the superposition of the hemispherical waves generated, respectively, from each emitter and interface elementary areas. In this work, as the defect is slightly concave, and the emitter/receiver aperture and the planar interface are parallels, the mode conversion at the reflector surfaces was not considered.

Considering an aperture with arbitrary radiating surface SA embedded in an infinite rigid baffle, the velocity

potential impulse response on each point Pi at the interface due to the aperture radiation is given by:

³



A S a ai ai i I

_dS

r

c

r

t

t

r

h

S

G

2

)

/

(

)

,

(

G

(7)

where rai is the distance from each radiating elementary area dSa to the point Pi.

Assume that the interface with defect is embedded in an infinite soft baffle and that it is large enough to intercept the main incident energy of the acoustic beam. The whole-extended interface and the defect are approximated by elementary areas dSi. In each of the receptor aperture elementary areas dSb, the velocity potential impulse response

is calculated from the impulse response obtained at the interface by:

i S ib i I ib ib b A

_dS

c

r

t

r

h

t

r

c

t

r

h

I

³



w

w

)

,

(

cos

2

1

)

,

(

G

T

G

S

i (8)

where SI is the surface of the interface with the defect, rib is the distance from the elementary area dSi located at rG

to the point located at rG_b in the aperture, andTib is the angle between the normal vector at Pi and the vector rib

G _.

Finally, the spatial acoustic pressure _p(r,t)

b

G _{over the surface of the finite receiver is calculated by the following}

temporal convolution: ) , ( ) ( ) , (r t v t h r t p _{b b} t G G w U ) (t v (9) w ) , (r t h b G

where is the excitation signal,U is the density of the propagation medium, and is defined by:

³

A S b b A b t h r t dS r h(G, ) (G, ) (10) 4. Results

Echo responses were obtained from a circular concavity defect to test the validity of both the impulse response and the discrete representation methods. Fig. 4 shows the geometry of such defect, which was produced on the plane surface of an aluminum solid sample. It also shows the six positions used to simulate the transducers displacements. Setting the center of the defect as zero (position 6), the positions 1 through 6 are displaced xoff= 20, 16, 12, 8, 4 and

0 mm, respectively. The transducers were placed 15 mm from the plane surface of the sample and were excited with short pulses by a pulser/receiver Panametrics 5072PR. All measurements were carried out in water (ȡ=1000 kg/m3_,

c=1480 m/s), using two 19-mm diameter transducers (1.6 MHz, BW 42%, Funbec, Brazil, and 2.25 MHz, BW 63%, Panametrics, USA).

Simulations were performed in Matlab using the same parameters described above for the experiments in order to allow the comparison of echo responses arising from the defect. The excitation signals used in the simulations were

acquired by a 0.6-mm-diameter needle hydrophone placed approximately 3 mm from the transducer faces, considering only the plane waves. For the impulse response method, the spatial sampling in the transducer aperture and in the solid sample are ǻx=0.005 mm. The concave geometry was approximated by concentric rings measuring ǻb=0.05mm. For the discrete representation method, the emitter aperture, the interface with the defect, and the receiver aperture discretizations (ǻx=ǻy) were, respectively, 0.15 mm, 0.2 mm, and 0.2 mm. For all cases, the sampling period was 16 ns.

.

.

.

.

.

.

xoff 4 5 6 Transducer 1 Aluminum 15 25 mm defect mm R=60 mm 2 3 1.3 mm

Fig.4 Experimental geometry. Positions 1, 2, 3, 4, 5 and 6 represent, respectively, the transducers displaced 20, 16, 12, 8, 4, and 0 mm from the defect axis.

The theoretical and experimental results shown in Fig. 5 and 6 were obtained from a 1.6 and a 2.25 MHz transducer, respectively. All signals were normalized by the maximum simulated and experimental amplitudes using the 1.6-MHz transducer at position 1.

21 22 23 24 25 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (μs) (c) xoff = 12 mm 21 22 23 24 25 -1 -0.6 -0.2 0 0.2 0.6 1 Time (μs) (a) xoff = 20 mm R ela ti v e A m p lit u d e 21 22 23 24 25 -1 -0.6 -0.2 0 0.2 0.6 1 Time (μs) (b) xoff = 16 mm 21 22 23 24 25 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Time(μs) R ela ti v e A m p lit u d e (d) xoff = 8 mm -0.15 -0.1 -0.05 0 0.05 0.1 0.15 21 22 23 24 25 26 27 28 Time(μs) (e) xoff = 4 mm 21 22 23 24 25 26 27 28 -0.15 -0.1 -0.05 0.05 0.1 0.15 Time(μs) (f) xoff = 0 mm 0

tal (solid lines) signals obtained by using the 1.6-MHz transducer displaced off the defect axis: (a) 20, (b) 16, (c) 12, (d) 8, (e) 4 and (f) 0 mm.

spatial samplings [3]. The greater the

ed for the defect geometry is different for both methods. A big difference exists in computational time for both methods. For instance, using a 2.4-GHz Intel Core 2 Duo computer, the processing time for the discrete representation method takes some hours (2-5h), while for the impulse response it

n).

The accuracy of the discrete representation method depends on temporal and

spatial discretization, the more the simulation approaches the exact impulse response method solution [5]. Moreover, greater temporal discretization leads to better resolution. However, the processing time increases. Thus, a balance between the best samplings and the computational time should be found.

Both methods presented show a similar solution for positions 1, 2, and 3, because in those positions there is a maximum reflected energy from a plane surface. For a plane surface reflection, the impulse response method applies the exact solution, whereas the discrete representation method approaches the exact solution when highly sampled.

Nevertheless, differences appear for positions 4, 5, and 6, that is, when the transducer radiates into the defect, because the discretization approach us

takes only some minutes (10-30 mi

20.5 21 21.5 22 22.5 23 -1 -0.6 -0.2 0.2 0.6 1 Time (μs) Rel at iv e A m pl it ud e (a) xoff= 20 mm 0 20.5 21 21.5 22 22.5 23 -1 -0.6 -0.2 0.2 0.6 1 Time (μs) (b) xoff = 16 mm 0 20.5 21 21.5 22 22.5 23 23.5 -0.4 -0.2 0.2 0.4 -0.6 Time (μs) (c) xoff= 12 mm 0 20.5 21 21.5 22 22.5 23 23.5 -0.4 -0.2 0 0.2 0.4 Time (μs) (d) xoff = 8 mm R ela ti v e A m p li tu d e 21 22 23 24 25 -0.15 -0.05 0.05 0.15 Time(μs) (e) xoff= 4 mm 0 21 22 23 24 25 -0.15 -0.05 0.05 0.15 Time(μs) (f) xoff = 0 mm 0

onse method (dotted lines), discrete representation method (dashed lines) and experimental (solid lines) signals obtained by using the 2.25-MHz transducer displaced off the defect axis: (a) 20, (b) 16, (c) 12, (d) 8, (e) 4 and (f) 0 mm.

was demonstrated that a good correlation between experimental and both theoretical pressure responses of a concave circular defect is possible, considering the surface of the defect is slightly curved. For the impulse response method, it was found that the concave circular geometry could be approximated by the addition of several concentric their axis. For the discrete representation method, the defect could be represented by adding up elementary areas. Although it can be done for any geometry, the processing time is certainly increased.

and simulated waveforms are due to the difficulty to experimentally adjust the parallelism between the plane interface and the transducer. The knowledge of the pressure response simulated for a given geometry allows accurate interpretations of the echoes generated from corrosions in a realistic pipeline provides good understanding of the spatial points where the transducers start and end traveling across

Ack

T

[1] piston in an infinite planar baffle,” J. Acoust. Soc. Am., vol. 49, pp.1629-1638, 1971. ] S. McLaren, J. P. Weight, “Transmit-receive mode response from finite-sized targets in fluid media,” J. Acoust. Soc. Am., vol. 82, n. 6, pp.

2102-2112, 1987.

[3] F. Buiochi, O. Martínez, L. G. Ullate and F. Montero de Espinosa, A computational method to calculate the longitudinal wave evolution caused by interfaces between isotropic media, IEEE Trans. Ultrason., Ferroelect., and Freq. Contr. Vol. 51, n . 2,,pp. 181-192, 2004. [4] B. Piwakowski and B. Delannoy, “Method for computing spatial pulse response: Time-domain approach,” J. Acoust. Soc. Am., vol. 86, n. 6,

pp. 2422-2432, 1989.

[5] B. Piwakowski and K. Sbai, “A new approach to calculate the field radiated from arbitrarily structured transducer arrays,” IEEE Trans. Ultrason., Ferroelect., and Freq. Contr., vol. 46, n. 2, pp. 422-440, 1999.

[6] D. E. Robinson, S. Lees, L. Bess, “Near field transient radiation patterns for circular pistons,” IEEE Trans. Acoust., Speech and Signal Processing, vol. 22, n. 6, pp. 395-403, 1974.

[7] J.P. Weight, “Ultrasonic beam structures in fluid media,” J. Acoust. Soc. Am., vol. 76, pp. 1184-1191, 1984. [8] G. R. Harris, “Transient field of a baffled planar piston having an arbitrary vibration amplitude distribution,” J.

Acoust.Soc.Am.,vol.70,pp.186-204, 1981. 5. Conclusion

It

rings displaced along

The effects of defect size and position on both amplitude and shape of the echo responses were investigated. The small differences among experimental

inspection. It the defect.

nowledgements

he authors thank the Brazilian government institutions Petrobras/ANP, FAPESP, and CNPq for the financial support that made this work possible.

References

P. R. Stepanishen, “Transient radiation from [2