simulation of ultrasonic fields and echoes obtained · pdf filet. kimura et al./ simulation of...

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E-Journal of Advanced Maintenance Vol.# (2010) 00-00Japan Society of Maintenology

ISSN – 1883 – 9894/10 © 2010 JSM and the authors. All rights reserved.

Simulation of Ultrasonic Fields and Echoes Obtained Using

Angle Beam Transducer by Hybrid FDTD Method Tomonori KIMURA1,* and Shusou WADAKA2 1 Mitsubishi Electric Corporation, 5-1-1 Ofuna, Kamakura, Kanagawa 247-8501, Japan 2 Ryoden Shonan Electronics Corporation, 25, Yamasaki, Kamakura, Kanagawa 247-0066, Japan

(Received; For the use of JSM) Abstract. A hybrid model to calculate an ultrasonic field and a received signal by the angle beam technique is presented. In this model, the field in a test object transmitted by an angle beam transducer is calculated using the Rayleigh integral with the geometrical optics approximation, and the field scattered by a flaw is calculated by the finite-difference time domain (FDTD) method. The signal received by the transducer is obtained by calculating the inner product of the transmitted field and the scattered field at each grid point used in the FDTD method, and then these products are integrated over a predetermined calculation area. Since the calculation area in the FDTD method can be limited to around the flaw, the calculation time and computer memory required can be reduced. In the angle beam technique, the transmission coefficient from a couplant to a test object becomes complex when the angle of incidence exceeds the critical angle. In order to calculate the transmitted field in this case, an analytic signal is introduced to deal with the complex transmission coefficient. The validity of the model is demonstrated by experiments using an angle beam transducer and a steel block with a side-drilled hole.

KEYWORDS: Ultrasonic waves, Nondestructive testing, Simulation model, Hybrid FDTD method, Angle beam technique

*Corresponding author, E-mail: [email protected]

1. Introduction

Recently the maintenance of aging structures such as steel bridges, energy plants, and so forth, has become an important issue. Nondestructive ultrasonic testing is expected to help maintain such structures effectively.

A suitable testing method depends on the type of structure. If a method is evaluated by experiments alone, the time required to apply the method becomes long, and the cost becomes high. Evaluation by simulation is very effective for reducing both the time and the cost.

Several simulation models for calculating ultrasonic fields in solids have been reported, for example, the finite difference method (FDM) [1]-[4], finite element method (FEM) [5]-[9], boundary element method (BEM) [10]-[12], finite-difference time domain (FDTD) method [13]-[16], and some hybrid models [17]-[19]. When the distance between a transducer and a flaw is very long compared with the wavelength, the calculation area is very wide and a large amount of calculation is required. As a result, the calculation time to obtain the ultrasonic field in the overall calculation area becomes long.

In order to reduce the calculation time, we have developed a hybrid FDTD method, and the calculation method and its validity were reported for the normal beam technique [20].

In this paper, the hybrid FDTD method is extended to the angle beam technique. A simulation model is presented for calculating the field in a test object transmitted by an angle beam transducer, the field scattered by a flaw, and the received signal. For the calculation of the transmitted field, the Rayleigh integral [21] and the geometrical optics approximation [22] are used. In the angle beam technique, the transmission coefficient from a couplant to a test object becomes complex when the

T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using Angle beam Transducer by Hybrid FDTD Method

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angle of incidence exceeds the critical angle. An analytic signal is introduced to deal with the complex transmission coefficient. For the calculation of the field scattered by a flaw, the FDTD method is used, and the received signal is calculated using the method in ref. [20].

The validity of the model is demonstrated by experiments using an angle beam transducer and a steel block with a side-drilled hole.

2. Field transmitted by angle beam transducer 2.1. Rayleigh integral with geometrical optics approximation

An angle beam transducer consists of a piezoelectric piston type transducer and a wedge, as shown in Fig. 1. Shear waves transmitted in the wedge are neglected, since it has been reported that longitudinal waves in a solid transmitted by a piezoelectric transducer are identical to those in an equivalent fluid medium and shear waves in a solid are negligible [23].

The particle velocity V0(ω) on the surface of a piezoelectric transducer is described by ( ) ( ) ( )ωωω TT EHV =0 , (1)

where HT(ω) is the transmitting frequency response of the piezoelectric transducer, ET(ω) is the frequency spectrum of the electric pulse used for excitation, and ω is the angular frequency. In the following, ET(ω) is set to a constant ET. This means that the excitation pulse is very short. Equation (1) becomes

( ) ( ) TT EHV ωω =0 . (2) A shear vertical (SV) wave is generated and transmitted in a test object following Snell’s law.

In Fig. 1, Q is a point on the surface of the piezoelectric transducer, R is an observation point, S is the point of incidence, at which Snell’s law is satisfied, α is the angle of incidence, θs is the angle of refraction of the SV wave, r1 and r2 are the distances shown in Fig. 1, V1l is the velocity of the longitudinal (L) wave in the wedge, V2s is the velocity of the SV wave in the test object, VT(m,n,ω) is the particle velocity at the observation point R(m,n), VTx and VTz are the x-axis and z-axis components of VT(m,n,ω), respectively, and dl is an infinitesimal element of the piezoelectric transducer.

A two-dimensional model is used in the following, that is, the wave fields are assumed to be uniform along the y-axis. Let

( )21

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

λω jG , (3)

whereλ1 is the wavelength in the wedge. Using the Rayleigh integral [21] and the geometrical optics approximation [22], the transmitted

SV wave field VT(m,n,ω) is given by

( ) ( ) ( )( )

∫−−

=⎥⎦

⎤⎢⎣

⎡=

l

VrVrtj

siTz

TxtjT dl

ReTTVG

VV

enmsl

10

2211

,,ω

ω ωωω dV , (4)

where Ti is the transmission coefficient from the wedge to the couplant, Ts is that from the couplant to the test object, d is the polarization vector [24], and R1 is the divergence factor. The polarization vector d is given by

⎥⎦

⎤⎢⎣

⎡−

=s

s

θθ

sincos

d , (5)

and the divergence factor R1 is given by

21

21

22

211coscos

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

sl

s

VV

rrRθα . (6)


3

Fig. 1. Arrangement of angle beam transducer and test object.

In Eq. (4) and Fig. 1, the thickness of the couplant is set to zero. This means that the ultrasonic

path delay in the couplant is neglected. Equation (4) is applicable in the case that the distance r1 exceeds several wavelengths. When r1 is very short compared with the wavelength λ1, the Hankel function should be used. Substituting Eq. (2) into Eq. (4), the following equation is obtained:

( ) ( )( )

∫−−

=l

tj

siTtj

T dlR

eTTHEenm1

'21

,,ττω

ω ωω dV , (7)

where ( ) ( ) ( )ωωω THGH =' , (8) and

sl V

rVr

2

22

1

11 , == ττ . (9)

The particle velocity in the time domain vT(m,n,t) is obtained by the inverse Fourier transform of VT(m,n,ω) as

( ) ( )∫

−−=

lsiTT dl

Rth

TTEtnm1

21'

,,ττ

dv , (10)

where

( ) ( ) ωωπ

ω deHth tj∫∞

∞−= ''

21 . (11)

2.2 Transmission coefficients

The transmission coefficient Ti from the wedge to the couplant is given by [25]

⎟⎟⎠

⎞⎜⎜⎝

⎛++

=

βαβθρρ

α

βαρρ

2sin2sin2coscoscos

2coscos2

1

21211

11

il

s

i

li

i

ii

li

VVV

VVV

VT , (12)

where ρ1 is the density of the wedge, ρi is the density of the couplant, Vi is the velocity of the L wave in the couplant, V1s is the velocity of the SV wave in the wedge, and β and θi are the angle of reflection for the SV wave in the wedge and the angle of incidence for the L wave in the couplant, respectively, as shown in Fig. 2(a). Point S1 in Fig. 2(a) is the point of incidence. Usually, V1l is greater than Vi; thus, Ti is real.

Piezoelectric transducer

r2

θs

Test object (V2s) R(m, n)

VT(m,n,ω)

VTx

-VTz

x(m)

z(n)

S

r1α

Wedge(V1l)

dl

QV0(ω)

Couplant

θs

Angle beam transducer


4

Fig. 2. Refraction and reflection.

The transmission coefficient Ts from the couplant to the test object is given by [25]

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−=

slli

ss

i

li

il

lil

s

s

VVV

VV

VV

T

θθθθρρ

θ

θθ

2sin2sin2coscoscos

2sincos2

2

22222

2

2

, (13)

whereρ2 is the density of the test object, V2l is the velocity of the L wave in the test object, andθl is the angle of refraction of the L wave in the test object as shown in Fig. 2(a). Point S2 in Fig. 2(a) is the point of incidence. When the thickness of the couplant is negligible, the location of point S2 is almost coincident with that of point S1 and that of point S in Fig. 1.

Figure 2(a) shows the case that the angle of incidence θi at the boundary between the couplant and the test object is less than the critical angle of incidence θcr, which is given by sin-1(Vi/V2l). In the case shown in Fig. 2(a), the following equations are obtained by Snell's law:

ii

ll V

V θθ sinsin 2= , (14)

2122 sin1cos

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−= i

i

ll V

V θθ . (15)

Figure 2(b) shows the case that θi > θcr. In this case, an evanescent wave Pevan is generated, given by

( )ll

lzx

Vj

tjlievan eeTTP

θθωω

cossin2

+−= , (16)

where the incident wave amplitude is assumed to be unity, and Tl is the transmission coefficient of the L wave. In Eq. (16),

2122 1sinsgncos

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛−= i

i

ll V

Vj θωθ , (17)

where

⎩⎨⎧

<−>

=01

01sgn

ωω

ωfor

for. (18)

The sign (plus or minus) in Eq. (18) is determined to ensure that Pevan given by Eq. (16) does not

θiCouplant(Vi)

Test object (V2l,V2s)

L

θlθs

(a) θi < θcr

θi

Couplant(Vi)

L

θs

Evanescent wave

(b) θi >θcrSV

L

SV

(Pevan)

αWedge (V1l, V1s)

αL

x

z

x

z

L

SVβ SV

Lβ

LL L

S1

S2

S1

S2

Wedge (V1l, V1s)

Test object (V2l,V2s)


5

become infinitely large for all frequencies (both positive and negative) as z→∞. Note that cosθl in Eq. (17) is complex.

Substituting Eq. (17) into Eq. (13), Ts becomes complex and can be expressed as follows: ωsgnjbaTs += , (19)

where a is the real part, and b is the imaginary part, for ω > 0. 2.3 Pulse waveform distortion

The effect of the complex transmission coefficient Ts on the transmitted field in the time domain vT(m,n,t) is discussed. Substituting Eq. (19) into Eq. (7), the transmitted field VT(m,n,ω) in the test object can be written as

( ) ( ) ( )( )

∫−−

+=l

tj

iTtj

T dlR

ejbaTHEenm1

'21

sgn,,ττω

ω ωωω dV

( )( )

( )( )

∫∫−−−−

+=l

tj

iTl

tj

iT dlR

ebTHjEdlR

eaTHE1

'

1

'2121

sgnττωττω

ωωω dd . (20)

Fig. 3. Pulse waveform distortion. The transmitted field in the time domain vT(m,n,t) is obtained by the inverse Fourier transform of VT(m,n,ω) as

( ) ( ) ωωπ

ω denmtnm tjTT ∫

∞

∞=

-Vv ,,

21,,

( )( )

( )( )

ωωωπ

ωωπ

ττωττωddl

RebTHEjddl

ReaTHE

l

tj

iT

l

tj

iT

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ∫∫∫∫−−∞

∞−

−−∞

∞− 1

'

1

'2121

sgn22

dd

( ) ( ) ( ) ( )∫ ∫∫ ∫ ⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=∞

∞−

−−∞

∞−

−−

l

tjiT

l

tjiT dldeH

RbT

jEdldeHR

aTE ωωω

πωω

πττωττω 2121 '

1

'

1sgn

21

21 dd

( ) ( )( )∫ ∫∫ ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−

−−=

∞

∞−l

iT

liT dld

th

RbT

EdlR

thaTE ξ

ττξξ

πττ

21

'

11

21' 1dd . (21)

The second term on the right side of Eq. (21) can be expressed using the Hilbert transform as

( ) ( )( ) ξ

ττξξ

πττ d

thth ∫

∞

∞− −−−=−−

21

'

21' 1;H . (22)

Using Eq. (22), the following equation is obtained:

α

θs

τ2

( )th '

( ) ( )21'

21' ; ττττ −−−−− thbtah H

R

ωsgnjbaTs +=

τ1

Wedge(V1l)

Test object (V2s)

Evanescent wave(Pevan)

SV wave

L wave Couplant

Q


6

( ) ( ) ( )∫∫

−−−

−−=

liT

liTT dl

Rth

bTEdlR

thaTEtnm

1

21'

1

21' ;

,,ττττ H

ddv

( ) ( )∫

−−−−−=

liT dl

Rthbtah

TE1

21'

21' ; ττττ H

d . (23)

The second term on the right side of Eq. (21) is introduced by the imaginary part b of the transmission coefficient Ts given by Eq. (19). This term causes the waveform distortion of the incident pulse, as shown in Fig. 3. 2.4 Analytic signal

The analytic signal ha’(t) is given by [26]

( ) ( ) ( )thjththa ;''' H+= . (24) Multiplying ha’(t) by the transmission coefficient Ts for ω > 0, the following equation is obtained: ( ) ( ) ( ) ( ){ }thjthjbathT as ;''' H++= ( ) ( ) ( ) ( ){ }tbhthajthbtah '''' ;; ++−= HH . (25) Comparing Eq. (25) with Eq. (23), we obtain

( ) ( ){ }∫

−−=

las

iTT dlRthT

TEtnm1

21'Re

,,ττ

dv . (26)

The particle velocity vT(m,n,t) can be expressed by the simple equation using the analytic signal ha’(t) .

3. Scattered field and received signal

The hybrid model reported for the case of the normal beam testing in ref. [20] is used to obtain

the scattered field and the received signal. The transmitted particle velocity in the test object can be calculated using Eq. (26). On the basis of the field, the field scattered by a flaw is calculated by the FDTD method.

In an isotropic medium, particle velocities (vx, vz) and stresses (Txx, Tzz, Txz) are calculated by Eqs. (27a)-(27e) using the FDTD method [14],

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂

Δ+∂+

∂Δ+∂

=Δ

−Δ+z

ttTx

ttTt

tvttv xzxxxx 221

2ρ (27a)

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂

Δ+∂+

∂Δ+∂

=Δ

−Δ+z

ttTx

ttTt

tvttv zzxzzz 221

2ρ (27b)

( ) ( ) ( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

∂∂

−+∂

∂=

ΔΔ−−Δ+

ztv

VVx

tvV

tttTttT z

slx

lxxxx 2

22

22

22 222

ρ (27c)

( ) ( ) ( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

∂∂

+∂

∂−=

ΔΔ−−Δ+

ztv

Vx

tvVV

tttTttT z

lx

slzzzz 2

22

22

22 222 ρ (27d)

( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

=Δ

Δ−−Δ+x

tvz

tvV

tttTttT zx

sxzxz 2

2222

ρ , (27e)

where Δt is the time step. Stresses around flaws are set to zero to satisfy the boundary conditions between a solid and a vacuum. Particle velocities and stresses are calculated like a leap-frog in the time domain as shown in Fig. 4, and a staggered grid is shown in Fig. 5, where Δh is the grid size used in the FDTD method. If the calculation is started from an arbitrary time T0, the particle


7

velocities at time T0 (vx(T0), vz(T0)) and the stresses at time T0-Δt/2 (Txx(T0-Δt/2), Tzz(T0-Δt/2), Txz(T0-Δt/2)) are required at each point in the grid shown in Fig. 5. Fig. 4. Calculation procedure in the time domain for the FDTD method.

vxvx

vx vx

vz

Txx Tzz

Txz

Txx Tzz

TxzΔh

x(m)

z(n)

Δh

Fig. 5. Two-dimensional staggered grid used in the FDTD method.

On the basis of the scattered field, the received signal can be calculated by [20]

( ) ( ) ( )tkTnmTnmtkTe R

M

m

N

nTR Δ+⋅−=Δ+ ∑∑

= =0

1 100 ,,,,2 vv , (28)

where vR(m,n,T0+nΔt) is the scattered field obtained by the FDTD method, k is the step number, and M and N are the grid numbers in the x and z directions in the calculation area, respectively, as shown in Fig. 6. Equation (28) shows that the received signal is obtained by calculating the inner product of the transmitted field at T0 and the scattered field at T0+ kΔt, and then integrating these inner products over the predetermined calculation area.

v(t)

T(t-Δt/2) T(t+Δt/2)

v(t-Δt) v(t+Δt)

Δt

Δt ΔtParticle velocities

Stresses

Time


8

Fig. 6. Calculation area and scattered field. 4. Comparison of simulated results with experimental results

In order to confirm the validity of the hybrid model, simulation and experiments were performed.

The material parameters used in the simulation are shown in Table I.

Table I Material parameters.

Velocity (L wave) Velocity (SV wave) Density

Wedge 2730 m/s 1430 m/s 1180 kg/m3

Couplant 1480 m/s - 1000 kg/m3

Test object 5930 m/s 3240 m/s 7700 kg/m3

4.1 Response of transducer The transmitting time response of a transducer h’(t) in Eq. (11) is needed to calculate the wave

fields and received signals in the hybrid model. The response is obtained by carrying out an experiment using an acryl block as a flat reflector. The experimental setup is shown in Fig. 7. A digital ultrasonic test instrument UI-25 [27] is used. The thickness of the acryl block is 10 mm. The size of the transducer is 6 mm, and the nominal frequency is 5 MHz. Figures 8(a) and (b) show a received signal reflected at the bottom surface of the acryl block and its frequency spectrum, respectively. Note that the frequency characteristics of the excitation pulse and the amplifier in the test instrument do not have to be specified since they are included in the spectrum shown in Fig. 8(b).

Ultrasonic test instrument

Bottom echo

Acryl block

10mm

Transducer

6mm

Fig. 7. Experimental setup to obtain time response of transducer.

M

N

vR(m,n,T0+kΔt)

Calculation area

Flaw

Piezoelectric transducer

r2

θs

Test object (V2s)

x(m)

z(n)

T

r1

α

Wedge(V1l)

dl

Q CouplantAngle beam transducer


9

Time [0.2μs/div]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

1.0

0.5

0Rel

ativ

e am

plitu

de [l

inea

r]

0 5 10Frequency [MHz]

(a) Received signal (b) Frequency spectrum Fig. 8. Received signal reflected at the bottom surface and its frequency spectrum.

The frequency spectrum shown in Fig. 8(b) includes the transmitting and receiving responses. The transmitting response h’(t) in Eq. (11) is extracted from the frequency spectrum EB(ω) shown in Fig. 8(b). EB(ω) can be approximately expressed as

( ) ( ) ( ) ( )ωλ

ωωλ

ω 22121

TRTB HjHHjE ⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛≅ , (29)

where HT(ω) is the transmitting response, HR(ω) is the receiving response, and the relation HT(ω)= HR(ω) is used. From Eq. (29), the following equation holds:

( ) ( ) ( ) ( ) ( )ωωωλ

ωλ

ωλ TTTB HGHjHjEj =⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

21212

2121

. (30)

Using Eq. (30) and Eq. (8), we obtain

( ) ( )2121

'

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛= ω

λω BEjH . (31)

The transmitted response h’(t) is calculated by the inverse Fourier transform of H’(ω) given by Eq. (31), and its analytic signal ha’(t), given by Eq. (24), is calculated using the Hilbert transform. Figure 9 shows the obtained analytic signal.

Time [0.2μs/div]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

Solid line: RealDashed line: Imaginary

ha’(t)

Fig. 9. Analytic signal. 4.2 Wave fields

The transducer shown in Fig. 7 is attached to an acryl wedge to form an angle beam transducer, as shown in Fig. 10. The angle of incidence is 36.6 degrees, and the propagation length from the transducer to the index point is 12 mm.

The relative locations of the angle beam transducer and the test object are shown in Fig. 11. The velocity of the SV wave in the test object is 3240 m/s, as shown in Table I; thus, the angle of refraction is 45 degrees according to Snell’s law. The thickness of the test object is 60 mm, and the diameter of the side-drilled hole (SDH) is 2 mm. In the simulation, received signals are calculated


10

for various horizontal distances L, shown in Fig. 11. The calculation area is set to be sufficient large (46.8 mm×24.4 mm) to include fields with significant amplitudes. The transmitted particle velocity field when L = 52 mm and T0 = 25μs is shown in Fig. 11. In this simulation, the couplant is water. The transmission coefficients Ti and Ts are shown in Fig. 12. Note that the horizontal axis in Fig. 12(a) is converted to θs from θi according to Snell’s law.

12mm36.6°

6mm

Acryl wedge

Index point

Transducer

Fig. 10. Angle beam transducer.

Angle beam transducer

46.8mm

24.4mm

Calculation area

Steel block

L = 52mm

60mm

SDH(φ2mm)

45°

Index point

7.5mm

T0=25μs

Couplant

Fig. 11. Relative locations of angle beam transducer and test object.

1.5

0

(a) Ti

0

0.2

-0.20 9030 60

1.0

0.5

0 9030 60Angle of refraction θs [deg]

(b) Ts

Am

plitu

de [l

inea

r]

Am

plitu

de [l

inea

r] Solid line: RealDashed line: Imaginary

Angle of refraction θs [deg]

Fig. 12. Transmission coefficients.

The scattered particle velocity fields obtained by the FDTD method are shown in Fig. 13.

Absorbing boundary conditions [16] are employed at the top side, the left side, and the right side of the calculation area. The grid size Δh is set to 28μm and the time step Δt is 2 ns. As shown in Fig. 13, the scattered fields are complicated owing to the propagation of the creeping wave along the surface of the SDH and the reflected wave at the bottom surface.


11

26μs 27μs

28μs 29μs

30μs 31μs

DirectCreeping

SDH

Bottom

Fig. 13. Scattered particle velocity fields obtained by the FDTD method. 4.3 Received signal

The received signal when L = 52 mm is shown in Fig. 14. As shown in the figure, there are five echo signals, referred to as “Direct”, “Creeping”, “I”, “II”, and “III’. The vertical axis is normalized by the amplitude of the “Direct” echo signal. The propagation paths of these echo signals can be estimated from the scattered fields shown in Fig. 13 and are shown in Fig. 15.

The experimental result is shown in Fig. 16. The echo signals shown in Fig. 16 are in good agreement with the simulated signals shown in Fig. 14. Fig. 14. Received signal obtained by the simulation (L = 52 mm).

Bottom Bottom Bottom

Path I Path II Path III

Bottom

CreepingDirect

Bottom Fig. 15. Propagation paths.

54 55 56 57 58 59 60Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

Direct

Creeping

III

III


12

Fig. 16. Received signal obtained by the experiment (L = 52 mm).

By scanning the angle beam transducer on the surface of the test object, the echo dynamic curve is simulated using the “Direct” echo signal. The result is shown in Fig. 17 by the dashed line. The experimental result shown by the solid line is in good agreement with the simulated curve.

Other examples of received signals are shown in Fig. 18 and Fig. 19. The simulated results are in good agreement with the experimental results. When L = 62 mm, the echo signal referred to as “IV” is received, as shown in Fig. 19, and the propagation path of this echo signal is shown in Fig. 20.

35 45 55 65 750

1.0

0.5

Rel

ativ

e ec

ho h

eigh

t [lin

ear]

L [mm]

Exp.

Sim.

Fig. 17. Echo dynamic curves.

50 51 52 53 54 55Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

49

(a) Simulation

50 51 52 53 54 55Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

49

(b) Experiment

Direct Direct

Creeping Creeping

Fig. 18. Received signal (L = 44 mm).

54 55 56 57 58 59 60Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0


13

60 61 62 63 64 65Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

59

(a) Simulation

60 61 62 63 64 65Time [μs]

Rel

ativ

e am

plitu

de [l

inea

r] 1.0

-1.0

0

59

(b) Experiment

Direct Direct

Creeping Creeping

I I

IV IV

Fig. 19. Received signal (L = 62 mm).

Bottom

Path IV

Fig. 20. Propagation path IV.

5. Conclusions

A hybrid FDTD method for angle beam technique is presented. The field transmitted by an

angle beam transducer is calculated using the Rayleigh integral with the geometrical optics approximation. An analytic signal is introduced to deal with the complex transmission coefficient from the couplant to the test object. The field scattered by a flaw is calculated using the FDTD method, and the received signal is calculated by integrating the inner products of the transmitted and scattered fields. Using the hybrid model, the calculation time and computer memory required can be reduced.

In order to confirm the validity of the model, experiments are performed using an angle beam transducer and a steel block with a side-drilled hole. The experimental results are in good agreement with those obtained by simulation.

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