E-Journal of Advanced Maintenance Vol.3, No.1 (2011) 11-24
Japan Society of Maintenology

Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle Beam Transducer by Hybrid FDTD Method
Tomonori KIMURA1,* and Shusou WADAKA2
1
2

Mitsubishi Electric Corporation, 5-1-1 Ofuna, Kamakura, Kanagawa 247-8501, Japan
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 Rayleighintegral with the geometrical optics approximation, and the field scattered by a flaw is calculated by thefinite-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 aside-drilled hole.
KEYWORDS

ARTICLE INFORMATION

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

Article history:
Received 29 December 2010
Accepted 25 April 2011

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 byexperiments 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, forexample, 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 somehybrid models [17]-[19]. When the distance between a transducer and a flaw is very long comparedwith 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 becomeslong.
In order to reduce the calculation time, we have developed a hybrid FDTD method, and thecalculation 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, theRayleigh integral [21] and the geometrical optics approximation [22] are used. In the angle beamtechnique, the transmission coefficient from a couplant to a test object becomes complex when the*

Corresponding author, E-mail: Kimura.Tomonori@da.MitsubishiElectric.co.jp
ISSN – 1883 – 9894/11 © 2011 JSM and the authors. All rights reserved.

1
11

T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Methodangle of incidence exceeds the critical angle. An analytic signal is introduced to deal with thecomplex 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 asteel 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
V0 (ω ) = H T (ω )ET (ω ) ,(1)
where HT(ω) is the transmitting frequency response of the piezoelectric transducer, ET(ω) is thefrequency spectrum of the electric pulse used for excitation, and ω is the angular frequency. In thefollowing, ET(ω) is set to a constant ET. This means that the excitation pulse is very short.
Equation (1) becomesV0 (ω ) = H T (ω )ET .
(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 thepoint of incidence, at which Snell’s law is satisfied, α is the angle of incidence, θs is the angle ofrefraction of the SV wave, r1 and r2 are the distances shown in Fig. 1, V1l is the velocity of thelongitudinal (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 beuniform along the y-axis. Let
12

⎛ j ⎞
G (ω ) = ⎜⎜ ⎟⎟ ,(3)
⎝ λ1 ⎠
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⎡V ⎤e jω (t − r1 V1l − r2 V2 s )VT (m, n, ω )e jωt = ⎢ Tx ⎥ = G (ω )V0 (ω )∫ TiTs d
dl ,
l
R1
⎣VTz ⎦(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⎡ cos θ s ⎤
d =⎢
(5)
⎥,
⎣− sin θ s ⎦and the divergence factor R1 is given by⎛
V cos 2 α ⎞⎟R1 = ⎜ r1 + r2 2 s 2⎜
V1l cos θ s ⎟⎠
⎝

12

.

(6)

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

dl

x(m)

Wedge
V (ω) (V1l)

Q 0
α
r1

Angle beamtransducer

Couplantz(n)SVT(m,n,ω)

-VTz

θs
r2

θs V
Tx
R(m, n)Test object (V2s)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 ultrasonicpath 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:∫

VT (m, n, ω )e jωt = ET H ' (ω ) Ti Ts d
l

where

e jω (t −τ 1 −τ 2 )
dl ,
R1

(7)

H ' (ω ) = G (ω )H T (ω ) ,

(8)

and
τ1 =

r1
V1l

, τ2 =

r2
.
V2s

(9)

The particle velocity in the time domain vT(m,n,t) is obtained by the inverse Fourier transformof VT(m,n,ω) ash ' (t − τ 1 − τ 2 )(10)v T (m, n, t ) = ET Ti Ts d
dl ,

∫

R1

l

where
h ' (t ) =

1
2π

∞

∫ H (ω )e dω .
'

j ωt

(11)

−∞

2.2 Transmission coefficients
The transmission coefficient Ti from the wedge to the couplant is given by [25]Ti =

ρ 1V1lρ i Vi2 cos α cos 2 β
⎛V
⎞
V2
ρ
cos α + 1 cos θ i ⎜ 1l cos 2 2 β + 1s sin 2α sin 2β ⎟⎜ Vi⎟
V1l Vi
ρi
⎝
⎠

,

(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 angleof 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 isgreater than Vi; thus, Ti is real.

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

β

SV

x

L α

Wedge (V1l, V1s)

z
S1

SV

L

L

θi
L

α

z

L
Wedge (V1l, V1s)

S1

Couplant
(Vi)

L

L
S2
Test object (V2l,V2s)

β

θi

S2
θs

L

θl

L

θs

Couplant(Vi)

Evanescent wave
(Pevan)Test object (V2l,V2s)
SV(a) θi < θcrSV(b) θi >θcr

Fig. 2. Refraction and reflection.

The transmission coefficient Ts from the couplant to the test object is given by [25]2Ts = −

V2s
cos θ i sin 2θ l
V 2l

⎞
⎛V
V2
ρ
cos θ l + 2 cos θ i ⎜ 2l cos 2 2θ s + 2 s sin 2θ l sin 2θ s ⎟⎟⎜ Vi
Vi V 2l
ρi
⎠
⎝

,

(13)

whereρ2 is the density of the test object, V2l is the velocity of the L wave in the test object, andθl isthe 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 isalmost 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:Vsin θ l = 2l sin θ i ,
(14)
Vi
12

2
⎧⎪ ⎛ V⎞ ⎫⎪2lcos θ l = ⎨1 − ⎜⎜sin θ i ⎟⎟ ⎬ .
(15)
⎠ ⎪⎭
⎪⎩ ⎝ Vi
Figure 2(b) shows the case that θi > θcr. In this case, an evanescent wave Pevan is generated,
given by

jωt

−j

ω
V2 l

( x sin θ l + z cos θ l ),(16)
where the incident wave amplitude is assumed to be unity, and Tl is the transmission coefficient of the
L wave. In Eq. (16),Pevan = Ti Tl e

e

12

2
⎧⎪⎛ V
⎫⎪
⎞
2l
cosθ l = − j sgn ω ⎨⎜⎜sin θ i ⎟⎟ − 1⎬ ,⎠⎪⎩⎝ Vi
⎪⎭

(17)

where

⎧ 1 for ω > 0
sgn ω = ⎨.
(18)⎩− 1 for ω < 0
The sign (plus or minus) in Eq. (18) is determined to ensure that Pevan given by Eq. (16) does not
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T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Method

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:
Ts = a + jb sgn ω ,(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 domainvT(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∫

VT (m, n, ω )e jωt = E T H ' (ω ) Ti (a + jb sgn ω )dl∫

= ET H ' (ω ) Ti adle jω (t −τ 1 −τ 2 )dl
R1

e jω (t −τ 1 −τ 2 )e jω (t −τ 1 −τ 2 )dl + jET H ' (ω ) Ti b sgn ωd
dl .
R1
R1
l

∫

(20)

h ' (t )
Wedge
(V1l)

Q
τ1
L waveαCouplant

Ts = a + jb sgn ωθsEvanescent wave
(Pevan)(ah ' (t − τ 1 − τ 2 ) − bH h ' ; t − τ 1 − τ 2τ2Test object (V2s))

R

SV wave

Fig. 3. Pulse waveform distortion.

The transmitted field in the time domain vT(m,n,t) is obtained by the inverse Fourier transform ofVT(m,n,ω) as∞

12π∫ V (m, n, ω )e dω⎧⎪Ee (H (ω )⎨ T ad
=
2π ∫
R
⎪⎩∫

v T (m, n, t ) =

-∞

∞

T

j ωt

T

jω t −τ 1 −τ 2 )

'

−∞

= ET

l

Ti a ⎧ 1
d⎨
l R1
⎩ 2π

∫
∫

= ET Ti ad
l

i

1

⎫⎪
E
dl ⎬dω + j T
2
π
⎪⎭

⎧⎪
e
H ' (ω )⎨ Tib sgn ωd
⎪⎩ l
−∞

∫

∞

∫

⎫
Ti b ⎧ 1
H ' (ω )e jω (t −τ 1 −τ 2 ) dω ⎬dl + jE T
d⎨
−∞
l R1
⎩ 2π
⎭

∫

∞

∫

Ti b ⎧⎪ 1
h (t − τ 1 − τ 2 )d⎨dl − ET
R1
⎪⎩ π
l R1
'

∫

jω (t −τ 1 −τ 2 )R1∞∫ sgn ωH (ω )e
−∞

'

⎫⎪
dl ⎬dω
⎪⎭jω (t −τ 1 −τ 2 )⎫dω ⎬dl⎭⎫⎪h ' (ξ )dξ ⎬dl .
⎪⎭− ∞ ξ − (t − τ 1 − τ 2 )∫∞The second term on the right side of Eq. (21) can be expressed using the Hilbert transform as
1 ∞h ' (ξ )H h' ;t − τ1 − τ 2 =dξ .
π −∞ ξ − (t − τ 1 − τ 2 )Using Eq. (22), the following equation is obtained:

(

)

∫

(21)

(22)

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

(

)

∫

h ' (t − τ 1 − τ 2 )H h ' ; t −τ 1 −τ 2dl − ET Ti bd
dl
R1
R1
l

∫

ah ' (t − τ 1 − τ 2 ) − bH h ' ; t − τ 1 − τ 2dl .
R1

v T (m, n, t ) = ET Ti adl∫(= ET Ti d
l

)

(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]( )ha' (t ) = h ' (t ) + jH h ' ; t .

(24)Multiplying ha’(t) by the transmission coefficient Ts for ω > 0, the following equation is obtained:{( )}= ah (t ) − bH (h ; t ) + j {aH (h ; t ) + bh (t )} .

Ts ha' (t ) = (a + jb ) h ' (t ) + jH h ' ; t
'

'

'

'

(25)

Comparing Eq. (25) with Eq. (23), we obtainv T (m, n, t ) = ET ∫ Ti dl{}

Re Ts ha' (t − τ 1 − τ 2 )
dl .
R1

(26)

The particle velocity vT(m,n,t) can be expressed by the simple equation using the analytic signalha’(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 obtainthe scattered field and the received signal. The transmitted particle velocity in the test object can becalculated 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],v x (t + Δt ) − v x (t ) 1 ⎛ ∂T xx (t + Δt 2) ∂T xz (t + Δt 2 ) ⎞=+⎟⎟⎜ρ 2 ⎜⎝
Δt
∂x
∂z
⎠

(27a)

v z (t + Δt ) − v z (t ) 1 ⎛ ∂T xz (t + Δt 2) ∂T zz (t + Δt 2) ⎞
=
+
⎜
⎟⎟
Δt
∂x
∂z
ρ 2 ⎜⎝⎠(27b)

T xx (t + Δt 2) − T xx (t − Δt 2 )∂v (t )∂v z (t ) ⎫⎧= ρ 2 ⎨V 22l x + V 22l − 2V 22s
⎬
Δt
∂x
∂z ⎭
⎩

(27c)

(

(27d)

(

)

∂v x (t )T zz (t + Δt 2 ) − T zz (t − Δt 2)∂v (t ) ⎫⎧= ρ 2 ⎨ V 22l − 2V 22s+ V 22l z ⎬
Δt
∂
x
∂z ⎭
⎩

)

T xz (t + Δt 2 ) − T xz (t − Δt 2 )⎛ ∂v (t ) ∂v (t ) ⎞= ρ 2V 22s ⎜⎜ x + z ⎟⎟ ,
Δt
∂x ⎠
⎝ ∂z

(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 thetime domain as shown in Fig. 4, and a staggered grid is shown in Fig. 5, where Δh is the grid sizeused in the FDTD method. If the calculation is started from an arbitrary time T0, the particle
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T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Method

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.
Particle velocitiesΔtΔtv(t-Δt)v(t+Δt)v(t)Time
T(t-Δt/2)

T(t+Δt/2)
Δt

StressesFig. 4. Calculation procedure in the time domain for the FDTD method.

x(m)
vx

Txx Tzz

vx

Txz

vz

Txz

z(n)

Txx Tzz

vx

Δh

vx

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]MN

e R (2T0 + kΔt ) = − ∑ ∑ v T (m, n, T0 ) ⋅v R (m, n, T0 + kΔt ) ,(28)

m =1 n =1

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 shownin Fig. 6. Equation (28) shows that the received signal is obtained by calculating the inner productof the transmitted field at T0 and the scattered field at T0+ kΔt, and then integrating these innerproducts over the predetermined calculation area.

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

Q

α

x(m)

Wedge
(V1l)
Couplant

r1

z(n)

T
θs

Test object (V2s)M

r2vR(m,n,T0+kΔt)
N
Flaw

Calculation area
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)

DensityWedge

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 anexperiment 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 thetest instrument do not have to be specified since they are included in the spectrum shown in Fig. 8(b).
Ultrasonic test instrument
6mm
Transducer
Bottomecho

10mm

Acryl block

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

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Relative amplitude [linear]

Relative amplitude [linear]

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

1.0

0

-1.0

1.0

0.5

0

Time [0.2μs/div]

5
Frequency [MHz]0(a) Received signal10(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
12

12

⎛ j⎞
⎛ j⎞
E B (ω ) ≅ ⎜ ⎟ H T (ω )H R (ω ) = ⎜ ⎟ H T2 (ω ) ,
(29)
⎝λ ⎠
⎝λ⎠
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:
12

12
12
⎧⎪⎛ j ⎞1 2⎫⎪
⎛ j 2 ⎞⎛ j⎞⎨⎜ ⎟ E B (ω )⎬ = ⎜ H T (ω )⎟ = ⎜ ⎟ H T (ω ) = G (ω )H T (ω ) .
⎝λ⎠⎝λ⎠⎪⎩⎝ λ ⎠⎪⎭
Using Eq. (30) and Eq. (8), we obtain

(30)

12

⎧⎪⎛ j ⎞1 2⎫⎪()
H ω = ⎨⎜ ⎟ E B (ω )⎬ .
(31)⎪⎩⎝ λ ⎠⎪⎭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.
Relative amplitude [linear]'

1.0ha’(t)
0
Solid line: RealDashed line: Imaginary-1.0Time [0.2μs/div]

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 thetransducer 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 ofrefraction is 45 degrees according to Snell’s law. The thickness of the test object is 60 mm, and thediameter of the side-drilled hole (SDH) is 2 mm. In the simulation, received signals are calculated
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T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Method

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.
6mm
Acryl wedge
Transducer

36.6°
12mm
Index point

Fig. 10. Angle beam transducer.
Angle beam transducer

Couplant

Index
point

45°

Steel block
46.8mmCalculation
area

SDH
(φ2mm)

T0=25μs

24.4mm

L = 52mm

60mm

7.5mm

Fig. 11. Relative locations of angle beam transducer and test object.
0.2
Amplitude [linear]

Amplitude [linear]

1.5
1.0
0.5
0
0

30
60
Angle of refraction θs [deg]90

Solid line: RealDashed line: Imaginary

0

-0.2
0

(a) Ti30
60
Angle of refraction θs [deg]90(b) TsFig. 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 thesurface of the SDH and the reflected wave at the bottom surface.

ISSN – 1883 – 9894/11 © 2011 JSM and the authors. All rights reserved.

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T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Method
26μs

27μs
SDH
Bottom29μs

28μs
DirectCreeping

31μs

30μs

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

Relative amplitude [linear]

4.3 Received signal
The received signal when L = 52 mm is shown in Fig. 14. As shown in the figure, there are fiveecho 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.
1.0

Direct

I

II

0
III

Creeping
-1.0
54

55

56
57 58
Time [μs]5960Fig. 14. Received signal obtained by the simulation (L = 52 mm).
DirectBottomCreeping

BottomPath I

BottomPath II

BottomPath III

BottomFig. 15. Propagation paths.
ISSN – 1883 – 9894/11 © 2011 JSM and the authors. All rights reserved.

21

Relative amplitude [linear]

T. Kimura et al./ Simulation of Ultrasonic Fields and Echoes Obtained Using
Angle beam Transducer by Hybrid FDTD Method
1.0

0

-1.0
54

55

56
57 58
59
60
Time [μs]Fig. 16. Received signal obtained by the experiment (L = 52 mm).

Relative echo height [linear]By scanning the angle beam transducer on the surface of the test object, the echo dynamic curveis 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 arein 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.
1.0
Exp.
0.5
Sim.
0

35

45

55
L [mm]

65

75

1.0
Direct0
Creeping
-1.0
49

50

51
52 53
Time [μs](a) Simulation

54

55

Relative amplitude [linear]

Relative amplitude [linear]Fig. 17. Echo dynamic curves.
1.0
Direct0
Creeping
-1.0
49

50

51
52 53
Time [μs](b) Experiment

54

55

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

ISSN – 1883 – 9894/11 © 2011 JSM and the authors. All rights reserved.

21

1.0

I
Direct0
Creeping
-1.0
59

60

IV

61
62 63
Time [μs](a) Simulation

64

65

Relative amplitude [linear]

Relative amplitude [linear]

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

1.0

I
Direct0
Creeping
-1.0
59

60

61
62 63
Time [μs](b) Experiment

IV
64

65

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

BottomFig. 20. Propagation path IV.

5.

Conclusions

A hybrid FDTD method for angle beam technique is presented. The field transmitted by anangle beam transducer is calculated using the Rayleigh integral with the geometrical optics
approximation. An analytic signal is introduced to deal with the complex transmission coefficientfrom 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 andscattered 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 beamtransducer and a steel block with a side-drilled hole. The experimental results are in good agreementwith those obtained by simulation.

References[1] A. H. Harker: Numerical modeling of the scattering of elastic waves in plates, J. Nondestr. Eval.
4 (1984), 89-106.
[2] A. Ilan and J. P. Weight: The propagation of short pulses of ultrasound from a circular source
coupled to an isotropic solid, J. Acoust. Soc. Am. 88 (1990), 1142-1151.
[3] J. A. G. Temple: Diffraction coefficients for flat-bottomed holes from 3-D finite difference
calculations, Ultrasonics 31 (1993), 3-12.
[4] H. Hatano, F. Hoki, and T. Matsuda: Finite deference simulation of elastic wave propagation inaxisymmetric models, J. Acoust. Soc. Jpn. 51 (1995), 755-762. (in Japanese)[5] R. Luding and W. Lord: A finite-element formulation for the study of ultrasonic NDT systems,
IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 35 (1988), 809-820.
[6] W. Lord, R. Ludwig, and Z. You: Developments in ultrasonic modeling with finite element
ISSN – 1883 – 9894/11 © 2011 JSM and the authors. All rights reserved.

23

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

analysis, J. Nondestr. Eval, 9 (1990), 129-143.
[7] C. Miki, K. Ikeda, K. Tateishi, and M. Takahashi: Numerical simulation of ultrasonic pulse-echo
inspection, Japan Society of Civil Engineers, Structural Eng. /Earthquake Eng. 9 (1993),
273s-246s.
[8] T. Xue, W. Load, and S. Udpa: Finite Elements Simulation and visualization of leaky Rayleighwaves for ultrasonic NDE, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 44 (1997), 557-564.
[9] S. Lin, T. Ito, K. Kawashima, and H. Nagamizo: Analysis of circumferential wave propagation inpipes excited by angle beam transducer with finite element method, J. Jpn. Soc. Nondestr.
Inspect. 49 (2000), 127-133. (in Japanese)[10] C. Y. Wang, J. D. Achenbach, and S. Hirose: Two-dimensional time domain BEM for scatteringof elastic waves in solids of general anisotropy, Int. J. Solids Structures 33 (1996), 3843-3864.
[11] P. P. Schafbuch, T. J. Rudolphi, F. J. Rizzo, and D. J. Shippy: A boundary element method foracoustic-elastic interaction with applications in ultrasonic NDE, J. Nondestr. Eval. 9 (1990),
101-112.
[12] P. J. Schafbuch, R. B. Thompson, and F. J. Rizzo: Application of the boundary element method
to elastic wave scattering by irregular defects, J. Nondestr. Eval. 9 (1990), 113-127.
[13] J. Virieux: SH-wave propagation in heterogeneous media: Velocity-stress finite difference
method, Geophysics 49 (1984), 1933-1942.
[14] J. Virieux: P-SVwave propagation in heterogeneous media: Velocity-stress finite difference
method, Geophysics 51 (1986), 889-901.
[15] P. P. Van’t Veen, E. J. M. Giling, M. Lorenz, and L. F. Van Der Wal: Application of afinite-difference modeling scheme for ultrasonic defect characterization, Review of Progress in
QNDE 12 (1993), 843-849.
[16] T. Kimura, K. Misu, S. Wadaka and M. Koike: Application of elastic FDTD method tosimulation of acoustic fields in ultrasonic angle beam testing, IEICE Technical Report (2006),
US2005-124. (in Japanese)[17] C, A. Brebbia and P. Georgiou: Combination of boundary and finite elements in elastostatics,
Appl. Math. Modeling 3 (1979), 212-220.
[18] K, Harumi: Computer simulation of ultrasonics in a solid, NDT International 19 (1986), 315-332.
[19] Y. Cho and J. L. Rose: An elastodynamic hybrid boundary element study for elastic guded wave
interactions with a surface breaking defect, Int. J. Solids and Structures 37 (2000), 4103-4124.
[20] T. Kimura, K. Misu, S. Wadaka, and M. Koike: A new hybrid model to calculate echoes andultrasonic fields scattered by flaws combining FDTD method with Rayleigh integral, IEICE
Trans. Fundamentals E90-A (2007), 1366-1375.
[21] V. M. Ristic, Principles of Acoustic Devices, John Wiley & Sons, New York, 1983, pp. 294-315.
[22] S. W. Lee, M. S. Sheshadri, V. Jamnejad and R. Mittra: Refraction at a curved dielectricinterface: geometrical optics solution, IEEE Trans. MTT 30 (1982), 12-19.
[23] L. W. Schmerr, Jr., and A. Sedov: An elastodynamic model for compressional and shear wave
transducers, J. Acoust. Soc. Am. 86 (1989), 1988-1999.
[24] L. W. Schmerr, Jr.: Fundamentals of Ultrasonic Nondestructive Evaluation, Plenum Press, New
York, 1998, pp. 115-119.
[25] pp. 285 – 289 in reference [21][26] J. W. Goodman: Statistical Optics, John Wiley & Sons, New York, Wiley Classics Library
Edition Published 2000, pp. 101 – 108.
[27] S. Sugimoto and M. Okamoto: Inspection Technology 9 (2004), 48-56. (in Japanese)

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