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Simulation of Ultrasonic and EMAT Arrays Using FEM and FDTD
Yuedong XIE 1, Wuliang YIN 1, Zenghua LIU2, Anthony PEYTON 1
1
School of Electrical and Electronic Engineering, University of Manchester; Manchester, United Kingdom
Phone: +44 7598 449793; e-mail: [email protected], [email protected],
[email protected]
2
College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology,
Beijing, 100124, China. E-mail: [email protected]
Abstract
This paper presents a method which combines electromagnetic simulation and ultrasonic simulation to build
EMAT array models. For a specific sensor configuration, Lorentz forces are calculated using the finite element
method (FEM), which then can feed through to ultrasonic simulations. The propagation of ultrasound waves is
numerically simulated using finite-difference time-domain (FDTD) method to describe their propagation within
homogenous medium and their scattering phenomenon by cracks. Radiation pattern obtained with Hilbert
transform on time domain waveforms is proposed to characterise the sensor in terms of its beam directivity and
field distribution along the steering angle.
Keywords: Ultrasonic Testing (UT), Phased array, Modelling and Simulation, Electromagnetic acoustic
transducer (EMAT), Finite difference time domain (FDTD) and Finite element method (FEM)
Highlights




We combine an EM model with an ultrasonic model for EMAT simulation
Frequency domain simulation (FEM) and time domain simulation (FDTD) linked together
Radiation pattern obtained with Hilbert transform on time domain waveforms
Various wave modes observed with the proposed method.
1. Introduction
There are a variety of non-destructive testing (NDT) techniques employed in industries, such
as magnetic particle inspection (MPI), electromagnetic methods (EM), eddy current methods,
and ultrasonic methods [1-4]. Due to its advantages of good penetration depth and mechanical
flexibility, the piezoelectric ultrasonic method is widely used for thickness measurement, flaw
evaluation and material characterization [5-10]. The transducer frequently used for the
conventional ultrasonic non-destructive testing is piezoelectric ceramics or crystals [9-11].
However, one primary disadvantage of the piezoelectric ultrasonic testing is the need to have
good sonic contact with the test piece, typically by means of a couplant for acoustic
impedance matching [12].
Electromagnetic acoustic transducers (EMATs) are becoming increasingly popular due to
their non-contact nature [13, 14]. An EMAT sensor typically consists of a permanent magnet
providing a large static magnetic field, a coil carrying an alternating current, which are placed
next to the test piece [15-17]. There are two EMAT interactions which can produce
ultrasound, magnetostriction for magnetic materials and the Lorentz force mechanism for
conducting metallic materials [14, 16, 18]. Because an EMAT generates ultrasonic waves
directly into the testing piece instead of coupling through the transducer, an EMAT has
advantages in applications where surface contact is not possible or desirable [19, 20]. Another
attractive feature of EMAT is a variety of waves modes can be produced based on different
combinations of coils and magnets [14, 21]. In this paper, only EMAT based on Lorentz force
mechanism to generate Rayleigh waves is discussed.
Considerable works were reported on EMAT modelling [22-26]. All of these papers used
finite element method (FEM) or finite-difference method to numerically model the EMAT,
and R.Jafari-Shapoorabadi proposed an improved FEM method by using complete equations
[27]. Because most of the previous work were 2-D, and did not consider the effect of the
dynamic magnetic field, Shujuan Wang proposed a 3D modelling method based on the finite
element method with a consideration of the dynamic magnetic field, which made the results
more reliable [15]. In addition, X. Jian combined analytical and numerical solutions together
to analyse EMAT performance, but the analytical solution was only available for simple
uniform force distributions [26].
Consequently, this paper proposes a method using the finite-difference time-domain (FDTD)
and finite element method (FEM) to model EMAT arrays. The FDTD approach is used to
describe the propagating behaviour of ultrasound waves, such as steering and focusing
phenomenon. The FEM method is employed to calculate the Lorentz stress distribution for a
given coil and DC biased magnet configuration, which can be fed through to ultrasonic
simulations to model EMAT arrays.
2. EMAT array modelling
An EMAT sensor consists basically of a coil carrying an alternating current, a permanent
magnet providing a large static magnetic field, and the test piece, as shown in Figure 1. There
are two EMAT mechanisms to generate ultrasound waves, magnetostriction for magnetic
materials and the Lorentz force mechanism for conducting metallic materials. In this work,
the test piece used is a stainless steel plate, which is mainly affected by Lorentz force
mechanism, so magnetostriction is not considered. The Lorentz force mechanism is: the coil
induces eddy currents 𝐽 in the surface layers of the testing material, and the interaction
between the static magnetic field 𝑩 and eddy currents 𝑱 produces a Lorentz stress 𝑭 based on
Equation (1), which in turn generates ultrasound waves propagating within the testing sample.
𝑭 = 𝑱 × 𝑩…………………..…….………….……. (1)
Figure 1. The configuration of a typical EMAT.
The receiving process of an EMAT is normally the reverse of the generation: in the presence
of a static magnetic field, the dynamic electric field are induced in the test piece due to
ultrasound waves (Equation (2)); induced eddy currents result in time varying magnetic fluxes
(Equation (3) and (4)), and in turn produce a voltage picked up by the receiving coil
(Equation (5)) [16] [28].
{
𝑬𝑒𝑑 = 𝒗 × 𝑩
…………………….………….……. (2)
𝑱𝑒𝑑 = 𝜎 ∗ 𝑬𝑒𝑑
2
∇ 𝑨0 = 𝑗𝜔𝜎0𝜇0𝑨0
2
∇ 𝑨1 = −𝜔2𝜀1𝜇1𝑨1
2
∇ 𝑨2 = 𝑗𝜔𝜎2𝜇2𝑨2
………………………….……. (3)
2
{
∇ 𝑨3 = 𝑗𝜔𝜎3𝜇3𝑨3
0
𝑨 = ∫−∞
𝑨1𝑑 (𝑧𝑒𝑑)…………………………….……. (4)
𝜕𝑨
{
𝑬𝑟 = − 𝜕𝑡
𝑽𝑟 = 𝑁 ∗ ℓ ∗ 𝑬 𝑟
…………………………….……. (5)
where 𝒗 is the velocity of particle vibration, 𝑩 is the static magnetic field, 𝜎 is the
conductivity of the test piece, 𝑬𝑒𝑑 and 𝑱𝑒𝑑 are the induced electric field and the induced eddy
current sheet due to the particle vibration at the depth 𝑧𝑒𝑑 respectively (as shown in Figure 2).
𝑨𝑖 , 𝜎𝑖 and 𝜇𝑖 are the magnetic vector potential, the conductivity and the permeability in zone
i respectively. 𝑨 is the total magnetic vector potential due to the induced eddy currents in the
whole test piece. 𝑁 and ℓ are the turn number and the length of the receiving coil used
respectively; 𝑬𝑟 and 𝑽𝑟 are the induced electric field and the induced voltage which can be
picked up by the receiving EMAT.
Figure 2. The geometry used for describing the reception of an EMAT sensor.
2.1 Electromagnetic simulation
Electromagnetic simulation is performed by the finite element method (FEM) and is used to
obtain the distribution of Lorentz stress; the testing sample used is a stainless steel plate with
a dimension of 400×400×80 mm3, and the permanent magnet used is NdFeB35, whose size is
60×60×30 mm3. The meander coil carries an alternating current with the peak of 50 A, the
operation frequency is 500 kHz, and the skin depth calculated is 0.679 mm. The Rayleigh
wave velocity is 3.033 mm/us in the stainless steel plate used, so the centre-to-centre distance
between two adjacent lines of the meander coil is 3.033 mm to form the constructive
interference. The meander coil used in this work has a dimension of 56×34.163×0.036 mm 3,
which is very small compared to the stainless steel plate used. In order to improve the
modelling time, only the area (90×90×10 mm3) where the meander coil has a major effect on
is picked to study the Lorentz stress distribution. Figure 3 describes the distribution of the
static magnetic flux density B and the induced eddy currents J. From these diagrams, the
maximum magnetic field occurs near the edges of the permanent magnet, and the induced
eddy currents are mainly distributed along the meander coil.
Figure 3. The distribution of the static magnetic field B (left) and the eddy currents J
(right).
Lorentz force density along x, y and z directions can be calculated from Equation (1). Based
on the left-hand rule, the Lorentz force density is mainly distributed along y direction, which
is the dominant reason why this setup produces Rayleigh waves; x component and z
component of the Lorentz force density are not considered in this work. Assuming the surface
of the stainless steel plate contains 301*301 space steps, the y component of Lorentz force
density is a 301*301 matrix as well. By averaging the value of every row in the matrix, the 3D model is simplified to a 2-D model; the subsequent ultrasonic simulations are carried out in
2D. Figure 4 shows the distribution of Lorentz force density in y direction on the surface of
the stainless steel plate. It can be seen that the Lorentz force density on the outmost lines is
larger than that on the inner lines; that’s because the maximum magnetic field occurs at the
places corresponding to the edges of the permanent magnet.
Lorentz force density: Fy
6
4
x 10
3
amlitude (N/m3)
2
1
0
-1
-2
-3
-4
0
50
100
150
200
250
300
350
space step
Figure 4. Lorentz force density distribution in y direction.
2.2 Elastodynamic equations and FDTD method
Elastodynamic equations are a set of partial differential equations describing how material
deforms and becomes internally stressed as shown in Equations (6) and (7), [29, 30].
𝜌(𝑥)
𝜕𝒗𝒊
𝜕t
(𝑥, 𝑡) = ∑𝑑𝑗=1
𝜕𝑻𝒊𝒋
𝜕𝑥𝑗
(𝑥, 𝑡) + 𝑓𝑖 (𝑥, 𝑡)………………….... (6)
𝜕𝑻𝒊
𝜕t
= ∑𝑑𝑗=1 ∑𝑑𝑖=1 𝑐𝑖𝑗𝑘𝑙 (𝑥)
𝜕𝒗𝑘
𝜕𝑥𝑙
+ 𝜃𝑖𝑗 (𝑥, 𝑡).……………………… (7)
where 𝜌 is the mass density and 𝑐𝑖𝑗𝑘𝑙 is the 4th stiffness tensor of the testing sample, 𝑓𝑖 and 𝜃𝑖𝑗
are the force source and strain tensor rate source respectively. The parameters to be calculated
are the velocity 𝒗𝒊 and stress tensor 𝑻𝒊 . Equation (6) is Newton’s Second Law: when a force is
applied to a testing sample, stress and deformation are generated, as well as particle
displacement. Equation (7) is, based on Hooke’s Law, describing the relationship of stress
tensor rate and strain tensor rate when deformation occurs.
The finite-difference time-domain (FDTD) method is a numerical method to solve differential
equations by discreting the differential form to the finite difference form [31]. In this paper,
forward difference and centre difference methods are used to calculate the unknown
parameters velocity 𝒗𝒊 and stress tensor 𝑻𝒊 [30]. The finite-difference time-domain (FDTD)
method is used to model ultrasound waves’ propagation in this work. The simulations of
phased array techniques based on the finite-difference time-domain (FDTD) method,
including focusing, steering, scattering from a crack and radiation pattern, are shown in the
appendix.
2.3 The combination of two simulations
As mentioned before, the calculated y component of Lorentz force density is used as the
excitation source to combine the electromagnetic simulation and ultrasonic simulation
together to model EMAT arrays (Figure 5). The 12 calculated alternating Lorentz stresses, S1,
S2, S3…S12, are added to the surface of the stainless steel plate as the force source (Equation
(6)) of the ultrasonic model to perform the EMAT array simulation; the unit of the Lorentz
stress is N/m3. Two receivers, R1 and R2, are placed inside of the stainless steel plate, on the
coordinate of (50, 79) and (50, 77) respectively, to inspect the arrival signals. Free surface is
employed on the top of the stainless steel plate, as shown in Figure 5; perfectly-matched layer
(PML) is used to absorb reflections from the left, right and bottom boundaries. In addition,
please note that, for a complete model, the EMATs result in the volume force for ultrasound
generation; in this work, we use an approximated model with only the surface source to
generate ultrasound waves, and this treatment will be validated with experiments.
Figure 5. Transformation from electromagnetic model to ultrasonic model.
The parameters used in the simulations of EMAT arrays are shown in Table 1; the spatial step
used is one thirtieth of the Rayleigh waves’ length, the time step is calculated from the
Courant-Friedrichs-Lewy (CFL) condition, and the spacing between two adjacent lines is one
half of the Rayleigh wavelength to perform the constructive interference.
Table 1. Parameters used in EMAT-Rayleigh modelling.
Parameters used in modelling EMAT-Rayleigh arrays
frequency
500 kHz
Rayleigh Velocity
3.03 mm/us
spatial step
0.2 mm
time step
0.0237 us
spacing
3 mm
turns
6
The propagation of EMAT-Rayleigh waves is shown in Figure 6: at 23 us after firing, both
the bulk waves, head waves and surface waves can be identified. Bulk waves contain
longitudinal waves and shear waves, which are obliquely propagating into the material; the
velocity of the longitudinal waves is larger than that of the shear wave, so the longitudinal
wave arrives earlier than the shear wave. Surface waves, which are Rayleigh waves in this
work, are propagating along the surface and the sub-surface of the material. The velocities of
Rayleigh waves and shear waves are slightly different; in most of situations, the velocity of
Rayleigh waves is 90 percent of that of the shear waves. So the propagation of Rayleigh
waves is slightly delayed than that of shear waves, as shown in Figure 7, where the Rayleigh
waves can be identified more clearly at 45 us.
Figure 6. After adding sources of Lorentz stresses, wave propagation at 23 us after firing.
Four waves types are produced: longitudinal waves, shear waves, head waves, and
Rayleigh waves.
Figure 7. After adding sources of Lorentz stresses, wave propagation at 45 us after firing.
Rayleigh waves’ propagation is slightly delayed than the shear waves’ propagation.
The receiving signals from R1 and R2 are shown in Figure 8; the first arrival signal is the
longitudinal wave arriving at about 25 us, and the later arrival signal is the Rayleigh wave at
about 49 us with a large amplitude compared to that of longitudinal waves. The Rayleigh
waves can be validated by the distance-of-flight calculation: the distance from R1 and the
EMAT sensor is a constant, so the flight distance of the longitudinal waves and the Rayleigh
waves should be the same. The velocities of the longitudinal waves and the Rayleigh waves
used are 5.9 mm/us and 3.033 mm/us respectively, and the arrival time of these two waves are
25.4 us and 49.8 us respectively, so the flight distance of the longitudinal waves and the
Rayleigh waves calculated is 149.86 mm and 150.89 mm, which are almost the same. In
addition, the amplitude of R1 is slightly larger than that of R2, which confirms that Rayleigh
waves mainly propagate along the surface of the material.
Figure 8. Receiving signals from receivers R1 and R2.
Figure 9 shows the displacement pattern along the depth of the stainless steel plate; the
displacement of Rayleigh waves decreases with the depth increasing; when the depth is larger
than the wavelength of Rayleigh waves, 6 mm, the displacement of Rayleigh waves is very
small. More specifically, the displacement of Rayleigh waves at depth 6 mm is 37% of that at
depth 1 mm; at depth 9 mm and 13 mm, the amplitude decreases to 13% and 12% of that at
depth 1 mm respectively. This confirms that Rayleigh waves mainly propagate along the
surface of the test piece.
particle displacement along the depth
1
1mm
4mm
6mm
9mm
13mm
0.8
0.6
Normalised displacement
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
10
20
30
40
50
60
70
time (us)
Figure 9. Displacement pattern at various depths of the stainless steel plate.
2.4 The radiation pattern of EMAT-Rayleigh waves
Based on the constructive interference, a Rayleigh wave is produced along the surface of the
testing sample. The radiation pattern of the EMAT-Rayleigh waves is shown in Figure 10(a),
where shows that Rayleigh waves are mainly distributed along the surface and sub-surface of
the material; longitudinal and shear waves are propagating obliquely into the material, and
shear waves have a larger magnitude than longitudinal waves. In this work, only Rayleigh
waves are of interest; the beam features of Rayleigh waves are studied by means of Figure
10(b).
a) The radiation pattern of EMAT-Rayleigh waves.
b) Radiation pattern for beam features.
Figure 10. The radiation pattern of EMAT-Rayleigh waves for studying beam features.
As shown in Figure 10(b), r denotes the radial distance from the middle of the sensor, 𝜃
denotes the steering angle; define the steering angle parallel to the surface is 00; in order to
minimise the effects of the longitudinal waves and shear waves, keep the radial distance r 140
mm and the steering angle 𝜃 from 00 to 200. The beam directivity calculated is shown in
Figure 11, where the Rayleigh waves is mainly distributed along the steering angles from 0 0
to 2.50. Because the radial distance r used is 140 mm, the depth d of the Rayleigh waves’
distribution can be calculated by:
𝑑 = tan(𝜃) ∗ 𝑟……………………….……….….……. (8)
The depth d calculated is 6.1 mm, and the wavelength of the Rayleigh waves used in this work
is 6.066 mm, which validates that the Rayleigh waves are mainly distributed within one
wavelength of the Rayleigh waves. In addition, when choose another value of the radial
distance r, the steering angle 𝜃 changes correspondingly, but the depth d of the Rayleigh
waves’ distribution is a constant.
Another beam feature, field distribution along the steering angle, is studied as well. The radial
distance r used is 200 mm from the centre of EMAT arrays, and the steering angle used is 00.
The field distribution along the steering angle 00 is shown in Figure 11, the length period of
the maximum magnitudes in this figure is about 21 mm, which is consistent with the
modelling geometry, where the distance from the centre to the end of the EMAT sensor is 20
mm. As a result, the maximum magnitude occurs at the places where the sensor is placed, and
decreases along the radial distance due to the attenuation of Rayleigh waves. The attenuation
of Rayleigh waves is smaller than that of longitudinal waves and shear waves; based on this
nature, Rayleigh waves are mainly used for long distance detection.
beam directivity
field distribution along the steering angle
1
1
0.9
0.9
0.8
Normalised Magnitude
Normalised Magnitude
0.8
0.7
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0
0.7
0
2
4
6
8
10
12
14
angle (degree)
16
18
20
0.1
0
20
40
60
80
100
120
140
160
180
200
r (mm)
Figure 11. The beam directivity and field distribution along the steering angle 00 of
EMAT-Rayleigh waves.
3. Experiments validation
Experiments were carried out to validate simulations. The schematic diagram and the picture
of the system setup are shown in Figure 12; the high power tone burst pulser and receiver,
RITEC RPR4000, is used to excite and receive EMAT signals; the impedance matching box
is used to match the impedance between the power amplifier and the coil to maximize the
power transfer; the transmitter and receiver consist of the permanent magnet and the meander
coil as described in the part of EMAT simulation; oscilloscope is to display and record signals.
Figure 12. System setup for validation of EMAT simualtions.
The magnet and the coil used have the same dimension and the same shape as described in the
part of EMAT modelling. The distance between the transmitter and the receiver is 150 mm;
the directly transmit Rayleigh waves are shown in Figure 13, where the blue curve and the
red curve represent experimental and simulation results respectively. From experimental
results, the directly transmit Rayleigh waves arrived later than the main bang, which is the
overloading of the EMAT receiver by the electrical interference produced by the high power
transmitter excitation. The maximum amplitude of Rayleigh waves occurs at 50.3 us after
firing; based on the transmitting distance and the receiving time, the calculated velocity of
Rayleigh waves is 2.98 mm/us, and the velocity of Rayleigh waves used in this work for
simulation is 3.033 mm/us. The relative error is 1.68%, which is within the noise and error
tolerances of the experiments.
By multiplying a scaling factor, simulation results have the same maximum amplitude with
the experimental signals as shown in the red curve of Figure 13; where the simulation shows
good consistence with the experiment. However, at the end of Rayleigh waves (as shown in
the “Note Area” of Figure 13), there is a slight difference between the simulation and the
experiment. Possible reasons are: 1), in this work, the simulated model is a simplified model
with only surface sources; 2), the numerical nature of FEM and FDTD: numerical
approximation due to finite mesh density and element interpolation are inevitable.
Figure 13. The receiving signal from the simulation and the experiment.
Changing the distance between the transmitter and the receiver from 140 mm to 190 mm with
a step of 10 mm, the maximum amplitude of the directly transmit Rayleigh waves from the
simulation and the experiment is shown in Figure 14, which shows a good agreement. From
Figure 14, the induced voltage decreases with the distance between the transmitter and the
receiver increasing; that is due to the attenuation of Rayleigh waves.
Induced Voltage
Normalised Voltage
1
0.999
0.998
0.997
0.996
0.995
14
15
16
17
18
19
Distance between the transmitter and the receiver (cm)
Experiment
Simulation
Figure 14. The maximum amplitude of the induced voltage at various distances between
the transmitter and the receiver.
4. Conclusion
A method combining electromagnetic and ultrasonic simulations together to model an EMAT
array is proposed. For ultrasonic simulation, FDTD is used to model ultrasound waves’
propagation. A new radiation pattern with Hilbert transform is employed to analyse the beam
features, including beam directivity and field distribution along the steering angle. The results
show that the maximum magnitude of ultrasound waves occurs along the steering angle and at
a radical distance slightly smaller than the focal length. For electromagnetic simulation, the y
component of the Lorentz force density is calculated by FEM. Finally, the y component of
Lorentz force density is used as the excitation source in the FDTD ultrasonic simulation.
Rayleigh waves are generated propagating along the subsurface of the testing sample using
the EMAT array described. Experiments are carried out to validate the simulation method
proposed, which shows a good consistence between the simulation and the experiment.
Appendix
1. Ultrasonic array modelling
1.1 Phased array techniques
Figure 15. Steering and focusing behaviours based on phased array techniques.
For a multiple elements transducer, phased array means ultrasound waves can be controlled to
steer at an arbitrary angle or focus at a specific point with prescribed time delays [32, 33].
Phased array techniques offers attractive features like mechanical flexibility, enlarged swept
area, and optimum beam shape [8]. Steering and focusing are two phased array phenomena as
shown in Figure 15Error! Reference source not found..
Figure 16Error! Reference source not found.(a) and (b) describes the geometry of the steering
and focusing simulation, where the testing sample is a steel plate, the ultrasonic array with 8
elements is placed on the top of the steel plate, ɵ is the steering angle, and the focal point is
placed at the centre of the steel plate. It is a 2D simulation; the length and the width of each
element is 0.5 mm and 0.1 mm respectively, the spacing between two adjacent elements is 4
mm, x and y axis are shown in space step, where one space step equals to 0.1 mm. The
excitation source used is an explosive source which is same with the one used in [30], and the
frequency used is 2 MHz .
Figure 16. The geometry of steering and focusing simulation.
Figure 17Error! Reference source not found. shows the steering behaviour: the wavefront can
be steered to an arbitrary angle, from these diagrams, steering allows the beam to be swept
without moving the transducer, so it enlarges the sweep area. Due to this feature, steering has
advantages in circumstances where mechanical moving is limited. Figure 18Error! Reference
source not found. describes the focusing behaviour at different times: the wavefront becomes
narrower before arriving at the focal point and becomes wider after arriving at the focal point;
in other words, the wavefront is focused on the focal point. Because focusing allows the beam
shape to be controlled and the beam to be concentrated on the expected defect location, it
further optimizes the detection capability. In addition, perfectly-matched layer (PML) is
applied to all of the boundaries of the simulation geometry to model an unbounded domain, so
there are no waves reflected from the boundary.
Figure 17. Steering behaviour: by firing elements at different times, the wavefront is
steered at 0 degree, 30 degree, 60 degree, and 90 degree, respectively.
Figure 18. Focusing behaviour: the wavefront is concentrated on the focal point 19 us
after firing.
1.2 Radiation pattern and beam features
In order to further study steering and focusing, it is necessary to analyse the beam features.
The radiation pattern, which defines the field distribution within the testing sample, is used
for analysing beam features [32, 33]. Most of the previous works calculate the radiation
pattern based on analytical equations[32, 34]. In this paper, we define the radiation pattern by
combing the FDTD method and the Hilbert transform. Focusing technique is used to describe
the radiation pattern; as shown in Figure 16Error! Reference source not found. (b), the
geometry of focusing simulation contains 1000*1000 points, where each point on the
geometry has a time series signal. The maximum amplitude of the time series signal indicates
the arrival time of the signal, and the Hilbert transform is used to calculate the envelop of the
signal (Equation (9), (10), and (11)), resulting in identification of the signal arrival time more
clearly, as shown in Figure 19Error! Reference source not found.(a) and (b).
1
ℎ(𝑡) = 𝑓(𝑡) ∗ 𝜋𝑡……………………………….…… (9)
𝑧(𝑡) = 𝑓(𝑡) + 𝑗ℎ(𝑡)……..…………………….…… (10)
𝑒(𝑡) = √𝑓(𝑡)2 + ℎ(𝑡)2 ………….…...…….….…… (11)
where 𝑓(𝑡) is the time series signal, ℎ(𝑡) is the signal after Hilbert transform, 𝑧(𝑡) is the
analytical signal, and 𝑒(𝑡) is the envelop of 𝑧(𝑡).
the time series signal
the envelop of the time series signal
0.025
0.025
0.02
0.02
0.015
y (envelop)
y (amplitude)
0.01
0.005
0
0.015
0.01
-0.005
-0.01
0.005
-0.015
-0.02
0
500
1000
1500
2000
2500
0
3000
x (time step)
0
500
1000
1500
2000
2500
3000
x (time step)
(a)
(b)
Figure 19. (a) The time series signal, (b) The envelop of the signal.
In Figure 18Error! Reference source not found., the coordinate of the centre of the element
array is (x1, y1), which is (240, 1000), and the coordinate of the focal point is (x2, y2), that is
(500,500), one space step equals to 0.1mm, so the focal length 𝑑 and steering angle 𝜃 is
calculated by Equation (12) and (13); the calculated the focal length 𝑑 and steering angle 𝜃 is
56.4 mm and 27.50 respectively.
𝑑 = √(𝑥1 − 𝑥2 )2 +(𝑦1 − 𝑦2 )2………………….…… (12)
𝑥 −𝑥
𝜃 = tan−1 |𝑦1 −𝑦2 |……..…………………….…… (13)
1
2
Figure 20Error! Reference source not found.(a) shows the radiation pattern of the focusing
behaviour as described in Figure 18Error! Reference source not found.. There are two beam
features to be analysed, beam directivity and field distribution along the steering angle. Beam
directivity is, at a radial length from the centre of the array, the velocity or stress distribution,
as shown in the red circle in Figure 20Error! Reference source not found.(b). Field distribution
along the steering angle is the velocity or stress distribution along the steering angle, as shown
in the yellow line in Figure 20Error! Reference source not found.(b).
Figure 20. (a) Radiation pattern with a steering angle of 27.50 and a focal length of 56.4
mm, (b) Radiation pattern used for analysing beam features.
The results of beam directivity and field distribution along the steering angle are shown in
Figure 21Error! Reference source not found.. From Figure 21Error! Reference source not
found.(a), the radiation pattern shows a good directivity and has the maximum magnitude
along the prescribed steering angle, 27.50. In Figure 21Error! Reference source not found.(b),
the simulated maximum magnitude occurs at the focal length slightly smaller than the
prescribed focal length, 56.4 mm, due to the effect of diffraction [32]. The simulated focal
length is 49.9 mm, which is 11.52% smaller than the prescribed focal length.
beam directivity
field distribution along the steering angle
1
(a)
Beam Directivity
Simulated Steering Angle
Presicribed Steering Angle
0.9
1.1
(b)
Field Distribution
Simulated Focal Length
Presicribed Focal Length
1
normalised magnitude
normalised mangitude
0.8
0.7
0.6
0.5
0.4
0.3
0.9
0.8
0.7
0.6
0.2
0.5
0.1
0
0
10
20
30
40
50
angle (degree)
60
70
80
90
0.4
0
10
20
30
40
50
60
70
80
radial distance (mm)
Figure 21. Beam features: (a) beam directivity, (b) field distribution along the steering
angle.
1.3 scattering simulation
The geometry of the scattering simulation is the same as the one used in focusing simulation
as shown in Figure 16Error! Reference source not found.(b). By firing elements at the
prescribed times, the beam is controlled to focus at the scatter. The ultrasound wave
propagation is shown in Figure 22Error! Reference source not found., where the ultrasound
waves are scattered when they encounter the defect. The scattered ultrasound waves are
received by the receiving array, which is placed at the top of the steel plate and is symmetrical
with the transmitting array. Number the receiving elements from left to right 1st receiver, 2nd
receiver ... and 8th receiver. Receiving signals from the 1st receiver and the 8th receiver are
shown in Figure 23Error! Reference source not found.: the directly transmit signal arrives first
and is followed by the scattered signal. The distance between the transmitter and the 1st
receiver is smaller than that between the transmitter and the 8th receiver, due to the
attenuation of ultrasound waves, the amplitude of the directly transmit ultrasound waves of
the 1st receiver is slightly larger than that of the 8th receiver. Perfectly-matched layer (PML)
is applied to all of the boundaries of the simulation geometry to absorb ultrasound waves, so
no waves are reflected from the boundary.
Figure 22. Wave propagation at different times: ultrasound waves are scattered by the
scatter at 20us.
the 1st receiver
-3
6
x 10
the 8th receiver
-3
4
x 10
3
4
velocity (mm/us)
velocity (mm/us)
2
2
0
-2
1
0
-1
-4
-6
-2
0
500
1000
1500
time step
2000
2500
3000
-3
0
500
1000
1500
2000
2500
3000
time step
Figure 23. Received signal from the 1st receiver and the 8th receiver respectively.
References
1.
2.
3.
4.
5.
6.
7.
Yin, W. and A. Peyton, Thickness measurement of non-magnetic plates using multi-frequency
eddy current sensors. NDT & E International, 2007. 40(1): p. 43-48.
Yin, W. and A.J. Peyton, Thickness measurement of metallic plates with an electromagnetic
sensor using phase signature analysis. Instrumentation and Measurement, IEEE Transactions
on, 2008. 57(8): p. 1803-1807.
Lovejoy, D., Magnetic Particle Inspection: A Practical Guide. David Lovejoy. 1993: Springer.
Park, M.H., I.S. Kim, and Y.K. Yoon, Ultrasonic inspection of long steel pipes using Lamb waves.
NDT & E International, 1996. 29(1): p. 13-20.
Cawley, P. Ultrasonic measurements for the quantitative NDE of adhesive joints-potential and
challenges. in Ultrasonics Symposium, 1992. Proceedings., IEEE 1992. 1992. IEEE.
Chassignole, B., et al. Characterization of austenitic stainless steel welds for ultrasonic NDT. in
REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION: Volume 19. 2000.
AIP Publishing.
Drinkwater, B. and P. Cawley, Measurement of the frequency dependence of the ultrasonic
reflection coefficient from thin interface layers and partially contacting interfaces. Ultrasonics,
1997. 35(7): p. 479-488.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
Drinkwater, B.W. and P.D. Wilcox, Ultrasonic arrays for non-destructive evaluation: A review.
NDT & E International, 2006. 39(7): p. 525-541.
Gallego-Juarez, J., Piezoelectric ceramics and ultrasonic transducers. Journal of Physics E:
Scientific Instruments, 1989. 22(10): p. 804.
Jaffe, H. and D. Berlincourt, Piezoelectric transducer materials. Proceedings of the IEEE, 1965.
53(10): p. 1372-1386.
Jaffe, B., Piezoelectric ceramics. Vol. 3. 2012: Elsevier.
Ribichini, R., Modelling of Electromagnetic Acoustic Transducer, in Department of Mechanical
Engineering2011, Imperial Colledge London.
Dixon, S., C. Edwards, and S. Palmer, High accuracy non-contact ultrasonic thickness gauging
of aluminium sheet using electromagnetic acoustic transducers. Ultrasonics, 2001. 39(6): p.
445-453.
Dhayalan, R. and K. Balasubramaniam, A hybrid finite element model for simulation of
electromagnetic acoustic transducer (EMAT) based plate waves. NDT & E International, 2010.
43(6): p. 519-526.
Wang, S., et al., 3-D modeling and analysis of meander-line-coil surface wave EMATs.
Mechatronics, 2012. 22(6): p. 653-660.
Hirao, M. and H. Ogi, EMATs for science and industry: noncontacting ultrasonic
measurements. 2003: Springer Science & Business Media.
Luo, W. and J. Rose, Guided wave thickness measurement with EMATs. Insight-NonDestructive Testing and Condition Monitoring, 2003. 45(11): p. 735-739.
Dhayalan, R. and K. Balasubramaniam, A two-stage finite element model of a meander coil
electromagnetic acoustic transducer transmitter. Nondestructive Testing and Evaluation,
2011. 26(02): p. 101-118.
Scruby, C. and B. Moss, Non-contact ultrasonic measurements on steel at elevated
temperatures. NDT & E International, 1993. 26(4): p. 177-188.
Edwards, R., S. Dixon, and X. Jian. Non‐Contact Ultrasonic Characterization of Defects Using
EMATs. in REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION. 2005.
American Institute of Physics.
Latimer, P.J. and D.T. MacLauchlan, EMAT probe and technique for weld inspection, 1998,
Google Patents.
Ludwig, R., Z. You, and R. Palanisamy, Numerical simulations of an electromagnetic acoustic
transducer-receiver system for NDT applications. Magnetics, IEEE Transactions on, 1993.
29(3): p. 2081-2089.
Thomas, S., et al., A coupled electromagnetic and mechanical analysis of electromagnetic
acoustic transducers. International Journal for Computational Methods in Engineering
Science and Mechanics, 2009. 10(2): p. 124-133.
Kaltenbacher, M., et al., Finite element analysis of coupled electromagnetic acoustic systems.
Magnetics, IEEE Transactions on, 1999. 35(3): p. 1610-1613.
Murayama, R. and K. Mizutani, Conventional electromagnetic acoustic transducer
development for optimum Lamb wave modes. Ultrasonics, 2002. 40(1): p. 491-495.
Jian, X., et al., A model for pulsed Rayleigh wave and optimal EMAT design. Sensors and
Actuators A: Physical, 2006. 128(2): p. 296-304.
Jafari-Shapoorabadi, R., A. Konrad, and A. Sinclair, Improved finite element method for EMAT
analysis and design. Magnetics, IEEE Transactions on, 2001. 37(4): p. 2821-2823.
Jian, X., et al., Electromagnetic acoustic transducers for in-and out-of plane ultrasonic wave
detection. Sensors and Actuators A: Physical, 2008. 148(1): p. 51-56.
Bossy, E., SimSonic Suite User’s guide for SimSonic3D. 2012.
Virieux, J., P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference
method. Geophysics, 1986. 51(4): p. 889-901.
Griffiths, D.F. and A.R. Mitchell, The finite difference method in partial differential equations.
1980: John Wiley.
32.
33.
34.
Azar, L., Y. Shi, and S.-C. Wooh, Beam focusing behavior of linear phased arrays. NDT & E
International, 2000. 33(3): p. 189-198.
Wooh, S.-C. and Y. Shi, Optimum beam steering of linear phased arrays. Wave motion, 1999.
29(3): p. 245-265.
Wooh, S.-C. and Y. Shi, A simulation study of the beam steering characteristics for linear
phased arrays. Journal of nondestructive evaluation, 1999. 18(2): p. 39-57.