Composites: Part B 54 (2013) 1–10
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Composites: Part B
journal homepage: www.elsevier.com/locate/compositesb
Non-reflecting boundary condition for Lamb wave propagation problems
in honeycomb and CFRP plates using dashpot elements
Seyed Mohammad Hossein Hosseini ⇑, Sascha Duczek, Ulrich Gabbert
Institute of Numerical Mechanics, Department of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39016 Magdeburg, Germany
a r t i c l e
i n f o
Article history:
Received 25 January 2013
Received in revised form 8 April 2013
Accepted 15 April 2013
Available online 29 April 2013
Keywords:
Non-reflecting boundary condition
D. Ultrasonics
A. Honeycomb
A. Carbon fibre
C. Finite element analysis (FEA)
a b s t r a c t
The paper’s objective is to introduce a new non-reflecting boundary condition using dashpot elements.
This is an useful tool to efficiently simulate Lamb wave propagation within composite structures, such
as honeycomb and CFRP plates. Due to the steadily increasing interest in applying Lamb waves in modern
online structural health monitoring techniques, several numerical and experimental studies have been
carried out recently. The proposed boundary condition poses the advantage of reducing the computational costs required to simulate the wave propagation in heterogenous materials. Different parameters
which can influence the functionality of such an artificial boundary are discussed and several applications
are presented. Finally, the results are also experimentally validated.
Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Structural health monitoring (SHM) in composite structures
using guided Lamb waves is a new technology in modern industries such as aviation and transportation. Piezoelectric (PZT) actuators and sensors are used to excite and to receive waves within
complicated composite structures [1,2]. This approach for SHM
applications is an interesting technique because of the low costs
of the required equipment, the possibility of an online monitoring
and the high sensitivity to detect small structural damages [1].
1.1. Lamb wave propagation in composite plates
Lamb wave propagation in composite plates has been studied in
several Refs. [1,3–12]. guided waves were used to detect sub-interface damages in foam core sandwich structures in [3]. A suitable
frequency was found that offers the highest sensitivity to detect
skin/foam core delaminations and to facilitate the interpretation
of the measured waveforms. Finally, delaminations were located
and characterized using an adapted signal processing. The numerical simulations were validated experimentally. Wave propagation
in light-weight plates with truss-like cores was investigated in [4].
It has been shown, that the vibrational behavior can be reduced to
equivalent plate models in the low frequency region where global
plate waves are dominant. An application example of a train floor
⇑ Corresponding author. Tel.: +49 3916711723.
E-mail addresses: [email protected] (S.M.H. Hosseini), sascha.duczek@st.
ovgu.de (S. Duczek), [email protected] (U. Gabbert).
1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compositesb.2013.04.061
section was tested to validate the theoretical dispersion characteristics. The Lamb wave propagation in particle reinforced composite
plates was studied in [5]. It has been reported that the volume fraction and the stiffness to density ratio of the particles are the main
parameters to affect the Lamb wave propagation properties in such
materials. In addition, a homogenization method was used to simplify the models. A reasonable agreement between the complex
model, incorporating many details of the real structure and the
simplified model in terms of the group velocity and the wavelength
has been observed, while tremendous savings in computational
costs were archived. Localized phase velocities in the frequency
range of 5–50 kHz were measured in honeycomb plates in [6]. It
has been reported that the proposed method is suitable to detect
delamination between the cover plate and the core in honeycomb
sandwich panels. In another study, a homogenization technique
was formulated for the analysis of vibration and the wave propagation problems in a honeycomb-like slender skeleton [7]. In addition, the effect of the cell size on the overall dynamic behavior of
a composite solid was characterized. The effect of the geometry
of the unit cells on the dynamics of the propagation of elastic
waves within the structure was studied in [8] using a two-dimensional finite element model and the theory of periodic structures.
The desired transmissibility levels in specified directions were
investigated for an optimal design configurations to obtain efficient vibration isolation capabilities. Debonding in sandwich CF/
EP composite structures with a honeycomb core was detected
using the anti-symmetric (A0) Lamb mode in [9] and the finite element modeling approach was validated with experimental results.
In a similar study theoretical, numerical and experimental
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S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
approaches were used to detect the delamination between the skin
and the honeycomb of a composite helicopter rotor blade in [10].
In addition, the Lamb wave propagation in honeycomb sandwich
panels was studied using a three-dimensional finite element approach in [1,11]. The results were compared with a simplified
model, where the core layer was replaced by a heterogeneous
material with a simple cubic geometry. The received results were
validated by experimental results in [1]. Continuous mode conversation of the Lamb wave propagation in CFRP composites was studied in [12] using numerical and experimental approaches.
These studies show the steadily growing research activities in
the field of structural health monitoring and highlight the need
for computationally efficient numerical tools to gain a deeper
understanding of the underlying physics of Lamb wave propagation in heterogenous materials.
boundary condition can be applied using all commercial finite element packages. In the following sections numerical and experimental results are presented. Different parameters are considered
to design an efficient non-reflecting boundary condition designed
by dashpot elements. Afterward, the capability of the proposed
method is compared with the approach of gradually damped artificial boundaries. Thereafter, the ability of dashpot non-reflecting
boundary condition is demonstrated to reduce the computational
costs. To this end, the model size is reduced in such a way that only
the minimum required solid medium between actuator and the
sensor is considered for the Lamb wave propagation analysis in a
structure. Finally, the results are validated numerically as well as
experimentally.
1.2. Non-reflecting boundary condition
A widely accepted approach is to use PZT patches to generate
the Lamb waves within a structure, cf. Fig. 1. As time dependent
excitation signal (Vin) a three and half-cycle narrow banded tone
burst [1] is applied as given in the following equation:
The application of a non-reflecting boundary condition was
studied in several references in order to reduce the computational
efforts [13–18]. A non-reflecting boundary condition was described
for the scalar wave equation in [13]. It has been mentioned that
this method is only applicable for wave propagation when uniform
medium at the boundary (any inhomogeneities should be avoided)
exist. In addition, the proposed method offers few choices for the
shape of an artificial non-reflecting boundary. Infinite elements
in Abaqus were used to design a non-reflecting boundary in [14].
However, it has been shown in [15] that this method is not satisfactory. In [15] a finite element approach for the analysis of the
wave propagation in an infinitely long plate was presented. To
avoid any spurious reflections generated by the finite boundary
of the finite element model a non-reflecting boundary condition
with a gradually damped artificial boundary was designed. The
length of the damping section was considered to be long enough
for gradual changes of the damping factor to avoid any spurious
reflection from any sudden damping. The proposed method was
implemented using the available finite element packages. In addition, the results in a plate with a horizontal crack were compared
with the strip element method and a good agreement has been reported. A similar design of non-reflecting boundary was introduced
in [16] using frequency domain analysis and absorbing regions. In
this paper the longest wavelength was suggested as a measure for
the length of the absorption region. It has been indicated in [15]
that the proposed non-reflecting boundary condition in [15,16]
can be computationally expensive depending on the model being
analyzed. Furthermore, the perfectly matched layer (PML) is introduced as a flexible and accurate method to simulate the wave
propagation in unbounded structures [17]. However, it is mentioned that this approach can be computational expensive [18].
Many additional unknowns insert in the standard PML formulations because the required wave equations stated in their standard
second-order form to be reformulated as first-order systems.
Additionally, it has been mentioned in [18] that local absorbing
boundary conditions among the available non-reflecting boundary
conditions are known as the simplest and most flexible approach
with a reasonable computational cost. They do not require any special functions and are capable to be coupled with standard finite
difference or finite element methods. Therefore, the aim of the
present paper is to introduce a novel local non-reflecting boundary
condition using dashpot elements which can reduce the computational efforts tremendously. The finite element method is known to
be a versatile and efficient numerical tool for a vast variety of engineering problems. Thus we decided to employ FEM to model the
guided wave propagation in heterogenous medium instead of utilizing other numerical approaches such as finite differences or the
boundary element method [19,20]. The proposed non-reflecting
2. Finite element modeling
2pfc t
sin 2pfc t:
V in ¼ V½HðtÞ Hðt 3:5=fc Þ 1 cos
3:5
ð1Þ
There t is the time, fc is central frequency and H(t) is the Heaviside
step function. A zero voltage is applied to the bottom surfaces of the
sensors and the actuator. Symmetric boundary conditions are applied to the inner borders of the plate to reduce the model size
and the computational costs, cf. Fig. 1. The piezoelectric sensor is attached parallel to the boundaries and located at a distance of
180 mm from the actuator.
Dashpot elements are used to damp the wave reflections from
borders of a structure, cf. Fig. 1. The viscous behavior of dashpots
in which the damping force (F) is proportional to the velocity, provides the ability of energy dissipation during cyclic loading [21].
F ¼ Cðu_2 u_1 Þ;
ð2Þ
F represents the force generated by the dashpot, C is the damping
factor, u_ 1 and u_ 2 are the velocity of two ends of the dashpot element
(in our case u_ 1 ¼ 0). To apply the new non-reflecting boundary condition, dashpot elements are connected only to one row and column
of the outer elements as a primary choice, cf. Fig. 1.
In this paper, each numerical model includes an ‘‘inner part’’
(shown by dashed lines in Fig. 1) which represents and captures
the main features of the micro- and macrostructure of the propagating medium. The sensor is considered to be attached to the
plate in this region. The rest of the structure (apart from the inner
part) is considered as outer borders of the plate. The definition of
the ‘‘frame’’ is used to indicate number of outer rows and columns
which are used in the border of a numerical model. A sketch is
shown in Fig. 1, each frame of elements includes a set of rows
and columns of elements from the outer borders of the plate. However, each frame may consist of several number of rows and columns (in this study number of rows and columns in each frame
is considered to be equal to the number of rows in the inner part).
The frame definition is also used to describe the model size and the
gradually increasing damping boundary condition. As an example
Fig. 1 shows a model with three columns and rows as frame size.
One can see the outer element frames (shown by gradually changing colors) in a numerical model in Fig. 2. It has to be mentioned
that each frame consists of all the nodes and elements across the
structure. For instance, in order to damp the waves in a honeycomb
sandwich plate, dashpot elements which are applied on each frame
are connected to all the nodes over the entire thickness of the
sandwich structure including the nodes on the cover plates and
the nodes in the honeycomb core.
S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
3
Fig. 1. A schematic representing of dashpot elements connected to a three frame-size plate. For the sake of visualization, the number of elements shown in the sketch of the
model has been reduced. Each element in the figure represents four elements in x and y directions for the original FE-model.
spring-damper option is provided by COMBIN14 elements. These
uniaxial tension–compression elements are available with up to
three degrees of freedom (i.e. x, y and z directions) at each node.
Symmetric boundary condition
3. Methodology
Sensor top
FEMAPMaterial2:asd
material7
FEMAPMaterial2:asd_1
FEMAPMaterial2:asd_2
FEMAPMaterial2:asd_3
FEMAPMaterial2:asd_4
FEMAPMaterial2:asd_5
FEMAPMaterial2:asd_6
FEMAPMaterial2:asd_7
FEMAPMaterial2:asd_8
FEMAPMaterial2:asd_9
FEMAPMaterial2:asd_10
FEMAPMaterial2:asd_11
FEMAPMaterial2:asd_12
FEMAPMaterial2:asd_13
FEMAPMaterial2:asd_14
FEMAPMaterial2:asd_15
FEMAPMaterial2:asd_16
FEMAPMaterial2:asd_17
FEMAPMaterial2:asd_18
FEMAPMaterial2:asd_19
FEMAPMaterial2:asd_20
FEMAPMaterial2:asd_21
FEMAPMaterial2:asd_22
FEMAPMaterial2:asd_23
FEMAPMaterial2:asd_24
FEMAPMaterial2:asd_25
FEMAPMaterial2:asd_26
FEMAPMaterial2:asd_27
Sensor bottom
FEMAPMaterial2:asd
material7
FEMAPMaterial2:asd_1
FEMAPMaterial2:asd_2
FEMAPMaterial2:asd_3
FEMAPMaterial2:asd_4
FEMAPMaterial2:asd_5
FEMAPMaterial2:asd_6
FEMAPMaterial2:asd_7
FEMAPMaterial2:asd_8
FEMAPMaterial2:asd_9
FEMAPMaterial2:asd_10
FEMAPMaterial2:asd_11
FEMAPMaterial2:asd_12
FEMAPMaterial2:asd_13
FEMAPMaterial2:asd_14
Z
Actuator
X
Y
Fig. 2. Schematic representation of the outer element frames in numerical model.
The PZT elements’ orientation and the symmetric boundary conditions are also
shown. The PZT actuator and sensors are modeled by SOLID5, coupled field
elements with displacement and voltage degree of freedoms in ANSYSÒ 11.0.
The Lamb waves propagate along the medium with different
wave forms, which are known as modes. Each mode can be either
a symmetrical (S) mode or an anti-symmetrical (A) mode. By subtracting (or adding) the signals on the top and bottom surfaces of a
plate one can identify the different modes propagating inside the
structure. However, this method is not suitable for thick sandwich
panels, where the arrival of the modes on the top and the bottom
surfaces differs. In this paper different modes and their reflections
are identified based on differences in the group velocity, the wavelength and the amplitude [11]. In addition, to verify the mode splitting, using B-scan images is also an alternative method to identify
different modes and reflections [22,11], cf. Fig. 3. The displacements of the nodes (located along the wave propagation direction)
in the time domain are shown in B-scan.
The energy transmission caused by the reflected and propagated waves is measured to show the functionality of the proposed
non-reflecting boundary condition. The transmitted energy is defined within this paper as the integral over the squared signal [23].
Etrans ¼
Location (mm)
0.1
Symmetric mode
0
Reflections
-0.1
Anti-symmetric mode
-0.15
-0.2
0
0.05
tend
V 2 ðtÞ dt:
ð3Þ
t start
0.05
-0.05
Z
0.1
0.15
0.2
Time (ms)
Fig. 3. The propagation of the Lamb modes and reflection waves are shown in a Bscan diagram. The Lamb wave propagation is considered in a honeycomb sandwich
plate. The geometrical properties of the plate are presented in Table 2, the central
frequency of the loading signal is 250 kHz.
To execute these simulations the commercial FEM software
ANSYSÒ 11.0 has been used. In this software, the longitudinal
To show the efficiency of the non-reflecting boundary, the ratio between the transmitted energy of the reflected waves and the Lamb
waves (Ereflected/ELamb) is calculated. As an example, one can see the
Lamb waves and the reflection in Fig. 5 which are separated with a
dashed line. The Ereflected and ELamb stand for the energy transmission caused by reflections and the Lamb waves (including both S0
and A0 modes), respectively. In this study an equal period of time
which is needed for propagation of the Lamb modes (which depends on the central frequency of the excitation wave) is considered
as a minimum period to measure the reflected waves.
The value of less than 1 for this ratio, shows an attenuation in
reflected waves and therefore one can distinguish the reflected
waves from the propagated Lamb modes. The value of bigger than
1 indicates that non-reflecting boundary is too weak and the reflected waves are sensed several times with sensor. As the ratio
tends to zero a better non-reflection boundary condition is indicated. In this study, the ratio of 0.009 and less is considered as
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S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
an ideal non-reflecting boundary condition. The post-processing
calculations described in this paper have been performed using
MATLABÒ.
4. Design parameters
Following parameters are considered to design an efficient
boundary condition using dashpot elements to avoid reflection
from borders.
Damping factor.
Direction of dashpot elements.
Number of dashpot elements.
4.1. Damping factor
Eq. (2) indicates that the magnitude of the generated force with
the dashpot elements is related to the damping factor and the nodal displacement. Therefore, the damping factor multiplied by the
group velocity (C Vg) is used to show the capabilities of the dashpot elements as non-reflecting boundary. Fig. 4 shows the energy
transmission values of the reflected waves over C Vg. Finding an
efficient value of C Vg and knowing the group velocity of the Lamb
waves (which depends on the central frequency of the excitation
signal, the material properties of the propagating medium and
the thickness of the plate) one can indicate the efficient damping
factor. Fig. 4 demonstrates that very small values of damping factor
do not generate enough force to avoid propagation of the reflected
waves effectively. On the other hand, the nodal displacements are
not big enough to move dashpot elements with very high damping
factors. A range of 28,000–280,000 of C Vg is found for the minimum value of the ratio between the transmitted energy of the reflected waves and the Lamb waves (Ereflected/ELamb). In this example
an aluminum plate with the thickness of 2 mm is considered and
the Lamb wave with central frequency of 200 kHz is generated.
The lowest group velocity belongs to the A0 mode and is approximately 2800 m/s [11], subsequently a damping factor between
10 and 100 (N s/m) is suggested.
4.2. Direction of dashpot elements
Different direction of dashpot elements according to Eq. (2) are
considered. It is observed that the dashpot elements in direction of
wave propagation have the best effect to reduce the reflection
waves. In this case one side of the dashpots are connected to the
1.2
fc = 200 kHz
ELamb = 7.44e-8 J.Ω
nodes on the plate and the other sides are fully fixed (with C Vg
equal to 2.8e5 N). In a particular example, an aluminum plate of
0.5 mm thickness is considered as an initial model, where the
nodes on the outer borders are connected to the dashpot elements
in direction of wave propagation (x). The Lamb wave is excited
with a central frequency of 150 kHz, cf. Fig. 1. In this case the transmitted energy by the reflected waves is 1.13 107 J X and the ratio between the transmitted energy of the reflected waves and the
Lamb waves (Ereflected/ELamb) is equal to 0.7. In the first case the
direction of dashpot elements are changed and set to the y direction. Subsequently, the energy transmission of the reflected waves
increase to 166% with the initial model (Ereflected/ELamb = 2.0). In the
second case the dashpot elements are to be considered to the
direction of z. It is observed that the energy transmission of the reflected waves increase by 240% in comparison to the initial model
(Ereflected/ELamb = 2.6). Fig. 5 compares the reflected waves in models with dashpot elements in x and z directions.
Furthermore, the combination of dashpot elements in all three
directions (x, y and z) is considered and only 3% reduction of the
energy transmission of the reflected waves (Ereflected/ELamb = 0.68)
in comparison to the initial model is observed. These results can
be explained by the fact that in a reduced-size model (cf. Fig. 1)
there are more nodes which are connected to the dashpot elements
in the direction of wave propagation (x) in comparison to the other
directions of y and z.
4.3. Number of dashpot elements
The influence of number of dashpot elements on the reflecting
wave is also considered. In a particular example an aluminum plate
with the thickness of 2 mm is considered and the Lamb wave with
central frequency of 200 kHz is excited. Initially 617 dashpot elements (with C Vg equal to 2.8e5 N and the dashpots are in x direction) are used to be connected to the first outer element row and
column from the outer borders of the plate (one frame of outer elements, cf. Fig. 1). In this case the transmitted energy by the reflected waves is 2.05 109 J X, where the energy of the Lamb
waves is 70.6 109 J X (Ereflected/ELamb = 0.03).
To show the influence of dashpot element number on the reflected waves, it is increased in two steps and the results are compared with the initial model. In the first case the number of
dashpot elements is increased to 1234 to be connected to two outer rows and columns (two frames of outer elements). In the second
case the dashpot elements are further increased to 2468 elements,
which are connected to four outer rows and columns (four frames
aluminum plate
thickness: 2 mm
0.15
0.10
Model size: 4 frames
Dashpots: 1 frame
Dashpots: 1 frame
0.6
Model size: 4 frames
0.15
0.05
0.00
0.10
-0.05
0.05
-0.10
0.00
0·105
1·10
5
2·10
5
3·10
5
5
4·10
5
5·10
5
6·10
Damping factor times the group velocity, C·Vg (N)
aluminum plate
thickness: 0.5 mm
A0
0.8
Voltage (V)
Ereflected / E Lamb (-)
1.0
S0
Lamb waves Reflections
Dashpot in x direction
Dashpot in z direction
-0.15
0.5·10-4
1.0·10-4
fc = 150 kHz
C · Vg = 2.8e5 N
1.5·10-4
2.0·10-4
Time (s)
Fig. 4. The ratio between the energy transmission of the reflected waves and the
Lamb waves for different damping factors. A model of an aluminum plate with
thickness of 2 mm is considered, and the Lamb wave is excited with a central
frequency of 200 kHz.
Fig. 5. Reflected waves in models with different dashpot element directions. A
model of an aluminum plate with thickness of 0.5 mm is considered, and the Lamb
wave is excited with a central frequency of 150 kHz.
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S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
6
Damping
Fig. 6. Increasing number of dashpot elements is shown schematically in a sixframe size model.
Ereflected / ELamb (-)
5
Dashpot
4
Top surface honeycomb
C · Vg = 2.8e5 N
3
Dashpots: 1 frame
2
Model size: 4 frames
1
0.15
0
Model size: 6 frames
Voltage (V)
0.10
aluminum plate
thickness: 2 mm
0.00
A0
617 dashpots (1 frame)
2468 dashpots (4 frames )
-0.15
0.5·10-4
1.0·10-4
200
250
300
350
400
Frequency (kHz)
Fig. 8. The values of energy transmission of the reflected waves are plotted over the
central frequency of the excitation signal. Two different kinds of non-reflecting
boundary conditions are considered. First a non-reflecting boundary condition
based on gradually damped artificial boundary with four frames of damping
materials on the borders is considered, labeled damping. Secondly a model with
dashpot elements on the borders is taken into account, labeled dashpot. A
honeycomb sandwich panel is considered, cf. Table 2 for the geometrical properties
and cf. Table 3 for the materials properties.
S0
-0.10
150
Lamb waves Reflections
0.05
-0.05
100
fc = 200 kHz
C · Vg = 2.8e5 N
1.5·10-4
2.0·10-4
Time (s)
Fig. 7. Reflected waves in models with different number of dashpot elements. A
model of an aluminum plate with thickness of 2 mm is considered, and the Lamb
wave is excited with central frequency of 200 kHz.
of outer elements). Fig. 6 represents increasing number of dashpot
elements schematically.
The ratio between the transmitted energy of the reflected waves
and the Lamb waves (Ereflected/ELamb) decreases in the first and the
second cases to 0.012 and 0.011, respectively. For all three cases
the obtained ratio between the transmitted energy of the reflected
waves and the Lamb waves is in an acceptable range and fairly far
from 1, cf. Section 3. Fig. 7 compares the reflected waves in a model
with 617 dashpot elements and a model with 2468 dashpot
elements.
The results in Fig. 7 and the Ereflected/ELamb values indicates that
the additional number of dashpot elements does not improve the
efficiency of the proposed boundary condition significantly. In
addition, considering the fact that the most of added dashpot elements (in the first and second cases) are connected to the nodes
along the outer rows one can conclude that the most of reflected
waves which are sensed with the sensor are reflected from the outer columns (shown on the right hand-side of the actuator in Fig. 1).
5. Applications
5.1. Influence of different central frequencies
The influence of changes in the central frequency of the excitation signal on the wave reflection in a honeycomb sandwich plate
using a non-reflecting boundary condition with dashpot elements
(labeled dashpot) is compared with the gradually damped artificial
boundary which is introduced in [15] (labeled damping). These
two approaches can be realized in commercial finite element software without difficulties. The geometrical properties and the
materials properties are given in Tables 2 and 3, respectively.
Fig. 8 shows that, as the central frequency increases less energy
is transmitted by the reflected waves from the non-reflecting
boundary condition with gradually damped artificial boundary. A
similar trend can be observed for the non-reflecting boundary with
dashpot elements. This phenomenon can be explained by the fact
that as the central frequency of the excitation Lamb wave increases, the energy which is transmitted by the Lamb waves, decreases. However, it is clear that the non-reflecting boundary
with dashpot elements is less sensitive to the changes in central
frequency of the exciting signal and it works enough good even
in lower frequency ranges. Table 1 shows the absolute values of
transmitted energy by the Lamb waves with different central
frequencies.
5.2. Reduced-size model
The major benefit to use dashpot elements is the possibility to
reduce the model size. This reduction in the model size results in
decreased computational costs. Within this study a non-reflecting
boundary condition which results in an attenuation of the reflected
waves is considered to be acceptable (Ereflected/ELamb < 1). This
attenuation behavior will help to distinguish the reflected waves
from the propagated modes in signal processing for structural
heath monitoring applications [11]. Fig. 9 shows how the model
reduction can effect the energy transmission of the reflected waves
from the borders. It is clear that the attenuation of the reflected
waves completely depends on the model size. This can be explained by the fact, that the damping factor of such a non-reflecting boundary is increasing gradually. Consequently, a bigger model
has a greater attenuation effect on the reflected waves.
However, the models with dashpots show less dependency on
the model size (in this example only one frame of outer elements
are connected to dashpot elements). This phenomena can also be
explained by the fact that number of dashpot elements does not
change the efficiency of the proposed boundary condition significantly, cf. Section 4.3 (by decreasing the model size, the number
of dashpot elements is also decrease).
5.3. Comparison with an ‘‘ideal’’ non-reflecting boundary
In order to verify the proposed non-reflecting boundary and
model reduction, Fig. 10 compares the Lamb wave propagation in
a model using an ‘‘ideal’’ non-reflecting boundary condition with
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S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
Table 1
Absolute ELamb values for different central excitation frequencies.
Central
excitation
frequency
(kHz)
100
ELamb (J X)
(106)
0.060
Ereflected / ELamb (-)
2.5
150
0.046
200
250
0.036
0.014
300
350
0.0048
400
0.0017
Dashpots: 1 frame
Table 2
The geometrical properties of the honey comb sandwich panel and the PZT actuator/
sensor (units are in mm).
0.00057
Skin plate (28 frames)
Honeycomb cell
PZT actuator and sensor
Length
Width
Thickness
Height
Thickness
Core size
Radius
Thickness
actuator/sensor distance
290
124
2
15
0.5
4.8
Damping
2.0
Dashpot
1.5
Aluminum plate
thickness: 2 mm
Honeycomb height
fc = 200 kHz
C ·Vg = 2.8e5 N
ELamb = 7.44e-8 J.W
1.0
Honeycomb core size
Z
0.5
Honeycomb thickness
0.0
0
4
3.17
0.7
180
8
12
16
20
X
Y
Fig. 11. Schematic representation geometrical properties of a honeycomb sandwich
panel, cf. Table 2.
Number of frames (model size)
Fig. 9. The values of energy transmission of reflected waves plotted over the model
size, cf. Fig. 1. Different non-reflecting boundaries including (a) damping materials
and (b) dashpot elements are compared. An aluminum plate with thickness of
2 mm is considered, and the Lamb wave are excited with a central frequency of
200 kHz.
0.100
0.075
Young’s
modulus (GPa)
Density (kg m3)
Poisson’s ratio (–)
Skin plate (Aluminum alloy T6061)
70
0.33
fc = 40 kHz
Top surface
C · Vg = 2.8e5 N
0.050
Voltage (V)
Table 3
Material properties of the plate and honeycomb cells [1].
0.025
2700
Honeycomb cell (HRH-36-1/8-3.0)
Ex = Ey
txy &tyz = txz (–)
Ez
(GPa)
(GPa)
Gxy
(GPa)
Gyz= Gxz
(GPa)
Density
(kg m3)
2.46
0.94615
1.154
50
3.4
0.3
0.000
-0.025
-0.050
-0.075
aluminum plate
thickness: 2 mm
-0.100
0·10-4
1·10-4
Table 4
Material properties of the twill CFRP (0°/90°) plate [12].
Ideal non-reflecting boundary
Dashpot non-reflecting boundary
2·10-4
3·10-4
4·10-4
Time (s)
Fig. 10. Comparison of the Lamb wave propagation in a model using an ‘‘ideal’’ nonreflecting boundary condition with twenty damping frames of gradually damped
artificial boundary (dashed line) and a reduced-size model with four frames and
one frame of dashpot elements (solid line). The excited Lamb wave is propagating
with a central frequency of 40 kHz in an aluminum plate of 2 mm thickness.
twenty damping frames of gradually damped artificial boundary
(Ereflected/ELamb is equal to 0.0027) and a reduced-size model (with
four frames) with one frame of dashpot elements (Ereflected/ELamb is
equal to 0.17). The time-of-flight is the same for both signals. But
the A0 mode in the model with dashpot elements has relatively
higher amplitude which can be explained by the amplification effect of reflections from the borders in a reduced-size model.
Ex
Ey
Ez
127.5 (GPa)
7.9 (GPa)
7.9 (GPa)
txy
tyz
txz
0.273 (–)
0.348 (–)
0.017 (–)
Gxy
Gyz
Gxz
5.58 (GPa)
2.93 (GPa)
2.93 (GPa)
an aluminum plate (thickness of 2 mm) from borders of the combined boundary increases by 23% in comparison to the model in
which only dashpot elements are used (where the transmitted energy by the reflected waves is 2.05 109 J X). The ratio between
the transmitted energy of the reflected waves and the Lamb waves
(Ereflected/ELamb) for the models with dashpot elements and combined boundary are 0.02 and 0.03, respectively. This increase can
be explained due to the fact that the material damping results in
a reduction in the nodal velocity and subsequently the dashpot
force decreases.
5.5. Wave propagation in composite structures
5.4. Combination of damping materials and dashpot elements
Furthermore, a combination of gradually damped artificial
boundary [15] and a dashpot boundary is considered, labeled combined boundary. The energy transmission by the reflected waves in
In this section two examples of application of the non-reflecting
boundary condition on the wave propagation in a honeycomb and
a CFRP plates are presented. Finally, the experimental results are
presented.
7
S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
0.15
Top surface honeycomb
fc = 200 kHz
C · Vg = 2.8e5 N
0.10
Maximum
Dashpots: 1 frame
Voltage (V)
Model size: 4 frames
0.05
0.00
S0
-0.05
3D laser sanning
vibrometer
A0
Damping
Lamb waves Reflections
-0.10
Dashpot
-0.15
0·10-4
0.5·10-4
1·10-4
1.5·10-4
Silicon for damping
Retro
-refle
2·10-4
Time (s)
ctive
Actuator Position
Fig. 12. Reflected waves are compared for a honeycomb reduced-size model with
four frames of damping materials on borders, labeled damping. Secondly the same
size model is considered with dashpot elements on borders, labeled dashpot. The
graph shows the voltage signal received from the sensor on the top surface of the
honeycomb model, the signal has a central frequency of 200 kHz, cf. Fig. 1 for
definition of frames and the reduced model size.
layer
te
te
osi
pla
mp
Co
Fig. 14. Setup for experimental test.
-9
Displacement (m)
4·10
Maximum
fc = 150 kHz
3·10-9
C · Vg = 2.8e5 N
2·10-9
A0
1·10-9
Without dashpot
With dashpot
S0
-9
0·10
-1·10-9
Model size: 4 frames
Twill CFRP (0/90°)
-4·10
0·10-4
0.5·10-4
1·10-4
1.5·10-4
2·10-4
Time (s)
Fig. 13. Propagated Lamb wave (nodal displacement signal) in twill CFRP (0°/90°)
plate (a) without non-reflection boundary and (b) with non-reflection boundary
using dashpot elements. The excited Lamb wave is propagated with a central
frequency of 150 kHz. The plate thickness is 1 mm, and the rest of geometrical
properties are presented in Table 2, cf. Table 4, also cf. Fig. 1 for definition of frames
and the reduced model size.
The geometrical properties of the honeycomb sandwich plate
and the PZT transducers are presented in Table 2. Fig. 11 represents
the geometrical dimensions of a honeycomb sandwich plate. The
same dimension is used for the other plates in the following sections. The material properties of the honeycomb sandwich panel
components and the aluminum plate are provided in Table 3, while
the material properties of the twill CFRP (0°/90°) plate are shown
in Table 4. It has been reported in [12] that the finite element model of a twill CFRP (0°/90°) plate without matrix provides results
which are in a better agreement with the experimental results.
The fibers are modeled using the material coordinate option in
the finite element model.
The dielectric matrix [e] and the piezoelectric matrix [e] are,
respectively [1]:
6
½e ¼ 4 0
0
5:2
3
0
and the stiffness matrix is
Dashpots: 1 frame
-3·10-9
6:45
0
0 12:7
-2·10
2
0
60
0
5:2 7
7
6
7
6
60
0
15:1 7
7ðC m2 Þ;
½e ¼ 6
60
0
0 7
7
6
7
6
4 0 12:7
0 5
Lamb waves Reflections
-9
-9
2
0
6:45
0
0
3
7
0 5109 ðC V1 m1 Þ;
5:62
2
6
6
6
6
½c ¼ 6
6
6
6
4
13:9
3
6:78
7:43
0
0
0
13:9
7:43
0
0
0
0
0 7
7
7
0 7 10
710 ðPaÞ:
0 7
7
7
0 5
11:5
sym:
3:56
0
2:56
2:56
The mass density of the PZT is 7700 kg m3 [1] and the mass density
of the twill CFRP (0°/90°) is 1550 kg m3 [12]. The skin plates of the
honeycomb sandwich plate are modeled using cubic 3D solid elements while 2-D shell elements are used to model the honeycomb
cells. Fig. 11 shows the connection of 2-D shell elements (core) and
the 3-D elements (cover plate), however, the thickness of core
structure is shown for a better visualization and explanation of geometrical dimensions. An accurate modeling approach is needed to
consider the correct stresses and deformation in the connection region where the solids elements links to the shells which is often the
weakest area. In the Lamb wave propagation study, where we are
only interested in the displacement field of the plate, multi-point
constraint equations are a reliable method to connect two portions
of a structure using solids and shells [24]. Constraint equations are
deployed to join the shell and solid elements within this study.
5.5.1. Honeycomb composite plate
Fig. 12 compares effects of different non-reflecting boundary
conditions (including artificial damping boundary and dashpot
boundary) on the wave propagation in a reduced-size honeycomb
model. In the reduced model four frames of outer boundary elements are used and the plate length is reduced to 220 mm and
the plates width is reduced to 24 mm (the dimensions of the original model are given in Table 2 which represents a model with
8
S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
40
Maximum value of the
CWT coefficients
S0
35
A0
Scale (-)
30
25
20
15
10
Time of the flight for S 0
150
200
250
300
Time of the flight for A 0
350
400
450
500
Time increment (-)
Fig. 15. Schematic representation of the absolute values of the CWT coefficients based on the Daubechies wavelet D10 in a contour plot. The Lamb wave is excited with a
central frequency of 200 kHz on a honeycomb sandwich plate.
agation in reduced-size CFRP model (cf. the dashed line in Fig. 13).
The ratio between the transmitted energy of the reflected waves
and the Lamb waves (Ereflected/ELamb) is equal to 1.2 (acceptable
range is less than 1) for the model without non-reflecting boundary condition.
Difference in group velocity (%)
Honeycomb top surface: twill CFRP (0/90°)
20
10
0
-10
S0 mode
-20
A0 mode
75
100
125
150
175
200
Frequency, fc (kHz)
Fig. 16. The group velocity values which are obtained from the experimental tests
are compared with simulation results at all tested frequencies. A honeycomb
sandwich plate (the honeycomb cell height is 8 mm and the cover plate is made of
twill CFRP (0°/90°) with 1 mm thicknesses, cf. Table 2 for the rest of geometrical
properties, cf. Table 3 and 4 for the material properties) is investigated.
twenty-eight frames of outer boundary elements, also cf. Fig. 1 for
definition of frames and the reduced model size).
The attenuation behavior of the reflected waves in the model
with non-reflecting boundary using dashpot elements is clearly
shown (cf. the solid line in Fig. 12). One can observe the amplification and attenuation effects of the reflected waves on the wave
propagation in reduced-size models with the gradually damped
artificial boundary. It is clear that the reflected waves in the model
with gradually damped artificial boundaries have almost the same
amplitude as the propagated A0 which can cause difficulties to
identify modes in a later performed signal processing (cf. the
dashed line in Fig. 12). The ratio between the transmitted energy
of the reflected waves and the Lamb waves (Ereflected/ELamb) is equal
to 1.1 (the acceptable range is less than 1) for the model with gradually damped artificial boundary.
5.5.2. CFRP composite plate
Fig. 13 presents an application example of using dashpot elements as a non-reflecting boundary in a twill CFRP (0°/90°) plate
as a complicated composite structures. The plate thickness is
1 mm and the rest of geometrical properties are presented in Table 2 and cf. Table 4 for material properties. It is clearly shown
how dashpot elements can reduce the amplitude of the reflected
waves from the borders (cf. the solid line in Fig. 13). In addition,
it is shown that the reflected waves may influence the wave prop-
5.5.3. Experimental validation
In addition to the numerical verification, results are also validated experimentally. The experimental setup is shown in
Fig. 14. The velocity of the nodes on the retro-reflective layer is
measured with scanning a laser vibrometer to evaluate the wave
properties [22,12].
The flight velocity of the propagated waves are calculated in reduced-size numerical models and compared with experimental results. The nodal displacement signal in the vertical direction u(t) is
transformed using the continuous wavelet transform (CWT) based
on the Daubechies wavelet D10 to evaluate the flight velocities
(the bar indicates complex conjugation).
1
WTða; bÞ ¼ pffiffiffi
a
Z
þ1
1
ta
dt:
uðtÞw
b
ð4Þ
The location of the maxima of the CWT coefficients gives the timeof-flight for each Lamb wave mode, cf. Fig. 15. Knowing the distance
and the time-of-flight one can calculate the group velocity [11]. The
time-of-flight is measured between the excitation and the arrival at
two different points on the top and the bottom surface, in order to
show the influence of the sandwich plate thickness and the core
material on the flight velocity of the wave propagation. Therefore,
the flight velocity on the bottom surface includes the effect of plate
thickness and differs from the flight velocity on the top surface
which is known as the group velocity. In this paper we use ‘‘group
velocity’’ to describe the flight velocity for both top and bottom
surfaces.
Because of the combination of the propagated modes and the
reflected waves in the reduced-size models without non-reflecting
boundary the location of the maxima of the CWT coefficients may
change up to 20% (in comparison to experimental results). Fig. 13
shows how the group velocity of the A0 mode changes in a model
without non-reflecting boundary condition. One can observe that
the maximum amplitude of the A0 mode occurs approximately
0.2 ms earlier in the model without non-reflecting boundary which
can be explained with amplification and attenuation effects of the
reflected waves on the wave propagation in the reduced-size model. A similar illustration is given in Fig. 12 where a weak nonreflecting boundary with gradually damped artificial in reduced
size model may effect the group velocity of the A0 mode. Therefore,
9
15
top surface
Aluminum honeycomb
10
5
0
-5
-10
-15
100
150
200
250
300
350
Difference in group velocity (%)
Difference in group velocity (%)
S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
bottom surface
15
10
5
0
-5
-10
S0 mode
A0 mode
-15
100
150
Frequency, fc (kHz)
200
250
300
350
Frequency, fc (kHz)
Fig. 17. The group velocity values which are obtained from the experimental tests are compared with simulation results for different excitation frequencies. The group
velocities are compared on both sides of a honeycomb plate which is made of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cell size is 6.4 mm and the honeycomb wall
thickness is 0.0635 mm.
the agreement between the group velocities in the simulation and
experimental tests is considered to show the capabilities of the
proposed reduced-size model using non-reflecting boundary based
on dashpot elements.
In the first example, the excited Lamb wave is propagated in a
honeycomb sandwich plate, where the honeycomb cell height is
8 mm and the cover plate is made of twill CFRP (0°/90°) with
1 mm thickness, cf. Table 2 for the rest of geometrical properties,
cf. Table 3 for the material properties of the honeycomb cells,
and cf. Table 4 for the material properties of the CFRP plate.
Fig. 16 compares the group velocity values of different modes
which are obtained experimentally and numerically, where the
central frequency of the excitation signal is increased from
50 kHz to 220 kHz. The relative difference in percent is calculated
as
Erel ¼ 100 Gsim Gexp
½%:
Gexp
ð5Þ
Gsim represents the group velocity values which are obtained from
the simulation test, and Gexp shows the group velocity values which
are obtained from the experimental investigations. For the proposed example we achieve a good agreement of the results. The
average of absolute differences (jErelj) is 2.30% and the absolute
maximum is 8.01%.
In another example the group velocity values of the propagated
waves in a honeycomb sandwich palate are compared on both
sides of the structure, cf. Fig. 17. The honeycomb plate is fully
made of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cell
size is 6.4 mm and the honeycomb wall thickness is 0.0635 mm.
The relative error is not exceeding 11% and the average of absolute
differences is 4.11%.
6. Summary
A non-reflecting boundary condition using dashpot elements is
introduced in order to reduce the reflections of the Lamb waves at
boundaries. It has been shown that by applying the proposed
boundary condition the computational costs for wave propagation
simulations can be reduced significantly. Different parameters
including the damping factor, the direction of dashpot elements
and the number of dashpot elements were examined to design
an efficient non-reflecting boundary. The influence of each parameter is summarized as follows:
Damping factor: It has been shown that very small damping factors do not generate enough force to suppress the propagation
of the reflected waves effectively. On the other hand, the nodal
displacements are not large enough to move dashpot elements
with very high damping factors. A range of 28,000–280,000 of
C Vg (damping factor times group velocity) is found as an
appropriate choice to significantly reduce the propagation of
the reflected waves.
Direction of dashpot elements: It has been observed that the
dashpot elements in direction of wave propagation are most
effective in reducing the reflections.
Number of dashpot elements: To apply the new non-reflecting
boundary condition, dashpot elements are connected only to
one row and column of the outer elements as a primary choice.
It has been indicated that the additional number of dashpot elements does not improve the efficiency of the proposed boundary condition significantly.
The application of the proposed method is demonstrated for the
Lamb wave propagation within different heterogenous materials
including a honeycomb sandwich plate with twill CFRP (0°/90°)
cover plate and an aluminum honeycomb sandwich plate. In addition, results were validated experimentally. Furthermore, combination of dashpot elements with gradually damped artificial
boundary were considered and it has been shown that this combination has only a minor effect on reflection of the waves from the
borders and causes extra efforts in the modeling process.
7. Conclusions
The advantages of the proposed non-reflecting boundary condition using dashpot elements to reduce the computational costs together with the possibility of implementing the proposed scheme
in commercial FEM packages, provide an efficient tool for researchers to numerically investigate the interaction of the Lamb waves
within complicated structures such as CFRP and honeycomb composites even with ordinary personal computers. These investigations are very important to design further health monitoring
systems using Lamb waves for composite structures. However, further investigations are required to extend the application of the
proposed non-reflecting boundary condition to study wave propagation in other heterogenous structures.
Acknowledgments
By means of this, the authors acknowledge the German Research Foundation for the financial support (GA 480/13). We would
also like to thank C. Willberg for that was invaluable in the execution of the experimental tests.
10
S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10
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