Yanfeng Shen1
Carlos E. S. Cesnik
Department of Aerospace Engineering,
University of Michigan,
Ann Arbor, MI 48109
1
This paper presents the local interaction simulation approach (LISA) for efficient modeling of linear and nonlinear ultrasonic guided wave active sensing of complex structures.
Three major modeling challenges are considered: material anisotropy with damping
effects, nonlinear interactions between guided waves and structural damage, as well as
geometric complexity of waveguides. To demonstrate LISA’s prowess in addressing such
challenges, carefully designed numerical case studies are presented. First, guided wave
propagation and attenuation in carbon fiber composite panels are simulated. The numerical results are compared with experimental measurements obtained from scanning laser
Doppler vibrometry (SLDV) to illustrate LISA’s capability in modeling damped wave
propagation in anisotropic medium. Second, nonlinear interactions between guided
waves and structural damage are modeled by integrating contact dynamics into the LISA
formulations. Comparison with commercial finite element software reveals that LISA can
accurately simulate nonlinear ultrasonics but with much higher efficiency. Finally,
guided wave propagation in geometrically complex waveguides is studied. The numerical
example of multimodal guided wave propagation in a rail track structure with a fatigue
crack is presented, demonstrating LISA’s versatility to model complex waveguides and
arbitrary damage profiles. This paper serves as a comprehensive, systematic showcase of
LISA’s superb capability for efficient modeling of transient dynamic guided wave phenomena in structural health monitoring (SHM). [DOI: 10.1115/1.4037545]
Introduction
The success of guided wave-based structural health monitoring
(SHM) systems substantially relies on the effective design of sensing array as well as the insightful interpretation of active sensing
signals. However, identifying the optimum design parameters,
such as transducer dimensions, interrogating frequency, preferable
wave mode, and sensing locations, is a challenging task. In addition, the multimodal, dispersive nature of guided waves as well as
their complex interaction dynamics with structural features and
damage further give rise to the difficulty of properly analyzing the
sensing signals. Thus, highly efficient computational models of
guided wave-based SHM procedures are of critical importance for
both SHM system design and signal interpretation. Several
demanding requirements for the computational models can be
identified: accuracy for high frequency, short wavelength, long
propagating distance waves, efficiency in terms of computational
time and computer resources, and versatility to explore a wide
range of design parameters such as different material and geometric scenarios. However, commercially available numerical modeling tools that are based on a finite element method (FEM) cannot
satisfy all these requirements, as ultrasonic wave propagation in
large-scale, complex structures usually results in a computationally prohibitive problem.
In order to satisfy the desperate needs for guided wave-based
SHM modeling tasks, research efforts in developing accurate, efficient, versatile modeling schemes have been widely pursued.
Shen and Giurgiutiu formulated the analytical expressions with
transfer functions for guided wave active sensing in simple
1
Corresponding author.
Manuscript received June 20, 2017; final manuscript received August 3, 2017.
Assoc. Editor: Mark Derriso.
isotropic plate structures, which was further consolidated as the
analytical framework called WaveFormRevealer [1]. Rahani and
Kundu [2] and Glushkov et al. [3] developed the Green’s
function-based distributed point source method and successfully
modeled wave generation, propagation, and scattering in a highly
efficient manner. Hybrid modeling techniques have also been proposed to develop efficient simulation schemes. The semianalytical finite element method has been used and combined with
local FEM to simulate wave interaction with damage in onedimensional wave propagation problems [4–6]. Moreau and Castaings have used orthogonally analytical relation to reduce the
size of FEM to obtain three-dimensional (3D) guided wave scattering features [7]. Gresil and Giurgiutiu investigated the hybrid
analytical/FEM modeling concept in the time domain and
achieved promising results [8,9]. Shen and Giurgiutiu further realized this hybrid concept in the frequency domain and conducted
experimental verifications [10]. A recent contribution from Hafezi
et al. put forward a peri-ultrasound technique for modeling ultrasonic response, which considered both linear and nonlinear wave
damage interaction scenarios [11].
Nevertheless, the aforementioned modeling schemes mainly
focused on relatively simple SHM configurations. Challenges
would generally arise when material anisotropy, damping, structural geometric complexity, and nonlinear wave–damage interactions need to be considered. A powerful modeling technique
based on the finite difference formulation and sharp interface
model, known as the local interaction simulation approach
(LISA), has been developed, acquiring more and more modeling
capabilities to handle complex simulation tasks. Delsanto et al.
first derived the one-dimensional, two-dimensional, and 3D LISA
formulations for isotropic, heterogeneous materials executed on
Connection Machines [12–14]. LISA underwent considerable progress during the past decade, with its application in metallic
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University of Michigan–Shanghai Jiao Tong
University Joint Institute,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: yanfeng.shen@sjtu.edu.cn
Local Interaction Simulation
Approach for Efficient Modeling
of Linear and Nonlinear
Ultrasonic Guided Wave Active
Sensing of Complex Structures
in a rail track with a fatigue crack will be presented, which demonstrates LISA’s versatility in handling wave propagation in geometrically complex waveguides and interaction with arbitrary
damage profiles.
2 Local Interaction Simulation Approach Framework
and Graphics Processing Units Implementation
Figure 1 presents the LISA framework, showing its derivation
procedure as well as the new features. LISA approximates the partial differential elastodynamic wave equations with finite difference quotient expressions. The coefficients in LISA iterative
equations depend only on the local physical material properties. A
sharp interface model was used to enforce the stress and displacement continuity between the neighboring cells and nodes. Therefore, changes of material properties in the cells surrounding a
computational node can be captured through these coefficients.
The details of formulation derivation can be found in Ref. [18].
Guided wave generation can be achieved with an efficient frequency domain local FEM. Details of this hybrid approach can be
found in Ref. [22]. Damping effects are considered based on the
3D Kelvin–Voigt viscoelasticity model. A viscosity matrix is
introduced for a generic lamina with arbitrary stacking angle to
capture the directional and coupling damping effects. It should be
noted that the iterative equations with damping effects require the
Fig. 1 LISA framework for modeling guided wave generation, propagation, and nonlinear interaction with damage in complex
structures
011008-2 / Vol. 1, FEBRUARY 2018
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structures [15,16], extension to general anisotropic materials
[17–19], coupled field capabilities [20], hybridization with other
numerical methods [21,22], and execution on powerful graphics
processing units (GPUs) with compute unified device architecture
(CUDA) technology [23–25]. In the authors’ recent study, a penalty method was deployed to introduce contact dynamics into the
LISA formulation to simulate contact acoustic nonlinearity [26].
The GPU implementation has substantially facilitated low-cost
supercomputing for ultrasonic elastic waves in large-scale
structures.
This paper presents the recent progress of the LISA framework
for efficient modeling of guided wave phenomena considering
material, structural, and wave–damage interaction complexities.
The LISA fundamentals and its GPU implementation will be
introduced. Three numerical case study examples will follow to
demonstrate LISA’s prowess in handling various aspects of the
modeling challenges. First, guided wave propagation in anisotropic composite panels will be modeled and compared with experiential measurements, which aims to show LISA’s capability in
considering material anisotropy and the guided wave attenuation
due to damping. Second, guided wave interaction with a fatigue
crack will be modeled and compared with the results obtained
from the commercial FEM package ANSYS, which would present
LISA’s ability to simulate nonlinear interactions between interrogating waves and the damage. Finally, guided wave propagation
3 Damped Wave Propagation in Anisotropic
Composite Plates
Composite materials are increasingly used in aerospace and
automotive industries due to their strong, light-weight, fatiguetolerant properties. However, such material advancement brought
considerable challenges for guided-wave SHM techniques,
because the wave characteristics (amplitude, velocity, and attenuation) are heavily direction dependent, which also imposes great
difficulty in accurately capturing such highly anisotropic wave
phenomena in computational models.
This section presents the numerical case study of damped guided
wave propagation in anisotropic composite panels using LISA as
well as the comparison with experimental measurements using
scanning laser Doppler vibrometry (SLDV). The multilayered
composite specimens used in this study were a 12-layer unidirectional [0]12T carbon fiber reinforced polymer (CFRP) composite
plate and a 12-layer cross-ply [0/90]3S CFRP composite plate.
They are manufactured from 0.125 mm thick pre-impregnated
composite tape made from IM7 fibers and Cy-com 977-3 resin.
Thus, the thickness of these composite panels is 1.5 mm. A 12.8mm diameter and 0.23-mm thick circular piezowafer was bonded
on the center of the plate surface for wave generation. It should be
noted that the fiber volume fraction does not enter the model
directly. It is considered via the parameters such as stiffness matrix,
damping matrix, and density. Details of the model and its parameters can be found in Refs. [18,22]. The material mechanical properties associated with our specimen are given in Table 1, and the
Journal of Nondestructive Evaluation, Diagnostics
and Prognostics of Engineering Systems
Table 1 Mechanical properties of different materials in the
experiment and simulations
Mech. prop.
E11 (GPa)
E22 (GPa)
E33 (GPa)
12
13
23
G12 (GPa)
G13 (GPa)
G23 (GPa)
q (kg/m3)
IM7/977-3
PZT 5A
147.00
9.80
9.80
0.41
0.41
0.69
5.3
5.3
3.31
1558
60.98
60.98
53.19
0.35
0.44
0.44
22.57
21.05
21.05
7750
Table 2 Transducer piezoelectric properties
Piezo. prop.
PZT 5A
e15 (C/m2)
e25 (C/m2)
e31 (C/m2)
e32 (C/m2)
e33 (C/m2)
j11 (nF/m)
j22 (nF/m)
j33 (nF/m)
12.29
12.29
5.35
5.35
15.78
8.13
8.13
7.32
Table 3 IM7/977-3 viscosity coefficients of unidirectional
CFRP lamina from model updating
Viscosity coefficients
Unit: (Pas)
D11
D22
D33
D44
D55
D66
328.6
525.7
525.7
722.8
394.3
394.3
piezoelectric properties are given in Table 2. The method of obtaining viscosity coefficients of solids itself is a challenging branch of
study. Available data are also limited for CFRP composite materials in current literature. Thus, in this study, the viscosity coefficients were obtained by updating our LISA model toward the
experimental measurements. The chosen viscosity matrix coefficient for the unidirectional CFRP lamina is given in Table 3.
To demonstrate the effectiveness of the anisotropic damping
formulation for an arbitrary lamination angle, the unidirectional
composite panel case was first considered. Figure 2 presents the
LISA simulation results of guided wave propagation and attenuation in the unidirectional carbon fiber composite plate at 75 kHz.
The damping effect was realized using the Kelvin–Voigt viscoelastic formulation, details of which can be found in Ref. [22].
The first row of Fig. 2 compares the simulation results without
damping effects and with damping effects. It was found that when
material damping is considered, guided waves undergo considerable attenuation. Then, the unidirectional plate was rotated with
various orientation angles, and the simulation results are shown in
the second row. The consistence of guided wave propagation and
attenuation profiles demonstrates LISA’s capability in modeling
wave propagation and attenuation in composite panels with an
arbitrary lamination orientation.
Figure 3 shows the comparison between LISA solutions and
SLDV measurements of guided wave propagation in composite
panels. The first row corresponds to the results of guided wave
propagation in the unidirectional [0]12T CFRP composite plate at
75 kHz. The second row corresponds to the results of guided wave
propagation in the cross-ply [0/90]3S CFRP composite plate at
75 kHz. The first column presents the group velocity directivity
curves of the fundamental wave modes S0, A0, and SH0 in these
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results from previous three time steps to determine the displacement result at current time step. A commercial preprocessor from
ANSYS was integrated seamlessly into the framework for computational grid generation and material properties allocation, which
enables easily modeling of structures with complex geometric features and material distribution. As mentioned previously, a penalty function method was implemented in the LISA formulation to
model the contact acoustic nonlinearity during wave damage
interactions. A Coulomb model was used to capture the stick–slip
motion at the crack interface. A nonreflective and absorbing
boundary condition was enabled to minimize the model size and
avoid undesired boundary reflections when required.
There are two major characteristics of the current LISA formulation that enables the computation to be expedited on GPUs.
First, LISA formulations are massively parallel. This is because
the computation of a general node or a contact node only depends
on the solutions of its 18 neighboring nodes at the previous three
time steps. Thus, the behavior of each node is independent from
the others at the target time step, i.e., the computation of each
node can be carried out individually in parallel. Second, the wave
propagation simulation tasks usually require dense discretization
of the structure, resulting in a computationally intensive problem.
GPUs, with their massive concurrent threading feature, are suitable to handle such large size problems by distributing the workloads among a large number of functional units and carry out
highly efficient parallel computing. In order to take advantage of
the nice parallelizable feature of LISA and the superb computational capability of the powerful GPU device, the LISA procedure
was implemented using CUDA. All the parameters are first established in the host memory (RAM). Then, a copy of these parameters is sent to the device memory (GPU global memory) for it to
be processed. The computation of each node is assigned to a functional thread, i.e., each thread will gather the displacements of its
18 neighboring nodes at the previous three time steps, process the
material properties in the eight surrounding cells, and execute the
kernel to compute the displacement of this node at the current
time step. Since one of the bottlenecks of a CUDA program is the
data transfer between the device memory and host memory, only
the required step results are gathered (every 20–30 steps depending on the frequency of the propagating waves) from the GPU to
the CPU to minimize such data transfer.
Fig. 3 Comparison of guided wave propagation between LISA solution and SLDV measurement in a unidirectional composite
panel (first row) and a cross-ply composite panel (second row)
two composite plates. The second column shows the wavefield
pattern from SLDV measurements, while the third column
presents the LISA simulation results. It can be observed that the
LISA simulation results agree well with experimental measurements, demonstrating LISA’s capability and accuracy in capturing
wave propagation in arbitrary lamination scenarios.
To further demonstrate the importance of considering wave
attenuation due to material damping, sensing signals at the coordinate location of (0 mm, 100 mm) in Fig. 3 for the unidirectional
011008-4 / Vol. 1, FEBRUARY 2018
composite panel case are presented in Fig. 4, comparing three situations: experimental measurement, LISA result with damping,
and LISA result without damping. The signal amplitudes are normalized to the maximum oscillation at the transducer. It can be
seen that the conventional LISA without damping effects would
overestimate the A0 mode amplitude. On the other hand, the
viscoelastic LISA formulation can capture the anisotropic damping behavior for guided wave attenuation. It should also be noted
that the damping effects have more influence on the A0 mode
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Fig. 2 Damped guided wave propagation in a unidirectional carbon fiber composite panel with varying lamination angles
Fig. 4 Simulation signal with and without damping effect and
the measured signal at a point 100 mm away from the source
normal to the fibers in a [0]12T composite laminate
4
Nonlinear Ultrasonics Modeling Capability
In addition to the modeling challenges from material anisotropy
and damping effects, the need for accurately capturing nonlinear
interactions between guided waves and structural damage would
further add to the complexity of the model. This is especially true
when the modeling endeavors aim at simulating nonlinear ultrasonic SHM techniques [27–29].
To demonstrate LISA’s capability in modeling contact acoustic
nonlinearity problems, a benchmark simulation described in Fig. 5
was conducted, and the solution was compared with the result
from ANSYS 14. The model consists of a 500 mm long, 20 mm
wide, and 5 mm thick aluminum strip. A 10 mm long throughthickness breathing crack is located in the center of the strip. A
pair of in-plane, in-phase line prescribed displacements (1 lm
peak to peak value) at both top and bottom of the plate surface
were used to generate 100 kHz 10-cycle tone burst S0 guided
waves into the structure. It should be noted that whether a certain
wave mode can be generated effectively or not depends on the
agreement between the excitation profile and the target wave
mode shape. S0 mode mainly undergoes in-plane motion which is
symmetric across the thickness. Thus, the in-plane, in-phase, top
and bottom surface excitation pair can generate S0 mode. On the
other hand, out-of-plane, in-phase, top and bottom surface excitation pair will generate A0 mode. In this study, the generated S0
waves will propagate along the structure, interact with the breathing crack, bring the crack information with them, and are finally
picked up at the sensing point. The out-of-plane displacement at
10 mm right after the crack was recorded.
The ANSYS contact model used SOLID45 eight node structural
element to discretize the geometry and CONTA173/TARGE170
contact elements to model the crack surfaces. In the LISA model,
the cell size was 1 mm in the in-plane direction and 0.5 mm across
the thickness. The mesh size le was checked against the shortest
wavelength kmin possible at the third harmonics, i.e.,
le kmin =20. A time step Dt of 0.125 ls was utilized for the time
marching Newmark-beta integration to ensure the accuracy up to
the third harmonic frequency, i.e., Dt 1=20fmax . The time
marching step was 0.06 ls obtained from the Courant–Friedrichs–
Lewy (CFL) number requirement [18]. It should be noted that a
denser mesh is used in the thickness direction as an effort to accurately depict the mode shapes of guided waves. The computational
grid on the mode shape dimensions imposes much influence on
Fig. 5 Schematic of a benchmark problem for nonlinear ultrasonic wave simulation
Journal of Nondestructive Evaluation, Diagnostics
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Fig. 6 Contact LISA solution compares well with the result
from commercial FEM package ANSYS: (a) time domain simulation signals and (b) frequency domain spectra
the guided wave modeling accuracy. A systematic study on this
topic has been conducted by Nadella [30].
Figure 6 shows the verification of the contact LISA model against
the ANSYS solution. It can be noticed that the signals agree with
each other in time domain with merely slight differences. The frequency spectra shown in Fig. 6(b) also demonstrated that the frequency components of fundamental excitation, low-frequency
direct current (DC) component, second, third, even fourth higher
harmonics compare very well with each other. Differences only
appeared at very high-frequency range, where the time step is not
sufficiently small enough to generate accurate results for meaningful comparison between the two methods. This numerical verification against commercial finite element code attested the capability
and validity of the new contact LISA model.
It should be noted that the new contact LISA model achieved
much higher computational efficiency over the conventional nonlinear FEM simulation. Both computational tasks were conducted
on an Asus ESC2000 G2 workstation with a 2.00 GHz Intel Xeon
E2-2650 processor, 32 GB of 1.60 GHz memory, and an Nvidia
GeForce GTX Titan graphics processor with 2688 CUDA cores.
The FEM simulation with 279,900 degrees-of-freedom (DOFs)
took around 19 s for each time step, resulting in a total computational time of 8 h for 1500 time steps. On the other hand, LISA simulation with 648,120 degrees-of-freedom merely consumes around
0.043 s for each time step, resulting in a total computational time of
2.15 min for 3000 time steps. Thus, it is apparent that the new contact LISA model is much more efficient than the conventional nonlinear FEM simulation, while achieving comparable results.
The efforts in capturing the nonlinear ultrasonic phenomena are
important, because the nonlinear ultrasonic signals may contain
rich diagnostic information to qualify and quantify the damage.
This aspect is further illustrated with Fig. 7, which presents the
simulation results and compares the sensing signals among the
pristine, notch, and the fatigue crack cases. The notch is modeled
by replacing a thin layer of material elements with air cells. The
fatigue crack is introduced using a discrete meshing strategy
which creates contact pairs at the crack interface. Detailed steps to
generate the fatigue crack can be found in Ref. [26]. It can be
noticed that the time trace of the notch case resembles the waveform of the pristine signal with an amplitude drop and balanced
positive and negative oscillatory amplitudes. In contrast, the time
trace of the fatigue crack case is heavily distorted. The
contact–impact between the crack surfaces induced highamplitude spikes. The unbalanced waveform results in a
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than the S0 mode. This is related to the motion characteristics of
their mode shapes and wavelengths.
concern in commercial FEM packages, but it remains a challenge
for many of the newly developed computationally efficient
schemes. LISA’s seamless integration with commercial preprocessors enables the handling of geometrically complex waveguides. To substantiate this aspect, this section presents the
numerical case study of utilizing LISA to model guided wave
propagation in a rail track.
low-frequency modulation on the entire wave packet. The waveform also shows zigzags unlike the smooth pattern present in the
notch crack case. The frequency spectra of these two signals show
great difference too. It can be observed that the pristine and notch
signal spectra show only the excitation frequency component at
100 kHz. No higher harmonics nor DC component exists. On the
other hand, the fatigue crack signal presents distinctive nonlinear
higher harmonics and DC component in its spectrum. The generation of higher harmonics and DC component is a classical nonlinear phenomenon, and it serves as the basis for many nonlinear
ultrasonic inspection methodologies. This numerical example
shows that LISA formulations with contact dynamics are capable
of simulating such nonlinear ultrasonic signals.
5 Wave Propagation in Geometrically Complex
Waveguides
The most commonly encountered modeling challenge comes
from the large, complex structural geometries. It may not cause a
Fig. 8
5.2 Guided Wave Propagation and Interaction With
Damage in the Rail Track. Figure 10 presents the LISA simulation result of guided wave generation, propagation, and interaction
with a fatigue crack in the rail track. Figure 10(a) shows the snapshots of guided waves generation by the line traction forces at
50 ls, guided wave propagation, mode development, and interaction with the crack at 150 ls, as well as guided wave absorption
by the absorbing layers with increasing damping boundaries at
300 ls. It can be seen that the guided wave modes are very
LISA model for guided wave propagation and interaction with a fatigue crack in a rail track
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Fig. 7 The interaction of guided waves with a fatigue crack
brings in distinctive nonlinear higher harmonics, while their
interaction with a notch does not: (a) time trace comparison
and (b) frequency spectrum comparison
5.1 LISA Model for a Rail Track. Figure 8 presents the
LISA numerical model for the investigation of guided wave propagation and interaction with a fatigue crack in a rail track. Two
lines of surface traction forces are utilized to simulate a pair of
transducers generating guided waves into the rail structure. A
100 kHz 10-cycle tone burst is used for wave generation. The
ultrasonic waves propagate along the rail track, interact with a
fatigue crack, and are finally picked up at the sensing location.
Absorbing layers with increasing damping is used on both ends of
the model to eliminate boundary reflections, enabling the simulation of wave propagation in an infinite long rail track with a finite
dimensional model. A 1 mm mesh size is adopted for the crosssectional plane, while a 2 mm mesh sized is deployed for the track
direction. The time step according to the CFL condition is
110.33 ns, which corresponds to a CFL number of 0.99. The 3D
LISA grid shows that the adopted discretization can capture the
complex geometric details of the rail track, i.e., LISA can handle
very complex structural geometries.
In order to depict the fatigue damage, a parametric fatigue
crack zone is used in the LISA model. Figure 9(a) shows the profile of a typical fatigue zone in a rail track. Figure 9(b) presents
the parametric fatigue zone characterized by the size of a. In this
study, fatigue cracks of progressive sizes are investigated, i.e.,
a ¼ 0 mm, 5 mm, 10 mm, 15 mm, and 20 mm, corresponding to
pristine case, incipient damage case, median damage case, and
sever damage case.
Fig. 9 (a) Typical fatigue zone in a rail track [31] and (b) parametric fatigue crack zone in the LISA model
Fig. 11 (a) Comparison of time domain sensing signals
between pristine and damaged cases, (b) time–frequency
domain presentation of the pristine case sensing signal, and (c)
time–frequency domain presentation of the damaged case
sensing signal. Note the appearance of superharmonic components in the damaged case signal.
This numerical case study demonstrated LISA’s capability in
handling guided wave propagation in geometrically complex
waveguides and nonlinear interactions with structural damage
with arbitrary profiles. It should be emphasized again that for this
case study, the model size reached 10,192,107 degrees-offreedom, which is computationally prohibitive for most conventional FEM packages. It is LISA’s parallel GPU implementation
that facilitates its highly efficient modeling capability.
5.3 LISA Model Data for Damage Quantification Analysis.
To further quantify the damage severity, a damage index (DI) can
Fig. 10 (a) Guided wave generation, propagation, and interaction with a fatigue crack in the rail track and (b) guided wave interacting with crack with crack open and close phenomenon
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complex in the rail track. Figure 10(b) shows the crack open and
close phenomenon during the wave damage interaction, which
introduces the contact acoustic nonlinearity into the sensing
signal.
Figure 11(a) presents the comparison of time domain sensing
signals between the pristine case (a ¼ 0 mm) and the severely
damaged case (a ¼ 20 mm). The signals are very complex with
multiple wave packets, arisen from the multimodal and dispersive
nature of guided waves. According to the conventional linear
ultrasonic techniques based on phenomena such as amplitude
change, phase shift, or scattering, it is very difficult to make an
appropriate assessment of the damage existence and severity. On
the other hand, from the perspective of nonlinear ultrasonic features, the damage signature is much more evident. Figures 11(b)
and 11(c) show the time–frequency domain sensing signals of the
pristine case and the severely damaged case, respectively. For the
pristine case, only the excitation frequency f0 exists. However, for
the damaged case, in addition to f0, superharmonics are present at
2f0, 3f0, 4f0, and so on. Such a unique nonlinear feature allows the
early detection of incipient fatigue damage in structures. Superharmonics is a classical nonlinear ultrasonic phenomenon which
has been widely reported using experiments in the literatures
[27–29]. Again, the focus of this research is on developing an efficient numerical approach to model such a nonlinear effect. Experimental verifications on this rail model should be conducted in a
future work.
be developed based on the nonlinear energy proportion in the
LISA model data. Figure 12(a) shows the 3D view of the
time–frequency sensing signal of the severely damaged case. The
magnitude represents the energy participation of each frequency
component along the time history. In order to consider all the
wave modes participating in the wave propagation, the time
domain integration is used at the fundamental and superharmonic
frequencies to construct the DI, which is given by
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uX
uX
ð
ð u N t
u N t
1
u
u
f0 dt
P
t;
nf
P
t;
n
þ
ð
Þdt
0
u
u
2
u n¼1 0
u n¼1 0
u
DI ¼ u ð t
ðt
t
t
Pðt; f0 Þdt
Pðt; f0 Þdt
0
(1)
0
where Pðt; nf0 Þ denotes the magnitude at the nth superharmonic
location at time t. It should be noted that the magnitude should be
elevated above zero by subtracting the minimum base value in the
spectral graph. The DI represents the nonlinear energy ratio, designating the degree of nonlinearity in the sensing signal. The first
term consists of the integration over the “peaks” of the spectrum,
while the second term is the integration over the “valleys” of the
spectrum serving as the inherent baseline. By subtracting the
inherent baseline, this DI becomes a indicative measurement of
signal nonlinearity. Figure 12(b) presents the damage index results
for various crack sizes. It can be noticed that the DI is of high sensitivity; even at incipient damage level (a ¼ 5 mm) the DI
increased noticeably. Thus, such high sensitivity will allow the
detection of small cracks at the early stage. In addition, the DI also
grows monotonically with the damage severity, which allows crack
growth monitoring and the quantification of the damage severity.
6
Concluding Remarks
This paper presented the local interaction simulation approach
for efficient modeling of linear and nonlinear ultrasonic guided
waves for active sensing of complex structures. Three major modeling challenges were addressed: material anisotropy with damping effects, nonlinear interactions between guided waves and
structural damage, as well as the geometric complexity of waveguides. First, guided wave propagation in unidirectional and
cross-ply carbon fiber composite panels was modeled and compared with experimental measurements. The LISA solutions
agreed well with the experimental results obtained by scanning
laser Doppler vibrometry, which demonstrated LISA’s capability
in modeling the directivity of guided wave propagation and
attenuation in composite materials. Second, the nonlinear
011008-8 / Vol. 1, FEBRUARY 2018
interactions between guided waves and structural damage were
considered. The verification against the commercial FEM package
ANSYS not only showed LISA’s accuracy in capturing the contact acoustic nonlinearity but also highlighted LISA’s superb computational efficiency over the conventional FEM approach. Third,
the case study of guided wave propagation and interaction with a
fatigue crack in a rail track was conducted. The simulation results
showed that LISA was capable of handling very complex structure
and damage geometries. The further analysis of LISA generated
sensing signals also allowed the development of a nonlinear damage index for the quantification of the fatigue damage. This paper
systematically demonstrated LISA’s prowess for efficient modeling of transient dynamic guided wave phenomena in SHM
procedures.
Acknowledgment
This work was also partially sponsored by the National Rotorcraft Technology Center (NRTC) Vertical Lift Rotorcraft Center
of Excellence (VLRCOE) at the University of Michigan, with
Mahendra J. Bhagwat as the program manager. Opinions, interpretations, conclusions, and recommendations are those of the
authors and are not necessarily endorsed by the U.S. Government.
Funding Data
The National Natural Science Foundation of China (Contract
No. 51605284).
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