# Environmental variation compensated damage classification and localization
in ultrasonic guided wave SHM using self-learnt features and Gaussian mixture
models

Shruti Sawant  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Mumbai 400076, MH India  Sheetal Patil  Department of
Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076,
MH India  Jeslin Thalapil  Department of Electrical Engineering, Indian
Institute of Technology Bombay, Mumbai 400076, MH India  Sauvik Banerjee
<EMAIL_ADDRESS>Department of Civil Engineering, Indian Institute of
Technology Bombay, Mumbai 400076, MH India  Siddharth Tallur
<EMAIL_ADDRESS>Department of Electrical Engineering, Indian Institute
of Technology Bombay, Mumbai 400076, MH India

###### Abstract

Conventional damage localization algorithms used in ultrasonic guided wave-
based structural health monitoring (GW-SHM) rely on physics-defined features
of GW signals. In addition to requiring domain knowledge of the interaction of
various GW modes with various types of damages, they also suffer from errors
due to variations in environmental and operating conditions (EOCs) in
practical use cases. While several machine learning tools have been reported
for EOC compensation, they need to be custom-designed for each combination of
damage and structure due to their dependence on physics-defined feature
extraction. Research on the use of deep learning tools such as convolutional
neural networks (CNNs) for automated feature extraction in GW-SHM is in a
nascent stage and has vast untapped potential for increasing the robustness of
GW-SHM systems. In this work, we propose a CNN-based automated feature
extraction framework coupled with Gaussian mixture model (GMM) based EOC
compensation and damage classification and localization method. Features
learnt by the CNNs are used for damage classification and localization of
damage by modeling the probability distribution of the features using GMMs.
The Kullback-Leibler (KL) divergence of these GMMs with respect to
corresponding baseline GMMs are used as signal difference coefficients (SDCs)
to compute damage indices (DIs) along various GW sensor paths, and thus for
damage localization. The efficacy of the proposed method is demonstrated using
FE generated GW-data for an aluminum plate with a network of six lead
zirconate titanate (PZT) sensors, for three different types of damages (rivet
hole, added mass, notch) at various temperatures (from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$), with added white noise and pink
noise to incorporate errors due to EOCs. We also present experimental
validation of the method through characterization of notch damage in an
aluminum panel under varying and non-uniform temperature profiles, using a
portable custom-designed field programmable gate array (FPGA) based signal
transduction and data acquisition system. We demonstrate that the method
outperforms conventional EOC compensation method using GMM with physics-
defined features for damage localization in GW-SHM systems prone to EOC
variations.

Convolutional neural networks (CNN), structural health monitoring (SHM),
ultrasonic non-destructive testing (NDT), Gaussian mixture model (GMM), damage
localization

††preprint: MSSP

##  I Introduction

Structural health monitoring (SHM) systems employed for detecting defects in
large infrastructure such as bridges, aircraft, railway tracks etc. consist of
large number of sensors mounted on the structure to perform a variety of
measurements to satisfy the vastly different requirements across different
structures and applications. Ultrasonic guided wave-based SHM (GW-SHM) is
popular approach for a variety of structures due to low cost of
instrumentation, sensitivity of high frequency waves to small defects, ability
of guided waves (GWs) to travel long distances on the structure without
attenuation etc.[1, 2, 3] However, a significant practical limitation of
damage detection algorithms in GW-SHM systems designed in laboratory
environment, is susceptibility to errors introduced through variations in
environmental and operational conditions (EOCs) such as temperature, moisture
absorption, loading etc.[4, 5, 6, 7, 8, 9] GW-based damage characterization
requires interpreting damage features of time-series corresponding to GW
signals recorded by sensor. Variation in EOCs, including temperature,
vibration, loading, moisture etc. result in change in phase and amplitude of
GW signals [10, 11]. Identifying effective damage features is crucial to the
success of damage characterization. Selection of suitable features usually
requires professional knowledge and varies for different structures and types
of damages. To relieve the burden of damage feature selection, researchers
have been investigating deep learning algorithms to automate damage
identification without explicit feature definition in advance. There are
number of studies reporting temperature compensation techniques [12, 13].
These can be broadly classified into two classes: baseline signal stretch
(BSS) and optimal baseline selection (OBS). In BSS, a single baseline signal
(e.g. recorded under nominal operating conditions at room temperature) is
stretched to match the signal obtained at different temperatures. In OBS
approach, baseline signals are recorded at different temperatures and
subsequent computations utilize selection of a baseline signal within the
database which is most similar to the monitoring signal recorded during
inspection. Other preprocessing techniques to remove the effect of EOC
variation recently reported in literature include homoscedastic nonlinear
cointegration [14], normalized smoothed envelope threshold (NSET) coupled with
high pass filtering [15], variational mode decomposition (VMD) algorithm for
denoising and removing seasonal patterns in guided wave signals [16] and
cointegration analysis [17, 18].

Machine learning (ML) tools prove useful in modeling complex nonlinear
phenomena particularly in the presence of uncertainty. In OBS approach, data-
driven techniques are used, wherein training of a model is performed using
data collected from prior experiments. In the absence of physical models,
these non-parametric methods are useful for successful damage
characterization. Data-driven techniques useful for such problems include
principal component analysis (PCA) [19], singular value decomposition (SVD)
[20] and independent component analysis (ICA) [21]. Among OBS techniques, data
driven probabilistic approaches are found to be most effective [22].
Abdeljaber et al. used convolutional neural network (CNN) for vibration-based
structural damage detection and localization [23]. Khan et al. demonstrated
CNN classifier for analyzing delaminations in composite structures wherein
CNNs are trained with 2-dimensional spectrograms obtained from time domain
vibration signal using Short Time Fourier Transform [24]. Bao et al. trained
deep neural network using gray scale images obtained using time-series
vibration signals [25]. CNNs were trained using stacked autoencoders and
greedy layer-wise training. To demonstrate the applicability of this
architecture, data from a long-span cable-stayed bridge in China was used. In
another work, Aria et al. applied deep learning to predict damage size and
remaining life of the structure using acoustic emission measurements [26].
Giannakeas et al. demonstrated a method to capture impact of EOC variation on
Lamb waves in composites through the Bayesian calibration of a finite element
(FE) model using experimental observations and compute the model assisted
probability of detection (MAPOD) [27].

While deep learning based algorithms for automated feature extraction have
been reported in vibration based SHM system, there are limited reports for GW-
SHM. Xu et al. extracted features from GW signals and these features were
processed by a CNN designed and trained to further extract high-level features
and implement feature fusion for crack evaluation [28]. Pandey et al. have
demonstrated a local interpretable model-agnostic explanations (LIME) tool for
1D CNN model to study damages (simulated as well as experimental) in aluminum
plates [29]. Rautela et al. recently presented extensive performance
evaluation of several deep learning approaches assisted with reduced-order
spectral finite element models for damage characterization [30]. Ren et al.
proposed Gaussian Mixture Model (GMM) based model for compensation of time
varying temperature [31, 32, 33]. This involves direct extraction of features
using time-domain signal recordings, without performing additional transforms.
Another example of OBS method for non-uniform temperature gradient on
experimental data acquired using sensor array placed on aluminum plate was
presented by Sun et al. [34]. Wang et al. explored the effect of such coupling
of two EOC parameters, namely loading and temperature, on GW characteristics
[35]. They utilized compensation strategy based on matching pursuit algorithm
wherein features calculated using amplitude and group velocity are used for
training the model. Khurjekar et al. have reported deep learning models for
feature extraction and damage localization in GW-SHM system wherein the
variations in environmental factors are modeled as uncertainties in the data
[36], albeit without any estimate of damage intensity. Liu et al. used
convolutional neural network based framework to represent damage
characterization problem as image classification problem [37]. There have been
limited reports of using deep learning approaches for damage characterization,
albeit in absence of variation in EOCs [38, 37]. It emerges that research on
application of deep neural networks for EOC compensation has been limited and
has extensive scope for performance evaluation and optimization.

Figure 1: Illustration of the proposed method using 1D-CNNs for feature
extraction and GMMs for temperature compensated damage localization in GW-SHM
systems.

Conventional algorithms for damage localization use manually computed features
based on time and/or frequency domain representation of the signals, and
physics-based analysis of impact of damage and EOCs on signal properties (e.g.
amplitude, phase, frequency etc.) of GW modes. In this work, we propose a
novel method for temperature compensated damage classification and
localization using ultrasonic GWs. The method utilizes CNNs to learn the
features of GW signals acquired on healthy and damaged structures, and to
classify the type of damage. The self-learnt features are then analyzed using
GMMs for EOC compensation. The GMMs obtained using monitoring data are
compared against those obtained with baseline data and the Kullback-Leibler
(KL) divergence values of the GMMs are used as signal difference coefficient
(SDC) for damage index assessment and damage localization. The utility of this
method is validated using GW data from a network of six lead zirconate
titanate (PZT) sensors on an aluminum plate at various temperatures (from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$) generated using FE simulations,
with added white and pink noise of various signal to noise ratios (SNR:
$2\text{\,}\mathrm{d}\mathrm{B}$, $4.5\text{\,}\mathrm{d}\mathrm{B}$ and
$10\text{\,}\mathrm{d}\mathrm{B}$) and three types of damage (rivet hole,
added mass and notch). Typically GW-SHM systems are designed for SNR of the
order of $20\text{\,}$-$50\text{\,}\mathrm{d}\mathrm{B}$ [39]. To maintain
such high SNR, large values of actuation voltage are used for the transducers.
We demonstrate that the proposed method works efficiently even for much lower
values of SNR. Finally, we present experimental validation of this technique
for notch damage in an aluminum panel using a custom-designed field
programmable gate array (FPGA) based signal transduction and data acquisition
system, and demonstrate that the method presented in this work outperforms
conventional damage localization algorithms in presence of non-uniform and
varying temperature conditions. The stochastic variation in the performance of
the algorithm due to added noise is significantly reduced by employing the
proposed method as compared to earlier reported approach for temperature
compensation using manually computed features and GMM [31, 32]. For damage
localization, features are extracted in automated fashion, without specifying
any statistical relation or domain knowledge such as group velocity, choice of
wave mode etc. The method presented here thus holds great promise approach for
damage localization and characterization in GW-SHM systems prone to EOC
variations.

##  II CNN+GMM based temperature compensated damage localization method

The proposed method mainly consists of four steps as illustrated in Figure 1:

  * •

Training: Instead of manually computing features from time-domain signals, we
employ a 1-dimensional (1D) CNN for damage classification and feature
extraction using time-domain GW signals. GW signals are recorded for various
sensor paths, denoted as pair of transmitter $i$ and receiver $j$. A CNN is
set up for each path, and trained using data for baseline (healthy) and
damaged conditions.

  * •

Feature extraction & damage classification: Once trained, the penultimate
layer of the trained CNN is used for automated feature extraction and damage
classification. This feature extraction layer is used to extract features from
monitoring data obtained from the structure under maintenance. These features
are used for damage classification by assigning labels to training data, as
well as damage localization as described in the following steps. Note that
several recordings are used for each sensor path in the monitoring stage, to
emulate data obtained under various EOCs over various times.

  * •

EOC compensation: Next, the features computed in the previous step are used to
form clusters using GMMs. This step is crucial for compensating the effect of
temperature and other uncertainties. The GMM for each path formed in
monitoring stage is compared against the corresponding GMM set up with
baseline data. The deviation of the monitoring GMM from baseline GMM is
quantified using the KL divergence (i.e. relative entropy of the two
probability distributions).

  * •

Damage localization: KL divergence of each path $i-j$ is used as the
corresponding SDC to compute the damage index and perform damage localization
using conventional GW SHM damage assessment method [40]. For a sensor network
comprising of $N$ sensors, we can obtain $N\times(N-1)$ SDC values considering
all possible sensor pairs, however not all paths may be necessary.

Figure 2: Illustration of CNN architecture.

###  II.1 CNN based feature extraction

Typical CNN architecture consists of convolutional layers, activation
function, pooling layers and a fully connected layer. Convolution operation
for 1D time-series data between neuron $i$ of layer $l-1$ and neuron $k$ of
layer $l$ is defined as follows (Figure 2):

$x_{k}^{l}=b_{k}^{l}+\sum conv(w_{ik}^{l-1},s_{i}^{l-1})$ (1)

Here, $conv$ represents the convolution operation, $w_{ik}$ is the weight
corresponding to the connection between neuron $i$ of layer $l-1$ and neuron
$k$ of layer $l$, and $b_{k}^{l}$ is the bias of layer $l$. The result of this
convolution operation $x_{k}^{l}$ is then processed by a nonlinear activation
function $f$. Typical choices of nonlinear activation functions include
rectified linear activation function (ReLU), sigmoid, tanh etc. The
intermediate output $y_{k}^{l}$ of this activation function is expressed as:

$y_{k}^{l}=f(x_{k}^{l})$ (2)

Next, pooling is performed to reduce dimensionality in order to reduce the
number of trainable parameters and computational cost in the network, thereby
controlling over-fitting. Popularly used pooling method is maxpooling wherein,
the highest-valued feature in a specified span is retained while other
features in the specified span are discarded. Other pooling methods include
average pooling or $L2-$norm pooling. All neurons in the fully connected layer
$l$ have connections to activation function outputs from the preceding layer
$l-1$. The set of features of layer $l$ obtained through these operations is
defined as:

$s_{k}^{l}=y_{k}^{l}\downarrow pf$ (3)

Here, $pf$ represents down-sampling factor used in the pooling operation. The
feature set of the final layer of the CNN is used for classification, using
the nonlinear sigmoid function to compute the probabilities of the input
belonging to various classes (e.g. healthy, or various types of damage such as
rivet hole, added mass, notch etc.) Equations (1)-(3) describe the forward
pass in CNN training. During back propagation step of training CNNs, a cost
function is computed using estimated targets and true targets. The network is
then iteratively optimised using stochastic gradient descent to obtain optimal
values of weights and biases for given time-series dataset (inputs). The
training is deemed complete when the cost function saturates, or reduces below
a pre-determined threshold. Once training is completed, these optimised
weights and biases are used directly for feature extraction, without having to
manually compute features using physics-based methods. The penultimate (fully
connected) layer of the trained CNN computes features as defined in equation
(3), and is therefore used as the feature extractor layer shown in Figure 1.
Suppose an $n-$dimensional time-series is given as input to the CNN, we obtain
an $m-$dimensional feature vector at the output of the feature extractor layer
($m<<n$). These features are used as inputs for the next step of the method
i.e. clustering using GMM for EOC compensation.

###  II.2 GMM based EOC compensation

GMM is soft probabilistic clustering model that describes the membership of
sample to a set of clusters using a mixture of Gaussian densities [41]. The
probabilistic nature of GMM makes it a suitable choice for compensating
uncertainties introduced by variations in EOCs. Unlike hard clustering, this
method assigns probabilities to a sample belonging to a finite number of
Gaussian distributions with unknown parameters. The optimum number of Gaussian
components $k$ is determined using silhouette criterion. The silhouette score
of the GMM takes into account following two components:

  * •

Mean distance between a sample and all other samples in the same cluster

  * •

Mean distance between a sample and all other samples in neighbouring cluster

The silhouette score varies in the range $[-1,1]$. A larger value of
silhouette score indicates well defined clusters. Suppose for baseline data
(GMM with $k$ clusters), the Gaussian components are represented as:

$p(x)=\sum_{i=1}^{i=k}\pi_{i}\mathcal{N}(x|\mu_{i},{\sigma_{i}})$ (4)

Here, the model parameters $\mu_{i}$, $\sigma_{i}$ and $\pi_{i}$ denote the
mean, standard deviation and weight of $i^{th}$ Gaussian component. These
parameters are obtained using expectation maximisation (EM) algorithm. The
probability distribution function (PDF) $p(x)$ for feature $x$ captures the
probabilistic nature of baseline time-series data obtained under various EOCs,
and represented by features obtained from the CNN. Similarly, the PDF of
monitoring data $q(x)$ is also obtained. After obtaining PDFs for both
baseline and monitoring data, we need to quantify their mutual difference
using an appropriate statistical relation. This is calculated using KL
divergence [42], defined as:

$KL(p||q)=-\sum_{x}x*p(x)*\log\left[\frac{q(x)}{p(x)}\right]$ (5)

Intuitively, if the probability of an event $x$ from distribution $p$ is
significantly different from that of one from distribution $q$, then the
resulting KL divergence magnitude will be large. In case of SHM application,
if the monitoring data deviates substantially from baseline data (due to
damage), the resulting KL divergence for that sensor path will be large.
Therefore, the KL divergence value is used as SDC for computing damage index
(DI) and performing damage localization.

###  II.3 Damage localization algorithm

This subsection presents a brief overview of the SDC based DI map computation
originally presented by Sikdar et al. [40]. For the arrangement shown in step
4 of Figure 1, $(x_{i},y_{i})$ and $(x_{j},y_{j})$ denote Cartesian
coordinates of transmitter $(i)$ and receiver $(j)$, respectively. The
distance between the transmitter and receiver is denoted as $d_{(i,j)}$. The
aluminum panel is represented by a $2-$dimensional grid, and at each grid
point $(x,y)$, the pixel value for DI map is computed by scaling the ellipse
$A_{i,j}(x,y)$ with focii as the transmitter $(i)$ and receiver $(j)$, and
corresponding SDC coefficient $SDC(i,j)$:

$DI(x,y)=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}SDC(i,j)\left[\frac{\beta-
A_{i,j}(x,y)}{\beta-1}\right]$ (6)

where $\beta$ is an empirically determined scaling parameter independent of
velocity of the propagating wave. The elliptical contour shaped spatial
distribution function with non-negative values, $A_{i,j}(x,y)$ is defined as:

$A_{i,j}(x,y)=\begin{cases}P_{(i,j)}(x,y),&P_{(i,j)}(x,y)<\beta\\\
\beta,&P_{(i,j)}(x,y)\geq\beta\end{cases}$ (7)

where

$P_{(i,j)}(x,y)=\\\
\frac{\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}}+\sqrt{(x-x_{j})^{2}+(y-y_{j})^{2}}}{d_{(i,j)}}$
(8)

The composite DI map is obtained by overlaying ellipses $A_{i,j}(x,y)$ for
various transmitter-receiver $(i-j)$ paths. Intersections of paths with damage
present in-line generate pixels with high intensity in the DI map, and
therefore are useful for damage localization. Higher contrast between values
of SDCs for damaged paths and undamaged paths yields better localization
results in the DI map.

##  III Generation of temperature-affected guided wave dataset for simulation
study

Propagation of Lamb waves in a plate depends on the thickness, density and
elastic material properties of the structure. For an isotropic plate of
thickness $h$, the Lamb wave dispersion equations are specified as
follows[43]:

  * •

Symmetric (S) mode:

$\frac{\tanh{\frac{\beta h}{2}}}{\tanh{\frac{\alpha
h}{2}}}=\frac{4k^{2}\alpha\beta}{\left(k^{2}+\beta^{2}\right)^{2}}$ (9)

  * •

Anti-symmetric (A) mode:

$\frac{\tanh{\frac{\beta h}{2}}}{\tanh{\frac{\alpha
h}{2}}}=\frac{\left(k^{2}+\beta^{2}\right)^{2}}{4k^{2}\alpha\beta}$ (10)

where $\alpha=\sqrt{k^{2}-\frac{\omega^{2}}{V^{2}_{P}}}$ and
$\beta=\sqrt{k^{2}-\frac{\omega^{2}}{V^{2}_{S}}}$, $k=\frac{\omega}{c_{p}}$ is
the wave number, $\omega$ is the angular frequency of the propagating wave
with a phase velocity $c_{p}$. $V_{P}$ and $V_{S}$ are the velocities of the
pressure (P)-wave and shear (S)-wave respectively, expressed as [43]:

$V_{P}=\sqrt{\frac{\lambda+2\mu}{\rho}}$ (11) $V_{S}=\sqrt{\frac{\mu}{\rho}}$
(12)

The terms $\lambda$ and $\mu$ are Lame’s constants, expressed as:

$\lambda=\frac{\nu E}{\left(1+\nu\right)\left(1-2\nu\right)}$ (13)
$\mu=\frac{E}{2\left(1+\nu\right)}$ (14)

The terms $E$, $\nu$ & $\rho$ denote the Young’s modulus of elasticity,
Poisson’s ratio and density of the material, respectively. The group velocity
$c_{g}$ of the wave modes can be expressed as [43]:

$c_{g}=c_{p}+k\frac{dc_{p}}{dk}$ (15)

Using equations (9) and (10) and wave number $k$, the phase velocity $c_{p}$
and group velocity $c_{g}$ can be calculated for symmetric and anti-symmetric
modes in a plate of a specified material ($E,\nu$ and $\rho$), thickness $h$
at frequency $f=\frac{\omega}{2\pi}$. Figure 3 shows phase velocity and group
velocity dispersion curves for a $2\text{\,}\mathrm{m}\mathrm{m}$ thick
aluminum 6061-T6 alloy plate at room temperature
($20\text{\,}\mathrm{\SIUnitSymbolCelsius}$). For our analysis,
$150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ is chosen as the excitation
frequency so that the fundamental symmetric mode ($S0$) and anti-symmetric
mode ($A0$) can be easily separated. The variation of material properties of
Al 6061-T6 with temperature is shown in Figure 4. The material properties are
obtained from COMSOL Multiphysics 5.5 material library. The variation of phase
velocity and group velocity with temperature ranging from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ are shown in Figure 5(a) and (b),
respectively.

Figure 3: Dispersion curves of (a) phase velocity, and (b) group velocity for
various GW modes in $2\text{\,}\mathrm{m}\mathrm{m}$ thick Al 6061-T6 plate at
$20\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$. Figure 4: Variation of
(a) Young’s modulus, (b) Poisson’s ratio, and (c) density of Al 6061-T6 with
temperature varying from $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ Figure 5: Variation of (a) phase
velocity, and (b) group velocity for $S0$ and $A0$ modes with temperature
varying from $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

The GW propagation in a $530\text{\,}\mathrm{m}\mathrm{m}$ $\times$
$530\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $2\text{\,}\mathrm{m}\mathrm{m}$
Al 6061-T6 alloy plate with surface mounted
$20\text{\times}20\text{\times}0.4\text{\,}\mathrm{m}\mathrm{m}$ PZT-5H
transducers was studied using FE simulations performed with COMSOL
Multiphysics 5.5, wherein the isotropic aluminum plate and PZT transducers
were modeled using Structural Mechanics Module coupled with Piezoelectric
Solid Interaction physics and analyzed using Time Implicit Study. The accuracy
of FE simulation of wave propagation is primarily determined by the mesh size
and time increment used in the simulation [44]. For excitation frequency ($f$)
of $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, the wavelengths of the $S0$
and $A0$ modes in the Al plate are approximately
$36\text{\,}\mathrm{m}\mathrm{m}$ and $17\text{\,}\mathrm{m}\mathrm{m}$,
respectively. Previous research on Lamb wave propagation simulations
recommends the largest mesh element size to be smaller than $\frac{1}{6}$ of
the mode wavelength [45]. Thus, the largest and smallest mesh sizes in the Al
plate are set to $5\text{\,}\mathrm{m}\mathrm{m}$ and
$2\text{\,}\mathrm{m}\mathrm{m}$ respectively. Physics controlled Extra Fine
mesh setting was applied for the surface mounted PZT-5H transducers. The time
step was chosen to be $0.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
$(>\frac{1}{60f})$ . The transmitter PZT was actuated using a 5-cycle sine
pulse shaped by a Hanning window as shown in Figure 6(a) expressed as:

$V=0.5\left(1-\cos\left(\frac{2\pi ft}{5}\right)\right)\sin(2\pi ft)$ (16)

Figure 6: (a) Input signal to transmitter:
$150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ five-cycle sine pulse shaped by a
Hanning window. (b) Setup for studying impact of temperature on group velocity
of GW modes: $530\text{\,}\mathrm{m}\mathrm{m}$ $\times$
$530\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $2\text{\,}\mathrm{m}\mathrm{m}$
Al 6061-T6 plate with $20\text{\times}20\text{\,}\mathrm{m}\mathrm{m}$ PZT
transmitter (1) and receiver (2) spaced apart by
$300\text{\,}\mathrm{m}\mathrm{m}$ distance.

For validation of the impact of temperature on group velocity of the GW modes
obtained from the FE model, a baseline (damage free) Al 6061-T6 alloy plate of
dimensions $530\text{\,}\mathrm{m}\mathrm{m}$ $\times$
$530\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $2\text{\,}\mathrm{m}\mathrm{m}$
with two surface mounted PZT-5H transducers as shown in Figure 6(b) was
modeled. The 5-cycle sine pulse shaped by a Hanning window was applied at the
transmitter PZT (1) and the output signal was recorded at the receiver (2),
separated by a distance of $300\text{\,}\mathrm{m}\mathrm{m}$ from the
transmitter. Low reflecting edge boundary condition was imposed on the four
free edges of the plate to avoid reflections of the $S0$ & $A0$ modes from the
edges. The total simulation time was set to
$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ with a time step of
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. The total number of
elements in the FE model were $155\,614\text{\,}$, with $914\,717\text{\,}$
degrees of freedom. The computation time for each simulation obtained on a
$64\text{\,}$-bit Intel® Core™-i7-7700HQ
$2.80\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ CPU with
$16\text{\,}\mathrm{G}\mathrm{B}$ RAM workstation was
$5700\text{\,}\mathrm{s}$. The time of flight for each wave mode was measured
using continuous wavelet transform of the output signal measured at the
receiver PZT (with Morlet wavelet used as mother wavelet), as shown in Figure
7. Table 1 shows excellent agreement between the group velocities at
$20\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ obtained from FE
simulations at to those obtained from analytical calculations (equation (15)).

Figure 7: (a) Input signal (blue trace) at transmitter (PZT-1) and GW signal recorded at receiver (PZT-2), showing $S0$ and $A0$ modes. (b) Time of flight for each mode is measured by the peak to peak distance of the wavelet envelopes of the corresponding signal at receiver from the time instant corresponding to peak value of input signal envelope. Table 1: Comparison of FE measured group velocities with analytically computed values for $S0$ and $A0$ modes at $20\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ Wave mode | FEM result [$\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$] | Analytical [$\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$]
---|---|---
S0 | $5.128$ | $5.342$
A0 | $2.671$ | $2.586$

To evaluate the performance of the damage identification and localization
method, FE simulations were carried out on an aluminum 6061-T6 alloy plate of
dimensions $530\text{\,}\mathrm{m}\mathrm{m}$ $\times$
$530\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $2\text{\,}\mathrm{m}\mathrm{m}$
with $6$ PZT-5H transducers and three types of damages (rivet hole, added mass
and notch; parameters listed in Table 2) introduced in the lower half of the
plate, as shown in Figure 8. Each time-series had a duration of
$150\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, obtained with a sampling
rate of $10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ for seven separate
transmitter-receiver paths (P12, P15, P24, P25, P26, P53 and P56, where Pxy
denotes the time-series output obtained with PZT x as transmitter and PZT y as
receiver). The damage is located at the intersection of the diagonal paths in
the lower half of the plate i.e. paths P15 and P24. The FE model had
$211\,069\text{\,}$ elements and $1\,210\,935\text{\,}$ degrees of freedom.

Figure 8: Layout of Al 6061-T6 plate with network of six PZT transducers used for FE simulations. Three types of damages are considered (listed in Table 2). Table 2: Types and parameters of damages studied for evaluation of the proposed damage classification and localization algorithm. The one-hot encoding based class labels used for CNN classification are also listed. Damage | Parameters | Label
---|---|---
Baseline | - | $1000$
Rivet hole |  Diameter $20\text{\,}\mathrm{m}\mathrm{m}$ | $0100$
Added mass |  Diameter $23\text{\,}\mathrm{m}\mathrm{m}$, weight $20\text{\,}\mathrm{g}$ | $0010$
Notch |  $60\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $7\text{\,}\mathrm{m}\mathrm{m}$ | $0001$

The computation time for each simulation (for the mentioned seven paths)
obtained on a $64\text{\,}$-bit Intel® Core™-i7-7700HQ
$2.80\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ CPU with
$16\text{\,}\mathrm{G}\mathrm{B}$ RAM workstation was
$15\,290\text{\,}\mathrm{s}$. Considering the long time required for
performing simulations, the following method was used to generate datasets at
various temperatures. Group velocities for the symmetric and anti-symmetric
modes were obtained from analytical solutions of equations (9) and (10), using
the expression in equation (15). FE simulations were performed for healthy
(damage-free) and various damage conditions at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ temperature. The temperature-
dependent elastic properties (Figure 4) were used to determine group
velocities of $S0$ & $A0$ modes for temperatures ranging from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Generally, both $S0$ and $A0$
modes are commonly used for damage detection in isotropic plates [46].
However, $S0$ mode is faster and has much lower attenuation than $A0$ mode and
therefore can propagate longer distances [47, 46]. In our present study, we
focused on interaction of $S0$ mode with the damage. For estimating the signal
at temperature $T$, the GW signal obtained from FE simulation for
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for any transmitter-receiver path on
the panel was time-shifted by an amount $\Delta t_{T}$:

$\Delta t_{T}=\frac{d}{c_{gT}}-\frac{d}{c_{g0}}$ (17)

where $d$ is the distance between the transmitter-receiver pair under
consideration, $c_{gT}$ is the analytically calculated group velocity of $S0$
mode at temperature $T$, and $c_{g0}$ is the analytically calculated group
velocity of $S0$ mode at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The
variation of signal amplitude with temperature was computed using the
amplitude of $S0$ wave packet for temperature ranging from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ obtained analytically using
Dispersion calculator, DC software [48]. The maximum amplitude value of $S0$
wave packet was observed to vary linearly with increasing temperature. An
analytical relation between $A_{f}$ (ratio of maximum amplitude of $S0$ wave
packet at temperature $T$ to that at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$) and temperature $T$ (specified in
units of $\text{\,}\mathrm{\SIUnitSymbolCelsius}$) ranging from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ is obtained using MATLAB Curve
Fitting Toolbox:

$A_{f}=1+\frac{T~{}[$\text{\,}\mathrm{\SIUnitSymbolCelsius}$]}{3550}$ (18)

The GW signal $s_{T}(t)$ at temperature $T$ can thus be obtained:

$s_{T}(t)=A_{f}s_{0}(t+\Delta t_{T})$ (19)

where $s_{0}(t)$ is the signal at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$
This technique helps in reducing the computational time required for obtaining
GW signals for all sensor paths at various temperatures, as this only requires
time-consuming FE simulations at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$
temperature. The data obtained at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$
was used to generate data at other temperatures obtained using equation (19).
To verify the accuracy of this method, we evaluated the cross-correlation
coefficient between the signal obtained using equation (19) and that obtained
from FE simulations performed at $50\text{\,}\mathrm{\SIUnitSymbolCelsius}$
and $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The cross-correlation
coefficient was observed to be $>$$99\text{\,}\mathrm{\char 37\relax}$, and
hence this method is justified to reduce computational complexity.
Representative temperature-adjusted time-series for path P15 at two different
temperatures: $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$, are shown in Figure 9(a). To
incorporate uncertainties caused by variation in other EOCs and experimental
uncertainties, we also added white noise and pink noise to the data:

$s_{n,T}(t)=s_{T}(t)+\beta\max(s_{T}(t))\left[w(t)+p(t)\right]$ (20)

where $s_{n,T}(t)$ is the signal at temperature $T$ with added noise,
$\max(s_{T}(t))$ is the peak amplitude of signal $s_{T}(t)$, $\beta$ is the
noise scaling factor, and $w(n),p(n)$ are white noise and pink noise
respectively, generated using normal distribution with zero mean and standard
deviation of $0.1$. We studied the performance of the damage assessment method
for three different signal-to-noise-ratio (SNR) values:
$2\text{\,}\mathrm{d}\mathrm{B}$, $4.5\text{\,}\mathrm{d}\mathrm{B}$ and
$10\text{\,}\mathrm{d}\mathrm{B}$. Figure 9(b) shows signal with added noise
for $SNR=$ $2\text{\,}\mathrm{d}\mathrm{B}$ for sensor path P15 at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

Figure 9: (a) Effect of temperature on baseline GW data corresponding to
sensor path P15. (b) Signal with added white noise and pink noise with
$4.5\text{\,}\mathrm{d}\mathrm{B}$ SNR, illustrated for sensor path P15 at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

##  IV Simulation results and discussion

Figure 10: Illustration of steps involved in training and monitoring phases.
(a) The training phase involves data generation for training CNNs. It also
involves modeling the baseline GMM. (b) Monitoring phase involves feature
extraction from monitoring data. The feature extractor and baseline GMM are
imported from training phase for computing SDC. Figure 11: (a) Accuracy, and
(b) loss curves for training and testing of 1D CNN corresponding to sensor
path P15 for $200$ epochs. Figure 12: For GMM-based clustering, optimum number
of clusters are selected using silhouette score. Silhouette scores for various
number of clusters for path P15 are shown for (a) baseline data, and (b) rivet
hole data. Distributions formed using optimum number of clusters $(4)$ are
shown for two prominent features for (c) baseline data, and (d) rivet hole
data. The two most prominent features were determined out of $16$ features
computed by the feature extractor layer in the CNN using PCA. Figure 13:
Trends in SDCs computed using equation (5) as described in CNN+GMM method. For
each of seven sensor paths, $10$ values of SDCs are computed for $10$ test
cases. Three different defects are shown in $3$ columns (rivet hole, added
mass and notch). Three different signal to noise ratio (SNR) levels as shown
in: (a) $10\text{\,}\mathrm{d}\mathrm{B}$, (b)
$4.5\text{\,}\mathrm{d}\mathrm{B}$, and (c) $2\text{\,}\mathrm{d}\mathrm{B}$.
Figure 14: DI maps obtained using SDC based damage localization algorithm.
Mean value of SDCs across $10$ test cases shown in Figure 13 was used to
represent SDC of each sensor path. Three different defects are shown in $3$
columns (rivet hole, added mass and notch). Three different signal to noise
ratio (SNR) levels as shown in: (a) $10\text{\,}\mathrm{d}\mathrm{B}$, (b)
$4.5\text{\,}\mathrm{d}\mathrm{B}$, and (c) $2\text{\,}\mathrm{d}\mathrm{B}$.

To demonstrate and study the performance of the temperature compensated damage
localization technique, the implementation was carried out in two phases:
training phase and monitoring phase, as shown in Figure 10. The training phase
involves data generation and training the CNNs (one CNN per path). The
temperature affected GW data set with added noise described in the previous
section was used for this purpose. The CNNs were trained using data
corresponding to healthy i.e. undamaged (baseline) condition, as well as
damaged conditions (rivet hole, added mass and notch). The penultimate layer
of the trained CNN was then imported as feature extractor. Using the feature
extractor, features for baseline data were computed and used for creating
baseline GMM. These two modules, feature extractor and baseline GMM were
imported as-is in the next phase i.e. monitoring phase. The monitoring phase
was designed such that it is readily deployable for real-time SHM. In this
phase, GW data is collected from the structure being monitored. Using the
feature extractor imported from training phase, features were computed for
monitoring data which were then used to form monitoring GMM. Finally, the KL
divergence between baseline GMM and monitoring GMM was computed. The values of
KL divergence computed for each path are used as SDC for damage localization
algorithm.

###  IV.1 Performance analysis of CNN+GMM based method for temperature
compensated damage localization

The time-series dataset constructed using method summarised in equation (20)
contained $200\times 11\times 4\times 3=$ $26\,400\text{\,}\mathrm{t}$ime-
series for each path ($200\text{\,}$ series generated for each of the
$11\text{\,}$ different temperature values:
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$10\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ $\dotsc$
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$, for each of the $4\text{\,}$
classes: baseline, rivet hole, added mass, notch) and $3\text{\,}$ different
SNR values: $2\text{\,}\mathrm{d}\mathrm{B}$,
$4.5\text{\,}\mathrm{d}\mathrm{B}$ and $10\text{\,}\mathrm{d}\mathrm{B}$). Out
of these $26\,400\text{\,}$ time-series, $13\,200\text{\,}$ were used for
training the CNNs and the remaining series were divided into $10\text{\,}$
sets that were used for performance analysis in monitoring phase. The training
dataset ($4400\text{\,}$ series for each SNR for each path) was split in the
proportion $75:20:5~{}(\%)$ resulting in $3300\text{\,}$ for training,
$88\text{\,}$ for testing, and $220\text{\,}$ for validation of the CNNs. Each
class was associated with a label generated with one-hot encoding, as listed
in Table 2. For each 1D CNN, learning rate of
$1\text{\times}{10}^{-7}\text{\,}$ and batch size of $64$ were used. Binary
cross-entropy loss was used as the cost function and Adam optimization scheme
was used to enable momentum and adaptive learning rate in the training
process. The networks were trained for $200$ epochs. The CNNs were trained on
a $64\text{\,}$-bit Intel® Core™-i7-10750H
$2.60\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ CPU workstation with
$16\text{\,}\mathrm{G}\mathrm{B}$ RAM. The models were developed in Python
using TensorFlow library and Keras environment. The architecture of each 1D
CNN is shown in Table 3.

Table 3: Summary of CNN architecture for binary classification of FE generated temperature and noise affected data Layer (type) |  Output shape |  Number of parameters
---|---|---
Conv1D |  $1499\times 16$ |  $64$
MaxPooling1D |  $749\times 16$ |  $0$
Dropout |  $749\times 16$ |  $0$
Flatten |  $11984$ |  $0$
Dense (ReLU) |  $16$ |  $191760$
Dense (Sigmoid) |  $4$ |  $68$
Total trainable parameters |  $191892$

Representative accuracy and loss curves (for $200$ epochs for path P15) are
shown in Figure 11. The training loss as well as testing loss decrease
smoothly with increasing number of epochs, thus indicating that the model is
not over-fitting to the data. For over-fit models, the training error is
extremely low while testing error increases sharply as the model tries to
over-fit to noise present in the training samples. Perfect classification
accuracy $(=1)$ was observed, which is not surprising for a homogeneous
material such as aluminum. The penultimate layer of the CNN, used as feature
extractor, yields $16\text{\,}$ features for each time-series consisting of
$1500\text{\,}$ samples. These features were then processed by GMMs. Firstly,
a baseline GMM for each path was created using features for $110\text{\,}$
time-series chosen at random across the $11\text{\,}$ different temperatures
for each SNR. In monitoring phase, we evaluated the performance of the method
using $10\text{\,}$ different test cases using time-series data that were not
used for training. Each of these $10\text{\,}$ test cases also contained
$110\text{\,}$ time-series. Using the feature extractor layer, we obtained
feature set of size $16\times 110$ that was used for creating monitoring GMM.
To determine the optimum number of clusters in the GMM, silhouette score was
computed by varying number of clusters for baseline features and monitoring
(rivet hole) features as illustrated for path P15 in Figure 12(a) and (b),
respectively. In both cases, the value of silhouette score dropped after
beyond $4$ clusters, therefore, for both GMMs, four clusters were formed as
shown in in Figure 12(c) and (d). Next, KL divergence value was computed
between baseline GMM and monitoring GMM using equation (5). The statistical
distribution of values of KL-divergence i.e. SDC for $10$ test cases and seven
sensor paths are shown in Figure 13 for the three different SNR values. Both
sensor paths containing the defect in-line (P15 and P24) show higher values of
KL divergence compared to other five sensor paths which do not contain the
defect. The corresponding DI maps are shown in Figure 14 for all SNR values.
The DI maps are constructed using the mean value of SDC (from $10\text{\,}$
test cases) for each path. The distinction between SDC values for paths
containing the damage (P15 and P24) and paths without damage is better
observed for higher SNRs, however the DI maps accurately highlight the
location of the damage even for $SNR=$$2\text{\,}\mathrm{d}\mathrm{B}$. Note
that $2\text{\,}\mathrm{d}\mathrm{B}$ SNR is unusually low in conventional GW
SHM systems, however the proposed method is effective even with such low SNR.
We observe that the distinction in SDC values for damaged and undamaged paths
was more apparent for rivet hole and added mass, as compared to notch defect.
This was attributed to the edge reflections in undamaged paths caused by the
notch (dimension $60\text{\,}\mathrm{m}\mathrm{m}$ $\times$
$7\text{\,}\mathrm{m}\mathrm{m}$). As shown in Figure 15, FE simulated GW
signals for path P56 (that does not contain the damage) corresponding to
baseline, rivet hole and added mass are very similar to each other
(overlapping curves), whereas significant phase shift was observed for notch.

Figure 15: GW signal corresponding to baseline, rivet hole, added mass and
notch for sensor path P56 at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$. No
significant difference from baseline is noticed for rivet hole and added mass
cases. The signal obtained for notch damage shows significant difference as
compared to the other signals. Figure 16: Denoising of noisy signal using
moving average filter with kernel size $=3$ for GW signal corresponding to
sensor path P15 at $0\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Figure 17:
Trends in SDCs computed using manually computed features. For each of seven
sensor paths, $10$ values of SDCs are computed for $10$ test cases for SNR =
$4.5\text{\,}\mathrm{d}\mathrm{B}$. Three different defects are shown in $3$
columns (rivet hole, added mass and notch). SDCs are computed using computed
using (a) KL divergence of GMMs modeled using physics-defined features
$DI_{1}$ and $DI_{2}$ (equations (21) and (22)), and (b) conventional SDC
definition using CCD metric (equation (23)) applied to signals denoised using
moving average kernel of size $3$. Figure 18:  DI maps obtained using SDC
based damage localization algorithm. Mean value of SDCs across $10$ test cases
shown in Figure 17 was used to represent SDC of each sensor path. Three
different defects are shown in $3$ columns (rivet hole, added mass and notch).
SDCs are computed using computed using (a) KL divergence of GMMs modeled using
physics-defined features $DI_{1}$ and $DI_{2}$ (equations (21) and (22)), and
(b) conventional SDC definition using CCD metric (equation (23)) applied to
signals denoised using moving average kernel of size $3$.

###  IV.2 Comparison of CNN+GMM method vs. GMM using physics-defined features

To evaluate the efficacy of the CNN+GMM method, we compared the performance
reported above to the method proposed by Ren et al. [31], that uses 2 features
($DI_{1}$ and $DI_{2}$) and GMM for EOC compensation:

$DI_{1}=1-\sqrt{\frac{\\{\int_{t_{0}}^{t_{1}}S_{b}(t)S_{m}(t)dt\\}^{2}}{\\{\int_{t_{0}}^{t_{1}}{S_{b}(t)}^{2}dt\int_{t_{0}}^{t_{1}}{S_{m}(t)}^{2}dt\\}}}$
(21)

$DI_{2}=\sqrt{\frac{\int_{f_{0}}^{f_{1}}\\{|S_{b}(f)|-|S_{m}(f)|\\}^{2}df}{\int_{f_{0}}^{f_{1}}|S_{b}(f)|^{2}df\int_{f_{0}}^{f_{1}}|S_{m}(f)|^{2}df}}$
(22)

Feature $DI_{1}$ quantifies the correlation coefficient deviation between
baseline signal $S_{b}(t)$ and monitoring signal $S_{m}(t)$, whereas feature
$DI_{2}$ quantifies the difference in the Fourier spectra of baseline
($S_{b}(f)$) and monitoring ($S_{m}(f)$) signals, normalized to their
respective energies. Here, $t_{0}$ is time of arrival for the $S0$ mode and
$t_{1}=t_{0}+t_{w}$ is the exit time, where $t_{w}=$
$33.3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ is pulse width of
actuation pulse (5-cycle sine wave with Hanning window in this case). Table 4
lists the values of $t_{0}$ and $t_{1}$ for various sensor paths. The values
of $f_{0}$ and $f_{1}$ were set to
$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and
$200\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ respectively, given that the
pulse actuation frequency is $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

Table 4: Time of arrival $(t_{0})$ and exit $(t_{1})$ of $S0$ mode for various sensor paths. Sensor Path |  $t_{0}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$] |  $t_{1}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$]
---|---|---
Vertical (P12, P56) | 28 | 61.3
Horizontal (P25) | 55.9 | 89.3
Diagonal (P15, P24, P26, P53) | 62.5 | 95.9

Using equations (21) and (22), we obtained feature vector of size $2\times
110$ for each of the test case datasets. The same test case datasets reported
above were used for this analysis. Using these features, we created two GMMs
for each path (baseline and monitoring, for each damage) and computed their KL
divergence. The trends in KL divergence values for $10$ test cases for $SNR=$
$4.5\text{\,}\mathrm{d}\mathrm{B}$ are illustrated in the top row in Figure
17. The distinction between damaged and undamaged paths is not as pronounced
for the CNN+GMM method (Figure 13). Figure 18(a) shows that for rivet hole and
added mass, the damage location was correctly identified in the DI map,
whereas large error was observed for notch. Thus, using GMMs in conjunction
with manually computed physics-defined features ($DI_{1}$ and $DI_{2}$) is not
as effective as the CNN+GMM method for countering EOC variations.

###  IV.3 Comparison of CNN+GMM method vs. conventional SDC algorithm using
denoised signals

Conventional SDC computation algorithms work extremely well in well-
controlled, laboratory environments where the noise levels are very low and
uniform temperature is maintained. Since the CNN used in the CNN+GMM method
involves filtering the input time-series signal, in this section we analyze
the performance of conventional SDC computation algorithm on denoised input
signals. The input signals with added noise and temperature variation
(equation (20)) are denoised (i.e. filtered) using moving average kernel of
size $3$. The filtering operation is similar to the one performed by the CNN.
Figure 16 illustrates a representative denoised time-domain signal for path
P15. The denoised signals are then used for SDC computation using the
conventional correlation coefficient deviation (CCD) definition [40]. Let
$S_{d}$ denote the signal received at receiver $(j)$, and $S_{b}$ denote the
baseline signal. Let $t_{a}$ denote the time of arrival for the desired GW
mode in the signal and let $t_{BW}$ denote the width of the time window
corresponding to the wave mode (i.e. time bandwidth) of the desired GW mode.
The SDC coefficient for sensor path $i-j$ is defined as measure of deviation
of received signal $S_{d}$ from baseline signal $S_{b}$ and expressed as CCD:

$SDC_{CCD}{(i,j)}=1-CC(i,j)$ (23) $CC{(i,j)}=\\\
\frac{1}{\sigma_{b}\sigma_{d}}\frac{\sum_{t_{a}}^{t_{a}+t_{BW}}(S_{b}-\mu_{b})(S_{d}-\mu_{d})}{\sum_{t_{a}}^{t_{a}+t_{BW}}\sqrt{(S_{b}-\mu_{b})^{2}}\sum_{t_{a}}^{t_{a}+t_{BW}}\sqrt{(S_{d}-\mu_{d})^{2}}}$
(24)

Here, $\mu_{b},\mu_{d}$ are mean values and $\sigma_{b},\sigma_{d}$ are
standard deviation values of the samples in baseline signal $S_{b}$ and
received signal $S_{d}$, respectively. The trends in SDC using equation (23)
computed for the same $10$ test cases for SNR =
$4.5\text{\,}\mathrm{d}\mathrm{B}$ are shown in the bottom row of Figure 17,
and corresponding DI maps computed using the mean values of $SDC_{CCD}$ are
shown in Figure 18(b). This conventional method produces large errors and is
thus entirely ineffective in countering EOC variations, thus validating the
impact of the novel CNN+GMM method proposed in this work. It is important to
note that the range of amplitudes of SDCs may vary depending on their
definition (CCD, KL-divergence etc.) However, the relative difference in the
values of SDCs for damaged paths and undamaged paths determines the accuracy
of damage localization. Higher contrast between values of SDCs for damaged
paths and undamaged paths yields better localization results. The DI maps are
normalized to the pixel with the highest DI value i.e. the highest intensity
in DI maps shown in Figures 14 and 18 is $1$.

##  V Experimental validation

###  V.1 Experimental setup

A portable embedded system was developed for transduction of guided wave
ultrasonic signals and data acquisition and storage. The system is based on a
low-cost field programmable gate array (FPGA) device (Xilinx Artix®-7
XC7A15T-1CPG236C, Digilent Cmod A7-15T module), and has provision for
connecting $8$ PZT channels. The FPGA is interfaced with a $12\text{\,}$ bit,
parallel input, multiplying digital to analog converter (DAC) - Texas
Instruments DAC7821, to generate the 5-cycle Hanning pulse actuation voltage
centered at $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. Since the DAC7821 is
an unbuffered current output DAC, a transimpedance amplifier constructed with
Texas Instruments TL072 operational amplifier IC is connected to the output of
the DAC, resulting in peak-to-peak amplitude of $4.8\text{\,}\mathrm{V}$ for
the 5-cycle Hanning pulse. The receiver PZT output is amplified using Texas
Instruments INA128 instrumentation amplifier and digitized using Maxim
Integrated MAX-1426 analog to digital converter (ADC) with $10\text{\,}$ bit
resolution and $10\text{\,}\mathrm{M}\mathrm{s}\mathrm{p}\mathrm{s}$ sampling
rate. The system is implemented on a printed circuit board (PCB), with the $8$
channels connected to the ADC and DAC using $8-$position switch banks. The
system is capable of storing $4096\text{\,}$ samples obtained at
$10\text{\,}\mathrm{M}\mathrm{s}\mathrm{p}\mathrm{s}$ i.e.
$409.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for each channel in the
memory (Block RAM) in the FPGA. A photograph of the PCB is shown in Figure 19.

An aluminum panel of dimensions
$530\text{\times}530\text{\times}2\text{\,}\mathrm{m}\mathrm{m}$ was used as
test structure in the experiment. Four PZT-5H transducers of dimensions
$20\text{\times}20\text{\times}0.4\text{\,}\mathrm{m}\mathrm{m}$ were attached
on the panel with center-to-center separation of
$90\text{\,}\mathrm{m}\mathrm{m}$ in horizontal and vertical directions, as
shown in the layout in Figure 20. Baseline data for all sensor paths were
recorded at room temperature on the panel without any damage. Next, $8$ data
sets were collected by applying heat to a portion of the panel using a heat
source (hair dryer), as shown in Figure 21(a). For each data set, the heat
source was held atop an arbitrarily chosen location on the panel for up to $2$
minutes, after which the heat source was removed and the data for all paths
were recorded sequentially (Figure 21(b) shows indicative data for path P34
corresponding to heat source position in Figure 21(a)). Next, a notch defect
of dimensions $10\text{\times}3\text{\,}\mathrm{m}\mathrm{m}$ was introduced
in the panel using a handheld electric drill (position shown in Figure 20,
photograph of defect shown in Figure 21(c)). After introduction of the notch
defect, $9$ data sets were collected - one at room temperature, and $8$ data
sets for various non-uniform temperature profiles using the heat source.
Figure 22 shows indicative photographs of the panel obtained with FLIR E60
thermal imaging camera during and after application of heat.

Figure 19: Photograph of the PCB embedded platform for transduction and data
acquisition used for performing experiments. Figure 20: Layout of
$2\text{\,}\mathrm{m}\mathrm{m}$ thick aluminum panel with 4 PZT transducers
used for experimental testing. A notch defect of dimensions
$10\text{\times}3\text{\,}\mathrm{m}\mathrm{m}$ was introduced in the panel.
Figure 21: (a) Photograph of aluminum panel (without damage) instrumented with
PZT transducers connected to PCB. A hair dryer was used to heat a portion of
the panel (example, above path P34 as shown) to induce non-uniform temperature
profile on the panel. (b) Indicative temperature affected data for panel with
notch defect at room temperature and in presence of non-uniform temperature
variation for path P34. (c) Photograph of notch defect introduced in the
panel. Figure 22: Photographs of panel captured with thermal imaging camera.
(a) Highest temperature reached during application of heat with hair dryer is
approximately $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$. (b) Non-uniform
heating of the panel is noticed in the image captured few seconds after
removal of heat source. The heat source was present atop the location
identified by the marker in the middle of the image.

###  V.2 Experimental results and discussion

Data for four sensor paths (P12, P13, P14, P24, for sensor positions as marked
in Figure 20) was considered for analysis. The data thus recorded were
denoised using $5^{th}$ and $6^{th}$ coefficients of $7-$level wavelet
decomposition, using Daubechies 10 (db10) as mother wavelet. This is essential
to remove electronic offset from the circuit components on the PCB. Next, each
of the $9$ data sets collected for undamaged panel and panel with notch defect
was used to generate noise-augmented data with
$4.5\text{\,}\mathrm{d}\mathrm{B}$ SNR (in a manner similar to equation (20)).
For each class (undamaged and notch), a total of $900$ noise-augmented time
series were generated per path for training CNN for binary classification, and
additional $90$ time series per path were generated for testing (monitoring).
Table 5 shows an overview of the CNN model used for analyzing the experimental
data. The model was trained for $200$ epochs, and produced perfect
($100\text{\,}\%$) accuracy with test data sets. Since the sensors are placed
closer to each other ($90\text{\,}\mathrm{m}\mathrm{m}$ separation along
horizontal and vertical paths) on the panel used in the experiment, as
compared to the FE generated data ($150\text{\,}\mathrm{m}\mathrm{m}$ and
$300\text{\,}\mathrm{m}\mathrm{m}$ separation between adjacent sensors along
vertical and horizontal paths respectively), we use the first
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ portion of each time
series, that contain the $S0$ mode. Accuracy curves (for $200$ epochs for path
P24) are shown in Figure 23.

Table 5: Summary of CNN architecture for binary classification of experimental data Layer (type) |  Output shape |  Number of parameters
---|---|---
Conv1D |  $997\times 16$ |  $64$
MaxPooling1D |  $498\times 16$ |  $0$
Dropout |  $498\times 16$ |  $0$
Flatten |  $7968$ |  $0$
Dense (ReLU) |  $16$ |  $127504$
Dense (Sigmoid) |  $2$ |  $34$
Total trainable parameters |  $127602$

Out of the $16$ features extracted from the penultimate layer of the CNN, the
$2$ most prominent features (determined using PCA) were used to construct
GMMs. The optimum number of clusters was determined to be $3$, based on the
silhouette score (Figure 24). KL divergence between the two GMMs was
calculated as described in equation (5). To illustrate the performance of
CNN+GMM method for experimental data, comparative analysis was conducted using
GMM with physics-defined features $DI_{1}$ and $DI_{2}$ as well as
conventional SDC algorithm (CCD) with denoised signals as explained in
subsections IV.2 and IV.3, respectively. For $90$ test samples, values of
$DI_{1}$ and $DI_{2}$ were computed for baseline and notch data sets. The time
of arrival and exit time for various sensor paths used for $DI_{1}$ and
$DI_{2}$ computation are listed in Table 6. KL divergence between the two GMMs
formed using the feature vector of size $90\times 2$ for each sensor path was
used for DI mapping. In case of conventional SDC algorithm using CCD, the
average of CCD values of the $90$ denoised test samples for each path was
considered for the computation. SDC values for noise augmented test data using
the three approaches are listed in Table 7 and DI maps obtained using these
SDC values are shown in Figure 25. We observe that only CNN+GMM method yields
higher SDC values for paths containing the damage (diagonal paths) as compared
to other paths, and therefore gives correct damage location for temperature
affected experimental data.

Table 6: Time of arrival $(t_{0})$ and exit $(t_{1})$ of $S0$ mode for various sensor paths in experimental data set. Sensor Path |  $t_{0}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$] |  $t_{1}$ [$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$]
---|---|---
Vertical (P12) | $16.8$ | $50.1$
Horizontal (P14) | $16.8$ | $50.1$
Diagonal (P13, P24) | $28$ | $61.3$
Figure 23: Accuracy curves for training and testing of 1D CNN corresponding to sensor path P24 in noise-augmented experimental data set for $200$ epochs. Figure 24: Silhouette scores for various number of clusters for path P24 in noise-augmented experimental data set, shown for (a) baseline (undamaged panel), and (b) notch defect. Figure 25:  DI maps for noise-augmented experimental data under varying non-uniform temperature profiles on aluminum panel with notch defect, obtained using SDC based damage localization algorithm. SDCs are computed using computed using: (a) the proposed CNN+GMM method as described in equation5, (b) KL divergence of GMMs modeled using physics-defined features $DI_{1}$ and $DI_{2}$ (equations (21) and (22)), and (c) conventional SDC definition using CCD metric (equation (23)) applied to signals denoised using moving average kernel of size $3$. Only the proposed CNN+GMM method is able to accurately localize the damage on the panel.  Table 7: Comparison of SDC values computed using various methods for noise-augmented experimental data. Only the CNN+GMM method shows higher SDC values for paths containing damage (diagonal paths) as compared to the vertical and horizontal paths. Numbers in parentheses indicate normalized value of SDC for each method. Sensor path |  Proposed CNN+GMM method |  GMM and physics defined features $(DI_{1}$ and $DI_{2})$ |  Conventional SDC algorithm (CCD) and denoised signal
---|---|---|---
P12 (Vertical) |  $0.02~{}(0.05)$ |  $6.62~{}(0.73)$ |  $0.64~{}(1)$
P14 (Horizontal) |  $0.05~{}(0.12)$ |  $1.54~{}(0.17)$ |  $0.41~{}(0.64)$
P13 (Diagonal) |  $0.41~{}(1)$ |  $1.44~{}(0.16)$ |  $0.24~{}(0.38)$
P24 (Diagonal) |  $0.17~{}(0.41)$ |  $9.11~{}(1)$ |  $0.39~{}(0.61)$

##  VI Conclusion and future work

The proof of concept implementation of the proposed method is illustrated for
GW-SHM system consisting six-sensor network mounted atop an aluminum plate,
with FE simulation based data generated at various temperatures from
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$, with added white noise and pink
noise with SNR varied from $2\text{\,}\mathrm{d}\mathrm{B}$ to
$10\text{\,}\mathrm{d}\mathrm{B}$, and experimental data with non-uniform
temperature variation and added noise. We have conclusively demonstrated that
CNN-based automated feature computation and analysis using KL divergence of
GMMs of features to represent SDC gives far superior performance as compared
to conventional methods reported previously in literature, using both
simulated as well as experimental data sets. Furthermore, the CNN-based
approach requires no domain knowledge (wave velocity, impact of damage on
amplitude and phase of wave modes etc.) and is therefore applicable broadly
for various structures. To illustrate the efficacy of the method for damage
classification and localization, the proposed algorithm was tested for
different types of defects (rivet hole, added mass and notch), albeit with the
same defect location, at the intersection of two diagonal sensor paths in the
network. The experimental validation with notch defect also required the
damage location to be at the intersection of the diagonal paths. This
constraint was necessary, since the SDC based damage localization algorithm
used to compute DI maps has limited spatial resolution and requires a grid of
sensors. This method is most sensitive to damages that are present at the
intersection of ellipses corresponding to different sensor paths. In future
work, we will explore alternate methods to improve the spatial resolution of
the damage localization using methods such as outer-tangent method [49] and
delay and sum algorithm [50], and also by employing deep learning coupled with
regression. Availability of extensive labeled data significantly limits the
applicability of supervised learning algorithms. Therefore we also seek to
explore unsupervised learning and transfer learning based approaches that have
more applicability and warrant further research. Additionally, we seek to also
explore development of light-weight deep learning models inspired by TinyML
framework [51], that could be directly embedded in the FPGA based embedded
system presented in this work, to truly design portable smart SHM systems.

## Acknowledgment

This work was partially supported by Indian Space Research Organization (ISRO)
[grant no. RD/0118-ISROC00-006]. The authors thank Ms. Sahar Almahfouz Nasser
and Prof. Amit Sethi at IIT Bombay for insightful discussions on deep
learning, and Mr. Rambabu Sugguna at National Centre for Photovoltaic Research
and Education (NCPRE), IIT Bombay for assistance with thermal imaging.

## Declaration of interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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