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\DOI{10.5802/crmeca.239}
\datereceived{2023-09-15}
\daterevised{2023-12-22}
\datererevised{2024-01-11}
\dateaccepted{2024-01-18}
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\dateposted{2024-02-01}
\begin{document}

\begin{noXML}

%\TopicEF{Research article}{Article de recherche}

\title{SGS method applied to damage location and~uncertainty modeling
for sensor grid in~the~ISHM}

\alttitle{M\'ethode SGS appliqu\'ee \`a la localisation des dommages et
\`a la mod\'elisation de l'incertitude pour la grille de capteurs dans
l'ISHM}

\author{\firstname{Paulo Elias Carneiro} \lastname{Pereira}\CDRorcid{0000-0001-6068-4692}\IsCorresp}
\address{School of Mechanical Engineering, Federal University of
Uberl\^andia, 2121~Jo\~ao~Naves de \'Avila Av., Uberl\^andia, Brazil}
\email[P. E. C. Pereira]{paulo\_elias\_carneiro@ufu.br}

\author{\firstname{Stanley Washington Ferreira} \lastname{de Rezende}\CDRorcid{0000-0003-3163-9862}}
\addressSameAs{1}{School of Mechanical Engineering, Federal University
of Uberl\^andia, 2121 Jo\~ao Naves de \'Avila Av., Uberl\^andia,
Brazil}
\email[S. W. F. de Rezende]{stanley\_washington@ufu.br}

\author{\firstname{Jos\'e dos Reis Vieira de} \lastname{Moura J\'unior}\CDRorcid{0000-0002-8701-8355}}
\address{Mathematics and Technology Institute, Federal University of
Catal\~ao, 1120~Dr.~Lamartine Pinto de Avelar Av., Catal\~ao, Brazil}
\email[J. R. V. Moura J\'unior]{zereis@ufcat.edu.br}

\author{\firstname{Roberto Mendes} \lastname{Finzi Neto}\CDRorcid{0000-0002-9340-3233}}
\addressSameAs{1}{School of Mechanical Engineering, Federal University
of Uberl\^andia, 2121 Jo\~ao Naves de \'Avila Av., Uberl\^andia,
Brazil}
\email[R. M. Finzi Neto]{finzi@ufu.br}

\keywords{\kwd{Structural health monitoring}\kwd{Electromechanical
impedance-based method}\kwd{Damage detection and
location}\kwd{Sequential Gaussian simulation}}

\altkeywords{\kwd{Surveillance de la sant\'e des
structures}\kwd{M\'ethode bas\'ee sur l'imp\'edance
\'electrom\'ecanique}\kwd{D\'etection et localisation des
dommages}\kwd{Simulation gaussienne s\'equentielle}}

\shortrunauthors

\begin{abstract} 
Damage location estimation is a critical aspect of condition-based
maintenance systems, particularly in the context of electromechanical
impedance monitoring. Various approaches have been developed to
estimate damage location, yet they often need more capability to assess
the reliability of data collected from sensor grids. In this paper, we
introduce a novel method based on Sequential Gaussian Simulation (SGS)
to pinpoint damage locations on aluminum plates and create maps that
illustrate the spatial uncertainty associated with damage index values
throughout the structure. Our proposed approach builds upon the SGS
method and encompasses the assessment of four different sensor grid
configurations to investigate how sensor spacing affects spatial
uncertainty. The findings demonstrate the technique's effectiveness in
accurately predicting damage positions. Moreover, by leveraging the
uncertainty information generated, we can identify specific areas
necessitating careful attention, thus offering valuable insights for
optimizing sensor grid design.\looseness=-1
\end{abstract}

\begin{altabstract}
L'estimation de la localisation des dommages est un aspect critique des
syst\`emes de maintenance bas\'es sur l'\'etat, en particulier dans le
contexte de la surveillance de l'imp\'edance \'electrom\'ecanique.
Diverses approches ont \'et\'e mises au point pour estimer la
localisation des dommages, mais elles n\'ecessitent souvent davantage
de capacit\'es pour \'evaluer la fiabilit\'e des donn\'ees collect\'ees
\`a partir des grilles de capteurs. Dans cet article, nous pr\'esentons
une nouvelle m\'ethode bas\'ee sur la simulation gaussienne
s\'equentielle (SGS) pour localiser les dommages sur les plaques
d'aluminium et cr\'eer des cartes qui illustrent l'incertitude spatiale
associ\'ee aux valeurs de l'indice de dommage dans l'ensemble de la
structure. L'approche propos\'ee s'appuie sur la m\'ethode SGS et
englobe l'\'evaluation de quatre configurations diff\'erentes de la
grille de capteurs afin d'\'etudier comment l'espacement des capteurs
affecte l'incertitude spatiale. Les r\'esultats d\'emontrent
l'efficacit\'e de la technique pour pr\'edire avec pr\'ecision les
positions des dommages. De plus, en exploitant les informations
d'incertitude g\'en\'er\'ees, nous pouvons identifier des zones
sp\'ecifiques n\'ecessitant une attention particuli\`ere, offrant ainsi
des indications pr\'ecieuses pour optimiser la conception de la grille
de capteurs.
\end{altabstract}

\maketitle
{\vspace*{-.3pc}}

\end{noXML}

\section{Introduction}

The Electromechanical Impedance-based Structural Health Monitoring has
been used for damage identification in several structures and materials
due to being a nondestructive, low-cost method and capable of detecting
small-scale phenomena~\cite{Giurgiutiu2014, Zagrai2009}.

Several researchers have demonstrated the application of Integrated
Structural Health Monitoring (ISHM) effectively. Nomelini
{\etal}~\cite{Nomelini2020}, Maruo \mbox{\etal}~\cite{Maruo2015}, and Finzi
Neto \mbox{\etal}~\cite{FinziNeto2011} successfully employed ISHM for
monitoring both aircraft structures and aluminum specimens, yielding
promising results. Furthermore, de Rezende \mbox{\etal}~\cite{deRezende2023}
and Cavalini~Jr \mbox{\etal}~\cite{Cavalini2015} extended the application of
impedance-based methods to rotating machines. Their innovative approach
involved integrating deep learning models to detect damage present in
these systems. These diverse applications encompass a wide range of
topics, with many investigations centered explicitly around the crucial
stage of damage detection.

However, the description of damage involves more than just identifying
it. Determining where the damage is located is crucial so that
maintenance efforts can be targeted effectively. Knowing the location
helps to make informed decisions about how to proceed with repairs or
other interventions.

Researchers have created multiple methods to predict the location of
damage in a material. One of these methods, known as the damage index
approach, was developed by Kim \mbox{\etal}~\cite{Kim2012}. They conceived a
metric that could indicate whether the damage was present and where it
was located by combining two measurements: the root-mean-square
deviation (RMSD) obtained from impedance-based measurements and the
RMSD obtained from guided wave-based measurements. Applying this new
index could generate polynomial curves that accurately located
debonding in a carbon-fiber-reinforced polymer (CFRP) layer attached to
a concrete beam. In their experiments, they found that the peaks of
these polynomial curves matched the positions of the debonding in the
material.

Cherrier \mbox{\etal}~\cite{Cherrier2013} employed Inverse Distance
Weighting (IDW) to generate damage localization maps for a beam and six
plates made of composite material. They assigned weights based on an
acoustic attenuation coefficient. Using IDW, the researchers could
forecast the probable locations of damage using probability maps. On
the other hand, the technique used by Cherrier
\mbox{\etal}~\cite{Cherrier2013} considers that the damage index values are
inversely proportional to the damage's distance. Based on this concept,
Castro \mbox{\etal}~\cite{Castro2020} developed an algorithm to indicate the
probability of damage in each pixel. The resulting probability depends,
among others, on the cross-correlation square difference (CCSD) index,
weighted by a linear inverse distance between each PZT patch and the
analyzed pixel. The technique was applied to an aluminum plate, whose
results showed high probability values near the damage's location.

Zhu \mbox{\etal}~\cite{Zhu2021} used a combination of the direct coupling
mechanical impedance (DCMI) methodology for signatures extraction and a
modified probability-weighted function to locate damage in a honeycomb
sandwich composite plate. After defining the optimal sensing radius of
the PZT patch, the raw signatures were processed using the DCMI
methodology and used to calculate RMSD indexes. The
probability-weighted algorithm could delineate the damage's position
once the higher probabilities were located near the real location of
the damage, especially in the results obtained from the DCMI
signatures.

The prediction of damage in a honeycomb sandwich composite plate was
explored also in Sikdar \mbox{\etal}~\cite{Sikdar2022}. Since conductance
signatures are affected by PZT patches-debonding distance, a debond
index was formulated, reasoned by the differences in the experimental
and numerical conductance signatures. Maps of debonding's probability
were generated from the proposed index, whose results showed good
compatibility between estimated and real damage's position.

Based on the spatial correlation among damage metric values in the
sensor grid, Gon\c{c}alves \mbox{\etal}~\cite{Goncalves2020} used the
ordinary kriging (OK) method to estimate the location of damage in
aluminum plates. Four damage scenarios were considered. From the
spatial information obtained by semivariogram, the RMSD values were
interpolated by OK, resulting in maps of RMSD values, in which the
higher ones were located at the respective damage positions.

The spatial correlation between damage metric values was also explored
by Gon\c{c}alves \mbox{\etal}~\cite{Goncalves2021}. The indicator kriging
(IK) technique was used to estimate the damage's probability of
occurrence. It considered five scenarios of sensor grids with different
spacing. The probability maps showed higher values located at the
plate's center, where the damage was positioned, being possible to
delineate the region of failure by the proposed approach.

Still, based on the relationship between damage metrics and distance
from the damage, Soman \mbox{\etal}~\cite{Soman2023} proposed an inverse
implementation to preview the damage location in a glass
fiber-reinforced polymer (GFRP) plate. An analytical model was
formulated, relating the damage metric and the PZT patch-damage
distance and relative angle, resulting in a dataset of damage index.
The experimental indexes were compared with the numerical ones, and the
scenario with the lowest mean absolute error (MAE) was defined as a
representative to determine the damage position. The results showed the
method's ability to estimate the damage's location.

Wang \mbox{\etal}~\cite{Wang2022} proposed the use of three frequency
intervals, based on finite element model (FEM) results, to detect and
locate a loosened bolt. The information collected was used in a Support
Vector Machine (SVM), resulting in a recognition accuracy of 81.25\%,
whereas using a single interval resulted in a maximum accuracy of
65.5\%.

Based on a FEM model of an aluminum beam, Fan and Li~\cite{Fan2020}
compared the sparse and Tikhonov regularization methods for damage
location. The FEM was adjusted, and the frequency shifts of the
Conductance spectra's resonance peaks were used in the regularization
methods to solve the inverse problem. The change in Young's modulus was
used as the damage index. The sparse regularization could estimate the
damage position in all scenarios considered, unlike the Tikhonov
regularization, which estimated the real position in just a few
scenarios.

Even with this, the existing methods have limitations in that they
solely focus on estimating the damage's location without offering
insights into the reliability of the data collected from the sensor
grids, even in cases following the OK approach. Therefore,
understanding the spatial uncertainty associated with a sensor grid
becomes imperative in designing an optimal sensor arrangement.

Here, the concept of a sensor grid is related to data, computing,
information, and knowledge discovery management. In contrast, the
sensor network concept is attributed to sensors' logical and physical
connectivity~\cite{Richards2006}.

This contribution presents an innovative approach rooted in Sequential
Gaussian Simulation (SGS). Its main objective is twofold: firstly, to
estimate the location of known damage and, secondly, to quantify the
spatial uncertainty of the Root Mean Square Deviation (RMSD) indices
through an aluminum plate. This approach opens new perspectives for
optimizing sensor grid designs by offering a detailed uncertainty map.
Furthermore, the impact of sensor spacing on spatial uncertainty is the
subject of studies, where regular and non-regular sensor grids with
varying spacing are meticulously evaluated. Finally, the effectiveness
of the SGS approach in accurately estimating damage positions is
subject to rigorous evaluation.

\section{Theoretical backgrounds} \label{sec2}

This section presents concepts related to the techniques and tools used
in this work. It includes explanations of the Electromechanical
Impedance method, Geostatistics, and the SGS method.

\subsection{Electromechanical impedance} \label{sec2.1}

The Electromechanical Impedance (EMI) technique is based on the
electromechanical coupling between the monitored structure and a
piezoelectric element, usually a PZT patch, attached to it. Due to this
interaction, the electrical impedance measured in the PZT patch's
terminals is related to the mechanical impedance of the hosted
structure, being a signature of its state~\cite{Giurgiutiu2014,
MartowiczChapter, Park2003}.

The PZT patch-structure can be considered as a spring-mass-damper
system of 1~degree of freedom, whose formulation was presented by Liang
\mbox{\etal}~\cite{Liang1997}. In this system, the PZT patch is excited by a
harmonic voltage signal resulting in an induced strain, affecting
locally the structure. Then, the dynamic response of this one is
transferred back to the sensor, represented by an electrical
signal~\cite{Bhalla_Soh, MartowiczChapter}.

The relation between the electrical impedance ($Z_E(\omega)$) and the
structure's mechanical impedance ($Z_s(\omega)$) can be described by
the mathematical model given by Equation~(\ref{eq1})~\cite{Liang1997},
which varies along a frequency ($\omega$) interval. In
Equation~(\ref{eq1}), $\tau$ is the geometric constant of the PZT
patch; $\mathrm{j}$, the imaginary unit; $\varepsilon_{33}^T$, the dielectric
constant for a given mechanical stress~$T$; $Z_p$ is the transducer's
mechanical impedance; $d_{3X}^2$, the dielectric constant, and;
$Y_{XX}^E$, the Young's modulus for a constant electric field~$E$. 
{\begin{equation} \label{eq1}
Z_E(\omega) = \frac{1}{\mathrm{j}\omega\tau}\left(\varepsilon_{33}^{T}-
\frac{Z_{s}(\omega)}{Z_{s}(\omega)+Z_{p}(\omega)}d_{3X}^{2}
Y_{XX}^{E}\right)^{-1}.
\end{equation}}\noindent

The application of an AC voltage signal to the PZT patch with a
frequency interval provides a signature of the electrical impedance,
which indicates the structural conditions due to the electromechanical
coupling. Thus, any changes in the impedance signatures can indicate
structural integrity changes~\cite{Park2003, Sun1995}.

Comparing the impedance signatures, usually their real parts
(Resistance), at any moment with those of the pristine state (baseline
signature) allows the detection of damages. Based on this, damage
indexes were developed to indicate it quantitatively. Among them, the
most used is the Root Mean Square Deviation (RMSD), defined in
Equation~(\ref{eq2}), and based on a Euclidean
norm~\cite{Giurgiutiu2014, Giurgiutiu1998}.
{\begin{equation} \label{eq2}
\text{RMSD} = \sqrt{\frac{\sum_{k=\omega_i}^{\omega_f}{[Z_E(k)
-Z_E^0(k)]^2}}{\sum_{k=\omega_i}^{\omega_f}{[Z_E^0(k)]^2}}}.
\end{equation}}\noindent

The RMSD index quantifies the changes in two signatures by calculating
the square root of the mean quadratic differences between the $k$th
electrical impedances at a particular moment ($Z_E(k)$) and at the
pristine condition ($Z_E^0(k)$), along a frequency interval defined by
its lower ($\omega_i$) and upper ($\omega_f$) limits.

The quantification of damage metrics for a sensor grid provides crucial
information for damage assessment, allowing better decision-making
regarding structure maintenance.

\subsection{On concepts of geostatistics} \label{sec2.2}

Geostatistics had its roots as an application of Georges Matheron's
Theory of Regionalized Variables~\cite{Matheron1963, Matheron1967,
Matheron1971, Matheron1973} and was first developed to address
geological and mining engineering problems, and is widely used in these
fields~\cite{Pereira2017, Oliveira2021, Ogunsanwo2021, Tayebi2023}. By
gathering a finite number of samples, geostatistical methods are
employed to create geological models by interpolating the sample
values. The interpolation is based on analyzing the distribution and
correlation of the sample values within the study area.

The type of variables treated by geostatistical techniques exhibit
spatial continuity within the studied area. This continuity results in
data points close to each other having similar values, leading to
reduced dissimilarities among them. Conversely, data points that are
distant from each other exhibit a higher contrast between the
corresponding values~\cite{Chiles2012}.

This behavior implies that not only the variable's values are used in
the analysis but also their relative positions. To quantify the spatial
continuity, it is used the semivariogram function
(Equation~(\ref{eq3})), which calculates half of the mean squared
differences between two values ($z(x_i)$ and $z(x_{i}+h)$) separated by
a vector~$h$~\cite{Journel1978, Abzalov}.
{\begin{equation} \label{eq3}
\gamma(h) = \frac{1}{2N(h)}\sum_{i=1}^{N(h)}{[z(x_i)-z(x_{i}+h)]^2}.
\end{equation}}\noindent

The calculation of the semivariogram for several classes of the
distances $h$ provides a quantification of the spatial continuity along
the direction under examination. Comparing the semivariogram values and
corresponding distances yields an experimental semivariogram
(Figure~\ref{fig1}), demonstrating a steadily increasing trend that
denotes a decrease in continuity with increasing separation distance
between pairs of sample data~\cite{Sinclair2002}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Example of an experimental semivariogram.}
\end{figure}

For use in the estimation or simulation processes, it is necessary to
fit the experimental semivariogram with theoretical functions, once the
calculated semivariogram is a discrete function with no semivariance
values between two successive distance classes. Spherical, exponential,
and Gaussian mathematical models (Figure~\ref{fig2}a) are commonly
employed to fit the computed semivariogram~\cite{Sinclair2002}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Some theoretical models for semivariogram
fitting~(a) and semivariogram model with its properties~(b).}
\end{figure}

A typical semivariogram model has three parameters (Figure~\ref{fig2}b)
that define its shape: Nugget Effect, Range, and Sill. The Nugget
Effect represents a discontinuity at the origin of the semivariogram
and indicates spatial variability occurring at a significantly smaller
scale than the data~spacing. The Range refers to the distance beyond
which no spatial correlation exists between sample pairs, while the
Sill corresponds to the data variance along the vector's $h$
direction~\cite{Pyrcz2014}.

The semivariogram model derived from fitting is utilized in various
geostatistical-based estimation and simulation techniques, such as the
SGS method.

\subsection{Sequential Gaussian simulation} \label{sec2.3}

The SGS is a stochastic simulation method characterized to have a
sequential nature, once the simulation of a given point in the space is
based on the sampled and previously simulated values within the search
neighborhood~\cite{Abzalov, Sinclair2002}.

The original random function (RF), which comprises a vector of the
sampled values at the analyzed domain, is transformed into a standard
normal distribution before the implementation of the SGS
algorithm~\cite{Chiles2012}. The use of another distribution, despite
honoring the mean, variance, and semivariograms (for fundamental
concepts, refer to Armstrong~\cite{Armstrong}, Isaaks and
Srivastava~\cite{Isaaks1989}, and Journel and
Huijbregts~\cite{Journel1978}), results in the incorrect global
distribution of simulated values, which is not a problem in a Gaussian
space~\cite{Pyrcz2014}.

The generation of a simulated value $Y_{\mathrm{SGS}}^*(u)$ at a given
node $u$ is done initially by estimating a value $Y_{\mathrm{SK}}^*(u)$
by simple kriging (SK) method (Equation~(\ref{eq4})) using the sampled
and previously simulated data at the $u_\beta$ locations, which have,
each one, a weight resulted from the solution of the SK
system~\cite{Pyrcz2014, Yamamoto2013}. Once the Gaussian mean ($m$) is
zero, the second term of the Equation~(\ref{eq4}) is zero, keeping only
the first one.
{\begin{equation} \label{eq4}
Y_{\mathrm{SK}}^* = \sum_{\beta=1}^{n}{\lambda_\beta Y(u_\beta)}+
m\left(1-\sum_{\beta=1}^{n}{\lambda_\beta}\right).
\end{equation}}\noindent

From SK it's obtained an estimated value $Y_{\mathrm{SK}}^*(u)$ at a
given node $u$ and its associated kriging variance
($\sigma_{\mathrm{SK}}^2(u)$), that are used for selecting a random
residual ($R(u)$) from the Monte Carlo simulation that yields a normal
distribution with null mean and variance equal to the SK variance. The
residual is then, added to the SK estimate (Equation~(\ref{eq5})),
resulting in the simulated value{\break} \cite{Chiles2012, Pyrcz2014,
Rossi2014}.
{\begin{equation} \label{eq5}
Y_{\mathrm{SGS}}^*(u) = Y_{\mathrm{SK}}^*(u) + R(u).
\end{equation}}\noindent

The simulated value is added to the conditioning data to ensure that
the covariance between all of the simulated values is reproduced, thus,
providing reproducibility of the input statistical distribution and the
covariance models, unlike the Kriging results, which is
smoothed~\cite{Lantuejoul2002}.

This process is repeated along the domain randomly to avoid artifacts,
and each node is visited only once, preserving those in the sampled
locations~\cite{Rossi2014}, resulting in conditional-to-data
simulations. Once a different seed is used for each realization,
providing different simulation paths, it results in equiprobable
scenarios, which can be used for spatial uncertainty analysis.

\section{Materials and methods} \label{sec3}

The spatial uncertainty of damage metric values was initially evaluated
using four regular grids, whose data sets were obtained from
Gon\c{c}alves \mbox{\etal}~\cite{Goncalves2021}. In this context, it was
considered the following spacings between sensors in the regular grids:
(1)~9.09~cm $\times$ 9.09~cm (10 $\times$ 10~PZTs); (2)~11.11~cm
$\times$ 11.11~cm (8 $\times$ 8~PZTs); (3)~14.29~cm $\times$ 14.29~cm
(6 $\times$ 6~PZTs), and; (4)~20.00~cm $\times$ 20.00~cm (4 $\times$
4~PZTs), all of them showed in Figure~\ref{fig3}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Maps depicting the positions of sensors under
regular grid conditions (first row) and non-regular grid conditions
(second row). The red circle designates the location of the actual
damage.}
\end{figure}

Based on the primary regular grids, the results derived from the SGS
approach were also analyzed under non-regular grid conditions. In each
sensor grid scenario, 10.0\% of the total information was randomly
removed, resulting in areas within the grid with no recorded data, as
shown in Figure~\ref{fig3}.

In all scenarios of sensor spacing, the damage of the same severity was
simulated in the center of the square aluminum plate of 100~cm $\times$
100~cm dimensions. In these cases, simulated damage was done by adding
mass corresponding to an M6 nut (3.1~g) glued to the center of the
plate. The procedures and tools used were consistent across all
conditions and are detailed below, with the corresponding flowchart
shown in Figure~\ref{fig4}.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Flowchart of the executed steps.}
\end{figure}

Once the SGS algorithm need the data to be transformed into a Gaussian
distribution, the original input data was first transformed to a
standard Gaussian distribution, which was done by using the
\textit{nscore} module of the Pygeostat package for
Python~\cite{DeutschPygeostat}. The Gaussian transformed data was
submitted to structural analysis by semivariograms with the aim of
identifying the preferential directions of continuity. The calculation
and theoretical fitting of the semivariograms were done using the
GeostatsPy Python Package~\cite{PyrczGeostatspy}.

The resulting semivariogram fitting was the basis for generating the
equiprobable realizations by SGS, executed in the \textit{sgsim}
algorithm adapted to Python language in the Pygeostat
\mbox{package}~\cite{DeutschPygeostat}. The generated realizations were
conditional, as the simulated values at the nodes with input data were
equal to these, to reproduce the variable's spatial continuity more
properly.

The parameters used in the \textit{sgsim} algorithm were the following:
(1)~file with data; (2)~number of the columns containing the $X$ and
$Y$ coordinates and the variable to be simulated; (3)~range of the data
to be considered in the simulations; (4)~the lower and upper limits of
the Gaussian distribution obtained by the \textit{nscore} process;
(5)~random seed; (6)~parameters of the grid-like model used to store
the realizations; (7)~number of realizations to be generated; (8)~the
minimum and maximum sample data and simulated nodes employed in each
simulation; (9)~semivariogram fitting outcomes; (10)~search ellipsoid
parameters, and; (11)~type of kriging to be used.

The input file comprised $X$ and $Y$ coordinates of each PZT patch in
the first and second columns, respectively, and normal
score-transformed RMSD values in the last column. The utilized format
adhered to GeoEAS standards, as required by the \textit{sgsim} process.

For the simulations, all the normal score-transformed RMSD values were
considered to fully map the entire structure instead of focusing on
specific values.

The random seed is the first value initializing the simulations and
corresponds to a large odd integer. To ensure unbiased results, a seed
was chosen from a random process for each simulation.

To store the realizations, a model consisting of unit cells with
dimensions of 4.00~cm $\times$ 4.00~cm was utilized, resulting in a
resolution of 4.00~cm. The dimensions of the specified cell were chosen
due to the following: (1)~being smaller than all semivariogram ranges;
(2)~being a multiple of the plate's size, contributing to aligning the
model with the plate's dimensions, and (3)~being smaller than the
sensor spacing in all scenarios evaluated.

The number of cells in each direction was determined by dividing the
plate size along the given axis by the corresponding unit cell
dimension, resulting in a grid-like model with the same dimensions as
the monitored structure.

The number of realizations adopted was 91. This value was the highest
number of realizations among all sensor grid scenarios at which the
variance of the RMSD values stabilizes. The stabilization criteria were
the value of 0.005 of the Jensen--Shannon distance calculated between
the variance's statistical distribution of two sequential numbers of
realizations. Figure~\ref{fig5} shows the Jensen--Shannon calculated
values and their respective theoretical fitting; the latter determines
the realization number at which the variance stabilizes.

\begin{figure}
{\vspace*{-.2pc}}
\includegraphics{fig05}
{\vspace*{-.2pc}}
\caption{\label{fig5}Calculated Jensen--Shannon values and theoretical
fitting for the 9.09~cm $\times$ 9.09~cm~(a), 11.11~cm $\times$
11.11~cm~(b), 14.29~cm $\times$ 14.29~cm~(c), and 20.00~cm $\times$
20.00~cm~(d) sensor spacing grids.}
{\vspace*{-.1pc}}
\end{figure}

The minimum and maximum sample data for the simulations were set as 1
and 6, respectively. This selection ensures that the search
neighborhood includes at least one PZT patch for node simulation and no
more than 6 PZT patches to reduce the impact of distant data that may
affect the local estimation. In addition, the number of simulated nodes
used for a given node was set to 5 to further reduce the influence of
distant data on the local estimation. The chosen values were utilized
to simulate all scenarios to provide a comparative analysis.

Once the semivariogram is a quantitative description of the spatial
continuity of the variable being simulated, the dimensions and
orientations of the search ellipsoid were set equal to the
corresponding range and directions of the semivariograms.

As stated in the section outlining the SGS methodology, the SK method
was the fundamental estimator in all simulations.

The realizations obtained for all scenarios had Gaussian distribution,
reproducing the histogram and spatial continuity of the respective
input data. Once each cell in the model had a standard normal
distribution of simulated RMSD values, it was calculated the lower
limit of the confidence interval (CI) for a significance level of
5.0\%, which was converted to original values through the Pygeostat
package's \textit{backtr} process~\cite{DeutschPygeostat}.

The lower limit of the CI was used to ensure that the lowest possible
values for the mean, for the considered significance level, would be
used in the later steps.

To visualize only the region of the damage location, a threshold equal
to the 95th percentile was selected, as seen in Soman
\mbox{\etal}~\cite{Soman2023}. In this way, each scenario had its specific
threshold, according to the RMSD sample data. From the selected
threshold, binary variables were created for the values of the
confidence interval's lower limit, according to Equation~(\ref{eq6}) so
that the ones indicate RMSD values higher than the specified threshold.
{\begin{equation} \label{eq6}
I_{95}(x) = \begin{cases}
1, & \text{if RMSD}(x) > t_{95}\\
0, & \text{otherwise}.
\end{cases}
\end{equation}}\noindent

In the Equation~(\ref{eq6}), $I_{95}(x)$ corresponds to the indicator
variable at a location~$x$; RMSD($x$), the lower limit for the RMSD
values at a location~$x$, and; $t_{95}$, the specified threshold. The
resulting maps were used to analyze the ability of the approach to
delineate the damage's position in each scenario.

Based on the indicator maps, which delineate the regions of possible
damage occurrence, the centroids of the areas with $I_{95}(x)$ equal to
1 were calculated, estimating the damage position. For this purpose,
the spatial coordinates of the binary values were obtained by means of
the Pygeostat package's \textit{addcoord}
module~\cite{DeutschPygeostat}, once these had no coordinate attributed
to them. The centroids obtained were compared to the actual damage
position in each scenario of sensor spacing using the Euclidean
distance as the parameter.

In all scenarios, the RMSD spatial uncertainty $U(x)$ at each location
$x$ was calculated as the standard error (Equation~(\ref{eq7})) of the
91 simulated values in each cell's model since each cell contains a
statistical distribution of RMSD indexes. For its calculation
(Equation~(\ref{eq7})), the standard deviation $s(x)$ was previously
defined at each location~$x$, considering the number of
realizations~($n$).
{\begin{equation} \label{eq7}
U(x) = \frac{s(x)}{\sqrt{n}}.
\end{equation}}\noindent

The nature of the standard error, as this parameter provides a
normalized value associated with data variability, allows the mapping
of areas with high variability, which provides several possibilities in
the design of sensor grids.

\section{Results and discussions} \label{sec4}

The maps in Figure~\ref{fig6}, which depict the confidence interval's
lower limit of the simulated RMSD values, show a gradual change in
values from high values near the center of the plate to low ones at the
borders, away from the damage, under both regular and non-regular grid
conditions. As the spacing between sensors increases, the RMSD values
become smaller, which is expected. Additionally, the area with
relatively high index values increases, as seen in the scenario with a
spacing of 20.00~cm $\times$ 20.00~cm.

\begin{figure}
\includegraphics{fig06}
{\vspace*{-.15pc}}
\caption{\label{fig6}Maps showing the lower confidence interval limit
of simulated RMSD values by SGS under regular (first row) and
non-regular (second row) grid conditions. A red circle indicates the
location of the actual damage.}
{\vspace*{-.15pc}}
\end{figure}

The resulting maps indicate that the random removal of PZT patches
resulted in noticeable changes in scenarios with fewer samples
(14.29~cm $\times$ 14.29~cm and 20.00~cm $\times$ 20.00~cm). However,
PZT patch removal had a smaller impact in scenarios with more data,
which was anticipated due to the higher information available.

However, in all scenarios, it is possible to identify the presence of
damage near the plate's center, as the highest values of RMSD are in
this region, which makes the proposed approach capable of mapping the
area with the occurrence of damage, even in the scenario with a little
amount of information (4 $\times$ 4~PZT patches).

The results obtained with the SGS method (Figure~\ref{fig6}) represent
the sample data spatially, meaning that regions with low sample values
had lower simulated values and vice versa. This behavior has been
observed in other works that have employed SGS~\cite{Falade2022,
Ali2023, Wang2022_geol, Karami2022, Ozen2022, Agyeman2023}, where the
simulated values follow the spatial variability of the sample values.

The variation of simulated RMSD values along the plate based on the
spatial distribution of sample data supports the potential use of the
method in other scenarios, including those where damage is located at
the border of the plate. In this regard, Gon\c{c}alves, Moura Junior
and Pereira~\cite{Goncalves2020} utilized Ordinary Kriging (OK), an
alternative geostatistical approach, to interpolate RMSD values on a
square plate with border damage, resulting in favorable outcomes.

In spite of this fact, this topic is outside the scope of this study,
which intends to introduce potential applications of the SGS method in
developing sensor grids for SHM. Subsequent research studies could
examine supplementary damage scenarios, encompassing multi-damage
situations, to evaluate the efficacy of the SGS method in detecting
structural damage.

Figure~\ref{fig7} displays the lower limit values higher than the 95th
percentile as green areas. The binary values were calculated utilizing
Equation~(\ref{eq6}) to delineate regions of damage occurrence.
According to the first two scenarios (10 $\times$ 10 and 8 $\times$
8~PZT patches) under regular and non-regular grid conditions, the green
area encompasses the center of the plate, allowing the damage location
to be confidently delineated.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Maps showing binary variables associated with the
threshold applied to the lower limit of confidence intervals for
simulated RMSD data for the regular (first row) and non-regular (second
row) data sets. The location of the actual damage is marked with a red
circle.}
{\vspace*{.4pc}}
\end{figure}

In contrast, the green area predominantly covers the center of the
plate in the regular grid condition with 6 $\times$ 6~PZT patches,
whereas the non-regular grid condition does not include the plate's
center. For the 4 $\times$ 4~PZT patches scenario with a regular grid,
the green area does not cover the plate's center, and the non-regular
grid condition fails to highlight an estimation of the damage location.

Also, Figure~\ref{fig7} indicates that the predicted area of damage
shifts away from the center of the plate as the spacing between the PZT
patches increases, which can also be seen in Figure~\ref{fig8}, where
the damage's estimated location (purple circles), referred to as the
centroids of the green areas in Figure~\ref{fig7}, becomes more distant
from the plate's center (red circles) as the sensor spacing widens.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Maps showing the actual location of the damage
(red circle) and the estimated location (purple circle) calculated
using the corresponding centroids on both uniform grid (first row) and
non-uniform grid (second row) conditions.}
{\vspace*{-.3pc}}
\end{figure}

Additionally, while it may be challenging to compare non-uniform
conditions once samples were removed randomly in each scenario,
Figure~\ref{fig8} reveals a general trend of increased deviation
between predicted and actual locations as sensor spacing increased.

Also, the findings indicate that, typically, the distances from the
actual damage location tended to increase within the same spacing when
eliminating PZT patches, and the damage location could not be estimated
for the 4 $\times$ 4~PZT patches scenario with non-regular grid
conditions.

The findings above are detailed in Figure~\ref{fig9}a, which displays
the Euclidean distance between the estimated and actual damage
locations. As per the outcomes, the highest distance from the actual
damage, within the regular grid condition, was 14.14~cm, attributed to
the 20~cm $\times$ 20~cm sensor spacing, which represents an increase
of 1350\% about the lowest one (0.98~cm), attributed to the 9.09~cm
$\times$ 0.09~cm spacing.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}Distance between the estimated and actual damage
positions~(a) and percentage of the distance between estimated and
actual positions about each plate's dimension for both regular grid~(b)
and non-regular grid~(c) conditions.}
\end{figure}

Figure~\ref{fig9}a indicates a general distance increase as the sensor
spacing widens in the non-regular grid condition. There is also an
increase in distance within the same spacing category about the regular
grid status. This effectively summarizes the findings drawn from the
maps mentioned previously (Figures~\ref{fig7} and~\ref{fig8}).

The 9.09~cm $\times$ 9.09~cm spacing scenario was an exception to this
overall behavior. In this one, removing samples reduced the distance
from the actual damage, with a 15.6\% reduction being detected. It is
possible that the random removal of some specific samples contributed
to this slight improvement in results, as the SGS method relies on the
special continuity among sampled points to provide the estimations.

Furthermore, the scenario that exhibited the most significant increase
in distance after removing samples was the 14.29~cm $\times$ 14.29~cm
spacing, whereby the distance was augmented by~27.7\%. This suggests
that scenarios with fewer samples could be more sensitive to reduced
available information. However, new research is needed to investigate
this issue.

Despite the highest distance between the estimated and real damage's
location being relatively high in value (14.14~cm), the decomposition
of the distances along $X$ and $Y$ directions reveals that even in the
scenario with the most increased sensor spacing in the regular grid
condition, the distance along $X$ and $Y$ represents only 10.0\% of the
respective plate's dimensions (Figure~\ref{fig9}b).

Considering the non-regular grid status (Figure~\ref{fig9}c), the
deviation from the plate's dimensions in both directions was similar to
that obtained with a regular grid configuration. However, determining
the damage location for a 20.00~cm $\times$ 20.00~cm spacing with a
non-regular grid was not feasible.

The results demonstrate that utilizing a consistent layout of sensors
allows for reasonable estimation of the damage location, even when
operating with the maximum spacing scenario considered (20.00~cm
$\times$ 20.00~cm spacing). Conversely, non-regular grid arrangements
face limitations when processing sparse data. Nonetheless, after
randomly eliminating PZT patch data, further research is required to
explore these issues in greater detail.

The absence of an estimation regarding the location of damage in the 4
$\times$ 4~PZT patches with a non-uniform grid, using the lower bound
of the confidence interval, can be ascribed to an extended range of
confidence intervals. This extension may have been caused by removing
samples that elevated variance among realizations.

The increasing trend in variance among realizations as the spacing
between sensors increases and samples are removed can be seen in
Figure~\ref{fig10}, which depicts the statistical distributions of the
standard error among realizations.

\begin{figure}
{\vspace*{-.3pc}}
\includegraphics{fig10}
{\vspace*{-.5pc}}
\caption{\label{fig10}Boxplots of the standard error for each sensor
spacing and grid type.}
{\vspace*{-.1pc}}
\end{figure}

Once the variance, and then the standard deviation, is related to the
standard error using Equation~(\ref{eq7}), these are correlated so that
the variance has the same behavior as the standard error.

Considering the regular grid condition, the findings show that the
standard error parameter increases as the sensor spacing widens. In the
scenario with the smallest spacing (9.09~cm $\times$ 9.09~cm), the mean
and maximum standard error values are 0.0275 and 0.0792, respectively.
As the sensor spacing increases from 9.09~cm $\times$ 9.09~cm to
11.11~cm $\times$ 11.11~cm, there is a 38.4\% increase in the mean and
a 52.9\% increase in the maximum value. The differences in mean and
maximum values are even greater when compared to the 14.29~cm $\times$
14.29~cm sensor spacing, with mean and maximum values being 198.7\% and
77.2\% higher than the respective ones in the 9.09~cm $\times$ 9.09~cm
spacing. In the highest spacing scenario (20.00~cm $\times$
20.00~cm),{\break} the~minimum standard error value was not zero, and
lower values were classified as outliers, unlike other scenarios.

Figure~\ref{fig10} shows that removing samples increased the
uncertainty parameters for each category of sensor spacing. However,
since the sample removal was random in each scenario, comparing
consecutive sensor spacing scenarios is difficult. Despite this
limitation, there is an overall increasing trend as the sensor spacing
widens, as observed in the regular grid condition.

When comparing the average uncertainty parameter between the
non-regular and regular grid states, it is evident that the 11.11~cm
$\times$ 11.11~cm spacing scenario was most affected by the sample
removal, as the average standard error increased by 91.9\%. Since the
samples were removed randomly, removing a particular group of samples
could have caused this behavior. In the other scenarios, the average
uncertainty increased by a similar magnitude.

The absolute difference in the standard error results between the
irregular and regular grid conditions (Figure~\ref{fig11}) shows that
the highest differences were also attributed to the 11.11~cm $\times$
11.11~cm spacing, whereas the 9.09~cm $\times$ 9.09~cm spacing was the
less affected scenario, likely due to the higher available information
in the 9.09~cm $\times$ 9.09~cm sensor spacing.

\begin{figure}
{\vspace*{-.3pc}}
\includegraphics{fig11}
{\vspace*{-.3pc}}
\caption{\label{fig11}Boxplots of the absolute difference in the
standard error between the regular and non-regular grid conditions.}
{\vspace*{-.1pc}}
\end{figure}

Once samples have been removed randomly, it becomes difficult to
compare different spacing scenarios. However, a preliminary analysis
indicates that removing samples generally does not significantly affect
conditions with a higher number of sensors. However, reducing the
number of sensors has a more significant impact on systems with limited
information. This aspect must be investigated in detail in future
research.

\looseness=-1
The mapping of the spatial distribution of the absolute difference
between the standard errors of the non-regular and regular grid
conditions (Figure~\ref{fig12}) shows an overall increase in the
uncertainty parameter near the sites where the samples were previously
located. This suggests that removing PZT patches increases the spatial
uncertainty of the damage index values in their vicinity.

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Maps showing the absolute difference in standard
error before and after sample removal across the plate. A red circle
indicates the location of the damage, while the gray circles represent
the positions of the PZT patches.}
{\vspace*{.4pc}}
\end{figure}

The uncertainty map in Figure~\ref{fig13} displays an increase in
parameter uncertainty for the regular grid condition as the sensor
spacing widens, resulting in the greatest standard error values in the
scenario with a 20.00~cm $\times$ 20.00~cm spacing. Additionally,
removing samples within the same spacing category increased the
uncertainty, as discussed previously.

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig13}Maps showing the standard error results for both
regular (first row) and non-regular (second row) data sets. The
location of the actual damage is marked with a red circle.}
\end{figure}

\looseness=-1
Figure~\ref{fig13} also highlights ring-shaped areas with higher
uncertainty values for a given sensor spacing, indicating a high
spatial variation of RMSD indexes in those regions. This may suggest a
transition zone from low to high RMSD values. Conversely, low
uncertainty values in certain areas suggest low spatial variability of
the damage metric, indicating stable RMSD data in those regions.

The behavior described above is most evident when using a 9.09~cm
$\times$ 9.09~cm and a 11.11~cm $\times$ 11.11~cm sensor spacings.
Standard error values are also low near the edges of the plate, where
damage metric values are uniformly low (as shown in Figure~\ref{fig3}).
However, as the RMSD values gradually increase from the edges to the
center of the plate, spatial variability in the damage metric values
becomes more pronounced, leading to higher standard error results.

According to the findings, the uncertainty maps help identify areas
with high spatial variability in the damage indexes. Based on the
spatial uncertainty results, it may be inferred that there is a lack of
sensor information in these regions, making it challenging to locate
damage accurately.

Alternatively, the spatial uncertainty results may be used to quantify
the level of uncertainty associated with a sensor grid. Therefore, the
results obtained from the SGS approach can be a valuable tool for
designing sensor grids that consider uncertainty.

\section{Conclusions} \label{sec5}

This study discusses the effectiveness of the SGS method in identifying
damaged positions within an aluminum plate. The investigation involved
analyzing four regular sensor grids with varying sensor spacings.
Additionally, the effect of sample removal was investigated for each
scenario of sensor spacing by randomly removing samples.

The results indicate that the method demonstrated proficiency in
estimating damage locations for the regular grid condition, with the
minimum and maximum distances between estimated and actual damage being
recorded at 0.98~cm and 14.14~cm, respectively. These measurements
correspond to the narrowest and widest sensor grids examined in the
study.

Removing samples increases the deviations from the actual damage
location in each scenario. Additionally, under the non-regular grid
conditions, predicting the damage location in the 20.00~cm $\times$
20.00~cm spacing scenario was impossible using the lower limit of the
confidence interval approach.

This absence of damage prediction for the non-regular-based 20.00~cm
$\times$ 20.00~cm sensor spacing may be due to increased variance among
realizations. This was confirmed by analyzing the standard error
distribution in each condition, which showed an overall increase with
sample removal for the same spacing class. Also, this increasing trend
was most evident near the position of the removed samples.

Additionally, the mapping of the standard error shows an overall
increase in uncertainty as the~sensor spacing widens. This provides
insight into the reliability of a sensor grid quantitatively.

Using the SGS method to map the spatial uncertainty of the damage index
along the structure provides valuable information for designing sensor
grids that consider this parameter. This information can be used to
identify areas that require careful treatment, offering new insights
into the design of sensor grids.

The SGS method applied to the ISHM can provide several possibilities.
Future research can be focused on (1)~determining an optimal sensor
arrangement based on uncertainty information, (2)~investigating the
capacity of the method to predict multiple damage positions, and
(3)~evaluating the industrial application of the proposed approach.

\section*{Declaration of interests}


The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Dedication}


The manuscript was written through contributions of all authors. All
authors have given approval to the final version of the manuscript.

\section*{Acknowledgments}

The first author is grateful to the Federal University of Catal\~ao in
provinding license to study doctorate at the Federal University of
Uberl\^andia.

\CDRGrant[UKRC]{2019-155900}

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