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\DOI{10.5802/crmeca.241}
\datereceived{2023-10-11}
\daterevised{2024-01-15}
\datererevised{2024-02-06}
\dateaccepted{2024-02-12}
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\dateposted{2024-03-14}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{A comparative analysis for crack identification in structural
health monitoring: a focus on experimental crack length prediction with
YUKI and POD-RBF}

\alttitle{Analyse comparative de l'identification des fissures dans le
cadre de la surveillance de la sant\'{e} des structures :
Pr\'{e}diction exp\'{e}rimentale de la longueur des fissures avec YUKI
et POD-RBF}

\author{\firstname{Roumaissa} \lastname{Zenzen}\CDRorcid{0009-0007-8921-9947}} 
\address{LMT Laboratory, Faculty of Sciences and Technology, University
of Jijel, Jijel, Algeria}
\email[Z. Roumaissa]{roumaissa.zenzen@univ-jijel.dz}

\author{\firstname{Ayoub} \lastname{Ayadi}} 
\address{University of Biskra, Laboratoire de G\'{e}nie Energ\'{e}tique
et Mat\'{e}riaux, LGEM, Faculty of Sciences and Technology, Biskra,
07000, Algeria}

\author{\firstname{Brahim} \lastname{Benaissa}\CDRorcid{0000-0002-9472-9331}\IsCorresp} 
\address{Design Engineering Laboratory, Toyota Technological Institute,
Nagoya, Japan}
\email[B. Benaissa]{benaissa@toyota-ti.ac.jp}

\author{\firstname{Idir} \lastname{Belaidi}\CDRorcid{0000-0003-3463-0580}} 
\address{LEMI Laboratory, Department of Mechanical Engineering,
University M'hamed Bougara Boumerdes, 35000 Boumerdes, Algeria}

\author{\firstname{Enes} \lastname{Sukic}\CDRorcid{0000-0002-0991-5480}} 
\address{Faculty of Information Technology and Engineering - FITI,
University Union - Nikola Tesla, 11070 Belgrade, Serbia}

\author{\firstname{Tawfiq} \lastname{Khatir}\CDRorcid{0009-0001-9553-4608}} 
\address{Artificial Intelligence Laboratory for Mechanical and Civil
Structures, and Soil, Institute of Technology, University Center of
Naama, 45000 Naama, P.O.B. 66, Algeria}

\shortrunauthors

\begin{abstract}
In recent years, substantial investments in structural construction
underscore the paramount importance of ensuring structural integrity
for safety and dependability. Structural Health Monitoring (SHM) has
emerged as a pivotal tool for assessing structural health, with an
emphasis on damage detection, localisation, and quantification,
particularly through vibration-based methods that exploit variations in
modal properties as precursors to structural damage. This study
presents an innovative methodology that synergistically combines Proper
Orthogonal Decomposition and Radial Basis Function interpolation for
predicting structural responses based on crack parameters.
Additionally, the YUKI algorithm, leveraging population clustering for
optimisation, is introduced. The approach is rigorously assessed
through experimental analysis of two distinct beams (Beam I and Beam
II) exhibiting varying crack depths. The results demonstrate the
effectiveness of the POD-RBF-YUKI approach, indicating a notable level
of accuracy and \mbox{consistency.} \mbox{Comparative} evaluations with conventional
optimisation algorithms, namely Cuckoo, Bat, and Particle Swarm
Optimisation, reveal similar Mean Percentage Error values but with
increased result variability, whereas Deep Artificial Neural Network
models with varied hidden layer sizes.
\end{abstract}

\begin{altabstract}
Ces derni\`{e}res ann\'{e}es, des investissements substantiels dans la
construction structurelle ont mis en \'{e}vidence l'importance
primordiale de l'int\'{e}grit\'{e} structurelle pour la
s\'{e}curit\'{e} et la fiabilit\'{e}. La surveillance de la sant\'{e}
des structures (SHM) est devenue un outil essentiel pour \'{e}valuer la
sant\'{e} des structures, en mettant l'accent sur la d\'{e}tection, la
localisation et la quantification des dommages, en particulier
gr\^{a}ce \`{a} des m\'{e}thodes bas\'{e}es sur les vibrations qui
exploitent les variations des propri\'{e}t\'{e}s modales en tant que
pr\'{e}curseurs des dommages structurels. Cette \'{e}tude pr\'{e}sente
une m\'{e}thodologie innovante qui combine de mani\`{e}re synergique la
d\'{e}composition orthogonale appropri\'{e}e (POD) et l'interpolation
de la fonction de base radiale (RBF) pour pr\'{e}dire les r\'{e}ponses
structurelles bas\'{e}es sur les param\`{e}tres des fissures. En outre,
l'algorithme YUKI, qui tire parti du regroupement de populations pour
l'optimisation, est pr\'{e}sent\'{e}. L'approche est rigoureusement
\'{e}valu\'{e}e par l'analyse exp\'{e}rimentale de deux poutres
distinctes (poutre I et poutre II) pr\'{e}sentant diff\'{e}rentes
profondeurs de fissures. Les r\'{e}sultats d\'{e}montrent
l'efficacit\'{e} de l'approche POD-RBF-YUKI, indiquant un niveau
notable de pr\'{e}cision et de coh\'{e}rence. Les \'{e}valuations
comparatives avec les algorithmes d'optimisation conventionnels, \`{a}
savoir Cuckoo, Bat et Particle Swarm Optimisation, r\'{e}v\`{e}lent des
valeurs d'erreur moyenne similaires mais avec une variabilit\'{e}
accrue des r\'{e}sultats, tandis que les mod\`{e}les de r\'{e}seaux
neuronaux artificiels profonds (ANN) avec des tailles de couches
cach\'{e}es vari\'{e}es.
\end{altabstract}

\keywords{\kwd{Crack identification}
\kwd{Model reduction}
\kwd{Experimental modal analysis}
\kwd{Inverse analysis}
\kwd{YUKI algorithm}}

\altkeywords{\kwd{Identification des fissures}
\kwd{R\'{e}duction du mod\`{e}le}
\kwd{Analyse modale exp\'{e}rimentale}
\kwd{Analyse inverse}
\kwd{Algorithme YUKI}}

\maketitle

\end{noXML}

\section{Introduction}


In recent years, substantial budgets have been invested in developing
and building structures in different areas, including the oil and gas
industry, as well as civil and aerospace engineering. Therefore,
ensuring the integrity of these structures during their predicted life
span is crucial for safe and reliable performance. In the last decade,
a great deal of attention has been directed toward preventing sudden
structural failure, which can lead to extensive casualties and property
damage. Structural Health Monitoring (SHM) has emerged as a powerful
tool for assessing and monitoring structural health in this context.
This technique, which considers damage detection a crucial concern,
involves four main stages: damage detection, damage localisation,
damage quantification, and damage extent~\cite{bib1}. However, in the
literature, the first three stages receive the most attention in the
assessment of structural integrity~\cite{bib2}. 

There has been a considerable and continuously growing body of research
dedicated to the examination of Structural Health Monitoring systems,
where numerous formulations and methods have been
proposed~\cite{bib3,bib4,bib5,bib6,bib7,bib8}. In this vein,
Nondestructive evaluation (NDE) is one of the most frequently stated
problems in the SHM~\cite{bib9,bib10,bib11}. Vibration-based structural
integrity approaches have gained new prominence as they offer an
efficient tool in practical applications~\cite{bib12,bib13,bib14}. This
concept is grounded in the principle that alterations in physical
characteristics can lead to variations in modal properties,
encompassing mode shapes, frequencies, and modal damping. These
variations can function as indicators for evaluating the structural
integrity~\cite{bib15,bib16}. Vibration-centric methodologies can be
expressed within either the temporal or frequency realms. Nevertheless,
it has been observed that the stability of frequency domain
characteristics surpasses that of the temporal and time-frequency
domains~\cite{bib15}. 

A pioneering study by Adams \etal~\cite{bib17} used vibration
measurements to evaluate the defect's location and magnitude. Numerous
papers have since summarised and reviewed early vibration-based damage
detection techniques~\cite{bib2,bib6,bib18,bib19}. Similarly, based on
modal parameters, Fan and Qiao reviewed and compared different damage
identification algorithms for plate and beam-type
structures~\cite{bib16}. Essentially, their research covered four
primary categories, which included investigations into curvature mode
shape, natural frequency, mode shape, and mode shape-frequency
techniques. As an alternative to natural frequencies, the measured
amplitudes of Frequency Response Functions (FRFs) are often used to
identify damage. The FRF-based damage detection method provides
enhanced insight into structural response dynamics by focusing on a
restricted set of modal data within precise frequency bands proximate
to resonance frequencies~\cite{bib20}. Furthermore, recent academic
review articles, have been written to address the theoretical aspect of
this research domain~\cite{bib15,bib21}.

Machine Learning (ML) algorithms are fast becoming a key instrument in
engineering applications due to the huge development in computational
capabilities and data availability. These techniques enable computers
to solve complex problems based on examples and experience provided
through data input~\cite{bib22}. In the field of structural
engineering, ML methods have found applications in various domains.
Nonetheless, research in ML within this domain is typically categorized
into the following areas: Structural Health Monitoring, identification
of structural systems, control of structural vibrations, structural
design, and predictive applications~\cite{bib23,bib24,bib25}.
Vibration-based damage detection methods based on ML techniques can be
either parametric or non-parametric~\cite{bib26}. 

Artificial Neural Networks (ANNs), being efficient pattern recognition
tools, have been employed by many researchers to determine damage
location and severity~\cite{bib27}. Lee \etal~\cite{bib28} introduced a
damage detection approach that integrates artificial neural networks
(AN) while considering the modelling error within the initial finite
element model utilised for training pattern generation. This method
relies on a back-propagation neural network for its implementation.
Mehrjoo \etal~\cite{bib29} introduced an approach to assessing the
severity of damage in truss bridge joints. Furthermore, Avci Onur
\etal~\cite{bib26} Offered an extensive examination of the recently
introduced methods in machine learning and deep learning applied to
structural health monitoring through vibration analysis.

Tran-Ngoc \etal~\cite{bib9} introduced an innovative machine-learning
methodology that draws inspiration from the evolutionary algorithm
known as Cuckoo Search (CS). This approach effectively addresses the
challenge of local minima in ML. In the training phase, their algorithm
operates concurrently with ML techniques. Given that local minima can
detrimentally impact the precision of ML models, numerous research
endeavours have been dedicated to surmounting this particular
constraint. Khatir \etal~\cite{bib30} demonstrated a solution to this
issue by utilising a combination of Particle Swarm Optimisation (PSO)
and Teaching Learning Based Optimisation (TLBO) in order to determine
the initial training parameters for the ANN. Working along similar
lines, Tran-Ngoc \etal~\cite{bib31} applied the CS algorithm to
determine the most suitable initial step. Their findings demonstrated
improved accuracy in comparison to traditional machine learning
techniques. In the research conducted by Zenzen \etal~\cite{bib32} 
they employed FRF data in conjunction with genetic and bat algorithms
to assess the detection and assessment of damage in both beam-like and
truss structures. Additionally~\cite{bib33} investigated composite
laminated beams and plates, utilising a damage assessment approach
centred around transmissibility and mode shape.

Model order eduction methods such as proper orthogonal decomposition
(POD), have pinpointed dependable and efficient strategies for damage
identification~\cite{bib34}. The POD technique is exploited for
processing substantial volumes of high-dimensional data, facilitating
computational analysis to anticipate the future dynamics of the
system~\cite{bib35}. Shane and Ratneshwar~\cite{bib36} introduced a
novel algorithm rooted in POD, incorporating proper orthogonal modes
(POM) as dynamical invariants. This comprehensive approach has been
specifically applied to composite beams, underscoring the algorithm's
potential and relevance in damage detection across diverse scenarios.
In their groundbreaking work, Eftekhar \etal~\cite{bib37} proposed an
innovative supervised learning method that integrates artificial neural
networks and POD. This approach was designed to differentiate
alterations in proper orthogonal modes attributed to structural damage
while isolating them from variations induced by varying applied load
conditions. The efficacy of this proposed methodology was demonstrated
through a series of simulated experiments. In a noteworthy study,
Khatir \etal~\cite{bib38} integrated the radial basis functions with
their method, referred to as POD-RBF, to identify the positions,
dimensions, and depths of cracks in composite structures made of carbon
fibre-reinforced polymer (CFRP). Their results underscore the efficacy
of the POD-RBF technique in combination with the Cuckoo search
algorithm. Subsequent research efforts have also delved into this
specific domain~\cite{bib39,bib40,bib42,bib43}. 

\subsection{POD-based radial basis functions}

Proper orthogonal decomposition (POD) is a powerful technique for
dimensionality reduction, widely employed in various data analysis
fields. Its fundamental principle lies in capturing the most essential
patterns or structures within high-dimensional datasets, thereby
enabling a more efficient representation while preserving critical
information. 

The system represents different points data snapshots, organised into a
data matrix $\mathbi{U}$. Each column of this matrix represents an
individual snapshot, while each row corresponds to a distinct variable
or data point. Where $\mathbi{U}$ is a $\mathbi{n}\times
\mathbi{s}$ matrix, where $\mathbi{n} $ denotes the number of
variables and $s$ signifies the number of collected snapshots.

To extract the underlying modes that constitute the predominant
variability in the dataset, we employ Singular Value Decomposition on
the centred data of $U$ matrix. The singular values in 
$\boldsymbol{\Sigma}$ are ordered in descending fashion, representing
the significance of the modes. By selecting the first few singular
vectors (modes) associated with the largest singular values, the
primary patterns within the data are captured. These modes encapsulate
the essential information for subsequent reduction. Radial basis
functions (RBFs) are a type of mathematical function used in this study
for multivariable interpolation for they are particularly advantageous
with scattered data points. RBFs are centred around a set of control
points, and their values at any given point in space are determined by
their distance from these centre points\cite{bib39}. 

In this research, the analysis utilises a combination of the POD
approach and the RBF interpolation methodology to deduce the structural
reaction in the framework of an inverse problem. The description of the
structural response relies on empirical information contained within
the $\mathbi{U}$ matrix.
{\begin{equation}
\label{eq1}
\mathbi{U}=\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}}
\begin{array}{c@{\hspace*{1pc}}c}
{{u}}_{1}^{1} & {{u}}_{1}^{2}\vspace*{5pt}\\
{{u}}_{2}^{1} & {{u}}_{2}^{2}
\end{array} & \cdots & \begin{array}{c}
{{u}}_{1}^{\mathrm{S}}\vspace*{5pt}\\
{{u}}_{2}^{\mathrm{S}}
\end{array}\\
\begin{array}{c@{\hspace*{1.5pc}}c}
\vdots & \vdots \end{array} & \ddots & \vdots \vspace*{2pt}\\
\begin{array}{c@{\hspace*{1pc}}c}
{{u}}_{\mathrm{N}}^{1} & {{u}}_{\mathrm{N}}^{2}
\end{array} & \cdots & {{u}}_{\mathrm{N}}^{\mathrm{S}}
\end{array}\right]
\end{equation}}\unskip
In this context, the dataset denoted as $\mathbi{U}$ comprises a
collection of   $\mathbi{N}$ snapshots, each characterised by a
snapshot vector of size  $\mathbi{S}$. The primary focus of this
research is to analyse the dimensions of the structural response
dataset. Additionally, matrix  $\mathbi{P}$ is employed to store
information pertaining to crack parameters. Following this, a set of
orthogonal vectors denoted as $\Phi $ is extracted. These vectors serve
as the basis for projecting the measurement data matrix $\mathbi{U}$,
ultimately yielding the amplitude matrix  $\mathbi{A}$:
{\begin{equation}\label{eq2}
\mathbi{A}=\boldsymbol{\Phi}^{{T}}\cdot \mathbi{U}
\end{equation}}\unskip
The matrix denoted as $\mathbi{A}$ offers an estimation of the
structural response data. It's worth mentioning that $\boldsymbol{\Phi}$
is determined via the POD process, which encompasses the derivation of
eigenvectors from the covariance matrix $C=U\cdot U^{T}$ using singular
value decomposition. Subsequently, a dimensionality reduction step is
performed to decrease the basis vectors within $\boldsymbol{\Phi}$,
resulting in a lower-rank variant $\widehat{\boldsymbol{\Phi}}$  obtained by
retaining only the initial k (where k ${\ll}$ S) columns associated
with the largest eigenvalues. As a result, a modified amplitude matrix,
referred to as $\widehat{\mathbi{A}}$, is introduced as follows:
{\begin{equation}\label{eq3}
\widehat{\mathbi{A}}=\widehat{\boldsymbol{\Phi}}^{{T}}\cdot \mathbi{U}
\end{equation}}\unskip
In the following stages, we create a connection between structural
response information and crack characteristics using Radial Basis
Function interpolation. This process entails the utilisation of
coefficient matrix $\mathbf{B}$ and interpolation matrix  $\mathbi{G}$
to define $\widehat{\mathbi{A}}$ as the product of $\mathbf{B}$ and 
$\mathbi{G}$. $\mathbi{G}$ is determined to be non-singular, the
expression can be formulated as follows:
{\begin{equation}\label{eq4}
\mathbf{B}=\widehat{\mathbi{A}}\cdot \mathbi{G}^{-1}
\end{equation}}\unskip
where $G(S) $ in this study, consider the normalised paramters values,
and represent the RBF distances of each sample point from all the other
points. $M$ is the number of sample points. 
{\begin{equation}\label{eq5}
G=\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}}
g_{1}(\|p^{1}-p^{1}\|) & \cdots & g_{1}(\|p^{j}-p^{1}\|) & \ldots 
& g_{1}(\|p^{M}-p^{1}\|)\vspace*{1pt}\\
\vdots && \vdots && \vdots \vspace*{1pt}\\
g_{1}(\|p^{1}-p^{i}\|) & \cdots & g_{1}(\|p^{j}-p^{i}\|) & \cdots & g_{1}(\|p^{M}-p^{i}\|)\vspace*{1pt}\\
\vdots && \vdots && \vdots \vspace*{1pt}\\
g_{1}(\|p^{1}-p^{M}\|) & \cdots & g_{1}(\|p^{j}-p^{M}\|) & 
\cdots  & g_{1}(\|p^{M}-p^{M}\|)
\end{array}
\right]
\end{equation}}\unskip
In this context, we evaluate the function, $\mathbi{g}_{\mathbi{i}}(p)$
for every parameter within the matrix  $\mathbi{G}$. Each parameter,
denoted as $p_{i}$, corresponds to  $\mathbi{U}_{\mathbi{i}}$ (where 
$i$ ranges from 1 to $N$). The magnitude of the difference $| p-p_{i}| $ 
signifies the input argument for the ${i}$th Radial Basis Function,
where  $\mathbi{p}$ represents the current parameters, and
$\mathbi{p}_{\mathbi{i}} $ denotes the reference parameters. Following
this, once the coefficient matrix $\mathbf{B}$ has been computed, we
introduce a reduced-dimensional model in vector format:
{\begin{equation}\label{eq6}
\mathbf{a}(\mathbf{p}_{\mathbf{new}})=\mathbf{B}\cdot 
\mathbf{g}(\mathbf{p}_{\mathbf{new}})
\end{equation}}\unskip

Using this method, an estimation of the structural reaction associated
with novel crack characteristics, denoted as 
$\mathbf{p}_{\mathbf{new}}$, is determined:
{\begin{equation}\label{eq7}
\mathbf{u}(\mathbf{p}_{\mathbf{new}})=
\widehat{\boldsymbol{\Phi}}\cdot \mathbf{a}(\mathbf{p}_{\mathbf{new}})
\end{equation}}\unskip
Hence, the POD-RBF model can replicate unknown structural responses for
various crack parameter sets ${\mathbf{p}}$. A summary of the POD-RBF
construction is depicted in Figure~\ref{fig1}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}POD-RBF algorithm.}
\end{figure}

\subsection{Problem formulation based on optimisation algorithms}

\subsubsection{The bat algorithm}

The Bat Algorithm, a nature-inspired optimisation algorithm, derives
its principles from the echolocation behaviour observed in
bats~\cite{bib44}. This algorithm emulates the hunting strategies of
bats, employing their echolocation capabilities for prey localisation.
The Cuckoo Search Algorithm, another nature-inspired optimisation
algorithm, is grounded in the reproductive conduct of cuckoo
birds~\cite{bib45}. It replicates the parasitic tendencies of certain
cuckoo species, specifically those that deposit their eggs in the nests
of other bird species. Within the algorithmic framework, optimisation
problem solutions are analogously represented as nests, and the
egg-laying strategy of cuckoos symbolises the exploration and
exploitation of the search space. The algorithm integrates random walk
and Levy flight steps to update candidate solutions iteratively, with
the overarching goal of identifying optimal or near-optimal solutions
for diverse optimization problems.

\subsubsection{Particle swarm optimisation (PSO)}

Particle Swarm Optimisation (PSO), a computational optimization
technique, takes inspiration from the social behaviour observed in
animal flocks~\cite{bib46}. It involves a population of particles
navigating through a defined search space, adjusting their positions
based on both individual and collective experiences to ascertain
optimal solutions for a given problem. Communication and information
sharing among particles facilitate the guidance of the search towards
regions within the solution space where more favourable solutions are
anticipated.

\subsubsection{YUKI algorithm}

This algorithm introduces an innovative approach to population
clustering, resulting in the formation of two distinct clusters. One
cluster is dedicated to extensive exploration, while the other focuses
on the exploration of the best regions so far. YUKI algorithm guides
this strategy by setting a consistent ratio across iterations,
determined by a user-defined parameter termed the exploration rate
(EXP), ranging from 0 to 1. The EXP parameter dictates the proportion
of the population dedicated to exploration~\cite{bib40}.

The algorithm establishes a local search area centred around the
current optimal solution, referred to as $\mathbf{X}_{\mathbf{best}}$.
The size of this area is determined by the distance between this
solution and the \textit{\textbf{MeanBest}} point, which serves as
the centroid of the cluster containing optimal points. The algorithm
calculates the local boundaries, denoted as  $\mathbi{L}\mathbi{T}$ and
$\mathbi{L}\mathbi{B}$, using the following equations:
{\begin{eqnarray}
\mathbi{D}&=&\mathbf{X}_{\mathbf{best}}-
\mathbf{X}_{{\mathbf{MeanBest}}}\label{eq8}\Seqnsplit
\mathbi{L}\mathbi{T}&=&\mathbf{X}_{\mathbf{best}}+\mathbi{D}\label{eq9}
\Seqnsplit
\mathbi{L}\mathbi{B}&=&\mathbf{X}_{\mathbf{best}}-\mathbi{D}\label{eq10}
\end{eqnarray}}\unskip

The exploration approach promotes diversification in the search for a
solution by expanding beyond the confines of the local search region.
This approach is mathematically represented as follows: In this
equation, ${\mathbi{X}}_{\mathbf{loc}}^{\mathbi{i}}$ represents the
local solution chosen for exploration, while
${\mathbi{X}}_{\mathbf{best}}^{\mathbi{i}} $  refers to the optimal
historical position associated with this specific location.
{\begin{equation}\label{eq11}
\mathbi{E}^{\mathbi{i}}={\mathbi{X}}_{\mathbf{loc}}^{\mathbi{i}}
-{\mathbi{X}}_{\mathbf{best}}^{\mathbi{i}}
\end{equation}}\unskip

The value denoted as $\mathbi{E}^{\mathbi{i}} $ plays a crucial role in
defining the extent of the exploration range applicable to the specific
point under consideration. To derive fresh solutions, we employ the
subsequent equation:
{\begin{equation}\label{eq12}
{\mathbi{X}}_{\mathbf{new}}^{\mathbi{i}}=
{\mathbi{X}}_{\mathbf{loc}}^{\mathbi{i}}
+\mathbi{E}^{\mathbi{i}}
\end{equation}}\unskip
The method guides other solutions to explore the vicinity of the search
centre by employing the subsequent equation. In this equation,
$\mathbi{F}^{\mathbi{i}}$ represents the distance from the chosen local
point to the optimal solution, and ``rand'' is a random value ranging
from 0 to 1, which is applied uniformly across all design variables.
{\begin{eqnarray}
\mathbi{F}^{\mathbi{i}}&=&{\mathbi{X}}_{\mathbf{loc}}^{\mathbi{i}}
-\mathbi{X}_{\mathbf{best}}\label{eq13}\Seqnsplit
\mathbi{X}_{\mathbf{new}}^{\mathbi{i}}&=&{\mathbi{X}}_{\mathbf{loc}}^{\mathbi{i}}
+\mathbf{rand}\times \mathbi{F}^{\mathbi{i}}\label{eq14}
\end{eqnarray}}\unskip

The algorithm pseudocode is written as follows:

\beqimg\begin{eqimg*}
%\begin{algorithm}[htp]
\begin{tabbing}
{Load search parameters}\\
{Initialize population X}\\
\hspace*{.5pc} {Evaluate fitness}\\
\hspace*{2pc} {Calculate X\_MeanBest and X\_best}\\
\hspace*{2pc} {for K $=$ 1 to K\_max}\\
\hspace*{2pc} {Calculate local boundaries}\\
\hspace*{2pc} {Generate random local population}\\
{If rand $<$ EXP (EXP $=$ 0.7)}\\
\hspace*{1pc}{Calculate exploration solutions}\\
\hspace*{1.3pc}{Else}\\
\hspace*{1pc}{Calculate focus solutions}\\
\hspace*{.5pc}{End}\\
\hspace*{1.3pc}{Update X\_MeanBest}\\
\hspace*{1.3pc}{Update X\_best if better solutions found}\\
{End}\\
\hspace*{1.3pc}{Return X\_best}
\end{tabbing}
%\end{algorithm}
\eeqimg\end{eqimg*}

\subsubsection{Objective function}

Natural frequencies are key parameters characterising the dynamic
behaviour of structural systems under external disturbances. These
frequencies depend on various factors, including material properties,
geometric configurations, and boundary conditions specific to the beam
structure. In an intact state, a beam's natural frequencies are
determined by its structural integrity. However, the presence of damage
alters these frequencies due to changes in stiffness and mass
distribution within the beam.

Typically, damage leads to a reduction in stiffness and potential
shifts in mass distribution, resulting in observable changes in the
beam's natural frequencies. Modal analysis techniques are commonly
employed to detect these deviations, involving either experimental
measurements or computational simulations to identify the altered
natural frequencies.

The utilisation of natural frequency changes for structural damage
identification is a well-established approach in structural health
monitoring. By comparing measured or computed natural frequencies with
those of an undamaged reference state, researchers can pinpoint the
location, extent, and severity of structural damage. 

The change of natural frequency is used as an objective function as
presented in the following equation:
{\begin{equation}\label{eq15}
OF={\sum }_{i}^{n}\left[\left({\omega }_{i}^{r}-{\omega }_{i}^{c}\right)^{2}
/\left({\omega }_{i}^{r}\right)^{2}\right]
\end{equation}}\unskip
where, $n$ is the number of modes, ${\omega }_{i}^{r}$ are the
frequencies calculated by the optimisation algorithm--POD,
and ${\omega}_{i}^{c}$ are the measured frequencies.

\section{Experimental analysis}

The purpose of this study is to develop a novel approach that combines
proper orthogonal decomposition (POD) and radial basis function (RBF)
interpolation for predicting structural responses based on crack
parameters while introducing the YUKI algorithm to optimise the process
and assess its accuracy in real-world crack length estimation as
presented in Figure~\ref{fig2}.


\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}(a) Cutting saw, and (b) Experimental setup.}
\end{figure}

In this paper, two beams are considred to predict notch and double
notch length based on modal analysis. An excitation hammer (Impact PCB
Hammer Type 086C03) with a force sensor whose sensitivity is 2:5 mv, a
PCB M352C66, Type ICP, sensitivity 96.9 mV/g accelerometer composed of
a mass, a data acquisition system and a PC were used. The notches
created using cutting saw as presented in Figure~\ref{fig2}a and
experimental setup is showed in Figure~\ref{fig2}b. 



In the initial investigation, a beam marked as ``beam I'' featuring
paired notches were introduced at the centre of the latter,
encompassing 25 distinct depths. These notches ranged from 1~mm to 25~mm
in extension length. Remarkably, a 1~mm deep crack was intentionally
introduced both at the upper and lower sections of the beam. Tabulation
of the mechanical attributes of the beam can be found in
Table~\ref{tab1}, while Figure~\ref{fig3} provides details regarding the
frequencies observed during the experimental analysis of both the
unaltered and double-notched beam.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Numerical frequencies of different crack depths (beam I).}
\end{figure}

%tab1
\begin{table}
\caption{\label{tab1}Dimensions and material characteristics of the
beam I}
\begin{tabular}{cc}
\thead
Item & Value \\
\endthead
Length (mm) & 800 \\
Width (mm) & 15 \\
Height (mm) & 50 \\
Density (kg/m$^{3}$) & 7850 \\
Poisson ratio (/) & 0.3 \\
Young modulus (GPa) & $2.1\times 10^{11}$
\botline
\end{tabular}
\end{table}

In the second instance, denoted as ``beam II'', a series of 20 crack
depths were intentionally induced within the central section of the
structure. These cracks were incrementally extended from 2~mm to 32~mm,
each step measuring 1~mm. The frequencies corresponding to each mode
shape resulting from these crack configurations have been compiled and
can be found in Figure~\ref{fig4}. Additionally, the mechanical
properties of beam II are detailed in Table~\ref{tab2}.

%tab2
\begin{table}
\caption{\label{tab2}Dimensions and material characteristics of the
beam II}
\begin{tabular}{cccccc}
\thead
Item & Value \\
\endthead
Length (mm) & 1000 \\
Width (mm) & 10 \\
Height (mm) & 40 \\
Density (kg/m$^{3}$) & 7850 \\
Poisson ratio (/) & 0.3 \\
Young modulus (GPa) & $2.1\times 10^{11}$
\botline
\end{tabular}
\end{table}

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Numerical frequencies of different crack depths
(beam II).}
\end{figure}


\section{Results and discussions}

In this section, we proposed a comparison between the results using Bat
algorithms, cuckoo search and YUKI and the results found
in~\cite{bib47}, where the ANN is used to predict the damage size in
two beams (Beam I and Beam II) and using different hidden layer sizes
(HLS), and the results indicate that the best regression is achieved
with HLS $=$ 8 for Beam I and HLS $=$ 10 for Beam II. In their research,
Seguini \etal~used the frequencies as an input and the damage size as
an output. Tables~\ref{tab3} and~\ref{tab4} compare their results with
results of the suggested approach for Beam~I and Beam~II respectively.

%tab3
\begin{table}
\caption{\label{tab3}Exact and estimated results using different
optimisation methods (beam I)}
\tabcolsep=3pt
\begin{tabular}{ccccccccc}
\thead
Cases & \parbox[t]{1.8cm}{\centering Actual \par  crack length~(mm)}
& Cuckoo & Bat & PSO & YUKI & 
\parbox[t]{1.5cm}{\centering ANN\par NHLS $=$ 8 \par\cite{bib47}} & 
\parbox[t]{1.7cm}{\centering ANN\par NHLS $=$ 10 \par\cite{bib47}} & 
\parbox[t]{1.7cm}{\centering ANN\par NHLS $=$ 12 \par\cite{bib47}}\vspace*{2pt}\\
\endthead
1 & \04  & \03.8815 & \03.8815 & \03.8815 & \03.882\0 & \03.80911 & \03.614833 & \03.84674\0 \\
2 & 10  & \09.4677 & \09.4677 & \09.4677 & \09.4781 & \09.54634 & \09.710795 & \09.66756\0 \\
3 & 20  & 19.1019 & 19.1019 & 19.1019 & 19.1166 & 19.86071 & 19.53756\0 & 19.848808
\botline
\end{tabular}
\end{table}

%tab4
\begin{table}
\caption{\label{tab4}Exact and estimated results using different
optimisation methods (beam II)}
\tabcolsep=3pt
\begin{tabular}{ccccccccc}
\thead
Cases & \parbox[t]{1.8cm}{\centering Actual\par crack \par length~(mm)}
& Cuckoo & Bat & PSO & YUKI & 
\parbox[t]{1.5cm}{\centering ANN\par NHLS $=$ 8 \par\cite{bib47}}
& \parbox[t]{1.7cm}{\centering ANN\par NHLS $=$ 10 \par\cite{bib47}} & 
\parbox[t]{1.7cm}{\centering ANN\par NHLS $=$ 12 \par\cite{bib47}} \vspace*{2pt}\\
\endthead
1 & \08  & \07.9389 & \07.9388 & \07.9388 & \07.9383 & \0\07.807456 & \08.18362 & \08.93587 \\
2 & 15  & 15.2842 & 15.2840 & 15.2840 & 15.2848 & 15.32850 & 15.22022 & 15.47560 \\
3 & 25  & 24.9308 & 24.9308 & 24.9308 & 24.9353 & 24.90070 & 25.19147 & 25.11821
\botline
\end{tabular}
\end{table}

Examination is made on the training points, Figure~\ref{fig5} shows the
absolute Errors for Different Methods (beam I). The discussion involves
a thorough examination of errors in identifying cracks, as presented in
Figure~\ref{fig6}.~That includes two important measures: Mean Error and
Standard Deviation, which are used to evaluate the accuracy and
consistency of various methods used for crack identification. Mean
Error represents the average difference between predicted and actual
values. In the context of crack identification, a lower Mean Error
indicates better accuracy. On the other hand, a lower Standard
Deviation suggests more consistent and predictable results.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Absolute errors for different methods (beam I).}
\end{figure}

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Mean error with standard deviations for different
methods (beam I).}
\end{figure}

When examining the data presented in the table, several noteworthy
observations can be made. To begin with, three methods---Cuckoo, Bat,
and PSO---display nearly identical Mean \mbox{Percentage} Error and Standard
Deviation values. Their Mean Percentage Error stands at approximately
4.26\%, indicating an average crack identification accuracy slightly
above 50\%. However, their Standard Deviation is relatively high, at
about 1.20, signifying significant variability in individual results.
The YUKI algorithm, on the other hand, demonstrates a slightly better
performance, with a Mean Percentage Error of 4.19, still falling within
the 50\% accuracy range. Its Standard Deviation is approximately 1.15,
suggesting a moderate level of consistency in its \mbox{predictions.}

Moving on to the Artificial Neural Network model with NHLS $=$ 8, it
exhibits a noticeable improvement with a Mean Percentage Error of 3.34.
This suggests enhanced accuracy compared to the earlier methods.
Additionally, its Standard Deviation is lower, at around 2.29,
indicating increased consistency in its predictions. The ANN model with
NHLS $=$ 10, while having a higher Mean Error than the NHLS $=$ 8 model
(4.94\%), compensates with an exceptionally low Standard Deviation of
4.07, implying highly consistent results. Lastly, the ANN model with
NHLS $=$ 12 performs the best, boasting a Mean Percentage Error of 2.64
and an impressively low Standard Deviation of 1.65. These values
collectively indicate both high accuracy and exceptional consistency in
its \mbox{predictions.}

When analysing the errors depicted in Figures~\ref{fig7}
and~\ref{fig8}, it becomes evident that the three methods, namely
Cuckoo, Bat, and PSO, demonstrate very similar Mean Error and Standard
Deviation values, all hovering around 0.98 and 0.83, respectively. This
suggests that, on average, these methods achieve a relatively high
level of accuracy in crack identification, with a minimal degree of
variability in their results. While the YUKI algorithm performs
slightly better, indicating a marginally improved accuracy, it still
falls within a similar range as the previous methods, suggesting
moderate consistency in its predictions.

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Absolute errors for different methods (beam II).}
\end{figure}

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Mean error with standard deviations for different
methods (beam II).}
\end{figure}

The Artificial Neural Network (ANN) model with NHLS $=$ 8 exhibits a
Mean Error of 1.66\%, indicating a slightly lower level of accuracy
compared to the previous methods. However, its Standard Deviation of
1.10 suggests a reasonable level of consistency in its predictions. 

For NHLS $=$ 10, the Mean Error improves to 1.51\%, indicating enhanced
accuracy compared to the NHLS $=$ 8 model. What's more, it's
remarkably low Standard Deviation of 0.77 suggests a high degree of
consistency and predictability in its crack identification results.
NHLS $=$ 12 stands out with a substantially higher Mean Error of
5.11\%, signifying relatively lower accuracy compared to other methods.
Furthermore, its Standard Deviation of 5.86 is notably higher,
indicating significant variability in its predictions and rendering it
less reliable compared to other\break methods.

In the evaluation of crack identification methods for both Beam I and
Beam II, we can observe certain similarities and differences between
the approaches. Firstly, when considering the Mean Percentage Error
values, the Cuckoo, Bat, and PSO methods exhibit comparable
performance. In~Beam I, they have an average error of approximately
4.26\%, while in Beam II, this value drops to around 0.98. This
suggests that these methods achieve a moderate to high level of
accuracy. However, it's important to note that these methods also
display significant variability in their results, as indicated by their
relatively high Standard Deviation values. In particular, Beam I show a
higher level of result variability for these methods with a Standard
Deviation of about 1.20, whereas Beam II exhibits a lower Standard
Deviation of approximately 0.83 for the same\break methods.

On the other hand, the YUKI algorithm outperforms the Cuckoo, Bat, and
PSO methods in terms of Mean Percentage Error in both Beam I and Beam
II, with an average error of around 4.20\% in Beam I and 0.98\% in Beam
II. While it demonstrates slightly better accuracy, it still falls
within the same range. Furthermore, the Standard Deviation values for
the YUKI algorithm are relatively consistent in both Beam I and Beam
II, indicating a degree of stability in its predictions. Thus, the YUKI
algorithm consistently delivers reliable results.

\section{Conclusion}

In this research paper, a comprehensive exploration of structural
health monitoring (SHM) techniques, particularly focused on
vibration-based damage detection methods, has been presented. The study
investigated the integration of nondestructive evaluation techniques
with vibration-based methodologies, aided by advanced machine learning
tools, to assess and monitor structural integrity. The research
emphasised the critical importance of ensuring structural stability and
safety across various industrial sectors, aiming to prevent
catastrophic failures. Key~methodologies employed in this investigation
include artificial neural networks, proper orthogonal \mbox{decomposition}
(POD), radial basis functions (RBF), and the innovative YUKI algorithm,
which introduces a population clustering approach to crack length
prediction.

The study's findings have highlighted the potential of these
methodologies in effectively detecting and localising structural
damage. Notably, the YUKI algorithm has shown promise in achieving
highly accurate predictions of crack lengths. However, it is essential
to acknowledge the inherent variability in predictive outcomes across
different scenarios and use cases, necessitating further comprehensive
investigations and rigorous validation procedures. Overall, this
research has contributed valuable insights into the application of SHM
techniques and their potential for enhancing structural safety. Future
research endeavours should focus on refining these methods and
assessing their suitability for real-world structural health monitoring
scenarios.

{\vspace*{2pt}}

\section*{Declaration of interests}

{\vspace*{2pt}}

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.

\back{}

{\vspace*{2pt}}

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