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                                                                P. Ghannadi et alii, Frattura ed Integrità Strutturale, 64 (2023) 51-76; DOI: 10.3221/IGF-ESIS.64.04 
 

51 
 

 

 
 
 
A review of the application of the simulated annealing algorithm  
in structural health monitoring (1995-2021) 
 
 
Parsa Ghannadi 
Department of Civil Engineering, Ahar Branch, Islamic Azad University, Ahar, Iran 
parsa.ghannadi@gmail.com, https://orcid.org/0000-0001-5441-9243 
 
Seyed Sina Kourehli 
Department of Civil Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran 
ss.kourehli@azaruniv.ac.ir, https://orcid.org/0000-0001-7599-8053 
 
Seyedali Mirjalili 
Centre for Artificial Intelligence Research and Optimisation, Torrens University, Adelaide, SA 5000, Australia 
Yonsei Frontier Lab, Yonsei University, Seoul, South Korea 
ali.mirjalili@torrens.edu.au, https://orcid.org/0000-0002-1443-9458 
 
 
ABSTRACT. In recent years, many innovative optimization algorithms have 
been developed. These algorithms have been employed to solve structural 
damage detection problems as an inverse solution. However, traditional 
optimization methods such as particle swarm optimization, simulated 
annealing (SA), and genetic algorithm are constantly employed to detect 
damages in the structures. This paper reviews the application of  SA in 
different disciplines of structural health monitoring, such as damage detection, 
finite element model updating, optimal sensor placement, and system 
identification. The methodologies, objectives, and results of publications 
conducted between 1995 and 2021 are analyzed. This paper also provides an 
in-depth discussion of different open questions and research directions in this 
area.  
 
KEYWORDS. Simulated Annealing, Inverse Problem, Structural Health 
Monitoring, Damage Detection 
 

 

 
 

Citation: Ghannadi, P., Kourehli, SS., 
Mirjalili, SA., A review of the application of 
the simulated annealing algorithm in structural 
health monitoring (1995-2021), Frattura ed 
Integrità Strutturale, 64 (2023) 51-76. 
 
Received: 13.09.2022  
Accepted: 06.01.2023 
Online first: 18.01.2023  
Published: 01.04.2023  
 
Copyright: © 2023 This is an open access 
article under the terms of the CC-BY 4.0, 
which permits unrestricted use, distribution, 
and reproduction in any medium, provided the 
original author and source are credited. 

 

 
 
INTRODUCTION  
 

he vital role of structures and infrastructures is undeniable in organizing voluminous human activities in large 
societies [1]. These structures are exposed to damage due to extreme conditions such as strong earthquakes, winds, 
or human-induced events [2,3]. Many structures have been constructed and are still in service despite expiration [4]. 

For instance, approximately 40% of bridges are more than fifty years old in the United States of America [5]. To provide 
structural safety and avoid human disasters and also financial losses, structural health monitoring (SHM) is a necessary 

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procedure for civil infrastructures and structures [6]. In 2007, I35W Bridge (Mississippi, Minneapolis, USA) collapsed. 
Unfortunately, 13 died, and also 145 were injured. In addition, significant financial losses were imposed. This incident and 
similar ones could be prevented by implementing suitable SHM systems and detecting damages at their early stages [7]. 
Structural damage detection is the central part of SHM systems, consisting of automatic procedures for identifying and 
quantifying existing damages. The schematic of this process is briefly illustrated in Fig. 1. Typical damage detection strategies 
include three phases. In the first phase, several accelerometers measure acceleration responses. In the second phase, data 
acquisition systems are employed to collect data. The measured data are processed in the third phase through different 
damage detection and quantification algorithms [8]. It should be noted that the acceleration signals for a large-scale structure 
such as Milad Tower (shown in Fig. 1) are usually measured under ambient excitation [9]. Local stiffness decreases due to 
structural damage [10]. This stiffness loss is reflected in dynamic characteristics such as natural frequencies and mode shapes. 
The variation of dynamic characteristics before the damage occurrence and damaged state can be analyzed through 
vibration-based damage identification methodologies for detecting the damage and quantifying its extent [11].  
Vibration-based methods are classified into two divisions: I) Response-based methods II) Model-based methods.  
Response-based methods are usually categorized as nonparametric approaches, and there is no need for finite element 
simulation as a baseline model. These methods typically require experimental response data and can only detect damaged 
elements. Response-based methods are a proper selection to establish a real-time SHM system because of their low 
computational cost [12]. In this regard, several signal processing techniques based on wavelet transformation and  
Hilbert–Huang transform have been introduced to address the structural damage detection problem more  
sensitively [13–18]. Model-based methods can identify both location and severity of the damaged elements. Experimental 
measurements and FEM of the structures are required to put model-based approaches into practice [12]. The following 
difficulties arise while using these techniques: 
a. The numerical models should accurately represent the behavior of the structures. Therefore, developing a high-fidelity 

FEM of the complex structures takes considerable effort [19]. To perform a dynamic analysis [20] of Milad Tower 
(shown in Fig. 1), a reliable FEM is carried out by Strand7 software [21], which can be implemented in future damage 
detection methodologies. 

b. There are some differences between the experimental results and those obtained by FEM due to the uncertainties in 
boundary conditions, material properties, and geometry [22]. Therefore, FEM updating is implemented as a crucial 
procedure to meet a good agreement between the measured and calculated modal characteristics [23]. A survey of FEM 
updating techniques in structural dynamics was presented by Mottershead and Friswell [24]. Recently, in 2015, another 
review paper in the area of FEM updating was published [25]. A comparative study of existing FEM updating methods 
has been conducted by Arora [26].  

c. In real-world SHM projects, the size of measured degrees of freedom (DOFs) does not match the full set of DOFs of 
the FEM [27] because measuring the mode shapes at all DOFs is not practical, and there is no economic justification 
for it. To overcome incomplete measurements, either FEM reduction or mode shape expansion methods can be utilized 
[28]. Ghannadi and Kourehli have investigated the efficiency of different FEM reduction techniques [29]. Dinh-Cong 
et al. presented a comparative study of different dynamic condensation methods in detecting damages in plate-like 
structures [30]. Some damage detection methodologies based on expansion techniques can be found in Refs. [31–34].  

d. Using complex FEM such as Milad Tower (shown in Fig. 1) is not practical for structural damage detection because of 
the extensive computational workload [35]. Hence, some simplified models are typically developed to represent the 
dynamic behavior of the structures. The simplified models can significantly reduce the computation time [36]. To detect 
damages, predict seismic responses, and optimize sensor locations, the FEMs of some famous structures such as 
Guangzhou New TV Tower [37], Shanghai Tower [36,38], MIT Green Building [39,40], and Dalian World Trade 
Building [41] were simplified. Pourkamali-Anaraki and Hariri-Ardebili have presented a two-step uncertainty 
quantification method that uses a simplified alternative model of Milad Tower [42]. The classification of vibration-based 
damage detection methods is illustrated in Fig. 2. 

In the recent two decades, model-based structural damage identification through an iterative optimization process has 
received significant attention [12]. The earliest damage detection methods have been developed based on the genetic 
algorithm (GA) [43–45]. Dynamic characteristics such as natural frequencies and mode shapes are employed to construct 
an objective function when using optimization-based damage detection methods [12]. In recent years, some novel 
optimization algorithms have been rapidly developed. For instance, several researchers have employed moth-flame [46], 
salp swarm [47], multiverse [48], whale [49], YUKI [50,51], wild horse [52] and slime mold [53] algorithms to solve the 
inverse problem of damage detection. However, conventional optimization methods such as simulated annealing (SA), 
particle swarm optimization (PSO) [54], and GA are still often used in damage identification problems. During the last two 
decades, the application of the SA algorithm is not limited to damage detection problems but also has other functions in 



 

                                                                P. Ghannadi et alii, Frattura ed Integrità Strutturale, 64 (2023) 51-76; DOI: 10.3221/IGF-ESIS.64.04 
 

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terms of SHM, such as optimal sensor placement, system identification, and FEM updating. This paper presents a tabulated 
review of the application of the SA algorithm in the field of SHM (1995-2021). This paper investigates roughly 30 previously 
published studies to discuss objectives, methodologies, types of structures, and overall results of recent articles.  
Some review papers in different disciplines were conducted between 1987 and 2018 (Tab. 1). The number of review papers 
on other fields is also depicted in Fig. 3. It can be observed that the present paper is the first review study on structural 
damage detection and families of SHM, such as FEM updating, system identification, and optimal sensor placement. 
 

 
Figure 1: The schematic of  vibration-based damage detection procedures (3D model courtesy of  Faham Tahmasebinia). 

 

Table 1: List of  review papers on applications of  SA algorithm in various fields. 

Ref. Year Title 

Aarts and van Laarhoven [55] 1987 Simulated annealing: a pedestrian review of the theory and some applications 

Koulamas et al. [56] 1994 A survey of simulated annealing applications to operations research problems 

Mavridou and Pardalos [57] 1997 Simulated annealing and genetic algorithms for the facility layout problem: A survey 

Suman and Kumar [58] 2006 A survey of simulated annealing as a tool for single and multiobjective optimization 

Nandhini and Kanmani [59] 2009 A survey of simulated annealing methodology for university course timetabling 

Hooda and Dhingra [60] 2011 Flow shop scheduling using simulated annealing: A review 

Pattanaik et al. [61] 2012 Simulated annealing based placement algorithms and research challenges: A survey 

Kaushik and Ghosh [62] 2014 A survey on Optimization Approaches to K-Means Clustering using Simulated Annealing 

Siddique and Adeli [63] 2016 Simulated annealing, its variants and engineering applications 

Sibalija [64] 2018 Application of simulated annealing in process optimization: A review 



 

P. Ghannadi et alii, Frattura ed Integrità Strutturale, 64 (2023) 51-76; DOI: 10.3221/IGF-ESIS.64.04                                                                   
 

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Figure 2: Classification of  vibration-based damage detection methods (3D model courtesy of  Faham Tahmasebinia). 



 

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Figure 3: Number of  review papers on different applications of  SA. 

 
 

SIMULATED ANNEALING (SA) ALGORITHM AND PROBLEM DEFINITION 
 

he SA is a widely used optimization technique that mimics the annealing procedure of solids [65,66].  
Kirkpatrick et al. [67] and Černý [68] each independently developed the SA algorithm. This procedure is a physical 
activity that produces high-quality materials by cooling them gradually from a high temperature [69]. Therefore, the 

initial solution randomly generates from a hot temperature. Then, the temperature slowly reduces, and the optimal solution 
achieves [65]. However, Metropolis et al. [70] introduced an algorithm for efficiently simulating the evolution of a solid to 
thermal equilibrium in 1953 for the first time [71]. After approximately 30 years, Kirkpatrick et al. [67] and Černý [68] 
realized that the optimization problems could be solved by implementing the Metropolis criterion. In other words, there is 
a significant analogy between minimizing the cost function of an optimization problem and the slow cooling of a solid till 
it reaches the ground state, which is little energy [71]. Finally, Kirkpatrick et al. [67] presented the SA algorithm by adjusting 
the cost for energy and performing the Metropolis algorithm at a series of gradually decreasing temperature levels [71]. The 
SA algorithm attempts to prevent entrapping in the local optimal solution through the Metropolis criterion and performs 
extra random searches in the neighborhood of the candidate solution [72]. Fig. 4 shows the flowchart of the SA algorithm.  
The Metropolis rule determines how a thermodynamic system changes from state Xold to state Xnew [73,74]. The acceptance 
probability is as follows: 
 

   
   

   
 

 
 


 
 

1

exp
new old

new old

E X E X

T
new old

if E X E X
p

if E X E X
                (1) 

 
where T is the temperature, E(Xnew) and E(Xold) represent the energy of  the system in states Xnew and Xold, respectively [65]. 
Different researchers have assessed the applicability of  the SA algorithm to various engineering problems, especially in 
structural engineering. One of  the earliest applications of  SA related to the weight optimization of  a 10-bar cantilever truss 
was published in 1988 [75]. In another work, Balling [76] implemented the SA for the optimal design of  three-dimensional 
steel frames. Shim and Manoochehri [77] introduced a combinatorial optimization procedure based on the SA algorithm to 
generate the optimal configuration of  structural members. Leite and Topping [78] studied the efficiency of  the parallel 

T 



 

P. Ghannadi et alii, Frattura ed Integrità Strutturale, 64 (2023) 51-76; DOI: 10.3221/IGF-ESIS.64.04                                                                   
 

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version of  SA in structural optimization problems. Over the past two decades, successful applications of  the SA have been 
reported several times.  For example, Bureerat and Limtragool [79], Lamberti [80], Sonmez and Tan [81], Tejani et al. [82], 
Kurtuluş et al. [83], Najafabadi et al. [84], and Goto et al. [85] are a few to be noted. 
It is necessary to define an objective function based on measured and calculated modal characteristics to solve the  
model-based inverse problem of  damage detection using optimization algorithms [86]. The inverse analysis using an 
optimization algorithm attempts to find the optimal design parameters for damage detection by minimizing the differences 
between the measured and calculated modal characteristics [87,88]. An inverse problem of  damage detection has the 
following mathematical formulation [89]: 

 
 
 

 

  



 



  

1 2F

t

ind , , ..

0

Min

,

imiz

.,

1 1, 2 ..o .

e

Subjec , t

Ne

e

f

e Ne

         (2) 

 
where  f  denotes the objective function, Ne represents the number of elements, and is a vector that includes  

stiffness reduction coefficients [89].  
The elemental mass matrix is typically assumed not to alter due to damage, and stiffness coefficients define structural  
damage [89,90]. If the stiffness coefficient of the eth element is considered to be e , the global stiffness matrix will be  
represented as the sum of the damaged and undamaged stiffness matrices [91] and can be expressed as: 
 

 


 
1

1
Ne

e e
e

K K            (3)  

 
where Ke denotes the stiffness matrix of the eth element and e is considered in the range of 0 to 1 and indicates the  
severity of the damage [92].  
For example, one of the frequently utilized objective functions in damage detection problems is given by Eqn. (4). 
 

   


 
   

 


2

1

1 Measured Calculated

Me
i

asure
i

d

m
i

i

f w
m

         (4) 

 

where m represents the number of  modes considered , w is a weighting factor, Meas
i

ured and Calcu
i

lated are the ith measured 
and calculated natural frequencies, respectively [89].  
 
 
ANALYSIS OF PAPERS ON STRUCTURAL HEALTH MONITORING USING THE SA ALGORITHM (1995-2021) 
 

he purposeless strategy for reading scientific articles is to read them like a textbook: start with the title and read 
through the list of references, taking in each word without any critical thought or analysis. The active reviewers 
should be able to identify the main structure and answer the following critical questions by skimming the article 

[93].  
• The objective of this study - Why was it conducted? 
• The methodology of this study - How was it conducted? 
• The result and finding of this study - What was found? 

This review paper enables the researchers to effectively comprehend the previously published papers' objectives, 
methodologies, and results. Tab. 2 presents a classified review of the SA algorithm's application in the context of FEM 
updating, system identification, optimal sensor placement, and especially structural damage detection. 

 
 
 
 

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Figure 4: Flowchart of  SA algorithm [65]. 
 
 
 
 
 



 

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Ref. Year Objective Methodology Structure Result and Finding 
Keilers 

and 
Chang 

[94] 

1995 This study presents a 
methodology to find the 
size and location of the 
delamination when built-in 
sensors are embedded in 
the laminated composite 
beams. 

A variant of SA is 
employed to minimize 
the weighted quadratic 
objective function and 
establish an agreement 
between the calculated 
and measured frequency 
responses.  Finally, the 
dimension of the 
delamination is estimated 
when the minimization 
process is over. 

Laminated 
composite 

beam 

This study showed the feasibility 
of the proposed delamination 
detection method based on built-
in sensors, actuators, and an 
optimization procedure for 
laminated composite beams.    
The optimization technique is a 
particular variant of the SA 
algorithm and enables parallel 
searching for numerous local 
minima until the global minimum 
is discovered. However, 
remarkable efforts are necessary 
for accurate damage detection 
under noisy conditions. 

Jeong and  
Lee [95] 

1996 This paper introduces a 
hybrid method known as 
the adaptive simulated 
annealing genetic algorithm 
(ASAGA). GA has a low 
capability in hill-climbing.  
In the opposite state, SA 
very well supports 
probabilistic hill-climbing.   
Therefore, ASAGA enjoys 
the merits of GA and SA 
simultaneously. Finally, the 
efficiency of the hybrid 
algorithm is demonstrated 
by a system identification 
example. 
 

Not  : System identification 
is an approach to 
developing a  mathematical 
model of a dynamic system 
through input and output 
measurements [96]. 

ASAGA is applied to 
estimate the model 
parameters of the  
auto-regressive moving 
average with exogenous 
excitation (ARMAX). 
Then, the outcomes were 
compared with those 
obtained by GA and a 
gradient algorithm. 

Discrete-time 
system 

The comparative results showed 
that the ASAGA is superior to the 
GA and a gradient algorithm.  
Besides, ASAGA improved GA's 
poor hill-climbing capability and 
accelerated the convergence.  
Therefore, the combination of SA 
and GA provides an efficient 
optimization algorithm. 

Levin and  
Lieven 

[97] 

1998 This study proposed the 
blended simulated 
annealing algorithm (BSA) 
for adjusting mass and 
stiffness during the FEM 
updating procedure. For 
additional investigation, the 
performance of BSA is also 
compared with GA. 

The optimization-based 
FEM updating is 
performed based on 
different objective 
functions in the 
frequency domain. 

Cantilever 
beam 

 
Flat plate wing 

It was concluded that the BSA 
provides better results than the 
GA in all examined cases. By 
increasing the discretization level, 
GA could improve the results. But 
the computation time increases 
considerably. 
The results of this study also 
clearly showed that the degree of 
agreement between the numerical 
model and experimental data 
depends on updating parameters.  
The updated model correlates 
sufficiently with experimental 
measurements when many 
parameters are taken. In contrast, 
limited updating parameters could 
not provide a reliable correlation. 



 

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Ref. Year Objective Methodology Structure Result and Finding 
He and  
Hwang 

[98] 

2006 GA encodes variables as  
finite-length  strings. 
Therefore,  excessive 
computation time is 
required. A new variant of 
GA called the  
real-parameter  genetic 
algorithm (RGA) was 
proposed to overcome this 
major drawback of GA.   
RGA saved much 
computation time by 
directly handling the 
variables without coding.   
However, GA is still weak 
in hill-climbing. A new 
hybrid algorithm based on 
SA and RGA was 
developed to address the 
low hill-climbing ability 
and slow convergence of 
GA, known as the adaptive  
real-parameter simulated 
annealing genetic algorithm 
(ARSAGA). 

Three objective functions 
are defined to formulate 
the damage identification 
problem as an 
optimization scheme. 
The first one is 
established on the 
discrepancy between the 
calculated and measured 
displacements.   The 
second objective 
function is based on 
natural frequency 
changes. The last is the 
sum of the first and 
second objective 
functions. 

Curved beam 
 

Cantilever 
beam 

 
Clamped -

Clamped beam 

The results indicated that the 
combination of SA and RGA 
(ARSAGA) modifies RGA's 
convergence speed and searching 
ability.  Additionally, the proposed 
algorithm is efficient for detecting 
the damages in  
beam-like structures with various 
boundary conditions, even under 
noisy and incomplete 
measurement conditions. 

Marwala 
[99] 

2007 Three optimization 
algorithms, including PSO, 
GA, and SA, are employed 
for FEM updating. Then, 
the correlation of updated 
mode shapes and natural 
frequencies is compared 
with corresponding 
measured ones.  
The computation time for 
the optimization 
algorithms was reported. 

This study considers the 
modulus of elasticity as a 
design parameter for 
FEM updating. The 
weighted objective 
function relying on the 
relative interval between 
the measured and 
calculated natural 
frequency is adopted for 
the minimization 
process. 

Irregular  
H-shaped 
structure 

The results of this study showed 
superior performance of PSO in 
terms of correlation between 
measured and updated dynamic 
characteristics. Moreover, PSO is 
the fastest optimization algorithm 
compared to GA and SA. 

He and  
Hwang 
[100] 

2007 This study suggests a  
two-step strategy through 
grey relation analysis 
(GRA) and ARSAGA for 
structural damage 
detection. The main 
contribution of this study is 
the use of narrowed search 
space and a reduced 
number of design variables 
as possible damage 
elements. 

The undamaged  
elements are first 
excluded using GRA. In 
the second step, 
ARSAGA solves the 
optimization problem by 
minimizing an objective 
function defined by the 
differences between the  
measured and calculated 
displacements. Finally, 
the damaged locations 
and their corresponding 
severities are determined. 

Plane truss 
 

Clamped -  
Clamped beam 

The results of this paper can be 
summarized as follows: 
I) Where the measured 

data includes a 5% error, 
the damage parameters 
of the clamped beam are 
accurately identified.   

II) Where there is a 5% 
error in the measured 
data, the predicted 
results of the plane truss 
are reasonable but not 
accurate. 

III) Under error-free 
conditions, the proposed 
technique efficiently 
identifies the damages in 
both structures. 

The GRA plays an undeniable role 
in this proposed damage detection 
strategy, as poor results are 
obtained if GRA is not utilized. 



 

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Ref. Year Objective Methodology Structure Result and Finding 
Jie and  
Aiqun 
[101] 

2008 To have a quick 
convergence,  
auto-optimizing network 
topology, and to avoid 
being entrapped into the 
local minimums, the 
backpropagation (BP) 
neural networks were 
improved by the SA 
algorithm, momentum 
item, bold driver 
technology, and stochastic 
hill-climbing algorithm. 

The learning process in 
modified BP neural 
networks avoids 
entrapment into the local 
minimums by 
implementing the SA 
algorithm. The topology 
of BP neural networks 
consists of input, hidden, 
and output layers 
optimized by the 
stochastic hill-climbing 
algorithm. 

Suspension 
bridge 

The Runyang Yangzi suspension 
bridge's hanger damage and its 
pattern could be successfully 
identified by the enhanced BP 
neural networks employing SA. 

Bayissa 
and  

Haritos 
[102] 

2009 Some difficulties, such as  
high-dimensionality of 
search space, nonlinearity, 
modeling error, and 
measurement noise, are 
encountered in the  
model-based damage 
detection method 
formulated as an 
optimization problem. This 
study presents a two-step 
approach to damage 
localization and 
quantification.  The 
presented methodology 
simultaneously applies  
non-model-based and 
model-based methods. The 
first step considers the  
damage-sensitive response 
parameters (DSRP) as a  
non-model-based method.   
The model-based 
technique minimizes an 
objective function via 
adaptive simulated 
annealing (ASA).   
ASA was employed as an 
optimization algorithm to 
address the challenge of 
extensive computation 
time in the standard SA. 

DSRP was initially 
determined by the 
statistical moments to 
find the damaged 
elements of the structure.   
In the second step, ASA 
is adopted using a  
model-based method to 
identify the damage 
severity by minimizing an 
objective function based 
on the DSRP. 

Simply 
supported 

beam 
 

I-40 bridge 

The results indicate that the 
presented technique could find 
the damaged elements using 
DSRP in the first step. By 
implementing ASA in the second 
step, the damage severities can be 
identified swiftly, even though 
incomplete and noisy data are 
utilized. 

Begambre 
and Laier 

[103] 

2009 A new algorithm based on 
the Nelder–Mead 
algorithm was developed 
and known as the particle 
swarm optimization 
–simplex algorithm (PSOS) 
to control the parameters 
of the PSO. Then, the 
performance of  PSOS  was 
benchmarked with the SA 
algorithm in different 
damage detection 
problems and benchmark 
functions. 

The inverse damage 
identification problem is 
formulated as the 
differences between the 
measured frequency 
response function (FRF) 
and the calculated FRF. 

Plane truss 
 

Free–Free 
beam 

Combining the standard PSO with 
the Nelder–Mead algorithm can 
improve the optimization's 
capability to find the global 
optimum in damage detection 
problems and mathematical 
benchmark functions. PSOS also 
performs better than the SA 
algorithm in all examinations. 



 

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Ref. Year Objective Methodology Structure Result and Finding 
Ferreira 

and  
Gomes 
[104] 

2009 A comparative study  
between the SA algorithm 
and GA was conducted on 
the localization and 
quantification of the 
damaged elements of the 
structure only by frequency 
datasets. 

Two objective functions 
based on natural 
frequencies are 
investigated for damage 
identification.  The first 
objective function is the 
multiple damage location 
assurance criterion 
(MDLAC), and the 
second is the normalized 
form of natural frequency 
differences. 

Cantilever 
beam 

 
Simply 

supported 
beam 

This paper can conclude with the 
following points: 
I) Both optimization 

algorithms, SA and GA, 
could detect the damages 
and their corresponding 
severities with similar 
accuracy and 
computation time. 

II) In most experimental 
examples, the damaged 
locations are identified 
by both GA and SA.  
However, neither 
algorithm accurately 
assessed the damage 
severity.   

Worden 
et al. [105] 

2009 This study compares the 
SA algorithm and GA for 
crack detection in  
beam-like structures. 

An objective function has 
been formed by 
combining the natural 
frequency and mode 
shape components. 

A numerical 
beam model 

 
An 

experimental 
cantilever beam 

The results obtained by numerical 
and experimental investigations 
showed that both algorithms  (SA 
and GA) could identify the correct 
extent and location of the 
damages. However, the GA needs 
several runs to provide a 
reasonable convergence rate. 

Marwala 
[106] 

2010 In this study, four 
optimization algorithms, 
including GA, PSO, SA, 
and the response-surface 
method (RSM), were 
employed in FEM 
updating. The efficiency of 
each algorithm was 
assessed by the agreement 
of updated and measured 
dynamic characteristics. 
Besides, some results and 
discussions were outlined 
on the computing speed. 

The differences between 
computed and measured 
natural frequencies and 
related mode shapes were 
used to create a weighted 
objective function.   
Young's modulus of 
elements in this study was 
considered as updating 
parameters. 

Free–Free 
beam 

 
Unsymmetrical 

H-shaped 
structure 

It was observed that PSO could 
provide more accurate results 
compared to SA, GA, and RSM.  
However, RSM is computationally 
more efficient than other 
algorithms. 

Marwala 
[107] 

2010 Comparing the capability 
of SA and PSO for FEM 
updating is the key aim of 
this research. 

The same methodology 
and objective function, as 
in Ref. [106], were 
applied once more in this 
study. 

Free–Free 
beam 

 
Unsymmetrical 

H-shaped 
structure 

The following results are obtained 
for the first example, Free–Free 
beam: 
I) When using PSO, the 

error between the 
measured and updated 
natural frequencies in 
the first to fourth modes 
are 0.0%, 1.8%, 0.0%, 
and 0.2%, respectively. 
When using SA, the 
above errors are 1.9%, 
0.2%, 0.5%, and 0.3%.  
Therefore, the PSO 
yielded better results 
with an average error 
rate of 0.5%. 

II) When using SA and 
PSO, the average  modal 



 

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Ref. Year Objective Methodology Structure Result and Finding 
assurance criterion 
(MAC) value between 
the measured and 
updated mode shapes 
was 0.9989. 

The following results are obtained 
for the second example, an 
unsymmetrical H-shaped 
structure: 
I) When using PSO, the 

error between the 
measured and updated 
natural frequencies in 
the first to fifth modes 
are 0.0%, 0.4%, 0.1%, 
0%, and 1.5%, 
respectively.   
When using SA, the 
above errors are    0.2%, 
1.3%, 0.6%, 0.1%, and 
2.1%.   Hence, the PSO 
provides accurate results 
with an average error 
rate of 0.4%. 

II) When using PSO and 
SA, the average MAC 
values between the 
measured and updated 
mode shapes were 
0.8434 and 0.8426, 
respectively. 

Generally, PSO presented 
accurate results in terms of FEM 
updating.  

Zhang 
and Sun 

[108] 

2011 This paper presents a  
two-step method to 
localize the damage and 
quantify its severity.  The 
proposed method relies on 
BP neural networks in the 
first stage and a hybrid 
optimization algorithm 
known as  
genetic-simulated 
annealing (GSA) in the 
second. 

In the first step, a  
three-layer feedforward 
neural network is 
employed to train 
samples and detect the 
damaged locations. Then, 
the damaged elements' 
extent was identified by 
minimizing an objective 
function based on 
displacement differences.   

Suspension 
bridge 

The results from the first step 
confirm that the BP neural 
networks can recognize the 
damage sites in the Nancha 
bridge.  It should be noted that the 
accuracy of BP neural networks 
mainly depends on having enough 
samples coming from finite 
element analysis or field 
measurements.  In the second step 
and during the process of 
estimating the  damage severity, 
the GSA performs a better 
convergence compared to the GA. 

Kourehli 
et al. [109] 

2013 In real-world health 
monitoring projects, the 
incompleteness of the 
measured data is a 
challenging problem.  
Consequently, this paper 
proposes an optimization-
based methodology to 
address the challenge of 
incomplete static and 
dynamic measurements.   

Three different objective 
functions with 
incomplete static and 
dynamic characteristics 
are established.  The first 
objective function is the 
dynamic residue force 
vector and accepts 
incomplete mode shapes 
and frequencies.  The 
second objective 

Simply 
supported 

beam 
 

Plane frame 
 

Spring-mass 
system 

The proposed method based on 
incomplete dynamic and static 
data and applying the SA 
algorithm could present 
promising results for numerical 
and experimental examples. 



 

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63 
 

Ref. Year Objective Methodology Structure Result and Finding 
This study applied the SA 
algorithm to optimize the 
objective functions. 

function is formulated 
using the differences 
between the measured 
and calculated 
incomplete 
displacements.  The last 
objective function is the 
weighted static residue 
force vector with 
incomplete 
characteristics. 

Al-Wazni 
et al. [110] 

2014 The SA algorithm was 
employed to minimize a 
hybrid objective function 
and determine the location 
and severity of the 
structural damages. 

A hybrid objective 
function was proposed, 
including weighted 
natural frequency and 
displacement 
components. It should be 
noted that the objective 
function only contained 
the first five vibration 
modes. 

Simply 
supported 

beam 

The presented study illustrates the 
efficiency of the SA algorithm and 
the proposed hybrid objective 
function for accurately detecting 
the damage in a simply supported 
beam with ten discretized 
elements. 

Tong et 
al. [111] 

2014 An improved version of 
the SA algorithm with 
search capability in 
multiple dimensions was 
developed. This modified 
version attempts to provide 
an optimal combination of 
sensor configurations. The 
performance of the 
improved SA algorithm 
was also compared with 
that of GA. 
 

Note: The optimal sensor 
placement is an  essential 
phase in the  
vibration-based SHM 
methods. 

Three objective functions 
were considered to solve 
the optimal sensor 
placement problem with 
two hundred sensor 
location candidates. The 
fisher information matrix 
(FIM), the mode shapes' 
mean square error 
(MSE), and the MAC as 
the sensor arrangement 
criteria are used to 
establish the first, second, 
and third objective 
functions. 

Rectangular 
concrete slab 

 
 

The results indicate that the 
proposed method outperforms 
GA and standard SA regarding 
optimal sensor placement.   
Besides, more minor mode shape 
errors were obtained using MAC 
and MSE as the objective 
functions. The results indicate that 
the MAC function performs 
better in the optimal arrangement 
of many sensors. 

Stutz et al. 
[112] 

2015 This paper presents a 
damage identification 
approach defined as an 
inverse problem through 
the minimization of an 
objective function using 
four optimization 
algorithms: differential 
evolution stochastic 
optimization (DESO), 
PSO, SA, and  
Luus–Jaakola. 

The differences between 
the measured and 
calculated flexibility 
matrix defined an 
objective function. It 
should be emphasized 
that the statically reduced 
stiffness matrix is applied 
to form the  
reduced-order flexibility 
matrix. 

Simply 
supported 

beam 

Results showed that the flexibility 
matrix is a sensitive index in 
damage identification, and DESO 
yielded satisfactory results 
compared with other optimization 
techniques, including SA, PSO, 
and Luus–Jaakola.  
The proposed method can work 
under noisy conditions and 
incomplete measured data. 

Astroza et 
al. [113] 

2016 The computational 
efficiency, accuracy, and 
convergence rate of the 
conventional SA method 
were enhanced by a hybrid 
technique known as the  
SA-unscented Kalman 
filter. This modified 

Two objective functions 
based on acceleration 
time series data and 
dynamic characteristics 
such as natural 
frequencies and mode 
shapes are established to 
solve the optimization 

Steel frame 
building 

The introduced SA-unscented 
Kalman filter can modify the 
accuracy, computational cost, and 
convergence rate of the 
conventional SA. Additionally, 
using an objective function based 
on acceleration time series data 
with six unknown parameters 



 

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Ref. Year Objective Methodology Structure Result and Finding 
version was employed in 
FEM updating. 

problem. For the first 
time, only two 
parameters are 
considered to be 
unknown. Then, the 
FEM updating problem 
is  performed by six 
unknown parameters. 

provides better results in FEM 
updating. 

Alalikhan 
et al. [114] 

2016 In this study, the 
application of the SA 
algorithm and Tabu search 
(TS) algorithm is 
demonstrated in the FEM 
calibration and damage 
detection of an 
experimental example. 

Two weighted objective 
functions were designed 
to be minimized by 
optimization algorithms.  
In the first objective 
function, there is a 
contribution of natural 
frequencies, MAC values, 
and mode shapes.  
However, the second 
objective function is 
based on the natural 
frequencies and mode 
shapes. 

Overhang steel 
beam 

The overall results revealed that 
both optimization algorithms (SA 
and TS) are pretty effective in 
FEM updating and damage 
identification. However, more 
investigations on the complex 
structural models are essential to 
approve the robustness of the 
proposed methodology. 

Guan et 
al. [115] 

2017 A two-step method is 
presented, including 
wavelet analysis and the 
application of optimization 
algorithms in  
vibration-based damage 
detection problems. The 
SA and GA are combined 
to find the global optimal 
solutions swiftly. 

The first step consists of 
the wavelet analysis to 
identify the damaged 
elements. Then, the 
severity of the damaged 
elements is estimated by 
an optimization 
procedure. For this 
purpose, the hybrid 
optimization algorithm 
(SAGA) is applied to 
minimize a weighted 
objective function 
defined through the sum 
of the differences 
between the measured 
and calculated 
frequencies and their 
corresponding mode 
shapes. 

Continuous 
beam 

In addition to accurately localizing 
the damaged elements, the 
proposed hybrid technique 
(SAGA) can also predict the 
damage's severity. 

Kourehli 
[116] 

2017 This study identifies the 
structural damage 
parameters by optimizing 
three kinds of objective 
functions with dynamic 
and static properties.  The 
SA is also adopted as an 
optimization algorithm.   
Natural frequencies are 
contaminated with a certain 
percentage of noise to 
simulate the real 
measurement conditions.   
To study modeling errors, 
perturbations in elemental 
stiffness and mass matrices 
are also implemented. 

The dynamic residue 
force vector, static 
residue force vector, and 
the discrepancy between 
the calculated and 
measured displacements 
are practiced as the 
objective functions. This 
paper's methodology and 
objective functions are 
similar to those presented 
by Kourehli [109], but 
this study applies 
complete measurements 
instead of incomplete 
ones. 

Plane frame 
 

Cantilever plate 
 

IASC-ASCE 
benchmark 

problem 

Another confirmation was made 
of the effectiveness of the damage 
identification approach, which is 
based on the SA algorithm and 
three objective functions [109] 
with static and dynamic 
properties. 



 

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65 
 

Ref. Year Objective Methodology Structure Result and Finding 
Mišković 
et al. [117] 

2018 After the successful 
application of SA and TS 
algorithms for both 
purposes of FEM updating 
and damage identification 
in simple structures [114], 
the capability of these 
algorithms was examined 
by a complex structure. 

The experimental modal 
analysis is carried out for 
an undamaged and 
damaged grid bridge. The 
bridge under the 
experiment was excited 
by ambient vibration, 
while eight 
accelerometers were 
placed on it to collect 
data. Then, ARTeMIS 
Modal software was used 
to extract the natural 
frequencies and mode 
shapes from the  
time-series data set. Two 
different damage 
scenarios with and 
without additional mass 
were considered. In the 
first scenario, the damage 
ratio of 0.4 was 
implemented on element 
54. The second damage 
scenario has an induced 
damage ratio of 0.4 on 
element 200. The 
objective function was 
optimized utilizing 
developed routines in 
MATLAB, while the 
FEM analyses were 
performed using the 
ANSYS software. The 
only frequency-based 
objective function was 
applied to FEM 
updating. Combining the 
characteristics of natural 
frequencies and mode 
shapes were considered 
to formulate the 
weighted objective 
functions for damage 
detection. 

Steel grid 
bridge 

The results showed that the SA 
and TS are practical tools for 
solving vibration-based damage 
detection problems.   The SA and 
TS could provide a good 
agreement between the 
experimental and calibrated 
models regarding FEM updating. 
Besides, TS and SA have fast 
convergence and high accuracy to 
explore a large search space and 
detect damaged elements and their 
extents. 

Xiao et al. 
[118] 

2019 This paper applies GA and 
SA to minimize an 
objective function relying 
on strain measurements for 
damage identification in a 
large-scale bridge. 

An objective function 
between the calculated 
strain and measured 
strain was defined. 
Changes in the structural 
elements' cross-sectional 
area are considered the 
design variables. 

Klehini river 
bridge 

Where GA is used, the objective 
function value after 51 iterations 
was 4.81131e-16. The objective 
function value after 51 iterations 
was also 9.84959e-10 when using 
the SA algorithm. It can be 
concluded that GA provides a 
better convergence rate compared 
to those obtained by SA. 

Boukellif 
and  

Ricoeur 
[119] 

2019 This paper compares GA 
and SA to solve the inverse 
problem of crack 
identification. 

Continuous distributions 
of dislocation densities 
were used to model the 
cracks and boundaries.  

Infinite and 
semi-infinite 

plate 

These results came from this 
paper: 
I. The arrangement of the 

strain gauges could not 
remarkably affect the 



 

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Ref. Year Objective Methodology Structure Result and Finding 
The presented 
methodology is also 
based on the strain 
measurements achieved 
from various locations on 
the surface of the plates. 

solution to the inverse 
problem.  

II. The results of the inverse 
problem can be 
significantly improved by 
increasing the number of 
measurement locations. 

III. SA is not a suitable 
algorithm for identifying 
several crack parameters. 
However, GA can be 
considered a robust tool 
for optimizing many 
variables. 

To accurately identify the crack 
parameters such as position, 
length, and inclination angles, the 
measurement errors in the strains 
should not exceed ±5−10%. 

Cui and 
Scalea 
[120] 

2019 This paper investigates the 
performance of the 
ultrasonic guided waves 
and the SA algorithm for 
the characterization of the 
elastic properties of 
laminated composites. 

The semi-analytical finite 
element (SAFE) method 
was utilized to predict the 
ultrasonic guided wave 
propagation in laminated 
composites. A cost 
function is formulated 
based on the discrepancy 
metric between true and 
predicted phase velocity 
curves. To correctly 
identify laminated 
composites' elastic 
properties, the SA 
algorithm should 
minimize the objective 
function. 
 

Note:  For the purpose 
of damage detection, it 
would be helpful to 
determine the elastic 
properties of laminated 
composites. 

Composite 
plates 

The three frequently used guided 
modes in SHM fields, including 
flexural mode (A0), axial mode 
(S0), and shear-horizontal mode 
(SH0), are taken into account to 
identify the properties of 
laminated composites during an 
optimization procedure with the 
assistance of the SA and phase 
velocity curves as an objective 
function. 
This study examined three 
different types of laminates, 
including quasi-isotropic, fully 
anisotropic, and unidirectional.  
The summary of results can be 
listed as follows:  

I. For all types of laminates, 
laminate axial stiffness 
(Ex) and longitudinal 
lamina modulus (E11) are 
effectively identified by 
modes A0 and S0 
propagating along x. 

II. For the anisotropic and 
quasi-isotropic laminates, 
Ex and E11 are  adequately 
determined by mode SH0 
propagating along x. 

III. The laminate transverse 
axial stiffness (Ey) and 
lamina transverse 
modulus (E22=E33) can be 
accurately recognized by 
modes A0 and S0 
propagating along x. 

IV. The in-plane shear 
stiffness of the laminate 
(Gxy) and in-plane shear 
modulus of the lamina 



 

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Table 2: A review of the application of the SA algorithm in SHM. 

Ref. Year Objective Methodology Structure Result and Finding 
(G12=G13) are best 
identified by mode SH0. 

V. The in-plane shear 
modulus for all types of 
laminates is appropriately 
identified by mode A0 
propagating along x. 

VI. For all laminates under 
consideration, transverse 
stiffness (Ey), longitudinal 
stiffness (Ex), transverse 
flexural rigidity (Ky), and 
axial rigidity (Kx) can be 
effectively determined by 
modes A0 and S0 
propagating along x. 

VII. The torsional rigidity 
(Kxy) of laminates is 
recognized by either SH0, 
A0, or S0. 

This study suggests experimental 
verification of numerical studies 
for future works. 

Cui and 
Scalea 
[121] 

2021 The central objective of 
this article is the 
experimental validation of 
the recently published 
methodology [120] based 
on the SA algorithm and 
ultrasonic guided waves for 
the nondestructive 
identification of elastic 
properties of composite 
plates. 

The SA algorithm 
minimizes an objective 
function in phase velocity 
curves, similar to Cui and 
Scalea [120]. The SAFE 
method predicts 
ultrasonic guided wave 
propagation in laminated 
composites. 

Composite 
plates 

The experimental study 
demonstrates that using ultrasonic 
guided wave data and SA 
algorithms as an optimizer can be 
considered a potential tool for the 
characterization of composite 
plates. 

Hu and  
Zhang 
[122] 

2021 This paper presents a  
two-step damage 
identification approach 
using the smooth 
orthogonal decomposition 
(SOD) method and an 
improved version of beetle 
antennae search algorithm 
(BAS). The fusion strategy 
of the SA algorithm was 
applied to BAS to establish 
a better optimization 
ability. However, improved 
BAS has some drawbacks, 
such as low accuracy and 
slow convergence for 
solving optimization 
problems in large search 
spaces. Therefore, this 
study attempts to reduce 
the search space by 
excluding undamaged 
elements in the first step. 

In the first step, the 
damaged members are 
identified by a damage 
localization technique 
called SOD. Finally, the 
frequency-based 
objective function was 
minimized by enhanced 
BAS to determine the 
extent of the damaged 
elements. 

Simply 
supported 

beam 
 

Cantilever 
beam 

The overall results of this paper 
can be expressed as follows: 
I. Where improved BAS is 

applied alone  for damage 
detection in symmetric 
structures, the damaged 
elements are identified 
wrongly. By employing the 
SOD method at the first 
step, the challenge of false 
identification of symmetric 
structures could be 
addressed. 

II. The proposed two-step 
method can function  
adequately even for noisy 
inputs (0.2% and 0.5%). 
However, the efficiency of 
this method should be 
investigated under high 
noise levels.  



 

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68 
 

 
Figure 5: Number of  publications in the field of  SHM. 

 

Fig. 5 displays the classified number of  publications by years. It is clear that optimization-based SHM approaches have 
advanced significantly in recent years. Fig. 6 shows the contribution of  the publications on different parts of  SHM, such as 
FEM updating, crack detection, the combination of  FEM updating and damage detection, system identification, and optimal 
sensor placement. It is obvious that the primary contribution of  previously published articles is damage detection. 
 

 
Figure 6: Contribution of  the publications in fields of  damage detection, FEM updating, crack detection, FEM updating+damage 
detection, system identification, and optimal sensor placement. 
 
The allocation of  employed structures to demonstrate the performance of  proposed methodologies is illustrated in  
Fig. 7. Beam-like structures are the most commonly used example to validate the many approaches in the domain of  SHM, 
as seen in Fig. 7. Fig. 8 depicts the classification of  used objective functions based on the number of  publications.  

0

2

4

6

8

10

1995 - 1998 2004 - 2009 2010 - 2015 2016 - 2021

N
u

m
b

er
 o

f 
P

u
b

li
ca

ti
on

s

Year



 

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69 
 

 
Figure 7: The allocation of  employed structures to illustrate the efficiency of  presented methodologies in the publications. 

 

 
Figure 8: The classification of  applied objective functions by the number of  publications. 

Frame
8%

Beam-like 
structures

43%

Plate-like 
structures

14%

Truss
5%

Bridge
14%

Other structures
16%

0 1 2 3 4 5 6 7

Natural Frequencies

Displacements

Natural Frequencies + Mode Shapes

Natural Frequencies + MAC

Residue Force Vector

Phase Velocity Curve

Natural Frequencies + Displacements

Strain

Frequency Response Function (FRF)

Acceleration Time Series Data

Damage-Sensitive Response Parameters (DSRP)

Fisher Information Matrix of Mode Shapes

Mean Square Error of Mode Shapes

Natural Frequencies + MAC + Mode Shapes

Modal Flexibility

Multiple Damage Location Assurance Criterion (MDLAC)

Modal Assurance Criterion (MAC)



 

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70 
 

DISCUSSION 
 
 

iscussion of  a subject must include asking and responding to questions. Asking questions and the attempt for 
solutions is the foundation of  science. Questioning helps bring about the true spirit of  science and plays a vital 
role in promoting scientists [123]. Tab. 3 presents several critical questions and their answers to make an efficient 

discussion on the application of  SA in SHM. 
 

 

Questions Answers 

0. Is SA the oldest algorithm among other 
traditional optimization techniques? 

No, the SA algorithm is not the oldest [66]. GA was proposed by Holland in 1975 [124]. 
Then, the SA algorithm was introduced by Kirkpatrick et al. in 1983 [67]. Another 
popular optimization algorithm, namely PSO, was developed by Kennedy and Eberhart 
in 1995 [125].  

 

 
1. Why are weighted objective functions 

applied in several studies 
[94,99,106,114,115,117]? 

The weighted sum method is a simple yet practical technique for solving multi-objective 
optimization problems. As shown in Eqn. (5), multiple objective functions are combined 
into a single objective function by multiplying every objective function by a weighting 
factor [126,127]:  

      1 1 2 2( ) ( ) ( ) ( )n nF x w f x w f x w f x                                                                (5)   
where w represents the weighting factor. 

2. Some studies propose two-step 
methods [100,102,108,115,122]; what is 
the necessity of  implementing these 
methodologies? 

The proposed two-step methods initially attempt to reduce the dimension of  search 
space by eliminating undamaged elements because optimization algorithms can function 
accurately in narrowed search space. Besides, the computation cost is dramatically 
reduced when optimizing a small number of  variables.  

3. What is the advantage of  hybrid 
algorithms [95,98] based on the SA 
algorithm and GA? 

GA is a powerful global optimization method. However, this algorithm is poor in  
hill-climbing. Therefore, the weak hill-climbing capacity of  GA and the problem of  slow 
convergence could be addressed by the combination of  GA and SA. 

4. Is there any variant of  the SA algorithm 
to reduce the computation time? 

To reduce the computation time in the optimization procedure conducted by the standard 
SA algorithm, a new variant, namely ASA, was proposed by Bayissa and Haritos [102]. 
ASA was employed as a part of  the damage assessment methodology, and both numerical 
and experimental examples validated its effectiveness. 

5. Is there any inspiration from SA to 
develop a new  algorithm? 

As there is a famous proverb, all new ideas are combinations of  old ones; it is possible to 
develop novel algorithms from old ones. In this regard, a new version of  BAS has been 
improved by the fusion procedure of  the SA algorithm [122]. 

 

Table 3: Several questions and answers to make a discussion on the application of  SA in SHM. 
 
 
CONCLUSIONS 
 

mplementing an optimization algorithm to minimize the objective function can be considered a widely used inverse 
solution for vibration-based damage identification problems. Developing novel optimization techniques has become a 
fast-growing research field in the recent decade, and the most successful nature-inspired optimizers, such as Grey Wolf, 

were introduced. However, traditional algorithms such as GA, PSO, and SA have been constantly utilized as global 
optimizers in damage detection problems. 
This paper comprehensively investigated previous studies between 1995 and 2021, and some utilized methodologies were 
discussed. A summary of  around 30 publications in the context of  SHM is as follows: 

 Most articles were published in the period from 2016 to 2021. 
 Beam-like structures make a considerable contribution than other types of  structures.  In contrast, the lowest 

contribution is related to truss structures. 

GA

(1975)

SA  

(1983)

PSO

(1995)

D 

I 



 

                                                                P. Ghannadi et alii, Frattura ed Integrità Strutturale, 64 (2023) 51-76; DOI: 10.3221/IGF-ESIS.64.04 
 

71 
 

 Numerous papers have presented approaches for damage detection (53.57%). The ratio of  other methodologies in 
SHM problems, such as FEM updating, and FEM updating + damage detection, are 25% and 7.14%, respectively. 
The articles in the field of  crack detection (7.14%), system identification (3.57%), and optimal sensor placement 
(3.57%) are also analyzed. 

 Over the past decades, natural frequencies and displacements have been the most utilized characteristics to define 
the objective function. 

 The hybrid algorithms based on GA and SA could address the weakness of  GA in hill-climbing and could reduce 
the computation time of  standard GA. 

 The weighted sum method was applied to minimize the multi-objective optimization problems by the SA algorithm. 
 The damaged elements are initially identified through different methods such as GRA, DSRP, BP neural networks, 

wavelet analysis, and SOD to improve the accuracy of  the SA algorithm for estimating the damage severity in the 
second step. In the second step, damage severities are predicted by minimizing an objective function based on the 
SA algorithm. 

 Two-step methods were provided as appropriate tools to reduce the computation time for the optimization process 
by the SA algorithm. Additionally, a new version of  the standard SA algorithm called ASA was presented in this 
regard. 

 
 
ACKNOWLEDGMENT 
 

e would like to express our deepest appreciation to Dr. Faham Tahmasebinia and his 
expert colleagues from the University of  Sydney for providing the 3D model of  the 
Milad Tower. 

 
 
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice