Computational methodologies for optimal sensor placement in structural health monitoring: A review

Abstract

Structural health monitoring plays an increasingly significant role in detecting damages for large and complex structures to ensure their serviceability and sustainability. Optimal sensor placement is critical in the structural health monitoring system as the sensor configuration directly impacts the quality of collected data used for structural health diagnosis. Therefore, this study presents a comprehensive review of computational methodologies for optimal sensor placement in structural health monitoring. The problem formulation of optimal sensor placement is first introduced, including commonly used evaluation criteria for sensor configurations. Then, various existing optimization methodologies for sensor placement are summarized and introduced in detail, especially for the evolutionary algorithms and their improved variants. Finally, the suitability of computational methods for specific structural health monitoring applications is also discussed. The main goal of this study is to deliver a comprehensive reference of computational methodologies for optimal sensor placement in structural health monitoring studies and applications. This article is concluded by highlighting the most widely utilized evaluation criteria and optimization methodologies for sensor configuration determination.

Keywords

Structure health monitoring optimal sensor placement evolutionary algorithm deterministic methodology

Introduction

Structural health monitoring (SHM) is defined as the process of implementing strategies to detect damages for the structures of civil, aerospace, and mechanical engineering^1–3 to guarantee their serviceability and sustainability. SHM has been widely used for the past decades as the increasing number of large and complex structures was constructed worldwide. Brownjohn⁴ and Karbhari and Ansari⁵ discussed the motivations for and the development of SHM applications to different categories of civil infrastructure in the 20th century. Case studies on specific types of structure are also provided for illustration.

A typical SHM system consists of sensors, data collection and transmission subsystem, a database for data management, and health diagnosis.⁶ One critical technology involved in an SHM system is sensing technology, which is not limited to SHM applications but has been applied in many fields such as structural design,⁷ construction progress monitoring,⁸ safety risk,^9–11 and smart operation and maintenance management.¹² Ou and Li⁶ reviewed the development of advanced sensing technologies and sensors in mainland China. Sensors are usually fundamental elements in the SHM system. The sensors in the SHM are utilized not only to monitor the structural status, such as stress and displacement, but also to monitor the environmental parameters, such as the temperature and wind speed which are impacting the structures.¹³ For example, Ansari¹⁴ introduced basic principles related to practical health monitoring of large civil structures with optical fiber sensors, where the sensor packaging, sensor placement, strain transfer mechanism, and sensor reliability were discussed.

Sensors are generally divided into two types, namely wired and wireless sensors. Wired sensors are usually limitedly applied since they require to be installed during structure construction, and the wiring might potentially impact the function of the structure with a limited number of sensors.¹⁵ Therefore, with the development of wireless sensors, wireless monitoring has emerged in recent years as a promising technology that could significantly impact SHM.^16,17 The development of wireless sensors and sensor networks for SHM was reviewed by Lynch and Loh.¹⁸ In this review, the hardware architecture is identified as a critical element in wireless sensors design for SHM. In addition, the advantages (e.g. automated detection through the collocation of computational power with the sensing transducer) and disadvantages (e.g. finite energy sources) of wireless sensor networks (WSNs) are also discussed.¹⁹ WSNs have already been deployed in several fields. For example, Xu et al.²⁰ described the design and deployment of a WSN for structural monitoring. Jang et al.²¹ illustrated the deployment and evaluation of a state-of-the-art WSN on a cable-stayed bridge named Jindo Bridge in South Korea, which was newly constructed with a 344 m main span and two 70 m side spans.

Sensor placement, also named as sensor deployment²² or sensor configuration, is always the basic process of constructing the sensor network for an SHM system. Besides, SHM allows an optimal usage of the structure, a minimized downtime, and the avoidance of catastrophic failures, helps improve the product design, and completely changes the approach to conducting maintenance activities. Therefore, optimizing the sensor configuration for SHM requires considering more aspects compared to other optimization problems. On one hand, the optimization of sensor placement needs to meet the requirements of SHM. One the other hand, an optimal sensor configuration requires minimizing the number of sensors required and increasing the accuracy of data collection and provides a powerful SHM system.²³ Several studies have explored the sensor placement optimization. For example, Younis and Akkaya²⁴ introduced the current state of studies on optimizing the sensor placement in WSN, where the placement strategies were divided into static and dynamic based on whether the optimization was conducted during the sensor deployment or at the operation stage of networks. Guerriero et al.²⁵ discussed the determination of sensor configuration by displacing sensors from an initial sensor configuration. In addition, Abdollahzadeh and Navimipour²² conducted a comprehensive review of deployment strategies in WSN. However, a comprehensive review of the optimal sensor placement (OSP) problem and related computational methodologies used for OSP is still lacking.

Considering the importance of SHM for the increasing number of large complex structures around the world and the impact of sensor configurations on SHM performance, this study aims at delivering a comprehensive review on the OSP problem and related computational methodologies used in sensor placement optimization, including both wired and WSN configurations. Furthermore, the reviewed results of this study are expected to provide an effective reference to sensor configuration determination for SHM system construction.

The rest of this study is organized as follows. Section “Problem formulation” formulates the sensor placement optimization problem, including problem description and the commonly used evaluation criteria. Optimization computational methodologies by category are introduced in section “Optimization methodologies.” Section “Conclusion” concludes this study.

Problem formulation

Degree of freedom and finite element model

As aforementioned, SHM mainly aims at monitoring and detecting structural damages, from materials to structural stability. Instability of a structure originates from the displacement at the joints of the structure. Degree of freedom (DOF) indicates the degrees of freedom (e.g. move or rotate) that any particular joint contains. A DOF of a certain structure denotes the number of directions that the structure can generate displacement at a joint. Therefore, the positions of joints or DOFs can be used as candidate locations to install proper sensors to enable data collection for structural damage monitoring and detection.

Methodologies of sensor configuration determination vary among different structures. For simple and regular structures or smaller number of DOFs, such as steel frame in real harsh environment,^26,27 cables in post-tensioned members,¹⁴ scaled-down transmission tower,²⁸ and simple building structures,^29,30 physical sensors (wired or wireless) can be installed for data collection and validate different sensor configurations for the target structures. However, SHM is more commonly applied in large and complex structures for damage detection or structural state monitoring. For these structures, such as large buildings, towers with an irregular shape, and long-span cable-stayed bridges (see Figure 1), the numerical experiment is preferred, because the unlimited sensor configuration can be efficiently and economically tested. Three-dimensional finite element (FE) models³¹ are usually required to be constructed for the numerical experiment to optimize the sensor placement.

Figure 1.

Illustration of real structures to FE models: (a) large building, (b) tower with irregular shape, and (c) long-span cable-stayed bridge.

The generated FE models may have thousands of DOFs given the complexity of the large structure. Therefore, model simplification and reduction methods are usually required to efficiently determine the sensor configuration. For example, Ni et al.³² established an equivalent reduced-order FE model for Canton Tower, which contained 37 beam elements and a total of 185 DOFs. Yi et al.³³ also used this established FE model for sensor configuration study on the Canton tower using various evolutionary algorithms.

To generate three-dimensional (3D) FE models for OSP, many computational applications can be utilized. The most widely used application among OSP studies is ANSYS.^7,28,33–41 For example, Yi et al.³³ used ANSYS to construct an FE model for Canton Tower, where the 3D full-order FE model contained 84,370 nodes and 505,164 DOFs. Xu et al.³⁹ created the FE model of the physical Tsing Ma Bridge using ANSYS. Chow et al.²⁸ used ANSYS to construct the FE model for the transmission tower. In addition to ANSYS, other FE analysis software, such as SAP,^31,42 SAFESA™,⁴³ Grillage,⁴⁴ Autodesk Simulation Multiphysics,⁴⁵ ETABS,⁴⁶ FLAC^3D,⁴⁷ and ABAQUS,²⁶ are also available for FE model generation.

OSP problem description

Since OSP mainly aims at determining the DOFs for sensor installation, most of the sensor placement optimization can be summarized as “given a set of $n$ candidate positions, find $m$ positions, where $m \leq n$ , which could maximize or minimize the value of the proposed evaluation criterion for the sensor configuration.”⁴⁸ The OSP problem is then converted to a kind of combinational optimization problem. The total number of all potential sensor configurations containing $m$ sensors can be obtained using equation (1)

c = \frac{n!}{m! (n - m)!}

(1)

where $n$ denotes the number of candidate positions and $m$ is the total number of sensors to be placed. One thing should be pointed out is that m in equation (1) is not a constant in many situations as m sometimes is also a parameter to be optimized (most time is to be minimized) to minimize the total cost of sensors and their installation, guaranteeing the evaluation criteria requirement.^49,50

For most studied structures, $C$ can be an extremely large number. Therefore, an exhaustive search over all potential sensor configurations to obtain the optimal configuration is very computationally expensive even for FE models with a relatively small number of DOFs.

In addition to global SHM, where low-frequency guided waves are mainly employed, as mentioned above, OSP has also been applied to ultrasonic testing, where nondestructive examination (NDE) techniques that use short, high-frequency ultrasonic waves to identify flaws in material or damages in a structural component.^51–54 For the ultrasonic NDE, usually the maximum value of coverage by the sensors or transducers is formulated as the fitness function to determine the sensor placement.⁵⁵ Actuator–sensor pairs usually obtain binary images of the sensor network system, where 0 and 1 values are included. Then, all these images are summed to evaluate the performance of the actuator–sensor network system. For example, Thiene et al.49 defined the coverage fitness function as

cov = \sum_{i = 1}^{nr pixel} c (i)

(2)

where $c (i)$ denotes to the indicator of the performance of the sensor’s configuration at pixel $i$ and $nr pixel$ is the number of pixels that the tested area contains.

Evaluation criteria

As mentioned in section “OSP problem description,” most OSP studies mainly aim at placing the required sensors on the candidate positions. During the sensor placement or sensor position optimization, a proper evaluation criterion is required. According to the reviewed OSP studies, quite a few evaluation criteria are available. This study summarizes some commonly utilized criteria, including modal assurance criterion (MAC),⁵⁶ Fisher information matrix (FIM),³⁶ information entropy (IE),⁵⁷ effective independence (EI),⁵⁸ effective independence driving-point residue (EFI-DPR),⁵⁹ and energy-based criteria.^43,60 Details of each criterion are introduced in the following sections.

MAC

Carne and Dohrmann⁶¹ attempted to determine sensor locations to ensure that all the inner products between shape vectors have relatively small cosines, where the shape vectors are easily distinguishable. Therefore, they utilized the commonly used MAC^62–64 to assess the cosine square between shape vectors as follows

MAC = \frac{{(Φ_{i}^{T} Φ_{j})}^{2}}{(Φ_{i}^{T} Φ_{i}) (Φ_{j}^{T} Φ_{j})}

(3)

where $Φ_{i}$ and $Φ_{j}$ represent the ith and jth column vectors in matrix $Φ$ , respectively, and the superscript $T$ denotes the transpose of the vector.

In equation (3), the element values in the MAC matrix range between 0 and 1, where 0 indicates that little or no correlation exists between the off-diagonal element $MA C_{ij} (i \neq j)$ (i.e. the modal vector easily distinguishable) and 1 indicates a high degree of similarity between the modal vectors (i.e. the modal vector fairly indistinguishable). As a result, the objective function can be formulated using the biggest total value of all the off-diagonal elements in the MAC matrix, which is represented as follows

f (x, c) = \max {MA C_{ij}}, i \neq j

(4)

where $f (x, c)$ denotes the total value of all the off-diagonal elements in MAC matrix; $i$ or $j$ are the number of columns in matrix $Φ$ .

FIM

FIM,^65,66 which is related to the information stored in the measured response, results from minimizing the covariance matrix of the estimated error for an efficient unbiased estimator from the statistics perspective. FIM can be obtained using equations (5) and (6)

F = \max | \det (FIM) |

(5)

FIM = Q_{0} = \sum_{i = 1}^{s} Φ^{i^{T}} Φ^{i} = \sum_{i = 1}^{s} Q^{i}

(6)

where $Φ^{i}$ denotes the ith row of the modal partition $Φ$ related to the ith DOF or sensor position and $s$ is the total number of sensors.

The procedure of using FIM for optimizing sensor placement is to unselect candidate sensor locations to maximize FIM. Three variants of FIM are usually used, the determinate, the trace, and the minimum singular value, to be maximized in order to increase the information or decrease estimate uncertainties. Although these are three different variants of FIM, different sensor placement approaches using these variants as the evaluation criteria should yield similar results.^67,68

IE

In order to accurately estimate the model parameters for structural damage detection, the quality of the collected information from the installed sensors requires to be guaranteed. Since IE⁶⁹ is efficient to quantify the parameter uncertainties, Papadimitriou et al.⁵⁷ used IE as a measure of minimizing the uncertainty in model parameter estimate when optimizing the sensor placement.⁷⁰ In the IE-based OSP studies, the model updating was usually realized based on a Bayesian statistical method. A probability density function (PDF) $p (θ | D)$ is used to quantify the uncertainties in the parameters $θ$ . The PDF is calculated based on the dynamic test data $D$ and prediction error. The calculation of IE is defined as follows

H (D) = E_{θ} [- \ln p (θ | D)] = - \int p (θ | D) \ln p (θ | D) d θ

(7)

where $E_{θ}$ is the mathematical expectation with respect to $θ$ , and $p (θ | D)$ can be obtained using equation (8)

p (θ | D) = c {[J (θ; D)]}^{- (N N_{0} - 1) / 2} π_{θ} (θ)

(8)

where $J (θ; D)$ is the fit measure between the measured and the model response time histories, $N$ is the number of sampled data, and $N_{0}$ is the number of observed DOF of the structural model.

EI

EI, as one of the most significant and widely used criteria in large structural dynamic testing and model updating, was proposed by Kammer.⁶⁵ An efficient unbiased estimator, the covariance matrix of the estimated error can be obtained using equation (9)

C = E [(q - \hat{q}) {(q - \hat{q})}^{T}] = [\frac{1}{σ^{2}} Φ_{m}^{T} Φ_{m}] = F^{- 1}

(9)

where C is the covariance matrix, and F is the FIM; $q \in R^{N}$ is the modal coordinate vector. $\hat{q} = [Φ_{m}^{T} Φ_{m}]^{- 1} Φ_{m}^{T} y_{m}$ is the vector of an efficient unbiased estimator of $q$ , and $m$ is the number of selected sensor locations.

According to the statistics theory,⁷¹ the effectiveness of the estimated parameters can be improved by maximizing the information in Fisher’s sense. To achieve this, the EI indicator is calculated using

E_{D} = diag (Φ {[Φ^{T} Φ]}^{- 1} Φ^{T}) = diag (Q Q^{T})

(10)

where $Q$ is an orthonormal matrix and $R$ is an upper triangular matrix. $P = Φ [Φ^{T} Φ]^{- 1} Φ^{T}$ denotes a projection matrix formed by the mode shape matrix $Φ$ , which can be decomposed as $Φ = QR$ to expedite the computation.

EFI-DPR

The EI provides an estimation of the linear independence of target modes and is efficient in terms of computation. However, sensor positions that have low energy can be chosen, which results in a reduction of the signal-to-noise ratio that makes mode shape identification difficult. To overcome this drawback, the EFI-DPR (driving-point residue) method,⁴³ which weights the EI indices with the associated driving-point residue (DPR),⁷² was used. DPR can be calculated as follows

DPR = Φ \otimes Φ Ω^{- 1}

(11)

where ⊗ is a term-by-term matrix multiplication. $Ω$ denotes a column vector consisting of circular frequencies corresponding to $Φ$ , and $Ω^{- 1}$ represents the inverse of each term in $Ω$ . DPR assesses the average contribution of each candidate position sensor to the mode shapes,⁷³ which is used to modify the EI indices as follows

E_{D} = diag (Q Q^{T}) \otimes DP R^{- 1}

(12)

where ⊗ denotes the same item to equation (11) and $Q$ is an orthonormal matrix.

Energy-based criteria

There are also some other evaluation criteria based on the energy for the sensor configuration. One commonly used criterion is the modal strain energy (MSE),^33,37,74 which helps to select the sensor positions with possible large amplitudes. It can be used to increase the signal-to-noise ratio that is important in real situations. The MSE can be calculated using equation (13)

H = \sum_{i = 1}^{m} \sum_{j = 1}^{m} \sum_{r \in n} \sum_{s \in n} | \emptyset_{rj} k_{rs} \emptyset_{sj} |

(13)

where the mode shape matrix is $Φ = [\emptyset_{1}, \emptyset_{2}, \dots, \emptyset_{m}]$ , $m$ denotes the number of mode shape vectors, $n$ is the number of measured points, $\emptyset_{rj}$ denotes the $r th$ element in the associated $i th$ mode shape, $\emptyset_{sj}$ is the $s th$ element in the associated $j th$ mode shape, and $k_{rs}$ denotes the stiffness coefficient between the $r th$ DOF and the $s th$ DOF.

The kinetic energy (KE) is another commonly used criterion.^43,60 It aims at obtaining a reduced configuration of sensor placements that maximizes the measure of the KE of the structure. Besides, energy efficiency had also been commonly used as the objective to determine sensor configuration on large and complex civil structures.^{25,30,75–77}

Optimization methodologies

Several optimization methodologies are available for sensor configuration determination. As mentioned in section “Problem formulation,” OSP mainly aims at assigning m sensors to n candidate positions (i.e. DOFs) on the target structures (e.g. buildings, bridges, and towers), which is usually treated as the combinational optimization problem. Therefore, evolutionary optimization methodologies that have the advantages of solving NP-hard problems can well address OSP problems. Besides, sequential sensor placement (SSP) algorithms, deterministic optimization methodologies, probability-based methodologies, and other optimization methodologies are also available for OSP problems. Details of each category of optimization methodologies are introduced in this section.

Evolutionary optimization methodologies

Evolutionary algorithms, which are based on a powerful principle of evolution (i.e. the survival of the fittest), constitute a critical category of heuristic search.⁷⁸ Evolutionary algorithms have been commonly utilized in science and engineering for addressing complex problems, such as the NP-hard problem of sensor placement.⁷⁹ Figure 2 illustrates the procedures in evolutionary algorithms. The evolutionary algorithms usually start from initializing a population with a certain number (i.e. population size) of candidate solutions. Then, the population is updated using evolutionary strategies (e.g. the genetic operators such as crossover and mutation in the genetic algorithm, position and velocity update mechanism for particles in particle swarm optimization (PSO)) based on the fitness function (i.e. the introduced evaluation criteria of sensor placement as mentioned in section “Evaluation criteria”). Finally, the evolution of population ends until a certain termination criterion is met. The best solution in the latest population is used as the optimal solution.

Figure 2.

Illustration of procedures in evolutionary algorithms.

There are quite a few commonly used evolutionary algorithms, including genetic algorithm, PSO, monkey algorithm, firefly algorithm, simulated annealing algorithm, and ant colony optimization. The evolutionary algorithms and their improved variants for OSP are summarized in Table 1. Details of each evolutionary algorithm are introduced in the following sections.

Table 1.

Evolutionary methodologies used in sensor placement optimization.

Algorithm category	Method name	Solution encode	Improved strategies	References
GA	GA	Decimal 2D array coding	Traditional GA processes	38
		Binary coding		27,28,51,55,57,77,85 –93
		Point array		26
	Generalized GA	Binary coding	Selecting an excellent method of 2/4	94,95
		Dual-structure coding	Punctuated equilibrium theory	96
			Elite conservation strategy Worst elimination policy	97
			Selecting an excellent method of 2/4	33
	Improved GA	Binary coding	Crossover based on the identification code Mutation based on two gene bits	98
	Improved GA	Dual-structure coding	Adaptive partial matching crossover and inversus mutation method	99
	FEM-GA	Binary coding	FEM problem formulation	100
	Virus co-evolutionary partheno GA		Virus infection operators	101
	Single parenthood GA		Subtree crossover	102,105
	Pareto GA	Dual-structure coding	Selection: Proportional Model Partially Matched Crossover	42
	MSE-AGA	Dual-structure coding	Initial sensor placement with the MSE method Final sensor placement with AGA uses adaptive adjustment processes	37
	Improved information matrix criterion GA	2D integer array coding	Embed Improved information matrix criterion	103
	Quantum GA	Quantum probability vector	Adopting quantum-rotating doors	104
MA	Modified MA	Integer coding	Euclidean distance increase diversity of the initial population Harmony search algorithm improves Climb process	106,107
	Asynchronous-climb MA	Dual-structure coding	Improved asynchronous-Climb process “Monkey King” as the current best to guide the search	35
	Niching MA		Chaos-based approach to initialize a population Niching generation operation for Climb	108
	Triaxial accelerometer MA		Harmony ladder climb Scanning watch-jump process	109
	Adaptive MA		Automatically control climb/watch–jump process Reflection and mutation somersault process	110
	Immune MA		Novel advanced climb processImmune selection, clonal proliferation, and immune renewal	50
	Distributed MA		Small subpopulations instead of a single large populationHarmony search for the final optimal solution	46
	Virus MA		Virus infection operatorVirus-evolutionary climb process	40
	Chaotic MA		Chaotic initialization, variable search step length, and adaptive watching time	111
PSO	PSO	Binary coding	Traditional PSO processes	52,112 –114
	Hybrid PSO		Self-configuring algorithmNelder–Mead algorithm to search in the better area	115
	Self-adaptive PSO		Self-adaptive inertia weight	116
	Improved discrete PSO		Gene exchange operator and the gene shift operator	117
	Improved PSO	Dual-structure coding	Improved inertia weight adjustment strategy	41
	Integer-encoding multi-swarm PSO	Integer coding	Parallel structure of multiple populations and grade evaluation	118
FA	Cluster-in-cluster FA	One-dimensional binary coding	Cluster-in-cluster strategy	74
	Hybrid discrete FA		Hybrid movement scheme including directive movement and nondirective movement	119
	Simple discrete FA		Hamming distance between two firefliesNewly developed movement scheme	120
	Multi-objective FA		Hybrid movement scheme	121
	Information-fusing FA	Dual-structure coding	Lévy flightsInformation fusion	122
SA	SA	Grid point system	Traditional SA processes	123,124
	Improved SA	Coordinate-based solution encoding	Improved solution encoding	125
	PSO-based SA	Uniform grids representation	Obtain global optima based on PSO results	126
ACO	ACO	Predefined rectangular area with candidate positions	Traditional ACO processes	49
	Siege ACO		Ant colony siege processPheromone total update process	127
Other evolutionary algorithms	Artificial bee colony algorithm	Integer array coding	Best-solution-guided updating strategy	45
	Distributed wolf algorithm	Dual-structure coding	Parallel subpopulationsImproved attacking process	128

GA: genetic algorithm; MSE-AGA: modal strain energy–based adaptive genetic algorithm; MSE: modal strain energy; MA: monkey algorithm; PSO: particle swarm optimization; FA: firefly algorithm; SA: simulated annealing algorithm; ACO: ant colony optimization; 2D: two-dimensional; FEM: finite element model.

Genetic algorithm

Among the evolutionary algorithms, genetic algorithm (GA) is the most widely applied in engineering problems.⁸⁰ GA was first introduced in 1975^81,82 which was inspired by the mechanism of natural evolution.⁸³ The algorithm usually starts from initializing a population with a certain number of solutions (i.e. encoded chromosomes), and then the evolution conducts based on genetic operators, including replication, crossover, and mutation. The fitness function is used to evaluate newly generated solutions during iteration until a certain termination condition is reached. More details of the original GA process can be referred to Davis.⁸⁴ As mentioned in section “Problem formulation,” the OSP problem is usually considered as an NP-hard problem. GA is one of the commonly used heuristics methodologies to solve NP-hard problems. Both original GA^{26–28,38,51,55,57,77,85–93} and improved GA^{33,37,42,94–104} had been used in OSP problems.

Solution coding of GA

Chromosome coding is an important component of GA. Since the widely used objective is to place $n$ sensors to $m$ potential positions of the converted FE model, the binary coding method has been commonly used.^{85,94,98,100–102} The coding length of a string is $m$ ; each bit position of the string is represented with 0 or 1. If the $i th$ bit position is 1, the $i th$ DOF is assigned with a sensor. In contrast, if the value of this position is 0, no sensor will be placed on the $i th$ DOF. For example, a string 0010011101 represents that five sensors require placing on 10 optional DOF, where the 3rd, 6th, 7th, 8th, and 10th DOF are placed with a sensor separately. If N sensors require to be placed and $p_{i}$ indicates the value of the $i th$ bit position of the coded string, one condition for the objective or fitness function of OSP problems is represented by equation (14)

p_{1} + p_{2} + \dots + p_{n} = N

(14)

where $p_{i}$ indicates the value of the $i th$ bit position and $N$ denotes the total number of sensors to be placed.

In addition to the binary coding method, dual-structure coding that contains two rows as presented in Table 2 has been used in GA-based OSP studies.^96,97 The first row $D_{i}$ is the append code that represents the indices of the optional DOFs for placing sensors. The second row is the variable code $S_{i}$ that relates to the append code $D_{i}$ , which plays the same role as coding in binary coding. More details of encoding solution using the dual-structure coding method is introduced in section “Monkey algorithm.”

Table 2.

An example of the dual-structure coding.

Append code $D_{i}$	8	3	5	2	1
Variable code $S_{i}$	0	1	0	1	0

Besides, other solution representations have been proposed in previous GA-based OSP studies. For example, Liu et al.³⁸ developed a decimal two-dimensional (2D) array coding method that was similar to the dual-structure coding method. Huang et al.²⁶ proposed a point array coding method that included both sensor locations and sensor type information. Zhan et al.¹⁰³ presented a 2D integer array coding method. Zhu et al.¹⁰⁴ used quantum probability vector to encode the solution, where the quantum information was encoded using pairs of complicated numbers.

Improved GA

To obtain a better evolution performance in OSP problems, such as more stable iteration and efficient convergence, GA has been improved using different strategies. Generalized GA,¹²⁹ which utilizes the selecting method of excellent of 2/4 when formulating a new generation, was used in previous works.^33,94,95 Selecting the excellent from both parent and offspring generations based on their fitness ensures a stable iteration. Besides, Yi et al.⁹⁶ adopted a “group in group” scheme to generate a leading population by selecting a small population with $k$ individuals from the initial population of N individuals according to the fitness. Roulette wheel selection is used in a leading population to obtain the fittest string for better offspring generation. Zhou et al.⁹⁷ utilized the two-quarter selection rule and “group in group” scheme to keep the stability of the population. In addition, partially matched crossover, swap, and inversion mutation were used in the iteration to make the generalized GA an excellent global optimizer.

Guo et al.⁹⁸ developed a new crossover and two-gene-bite mutation to generate new binary string named identification code to guarantee that the new offspring does not violate the constraints. Shan et al.⁹⁹ used the adaptive GA approach¹³⁰ to improve the capability of global search and change the probabilities of crossover $p_{c}$ and mutation $p_{m}$ based on the value of fitness function, where $p_{c}$ and $p_{m}$ can be automatically updated using equations (15) and (16)

p_{c} = {\begin{matrix} p_{c 1} - \frac{(p_{c 1} - p_{c 2}) (f_{1} - f_{avg})}{f_{\max} - f_{avg}}, & f_{1} \geq f_{avg} \\ p_{c 1}, & f_{1} < f_{avg} \end{matrix}

(15)

p_{m} = {\begin{matrix} p_{m 1} - \frac{(p_{m 1} - p_{m 2}) (f_{2} - f_{avg})}{f_{\max} - f_{avg}}, & f_{2} \geq f_{avg} \\ p_{m 1}, & f_{2} < f_{avg} \end{matrix}

(16)

where $f_{\max}$ denotes the solution with the maximum fitness value in the population, $f_{avg}$ means the average fitness value of population, $f_{1}$ denotes the fitness of the individual with bigger fitness value of the two individuals for crossover, and $f_{2}$ denotes the fitness of the individual for mutation. Other parameters are predefined constants, $p_{c 1} = 0.9$ , $p_{c 2} = 0.6$ , $p_{m 1} = 0.1$ , and $p_{m 2} = 0.001$ .

Kang et al.¹⁰¹ involved parthenogenesis¹³¹ in their proposed partheno genetic algorithm (PGA), which contained three basic partheno-genetic operators, namely, swap, reverse, and insert.^132–134 In order to improve the operation, virus infection operators^135–138 were also introduced in the PGA. Zhang et al.⁴² adopted a proportional selection model to guarantee the quality of the population, and the partially matched crossover was implemented to improve the evolution performance. In addition, Zhan et al.¹⁰³ embedded an improved information matrix into GA for sensor placement design. Zhu et al.¹⁰⁴ adopted the quantum theory to improve GA performance, where chromosomes are represented with quantum probability vectors, and quantum-rotating doors were used to update and optimize the population by globally searching for an optimal solution. Vosoughifar et al.¹⁰⁰ developed a hybrid FE-approach to select the best sensor position for a cold-formed steel structure based on the blast loading response of the system. He et al.³⁷ proposed a hybrid optimization approach named modal strain energy–based adaptive genetic algorithm (MSE-AGA) to address the OSP problems, where MSE methodology was used to conduct initial sensor placement to determine the candidate sensor locations, and AGA was used to determine the final sensor placement.

Monkey algorithm

Monkey algorithm (MA) is originally designed by Zhao and Tang¹³⁹ to solve the optimization of multivariate systems. MA simulates the mountain-climbing processes of monkeys to find the highest mountaintop, which was used as the optimal value of the objective function.¹³⁹ The mountain-climbing processes are used to conduct the evolution or iteration during optimization. The evolution processes are usually decomposed into four main steps: climb, watch, jump, and somersault. A monkey (i.e. one potential solution) climbs from the current position (i.e. initial position) to the top of the mountain where it locates (i.e. usually treat as local optimum), then it watches around to find out whether any neighbor mountaintop is higher than the current one. If yes, it jumps to the higher mountaintop. In order to find a much higher mountaintop, a monkey somersaults to a total new mountain area (i.e. new search domain). The detailed mathematical representations of MA solution and the four mountain-climbing steps can be referred to Zhao and Tang.¹³⁹

Solution coding of MA

Since MA was first applied in sensor placement problem Yi et al.,¹⁰⁶ quite a few studies explored original MA and its improved versions for OSP. As an emerging evolutionary algorithm, solution representation or encoding usually requires to be determined before evolution implementation. MA is originally designed to solve optimization problems with continuous variable.¹³⁹ However, OSP aims at determining discrete sensor locations, which is usually treated as an optimization problem with discrete variables as mentioned in section “Problem formulation.” Instead of the traditional one-dimensional binary coding, which requires increased string length and computational time when sensor number increases, Yi et al.¹⁰⁷ proposed an integer coding. For one monkey (i.e. one potential solution), its location is defined as a vector $x = (x_{1}, x_{2}, \dots, x_{n})$ , and $n$ is the number of sensors to be placed. When there are $n$ sensors to place in the total m DOFs (i.e. the number of candidate sensor positions), the coding length of the string is $n$ . Every element of the vector is the sorted DOF on which the sensor is located. For example, (4, 7, 13, 16, 22, 27) is a vector, which denotes that the sensors are uniquely placed on the 4th, 7th, 13th, 16th, 22nd, and 27th DOFs.

In addition to the integer-coding method, a developed dual-structure coding method is more common in the MA-based OSP problem.^{35,40,46,50,108–111} In this dual-structure coding method, an ordered pair $(x, c)$ represents possible solutions for each monkey, where $x$ denotes the position vector in the improved MA and $c$ represents the binary vector that stands for the location of the sensor. The dual-structure coding method of the solution representation is given as follows:

Suppose there be $f$ candidate sensor positions (i.e. the total DOFs of the developed FE model), thus the $f$ integers from 1 to $f$ can be obtained.

For ith monkey in the initial population, its solution of the proposed optimization problem is denoted as $x_{i} = (x_{i}, c_{i}) = {(x_{i, 1}, c_{i, 1}), (x_{i, 2}, c_{i, 2}), \dots, (x_{i, f}, c_{i, f})}$ , where the element of the position vector $x_{i}$ is the real number selected randomly from the interval [down, up], where down =-5 and up = 5, and $c_{i}$ is the binary vector, which can be calculated using the following equation

c_{i, j} = sig (x_{i, j}) = \frac{1}{1 + e^{- x_{i, j}}}

(17)

where $e$ is the universal constant.

When using equation (17), a threshold $ε$ should be determined first. That is if $sig (x_{i, j}) \leq ε$ , then $c_{i, j} = 0$ (i.e. no sensor is located on this DOF); if $sig (x_{i, j}) > ε$ , then $c_{i, j} = 1$ (i.e. a sensor is located on this DOF), here $j \in {1, 2, \dots, f}$ . Besides, $ε$ is defined as 0.5, thus when selecting each component $x_{i}$ randomly from the interval [–5, 5], it can be found that $0.0067 \leq sig (x_{i}) \leq 0.9933$ and $sig (x_{i}) = 0.5$ which proves that the threshold defined is reasonable.

Improved MA

To improve the evolution performance, several strategies are proposed to improve MA in different perspectives. Since MA is swarm intelligent algorithm, which evolves based on population, increasing the quality and diversity of the population can avoid generating similar monkey positions and improve the evolution. For example, Yi et al.¹⁰⁷ adopted Euclidean distance that could increase the diversity of the positions of monkeys, reducing the possibility of being trapped in local optima. Euclidean distance (Ed) is defined as

| x_{i} - x_{j} | = \sqrt{\sum_{s = 1}^{n} {(x_{is} - x_{js})}^{2}}

(18)

where $x_{is}$ and $x_{js}$ are the sth component in the vectors $x_{i}$ and $x_{j}$ , respectively. As Ed and the similarity between the positions of two monkeys are negatively correlated, a threshold hm is set for Ed. Ed should be bigger than or equal to hm when initializing the population. Yi et al.¹⁰⁸ suggested a chaos-based approach to initialize the population. Chaos is a general natural phenomenon and can be obtained using the following logistic equation¹⁴⁰

h_{i} = μ \times h_{i - 1} \times (1 - h_{i - 1})

(19)

where $h_{i}$ $(i = 1, 2, 3, \dots)$ is a chaos variable, especially it works as a chaotic dynamics when $μ = 4$ and $h_{0} \notin {0, 0.25, 0.5, 0.75, 1}$ ; $μ$ is a control parameter. Therefore, for the monkey $i$ with the position $x_{i}$ , the procedures of initializing population using the chaos-based approach are as follows:

Determine the initial chaos variable $h_{0}$ ;

Generate the chaos variable $j$ of the ith monkey using equation (19);

Obtain the jth variable in the position vector of the ith monkey using equation (20)

x_{i, j} = down + h_{j} \times (up - down)

(20)

where $j \in {1, 2, \dots, f}$ .

4. Repeat Steps 1–3, until $S = M \times N$ monkeys are generated ( $S$ is the population size; N and M represent numbers of niches and monkeys in each niche, respectively).

The chaotic-based approach was applied Zhang et al.¹¹¹ Yi et al.⁴⁶ also introduced a distribution mode of multiple monkey populations, where small subpopulations were used instead of a single large population. Original MA is executed on each subpopulation separately. Then, the harmony search algorithm (HSA),¹⁴¹ which is integrated with the dual-structure coding method, is used to obtain the final optimal solution.

In addition to the population, the performance of MA is also improved by improving the evolution process. The climb is the main process to change the positions of monkeys from the initial positions to the new ones with the improved fitness values. Therefore, to improve the climbing process results in a better evolution, Yi et al.¹⁰⁷ used the stochastic perturbation mechanism of HSA after a large-step climb process to keep the diversity of the monkeys. The pitch adjusting rate in the HSA is adapted to the new components $x_{ij}$ of the vector $x_{i}$ after the large-step climb using equation (21)

x'_{ij} = x_{ij} + round (2 * u * rand - u)

(21)

where $x_{ij}$ is the j component of the vector $x_{i}$ , $u$ is the arbitrary distance bandwidth, $rand$ denotes a random number between 0 and 1, and $round$ indicates the rounding operation.

Yi et al.³⁵ developed an asynchronous-climb process, which is motivated by the simulation of the social behavior of monkeys. In the asynchronous-climb process, monkeys move around the search space and exchange information with others using equation (22)

x_{i, j}^{new} = x_{i, j} + round (m_{1} r_{1} (c p_{i, j} - x_{i, j}) + m_{2} r_{2} (h w_{i, j} - x_{i, j}))

(22)

where $i \in {1, 2, \dots, M}$ , and $j \in {1, 2, \dots, f}$ . Constants $r_{1}$ and $r_{2}$ are uniformly generated random numbers between ‒1 and 1. $m_{1}$ and $m_{2}$ are positive constants to represent cognitive and social parameters. Besides, $m_{1}$ and $m_{2}$ can be properly selected using the following equations

m_{1} = m_{\max} - loop \times (m_{\max} - m_{\min}) / Nc 1

(23)

m_{2} = m_{\min} + loop \times (m_{\max} - m_{\min}) / Nc 1

(24)

where $loop$ is the maximum number of iterations of the initial climb; $Nc 1$ is the maximum number of iterations of asynchronous-climb, $m_{\min}$ and $m_{\max}$ are the boundaries of the ranges of $m_{1}$ and $m_{2}$ .

In addition, Yi et al.¹⁰⁸ significantly modified the climbing process using the niching techniques. Jia et al.¹⁰⁹ proposed a harmony ladder climb process, where stochastic perturbation mechanism of the HSA was used to avoid the problem of premature convergence; Yi et al.¹¹⁰ designed a novel adaptive scheme to automatically control the implementation of climb process. Yi et al.⁵⁰ also improved the climbing process by dividing the entire climb process into initial and advanced climb process. Yi et al.⁴⁰ incorporate virus theory¹⁴² into the climbing process to search the solution in a better local area.

Other processes of MA, such as watch and jump, were also improved. For example, in the original MA, eyesight constant is used to watch around for a higher mountaintop (i.e. a better solution). Either too large or too small values can lead to a miss of better solution nearby. Therefore, Jia et al.¹⁰⁹ proposed a scanning watch-jump process that reduced the probability of missing a nearby higher mountain. Besides, Yi et al.¹¹⁰ developed a novel adaptive scheme to control the implementation of the watch-jump process automatically.

PSO

PSO algorithm, which imitates the social behavior of the flocking of birds, was originally introduced in 1995 by Eberhart and Kennedy.¹⁴³ In PSO, each solution is a point in the D-dimensional search domain that can be treated as a bird or a particle. All particles, which have fitness and velocities, fly through the search space by learning from the best experiences among all the particles. The velocity and position of each particle are updated using equations (25) and (26)¹⁴³

v^{i + 1} = w v^{i} + c_{1} r_{1} (pbes t^{i} - x^{i}) + c_{2} r_{2} (gbest - x^{i})

(25)

x^{i + 1} = x^{i} + v^{i + 1}

(26)

where $w$ ¹⁴⁴ denotes the inertia weight factor, $v$ is the velocity of the particle, $x$ is the position of a particle, $pbest$ is the personal best solution, $i$ means ith generation, $c_{1}$ and $c_{2}$ are the construction factors, and $r_{1}$ and $r_{2}$ are random values range from 0 to 1.

Several OSP studies have utilized traditional PSO processes to obtain the best locations of sensors in SHM systems for different large structures, such as long-span cable-stayed bridges,^112,113 truss structure,¹¹⁴ and an aluminum plate.⁵² As for the coding of the PSO solution, two commonly used methodologies, binary and dual-structure coding methods, are the same as introduced in GA and MA sections. In addition to applying the traditional PSO process for OSP problems, researchers also tried to improve its performance during sensor placement optimization.

Improved PSO

Since the introduction of PSO in 1995, PSO obtains increasing attention for the past decades. Quite a few researchers have attempted to improve its performance with different efforts and to develop several interesting variants.^145–148 In the PSO-based OSP studies, various strategies are proposed by different researchers to improve the performance of PSO when optimizing the sensor locations. For example, Rao and Anandakumar¹¹⁵ tuned the behavior of the PSO algorithm qualitatively and quantitatively by adjusting the PSO parameters dynamically, which achieved a large diversity of generation. In addition, the Nelder–Mead algorithm,¹⁴⁹ which was used to conduct a local search, was used to obtain a better local area. According to equation (25), $w$ takes charge of balancing the local and global search. When $w$ is big, it is better for the global search; when $w$ is small, it is better for local search. Therefore, adjusting $w$ is one important direction of improving PSO performance. Shi and Eberhart¹⁴⁴ proposed an approach to algebraically linear reduce $w$ during the search, where $w$ was linearly reduced from 0.9 to 0.4 with better experiment results. In order to reflect the particle status during parameter adjusting, Wang et al.¹¹⁶ analyzed the relationship between the personal best solution and the global best solution during iteration and the adjusted $w$ using equation (27)

w (t + 1) = w_{0} + \sqrt{\frac{\sum_{i = 1}^{n} {(fitness (x_{i} (t)) - fitness (g))}^{2}}{n}}

(27)

where $w (t + 1)$ denotes the inertia weight of particles from the $t$ generation to $(t + 1)$ generation, $fitness (x_{i} (t))$ is the fitness of the ith particle in $t$ generation, $fitness (g)$ is the fitness of the global best solution in $t$ generation, and $w_{0}$ is a constant. Besides, Zhang et al.⁴¹ proposed an inertia weight, which nonlinearly decreases from $w_{\max}$ to $w_{\min}$ according to the cosine function feature. The strategy of the nonlinear inertia weight can be mathematically represented as equation (28)

w (t) = w_{\min} + {(\frac{1 + \cos (t π / t_{\max})}{2})}^{k} (w_{\max} - w_{\min})

(28)

where $w_{\min}$ and $w_{\max}$ are the boundaries of the $w$ values, $t_{\max}$ is the maximum number of iterations, $k$ is a positive constant that can adjust the curve slope of inertia weight decreasing, and $t$ is the current iteration.

In addition, to avoid PSO being trapped in a local extreme value and the phenomenon of prematurity, Lian et al.¹¹⁷ proposed a hybrid approach of clonal selection and discrete PSO. Clonal selection approach (CSA) is a clonal selection theory-based immune optimization algorithm, which guarantees the diversity of antibodies in the system. Besides, He et al.¹¹⁸ proposed a parallel structure-based integer-coding multi-swarm particle swarm optimization (IMPSO) approach, which simulated biogenic accumulation forms, using grade evaluation to guarantee the diversity of the population, avoiding being trapped in the local optimum.

Firefly algorithm

Firefly algorithm (FA), which is a newly developed swarm intelligence algorithm,^150,151 mimics the social behaviors of fireflies, such as communication, search for food, and mates finding. FA optimization process is conducted in the way that a firefly (i.e. a solution) with a low light intensity (i.e. low fitness) will be attracted and moved to a firefly with a high light intensity (i.e. high fitness). Therefore, to model this process with mathematical representation, three hypotheses are applied: (1) Attractive action among fireflies is inspired by light intensity; (2) the firefly light intensity that is based on the location of a firefly is proportional to the objective function; and (3) the light intensity weakens when the distance increases. Then, the movement of firefly $i$ toward $j$ can be represented as

X_{i}^{t + 1} = X_{i}^{t} + β_{0} e^{- γ r_{ij}^{2}} (X_{j}^{t} - X_{i}^{t}) + α ε_{i}^{t}

(29)

where $X_{i}$ and $X_{j}$ denote the locations of ith and jth fireflies, respectively; $t$ is the number of iteration or time; $γ$ denotes the absorption coefficient of light; $r_{ij}$ is the distance between the ith and jth fireflies; $β_{0}$ is the attractiveness when $r$ equals 0; α is a random parameter locates in the interval [0, 1]; and $ε_{i}$ denotes a random number vector according to a Gaussian distribution.

Like other swarm-based intelligence algorithms, FA encodes the solution using one-dimensional binary coding and dual-structure coding methodologies as mentioned previously. More details of both coding methods can be found in the sections of GA, MA, and PSO.

Improved FA

Since the original FA is proposed to solve continuous variable problems, FA requires being modified and improved to fit the OSP problem that is usually treated as a knapsack problem, selecting specified DOFs for sensor placement to effectively describe the structural performance. There are several studies aiming at improving FA for OSP problems. For example, in order to improve the speed of convergence and avoid trapping in a local extreme value, Zhou et al.⁷⁴ introduced the cluster-in-cluster strategy, which divided the fireflies into several clusters. The firefly cluster is usually formulated based on the Hamming¹⁵² distances. The firefly of the highest light intensity of each cluster is selected as the cluster-head. Fireflies only move with their own cluster to shorten the movement time. Besides, the Hamming distance was also applied by Zhou et al.¹¹⁹ in order to describe the distance among fireflies. They proposed the hybrid discrete firefly algorithm (HDFA), which used a hybrid movement scheme, including directive and nondirective movements, to enhance the robustness of the HDFA to obtain the best sensor configuration. Zhou et al.¹²⁰ also used the Hamming distance to represent the distances among fireflies, and they developed a movement scheme that involves stochastic searching, where the distance $d_{ij}$ of movement from the $i th$ firefly to the jth firefly is defined as

d_{ij} = random (1, 0.5 r_{ij})

(30)

where $r_{ij}$ is the distance between the $i th$ and the jth fireflies. Then, s tree-step semi-oriented movement scheme is developed as follows:

Obtain the difference between the $i th$ firefly and the jth firefly

Δ X_{ij} = X_{j} - X_{i}

(31)

2. Select $d_{ij}$ elements from $Δ X_{ij}$ randomly with a value of 1 and update these elements to −1; and then, select $d_{ij}$ elements from $Δ X_{ij}$ randomly with a value of −1 and update them to 1. $Δ X_{ij}$ changes to $[Δ X_{ij}]$ .

3. The string of $i th$ firefly can be updated by

X_{i} \leftarrow X_{i} + [Δ X_{ij}]

(32)

where $Δ X_{ij}$ denotes the light intensity difference between the $i th$ and the $j th$ fireflies.

In addition, Hao and Zhang¹²¹ proposed a multi-objective FA that also used a hybrid movement scheme to optimize wireless sensor placement trade-off energy efficiency and modal independency while guaranteeing the connectivity of the sensor network. Zhou et al.¹²² utilized Lévy flights,^153–155 which were the typical flight behaviors among insects and animals, to drive fireflies flying to fireflies with a higher light intensity, where the computational performance was improved. Besides, information fusion was used to improve the original FA by speeding up the convergence and avoid being trapped in a local optimum.

Simulated annealing algorithm

Simulated annealing (SA) algorithm, which is based on the studies of Metropolis et al.¹⁵⁶ and Kirkpatrick et al.,¹⁵⁷ is an optimization process simulates thermodynamic physics process of metal cool and anneal.¹⁵⁸ The actual implementation for OSP problems is defined by the fitness function, perturbation operator, parameter setting, and constraints.¹⁵⁹ The annealing process (i.e. iteration) is driven by perturbation operators. During the SA iteration, every variable is selected for perturbation randomly. Then, the operator uses direction cosines for each variable to obtain a random direction. Constraints guarantees that the iteration is conducted within the search domain until an optimal solution is reached. Chattopadhyay and Seeley¹⁶⁰ used SA for both discrete and continuous issues when designing intelligent structures, and SA outperforms previous nonlinear programming approach in computational performance. Therefore, SA can also be implemented to solve the OSP problem, which is usually treated as the combinational problem. Chiu and Lin¹²³ used SA to address the OSP problem for target location considering limited cost and complete coverage. Gou and Cui¹²⁴ proposed a mathematical model to minimize the total energy of sensor placement on a simple beam based on both GA and SA.

Improved SA

In addition to implementing traditional SA, some studies attempted to improve SA to better address the OSP problem. For example, Tong et al.¹²⁵ developed a coordinate-based solution encoding (CSA) to involve more spatial information that was usually limited in the traditional one-dimensional coding method. CSA allows conducting a search in all directions simultaneously, resulting in a better chance to avoid being trapped by a local extreme value. Wang et al.¹²⁶ integrated PSO with SA to improve the chance to obtain the optimal configuration of the WSN. PSO was used to get several local best locations of particles during the iteration; then SA was applied to these local optimal solutions to obtain the best global solution. Besides, SA is integrated with the adaptive mechanism in order to guarantee the quality of the solution and to improve the convergence speed of GA.¹⁶¹

Ant colony optimization

The ant colony optimization (ACO) algorithm utilizes an artificial ant colony, where all ants cooperate to search (i.e. iteration) and reinforce pathways (i.e. solutions) to obtain the shortest pathway (i.e. optimal solution).¹⁶² In order to use ACO for OSP problems, the modification is required as ACO usually works on a travel salesman problem.¹⁶³ Fidanova et al.⁴⁹ used the MAX–MIN ant system¹⁶⁴ to deploy the WSN that can fully cover the sensor field with the minimum number of sensors and maintain the network connectivity. In addition, Feng et al.¹²⁷ improved ACO by proposing an ant colony siege process to improve the efficiency of the sensor configuration. Since pheromone updating, which is the basic updating method used to find better pathways in original ACO, does not work in OSP problem, a novel approach named pheromone total update process was presented. More details of this novel approach can be referred to Feng et al.¹²⁷

Other evolutionary algorithms

In addition to the commonly used algorithms as mentioned above, other evolutionary algorithms have also been proposed for OSP problems. For example, Sun and Büyüköztürk⁴⁵ used the artificial bee colony (ABC) algorithm, which is inspired by imitating the foraging behavior of honey bees, to solve the OSP problem. A new solution updating approach that transforms continuous solution into an integer version was proposed in the developed discrete ABC approach. Yi et al.¹²⁸ proposed a distributed wolf algorithm (DWA), which was inspired by wolf behaviors for food hunting, for OSP problems with faster convergence and a higher capability of searching.

SSP algorithm

In addition to evolutionary algorithms, another efficient heuristic algorithm named SSP algorithm⁸⁶ has also been proposed for OSP problems. SSP is usually divided into two types, namely forward sequential sensor placement (FSSP) and backward sequential sensor placement (BSSP). As for FSSP, the positions of all the sensors required to be placed are sequentially computed by placing one sensor each time on the structure at a position, which reduces the maximum off-diagonal element of MAC most^165,166 or leads to the smallest increase in IE.^70,167–169 Usually, the position of the first sensor is selected as the one with the optimum value of the criteria used for sensor configuration evaluation for the one-sensor scenario. Then, the position of the first sensor is fixed. The position of the second sensor will be selected as one that also leads to the optimum value of the criteria for the two-sensor scenario. Such sensor placement continues until all sensors have been placed on the target structure. The SSP algorithm can also be used inversely. Starting with all sensors placed at all DOFs of the target structure and successively removing one sensor each time from the position. This algorithm is termed as BSSP.

Deterministic optimization methodologies

Several constrained (linear and nonlinear programming)¹⁷⁰ and unconstrained (Newton methods)¹⁷¹ deterministic optimization methods have been proposed for OSP problems. Usually, simple deterministic approaches are sufficient to conduct a local search, and the constrained optimization owns a high degree of complexity.¹⁷² For SHM with simple and regular shapes such as beam and plates, optimal sensor configuration can be determined directly using constrained deterministic techniques like recursive quadratic programming approach,^173,174 as their mode frequencies and shapes can be represented directly with an analytical expression. OSP problems are usually defined as integer programming problems that are more complicated and costly to be addressed, compared to continuous optimization problems. Therefore, discrete variables are converted as continuous variables and then addressed with deterministic optimization techniques. For example, Sepulveda et al.¹⁷⁵ proposed a control-augmented structural synthesis approach, which treats the placement of actuators and sensors as mixed 0-1 continuous variable design problem.¹⁷⁶ Sunar and Rao¹⁷⁷ well addressed the thermopiezoelectric sensor placement on the cantilever beam-like structures based on quasistatic thermopiezoeletricity equations.

Probability-based methodologies

Some studies also used Bayesian theory to determine the sensor configuration. For example, Capellari et al.¹⁷⁸ developed an OSP approach by integrating Bayesian inference with a stochastic search algorithm. Flynn and Todd⁵³ proposed a Bayesian probabilistic framework to place active sensors for crack detection on unknown positions. Yuen et al.¹⁷⁹ presented an IE-based framework to select optimal sensor configuration using the Bayesian statistical method. Besides, Azarbayejani et al.¹⁸⁰ developed a probabilistic approach, which used the weights of a neural network that was trained based on simulations to produce a normalized probability distribution function to determine the optimal sensor configuration.

Other optimization methodologies

In addition to the aforementioned optimization methodologies, various researchers also used different techniques to determine the sensor configuration. For example, Guratzsch¹⁸¹ used Snobfit,¹⁸² which was an optimization strategy designed for bound-constrained optimization of noisy objective functions, to determine the sensor layout.^183,184 Using Snobfit, no candidate sensor positions require being previously determined. Kalman filter algorithm,¹⁸⁵ which estimates the state vector and its corresponding error variances, is integrated with sequential placement strategies to address OSP problems.^39,186 Besides, many other methodologies, such as backup sensor-based fault-tolerance SHM method,³⁰ energy-efficient sensor deployment,⁷⁶ frequency domain–based OSP technique,¹⁸⁷ mixed variable pattern search algorithm,¹⁸⁸ Gram–Schmidt orthogonalization procedure,⁷² three-phase sensor placement approach,²⁹ e-Estimator algorithm,⁷⁵ and wave propagation-based local interaction simulation approach,¹⁸⁹ are also available for OSP. More development details of these methodologies can be referred to the associated studies.

Besides, there is also a trend to use machine learning (ML), which automatically improves or learns from the study or experience, and acts without being explicitly programmed,^190,191 to optimize sensor placement. ML is a method of data analysis that automates analytical model building. To a certain degree, ML with its iterative feature helps improve the efficiency of the formulation and training of the sensor placement optimization model. According to the existing studies of applying ML algorithms for OSP problems, a supervised learning algorithm takes the instantaneous signals (pressure or wall shear stress) from the civil structures as input variables and function characterizing the instantaneous flow state as the output and trains a ML model to generate predictions for the response to new input data. The optimal sensors are usually determined as the input signals (from their corresponding locations) with the highest relevance to the trained model. For example, Semaan¹⁹² introduced a new ML-based approach, which first formulates ML models based on a range of input sensors to predict a response function. Then, the optimal sensor positions are determined as the most relevant features according to variable importance ranking. Amin and Riza¹⁹³ presented an ML-enhanced optical distance sensor, where a regularized polynomial regression–based supervised ML algorithm is used to increase the accuracy of the sensor. Yoganathan et al.¹⁹⁴ proposed a novel data-driven approach that integrated clustering algorithms, information loss approach, and Pareto principle to optimize the sensor placement in an office building. In addition, Praveen Kumar et al.¹⁹⁵ reviewed various ML-based algorithms for WSNs. The advantages and drawbacks of these algorithms were discussed. Compared to the existing approach, ML-based solutions for OSP usually have the advantages of simplicity, adaptivity, and low computational cost.¹⁹²

Comparison of the main computational methodologies for OSP

After reviewing the summarized computational methodologies for OSP, it can be concluded that quite a few computational methodologies can be applied for sensor configuration determination in different structures. Each methodology has its own advantages or disadvantages when being used to solve the OSP problem of SHM. As the most widely used methodologies, evolutionary algorithms show their advantages on sensor placement compared to traditional optimization approaches as the OSP is usually treated as the combinational problem (i.e. NP-hard problem). Although different algorithms with their variants show different computational performance when addressing OSP, how to improve the original algorithm with different improving strategies is shared by all the evolutionary algorithms as presented in Table 1, including new solution coding system, new operators, improved movement scheme, and iteration strategies. In addition to introducing the improved algorithms, the advantages and disadvantages of mainly review computational methodologies as well as the suitable applications for each of them have been summarized in Table 3.

Table 3.

Comparison of OSP computational methodologies.

Computational methodologies	Advantages	Disadvantages	Applications
Evolutionary algorithms
GA	Designed for combinational problemEasy for understanding and implementation	Low globally searching ability with large searching space	Better for civil structure with limited DOFs
MA	Good global searching abilityEfficient local searching abilityGenerating more stable optimal solutions	Originally designed for continuous variable problems	Large structuresPerformance tradeoffs are not unbearableNumber of combinations is too large to preclude enumeration
PSO	Simple, easy to understand and implementStronger global searching ability and faster convergence	Sensitive to the random initial configuration	Large and complex civil structures
FA	Higher computational efficiencyBetter global searching abilityHigher reliability	Originally designed for continuous variable problem	Particularly suited for parallel implementation
SA	Good global searching abilityHigher positioning accuracyLarger sensor fields	Original SA has poor random search mechanism	Better to integrate with other algorithms (e.g. PSO)
ACO	Good global convergence property with improved variantsStable optimal results	Original ACO has low computational performance and accuracy	Suitable for minimizing sensor numberSimple civil structures
Deterministic optimization	Efficiently local search abilityEffective on continuous variables	Discrete OSP problem requires converting to continuous variable	Simple and regular shapes
Probability based	Situation of excitation is not measuredHigher accuracy of approximation	Difficult to implement	Simple structures with a limited number of sensor configuration
Machine learning	Automatically improves or learns from the study or experienceActs without being explicitly programmedLow computational cost	A limited number of training samples	Suitable for existing sensor placement improvement

GA: genetic algorithm; MA: monkey algorithm; PSO: particle swarm optimization; FA: firefly algorithm; SA: simulated annealing algorithm; ACO: ant colony optimization; OSP: optimal sensor placement; DOF: degree of freedom.

Specific SHM applications

In addition to comparing different computational methodologies utilized in OSP of SHM, the studies of specific applications, including the most widely used vibration-based and ultrasonic-based damage detection methods in SHM, had also been explored. Vibration-based damage detection approaches monitor changes in natural frequencies, vibration modes, and related derivatives.⁹⁰ Besides, as most of the structural elements of civil infrastructures are natural waveguides, ultrasonic-based damage detection methods are also utilized in SHM.⁵⁴ For these two specific damage detection methodologies in SHM, evolutionary algorithms are mainly used to obtain the optimal sensor configuration. For example, Rao and Anandakumar¹¹⁵ proposed a PSO-based positioning algorithm for sensor in large-scale structures, where the best identification of frequencies and modals was achieved. Gomes et al.⁹⁰ employed GA to locate the best sensor configuration to cover a specific number of low-frequency modes for laminated composite plates under vibration. As for ultrasonic-based SHM application, Gao and Rose⁵⁴ used covariance matrix adaptation evolutionary strategy (CMAES) to determine the sensor number and sensor distribution in SHM. Guo et al.⁹⁸ adopted GA to search for the optimal locations of sensors. According to the mentioned studies of specific SHM applications, it can be found that evolutionary algorithms are more commonly employed to obtain the optimal sensor configuration.

Conclusion

This study provides a comprehensive review of the computational methodologies for OSP in SHM. First, the problem formulation of OSP is presented. To determine the sensor configuration for various types of structures (e.g. buildings, bridges, and towers), associated 3D FE models consisting of a certain number of DOFs are usually constructed for the numerical experiment as physically testing different sensor configurations are impossible. The OSP is treated as a combinational problem, which aims at assigning sensors to candidate DOFs of the target FE model. Commonly used criteria, including MAC, FIM, IE, EI, and EFI-DPR, are summarized and introduced in detail. Then, optimization methodologies for OSP are introduced.

This study covers most of the existing optimization methodologies, including evolutionary algorithms, SSP methodologies, deterministic optimization, and probability-based methodologies. Since evolutionary algorithms outperform other methods when addressing combinational methodologies, the most widely used evolutionary algorithms (e.g. GA, MA, PSO, FA, SA, and ACO) and their improved variants are summarized in detail. ML algorithms are also identified as the emerging techniques for OSP considering their advantages over traditional approaches. Besides, the advantages and disadvantages of the main computational methodologies as well as the suitable applications for each of them are also summarized and compared. Finally, the study briefly discussed the suitability of computational methods for specific SHM applications, where evolutionary algorithms are more commonly utilized. With the reviewed problem formulation, commonly used optimization methodologies, and the summarized pros and cons of each methodology, this study provides an effective reference for OSP in SHM. Researchers are expected to not only better understand the OSP problem but also conduct the OSP study in a more efficient manner.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the Start-Up Grant at Nanyang Technological University, Singapore (No. M4082160.030), the Ministry of Education Tier 1 Grant (No. M4011971.030), and Start-Up Grant at Shenzhen University, China (No. 2019092).

ORCID iDs

Yi Tan

Limao Zhang

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