A Novel Hybrid Whale-Chimp Optimization Algorithm for Structural Damage Detection

Abstract

Damage detection of structures based on swarm intelligence optimization algorithms is an effective method for structural damage detection and key parts of the field of structural health monitoring. Based on the chimp optimization algorithm (ChOA) and the whale optimization algorithm, this paper proposes a novel hybrid whale-chimp optimization algorithm (W-ChOA) for structural damage detection. To improve the identification accuracy of the ChOA, the Sobol sequence is adopted in the population initialization stage to make the population evenly fill the entire solution space. In addition, to improve the local search ability of the traditional ChOA, the bubble-net hunting mechanism and the random search mechanism of the whale optimization algorithm are introduced into the position update process of the ChOA. In this paper, the validity and applicability of the proposed method are illustrated by a two-story rigid frame model and a simply supported beam model. Simulations show that the presented method has much better performance than the ChOA, especially in dealing with multiple damage detection cases. The W-ChOA has good performance in both overcoming misjudgment and improving computational efficiency, which should be a preferred choice in adoption for structural damage detection.

1. Introduction

Structures usually have a long service life. During their service period, the structural performance will continue to decline due to factors such as temperature changes, chemical erosion, and component aging [1]. Therefore, a timely and accurate assessment of the overall performance of the structure is essential to ensure the safety and long-term service of the building structures [2]. Structural health monitoring technology is used to evaluate the safety and durability of the structure during service to determine the service performance and remaining life of the structure [3,4]. Structural damage detection based on structural properties is a core research topic in structural health monitoring. It has always been a hot topic in academic discussions and research [5,6].

Many scholars have proposed various damage detection methods, which are mainly divided into the following two types: methods based on static characteristics and methods based on dynamic characteristics. Static characteristics of the structure can reflect the stiffness change of the structure more intuitively, which has been studied by many scholars. Pitchai et al. [7] calculated the structural boundary conditions and material properties through the change of structural characteristics under static load, to achieve the goal of identifying the operating state of the structure. Le et al. [8] established an objective function for structural damage detection based on the change of static deflection and carried out structural damage detection by solving the optimal value of the objective function. Emadi et al. [9] performed an inversion analysis of structures including shear effects based on vertical deflection data to identify structural damage. Yang and Sun [10] proposed a method to determine structural damage based on static response equations. Abdo [11] developed a method based on displacement curvature changes to identify structural damage. Boumechra [12] proposed a strategy to identify structural damage based on the deflection change of bridges to update the finite element model. Fang and Huang [13] introduced a model-free structural damage detection method using static measurement data and modal interval analysis to detect damage. Liu et al. [14] presented a method for structural damage detection of statically loaded structures based on the change of internal force and displacement of the finite element model. Caddemi et al. [15] successfully identified the local damage of Euler-Bernoulli beams based on static loads and boundary conditions under the interference of environmental noise. However, these studies based on static characteristics still have some shortcomings, such as the cost being high and the response data being easily disturbed by external factors. Therefore, some researchers begin to pay attention to structural damage detection methods based on dynamic characteristics.

Damage detection methods based on structural dynamic characteristics can be divided into damage detection methods based on dynamic fingerprints [16] and damage detection methods based on model updating [17]. The dynamic fingerprint methods, also known as the modal parameter method, use the modal parameters of the structure as damage factors of the structure, such as natural frequency [18,19], mode shape [20,21,22], flexibility matrix [23,24], modal curvature [25,26] and modal strain energy [27,28]. Usually, when a structural element is damaged, the stiffness of the corresponding element will change, and the modal information will also change. However, obtaining complete modal information for structures requires a large number of sensors, which is difficult to achieve in practical engineering. In addition, the dynamic fingerprint measurement process is easily affected by noise, which limits the development of such methods to a certain extent.

By continuous optimization and adjustment of the physical parameters of the benchmark finite element model, the model updating method is widely used based on the static or dynamic test data of the structure [29,30]. Depending on this, a nonlinear objective function is established according to the parameter changes before and after the updating of the finite element model, which is transformed into a mathematical function optimization problem. Then, the position and degree of structural damage can be detected by optimizing the mathematical function [31]. Structural damage detection based on model updating requires establishing a precise benchmark finite element model, which usually contains many degrees of freedom and multiple correction parameters, resulting in a high-dimensional multi-extremum optimization function [32]. Traditional mathematical function optimization methods are challenging to solve such complex mathematical models. To optimize such mathematical models quickly and accurately, some scholars have introduced the heuristic intelligent algorithm into structural damage detection based on model updating [33,34]. Wang et al. [35] developed a new method of densely connected convolutional networks, which improved the accuracy of structural damage identification. Li et al. [36,37] introduced a variational Bayesian inference approach and phase space embedding with the stochastic Koopman operator to identify structural damage. Sadeghi et al. [38] combined modal strain energy changes and general neural networks to identify structural damage. He et al. [39,40] proposed an indirect structural damage detection strategy and a two-stage structural damage identification method.

Chen et al. [41] adjusted the inertia weight factor of particle swarm optimization by sigmoid function, which improved the identification accuracy of structural damage identification. Gomes et al. [42] introduced the sunflower algorithm into structural damage detection to enhance the robustness of structural damage detection. Pan et al. [43] combined the firefly algorithm with the Nelder-Mead algorithm to detect structural damage. Tiachacht et al. [44] used the modal strain energy ratio as a damage factor to identify the location and degree of structural damage in stages through the slime mould algorithm. Guedria [45] proposed a structural damage identification strategy based on an accelerated differential evolution algorithm. Cao et al. [46] implemented structural damage identification in stages by auto-associative neural networks and Levenberg-Marquardt neural networks. Guo et al. [47] combined partial modal mode shapes and natural frequencies to identify structural damage through artificial intelligence neural networks. Jafarkhani and Masri [48] proposed a structural damage detection strategy based on an adaptive covariance matrix. Ahmadi-Nedushan and Fathnejat [49] combined the flexibility matrix with the modal strain energy to offer an improved teaching-learning optimization algorithm. Fu and Jiang [50] combined the probabilistic neural network and data fusion technology with correlation fractal dimension to identify structural damage. However, for some high-dimensional swarm intelligence algorithms, the convergence speed and identification accuracy can still be improved. Then, the chimp algorithm is introduced as an attempt to solve the optimization problem of structural damage detection.

Inspired by the hunting behavior of the chimp population, the Chimp Optimization Algorithm (ChOA) was first proposed by Khishe and Mosavi in 2020 [51]. ChOA is a heuristic intelligent algorithm with a fast global search ability, which can be used to solve optimization problems. Compared with other similar algorithms, ChOA has fewer parameters and is easier to improve. However, ChOA still suffers from the problem of unbalanced exploration, which leads to premature traps and insufficient identification accuracy. In this regard, some scholars have proposed a few improvement strategies for this method. For instance, Jia et al. [52] improved the optimization ability of the chimp algorithm by using three strategies of highly damaged polynomial edge, Spearman’s rank correlation coefficient, and beetle antennae operator, respectively. Kumari et al. [53] proposed a boosted ChOA by studying different variants of the ChOA. Dhiman [54] proposed a hybrid ChOA based on the Sine-cosine function and the spotted hyena optimizer attack strategy. The paper proposes an improved hybrid whale-chimp optimization algorithm (W-ChOA). Firstly, the Sobol sequence [55] is introduced to initialize the population to enhance the initial convergence speed of the algorithm. Secondly, the multi-strategy hunting mechanism of the whale optimization algorithm [56] is integrated with the hierarchical hunting mechanism of the ChOA to avoid falling into the local optimum. Simulations show that the W-ChOA has good performance in structural damage detection. The article is divided into five sections. The presented W-ChOA is deduced in Section 2. Then, Section 3 details the mathematical model of structural damage detection by the W-ChOA. Numerical simulations of the W-ChOA for structural damage detection and the comparison results with the ChOA are demonstrated in Section 4. Finally, some conclusions are discussed in Section 5.

2. Hybrid Whale-Chimp Optimization Algorithm

2.1. Chimp Optimization Algorithm

The ChOA is a heuristic intelligent algorithm with global fast searchability, which was jointly proposed by Khishe and Mosavi in 2020 [51]. The idea of the ChOA comes from the hunting behavior of the chimp population. Chimps divide labor according to individual abilities into four types: chaser, barrier, driver, and attacker. The four types of chimp groups divide labor and cooperate, perform their duties, and complete hunting activities together. Hunting activities are mainly divided into the exploration stage and exploitation stage. In the exploration stage, barrier, driver, and chaser are mainly used to surround the prey. In the exploitation stage, the attackers mainly complete the final hunting process. The mathematical model can be expressed as

$$d=\left|c·X_{\mathrm{prey}}\left(t\right)-m·X_{\mathrm{chimp}}\left(t\right)\right|$$

(1)

$$X_{\mathrm{chimp}}\left(t+1\right)=X_{\mathrm{prey}}\left(t\right)-a·d\mathbf{}$$

(2)

where $\Vert$ represents the absolute value and $·$ represents the multiplication of vectors or matrices. $d$ is the distance vector between the prey and the chimp predator. $X_{\mathrm{prey}}$ and $X_{\mathrm{chimp}}$ represent the current position vector of the prey and the chimp predator, respectively. $t$ is the current iteration number of the population. $m$ is a chaotic vector in the chaotic cloud map to simulate the influence of chimp sexual motivation in the later stage of exploitation. $a$ and $c$ represent the correlation coefficient vectors, which can be expressed as

$$a=2f·r_1-f;c=2r_2$$

(3)

where $a$ is a random vector that controls the distance between the chimp and the prey, and its elements are random values between $\left[-f,f\right]$. $f$ is the ability of global search and local search, and its element values decrease nonlinearly from 2.5 to 0 with the increase of the number of iterations. $r_1$ and $r_2$ are random vectors in the range of [0, 1], which make the population have diverse characteristics. $c$ is a control coefficient vector of chimps intercepting and chasing prey, and its elements are random values between [0, 2].

In the ChOA, different independent groups can be set to perform a global search and local search simultaneously to accomplish the common goal. The groups of chimps in various leadership levels are established by setting $f$ with successively changes. The dynamic change formulas of $f$ are as follows:

$$\begin{gathered} f_1=1.95-2\times \frac{t^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{T_{\max }{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} \\ {f}_2=1.95-2\times \frac{t^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{T_{\max }{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}} \\ {f}_3=\left(-3\times \frac{t^3}{T_{\max }{}^3}\right)+1.5 \\ {f}_4=\left(-2\times \frac{t^3}{T_{\max }{}^3}\right)+1.5 \end{gathered}$$

(4)

where $T_{\max }$ represents the maximum number of iterations of the population. $f_1$, $f_2$, $f_3$, and $f_4$ represent the searchability of the attacker, barrier, chaser, and driver, respectively. Vector $f$ is the ability of global search and local search, which is composed of $f_1$, $f_2$, $f_3$, and $f_4$.

To establish a more accurate mathematical model for simulating chimp hunting, the chaser, barrier, driver, and attacker of population initialization are assumed to be located in the positions of the four putative prey species. The other chimps in the population will update their location based on the positions of the four putative prey species. The location update rules are as follows:

$$\begin{gathered} X_{\mathrm{c}1}=X_{\mathrm{attacker}}-a_1·\left|c_1·X_{\mathrm{attacker}}-m_1·X_{\mathrm{chimp}}\left(t\right)\right| \\ {X}_{\mathrm{c}2}=X_{\mathrm{barrier}}-a_2·\left|c_2·X_{\mathrm{barrier}}-m_2·X_{\mathrm{chimp}}\left(t\right)\right| \\ {X}_{\mathrm{c}3}=X_{\mathrm{chaser}}-a_3·\left|c_3·X_{\mathrm{chaser}}-m_3·X_{\mathrm{chimp}}\left(t\right)\right| \\ {X}_{\mathrm{c}4}=X_{\mathrm{driver}}-a_4·\left|c_4·X_{\mathrm{driver}}-m_4·X_{\mathrm{chimp}}\left(t\right)\right| \\ {X}_{\mathrm{chimp}}\left(t+1\right)=\left(X_{\mathrm{c}1}+X_{\mathrm{c}2}+X_{\mathrm{c}3}+X_{\mathrm{c}4}\right)/4 \end{gathered}$$

(5)

where $X_{\mathrm{attacker}}$, $X_{\mathrm{barrier}}$, $X_{\mathrm{chaser}}$, and $X_{\mathrm{driver}}$ represent the position vectors of the attacker, barrier, chaser, and driver, respectively. The subscripts 1, 2, 3, and 4 represent parameters corresponding to the attacker, barrier, chaser, and driver, respectively. For instance, $X_{\mathrm{c}1}$ represents the new position vector with the chimp updated according to the positions of the attacker; $a_1$ represents the ability vector of the attacker to control the distance between itself and the prey; $c_1$ represents the correlation coefficients vector preventing the attacker from the prey; $m_1$ represents the influence of the sexual motivation vector of the attacker during its hunting behavior.

However, food gratification and subsequent social motivation may cause chimps to neglect their hunting duties. Then the chaotic behavior of the chimp population in the final stage is helpful in solving the problem of precociousness and slow convergence when the algorithm faces multi-extremum non-differentiable problems. The location update formula can be improved as follows:

$$X_{\mathrm{chimp}}\left(t+1\right)=\left\{\begin{array}{c}{X}_{\mathrm{prey}}\left(t\right)-\mathbf{a}·\mathbf{d}\hspace{1em}if \mu <0.5\\ X_{\mathrm{chaotic}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}if \mu \ge 0.5\end{array}\right.$$

(6)

where $\mu$ is a random value in the interval of [0, 1]. $X_{\mathrm{chaotic}}$ is the mapped value in the chaotic cloud graph.

2.2. Initialize Population with Sobol Sequence

When using a swarm intelligence algorithm to solve the multi-extremum and non-differentiable problems, the initialization of the population has a great influence on the convergence speed and optimization accuracy. The ChOA uses the random function to initialize the population. The population initialized by random function has good randomness, while lacking ergodicity of the population and easy to cause individual aggregation of the population. Given this, the Sobol sequence [55] is introduced to initialize the chimp population in this study. Compared with the pseudo-random sequence generated by the random function, the Sobol sequence generates a deterministic quasi-random sequence through the Quasi-Monte Carlo method [29]. Assume that the value range of the independent variables in the solution space is $\left[X_{\min },X_{\max }\right]$. The mathematical model with chimp population initialization can be shown as:

$$X_{\mathrm{position}}^e=X_{\min }+S_e·\left(X_{\max }-X_{\min }\right)$$

(7)

where $X_{\mathrm{position}}^e$ is the position of the chimp in the $e$-th dimension. $S_e$ is $e$-th the random value generated by the Sobol sequence in the interval of [0, 1].

Assume that the range of the solution space is [0, 1], the search dimension is two and the population number is set to 80. Figure 1 illustrates the spatial distribution of the pseudo-random sequences and the Sobol sequence, respectively. The horizontal and vertical coordinate axes represent the value of the independent variable in each dimension, respectively. As shown in Figure 1, the spatial distribution of the Sobol sequence is more uniform than the pseudo-random sequence. Moreover, the individual population agglomeration phenomenon of the Sobol sequence is significantly lower. Simulation result shows that the spatial distribution of the Sobol sequence has stronger ergodicity than the pseudo-random sequence. Then, the Sobol sequence will be introduced to initialize the chimp population in this study.

2.3. Whale Optimization Algorithm

The whale optimization algorithm is a swarm intelligence optimization algorithm by simulating the foraging behavior of humpback whales [56]. Due to the small diameter of the esophagus, whales only catch krill and small swarming fish for food. Therefore, a unique way of hunting named bubble-net hunting is formed. The whale optimization algorithm realizes the optimization of the objective function by constructing the mathematical model of bubble-net hunting. The whole hunting process can be divided into three stages: shrink and surround the prey, spiral bubble-net attack, and random search mechanism.

The whale population continues to shrink and surround the prey, which reflects the local searchability of the algorithm. The mathematical model of shrinking and surrounding can be shown as follows:

$$D=\left|C·X^{\ast }\left(t\right)-X\left(t\right)\right|$$

(8)

$$X\left(t+1\right)=X^{\ast }\left(t\right)-A·D\mathbf{}$$

(9)

where $X^{\ast }$ indicates the current position vector of the prey. $X$ is the position vector of the humpback whale. $t$ is the current iteration number of the population. $D$ is the distance vector between the prey and the whale predator. $A$ and $C$ represent the correlation coefficient vectors, which can be expressed as:

$$A=2u·r_3-u;C=2r_4$$

(10)

where $u$ is the convergence coefficient vector, which decreases linearly from 2 to 0 during the iterative process. $r_3$ and $r_4$ are random vectors in the interval of [0, 1].

The whale population builds a bubble-net while spiraling upwards. To simulate the mechanism of the spiral bubble-net, it is necessary to simulate the distance between the individual population and the food. Then Equation (9) can be rewritten by the bubble-net hunting mechanism as:

$$Xt+1={\mathrm{e}}^{ql}\left(\cos 2\mathsf{\pi }l\right)D\prime +X^{\ast }\left(t\right)$$

(11)

where $D\prime$ represents the distance between the whale with an optimal position and other whales. $q$ represents the constant of the logarithmic spiral equation. $l$ is a random value in the interval of [−1, 1].

It is worth noting that the optimization strategy of the whale optimization algorithm is based on the calculation of the coefficient vector $A$. If $A$ exceeds the range of [−1, 1], it indicates that the individual whale is outside the surrounding area of the population bubble-net. At this time, the whale needs to search randomly according to its position and update its position. The optimization strategy reflects the global search performance of the whale optimization algorithm. It is assumed that each whale population has a 50% probability to choose one of the position update mechanisms. Then Equations (8) and (9) can be rewritten by the random search mechanism as:

$$D=\left|C·X_{\mathrm{random}}-X\left(t\right)\right|$$

(12)

$$X\left(t+1\right)=X_{\mathrm{random}}-A·D\mathbf{}$$

(13)

where $X_{\mathrm{random}}$ is the random position of whales.

2.4. Hybrid Whale-Chimp Optimization Algorithm

The ChOA has a clear theory and few parameters, which has advantages for solving complex optimization problems. However, the algorithm has a fast convergence speed and a simple position update method, which is prone to premature phenomena in the optimization process. In this regard, the location update strategy of the whale optimization algorithm is introduced. To improve the ability of the ChOA to jump out of the local extreme value, it is necessary to update the position of the chimp in the population according to the bubble-net hunting mechanism or the random search mechanism in each iteration process. The specific steps of the W-ChOA are as follows

Step 1: Set relevant parameters, population size $N$, and the maximum number of iterations $T_{\max }$, convergence factor $f$, correlation coefficient vectors $A$, $C$, chaos factor $m$, etc. Set the limit of the individual position of the population in the interval of [0, 0.9] and generate the initial population $X_k$, $k=1, 2,\dots , N$.

Step 2: Calculate the fitness value of the individual and determine the first four optimal solutions ${\mathbf{X}}_{\mathrm{attacker}}$, $X_{\mathrm{barrier}}$, $X_{\mathrm{chaser}}$, and $X_{\mathrm{driver}}$.

Step 3: Update the population position according to the size of random value $p$ and correlation coefficient vector $A$. If $p<0.5$ and $\left|A\right|\ge 1$, the random search mechanism is used for updating the population position according to Equation (13). If $p<0.5$ and $\left|A\right|<1$, the chimp individuals update their population positions with the traditional ChOA according to Equation (5). If $p\ge 0.5$, the bubble-net hunting mechanism is used for updating the population position according to Equation (11).

Step 4: Determine whether the algorithm reaches the convergence conditions or the maximum number of iterations. If so, the end of the algorithm and the optimal position $X_{\mathrm{attacker}}$ is output; otherwise, step 2 and step 3 are executed to start a new round of iteration.

The W-ChOA combines the advantages of these two algorithms. Compared with the ChOA, the search accuracy and convergence speed of the W-ChOA have been significantly improved. The flowchart of the W-ChOA is shown in Figure 2.

2.5. Verification by Benchmark Functions

To demonstrate the effectiveness of the W-ChOA, three classical benchmark functions are used for verification. The search ranges, global optima, and function graphs of these benchmark functions are shown in

Table 1. Three classic benchmark functions.

Function	Function Expression	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$	Landscape
Rastrigin	$f_1\left(x\right)={\displaystyle {\displaystyle \sum }_{g=1}^{30}}\left[x_g^2-10\cos \left(2\mathsf{\pi }{x}_g\right)+10\right]$	$\left[-5.12,5.12\right]$	0
Quartic	$f_2\left(x\right)={\displaystyle {\displaystyle \sum }_{g=1}^{30}}g{x}_g^4+\mathrm{random}\left[0,1\right.)$	$\left[-1.28,1.28\right]$	0
Griewank	$f_3\left(x\right)=\frac{1}{400}{\displaystyle {\displaystyle \sum }_{g=1}^{30}}{x}_g^2-{\displaystyle {\displaystyle \prod }_{g=1}^{30}}\left(\frac{x_g}{\sqrt{g}}\right)+1$	$\left[-600,600\right]$	0

. All three benchmark functions are high-dimensional multimodal functions. Both the ChOA and the W-ChOA are verified by these benchmark functions with the same input parameters. The population size $N=50$ and the maximum number of iterations $T_{\max }=400$. The chaotic vector $m$ is selected as Gauss chaotic cloud map. Repeat each calculation ten times and the calculation results are shown in

Table 2. Test results of benchmark functions.

Function	Calculated Performance	ChOA	W-ChOA
Rastrigin	Optimal value	$5.68\times {10}^{-14}$	0
	Average value	$3.72\times {10}^{-9}$	$2.96\times {10}^{-13}$
Quartic	Optimal value	$3.34\times {10}^{-4}$	$1.57\times {10}^{-4}$
	Average value	$1.66\times {10}^{-3}$	$6.27\times {10}^{-4}$
Griewank	Optimal value	$5.68\times {10}^{-14}$	0
	Average value	$2.59\times {10}^{-12}$	$5.80\times {10}^{-14}$

and Figure 3.

Table 2 lists the optimal fitness values and average fitness values of three benchmark functions by the ChOA and the W-ChOA, respectively. As shown in Table 2, when adopted to solve a high-dimensional multimodal function, the ChOA is prone to fall into local traps and easy to obtain unsatisfactory results. By introducing the bubble-net hunting mechanism and the Sobol sequence in the ChOA, the W-ChOA can obtain much better optimal values of these functions whether the average values or the minimum values. As shown in Figure 3, the W-ChOA has a significantly faster convergence speed than the ChOA. Test function verification results show that the W-ChOA can be used to enhance the optimization ability and calculation efficiency of the ChOA.

3. Mathematical Model of Structural Damage Detection

3.1. Definition of Structural Damage

Damage detection of structures based on model updating is a common theoretical method in the field of structural damage detection. Generally speaking, without considering the damping of the structure, the vibration differential equation of the structure can be expressed as:

$$\left(K-{\omega }^{2}M\right)\Phi =0$$

(14)

where $K$ and $M$ represent the global stiffness matrix and the global mass matrix of the structure, respectively. $\omega$ and $\Phi$ represent the frequency and the mode shape matrix of the structure, respectively.

The location and degree of structural damage can be detected by identifying changes in structural modal parameters, $\omega$ and $\Phi$. The modal parameters of the structure are only related to the stiffness matrix and mass matrix of the structure as shown in Equation (14). Therefore, the damage of the structure can be simulated by changing the stiffness or mass of the structure. In this study, the structural damages are simulated by reducing the stiffness of the structure.

Based on the finite element theory, the model of the detected structure can be divided into $Q$ elements. The stiffness matrix of the structure with damages can be expressed as:

$$K_d={\displaystyle \sum }_{j=1}^Q\left(1-{\alpha }_j\right)K_e^j$$

(15)

where $K_d$ represents the global stiffness matrix of the structure under damaged condition and $K_e^j$ represents the intact condition of the $j$-th element, respectively. The damage factor ${\alpha }_j$ belongs to (0, 0.90) in this paper, which represents the stiffness reduction coefficient of the $j$-th element.

3.2. Definition of Objective Function

Inspired by the model updating method, the measured responses of the structure and the finite element model information of the structure are used to establish the objective function. The frequency and mode shape can be calculated from the finite element model of the structure and also can be measured from the structure in the laboratory, respectively. Then, the modal information under the two conditions is compared by the modal guarantee criterion and the objective function. Finally, structural damage detection can be transformed into a function optimization problem with constraints. The objective function of this study can be expressed as:

$$\min f\left(\alpha \right)={\displaystyle \sum }_{i=1}^s\left(\left(1-MAC\left({\Phi }_i^m,{\Phi }_i^c\right)\right)+ER\left({\omega }_i^m,{\omega }_i^c\right)\right)$$

(16)

where ${\Phi }_i^m$ and ${\omega }_i^m$ represent the measured $i$-th mode shape vector and frequency of the structure, respectively. ${\Phi }_i^c$ and ${\omega }_i^c$ represent the calculated $i$-th mode shape vector and frequency of the finite element model of the structure, respectively. $s$ represents the selected structural modal order. $\alpha$ represents the damage factor of each element. $MAC\left({\Phi }_i^m,{\Phi }_i^c\right)$ and $ER\left({\omega }_i^m,{\omega }_i^c\right)$ represent change rate of the modal confidence and frequency of the structure, respectively. The functions can be defined as:

$$MAC\left({\Phi }_i^m,{\Phi }_i^c\right)=\frac{{\left|{\left({\Phi }_i^m\right)}^{\mathrm{T}}{\Phi }_i^c\right|}^2}{\left|{\left({\Phi }_i^m\right)}^{\mathrm{T}}{\Phi }_i^m\right|\left|{\left({\Phi }_i^c\right)}^{\mathrm{T}}{\Phi }_i^c\right|}$$

(17)

$$ER\left({\omega }_i^m,{\omega }_i^c\right)=\left|\frac{{\omega }_i^m-{\omega }_i^c}{{\omega }_i^m}\right|\times 100\%$$

(18)

where superscript $\mathrm{T}$ represents the transpose of a matrix.

By defining the structural damage and the objective function, structural damage detection can be converted to solve the independent variable $\alpha$ corresponding to the optimal value of the objective function. Then, the ChOA and the proposed W-ChOA can be used to identify the location and degree of structural damage.

4. Numerical Simulations

4.1. Structural Damage Detection of Two-Story Rigid Frame

A two-story rigid frame structure is divided into 18 elements, as shown in Figure 4. The numbers in the circles represent the element number and the other numbers represent the node number. To prevent the interference of other factors, the nodes of the two-story rigid frame are all set as steel nodes. The overall degree of freedom of the structure is 16 and the first type of constraint is used to prevent rigid body displacement. Physical parameters of the two-story rigid frame structure are listed in

Table 3. Physical parameters of rigid frame structure.

Physical Parameter	Column	Beam
Elasticity modulus (N/m2)	2.0 × 1011	2.0 × 1011
Moment of sectional inertia (m4)	1.26 × 10−5	2.36 × 10−5
Sectional area (m2)	2.98 × 10−3	3.20 × 10−3
Density (kg/m3)	8590	7593

as follows:

To simplify the comparison process, both the ChOA and the W-ChOA use the same parameters. The parameters are set as: population size $N=50$ and maximum iterations $T_{\max }=200$. Modal order is set to five and search dimension is set to 18. The optimal value is selected by running five times for each case. Excluding the set of real damage elements, damage detection values greater than 10⁻³ are considered misjudgments.

In structures, the failure of the frame structure often occurs in the mid-span where it is subjected to a big bending moment. Therefore, the damage simulation is set in the middle element of the beam. The damage degree is realized by reducing the stiffness of the element. As listed in

Table 4. Different damage cases of two-story rigid frame.

Case	Damages	Damage Elements	Degree of Damage
1	Single damage	17	10%
2	Double damages	8, 17	10%, 10%
3	Triple damages	5, 8, 17	10%, 10%, 10%

, the stiffness of the elements is reduced by 10% to simulate different damage situations of the frame structure. In addition, to accurately simulate the effects of measurement errors on natural frequency and mode shape, random noise is added to the measured modal parameter, as shown:

$$S_{\mathrm{noise}}=S_{\mathrm{original}}\left(1+R_p{N}_{\mathrm{variable}}\right)$$

(19)

where $S_{\mathrm{noise}}$ is the modal parameter with noise variable. $S_{\mathrm{original}}$ is the original modal parameter of the rigid frame. $R_p$, $N_{\mathrm{variable}}$ represents the noise level and the type of noise subject to the standard normal distribution, respectively.

Figure 5 shows the single damage results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 5a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 17 without misjudgment. In contrast, the identification results of the ChOA have a misjudgment at element 2. As shown in Figure 5b, if the damage percent is 10% and the noise level is 10%, both the ChOA and the W-ChOA have a misjudgment at element 2 and the misjudgment value of the ChOA is greater than the W-ChOA.

Figure 6 shows the double damage detection results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 6a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 8 and element 17 without misjudgment. In contrast, the identification results of the ChOA have a misjudgment at element 3. As shown in Figure 6b, if the damage percent is 10% and the noise level is 10%, the ChOA results have three misjudgments in element 2, element 10, and element 11. The identification results of the W-ChOA are much better with only one misjudgment at element 11.

Figure 7 shows the triple damage detection results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 7a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 5, element 8, and element 17 with only one misjudgment at element 11. In contrast, the identification results of the ChOA have four misjudgments in element 2, element 3, element 7, and element 13. As shown in Figure 7b, if the damage percentage is 10% and the noise level is 10%, the ChOA results have nine misjudgments. The identification results of the W-ChOA are much better with only one misjudgment at element 11.

With the increase in noise level, the identification accuracy of the ChOA reduces significantly. Compared with the ChOA, the W-ChOA shows remarkable robustness. In addition, with the increase of damage elements, the misjudgments of the ChOA increase significantly. As shown in Figure 7b, the number of misjudgment damage elements of the ChOA even reaches nine. Compared with the ChOA, the W-ChOA shows perfect case stability. With three different damage cases, the maximum number of misjudgment damage elements of the W-ChOA is only one.

The mathematical model of multi-damage detection has multiple global optimal values, which causes the optimization algorithm to be easily disturbed during the iterative optimization process. By introducing the Sobol population initialization strategy, the bubble-net hunting mechanism, and the random search mechanism of the whale optimization algorithm, the W-ChOA shows significantly better performance than the ChOA algorithm in multi-damage detection cases. Simulation results of the two-story rigid frame show that both the ChOA and the W-ChOA have good performance in the single-damage detection case with a low noise level. When the noise level is high or multi-damage exists in the structure, the W-ChOA shows its superiority over the ChOA algorithm regarding damage detection accuracy and robustness.

4.2. Structural Damage Detection of Both Asymmetric Damage and Symmetric Damage

To verify the universality of the proposed method in both asymmetric damage and symmetric damage, a simply supported beam is shown in Figure 8. Physical parameters of the beam are as follows: The beam length $L=2.5 \mathrm{m}$, the section width $b=0.15 \mathrm{m}$, the section height $h=0.02 \mathrm{m}$, the elastic modulus $E=7\times {10}^{10}\mathrm{Pa}$, and the material density $\rho =2750 \mathrm{kg}/{\mathrm{m}}^3$. The simply supported beam is divided into 10 elements, each element has two nodes and four degrees of freedom. Assume that the axial deformation is ignored and only the vertical degrees of freedom of each element are extracted from the modal shape during damage detection.

In this simulation example, the search dimension of the W-ChOA is set to 10 and other parameter settings are the same as in the previous simulation example. As shown in

Table 5. Different damage cases of the simply supported beam.

Case	Type	Element	Degree of Damage
1	asymmetric	3	10%
2		3, 7	10%, 10%
3		3, 7, 9	10%, 10%, 10%
4	symmetric	2, 9	10%, 10%

, the stiffness of the elements is reduced by 10% to simulate different damage cases of the simply supported beam. Random noise is also added to the measured modal parameters as shown in Equation (19).

Figure 9 shows the single structural damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. In this case, both the ChOA and the W-ChOA have perfect damage identification results without any misjudgment.

Figure 10 shows the double damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. As shown in Figure 10, if the damage percent is 10% and the noise level is 5% or 10%, the W-ChOA can accurately identify damage location and damage degree at element 3 and element 7 without any misjudgment. In contrast, the identification results of the ChOA have more than two misjudgments in this case.

Figure 11 shows the triple damage detection results of the simply supported beam by the ChOA and the W-ChOA under different noise levels. As shown in Figure 11a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 3, element 7, and element 9 with only one misjudgment at element 4. In contrast, the identification results of the ChOA have four misjudgments in element 1, element 2, element 4, and element 5. As shown in Figure 11b, if the damage percentage is 10% and the noise level is 10%, the ChOA results have five misjudgments. The identification results of the W-ChOA are much better with only two misjudgments in element 1 and element 6.

Figure 9, Figure 10 and Figure 11 show the damage detection results of the simply supported beam in asymmetric damage cases. To satisfy the integrity of numerical simulation, Figure 12 illustrates the damage detection results of the simply supported beam in the symmetric damage case.

As shown in Figure 12, if the damage percent is 10% and the noise level is 5% or 10% in the symmetric damage case, the W-ChOA can accurately identify damage location and damage degree at element 2 and element 9 without any misjudgment. In contrast, the identification results of the ChOA have more than two misjudgments in this case.

Based on the simulation results of the simply supported beam, some conclusions similar to the previous example can be drawn. That is, the W-ChOA shows better case stability and higher robustness compared with the ChOA. Especially when the noise level is high or multi-damage exists in the structure, the W-ChOA shows its superiority over the ChOA algorithm both in terms of asymmetric damage cases and symmetric damage cases.

5. Conclusions

In this study, a hybrid whale-chimp optimization algorithm (W-ChOA) is proposed for structural damage detection. Numerical simulations of a two-story rigid frame and a simply supported beam model are adopted to verify the effectiveness of the novel algorithm, and the following conclusions can be drawn:

(1)

With these three high-dimensional multimodal test functions, similar conclusions can be drawn. The calculation accuracy and convergence efficiency of the W-ChOA are significantly better than those of the basic ChOA. Test function verification results show that the hunting mechanism of the whale optimization algorithm and the Sobol sequence introduced in the ChOA are practical and successful.

(2)

Validated by a two-story rigid frame model and a simply supported beam model, the W-ChOA can effectively identify the damage location and damage degree in both single-damage and multi-damage condition. In cases with extensive noise interference or multi-damage existing in the structure, the high misjudgment rate of the ChOA has been effectively reduced and improved by adopting the W-ChOA.

(3)

Compared with the ChOA, the proposed W-ChOA has much better performance such as fast convergence speed, high identification accuracy, good case stability, and strong robustness when applied to structural damage detection both in asymmetric damage cases and symmetric damage cases. Due to the excellent performance in different simulations, the W-ChOA is strongly recommended for structural damage detection.

(4)

For actual structures, the influence of external environmental factors such as temperature and humidity on the modal parameters of the structure is sometimes even greater than the effect of noise. Therefore, the proposed method still needs to consider environmental influences to be better applied to the actual structure. In addition, the parameters of the W-ChOA need to be further refined for appropriate application in the field of structural damage detection.