A Particle Swarm Optimization Algorithm with Sigmoid Increasing Inertia Weight for Structural Damage Identification

Abstract

In this study, a particle swarm optimization with a sigmoid increasing inertia weight (SIPSO) algorithm is proposed for structural damage identification based on the optimization of structural vibration response constraints. In view of the existing problems for particle swarm optimization algorithms used for structural damage identification, such as low accuracy of damage identification and easy misjudgment of damage location, the sigmoid increasing inertia weight is introduced to improve the global and local search ability of the algorithm. Simulation results show that the parameters of the sigmoid increasing inertia weight have a significant effect on the performance of the SIPSO algorithm for structural damage identification. Compared with similar improved particle swarm optimization algorithms, the SIPSO algorithm has some advantages of fast convergence speed, high identification accuracy, and strong robustness ability in structural damage identification.

1. Introduction

With the accumulation of service time, structures will be damaged to a certain extent, resulting in potential safety hazards. If structural damages can be found as early as possible during service, the service function and remaining life of the structure can be effectively improved. The existence of structural damage will generally cause changes in structural dynamic characteristics such as structural stiffness, frequency, or vibration mode. Structural damage identification is one of the inverse problems of structural health monitoring (SHM) [1]. Li et al. [2,3] briefly reviewed the field of SHM and discussed the future development trend and proposed a nested attribute-based few-shot meta-learning paradigm for structural damage identification.

Structural damage identification has developed rapidly in recent decades. Curadelli et al. [4] proposed the use of the instantaneous damping coefficient as a damage detection index combined with a wavelet transform; the method was verified in numerical simulations and laboratory tests. Li et al. [5,6,7] put forward some new methods for structural damage detection by developing a novel approach based on densely connected convolutional networks, using the higher-order response and phase space topology technology based on singular spectrum analysis, and improving the artificial bee colony algorithm. Ren et al. [8] deduced a formula for calculating the first-order sensitivity of element modal strain energy and successfully applied to detect damage of numerical models. Chan et al. [9,10,11,12] put forward a series of new damage detection methods, respectively, using a multi-layer genetic algorithm, developing a geometric modal strain energy–eigenvalue ratio, and a double criteria method based on vibration characteristics to detect and locate the damage of various kinds of structures. Kong and Cai [13,14,15] reviewed the state-of-the-art approaches for the framework of vibration-based damage identification and proposed new methods for damage identification by using vehicle–bridge interaction analysis.

Noh et al. [16] introduced a damage diagnosis algorithm by using a sequential change-point detection method. Dessi and Camerlengo [17] compared several structural damage identification methods based on modal curvature, natural frequency, and modal strain and pointed out the identification ability of each method on the location and degree of damage. Shadan and Khoshnoudian [18] utilized the sensitivity of the frequency response function to effectively identify the location and degree of damage. Bonessio et al. [19] proposed a multi-mode multi-directional damage index method for multi-directional damage detection in frame structures. Entezami et al. [20] proposed a new iterative regularization method and an improved sensitivity function for the identification of structural damage. Zhu et al. [21,22] proposed two new methods to detect damage, respectively, based on the Bayesian probabilistic approach and a time series analysis with sparse regularization. Yang et al. [23] proved that the square of the instantaneous amplitude is a good damage detection index for a simply supported beam on an elastic foundation. Giordano et al. [24] presented a damage detection method for masonry arches, and the frequency variations are used in this new method. Li et al. [25] presented a multi-scale reconstructed attractors-based method for the identification of structural damage. Wang et al. [26] developed a novel structural damage detection method using factor analysis in the sparse Bayesian learning framework. Xu et al. [27] proposed a novel damage identification approach for longitudinally vibrating structures. Li et al. [28] developed a damage identification approach based on strain mode differences using the inverse finite element method. Gardner et al. [29] used kernelized Bayesian transfer learning with the population-based SHM. Hou and Xia [30,31] simulated space frame structures with semi-rigid connections for structural damage detection studies, then reviewed the vibration-based damage identification methods in the last ten years and discussed the advantages and disadvantages of these methods. It is helpful for researchers in the field to make better use of existing methods.

Moreover, intelligent algorithms are also developing rapidly in the field of structural damage identification. Meruane and Heylen [32] implemented a hybrid real-coded genetic algorithm with damage penalization to locate and quantify structural damage, which can effectively avoid the false damage detection caused by experimental noise or numerical errors. Wang et al. [33,34] proposed an iterative two-stage structural damage identification method combining the modal strain energy decomposition method with the multi-objective particle swarm optimization algorithm to detect damage and estimate its severity. Ding et al. [35] used an artificial bee colony algorithm with a hybrid search strategy to detect damage. Cao et al. [36] developed a hierarchical wavelet-aided neural intelligent method to identify structural damage. Abdulkareem et al. [37] presented a non-probabilistic wavelet method to consider uncertainties in structural damage detection. Guo et al. [38] used partial modal results and artificial neural networks to identify damage. Ghiasi et al. [39] proposed an improved bat algorithm to improve the accuracy of structural damage identification. Fu and Jiang [40] combined the probabilistic neural network and data fusion technology with the correlation fractal dimension to identify structural damage. Standoli et al. [41] developed a finite element model calibration based on ambient vibration tests and machine learning processes. Ho et al. [42] used a marine predator algorithm and a feedforward neural network to detect structural damage. Shang et al. [43] proposed a damage detection strategy based on an unsupervised deep neural network. Li et al. [44] presented an approach based on the combination of statistical indicators to characterize acceleration measurements in the time domain and computational intelligence techniques to detect damage.

Kennedy and Eberhart [45] formed the particle swarm optimization (PSO) algorithm that is inspired by the foraging behavior of birds. Due to the simple implementation process and few parameters, the PSO algorithm has been widely used in many fields; however, the PSO algorithm also has some shortcomings, such as premature convergence and falling into local optimum, especially in the process of multimodal function optimization [46]. Shi and Eberhart [47] introduced the inertia weight parameter and controlled the speed and direction of particles during flight to improve the optimizer, which was named the standard PSO algorithm. Inertia is the most important parameter among the parameters of the PSO algorithm and many scholars consider the strategy of improving from the inertia weight parameter. Chatterjee et al. [48] put forward a parabolic form of decreasing the inertia weight parameter. Melo et al. [49] added a Gaussian disturbance strategy to the PSO algorithm, which enhanced the ability to escape from the local optimum and improved the convergence speed of the algorithm. Malik et al. [50] put forward the strategy of decreasing and increasing the inertia weight parameter of the sigmoid. Huang et al. [51] also put forward the nonlinear decreasing inertia weight parameter based on the inverted S-shaped curve. By controlling the adjustment factors, the convergence degree of the algorithm is greatly improved. The modal assurance criterion (MAC) is a valid indicator of structural damage and an intelligent group algorithm, which can effectively detect structural damage. Allemang [52] reviews the development and other related assurance criteria that have been proposed of the original MAC. Huang et al. [53] combined the cuckoo algorithm and PSO algorithm to carry out structural damage identification under the influence of temperature. Gerist and Maheri [54] proposed a multi-stage damage identification method based on the PSO algorithm. Wei et al. [55] developed an improved PSO algorithm and introduced a disturbance in the evolution process to avoid the occurrence of premature.

In this study, a particle swarm optimization with a sigmoid increasing inertia weight (SIPSO) algorithm is proposed to improve the optimization of structural damage identification. Three damage identification cases, including single damage identification, double damages identification, and triple damages identification, were used to verify the identification accuracy of the SIPSO algorithm. This paper is organized as follows. The background of PSO and its improved theory is demonstrated in Section 2. The influence of the parameters of the SIPSO algorithm is illustrated in Section 3. Then, Section 4 presents the detailed performance of the SIPSO algorithm in structural damage identification. Finally, from the numerical simulation results, some conclusions and recommendations are provided for practically applying the new technique in this field.

2. Particle Swarm Optimization with Sigmoid Increasing Inertia Weight Algorithm

2.1. Theory of Particle Swarm Optimization

The PSO algorithm is a random swarm intelligent search algorithm based on the process of birds foraging. Birds flying for food are regarded as particles and food is regarded as the optimal solution. In each iteration process, the particle changes its speed and position by tracking individual extremum and global extremum. Individual extremum represents the optimal solution found by the particle itself, defined as $pbest$. Global extremum represents the optimal solution found by particles in the whole particle swarm at present, defined as $gbest$. The velocity and position formulas of the standard PSO are as follows,

$$v_{k+1}=v_k+c_1{r}_1\left(pbes{t}_k-x_k\right)+c_2{r}_2\left(gbes{t}_k-x_k\right)$$

(1)

$$x_{k+1}=x_k+v_{k+1}$$

(2)

where $k$ is the number of iterations; $v_k$ is the velocity vector of particles; $x_k$ is the position of the current particles; $pbes{t}_k$ indicates the position of the optimal solution found by the particle itself; $gbes{t}_k$ represents the random number evenly distributed among the positions of the optimal solution found by the whole population at present. $c_1$ and $c_2$ are learning factors, which, respectively, represent the cognitive ability and social ability of particles, $c_1=c_2=2$; $r_1$ and $r_2$ are random numbers between 0 to 1. From a sociological point of view, the first part of Equation (1) (on the right side) is called a memory term and the second part is called a self-cognition, which means that the action of the particles comes from their own experience. The third part of Equation (1) is called a group cognition term, which indicates the cooperation and knowledge sharing among particles. To control and improve the search ability of the PSO algorithm, an inertia weight factor $\omega$ can be added to Equation (1) as,

$$v_{k+1}=\omega v_k+c_1{r}_1\left(pbes{t}_k-x_k\right)+c_2{r}_2\left(gbes{t}_k-x_k\right)$$

(3)

where $\omega$ indicates the influence of the speed of the previous generation of particles on the speed of the next generation of particles. If $\omega$ is large enough, it has strong global search ability; otherwise, it tends to search locally, which can be calculated as,

$$\omega ={\omega }_{max}-iter\times \frac{{\omega }_{max}-{\omega }_{min}}{ite{r}_{max}}$$

(4)

where $iter$ represents the current iteration steps, $ite{r}_{max}$ represents the maximum iteration steps.

2.2. Theory of SIPSO

The PSO algorithm shows strong global optimization ability and poor performance for local optimization due to the fact that inertia weight ω directly affects the velocity and the position of the particles. Adjusting ω can effectively adjust the optimization ability of the PSO algorithm; therefore, it is necessary to select the appropriate value or function for the calculation of the inertia weight according to the situation. Two kinds of inertia weight can be selected according to the whole optimization process. The first one is the fixed inertia weight, which keeps the exploration and development ability of the flight particles consistent. The other is the time-varying inertia weight, which changes within a certain range, which means that the particles have different exploration and development abilities at different times.

The PSO algorithm is able to search globally first and then locally. Due to the global search in the early stage being random and essentially non-directional, the local search in the later stage is based on the found global solution, which makes it easy to form the common local optimal solution by using the strategy of the increasing curve and taking advantage of the parallel characteristics of individual and global extremum of the particle swarm. In essence, the slow increase in inertia weight in the early stage can ensure sufficient local search ability of the particles of the algorithm. With the increase in inertia weight in the later stage, the global search ability of the particles will increase. At the moment, there will be some beneficial perturbations due to the existence of individual and global extremum—this helps the particles escape from the trap of the local optimum in the later stage of the algorithm. For this reason, an improved algorithm is proposed based on the following two aspects:

(1)

Each particle still maintains its extremum at the end of the algorithm. In other words, the new algorithm not only finds the optimal solution but also obtains several better sub-optimal solutions.

(2)

The unique memory is insensitive to the population size. Even if the population number decreases, the performance will not decrease obviously.

In this study, the improved sigmoid increasing inertia weight can be defined as,

$$\omega ={\omega }_{min}+\frac{{\omega }_{max}-{\omega }_{min}}{1+e^{a-b\times \frac{iter}{ite{r}_{max}}}}$$

(5)

where a and b are parameters for adjustment, which are carefully chosen through numerical simulations. The improved algorithm results in the inertia weight showing an increasing sigmoid trend in the whole iterative process.

To verify the new algorithm, two PSO-based methods are introduced. The first one is the sigmoid-function-based new particle swarm optimization algorithm (SNPSO) with increasing inertia weight proposed by Malik et al. [50], and the second one is the sigmoid-function-based adaptive particle swarm optimization algorithm (SAPSO) with decreasing inertia weight proposed by Huang et al. [51] The inertia weight formula of the SNPSO algorithm is,

$$\omega ={\omega }_{min}+\frac{{\omega }_{max}-{\omega }_{min}}{1+e^{u\times \left(iter-0.75\times ite{r}_{max}\right)}}$$

(6)

where $u={10}^{\left(\log \left(ite{r}_{max}\right)-2\right)}$.

The inertia weight formula of the SAPSO algorithm is,

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{1+e^{3.4-0.7iter}}$$

(7)

2.3. Damage Simulation of Rigid Frame Structure

The structure could be expressed as a dynamic system composed of stiffness matrix, mass matrix, and damping matrix. The equation of motion of the structure at unit $i$ can be expressed as,

$$(\mathit{k}-{\omega }_i^2\mathit{m}){\Phi }_i=\left[0\right]$$

(8)

where $\mathit{k}$ and $\mathit{m}$ are the global stiffness matrix and the mass matrix of the structure, ${\Phi }_i$ and ${\omega }_i$ are the $i$-th mode shape and natural frequency of the structure.

In general, for structural damage identification, the stiffness reduction in the structure is considered while the structural mass change is ignored, then,

$${\mathit{k}}_d={\displaystyle \sum }_i^n(1-x_i)k_{ih}$$

(9)

where $k_d$ is the structural stiffness of the damaged system, $k_{ih}$ is the $i$-th stiffness of the intact system, $n$ presents the number of total finite elements, $x$ is the $i$-th stiffness damage coefficient, the expression of damage element stiffness can be expressed as,

$$x=1-\frac{k_{id}}{k_{ih}}$$

(10)

where $k_{ih}$ and $k_{id}$ are the stiffness of the $i$-th element with intact and damaged conditions, respectively.

Structural damage detection is performed to determine the location and degree of damage. In this paper, structural damage detection is transformed into a parameter constrained optimization problem. The damaged conditions are set and the dynamic parameters corresponding to the damaged condition are established. Then, the measured dynamic parameters of the structure are compared with the set damaged condition. If the structure is damaged, the stiffness of the structure will be decreased. In this study, the stiffness change of the structure is taken as the parameter representing structural damage identification. The finite element model of the damaged structure is established by the relationship ratio of the frequency and mode shape before and after the structure is damaged. The objective function is as follows,

$$f\left(x\right)={\displaystyle \sum }_{i=1}^s\left(\left(1-MAC\left({\varnothing }_i^h,{\varnothing }_i^d\right)\right)+ER\left({\omega }_i^h,{\omega }_i^d\right)\right)$$

(11)

$$MAC\left({\varnothing }_i^h,{\varnothing }_I^d\right)=\frac{{\left|{\varnothing }_i^{hT},{\varnothing }_i^d\right|}^2}{\left|{\varnothing }_i^{hT},{\varnothing }_i^h\right|\left|{\varnothing }_i^{dT},{\varnothing }_i^d\right|}$$

(12)

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

(13)

• $MAC$—modal assurance criterion;
• $ER$—frequency change rate;
• $\left({\varnothing }_i^h,{\varnothing }_i^d\right)$—vibration modes of the structure under intact and damaged conditions;
• $\left({\omega }_i^h,{\omega }_i^d\right)$—frequencies of the structure under intact and damaged conditions;
• $s$—modal number of structure;
• $x$—damage value of element stiffness.

where the value of $s$ is five, and the variation range of x is 0 to 0.9 in this study. Then, the structural damage detection can be described as a typical constrained optimization problem by obtaining the minimum value of Equation (11). The objective function can be solved iteratively by the proposed algorithm. For the PSO-based algorithm, the fitness value is to obtain the minimum value of Equation (11). When the objective function reaches the minimum value, the corresponding $x$ can be obtained and defined as a damage index.

As shown in Figure 1, a finite element model of a two-story rigid frame structure is adopted in this study. The height and width of each layer of the rigid frame is 1.41 m. The total element number of the structure is 18 and the degrees of freedom number (DOF) is 16. The structural physical parameters of the beam and the column are listed in

Table 1. 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

.

We assume that the nodes are rigid connections and that there is no need to consider the damage of the nodes. The numbers inside the circle represent the location of the elements, and the rest of the numbers represent the location of the nodes. For the single damage case, the damage section is set at element 17. For the double damages case, the damage sections are set at element 8 and element 17. For the triple damages case, the damage sections are set at element 5, element 8, and element 17.

The model modification method was used to simulate the local damage of the structure by reducing the stiffness of the finite element model. The first five modes of the healthy structure were 41.45 Hz, 129.27 Hz, 288.71 Hz, 317.11 Hz, and 392.61 Hz, respectively. The influence of environmental noise on the identification results was simulated by adding white noise to the measured modal parameters, which can be expressed as,

$$m_n=m_{cal}\left(1+L_{n}rand\left(n\right)\right)$$

(14)

where $m_n$ and $m_{cal}$ are the modal parameter with and without noise interference, respectively. $L_n$ is the noise level and $rand\left(n\right)$ is an $n\times n$ standard normal distribution matrix.

3. Choosing Optimal Parameters for SIPSO Algorithm

The PSO algorithm can easily fall into the local optimum due to particle convergence. To solve this problem, a SIPSO algorithm is proposed based on structural dynamics and applied mathematics. As mentioned above, the parameters a and b of the SIPSO algorithm in Equation (5) have significant effects on its performance. In this subsection, three functions named the Rosenbrock criterion function, the Rastrigin criterion function, and the Griewank criterion function [56,57] are used to choose the optimal parameters for the SIPSO algorithm. The three criterion functions are given as below:

Rosenbrock criterion function [56],

$$f\left(x\right)={\displaystyle \sum }_{i=1}^n\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right)$$

(15)

Rastrigin criterion function [57],

$$f\left(x\right)={\displaystyle \sum }_{i=1}^n\left(x_i^2-10\cos \left(2\pi x_i\right)+10\right)$$

(16)

Griewank criterion function [57],

$$f\left(x\right)=\frac{1}{4000}{\displaystyle \sum }_{i=1}^n{x}_i^2-{\displaystyle \prod }_{i=1}^n\cos (\frac{x_i}{\sqrt{i}})+1$$

(17)

The initialization ranges of the Rosenbrock, Rastrigin, and Griewank functions are [−30,30], [−5.12, 5.12], and [−600, 600], respectively. The Rosenbrock function is an ill-conditioned quadratic function with one peak and the optimal value is 1. The Rastrigin function is a multimodal function with a large number of local extremum points and the optimal value is 0. The Griewank function is another multimodal function with the mutual influence of independent variables and the optimal value is also 0.

3.1. Choosing Optimal Parameter a

Ordinary parameters of the SIPSO algorithm are set as follows: the number of particle swarms is set as 30 and the maximum number of generations is set as 200. In Equation (5), ${\omega }_{max}$= 0.9 and ${\omega }_{min}$= 0.4. To choose the optimal value, parameter a is set as 1, 1.5, 2, and 2.5, respectively. Each case is calculated 50 times and the average value is taken for comparison as shown in

Table 2. Mean optimal function value of three criterion functions with different values of parameter a.

Function	a = 1	a = 1.5	a = 2	a = 2.5
Rosenbrock	1.0061	1.0048	0.9995	0.9982
Rastrigin	2.1169 × 10−2	−3.7135 × 10−4	1.0281 × 10−9	1.2809 × 10−7
Griewank	0.0673	0.0656	0.0654	0.0638

.

Table 2 tabulates the mean optimal function value of the three criterion functions with different values of parameter a. Figure 2 shows the influence of parameter a on the SIPSO algorithm. As shown in Table 2, with the Rosenbrock criterion function, the parameter a = 2 corresponds to the optimal function value. As shown in Figure 2a, with the Rosenbrock criterion function, the convergence curve of parameter a = 2 is located at the bottom and converges rapidly, which is obviously better than other values. When the Rastrigin criterion function is adopted, the parameter a = 2 also corresponds to the optimal function value, and the convergence curve of parameter a = 2 is much better than other values, as shown in Table 2 and Figure 2b. As shown in Table 2, with the Griewank criterion function, the parameter a = 2.5 corresponds to the optimal function value and the parameter a = 2 corresponds to the second-best function value. As shown in Figure 2c, with the Griewank criterion function, when the number of generations is less than 50, the convergence curve of parameter a = 2 is located at the bottom and converges rapidly, which is better than other values. In addition, when the number of generations is more than 50, the convergence curves of parameter a = 2 and parameter a = 2.5 are very close and it is hard to tell which one is better.

The most direct and effective method to select the optimal parameter value a of the SIPSO algorithm is to apply it to structural damage identification. In this study, each damage identification case was calculated five times and the average value was taken as the final identification result. The calculation principle is consistent with this in subsequent studies. Figure 3 shows the structural damage detection results under different values of parameter a of the SIPSO algorithm. There were two damaged sections, namely element 8 and element 17. The damage index is set as 10% and the noise level is also set as 10%. As shown in Figure 3, the SIPSO algorithm with four values of parameter a can accurately identify the damage location and degree of damage in element 8 and element 17, but the misjudgment results of each parameter are significantly different. When parameter a = 1, there are four misjudged-as-damaged sections and the identification result is the worst among the four parameters. When parameter a = 1.5 and a = 2.5, there are two misjudged-as-damaged sections. When parameter a = 2, there is only one misjudged-as-damaged section and the identification result is the best among the four parameters. After comprehensively considering the results of the three criterion functions and structural damage identification, parameter a = 2 is selected as the optimal parameter a of the SIPSO algorithm. In this subsection, optimal parameter b is adopted by default and the problem of how to select the optimal parameter b of the SIPSO algorithm is detailed in the next subsection.

3.2. Choosing Optimal Parameter b

The ordinary parameters of the SIPSO algorithm are the same as the previous subsection. To choose the optimal value, parameter b is set as 0.1, 0.3, 0.5, and 0.8, respectively. Each case was calculated 50 times and the average value was taken for comparison, as shown in

Table 3. Mean optimal function value of three criterion functions with different values of parameter b.

Function	b = 0.1	b = 0.3	b = 0.5	b = 0.8
Rosenbrock	1.0451	0.9900	0.9995	1.0335
Rastrigin	4.1431 × 10−8	1.5700 × 10−9	1.0281 × 10−9	2.0728 × 10−8
Griewank	0.1279	0.0644	0.0654	0.0671

.

Table 3 tabulates the mean optimal function value of the three criterion functions with different values of parameter b. Figure 4 shows the influence of parameter b on the SIPSO algorithm. As shown in Table 3, with the Rosenbrock criterion function, the parameter b = 0.5 corresponds to the optimal function value. As shown in Figure 4a, with the Rosenbrock criterion function, the convergence curve of parameter b = 0.5 is located at the bottom and converges rapidly, which is obviously better than other values. When the Rastrigin criterion function is adopted, parameter b = 0.5 also corresponds to the optimal function value and the convergence curve of parameter b = 0.5 is much better than other values, as shown in Table 3 and Figure 4b. As shown in Table 3, with the Griewank criterion function, parameter b = 0.3 corresponds to the optimal function value and parameter b = 0.5 corresponds to the second-best function value. As shown in Figure 4c, with the Griewank criterion function, when the number of generations is less than 50, the convergence curve of parameter b = 0.5 is located at the bottom and converges rapidly, which is better than other values. In addition, when the number of generations is more than 50, the convergence curves of parameter b = 0.3 and parameter b = 0.5 are very close and it is hard to tell which one is better.

Figure 5 shows the structural damage detection results under different values of parameter b of the SIPSO algorithm. There are two damaged sections, namely element 8 and element 17. The damage index is set as 10% and the noise level is also set as 10%. As shown in Figure 5, the SIPSO algorithm with four values of parameter b can accurately identify the damage location and degree of damage in element 8 and element 17, but the misjudgment results of each parameter are different. When the values of parameter b = 0.1, b = 0.3, and b = 0.8, there are two misjudged-as-damaged sections. When parameter b = 0.5, there is only one misjudged-as-damaged section and the identification result is the best among the four parameters. After comprehensively considering the results of the three criterion functions and structural damage identification, parameter b = 0.5 is selected as the optimal parameter b of the SIPSO algorithm. In this subsection, the optimal parameter a is adopted by default.

4. Structural Damage Identification Based on SIPSO Algorithm

To verify the effectiveness of the proposed SIPSO algorithm in structural damage identification, the three criterion functions mentioned above are used again. Moreover, three similar methods are also used to compare the identification accuracy and the convergence speed of the SIPSO algorithm: the standard PSO algorithm, the SNPSO algorithm, and the SAPSO algorithm.

Table 4. Mean optimal function value of three criterion functions with different PSO-based algorithms.

Function	PSO	SNPSO	SAPSO	SIPSO
Rosenbrock	1.1758	1.0075	1.0016	0.9995
Rastrigin	5.9698 × 10−2	1.0281 × 10−9	2.1207 × 10−2	7.8939 × 10−10
Griewank	0.3926	0.0654	0.0688	0.0651

tabulates the mean optimal function value of the three criterion functions with four PSO-based algorithms. As shown in Table 4, the SIPSO algorithm corresponds to the optimal function value with all three criterion functions. Figure 6 shows the comparison of convergence characteristics of the four PSO-based algorithms. As shown in Figure 6, the convergence curve of the SIPSO algorithm is located at the bottom and converges rapidly, which is better than the other three methods. To further verify the identification effects of the four algorithms under various damage cases, numerical simulations of single damage, double damage, and triple damage identification were carried out, respectively.

4.1. Single Damage Identification Case

In this condition, there is one damaged section located at element 17. The damage index is set as 10% and 40% and the noise level is set as 0, 5%, and 10%, respectively.

As shown in Figure 7a, when the damage index is set as 10% and the noise level is set as 0, the PSO algorithm can identify the damage at element 17 except for two obvious misjudged-as-damaged sections at element 2 and element 5. On the contrary, the identification results of the SNPSO, the SAPSO, and the SIPSO algorithm are very accurate. Namely, the identification damage index of the three improved PSO algorithms is 10%, which is equal to the actual damage index. In addition, none of these three algorithms has a misjudged-as-damaged section.

As shown in Figure 7b, when the damage index is set as 10% and the noise level is set as 5%, the PSO algorithm can identify the damage at element 17 except for three misjudged-as-damaged sections at element 2, element 8, and element 9. On the contrary, the identification results of the SNPSO, the SAPSO, and the SIPSO algorithm are much better with only one misjudged-as-damaged section at element 2.

As shown in Figure 7c, when the damage index is set as 10% and the noise level is set as 10%, the PSO algorithm and the SAPSO algorithm can identify the damage at element 17 except for four misjudged-as-damaged sections at element 2, element 3, element 11, and element 12. On the contrary, the identification results of the SNPSO and the SIPSO algorithm are much better with only one misjudged-as-damaged section at element 2.

As shown in Figure 7d, when the damage index is set as 40% and the noise level is set as 5%, all four PSO-based algorithms have one misjudged-as-damaged section at element 2. As shown in Figure 7e and

Table 5. Numbers of misjudgments in damage detection of four PSO-based algorithms with single damage.

Condition	Damage Index	Noise Level	PSO	SNPSO	SAPSO	SIPSO
1	10%	0	2	0	0	0
2	10%	5%	3	1	1	1
3	10%	10%	4	1	4	1
4	40%	5%	1	1	1	1
5	40%	10%	2	1	2	1

, when the damage index is set as 40% and the noise level is set as 10%, the identification results of the SNPSO and the SIPSO algorithm are the best with only one misjudged-as-damaged section at element 2. On the contrary, the other two PSO-based algorithms have two misjudged-as-damaged sections.

Considering the numerical simulation results of the five conditions with a single damage section, the identification results of the PSO algorithm are the worst while the three improved PSO algorithms are all effective compared with the PSO algorithm. In addition, as shown in Figure 7 and Table 5, the SNPSO algorithm and the SIPSO algorithm are the best to be used for single damage section identification, with only one misjudgment element in four conditions containing noise interference.

4.2. Double Damages Identification Case

In this condition, there are two damaged sections located at element 8 and element 17. The damage index is set as 10% and 40% and the noise level is set as 5% and 10%, respectively.

As shown in Figure 8a and

Table 6. Numbers of misjudgments in damage detection of four PSO-based algorithms with double damages.

Condition	Damage Index	Noise Level	PSO	SNPSO	SAPSO	SIPSO
6	10%	5%	7	2	3	1
7	10%	10%	8	2	5	1
8	40%	5%	3	1	3	1
9	40%	10%	3	1	3	1

, when the damage index is 10% and the noise level is set as 5%, the PSO algorithm can identify the damages at element 8 and element 17 except for seven obvious misjudged-as-damaged sections. In this condition, the SNPSO algorithm has two misjudged-as-damaged sections and the SAPSO algorithm has three misjudged-as-damaged sections. The identification result of the SIPSO algorithm is the best, with only one misjudged-as-damaged section at element 2.

As shown in Figure 8b, when the damage index is set as 10% and the noise level is set as 10%, the PSO algorithm can identify the damages at element 8 and element 17 except for eight misjudged-as-damaged sections. In this condition, the SNPSO algorithm has two misjudged-as-damaged sections and the SAPSO algorithm has five misjudged-as-damaged sections. The SIPSO algorithm is the best with only one misjudged-as-damaged section at element 2.

As shown in Figure 8c, when the damage index is set as 40% and the noise level is set as 5%, the PSO algorithm and the SAPSO algorithm can identify the damages at element 8 and element 17 except for three misjudged-as-damaged sections. The SNPSO algorithm and the SIPSO algorithm are the best with only one misjudgment element. As shown in Figure 8d, when the damage index is set as 40% and the noise level is set as 10%, the PSO algorithm and the SAPSO algorithm have three misjudged-as-damaged sections, and the SNPSO algorithm and the SIPSO algorithm have one misjudged-as-damaged section.

Considering the numerical simulation results of the four conditions with double damage sections, the identification results of the PSO algorithm are the worst while those of the SIPSO algorithm are the best among the four PSO-based algorithms. When the SIPSO algorithm is used for double damage sections identification, the number of misjudgment elements is only one in all conditions. Moreover, the SNPSO algorithm and the SAPSO algorithm have some improvements compared with the PSO algorithm.

4.3. Triple Damages Identification Case

In this condition, there are triple damaged sections located at element 5, element 8, and element 17. The damage index is set as 10% and 40% and the noise level is set as 5% and 10%, separately for the two damage indexes.

As shown in Figure 9a and

Table 7. Numbers of misjudgments in damage detection of four PSO-based algorithms with triple damages.

Condition	Damage Index	Noise Level	PSO	SNPSO	SAPSO	SIPSO
10	10%	5%	5	1	3	1
11	10%	10%	10	1	4	1
12	40%	5%	2	1	1	1
13	40%	10%	3	2	2	1

, when the damage index is set as 10% and the noise level is set as 5%, the PSO algorithm can identify the damages at element 5, element 8, and element 17 except for five misjudged-as-damaged sections. In this condition, the SAPSO algorithm has three misjudged-as-damaged sections. The identification results of the SNPSO algorithm and the SIPSO algorithm are the best with only one misjudged-as-damaged section at element 11.

As shown in Figure 9b, when the damage index is set as 10% and the noise level is set as 10%, the PSO algorithm can identify the damages at element 5, element 8, and element 17 except for ten misjudged-as-damaged sections. In this condition, the SAPSO algorithm has four misjudged-as-damaged sections. The identification results of the SNPSO algorithm and the SIPSO algorithm are the best with only one misjudged-as-damaged section at element 11.

As shown in Figure 9c, when the damage index is set as 40% and the noise level is set as 5%, the PSO algorithm has two misjudged-as-damaged sections, and the other three algorithms have one misjudged-as-damaged section. As shown in Figure 9d, when the damage index is set as 40% and the noise level is set as 10%, the PSO algorithm has three misjudged-as-damaged sections while the SNPSO algorithm and the SAPSO algorithm have two misjudged-as-damaged sections. In this condition, the identification result of the SIPSO algorithm is the best with only one misjudged-as-damaged section at element 2.

Considering the numerical simulation results of the four conditions with triple damaged sections, the identification results of the PSO algorithm are the worst and those of the SIPSO algorithm are the best among the four PSO-based algorithms. When the SIPSO algorithm is used for triple damage sections identification, the number of misjudgment elements is only one in all conditions of this case.

5. Conclusions

In this study, a particle swarm optimization with a sigmoid increasing inertia weight (SIPSO) algorithm is proposed and the parameters of the sigmoid increasing inertia weight have been investigated. Based on the simulation results, conclusions can be drawn as follows:

(1)

After comprehensively considering the results of the three criterion functions and structural damage identification, the parameters a = 2 and b = 0.5 are selected as the optimal parameters of the SIPSO algorithm.

(2)

Compared with the other three PSO-based algorithms, the SIPSO algorithm corresponds to the optimal function value with all three criterion functions. The convergence curve of the SIPSO algorithm is located at the bottom and converges rapidly, which is better than the other three methods. After verifying with the three criterion functions, the SIPSO algorithm is the best method among the four PSO-based algorithms.

(3)

Considering the numerical simulation results of the three cases of structural damage identification, the identification results of the PSO algorithm are the worst, while the identification results of the SIPSO algorithm are the best among the four PSO-based algorithms. Meanwhile, the SNPSO algorithm and the SAPSO algorithm have some improvements compared with the PSO algorithm.

(4)

Finally, the SIPSO algorithm has many advantages in structural damage identification, such as fast convergence rate, high identification accuracy, and the ability to address the noise interference problems excellently. The SIPSO algorithm is preferred for structural damage identification in the field due to its remarkable performance.