As more and more large-scale space engineering structures are put into use, it has become a hot topic for engineers and scholars in various countries to establish a matching structural health monitoring system for all-around safety assurance, and one of the most important aspects is how to effectively identify the damage of space structures, including damage localization and quantitative analysis of the damage degree.

Damage identification based on the difference in the dynamic properties of the structure before and after damage has been a hotspot of concern in the engineering community. This type of strategy determines the location and extent of damage by evaluating the correlation of the dynamic parameters of the healthy and damaged structure, which is essentially an inverse problem, and the finite element model updating technique is often applied to solve this type of inverse problem.

Damage identification methods based on finite element model updating can transform the damage identification problem into a constrained optimization problem, which generally starts with the establishment of an objective function for evaluating the structural relevance of health and damage, and is subsequently solved using an optimization algorithm to find an optimal solution. However, traditional optimization algorithms, such as the most rapid descent method and the conjugate gradient method, are not only slow to converge but also prone to fall into local optimum. Therefore, developing or searching for more powerful optimization algorithms for damage identification is of great significance for the development of this field.

With the rapid development of intelligent optimization algorithms in recent times, more and more scientists have studied this in depth and used it to solve optimization problems in damage identification. Qian C Y et al [1] utilized an improved cuckoo algorithm combining frequency and vibration factors for damage identification of Benchmark frames with significant results. Vaez S R H et al [2] proposed a hybrid genetic and particle swarm algorithm for damage identification of damaged thin plate models and achieved good results. Wan Z H et al [3] proposed a two-stage damage identification method based on augmented whale algorithm to successfully identify multiple damages in simply supported beams with thin plates. Mohan S C et al [4] successfully localized and quantified the damage on beams and planar frames using damage detection technique based on frequency response function and particle swarm algorithm. Miguel L F F et al [5] proposed a hybrid stochastic/deterministic optimization algorithm for damage identification, which is more accurate and efficient in identifying damages when compared to genetic, harmonic and particle swarm algorithms. Ding Z H et al [6] proposed a damage identification method based on artificial bee colony algorithm with hybrid search strategy and modal data and successfully identified the damage on truss model and plate model. Pan C D et al [7] proposed a damage identification method based on hybrid adaptive firefly algorithm and verified the validity of the proposed method by two-story rigid frame structure and beam model. Chen Z P et al [8] embedded the Nelder-Mead algorithm into a particle swarm algorithm to obtain a hybrid particle swarm algorithm and used it for damage identification, and verified the effectiveness and superiority of the method through numerical simulation of simply supported beams as well as experiments on rectangular cross-section steel beams. Ding Z et al [9] proposed a damage identification method based on sparse regularization, Bayesian inference of improved objective function and improved Jaya algorithm, and verified the feasibility of the method through truss structure simulation and reinforced concrete bridge experiment. Guilherme F G et al [10] proposed an optimization algorithm based on sunflower motion and successfully identified the damage in composite laminates by this algorithm. Alkayem N F et al [11] proposed a damage identification method based on social swarm algorithm, modal strain energy, and modal vibrational curvature and verified the effectiveness of the method by numerical simulation. Huang M S et al [12] proposed a damage identification method based on modal frequency strain energy assurance criterion, modal flexibility and enhanced moth flame optimization algorithm and verified the effectiveness of the method by simply supported beams and triple shear frames experiments. Minh H L et al [13] proposed an augmented particle swarm algorithm for solving the damage identification problem and successfully applied the method to the damage identification of power transmission towers. Li X L et al [14] proposed a multi-component particle swarm algorithm with cooperative learning for damage identification, which fuses four particle swarm algorithm variants to build a pool of strategies, and the numerical simulation results show that the algorithm possesses higher accuracy for damage identification compared to a single variant and other algorithms. Zhang G C et al [15] proposed a new hybrid algorithm based on Jaya and differential evolutionary algorithm, and at the same time, established the objective function through the adjacent acceleration correlation function to identify the damage of cantilever beams and other models, and the results show that the method has high accuracy and computational efficiency. Li Y F et al [16] proposed a hybrid K-clustering mean optimization algorithm and used it for the identification of damage of small experimental dams, and the results show that this algorithm has high accuracy and computational efficiency for the identification of large and complex structures. The results show that the algorithm has high identification accuracy and computational efficiency, and has high potential for identifying the damage of large and complex structures. Thanh S T et al [17] proposed the goby joint search algorithm and used it for damage identification of Guangzhou Tower, the results show that this algorithm has higher damage identification accuracy compared to GAs, genetic particle swarm hybrid algorithms and so on. Muhammad I S et al [18] proposed a damage identification method based on hybrid YUKI-ANN algorithm with modal strain energy change ratio and successfully applied it for damage identification in plywood.

In this paper, the improved non-uniform mutation approach, improved mutation rate and crossover rate variation approach are added to the GA [19] to obtain the RGA and a high-performance optimization algorithm—INFO [20] is introduced. Next, an objective function based on the dynamic response transfer-ratio is established, which balances the sensitivity of each part of the function to damage by weighting factors calculated from the damage classification. Finally, taking simply supported beam and cantilever plate as objects, GA, RGA and INFO are utilized to identify a variety of damage cases under different levels of noises and different frequency spacing of pickup points, to verify the effectiveness of the methods and compare the damage identification effect of the algorithms.

The GA has strong global search performance, high efficiency and robustness, but subsequent studies have also shown that the GA performs poorly in local search ability, and the algorithm itself is prone to immature, slow convergence and degradation and other problems. To address these problems, this paper improves the mutation operator, crossover and mutation rate. The operations of the RGA are as follows (Figure 1 depicts the flow of the RGA).

(1) Selection and crossover

The selection operation in the paper adopts the roulette wheel approach, while adding the elite retention strategy. And the crossover operation adopts arithmetic crossover, which selects any two bodies in the parent generation for linear combination to produce a new individual, which can be expressed as Where ${\mathit{G}}_{A1}^{gt}$ and ${\mathit{G}}_{A2}^{gt}$ are the individuals of the $gt$ ^th generation, ${\mathit{G}}_{A1}^{gt+1}$ and ${\mathit{G}}_{A2}^{gt+1}$ are the individuals of the $gt$ +1^th generation, and $\lambda$ is a random number within the range $\left[0,1\right]$ . To ensure that the crossover individuals do not exceed the specified range, a bounding judgment is added. Figure 2 displays the arithmetic crossover.

In this paper, the RGA uses a modified non-uniform mutation approach, randomly selecting individuals and generating a non-uniform perturbation to them that varies according to the number of evolutionary generations, to improve the algorithm's ability to search for the focal region, which can be expressed as where ${\mathit{G}}_{old}^{gt}$ and ${\mathit{G}}_{new}^{gt}$ denote the unmutated and mutated individuals in the $gt$ ^th generation respectively, $Maxg$ are the maximum number of generations, ${\lambda }_1$ and ${\lambda }_2$ are random number within the range $\left[0,1\right]$ . $tb$ is a suitable perturbation parameter, which is taken as 3.7 in this paper. To ensure that the new individuals after mutation will not exceed the specified range, this step also requires boundary judgment.

(3) Mutation rate and crossover rate

In this paper, the RGA adopts a modified crossover rate and mutation rate change method, in which, on the one hand, the exponential function is used to increase the strength of crossover and mutation to increase the diversity of the population, so as not to let the population tend to a single dominant individual and fall into the local optimum, and on the other hand, the good individuals will be retained due to the small crossover rate and mutation rate, and the poor individuals will disappear rapidly due to the large mutation rate and crossover rate. The modified method can be expressed as follows Where $P{I}_c$ and $P{I}_m$ represent the crossover rate and mutation rate respectively, $p{i}_{c1}$ and $p{i}_{c2}$ are the crossover calculation coefficients with values of 0.9 and 0.6 respectively, $p{i}_{\mathrm{m}1}$ and $p{i}_{\mathrm{m}2}$ are the variation calculation coefficients with values of 0.1 and 0.01 respectively, $fi{t}_{\max }$ and $fi{t}_{\min }$ denote the maximum and minimum fitness values of individuals in population respectively, $fi{t}_{\mathrm{avg}}$ denotes the mean value of population fitness, $fi{t}_{\mathrm{wb}}$ denotes the larger fitness value of the two individuals that are about to cross, and $fi{t}_{\mathrm{w}1}$ denotes the fitness value of the individual that is about to take part in the mutation, $po{p}_1$ and $po{p}_2$ denote the number of individuals in the population with higher and lower fitness values than the population mean respectively, and $eps$ is an infinitesimal positive integer.

INFO is a new type of intelligent algorithm proposed by Dr. Ahmadianfar et al. in 2022. The algorithm applies the improved weighted mean idea to the entity structure, and updates the solution vector through three core steps: updating rules, vector combination and local search. It has the advantages of strong optimization ability and fast convergence speed. The specific implementation process is described as follows (Figure 3 depicts the flow of the INFO).

(1) Updating rule stage

INFO's updating rule operator uses the weighted mean of the vectors to create new vectors and performs an efficient global search in this way. Firstly, INFO has to randomly select three difference vectors from all solution vectors of each generation to calculate the vector-weighted mean value $\mathit{W}\mathit{M}{1}^{it}$ , and at the same time select the best, better and worst solution vectors of the total solution vectors to calculate the vector-weighted mean value $\mathit{W}\mathit{M}{2}^{it}$ , which can be expressed respectively as where $r{h}_1=\cos \left(obj\left({\mathit{u}}_{gx}^{it}\right)-obj\left({\mathit{u}}_{gy}^{it}\right)+\mathsf{\pi }\right)\cdot \exp \left(-\left|{\displaystyle \frac{obj\left({\mathit{u}}_{gx}^{it}\right)-obj\left({\mathit{u}}_{gy}^{it}\right)}{\max \left(obj\left({\mathit{u}}_{gx}^{it}\right),obj\left({\mathit{u}}_{gy}^{it}\right),obj\left({\mathit{u}}_{gz}^{it}\right)\right)}}\right|\right)$ ,

$r{h}_6=\cos \left(obj\left({\mathit{u}}_{bt}^{it}\right)-obj\left({\mathit{u}}_{ws}^{it}\right)+\mathsf{\pi }\right)\cdot \exp \left(-\left|{\displaystyle \frac{obj\left({\mathit{u}}_{bt}^{it}\right)-obj\left({\mathit{u}}_{ws}^{it}\right)}{obj\left({\mathit{u}}_{ws}^{it}\right)}}\right|\right)$Where $obj\left(\mathit{u}\right)$ is the value of the objective function, ${\mathit{u}}_{gx}^{it}$ , ${\mathit{u}}_{gy}^{it}$ , and ${\mathit{u}}_{gz}^{it}$ are individuals in the $it$ ^th generation. $gx$ , $gy$ , and $gz$ are different integers randomly chosen from the range $\left[0,1\right]$ , $NP$ is the population size, $ep$ is an infinitesimal constant, $ran{d}_1$ and $ran{d}_2$ are random numbers within the range $\left[0,1\right]$ , $\mathit{o}\mathit{e}$ is a vector whose all elements are 1, ${\mathit{u}}_{bs}^{it}$ , ${\mathit{u}}_{bt}^{it}$ and ${\mathit{u}}_{ws}^{it}$ are the best, better, and worst individuals in the $it$ ^th generation respectively, $sc$ is a scaling factor, which can be expressed as follows Where $ran{d}_3$ is a random number within the range $\left[0,1\right]$ , $Maxit$ is the maximum number of evolutionary generations.

Next, INFO uses the $\mathit{W}\mathit{M}{1}^{it}$ and $\mathit{W}\mathit{M}{2}^{it}$ constructs the $\mathit{MeanRule}$ , which can be expressed as follows Where $\theta$ is a random number within the range $\left[0,0.5\right]$ . Finally, to further improve the global search capability of the algorithm, an additional convergence acceleration module is added, which makes full use of the optimal solution vectors in each generation to change the current solution vectors in the search space. Therefore, the final update rule formula for INFO is expressed as follows where $\mathit{cz}{1}_{n\mathit{u}}^{it}$ and $\mathit{cz}{2}_{n\mathit{u}}^{it}$ are the new solution vectors of the $it$ ^th generation, $nu=1$ , $2$ ,…, $NP$ ,and $nu\ne gx\ne gy\ne gz$ , $ran{d}_4$ is a random number within the range $\left[0,1\right]$ , $rand{n}_1$ , $rand{n}_2$ , $rand{n}_3$ and $rand{n}_4$ are standard normally distributed random numbers. $\sigma$ is the scaling factor, which can be expressed as follows

where $ran{d}_5$ is a random number within the range $\left[0,1\right]$ .

(2) Vector combining stage

To improve the local search capability of INFO, the solution vectors $\mathit{cz}{1}_{n\mathit{u}}^{it}$ and $\mathit{cz}{2}_{n\mathit{u}}^{it}$ computed in the updating rule stage are combined to generate a new solution vector $\mathit{c}{\mathit{v}}_{n\mathit{u}}^{it}$ , which can be expressed as follows where $ns=0.05rand{n}_5$ , $rand{n}_5$ is a standard normally distributed random number, $ran{d}_6$ and $ran{d}_7$ are random numbers within the range $\left[0,1\right]$ .

(3) Local search stage

The local search stage will create a new solution vector $\mathit{c}\mathit{v}{\mathit{s}}^{it}$ based on the mean rule to further enhance the local search capability of INFO, and the specific implementation formula can be expressed as

Where ${\mathit{u}}_{\mathrm{rnd}}^{it}=\varphi \cdot {\displaystyle \frac{\left({\mathit{u}}_{gx}^{it}+{\mathit{u}}_{gy}^{it}+{\mathit{u}}_{gz}^{it}\right)}{3}}+\left(1-\varphi \right)\cdot \left(\varphi \cdot {\mathit{u}}_{bt}^{it}+\left(1-\varphi \right)\cdot {\mathit{u}}_{bs}^{it}\right)$

$\mathit{v}{\mathit{s}}_1=\left\{\begin{array}{c}2ran{d}_{10}\hspace{1em}\hspace{1em}\hspace{1em}\eta >0.5\\ 1\hspace{1em}\hspace{1em}\hspace{1em}\eta \le 0.5\end{array}\right.$, $v{s}_2=\left\{\begin{array}{c}ran{d}_{11}\hspace{1em}\hspace{1em}\hspace{1em}\eta <0.5\\ 1\hspace{1em}\hspace{1em}\hspace{1em}\eta \ge 0.5\end{array}\right.$

where $\varphi$ , $ran{d}_8$ , $ran{d}_9$ , $\eta$ , $ran{d}_{10}$ and $ran{d}_{11}$ are random numbers within the range $\left[0,1\right]$ , $rand{n}_6$ , $rand{n}_7$ , $rand{n}_8$ and $rand{n}_9$ are standard normally distributed random numbers.

The use of intelligent algorithms to localize and quantify structural damage requires the establishment of a corresponding objective function. Dynamic response transfer-ratio is combined with modal assurance criterion [21] to define the following assurance criterion function of transfer-ratio $TFAC\left(\omega \right)$ ,dynamic response transfer-ratio is combined with amplitude correlation coefficient [22] to define the following amplitude correlation function of transfer-ratio $TFAL\left(\omega \right)$ , which can be expressed as where $\mathit{T}\mathit{F}\left(\omega \right)$ is the dynamic response transfer-ratio vector measured, and $\mathit{T}\mathit{F}\left(\omega ,\mathsf{\alpha }\right)$ is the dynamic response transfer-ratio vector calculated based on the finite element numerical modeling theory. The objective function is established based on $TFAC\left(\omega \right)$ and $TFAL\left(\omega \right)$ , which can be expressed as where

$B{J}_1={\displaystyle \sum _{z=1}^{num}\left(1-TFAC\left({\omega }_z\right)\right)}$ $B{J}_2={\displaystyle \sum _{z=1}^{num}\left(1-TFAL\left({\omega }_z\right)\right)}$

where $num$ is the number of selected frequency point, $TFAC\left({\omega }_z\right)$ is the assurance criterion function value of transfer-ratio at the given frequency ${\omega }_z$ , and $TFAL\left({\omega }_z\right)$ is the amplitude correlation function value of transfer ratio at the given frequency ${\omega }_z$ . ${\eta }_1$ and ${\eta }_2$ are the different weighting factors.

Since the assurance criterion function and the amplitude correlation function of the dynamic response transfer-ratio have different sensitivities to damage, it is likely that damage will lead to large differences in the values of $B{J}_1$ and $B{J}_2$ . This can easily lead to intelligent algorithms taking one part of the objective function as the dominant part of the constrained optimization problem when solving the problem, which in turn makes the role of the other part of the objective function significantly weakened.

In this paper, according to the method of literature [23], the weighting factors are calculated and substituted into the objective function, so as to weaken the numerical difference between $B{J}_1$ and $B{J}_2$ to make the intelligent algorithm solution more accurate. The specific calculation method is expressed as follows Where $nx$ is the number of units in the structure division, $ny$ is the number of damage grading, $B{J}_1^{na,nb}$ and $B{J}_2^{na,nb}$ are the values of $B{J}_1$ and $B{J}_2$ for the $na$ ^th unit at damage level $nb$ respectively. A total of ten levels of damage grading were chosen for this paper, with unit damage levels of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.99 in descending order.

In order to evaluate the accuracy of the intelligent algorithm in identifying the damage degree of the preconfigured damage units (the target damage units in this paper refer to the preconfigured damage units in the damage cases) in each damage case in the simulation, the mean error $Em$ will be introduced, which is denoted as Where ${\alpha }_{\phi }^{th}$ and ${\alpha }_{\phi }^{re}$ denote the preconfigured damage degree of the $\phi$ ^th preconfigured damage unit and the damage degree identified by the algorithm under a certain case respectively, and $ne$ denotes the number of preconfigured damage units in the case.

In order to verify the identification accuracy of the damage identification method based on the RGA and INFO under different levels of noise and different frequency point spacing, a numerical case study was carried out in MATLAB for simply supported beams and cantilever plates (the damage identification results in this paper are the average values obtained after ten runs). In addition, the mutation rate and crossover rate of the standard GA as a comparison are set to 0.2 and 0.4 respectively, and the number of populations and iterations of the three algorithms are set to 120 and 800 respectively. The acceleration response signal added white noise into can be expressed as follows where $Ac{c}_0$ and $Ac{c}_1$ are the acceleration response signals before and after the noise interference, $noi$ is the noise level, and $rdn$ is a standard normally distributed random number.

In this paper, RGA was obtained by improving GA with improved variation rate and crossover rate and non-uniform variation operator, which significantly improves its optimization ability, and INFO was first introduced as a high-performance optimization algorithm to identify the structure damage. Then the objective function based on dynamic response was established for each intelligent algorithm for damage identification by combining the finite element theory of structural dynamics, and the objective function balances the sensitivity of each part of the function to the damage through the weighting factors calculated by the damage classification. Finally, GA, RGA and INFO were respectively utilized in the numerical simulation to identify the damage cases of simply supported beams and cantilever plates with different levels of noise and different spacings of the frequency points. The simulations verify the effectiveness of the method and compare the effect of the algorithms for damage identification, and the following conclusions are obtained: