1. Introduction

2. Overview of Bio-Inspired Optimization

2.1. Genetic Algorithm

Genetic algorithms (GA) [17,18], one of the types of evolutionary-based optimization, are a kind of stochastic optimization technique inspired by the principles of natural evolution and heredity. GA performs well in exploring complex optimization problems in parallel and globally, and finding optimal solutions. Applying Darwin’s theory of natural selection, it goes through a process where good solutions survive and bad solutions disappear in a set of candidate solutions. GA goes through repetitive search processes to find the optimal solution. In the initialization phase, which is performed once at a time, GA randomly creates a set of individuals called chromosomes. At this time, each entity represents a point in the solution space. After that, GA performs three important actions at each stage that imitate the behavior of the genetics of initialization, selection, crossover, and mutation. Each action mimics the genetic principle as follows.

• Selection: Select an individual from the current group. This is the process of becoming the parent of the next generation, and preferably a good parent should be selected to produce offspring. The selection phase includes the selection of Roulette wheel, Tournament selection, Ranking-based selection, and Elitism. Each method in common increases the likelihood of being transmitted to future generations.

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  In Roulette wheel selection, the chromosomes are compared to each other to select the chromosome to be passed on to the next generation in the current set of objects. In the Roulette wheel selection, the probability of selection for the i-th chromosome is expressed mathematically as $p\left(x_i\right)=f\left(x_i\right)/{\sum }_{j=1}^{N}f\left(x_j\right)$, where f is the fitness function and N is the number of chromosomes present in a generation.

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  Tournament selection produces a random number between 0 and 1 after randomly selecting two chromosomes. If this value is less than threshold, choose the one with the higher fit, or choose the one with the lower fit.

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  In Ranking-based selection, chromosomes are selected based on ranking rather than fit. Ranking each chromosome in an individual set is done, and the better the ranking, the better the probability of being selected.

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  In Eliteism, Some of the best chromosomes are selected (considering elitism rate $\beta$), and the selected chromosomes are passed on to the next generation without any change. This maintains good information and helps improve performance.

• Crossover: The characteristics of the two solutions are crossed and partially combined to create one new characteristic. Methods of crossover include single point, two point, and uniform. The single point and two point mean the number of intersections and combine the characteristics of the two chromosomes based on each intersection. The unit generates a random number at each genetic factor location on the chromosome and intersects the characteristics between the two chromosomes compared to threshold.
• Mutation: In this phase, mutation is applied to some chromosomes probabilistically (considering mutation rate $\gamma$) across the entire chromosome. Randomly changing the genetic factors of chromosomes to cause other harm, allowing the children to have features that are not present in the parents. This prevents the entire generation from converging to the local minimum.

Algorithm 1: Genetic Algorithm

2.2. Particle Swarm Optimization

Swarm-based optimization is a type of mathematical optimization in which multiple optimizers exchange information to perform optimization. Compared to single-agent optimization, it requires more computational resources but is more likely to converge to the global optimum. One of the well-known algorithms in optimization inspired by swarm-based is Particle Swarm Optimization (PSO), which was proposed in 1995 [19]. PSO draws inspiration from the social behavior patterns seen in bird flocks and fish schools. The method finds the optimal solution by considering both the individual optimal found in each particle and the global optimal found in the entire swarm. In Heppner’s bird simulation [20], the introduction of a two-dimensional comfield vector is proposed to model the dynamic behavior of the flock. Each agent within the simulation evaluates its current position $(x_i,y_i)$ using $\sqrt{{(x-x_i)}^2}+\sqrt{{(y-y_i)}^2}$ with a value of 0 when the current position is (100, 100).

The optimal solution is determined as Equation (1):

$$v_i(t+1)=v_i\left(t\right)+2·\mathrm{rand}\left(\right)·({\mathrm{pBest}}_i-x_i\left(t\right))+2·\mathrm{rand}\left(\right)·(\mathrm{gBest}-x_i\left(t\right))$$

(1)

where $v_i\left(t\right)$ and $x_i\left(t\right)$ are velocity and position of particle i at point t, "$pBes{t}_i$ represents the optimal position discovered by particle i to date, while $gBest$ indicates the best position indentified as the entire swarm. The decision to subtract or add g with a random constant applied to $v_i$ is determined by comparing $x_i$ with $gBest$, where the g is a weight parameter.

$${\mathrm{v}}_x\leftarrow \left\{\begin{array}{cc}{\mathrm{v}}_x-\mathrm{rand}\left(\right)·g\hfill & \mathrm{if}{\mathrm{X}}_x>{\mathrm{pBest}}_x\left[\mathrm{gbest}\right]\hfill \\ {\mathrm{v}}_x+\mathrm{rand}\left(\right)·g\hfill & \mathrm{if}{\mathrm{X}}_x<{\mathrm{pBest}}_x\left[\mathrm{gbest}\right]\hfill \end{array}\right.$$

$${\mathrm{v}}_y\leftarrow \left\{\begin{array}{cc}{\mathrm{v}}_y-\mathrm{rand}\left(\right)·g\hfill & \mathrm{if}{\mathrm{X}}_y>{\mathrm{pBest}}_y\left[\mathrm{gbest}\right]\hfill \\ {\mathrm{v}}_y+\mathrm{rand}\left(\right)·g\hfill & \mathrm{if}{\mathrm{X}}_y<{\mathrm{pBest}}_y\left[\mathrm{gbest}\right]\hfill \end{array}\right.$$

Algorithm 2: Particle Swarm Optimization (PSO)

2.3. Sea Lion Optimization Algorithm

The Sea Lion Optimization (SLnO) algorithm [21], like PSO, is an optimization method inspired by swarm activity. SLnO is a metaheuristic optimization method modeled after hunting behavior of sea lions. Sea lions utilize their whiskers to recongnize the location of their prey., communicate with each other through vocalizations, and hunt in groups. These behaviors are mathematically modeled as follows to solve optimization problems effectively.

First, in the Detecting and tracking phase:

$$\overrightarrow{D}=|2·\overrightarrow{B}·\overrightarrow{P\left(t\right)}-\overrightarrow{X\left(t\right)}|$$

(2)

$$\overrightarrow{X(t+1)}=\overrightarrow{P\left(t\right)}-\overrightarrow{D}·\overrightarrow{C}$$

(3)

equation (2) indicates the distance between the sea lion $\overrightarrow{X}$ and the target prey $\overrightarrow{P\left(t\right)}$ at point t, where $\overrightarrow{B}$ is a random vector in $[0,1]$, which is multiplied by 2 to expand the solution area and assist in identifying the optimal solution efficiently. Equation (3) represents the process of approaching the target prey, where $\overrightarrow{C}$ decreases linearly from 2 to 0.

Second, in the Communication phase, when the sea lions detect prey, they exchange information through vocalization. This behavior is modeled by the following Equation (4) to enhance the algorithm’s performance by ensuring that all agents work together to find the optimal solution.

$$V_{\mathrm{leader}}=\left|{\displaystyle \frac{V_w(1+V_a)}{V_a}}\right|$$

(4)

where $V_{\mathrm{leader}}$ is the velocity of sound of the sea lion leader, $V_w$ and $V_a$ are the speeds of sound in water and air, respectively, each having values of $sin\theta$ and $sin\varphi$.

Third, the attacking (hunting) phase is crucial for refining the search process and honing in on the optimal solution. This phase involves two main phases: Dwindling encircling technique and Circle updating position. In the Dwindling encircling technique, the gradual encircling of the prey by sea lions is mathematically modeled. In Equation (3), the value of $\overrightarrow{C}$ is decreased linearly from 2 to 0, which helps to narrow the search space around the prey. The Circle updating position, where sea lions encircle and attack the prey from all directions, is mathematically modeled as follows in Equation (5).

$$X(t+1)=\left|P\right(t)-X(t\left)\right|·cos\left(2\pi m\right)+P\left(t\right)$$

(5)

where m is a random number in $[-1,1]$, and $cos\left(2\pi \right)$ is used because the sea lions swim along a circular path to hunt the prey.

Fourth, in the Searching for prey phase, the exploration is diversified using a random value $\overrightarrow{C}$ to enhance the search for the global optimum. If $\overrightarrow{C}$ is smaller than 1, the position of the search agent is updated. If $\overrightarrow{C}$ is bigger than 1, a search agent is selected randomly. If the optimal solution is found, the SLnO algorithm ends.

Algorithm 3: Sea Lion Optimization (SLnO) Algorithm

2.4. Coral Reefs Optimization Algorithm

The Coral Reefs Optimization (CRO) [22] algorithm is a novel combination of biological inspiration and meta-heuristic algorithm designed to solve complex optimization problems by simulating the natural processes of coral reef development and coral reproduction. Coral reefs, during their growth process, engage in a competitive struggle to occupy available space. This competition, combined with the unique reproductive mechanisms of corals, leads to the development of a robust and efficient optimization algorithm. CRO algorithm models the reef as a grid ($\Lambda$) of size $N\times M$. Each grid cell can host a coral (solution) or be empty. The initial proportion of occupied cells is denoted as ${\rho }_0$. Each coral has an corresponding health function $f\left(X_{ij}\right)$ that denotes the objective function value of the optimization problem. CRO algorithm simulates both sexual and asexual reproduction of corals:

• Broadcast Spawning (External Sexual Reproduction): A portion of corals are selected to release gametes into the water. These gametes combine to form larvae that try to settle on the reef.
• Brooding (Internal Sexual Reproduction): A portion of corals reproduce by internal mutation, producing larvae that are released into the water.
• Larvae Setting: Larvae attempt to settle in the reef. When a larva attempts to settle in a cell that is occupied, it replaces the existing coral when its health function has improved. Each larva has a limited number of attempts before being depredated.
• Asexual Reproduction (Budding or Fragmentation): The corals currently present in the reefs are sorted based on their health levels as determined by a certain criterion. A portion of the best corals replicate and attempt to establish themselves in various parts of the reef.
• Depredation in Poly Phase: With a small probability, the worst corals are removed from the reef, freeing up space for new corals.

Algorithm 4: Coral Reefs Optimization (CRO) Algorithm

3. Estimation of Damage Detection in 2-Bay Truss Structure Using Bio-Inspired Optimization Methods

Table 1. Average errors for different optimization methods. The table shows the performance of each method in terms of average error percentage.

4. Vision Based Structural Health Monitoring System Using Phase-Based Motion Magnification

5. Estimation of Damage Detection in Beam Structure Using Natural Frequency and Rotational Spring Method

6. Damage Detection Using Bio-Inspired Optimization Methods

7. Conclusions

References

1. Yan, K.; Zhang, Y.; Yan, Y.; Xu, C.; Zhang, S. Fault diagnosis method of sensors in building structural health monitoring system based on communication load optimization. Smart Mater. Struct. 2020, 29, 105034. [CrossRef]
2. Hassani, S.; Dackermann, U. A Systematic Review of Optimization Algorithms for Structural Health Monitoring and Optimal Sensor Placement. Sensors 2023, 23, 1050.
3. Sun, H.; Büyüköztürk, O. Optimal sensor placement in structural health monitoring using discrete optimization. Smart Mater. Struct. 2015, 24, 125034. [CrossRef]
4. Guo, H.Y.; Zhang, L.; Zhang, L.L.; Zhou, J.X. Optimal placement of sensors for structural health monitoring using improved genetic algorithms. Smart Mater. Struct. 2004, 13, 528. [CrossRef]
5. Eltouny, K.; Gomaa, M.; Liang, X. Unsupervised Learning Methods for Data-Driven Vibration-Based Structural Health Monitoring: A Review. Sensors 2023, 23, 3290.
6. Shabbir, K.; Umair, M.; Sim, S.-H.; Ali, U.; Noureldin, M. Estimation of Prediction Intervals for Performance Assessment of Building Using Machine Learning. Sensors 2024, 24, 4218.
7. Choi, A.J.; Han, J.-H. Frequency-based damage detection in cantilever beam using vision-based monitoring system with motion magnification technique. J. Intell. Mater. Syst. Struct. 2018, 29, 3923-3936. [CrossRef]
8. Zhou, L.; Huang, P.; Chi, S.; Li, M.; Zhou, H.; Yu, H.; Cao, H.; Chen, K. Structural health monitoring of offshore wind power structures based on genetic algorithm optimization and uncertain analytic hierarchy process. Renewable Energy 2020, 147, 1783-1795. [CrossRef]
9. Barontini, A.; Masciotta, M.-G.; Ramos, L.F.; Amado-Mendes, P.; Lourenço, P.B. An overview on nature-inspired optimization algorithms for Structural Health Monitoring of historical buildings. Journal of Cultural Heritage 2021. [CrossRef]
10. Das, S.; Saha, P.; Satapathy, S.C.; Jena, J.J. Social group optimization algorithm for civil engineering structural health monitoring. Appl. Soft Comput. 2020, 96, 1651-1670. [CrossRef]
11. Loh, K.J.; Ryu, D.; Lee, B.M. Bio-inspired Sensors for Structural Health Monitoring. In Bio-inspired Sensors for Structural Health Monitoring; Springer: New York, NY, USA, 2020; Chapter 11.
12. Vakil-Baghmisheh, M.-T.; Peimani, M.; Sadeghi, M.H.; Ettefagh, M.M. Crack detection in beam-like structures using genetic algorithms. Appl. Soft Comput. 2008, 8, 1150–1160.
13. Boonlong, K. Vibration-Based Damage Detection in Beams by Cooperative Coevolutionary Genetic Algorithm. Adv. Mech. Eng. 2014, 6, 1–12.
14. Chen, J.G.; Wadhwa, N.; Cha, Y.J.; Durand, F.; Freeman, W.T.; Buyukozturk, O. Modal identification of simple structures with high-speed video using motion magnification. J. Sound Vib. 2015, 345, 58–71.
15. Yang, Y.; Dorn, C.; Mancini, T.; Talken, Z.; Kenyon, G.; Farrar, C.; Mascarenas, D. Blind identification of full-field vibration modes from video measurements with phase-based video motion magnification. Mech. Syst. Signal Process. 2017, 85, 567–590.
16. Poozesh, P.; Sarrafi, A.; Mao, Z.; Avitabile, P.; Niezrecki, C. Feasibility of extracting operating shapes using phase-based motion magnification technique and stereo-photogrammetry. J. Sound Vib. 2017, 407, 350–366.
17. Mitchell, M. An Introduction to Genetic Algorithms; MIT Press: Cambridge, MA, USA, 1998.
18. Algedir, A.A.; Refai, H.H. Energy Efficiency Optimization and Dynamic Mode Selection Algorithms for D2D Communication Under HetNet in Downlink Reuse. IEEE Access 2020, 8, 95251–95265.
19. Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95 - International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948.
20. Heppner, F.; Grenander, U. A stochastic nonlinear model for coordinated bird flocks. In The Ubiquity of Chaos; Krasner, S., Ed.; AAAS Publications: Washington, DC, USA, 1990.
21. Masadeh, R.; Mahafzah, B.A.; Sharieh, A. Sea Lion Optimization Algorithm. Int. J. Adv. Comput. Sci. Appl. 2019, 10, 1–10.
22. Salcedo-Sanz, S.; Del Ser, J.; Landa-Torres, I.; Gil-López, S.; Portilla-Figueras, J.A. The Coral Reefs Optimization Algorithm: A Novel Metaheuristic for Efficiently Solving Optimization Problems. The Scientific World Journal 2014, 2014, 15 pages. [CrossRef]
23. Wadhwa, N.; Rubinstein, M.; Durand, F.; Freeman, W.T. Phase-based video motion processing. ACM Trans. Graph. 2013, 32, 1–10.
24. Adams, R. D.; Walton D.; Flitcroft, J. E. Short, D. Vibration testing as a nondestructive test tool for composite materials. J. Compos. Mater 1975, 13, 161–175.
25. Rizos, P. F.; Aspragathos, N. A.; Dimarogonas, A. D. Identification of Crack Location and Magnitude in a Cantilever Beam From the Vibration Modes. J. Sound Vib. 1990, 138, 381–388.
26. Diamarogonas, A. D.; Paipetis, S. A. Analytic Methods in Rotor Dynamics, 2nd ed.; Springer: New York, USA, 1983.