Social Parameter’s Role in BEM-PSO-Based Inverse Analysis for Locating Corrosion of Reinforced Concrete

The objective of this research is to investigate the effect of social parameter on the performance of boundary element inverse analysis (BEIA) to determine the location of reinforcing steel corrosion in concrete. BEIA was constructed by integrating the Boundary Element Method (BEM) and Particle Swarm Optimization (PSO) (PSO). BEM was used to calculate the corrosion potential on the whole surface of reinforced concrete. The cost function for detecting corrosion of reinforcing steel in concrete was evaluated using PSO. BEIA was conducted with 15 electrical potential data on the surface of reinforced concrete as a reference, such as from half-cell potential measurement. Numerical simulations of reinforced concrete with a single reinforcement reveal a difference in the speed of BEIA in locating corrosion sites. The higher the social parameter value, the faster the particles tend to migrate. However, if the parameter value is too high, the number of iterations required for the particles to converge increases, which has a negative effect on locating the corrosion point. Consequently, the variation of the social parameter affects the performance of BEIA in detecting the location of corrosion in reinforced concrete.


Introduction
Corrosion of reinforcing steel in concrete is a severe issue in concrete constructions.Corrosion is the primary cause of reinforced concrete infrastructure's premature failure [1].According to research conducted in the United States, corrosion has cost hundreds of billions of dollars.Several governments have investigated the cost of corrosion.According to research by Schmitt et al. [2], corrosion costs account for 3 to 4% of gross domestic product (GDP).According to Jackson [3], the cost of corrosion might reach 6.2% of the gross domestic product.Therefore, in order to save expenses, it is necessary to limit corrosion.Corrosion detection of reinforced concrete, which must be undertaken as soon as feasible in order to take corrective action, is one method for achieving this goal.
Half-cell potential mapping is a typical technique for the early diagnosis of reinforced concrete deterioration.ASTM C876 [4] is the applicable standard for this procedure.This method determines the potential corrosion values on the concrete surface, which are subsequently utilized as a reference for the occurrence of steel corrosion in reinforced concrete [1].However, this method has a few limitations, including the need for caution in data interpretation [5], the fact that it only determines the possibility of corrosion [6], and the fact that some factors that affect potential are not taken into account [7][8], which can lead to potential misinterpretation of the data due to ill-posed problems [9].Therefore, it is essential to devise a new strategy to overcome the restrictions.
The Boundary Element Inverse Analysis (BEIA) based on the Boundary Element Method (BEM) and Particle Swarm Optimization (PSO) were used to develop a method for local corrosion detection of reinforced concrete [10].In BEIA, PSO is used to detect corrosion by comparing measurement result data with calculation result data (simulation) obtained from BEM [11].Inverse analysis can be used to predict the location of corrosion occurrences by knowing the corrosion potential on the surface of concrete.
In previous research [10,11], BEIA was performed, and the approach identified the precise area of corrosion with 95.65% accuracy [12].However, the use of PSO in computation-based inverse analysis for the detection of corrosion locations is not yet optimal.Several factors in the PSO parameter, such as learning rates for social parameters, have not yet been investigated.In order to detect corrosion more efficiently, it is necessary to do additional research on the relevant parameters.Consequently, the purpose of this study is to investigate the effect of social parameter on the performance of BEIA for detecting the location of corrosion in reinforcing steel in concrete.

Modelling Corrosion in Reinforced Concrete
Corrosion is a naturally occurring process of material deterioration; it cannot be stopped but can be regulated.According to Fontana [13], corrosion is the deterioration of material quality caused by electrochemical reactions with the surrounding environment.In reinforced concrete, the concrete layer is contaminated by the surrounding environment, causing the reinforcing steel to begin reacting with its surroundings and a corrosion process to take place.
To represent the corrosion of reinforcing steel in concrete, the concrete is assumed to be a closed system.There are no ions entering or leaving the system.Figure 1 is a model of corrosion in reinforced concrete.The concrete contains a single corroded steel reinforcement.The corroding (anode) and noncorroding (cathode) portions of the steel are positioned as shown in the figure.The Laplace equation, written as Equation (1) [5,9], governs the electrical corrosion potential (ϕ) for the reinforced concrete region (Ω) in this model.

Figure 1. The model of reinforcing steel corrosion in a concrete structure
In this situation, the current density (i) and electrical potential (ϕ) have a correlation given by Equation ( 2) for the domain [5,9].

ISAIME-2022
Journal of Physics: Conference Series 2739 (2024) 012037 where κ, n, and ∂⁄∂n are the domain conductivity, the outward normal unit, and the derivative in the normal direction, respectively.
BEM is used to solve the Laplace equation.BEM requires the knowledge of several boundary conditions.Equations (3) through ( 5) are the boundary conditions for the corrosion model of reinforcing steel shown in Figure 1.The boundary condition of the concrete surface (Γ1) is described by Equation (3).Due to the low conductivity of concrete, the current density (i0) is equal to zero in this circumstance.The polarization curves of the anode and cathode then represent their respective boundary conditions.The empirically determined correlation between the electrical potentials (ϕ) and current density (i) is given by Equation ( 4) for the cathode (Γ 2 ) and Equation ( 5) for the anode (Γ 3 ) [5].Brebbia and Domiguez [14] provide a more extensive description of the formulation of BEM.Following the formulation approach, it is possible to calculate the unknown values of ϕ and i for the entire concrete domain.

Particle Swarm Optimization (PSO)
PSO was established by Kennedy and Eberhart in 1995 [15].The PSO mechanism draws inspiration from the characteristics of animals such as birds and fish.PSO has the advantages of being straightforward to program and simple in terms of mechanism, yet effective.Each iteration of PSO employs a collection of particles that evaluate objective functions.The particles are then categorized based on their conformity with the objective function.The particle with the highest objective function value will serve as the standard.Then, during the subsequent iteration, the position of each particle will shift relative to the best particle.

Boundary Element Inverse Analysis (BEIA)
BEIA is a technique for detecting corrosion of reinforcing steel in concrete that employs BEM and PSO.Previous studies [10][11] have conducted research on BEIA.Inverse analysis is accomplished by minimizing the cost function, which is the difference between the corrosion potential value derived from field measurements (such as the potential mapping technique) and the calculated corrosion potential value by using BEM.Using PSO, the cost function is evaluated throughout simulation procedures.In this situation, the results are utilized to determine the corrosion profile, or the location of corrosion.Therefore, the BEM is used to determine the corrosion potential value, while the PSO is employed as an optimization technique to minimize the cost/objective function in the BEIA. Figure 2 depicts the algorithm used to construct BEIA.
The initial stage in executing the BEIA is to specify the necessary parameters, including Z, jmax, ε0, a1, a2, and W. Z is the number of particles, jmax is the maximum number of iterations, a1 and a2 are constant values reflecting the particle learning rates, and W is the inertia weight value.The term "particle" is used for candidate solutions in the process.The corrosion profile Xj and particle velocity Vj are then determined at random on the steel.BEM is utilized to calculate the potential value of each particle or candidate solution on the concrete surface.
The subsequent step is to insert the N data of corrosion potential ( ̅ ).on the surface of the concrete derived from field measurements.Equation ( 6) yields the value of the cost function (ε) for each particle.

𝜀 (𝑋) = ∑ [(
In the equation, X represents the corrosion profile on the reinforcing steel and N is the number of potential values measured for BEIA. and  ̅ are corrosion potential values computed with BEM and corrosion potential data collected with techniques such as half-cell potential mapping, respectively. ̅  represents the maximum corrosion potential among N data.Under the provided parameters, the value of the cost function is assessed for each particle of each iteration. The simulation has concluded when the maximum number of iterations has been reached.If it is not reached, however, the iteration continues by updating Xj and Vj.The changes in position and velocity of each particle throughout each iteration are determined by Equations ( 7) and (8).In these equations, X(j+1) is the next particle position, Xj is the current particle position, V(j+1) is the next particle velocity, Vj is the current particle velocity, W is the inertia weight, a1 (cognitive parameter) and a2 (social parameter) are constants, r1 and r2 are random numbers (0-1), pbest is the best local particle position, gbest is the best global particle position, and j is the number of the iteration.The words "best local" and "best global" used to indicate the best location experienced by each particle and all particles, respectively, during the iterations.The value of is assessed until the location of each particle reaches the same point or the maximum number of iterations is reached, whichever occurs first.In Figure 3, the presumed location of corrosion is indicated by the red section.However, the location of corrosion is unknown in this instance.The location of corrosion is then determined using the BEIA technique.The BEIA simulation requires N measured potential data, Z, W, jmax, a1, and a2 as input parameters.In this instance, the N measured potential values were derived from a direct BEM simulation with the 2D model depicted in Figure 3(b).Under the assumption that corrosion occurs at X = 50 cm and a size of 2 mm, the corrosion potential data from the BEM simulation were used as field measurement potential data.
The BEIA was then executed with the parameters N = 15, Z = 5, W = 0.25, and , jmax = 20 to locate the corrosion.In this study, the simulation was repeated four times; the variable a 2 was assigned a low value in the first simulation, and its value was increased in each succeeding simulation.PSO includes a2 as one of its variables.Other variable required by BEIA to determine the location of corrosion was a1, which was adjusted at 0.5 based on prior studies [10][11].Table 1 displays the values of a1 and a2 for all simulations.The simulation depicted in Figure 5 has an a2 value of 0.80.The results of the simulation demonstrate that the position of corrosion can be determined with an inaccuracy of <5% and that only nine iterations are required for all particles to converge on the location.In these settings, it may be hypothesized that as the value of a 2 increases, so does the speed at which particles discover the solution.Figure 6 is simulation result for a2 = 1.5 demonstrates that the particle velocity increases proportionally to the value of a2.It can be seen that the initial location of the particles was more away from the corrosion location, the faster the particles moved, and they even passed the actual area of corrosion.The features of particle movement are analogous to waves whose amplitude decreases as they approach the site of corrosion, at which point they cease to move. Figure 7 depicts the simulation outcome when a2 = 3 It demonstrates that the initial distribution of the five particles is random.However, particle mobility is faster than in all previous simulations, which reduces the particles' capacity to migrate smoothly to the actual location of corrosion.Therefore, it takes a considerable amount of time for all particles to converge at the site of corrosion.Upon entering the seventeenth iteration of this simulation, all particles begin to stop moving.Figure 8 is a summary of the whole simulation that illustrates the relationship between the social parameter and the iteration in which all of the particles stop moving (converging).As the value of a2 approaches 0.8, the number of iterations necessary for the particles to converge reduces as depicted in the figure.Otherwise, it increases when a2 is greater than 0.80.It is believed that this phenomenon arises as a result of a rise in particle velocity when the social parameter value increases.By increasing the value of a2, the velocity of particles is accelerated.When a2 reaches a particular value, however, it appears difficult for all particles to converge because certain particles travel very quickly.Consequently, the required number of iterations to achieve convergence grows.The value of a2 must be considered so that the particles travel neither too slowly nor too quickly.On the basis of all performed BEIA simulations, it can be concluded that the value of a2 may impact the movement of particles when determining the location of reinforcing steel corrosion in concrete.

Conclusion
From the results of all simulations, it can be concluded that the value of the social parameter has a considerable impact on the speed of particles in locating the area of corrosion.Therefore, it could affect BEIA's performance.

Figure 3 .
Figure 3. Model of reinforced concrete with one reinforcement

Figure 8 .
Figure 8. Relationship between a 2 and the number of iterations required for particles to converge

Table 1 .
The cognitive and social parameter values for each simulation.Figures 4 to 7 depict the results of BEIA simulations achieved by altering the value of a2.Figure4depicts the motion of each particle with an a2 value of 0.2 for each conducted iteration.Up to the twentieth iteration, none of the particles have congregated or converged, as depicted in the figure.The site of corrosion cannot be determined until the twentieth iteration.Nevertheless, based on the features of particle mobility in each iteration, it is evident that each particle continues to advance steadily toward the region of corrosion.It is projected that the location of corrosion will be determined if the iterations continue.