Erosion depth prediction of chloride ions under stray current using FEM based CNNs

Stray currents can accelerate the transport of corrosive ions, especially Cl−, in concrete materials, which is very detrimental to structural safety. Effectively predicting the erosion depth of Cl− is crucial for evaluating structural safety. This article is based on a finite element model and verifies the erosion depth of Cl− under different voltages, Cl− concentrations, and corrosion time through experimental data. A polynomial was used to fit the quantitative relationship between erosion depth, Cl− concentrations, and corrosion time under single voltage condition. However, this formula only applies to a single voltage and has too many parameters. Therefore, this article also established a CNNs regression model to predict the depth of Cl−, and the results showed the multiple regression ability of CNNs. It has been proven that CNNs can accurately predict the erosion depth, which helps to accurately evaluate structural safety. After comparing experimental values, CNNs, ResNet, and ResNet-attention, it was found that residual networks and attention mechanisms did not significantly improve the prediction accuracy of deep networks, which may be related to insufficient data volume. After expanding the dataset, ResNet performed the best overall, and ResNet-attention had better testing performance, which is related to the powerful feature extraction ability of the attention mechanism.


Introduction
Stray current refers to the current that does not flow along the expected path, usually originating from leaks in electrical systems such as underground pipelines, railways, and power lines [1,2].When this current is generated in concrete structures, it will accelerate the erosion of metal materials (such as steel bars) inside.Due to the porosity of concrete and the conductivity of pore solutions, stray currents can form electrochemical corrosion on the surface of steel bars [3][4][5], causing them to gradually lose their strength and load-bearing capacity.In addition, the presence of corrosive media such as Cl − can further exacerbate this process, posing a serious threat to the durability and safety of concrete structures [6].This erosion not only affects the normal service life of the structure, but may also cause serious engineering accidents, posing a huge threat to people's life and property safety.Therefore, studying and controlling the erosion effect of stray currents on concrete structures is of great significance.
Under stray currents, ion migration and diffusion dominate, with Cl − reaching the critical concentration faster [7][8][9].Cl − will accumulate around the steel bars, accelerating corrosion and deterioration [10].At the same time, internal ions will also accelerate outward migration, leading to the decomposition of hydration products.Scholars have conducted a large number of related experiments and studies.Tang [3,11] mainly studies the effect of stray current on the corrosion of steel fibers in steel fiber reinforced concrete.The results indicate that concrete containing discrete steel fibers has certain resistance to stray AC corrosion.The chloride threshold level of steel fiber reinforced concrete is significantly higher than that of traditional steel reinforcement, which is about 4% NaCl (calculated by cement mass).However, when the pore solution contains high concentrations of Cl − , this corrosion resistance will decrease.Eichler and Isecke [12] investigated the corrosion behavior of reinforced concrete structures under the action of DC stray currents through numerical calculations.They utilized simplified geometric structures to extract key parameters related to structures affected by stray currents, and conducted in-depth research on the interactions between these parameters.Han et al [13] focused on the issue of stray current corrosion in reinforced concrete infrastructure in high-speed electrified rail transit.The model they established takes into account the spatial distribution of oxygen, time consumption, and pore saturation.Based on the fluctuation of oxygen concentration, they proposed the concept of quasi steady state and systematically compared the corrosion behavior of different pore saturations.Liu et al [14] used various methods such as corrosion morphology observation, weight loss experiments, and electrochemical testing to deeply explore the influence of stray currents with bipolar characteristics on the corrosion behavior of steel fibers in simulated pore solutions and concrete environments.They found that the short-term corrosion effect of stray currents on steel fibers is relatively limited.However, when Cl − is introduced, the self-healing ability of the passivation film on the surface of steel fibers weakens or even disappears, accelerating the pitting process on the surface of steel fibers.The preliminary work [4,15] of this article delves into the effects of stray currents and saline alkali environments on steel fiber reinforced concrete.The experiment found that under the action of an electric field, the transfer speed of Cl − is significantly faster than that of SO 4 2− , accelerating the corrosion process of steel fiber reinforced concrete.In addition, the strength loss of steel fiber reinforced concrete is mainly caused by the erosion and expansion of steel fibers.It is worth noting that products such as gypsum and ettringite are only produced in the cathode area, and the formation of these products further reduces the strength of steel fiber reinforced concrete.It can be seen that the coupling effect between stray current and saline alkali environment poses great harm to reinforced concrete materials.
Currently, advanced data processing methods are bringing revolutionary changes to the engineering field, especially in the assessment of concrete structure strength [16][17][18].It is worth noting that the strength of concrete structures is inversely proportional to the depth of Cl − erosion.In order to overcome the limitations of traditional evaluation methods with multiple empirical parameters and low prediction accuracy, we utilized advanced technologies such as machine learning to accurately predict the depth of Cl − erosion, thereby effectively evaluating the strength and durability of concrete structures.We have developed a CNNs model for the key parameter of Cl − erosion depth.In terms of data acquisition, we used finite element simulation to generate the dataset and rigorously validated it through experimental data.This data foundation ensures the accuracy and reliability of model training.In order to further improve the prediction accuracy, we have introduced residual networks and attention mechanisms, enabling the model to better capture subtle changes in the data and make more accurate predictions.The implementation of these works not only helps us accurately evaluate the strength and durability of concrete structures, but also provides strong support for fault diagnosis, state maintenance, and process control.By introducing control and automation technologies, we can achieve real-time monitoring and early warning of the state of concrete structures, providing data support for decision-making science.The application of these technologies is of great significance for ensuring the safety of concrete structures and extending their service life, while also demonstrating the enormous potential of advanced data processing methods in the engineering field.

Finite element model and validation
Firstly, a two-dimensional two-phase finite element model for Cl − transmission under stray currents was established based on the theories of electromigration and diffusion, and the final simulation parameters were determined through experimental data validation.Based on this set of simulation parameters, erosion depths of Cl − with different voltages, Cl − concentrations, and corrosion time were simulated, expanding the erosion depth dataset and avoiding a large number of repetitive experiments.

Transmission theory
The transmission of Cl − can be obtained based on the Nernst Planck equation [19,20] and Fick's first law [21,22].The flux expressions for both are shown in equations ( 1) and (2): In the formula, J i,e is the ion flux during electromigration and mass transfer processes, in mol•s m −2 ; z i is the electricity price of corrosive ions; F is the Faraday constant, with a value of 9.648 × 10 −4 C mol −1 ; R is the gas constant, with a value of 8.314 J (mol K) −1 ; T is the absolute temperature, in K; as well as ψ It is the voltage, measured in V. J i,d is the flux during the diffusion process of corrosive ions, expressed in mol•s m −2 ; D a i is the apparent diffusion coefficient, in m 2 s −1 ; C i is the concentration of corrosive ions, in mol m −3 ; ∇ is Hamiltonian operators.
For transient mass transfer processes, according to Fick's second law [23,24], the ion transport is equation (3): (3) In the formula, t is the corrosion time, in hours.

Model and calculation parameters
This transmission model takes into account the multiphase nature of concrete materials and establishes a two-dimensional model based on the cross-section of actual steel fiber reinforced concrete specimens to distinguish between coarse aggregates and cement mortar.Figure 1 shows the geometric model and mesh partitioning of a two-dimensional multiphase ion transport model.The grid is divided into 84 173 units using a free triangle network.
Cl − erosion is the main factor leading to a decrease in the strength of steel fiber reinforced concrete [4,25,26].In order to further expand the database of Cl − erosion depth, simulations were conducted on the transport process of Cl − in steel fiber reinforced concrete under different voltages, chloride ion concentrations, and corrosion time.The characteristic of the decrease in electrical resistance and increase in electrical conductivity of concrete caused by steel fibers is considered to be an increase in the relative dielectric constant of mortar.The calculation parameters are the same as those used in literatures [8,15,27].

Parameter validation
Figure 2 shows the comparison between experimental and simulated values of cathodic erosion depth.The experimental data is sourced from the [15].As shown in the figure, the two-dimensional two-phase finite element model for Cl − transmission under stray currents simulates the depth of cathodic erosion, and the simulated values are very close to the experimental values.The Pearson correlation coefficient between the experimental and simulated values was 0.9935.This verifies the correctness of the model for Cl − transmission.

CNNs architecture and training parameters
CNNs can play an important role in regression prediction tasks [28][29][30][31].By using sequence or image data as input, CNNs can extract key features and map these features to a continuous output space through fully connected layers, thereby achieving regression prediction.Compared with other methods, CNNs have significant advantages [32,33] in regression prediction tasks.Firstly, CNNs can automatically extract features from input data without the need for manual feature engineering, thereby reducing the complexity of model design.Secondly, CNNs significantly reduce the number of model parameters and reduce the risk of overfitting by utilizing local perception and weight sharing.Finally, CNNs perform well in processing complex inputs such as images and sequence data, capturing subtle changes in the data and improving prediction accuracy.Therefore, CNNs have broad application prospects in regression prediction tasks.
The architecture of CNNs is shown in figure 3. It starts from input data of [3,1,1] size and extracts key features through two convolutional layers.Each convolutional layer uses a 3 × 1 kernel to generate 16 and 32 feature maps, respectively.Following each convolutional layer, there is a batch normalization layer to accelerate training and improve model performance, as well as a ReLU activation layer to introduce nonlinearity.In addition, there is a max pooling layer used for down-sampling and feature extraction.After passing through these layers, the network uses dropout layers to prevent overfitting and ultimately maps features to the final predicted values through fully connected layers.The last layer of this architecture is the regression layer, which is responsible for outputting continuous prediction results.The entire network structure is compact and efficient, suitable for extracting features from small datasets and performing continuous value prediction tasks.
The random gradient descent method with momentum is used as the optimization algorithm, and the batch size is set to 32 to balance computational efficiency and model generalization ability.The maximum number of training iterations is 1200, ensuring that the model fully learns data features.The initial learning rate is set to 0.01 to facilitate rapid convergence of the model in the early stages of training.The learning rate reduction strategy adopts the segmented constant method, which reduces the learning rate every 800 training cycles with a reduction factor of 0.1, which helps the model finely adjust weights in the later stage.At the beginning of each training cycle, the data will be shuffled again to reduce the risk of overfitting.These parameters work together in model training to improve model performance and convergence.

Comparison between polynomial fitting and CNNs
Polynomial fitting and CNNs both aim to approximate functions from given data, but they differ significantly in their approach and capabilities.Polynomial fitting involves finding the best-fitting polynomial curve to a given dataset [34,35], typically through techniques like least squares regression [36,37].This approach is simple and computationally efficient, but it assumes a prior knowledge of the functional form (polynomial) and may not capture complex, nonlinear relationships well.On the other hand, CNNs are deep learning models specifically designed to handle complex pattern recognition tasks.They automatically learn hierarchical representations of data, making them excellent at capturing intricate dependencies and nonlinearities.CNNs require a larger amount of data and computational resources but can often achieve superior performance, especially in tasks where the underlying structure of the data is unknown or highly nonlinear.In summary, polynomial fitting offers a simple and efficient solution for specific tasks, while CNNs provide a more flexible and powerful approach for complex pattern recognition and prediction tasks.

Construction of datasets
The analysis of the simulation results in the previous section verified the correctness of the Cl − transmission model.The polarization potential caused by stray currents in real engineering environments is often within 200 mV, and the Cl − concentration in service environments is set within the range of 0.6-1.8mol l −1 .This section uses the numerical model to simulate the variation of erosion depth with corrosion time under different voltages and Cl − concentration.Figure 4 shows the contour map of 0.1 mol l −1 Cl − concentration simulated by a two-dimensional multiphase ion transport model, which is in good agreement with the erosion depth of steel fiber reinforced concrete in color rendering experiments.Therefore, in this chapter, the contour line of 0.1 mol l −1 will be used as the criterion for determining the erosion depth in steel fiber reinforced concrete.

Polynomial fitting
In this section, a quadratic polynomial was used to fit the quantitative relationship between erosion depth, Cl − concentration, and corrosion time under the same voltage conditions.Figure 5 shows the fitting surfaces of erosion depth, Cl − concentration, and corrosion time under different voltages.Table 1 shows the parameter values for polynomial fitting.polynomial fitting method has a good fitting effect on the dataset.Figure 6(d) shows the residual plot of CNNs.Similar to polynomial fitting residual plots, by observing the residual size, sign, and distribution of each data point, we can understand the fitting effect and predictive ability of CNNs on the dataset.Compared with polynomial fitting, CNNs can accurately predict erosion depth except for a few points.

Comparative analysis
In addition to the traditional CNNs mentioned above, we also introduce residual networks and channel attention mechanisms.Figure 7 shows the architecture of ResNet and ResNet-Attention.Figure 8 shows the comparison between experimental values, CNNs, ResNet, and ResNet-Attention.After comparing experimental values, CNNs, ResNet, and ResNet-Attention, it was found that residual networks and attention mechanisms did not significantly improve the prediction accuracy of deep networks, which may be related to insufficient data volume.Meanwhile, this also indicates that traditional cellular neural networks can meet the requirements of predicting erosion depth of Cl − .
After expanding the dataset (0.025 V/0.075 V/0.125 V), compare the traditional CNNs, ResNet, and ResNet attention architectures again.Table 2 shows the performance of three architectures.From the table, it can be observed that after dataset expansion, the performance of ResNet and ResNet attention has significantly improved compared to before.Compared with traditional CNNs, ResNet performs better on both the training and testing sets.However, the ResNet-attention architecture performs worse on the training set than the previous two, but performs better on the testing set than traditional CNNs.This is related to the fact that the attention mechanism itself can improve the feature extraction ability of the model and strengthen the feature extraction of key features.In summary, this article investigates the effect of stray currents on the erosion depth of Cl − in concrete and establishes a prediction model based on CNNs.This model provides a new method for the safety evaluation of concrete structures, which helps guide the design of protective measures in engineering practice.

Figure 2 .
Figure 2. Comparison between experimental and simulated values of cathodic erosion depth.

Figure 6 (
Figure 6(a) shows the training effect of CNNs.Overall, based on the given training parameters, CNNs have received sufficient training to fully capture the features of the training set data. Figure 6(b) shows the comparison between CNNs and polynomial fitting.This graph compares the fitting and predictive performance of CNNs and polynomial fitting methods on the same dataset.By comparing the curves of different methods, we can clearly see the differences between CNNs and polynomial fitting in data fitting and prediction.Due to their powerful feature extraction and learning capabilities, CNNs are often able to accurately fit complex data patterns and exhibit good performance in prediction.Although CNNs are slightly weaker in fitting than polynomial fitting, this is because polynomial fitting has all prior knowledge.Figure 6(c) is the residual plot of polynomial fitting.This figure shows the residual distribution of the polynomial fitting method.By observing the residual plot, we can observe that the residual distribution is random and very small, indicating that the

Table 2 .
Performance of three different architectures: CNNs, ResNet, and ResNet-attention.ConclusionIn concrete structures, stray currents can accelerate the transport of corrosive ions such as Cl − , leading to issues such as steel corrosion.In order to effectively predict the erosion depth of Cl − , we have established a reliable prediction model.The main content is as follows:(1) Firstly, the finite element model was used to simulate the erosion process of Cl − .The accuracy of the model was verified by comparing experimental and simulation results.Then, based on this model, we studied the erosion depth under different voltages, Cl − concentrations, and corrosion time.Under a single voltage condition, we found a quantitative relationship between erosion depth, Cl − concentration, and corrosion time.To describe this relationship, we use polynomials for fitting.However, this method has some limitations.Firstly, it only applies to a single voltage condition and requires re fitting the formula to other voltage conditions.Secondly, polynomial fitting requires a large number of parameters, which increases the complexity of the model.(2) To overcome these limitations, we established a CNNs regression model to predict erosion depth.We used voltages, Cl − concentrations, and corrosion time as input data and constructed a mapping relationship between the input data and the erosion depth of Cl − by training a CNNs model.Finally, we used the trained model to predict the erosion depth of Cl − .The results indicate that the CNNs model has good predictive performance.This is because CNNs have powerful feature extraction and learning capabilities, which can automatically extract useful information from input data and establish complex nonlinear mapping relationships.In addition, CNNs models also have good generalization ability and can adapt to predicting erosion depth under different conditions.(3) After comparing experimental values, CNNs, ResNet, and ResNet-attention, it was found that residual networks and attention mechanisms did not significantly improve the prediction accuracy of deep networks, which may be related to insufficient data volume.Meanwhile, this also indicates that traditional CNNs can meet the requirements of erosion depth prediction.(4) After expanding the dataset, ResNet performed the best overall, and ResNet-attention had better testing performance, which is related to the powerful feature extraction ability of the attention mechanism.