The Limit of Incomplete Information in Inverse Analysis for Hidden Corrosion Inference in Reinforced Concrete

One recurring problem in reinforced concrete structures is hidden corrosion. Detection of hidden corrosion is difficult due to incomplete information that can be measured. To overcome this problem, inverse analysis can be employed. Inverse analysis uses incomplete information from field measurement, and employs optimization to infer further information. Previous research has demonstrated the inference capability of inverse analysis for hidden corrosion detection in reinforced concrete. However, question remains on the limit of incompleteness of measurement data. The aim of this study is to simulate and examine the limit of incomplete information that can be allowed for an effective inverse analysis. The inverse analysis of hidden corrosion in reinforced concrete is formulated as a PDE-constrained optimization. The objective function of the optimization is the distance function between measurement data and FEM simulation. The search space is the PDE boundary conditions configuration. A FEM simulation is used to simulate measurement data by half-cell potential mapping on a concrete block of 2 by 1 meters. To simulate incomplete information, the number of measurement points is varied from 2 to 20 points on the surface of the concrete block. Results show that 20 measurement points are needed to successfully infer hidden corrosion profile. However, when the number of measurement points are below this, the effectiveness of inverse analysis varies and does not seem to correlate to the amount of information (i.e., number of measurement points). Further study is needed if this is due to a fluke or choice of measurement point’s location.


Introduction
One of the key goals of the Sustainable Development Goals (SDG), notably SDG 9.1, 9.a, and 11.b [1], is to ensure infrastructure stability and resilience.Development of infrastructure and the policies that go along with it are therefore supported.Monitoring and evaluating infrastructure require intricate interactions between data, computing, and field measurement.Therefore, using machine learning techniques is an obvious answer.A main mechanism of reinforced concrete deterioration is the corrosion of steel rebar which is hidden inside the concrete structures [2].The corrosion of steel rebar progresses quietly under the concrete and leads to premature failure of the concrete structures.Half-cell potential mapping is a method for assessing hidden corrosion in reinforced concrete structures [3].However, the method can only provide partial information of the corrosion condition, such as the general location of the corrosion inside the concrete structures [4].To obtain a more complete examination of the corrosion condition, inverse analysis can be employed [4,5].Inverse analysis uses partial information from in-situ examination to build a more complete picture of the system.For example, half-cell potential mapping, as a non-destructive evaluation technique, can only provide the electrochemical corrosion potential on the outside wall of the concrete structure, while the corrosion itself occurs inside the concrete.By using this data, inverse analysis can provide inference of the corrosion condition inside the concrete structure.Inverse analysis works by combining in-situ examination data with computational simulation and optimization technique.In-situ examination data is incomplete by nature, due to the physical challenge of measurement, environmental factors and human errors.Question remains on how much in-situ examination data is needed to maintain effectiveness of inverse analysis.This study aims to examine the limit of incomplete data that can be tolerated by the inverse analysis of hidden corrosion in reinforced concrete.

The Hidden Corrosion Problem and Inverse Analysis
Rebar corrodes by an electrochemical process [6].Because concrete is a porous material, it may be penetrated by organisms that can cause corrosion through an electrochemical process.Corrosion response may take place after the species reach the rebar.Potential fields are created over the concrete mass as a result of the corrosion process.The concrete's surface may be used to quantify this potential field, Figure 1 illustrates its mechanism This potential distribution inside the concrete can be modeled using Laplace's equation [7], where ∇ is Laplacian in terms of Cartesian coordinate, q is the current density, ϕ is electrochemical potential, n is normal vector, κ is conductivity and Eq. ( 3), ( 4) and ( 5) are the boundary conditions.Eq. ( 3) is non-electrode boundaries of the system that are modeled as insulation, where the current density is zero due to discontinuity or distant from source.Eq. ( 4) is a boundary condition representing the anode or the corroded part of the rebar, while Eq. ( 5) represents the rest of the rebar.To capture the polarization phenomenology in corrosion, Eq. ( 4) and ( 5) are functional boundary conditions, where the function fa (for anode polarization) and fc (for cathode polarization) are obtained from curve-fitting of experimental polarization data (which depends on the electrodes and electrolyte).As can be seen in Fig 1(a), the half-cell potential mapping provides measurement data on the surface of the concrete that can be used to infer the condition of rebar corrosion under the concrete.This measurement data is hence an incomplete information on the hidden corrosion.In inverse analysis, the incomplete information is used to guide an optimization or a search algorithm to find a computational model that yields results that best fit the measurement data.Hence, the objective function of the optimization or the search is a distance function [8], subject to Eq. ( 1) to ( 5), ϕm is experimentally/field measured potential on the concrete surface, ϕs is simulated potential on the concrete surface.In this study, Bayesian optimization is used as the optimization method.

Bayesian Optimization
Bayesian optimization is a model-based, black-box optimization method.It searches for the optima by sampling the objective function.To sample the objective function, it builds a model of the entire function that it is optimizing, that is a surrogate model.This model includes both the current estimate of that function and the uncertainty around that estimate.It then chooses the next point by computing a posterior distribution of the objective function using the likelihood of the data already acquired and a prior on the type of function [9].The surrogate model that is directly compatible to Bayes theorem is the Gaussian Process (GP).To model the objective function, f(x), using GP, a number of test points, xi, were selected over the domain of f(x).The objective function, f(x), as a GP is then, where m(x) is mean function, and k(xi, xj) is covariance function.Once a surrogate model has been built, the next task is to select the best candidate solutions through an acquisition function.The best solution is selected through an acquisition function or infill criterion.A common acquisition function is Probability of Improvement and (PI, Eq. 8).

− (8)
For the case of topological optimization in this study, the objective function is Equation ( 6) where the variable is ϕB which is obtained from the constraints of the objective function (Eq.( 7) to Eq. ( 10)).

Implementation and Case Study
A case study is presented that models a half-cell potential mapping on a block of concrete (as illustrated in As illustrated in Figure 1(a), the half-cell potential mapping is based on the number of measurement points.It is expected that more measurement points will give a more complete information and increase the effectiveness of inverse analysis.In this study, a study case is presented where the number of points of half-cell potential measurement were varied from 21 points to 2 points (Figure 3).As in Figure 3, there are 5 cases in total.The interval between points is the width of the concrete block (2 m) divided by the number of points to get uniform interval along the width.For example, the interval for the case of 10 points of measurement is 2/10 m.Consequently, the number of ϕm in Eq. ( 6) were varied following the number of in Figure 3.The results show a somewhat non-linear relationship between the amount of information (the number of measurement points) with effectiveness of the inverse analysis (Fig 5).However, a hard limit of inverse analysis effectiveness can be determined to be at 20 measurement points for this case, since this is the number of measurement points (information) required to recover the exact corrosion profile.Below this number, the inverse analysis performs differently.
Higher amount of information (15 points) does not guarantee that the inverse analysis successfully recovers the corrosion profile.However, lower amount of information (2 and 10 points) was shown to be good enough for the inverse analysis to recover the corrosion profile.The good performance of the inverse analysis with low amount of information (2 points) can be due to the location of the measurement points.In this case, the performance of inverse analysis is also influenced by choosing the right location of half-cell measurement.On the other hand, the bad performance with higher amount of information (15 points) may be due to a fluke.Further study is needed to determine whether the truth of the latter two prepositions.

Conclusion
Inverse analysis can be used for detection of hidden corrosion in reinforced concrete.Inverse analysis uses incomplete information of reinforced concrete to infer a corrosion profile inside the reinforced concrete.The incomplete information is in the form of half-cell potential measurement.In this study, the limit to the number of half-cell potential measurement needed to perform inverse analysis is examined.A simulated case study of half-cell potential measurement with known corrosion profile (or the "source corrosion profile") is presented.The number of half-cell potential measurement is varied to It is expected that more information (more points) would result in a more effective inverse analysis.Inverse analysis using 20 measurement points is shown to be able to recover the source corrosion profile.However, inverse analysis that uses 15 points performs worst than the one using lower number of points (1 and 10 points).Further study is needed to determine if this is due to a fluke or the choice of measurement point's location.

Figure 1 .
Figure 1.(a) Schematic of corrosion mechanism in reinforced concrete and half-cell potential mapping for its assessment, (b) PDE modelling of reinforced concrete corrosion.
Fig 1(a)) with 2 m in width and 1 m in height.The hidden corrosion is located 0.6 m from the left side of the block (Fig 2(a).).The length of the hidden corrosion is 0.4 m.This corrosion profile (the location, size and number) is used as the source of a simulated half-cell potential measurement; hence it will be termed as the "source corrosion profile".The model is discretized with 302 triangular elements (Fig 2 (b)).The boundary ΓB is the simulated location of half-potential measurement.By solving the aforementioned Laplace's equation and boundary conditions, potential values on ΓB can be obtained that simulates half-cell potential measurement on the surface of the concrete (Fig 2(c)).Fig 2(c) shows 21 points of simulated half-cell potential measurement.Figure 2(d) shows the potential distribution in the concrete due to the rebar corrosion in Figure 2(a).

Figure 2 .
Figure 2. (a) A study case of a rebar corrosion, where the corrosion location and size (highlighted red) are known (i.e., the "source corrosion profile") (b) FE mesh for the simulation of the study case to obtain the simulated half-cell potential measurement (c) The potential value on the surface ΓB which is the simulated half-cell potential measurement.(d) FE simulation of the potential distribution on the body of the concrete due to rebar corrosion.

Figure 3 . 3 .
Figure 3. Various schemes of simulated half-cell measurement on a concrete block.As illustrated in Fig 1(a), half-cell potential mapping is based on measurement points.Five schemes are presented here with variations in number of measurement points.

Figure 4 .
Figure 4. Two samples of Bayesian optimization iterations (red lines) for the inverse analysis of hidden corrosion with 2 points and 20 points of half-cell measurement data.The blue lines indicate the smoothing of the curve by moving average.

Fig 5
Fig 5  shows the discovered corrosion profiles by inverse analysis using different amount of information (i.e., number of measurement points).The amount of information clearly affects the performance of inverse analysis.The inverse analysis can recover the complete corrosion profile (indicated by red rectangle) as in Fig2(a) by using 20 measurement points.Hence, this can be seen as the limit of incomplete information for the inverse analysis of this hidden corrosion case.However, when the measurement points are reduced, the effectiveness of the inverse analysis varied.It can clearly be seen that using 15 points and only 4 points, the inverse analysis completely failed to recover the corrosion profile (indicated by red rectangle).Instead of a single corrosion near the middle of rebar, the inverse analysis found multiple corrosions situated in the opposite side of the rebar (Fig5).Curiously, the inverse analysis carried out with 2 and 10 measurement points were capable of recovering corrosion profiles that are almost similar to the source corrosion profile (Fig2(a)).It is as if the inverse analysis found the almost-correct corrosion profile.To quantify the effectiveness of the inverse analysis, a similarity measure is calculated between the source corrosion profile (Fig2(a)) and each of the recovered corrosion profile (Fig5).Jaccard dissimilarity is used to accomplish this[10].Fig6shows Jaccard Dissimilarity index for each inverse analysis scenario with the source corrosion profile (Fig2(a)).It can be seen that the inverse analysis using 20 points has 0 dissimilarity, which means that the recovered corrosion profile is exactly similar to the source corrosion profile in Fig2(a).The inverse analyses using 4 and 15 measurement points have the highest dissimilarity indices due to how far the recovered corrosion profile are from the source corrosion profile.On the other hand, the inverse analyses carried out with 2 and 10 measurement points have lower dissimilarity indices, which points to more similar corrosion profiles to the source corrosion profile (which are illustrated inFig 5).The results show a somewhat non-linear relationship between the amount of information (the number of measurement points) with effectiveness of the inverse analysis (Fig5).However, a hard limit of inverse analysis effectiveness can be determined to be at 20 measurement points for this case, since this is the number of measurement points (information) required to recover the exact corrosion profile.Below this number, the inverse analysis performs differently.

Figure 5 .
Figure 5. Two samples of Bayesian optimization iterations (red lines) for the inverse analysis of hidden corrosion with 2 points and 20 points of half-cell measurement data.The blue lines indicate the smoothing of the curve by moving average.

Figure 6 .
Figure 6.Dissimilarity index between the source corrosion profile (Fig 2(a)) and the recovered corrosion profile inferred by the inverse analysis using different amount of incomplete information (number of measurement points).