# Detection of Cracks in Arch Dams Using Monitoring Data and Numerical Models
## 2 Methods  
### General Approach  
The proposed method for crack detection (Figure 1) applies to any specific dam with known geometry, material properties, and records of monitoring devices and reservoir level (current and from a previous year). For real cases, the first phase of the methodology is performed before analyzing current dam records and involves four steps. Step 1: Select calibration reservoir levels and simulate, for each one, the displacements at the monitoring points. Step 2: Locate the area where cracks are most likely to appear based on tensile stresses, determine (by expert judgment) the most probable crack shape, and select calibration cracks. Step 3: Simulate, for each crack and reservoir level, the displacements at the monitoring points and calculate their difference against the displacement without any crack. Step 4: For each calibration crack and each monitoring device, determine an equation indicating the incremental value for any given reservoir level, based on the interpolation of the simulated levels. Steps 2 and 3 may be repeated as many times as desired in order to have a good collection of possible cases from which to interpolate. This whole process, which requires computation resources, is only performed once and may be used throughout the life of the dam.  
The second phase takes place when data from a specific time of the dam is to be analyzed. This phase consists of three more steps. Step 5: Gather the current reservoir level and displacement data and calculate increments in relation to the dam without cracks, assumed as the same reservoir level in the same month of the previous year. Step 6: Estimate, in accordance with the equations of step 4, for each calibration crack, the displacement increments under the actual reservoir level by interpolating the incrementsof the representative reservoir levels. Step 7: Under the hypothesis that the dam was not cracked, calculate the error of the real data, then re-calculate, changing the hypothesis to each calibration crack. Finally, the minimum error is interpolated to estimate the real crack dimensions.  
An alternative methodology will be analyzed under the variation that step 4 is omitted and that in step 6, new FEM simulations are performed for each calibration crack with the current reservoir level. This methodology requires more calculation time each time an actual situation is to be monitored; therefore, it is not suitable for continuous monitoring systems. The article focuses on whether these methodologies are valid for detecting cracks or not, compared with the precision of the monitoring devices, rather than on how accurate these are simulated.  
Throughout this paper, a numerical experimentation campaign on a case study was performed to test these methodologies for the detection of cracks in arch dams. It should be clarified that it was decided to use three calibration reservoir levels, although the method would be valid for any number of levels. To verify that the methodologies work properly,  
Figure 1: Proposed method for crack detection.  
nine calibration cracks and four validation cracks were simulated, allowing the number of cases in which the identification works correctly to be tested.  
To calculate errors, the results were contrasted using two measures. The first was a plain Mean Absolute Error (_MAE_) while the second was the Root Mean Squared Error (_RMSE_). _RMSE_ was included because it increases the importance of large errors and reduces the importance of small errors, which may be due to measurement inaccuracy. Relative error measures were not used in this study because, at points where the displacement is quite low, the measurement uncertainty can lead to exceedingly high relative errors.  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}-y_{i}\right)^{2}} \tag{1}\]  
(1) Here, \(x\)--calculated value; \(y\)--expected value; and \(N\)--number of monitoring points.