# Non-Deterministic Methods and Surrogates in the Design of Rockfill Dams
## 2 Problem Description  
Figure 1 depicts a two-dimensional cross section of the Romain-2 dam using Plaxis. The height of the dam is 112 m, and it has an asphalt core with a grouted rock foundation. Crushed stones that act as supports with a maximum size of 80 mm surround the core. Next to the support region lies the transition zone (N) composed of crushed stones that have a maximum size of 200 m. The internal shell zone (O) is comprised of particles with a maximum size of 0.6 m, and the external shell zone (P) is made of materials that reach a size of 1.2 m [6]. This investigation only considers the variations that occur in zone P, the section that covers the maximum portion of the Romain-2 dam. Data from the vertical  
inclinometer INV01 located upstream (4.6 m to the left of the central axis) was used to construct the error function. The INV01 measurements used in this study represent the displacements during construction without the impoundment effects. Constitutive laws relate external factors to the responses of a rockfill material governed by its internal constitution. Fundamentally, they describe the stress-strain characteristics of the soil. Mohr-Coulomb, Duncan-Chang, and Hardening soil are some of the main constitutive models applied in rockfill scenarios. The Mohr-Coulomb approach in general considers seven soil properties, whereas the Duncan-Chang and Hardening soil models involve 10 and 12 parameters, respectively. Numerous studies indicate that the Hardening soil method can accurately represent a wider set of geotechnical problems compared to the other two [9]. However, this study considers the Mohr-Coulomb approach because of its small number of required parameters. Nonetheless, it is possible to achieve a satisfactory level of accuracy with the Mohr-Coulomb with which to model the present scenario, and thus avoid the use of more complex methods [10]. The differences between the measured displacement values and those predicted by the numerical model in Plaxis are considered as errors. A simple error/objective function that incorporates this idea could be defined using the least squares method, and in the matrix form it is given by  
\[\mathrm{E_{obj}}=\left[\mathbf{x^{meas}}-\mathbf{x^{calc}}\right]^{\mathrm{ T}}\left[\mathbf{x^{meas}}-\mathbf{x^{calc}}\right] \tag{1}\]  
where \(\mathbf{x^{meas}}\) is a measured vector composed of measurements performed at different locations of the dam, and \(\mathbf{x^{calc}}\) is a calculated vector that contains simulated displacement values at locations identical to those of the measured ones. An alternate generalized version of the above equation that considers an additional weighted term is given by  
\[\mathrm{E_{obj}}=\left[\mathbf{x^{meas}}-\mathbf{x^{calc}}\right]^{\mathrm{ T}}\mathbf{W}\left[\mathbf{x^{meas}}-\mathbf{x^{calc}}\right] \tag{2}\]  
Various strategies could be applied to define the weighted term \(\mathbf{W}\) (diagonal matrix) in Equation (2), depending upon the type of error. The effects due to a fault in the instrument or because of human factors, such as an error introduced by the procedure applied for extracting measurements, are two examples of error types. These aspects could be included in the function through \(\mathbf{W}\). In order to estimate such discrepancies between the measured and the simulated values, the maximum likelihood approach is a preferred choice [11]. Figure 1: 2D cross-section model of the Romaine-2 dam in Plaxis. Different zones of the dam are highlighted by letters. The vertical inclinometer INV1 is found in section PM560. The following equation describes the maximum likelihood method, which has a structure similar to that of Equation (2).