# Non-Deterministic Methods and Surrogates in the Design of Rockfill Dams
## 4 The Surrogate-Assisted Non-Deterministic Algorithm
### Surrogate Model of Rockfill Dams  
The execution of computer codes is time-consuming, hence identifying the inputs that have the potential to influence results the most is important. The design of experiments attempts to address this aspect by providing a sampling plan for relevant inputs in the design space. This phase forms the first stage of the surrogate modeling process, as shown in Figure 8. This work uses the open-source library OpenTURNS to perform sampling. Special techniques are considered to ensure that the input points are spread evenly throughout the design space without sacrificing efficiency, including the Monte Carlo (MC) method, the _Hammersley_, _Halton_, _Faure_, and _Sobol_ sequences (family of low-discrepancy sequences), and _Latin hypercube_ sampling (LHS) [27]. Over the years, MC methods have become a standard approach for computer-based simulations, and they have been applied successfully in a wide range of problems. However, such techniques are based on random sampling, which may lead to the clustering of sample (input) points, resulting in extraneous and expensive computational runs. Clustering occurs because the points generated randomly lack uniformity and do not always occupy the spaces between already-sampled points. Quasi Monte Carlo (QMC) techniques, also known as low-discrepancy sequences (LDSs) and LHS, are better equipped to generate uniform point distribution. The variance reduction LHS method has been effective in numerous examples, specifically involving lower dimensions (n < 20). Depending on the problem, modified designs of LHS, using optimal space-filling criteria such as the minimax, maxmin approach and the minimum spanning tree strategy, may show some improvement in LHS efficiency. QMC involves deterministic sequences with no random components; the points are generated in a manner that rigorously imposes the concept of uniform coverage of the sampling space. Among the sequences in this category, Sobol LDSs are the most effective, and this work considers them to generate the input parameters for zone P, as shown in Table 1. In a relatively low dimension, studies have demonstrated that the QMC method displays lower discrepancy and higher efficiency when compared with the MC and standard LHS methods [28]. Our investigations in [29] highlight the standard deviation patterns involving different sample sizes using Sobol and LHS for the Romaine-2 problem. The table below presents the different parameter bounds for zone P. Zones O and N have the constant values as specified; for more details refer to [6;10]. These estimates provided a good approximation of the dam's behavior. In order to create the surrogate model, 300 sample sets (in Zone P) were generated, where an individual set is comprised of five parameters, each having a value that lies within the specified bounds. The study in [29] indicated that 300 samples are adequate to provide a good representation of the dam's response (a portion of this sample is provided in Appendix A). Every sample for zone P, along with the parameters for zones O and N, were then used to simulate and solve the Romaine-2 problem in Plaxis, and the corresponding displacements along the coordinates of INV01 were recorded. Figure 9 provides the surface plots of the simulated displacement values displaying the troughs and crescents. The top, middle, and bottom sections are situated at altitudes of 242 m, 190 m, and 144 m, respectively. For additional details, refer to Section 2 and Figure 1.  
Figure 8: Steps involved in generating a surrogate model of a rockfill dam. Thirty two evenly distributed locations along the inclinometer were considered for the analysis. A set of five parameters was used to generate the 300 input points. These inputs were used to create the regression models for the 32 locations using the Sckit-learn package and the actual displacement values at these locations.