# Non-Deterministic Methods and Surrogates in the Design of Rockfill Dams
## 1 Introduction  
The design of Rockfill dams presents significant challenges to geotechnical engineers because of the uncertainties and the complex structural and material characteristics involved [1]. Computational methods have played an important role in addressing the associated difficulties. These methods have been successful in providing better dam designs that are more reliable and have helped in reducing the cost and time of dam construction. Such approaches usually involve the development of numerical models, and the application of the finite element method (FEM) in this discipline has become the norm. This successful use of the FEM could be attributed to its ability to provide a high degree of accuracy and to effectively deal with complex geometries and boundary conditions as well as material (rock/soil) nonlinearities [2, 3]. However, the formulation of numerical models involves the approximation and simplification of the physical problem. In addition, the lack of information pertaining to material properties, external factors such as weather, andshortcomings in the applied constitutive law limits such models in the precise representation of actual dam behavior. In such a scenario, inverse analysis procedures can help to estimate associated parameters that are reliable and thereby improve the design of the dam under consideration. Inverse analysis in essence is an optimization process. The approach involves two major components: an optimization algorithm that attempts to minimize an error/objective function, and a numerical model/solver. Instrumentation data, which may include the deformations, stresses, seepage, and other performance measurements of a dam, are used to define the error function. The optimization algorithm, together with the numerical solver, searches for optimum parameter values in order to minimize the objective function. Earlier, methods such as the conjugate gradient, steepest descent, and trust region were commonly used for these problems as optimization/search approaches [4]. However, soil and rocks present highly nonlinear behavior, and such methods are not always adequate to deal with them. Those methods may lead to the solution being trapped in local optima, or even fail to converge. To avoid such scenarios, non-deterministic techniques, also known as stochastic methods, are preferred [5]. Non-deterministic approaches are robust in nature and can avoid the pitfalls of non-convergence and local optima. The advent of powerful computer hardware and sophisticated software have mitigated several of the bottlenecks associated with non-deterministic methods, and they now find widespread use. A major drawback of using such an algorithm in an inverse analysis is the large number of function evaluations required, as this causes multiple calls to the FEM-based numerical solver, which significantly increases the computational cost. An ingenious approach to address this issue involves the use of regression models also known as Surrogate models, which are used as a \"lookup table\" in place of the actual numerical model. This paper performs an inverse analysis for the real-world scenario of the Romain-2 rockfill dam (recently constructed in the province of Quebec, Canada) [6]. Five soil parameters: shear modulus (G), Poisson coefficient (v), friction angle (\(\varphi\)), cohesion (c), and specific weight (\(\gamma\)), are identified using the displacement values recorded near the core of the dam (from a vertical inclinometer). An appropriate error function between the measured and simulated values is prescribed to calculate these five parameters. A framework involving a surrogate-assisted non-deterministic approach for the optimization/search segment coupled with the commercially available PLAXIS package as the solver is proposed [7]. The surrogate-aided non-deterministic strategy is implemented using open-source tools such as Python, OpenTURNS [8], Scikit-learn, Pandas, NumPy, and Matplotlib. In the scientific community, there are numerous variants of non-deterministic approaches.