Investigation of the potential of using surrogate models in the gear design process

Abstract

State of the Art Surrogate models, also known as response surface models or metamodels, are approximation models, which are based on mathematical functions (Ref.1). In engineering, surrogate models are used to correlate the input and output variables of experiments and simulations (Refs. 2–10). This is especially true for very time-consuming, costly or high number of experiments/ simulations. In this case, the surrogate model can be evaluated much faster in comparison to the experiment or complex simulation. This is most important for design space exploration or optimization where a high number of experiments of simulations is necessary (Ref. 5). In order to reduce the time effort, the extensive simulation is only performed for a reduced number of parameter sets. These initial parameter sets are defined by means of methods of design of experiment (DOE), e.g., fullfactorial sampling or latin hypercube sampling (Ref. 11). For computational problems a latin hypercube sampling or the Monte-Carlo approach (random sampling) is often used to identify the initial parameter sets. Once the initial parameter sets are identified, the simulation is performed at these given points. The results of the simulation are used to fit a surrogate model to the given input variables in order to approximate the system behavior of the engineering system. Possible approximation types for surrogate models are shown in Figure 1. The most common modeling types are models based on radial basis functions (RBF), kriging models, also known as Gaussian process models, and models based on multivariate adaptive regression splines (MARS). RBFs are functions whose value only depends on the Euclidian distance from the origin (Ref. 12). An approximation model consists of a number of different radial basis functions, which are weighted accordingly. The weights of each basis functions are tuned in order to improve the quality of the approximation for the given number of data points. In the example in Figure 1 the function f(x) = 1 + sin(x2) was evaluated at six test data points and approximated by the usage of an RBF surrogate model consisting of Gaussian basis functions, as a type of RBF. The approximation follows the trend of the sine function but is not able to predict any of the test data points in high accordance. Kriging or Gaussian process models originate from geosciences and are usually used to predict the location of certain commodities like oil or gold for which only a finite number of boreholes exists (Ref. 13). The Gaussian process consists of two parts, one global and one local part. The global part can be