Multi-layer designs and composite Gaussian process models with engineering applications

The modern era witnesses the prosperity of computer experiments, which play a critical role in many fields of technological development where the traditional physical experiments are infeasible or unaffordable to conduct. By developing sophisticated computer simulators, people are able to evaluate, optimize and test complex engineering systems even before building expensive prototypes. Since the computer experiments are usually time-consuming to run, surrogate models are often fitted to approximate these computationally expensive simulations. Because the fitted surrogate models are much faster to run, they can be readily used to provide instant predictions and facilitate the analysis of the underlying system.;In building the surrogate model for computer experiments, there are two important research topics. The first one is how to efficiently select a set of input values to run the computer simulation for a finite number of times, and this is called the design of computer experiments. After we obtain the simulation outputs, the second question is how to model these data in order to accurately approximate the unknown response surface generated by the simulator.;This thesis consists of three chapters, covering topics in both the design and modeling aspects of computer experiments as well as their engineering applications. The first chapter systematically develops a new class of space-filling designs for computer experiments, and the second chapter proposes a novel modeling approach for approximating computationally expensive functions that are not second-order stationary. The third chapter is devoted to a two-stage sequential strategy which integrates analytical models with finite element simulations for a micromachining process.;In computer experiments, space-filling designs such as Latin hypercube designs (LHDs) are widely used. However, finding an optimal LHD with good space-filling properties is computationally cumbersome. On the other hand, the well-established factorial designs in physical experiments are unsuitable for computer experiments owing to the redundancy of design points when projected onto a subset of factor space. In the first chapter, we present a new class of space-filling designs developed by splitting two-level factorial designs into multiple layers. The method takes advantages of many available results in factorial design theory and therefore, the proposed Multi- layer designs (MLDs) are easy to generate. Moreover, our numerical study shows that MLDs can have better space-filling properties than optimal LHDs.;In the second chapter, a new type of non-stationary Gaussian process model is developed for approximating computationally expensive functions. The new model is a composite of two Gaussian processes, where the first one captures the smooth global trend and the second one models local details. The new predictor also incorporates a flexible variance model, which makes it more capable of approximating surfaces with varying volatility. Compared to the commonly used stationary Gaussian process model, the new predictor is numerically more stable and can more accurately approximate complex surfaces when the experimental design is sparse. In addition, the new model can also improve the prediction intervals by quantifying the change of local variability associated with the response. Advantages of the new predictor are demonstrated using several examples.;Chapter three considers the problem of integrating analytical models with finite element simulations. We show that computationally cheap analytical models can be used to perform a sensitivity analysis which can reveal critical information about the underlying system prior to conducting the computationally intensive simulation study. We propose a two-stage sequential strategy, which can efficiently absorb the prior information from the sensitivity analysis and assign a customized number of levels for each input variable in the finite element simulations. The method is also broadly applicable for integrating other types of models having different levels of accuracy and speed. A case study for developing force metamodels in micromachining is presented to illustrate the effectiveness of the proposed method.