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This paper is concerned with modeling computer experiments with multiple responses. It has been common that one may collect multiple responses from a physical or computer experiment. However, to our best knowledge, there is little work to model computer experiment with multiple responses. In this paper, we propose a modeling procedure for such computer experiment by using a multivarate kriging model, a natural extension of the ordinary kriging model. We further extend functional ANOVA of single response to multiple response and apply it for analyzing the effect of each design variable. The proposed methodology is demonstrated by an analysis of data collected from a case study concerned with the design of the engine structure to minimize the radiated noise.

Metamodeling approach has been widely used due to the high computational cost of using high-fidelity simulations in engineering design. Interpretation of metamodels for the purpose of design, especially design under uncertainty, becomes important. The computational expenses associated with metamodels and the random errors introduced by sample-based methods require the development of analytical methods, such as those for global sensitivity analysis and uncertainty propagation to facilitate a robust design process. In this work, we develop generalized analytical formulations that can provide efficient as well as accurate global sensitivity analysis and uncertainty propagation for a variety of metamodels. The benefits of our proposed techniques are demonstrated through vehicle related robust design applications.

To address the conflicting goals of minimal piston friction and minimal piston noise, a dynamic power cylinder model was developed to predict piston motion and side loads within the cylinder. This correlated model was the basis of a comprehensive analytical design of experiments (DOE) where both piston noise and piston friction were monitored. The results of the DOE were used to generate metamodels for piston friction and for piston noise. To insure design robustness, variability was introduced into the surrogate models via First Order Reliability Method (FORM). A Pareto curve using 99% probability was constructed and a piston robust to both noise and friction was selected.

There is a trend in the automotive industry to reduce the number of physical prototypes and to rely more on Computer Aided Engineering (CAE) for sizing and final design of vehicle structures. The traditional deterministic approach does not necessarily clarify the degree of variability and conservatism. With small variability in influence parameters and a design factor for final design, the approach may be over conservative resulting in weight and cost penalty. On the other hand, with large variability and the same design factor, the deterministic approach may not satisfy durability requirements. It is important to identify the variability of all factors including road loads and sensitivities of the control parameters, and to minimize their effects on durability so that fatigue life distribution meets the durability requirements.

When conducting computer experiments, engineers frequently encounter functional data for which the response data are defined over an operating cycle, time, or spatial interval. There is little work in the literature to model functional response from computer experiment. In this paper, we propose functional linear model for the mean function in the Gaussian kriging model. We further propose a two-step estimation procedure for the functional linear model using penalized spline approach. The proposed approach is applied to the design of a valvetrain system.

Sensitivity analysis plays an important role to help engineers gain knowledge of complex model behaviors and make informed decisions regarding where to spend engineering effort. In design under uncertainty, probabilistic sensitivity analysis (PSA) is performed to quantify the impact of uncertainties in random variables on the uncertainty in model outputs. One of the most challenging issues for PSA is the intensive computational demand for assessing the impact of probabilistic variations. An efficient approach to PSA is presented in this article. Our approach employs the Kolmogorov-Smirnov (KS) distance to quantify the importance of input variables. The saddlepoint approximation approach is introduced to improve the efficiency of generating cumulative distribution functions (CDFs) required for the evaluation of the KS distance.