Metamodelling with the Multi-dimensional Hermite Polynomial Chaos Expansion

Reliability-based design relies on models, abstract representations of reality. Models of models are simply designated metamodels, further abstractions which are typically easier to generate and to evaluate. It has been suggested to utilise approximate response functions in order to reduce computational costs. Among the approximate methodologies developed so far the polynomial chaos expansion may be utilised for the functional representation of stochastic variability. The stochastic response surface methodology is herein presented in the form of a multi-dimensional Hermite polynomial chaos expansion proper for gaussian stochastic processes. The terms are listed for the order and the number of standard normal random variables considered truncated full and truncated full sparse forms. In this latter case a truncated full polynomial chaos expansion is supplemented by a group of sparse terms on higher order used whenever approximation difficulties are significant, namely whenever nonlinear behaviour and convergence difficulties are present.