Simulation of non-Gaussian, non-stationary stochastic processes with evolutionary power

Abstract

A methodology is presented to determine the Evolutionary Spectrum (ES) for non-stationary processes from a prescribed or measured non-stationary Auto-Correlation Function (ACF). The classic integral expression put forth by Priestely, providing the ACF from the ES, is not invertible in a unique way so that the ES could be determined from a given ACF. However, the benefits of an efficient inversion from ACF to ES are vast. Consider for example various problems involving simulation of non-stationary processes or non-homogeneous fields, including non-stationary seismic ground motions as well as non-homogeneous material properties such as those of functionally graded materials. In such cases, it is sometimes more convenient to estimate the ACF from measured data, rather than the ES. However, efficient simulation depends on knowing the ES. Even more important, simulation of non-Gaussian and non-stationary processes depends on this inversion, when following a spectral representation based approach. This work briefly examines the existence and uniqueness of such an inversion from the ACF to the ES under a set of special conditions and assumptions (since such an inversion is clearly not unique in the most general form). It then moves on to efficient methodologies of computing the inverse. Some established optimization techniques are explored, and a novel methodology is proposed. Its application within the framework of translation models for simulation of non-Gaussian, non-stationary processes is discussed. Numerical examples are provided demonstrating the capabilities of the methodology.