A novel model and simulation method for multivariate Gaussian fields involving nonlinear probabilistic dependencies and different variable-wise spatial variabilities

Abstract

The inherent randomness of engineering structures significantly influences the analysis of structural stochastic responses and safety assessments. It is critical to quantify the three aspects of random fields, including the randomness of individual variables, the probabilistic interdependence among multiple variables, and the spatiotemporal correlation of fields. This paper introduces a novel modeling framework for multivariate fields that accommodates both nonlinear probabilistic dependencies captured through copula, and the distinct spatial variability of individual fields described by correlation functions. Specifically, the framework defines a new analytical function, termed the bridge function, which establishes the relationship between the correlation functions of two fields governed by any copula structure. This proves the consistency of the new model, i.e., the copula function, as a between-variable constraint, allows the spatial correlation function of different variables to be freely selected, either with different correlation length or even with different shape. Further, to facilitate simulation, by the bridge function samples from multiple independent Gaussian fields can be onverted into those of multivariate fields that involve the specified vine copula dependencies and individual correlation functions. This approach addresses the challenge of simultaneously satisfying nonlinear dependencies and spatial variability in multivariate field simulations. The paper details the analytical expressions and numerical solution procedures for the bridge function, along with a comprehensive simulation method that integrates vine-copula-based conditional sampling and stochastic harmonic functions. The effectiveness of the proposed method is validated through various engineering application case studies, demonstrating its potential for accurate uncertainty quantification in complex engineering scenarios.