Quasi-weak and weak formulation of stochastic finite elements on static and dynamic problems—a unifying framework

Abstract

To model uncertainty of spatial and/or temporal variations widely present in synthetic and natural media, a variety of displacement-based stochastic finite element methods (SFEMs) have been formulated using the standard displacement-based finite elements. In this paper, by distinguishing a quasi-weak form from a weak form in both real and random space, a unifying framework of variational formulation is presented covering both the displacement-based SFEMs and the recently proposed Green-function-based (GFB) SFEM. The study shows that Monte Carlo, perturbation, and weighted integral SFEMs correspond to the quasi-weak form, while the weak form results in spectral SFEM, pseudo-spectral SFEM, and GFB-SFEM. Within the unifying framework, dynamic problems are further addressed especially to demonstrate the unique feature of GFB-SFEM on problems with inputs characterized as random fields or random processes.