  For decades, uncertainty quantification techniques based on the spectral
approach have been demonstrated to be computationally more efficient than the
Monte Carlo method for a wide variety of problems, particularly when the
dimensionality of the probability space is relatively low. The time-dependent
generalized polynomial chaos (TD-gPC) is one such technique that uses an
evolving orthogonal basis to better represent the stochastic part of the
solution space in time. In this paper, we present a new numerical method that
uses the concept of 'enriched stochastic flow maps' to track the evolution of
the stochastic part of the solution space in time. The computational cost of
this proposed flow-driven stochastic chaos (FSC) method is an order of
magnitude lower than TD-gPC for comparable solution accuracy. This gain in
computational cost is realized because, unlike most existing methods, the
number of basis vectors required to track the stochastic part of the solution
space, and consequently the computational cost associated with the solution of
the resulting system of equations, does not depend upon the dimensionality of
the probability space. Four representative numerical examples are presented to
demonstrate the performance of the FSC method for long-time integration of
second-order stochastic dynamical systems in the context of stochastic dynamics
of structures.
