Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems

TL;DR

The paper introduces the flow-driven spectral chaos (FSC) method, a computationally efficient technique for long-time integration of second-order stochastic dynamical systems that maintains accuracy without increasing basis size with probability space dimensionality.

Contribution

The FSC method uses enriched stochastic flow maps to reduce computational cost and decouple basis size from probability space dimension, improving long-term stochastic system simulations.

Findings

FSC achieves an order of magnitude lower computational cost than TD-gPC.

FSC maintains accuracy without increasing basis vectors with probability space dimension.

Numerical examples demonstrate FSC's effectiveness in stochastic structural dynamics.

Abstract

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.