A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.