A family of weak stochastic Newmark methods for simplified and efficient Monte Carlo simulations of oscillators

Engineers are often more concerned with the computation of statistical moments (or mathematical expectations) of the response of stochastically driven dynamical systems than with the determination of path-wise response histories. With this in perspective, weak stochastic solutions of dynamical systems, modelled as n degrees-of-freedom (DOF) mechanical oscillators and driven by additive and/or multiplicative white noise (or, filtered white noise) processes, are considered in this study. While weak stochastic solutions are simpler and quicker to compute than strong (sample path-wise) solutions, it must be emphasized that sample realizations of weak solutions have no path-wise similarity with strong solutions. However, the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is ‘close’ to that of the true response (if it exists) within a certain order of a given time step size. Computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems (oscillators) of engineering interest. To efficiently generate such weak responses, a novel class of weak stochastic Newmark methods (WSNMs), based on implicit Ito–Taylor expansions of displacement and velocity fields, is proposed. The resulting multiple stochastic integrals (MSIs) in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities. In fact, yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher-level MSIs. Estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided. Numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy, simplicity and higher computational speed of WSNMs vis-à-vis a few other popularly used stochastic integration schemes as well as the path-wise versions of the stochastic Newmark scheme