Efficient bounds for the Monte Carlo–Neumann solution of stochastic systems

The numerical solution of uncertainty propagation problems in mechanics can be computationally demanding. In this article, a well-known property of the Neumann series is explored in order to derive lower and upper bounds for expected value and autocorrelation of stochastic system responses. Uncertainties in system parameters are represented as parameterized stochastic processes. Monte Carlo simulation is employed to obtain samples of system response, from which lower and upper bounds of expected value and autocorrelation are computed. The proposed methodology is applied to two example problems, involving beam bending and axial thermo-elasticity. It is shown that accurate and efficient bounds can be obtained, for a proper choice of operator norm, with as few as one or two terms in the Neumann expansion. The Monte Carlo–Neumann bounding scheme proposed herein is shown to be an efficient alternative for the solution of uncertainty propagation problems in mechanics.