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Abstract A methodology based on the Wiener path integral (WPI) technique is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the satisfactory performance of the methodology. 

A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this a priori marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the ‘curse of dimensionality’ in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.