Multi-scale and multi-uncertainty modeling of energy harvesters under environmental vibrations

Localization and homogenization conditions for electro mechanically coupled problems are recently derived by Ref. 1. Moreover, two scale homogenization procedures were also proposed for electromechanical solids at both small deformation (Ref. 2) and finite strains (Ref. 3). However, these approaches are limited to the static response and unfortunately, most of electromechanical devices operate in a dynamic regime. Extension of the multilevel finite element techniques to dynamics leads to a huge computational cost not available in the current practise. At the same time, a lot of research focuses on development of reduced order modeling procedures for mems and nems applications. These approaches present several advantages: mainly they allow to include effects of parameter uncertainties in the design and to optimize energy harvester performances (Ref. 4 and Ref. 5). In this paper, a computational approach for multi-scale and multi-uncertainty modeling of energy harvesting devices under environmental vibration is proposed. Three different levels are considered in a bottom-up approach: micro, macro and system level scales. Transition from micro to macro quantities is based on Hill's energy principle while we show that transfer from macro to system level variables is possible if special conditions are satisfied at the macroscale. A vibration based energy harvesting device is used to assess the capability of the proposed computational procedure. Stationary and non-stationary base excitations are considered and the uncertainty levels due to intrinsic randomness are assessed in terms of displacement and output voltage responses. At the same time, since the overall state space description is based on lumped coefficients enriched with information derived by the multiscale simulations, micro and macro stress/strain and electric potential distributions are also predicted with a good accuracy.