I want you to act as an academic journal editor Please rephrasepolish the paragraph from an academic angle based on the writting style of the nature journalThe localization of elastic wave at defect i

The use of localized elastic wave defects in phononic crystals (PnCs) has been a promising strategy for designing efficient piezoelectric energy harvesting (PEH) devices. However, previous studies based on classical linear elasticity theory have failed to accurately predict the structural behavior at the micron scale. In this work, we present a novel approach that incorporates a modified couple stress theory (MCST) and a mixed finite element method to simulate the microstructure-dependent size effect in PnC-based PEH devices. Using the physics builder in COMSOL® Multiphysics, we demonstrate the size dependence of our model through three sets of simulations at different sizes. Our numerical results reveal that the inclusion of size effect in the model leads to an increase in the frequencies of the bandgap and defect bands as the model size decreases, in contrast to the predictions of classical theory. Moreover, the size reduction induces changes in the internal stiffness ratio, which in turn affects the displacement amplitude, output voltage and output electrical power of the PEH device. Our findings indicate that size effects are inevitable in micron-level models, and can be leveraged to design small-scale PnC-based PEH devices that offer enhanced energy collection