A refined Bloch mode synthesis framework for fast and accurate analysis of electroelastic metamaterials with piezoelectric resonant shunt damping

Abstract

Bloch mode synthesis (BMS) is a model order reduction method that was originally formulated to accelerate band-structure calculations of periodic elastic materials. In this paper, the BMS framework is expanded to offer a platform suitable for analysis of electroelastic metamaterials with piezoelectric resonant shunt damping. First, a transformation of coordinates formed by a set of dominant fixed-interface modes and a set of static-constraint modes is used in combination with Bloch periodicity boundary conditions to derive the classical BMS method in the context of electroelastic metamaterials. Then, a set of residual modes that is simply truncated in the classical BMS, is augmented into the reduction process, leading to an enhanced BMS method, denoted RMH-BMS.Currently available model order reduction techniques for electroelastic metamaterials adopt a reduction basis obtained from one of the limiting eigenvalue problems associated with short-circuit (SC) and open-circuit (OC) boundary conditions. In either case, the electrical circuit parameters are ignored during modal reduction. In this paper, it is shown that such approximation leads to inaccurate band-structure predictions for electroelastic metamaterials equipped with periodic array of resonant shunt circuits. To this end, two versions of the proposed BMS methods are evaluated and compared. In the first version, the reduced order model is constructed following the widely used OC basis approach. The second version, on the other hand, exploits a coupled basis approach, which requires knowledge of the electrical circuit parameters prior to the interior modal reduction step. A comprehensive error analysis is conducted to assess the computational performances of the BMS methods, and the results revealed that whether the classical BMS or the RMH-BMS is employed to construct the reduced order model, the coupled basis approach produces significantly more accurate results compared to the OC basis approach. The results presented in this work expose the limitations of implementing the OC basis approach in wave propagation analysis of electroelastic metamaterials with piezoelectric resonant shunt damping and shed light on the associated consequences in predicting the bandgap properties of such metamaterials.