## 1 Introduction
first investigated one kind of piezoelectric metamaterial plate (PMP) for vibration energy harvesting and analyzed the effects of geometric and material parameters on vibration bandgaps by finite element simulations [19]. Later, [PERSON] et al. continued to build an elastic-electro-mechanical model of the PMP with interface circuits (PMPICs) [20]. [PERSON] et al. proposed an acoustic metamaterial that had dual functionality of sound insulation and energy harvesting simultaneously [21]. Existing studies have shown that PMPs provide a promising way of harvesting structural vibrations. Bandgap characteristics of periodic structures are highly dependent on the topology of the unit cell. In recent years, many studies have been conducted to design and optimize the unit cell in order to obtain targeted bandgaps. In particular, topology optimization has been identified as a powerful computational tool for the design of bandgap material [22]. [PERSON] et al. further improved the above topology optimization method by considering both scalar and coupled problems [23]. A 2D phononic crystal with an arbitrarily asymmetric lattice was also obtained by combining the genetic algorithm with the finite element model [24]. [PERSON] et al. maximized the bandgap width of a phononic crystal by using a GA combined with the fast plane wave expansion method [25]. [PERSON] et al. proposed an improved fast plane wave expansion method for topology optimization of phononic crystals [26]. In 1993, [PERSON] and [PERSON] first attempted to carry out topology optimization of continuum structures using binary {0, 1} variables [27]. In 2018, [PERSON] and [PERSON] proposed the topology optimization of binary structures (TOBS) method, where the material layout is defined by a set of binary {0, 1} design variables [28]. By now, the TOBS method has attracted much attention due to the advantage of suiting well separate physics domains. The current challenge for structural topology optimization is to develop reliable techniques to account for different physics interactions. In 2020, [PERSON] and [PERSON] further extended the TOBS method to design topologies by considering fluid-structure interaction loads [29], pressure, and thermal loads simultaneously [30]. A PMPIC is composed of a piezoelectric metamaterial plate and an interface circuit. The aim of PMPIC design is to obtain proper bandgaps to meet the needs of low frequency and broadband. Bandgaps are strongly related to elastic wave propagation and the interface circuit; therefore, elastic-electro-mechanical coupling effects should be considered in topology optimization. To the best of the author's knowledge, this is the first work to optimize the topology of a PMPIC by considering elastic-electro-mechanical coupling behaviors. In order to optimize the topology of the PMPIC, a binary topology optimization process is first proposed based on an extended plane wave expansion (PWE) method in this paper. The remainder of this paper is organized as follows. In Section 2, the problem of topology optimization of a PMPIC is outlined, and a binary-like topology optimization scheme is presented. In Section 3, a unified motion equation is derived for the elastic plate and the piezoelectric patch, and an electromechanical coupling model is built. Then, the bandgap character of the PMPIC is modeled based on an extended plane wave expansion method and the Bloch theorem is used to solve the above model. In Section 4, an extended genetic algorithm is applied for the topology optimization of the PMPIC. Numerical and finite element simulations are carried out in Section 5. Finally, the conclusions are summarized in Section 6.