## 1 Introduction  
Piezoelectric energy harvesters have attracted substantial attention in the last decades. Energy harvesters can generate sustainable power and, hence, are promising as an alternative to batteries, which have a limited life span, while requiring costly maintenance [1, 2]. Piezoelectric energy harvesters, therefore, can be an excellent power source for self-powered devices, such as MEMS (microelectromechanical systems) and WSNs (wireless sensor networks). Among the different vibration-based energy resources, wind flow has been suggested as one of the most reliable energy resources, since the wind-induced vibrations of bluff bodies can have considerable amplitudes [3, 4]. Consequently, piezoelectric energy harvesters based on wind-induced vibrations of bluff bodies can produce considerable electric energy. Wind-induced vibrations commonly utilized in piezoelectric wind energy harvesters (WEHs) include vortex-induced vibration (VIV) [5, 6], galloping [7, 8, 9, 10], flutter [11, 12, 13, 14], and mixed aerodynamic phenomena [15, 16, 17]. Several comprehensive reviews on harnessing energy from wind flow have been published recently [18, 19, 20, 21, 22, 23, 24]. Various mathematical methods have been employed to promote harvesting performance [24]. For instance, [PERSON] et al. [25] developed a coupled model of galloping-based WEH. Their results showed that the electrical load resistance and the Reynolds number play an essential role in determining the level of the harvested power and the onset wind velocity of galloping. [PERSON] et al. [26] compared the lumped variable model with the distributed variable model. The effects of load resistance, wind exposure area of the bluff body, mass of the bluff body, and length of the piezoelectric sheets on the power output were investigated. [PERSON] et al. [27] solved electromechanical coupled distributed variable equations and theoretically examined the effects of electrical damping and mechanical damping on the performance of the WEH. The parameters were optimized based on mathematical methods for improving the performance of a WEH. The mathematical models indicate that the geometric shape of the bluff body is one of the critical parameters determining the wind force on the WEH and governing the performance of the WEH [28; 29]. [PERSON] et al. [30] compared the effects of bluff bodies with different sections on the performance of a galloping-based WEH. The performance of square, rectangular, triangular, and D-shaped galloping-based WEHs was evaluated through experiments and analytical models. The test results showed that the square section had the lowest onset wind velocity of 2.5 m/s and the highest wind energy power of 8.4 mW. [PERSON] et al. [31] evaluated the effects of geometric shape and size of the tip body of a transversely galloping-based WEH, experimentally and theoretically. The results showed that the optimal combinations of lateral aerodynamic force coefficients minimized the critical wind velocity for instability and increased the amplitude. [PERSON] et al. [32] investigated the effects of section geometry, resistance, and wind speed on an energy harvester. The results showed that the section geometry greatly influenced the initial galloping wind velocity. However, the efficiencies of WEHs were limited by the typical bluff body. Aerodynamic shape optimization is an effective and simple method to improve the efficiency of a WEH. [PERSON], [PERSON], and [PERSON] [7] fitted fins to various corners of a square prism to form the kinetic energy source of a galloping-based piezoelectric energy harvester. They found that attaching fins to the leading corners improved the efficiency of the energy harvester by up to 2.5 times. [PERSON] et al. [33] and [PERSON] et al. [34] investigated the WEH with two small-size rod-shaped attachments on the main circular cylinder. The harvester harnessed the wind energy beyond the defined onset wind velocity and could still work efficiently over the range of lock-in wind velocity. [PERSON] et al.