## 1 Introduction
They validated the numerical optimization results with experimental prototypes fabricated via laser cutting and piezoelectric lamination, showing a close match between predicted and measured outputs. This study provides a rare full-cycle validation of optimization theory, simulation, and practical deployment. In more advanced geometrical work, [PERSON] and [PERSON] [11] presented a comparative study of multiple beam shapes, including exponential, trapezoidal, and sinusoidal tapers. Their results showed that even minor profile changes can result in significant shifts in resonant frequency and strain energy distribution. This kind of shape-function tuning is critical in applications where ambient vibration characteristics are narrowband and require precise alignment with harvester dynamics. [PERSON] et al. [12] used topology optimization approaches to nanoscale piezoelectric structures, demonstrating how surface effects and scale-dependent material behavior significantly influence electromechanical coupling at micro- and nanoscales. Similarly, [PERSON] et al. [13] applied topology optimization with stress norm constraints to achieve uniform strain distribution in unimorph harvesters. Their method created hybrid geometries that offered superior performance in simulated environments, though they highlighted practical challenges related to stress concentrations at sharp corners and the fabrication of irregular contours. Electrode layout and poling optimization have also been shown to significantly influence performance. Rather than relying on uniform polarization, [PERSON] et al. [14] introduced a method to spatially vary the electrical boundary conditions and polarization fields within the piezoelectric domain, allowing the device to more efficiently convert mechanical strain into electrical output. Their results showed that this tailored configuration leads to significantly better energy harvesting performance compared to conventional electrode arrangements. [PERSON] et al. [15] and [PERSON] et al. [16] analyzed the effect of modal strain nodes on electrode placement, showing that misalignment can cause charge cancellation and loss of output. Alternative poling strategies, such as d15-mode (shear mode) operation and interdigitated electrode configurations, were examined in [17; 18]. These methods offer higher electromechanical coupling but often introduce additional complexity in fabrication and modeling. [PERSON] et al. [17] showed that shear mode operation increased power by more than 60% under the same excitation conditions as conventional d31 mode designs. Performance metrics reported in the literature vary but typically include tip deflection, average and peak strain, resonant frequency, voltage output, load-matched current, and power density. [PERSON] et al. [19] proposed arc-shaped piezoelectric elements, demonstrating experimentally that such curvature leads to a 200% increase in power density and 60% greater conversion efficiency compared to standard flat beams. In contrast, [PERSON] et al. [20] introduced a two-degree-of-freedom harvester capable of operating over a broader frequency range, with output improvements attributed to modal superposition and strain field enhancement. In the context of non-linear energy harvesting, [PERSON] and [PERSON] [21] explored optimal charging conditions in bistable PEHs under harmonic excitation, offering insights into how system dynamics and electrical loading interact. Complementarily, [PERSON] and [PERSON] [22] focused on the electrical side, optimizing power conditioning strategies for such systems and demonstrating how careful matching between mechanical response and load impedance maximizes harvested power. [PERSON] et al. [23] proposed a unified performance index termed \"effectiveness\", which incorporates both mechanical quality factors and electrical transmission efficiency. This metric provides a comprehensive means of evaluating and comparing PEH designs beyond raw output values, capturing the dynamic interaction between structure and energy transduction.