## 1 Introduction  
Fossil fuel energy reserves are declining and will fade out of human history in the near future. Consequently, new sustainable energy sources are becoming more and more popular for researchers. State-of-the-art, small-scale electronic devices have promoted the need for portable, small, efficient, self-powered energy generators. Such small-scale electronic devices are hard to charge considering their size and commonly far-away location. Therefore, the self-powering property is an intriguing solution to this [1, 2]. Piezoelectric material can exactly fulfill these requirements. By capturing the pressure or movement of the natural world, a piezoelectric harvester could generate electric power. Moreover, it could capture outside disturbance, which could also serve as a sensor. Therefore, the design of the piezoelectric harvester is the key question explored here. The designs of piezoelectric actuators and energy harvesters are complicated [3, 4, 5, 6]. In recent years, topological optimization and other systematic design methods have been successfully applied to solve the optimization problem, and the optimal design has been obtained. By combining the homogenization method with the topology optimization method, the unit cell topology could be optimized. Usingthe Sequential Linear Programming (SLP) method, the material topology is realized [7; 8]. The design variables describe the allocation of piezoelectric material and polarization degrees at each finite element unit, and this can be applied to the design of sensors, actuators, and energy harvesters [9; 10]. The geometric design freedom and multi-material parallelism of 3D printing make it possible to manufacture topology-optimized flexible mechanisms. [PERSON] et al. researched a 2D piezoelectric topology problem, and discovered the best possible layout to overcome the charge cancellation problem [11]. [PERSON] et al. researched the topology optimization problem surrounding hydrophones. Using the level set method, the hydrostatic charge efficiency is enhanced [12]. [PERSON] et al. propose the bi-objective optimization for the installation of pumps operating as turbines (PATs) in systems of transmission mains [13]. [PERSON] et al. [14] used topology optimization to design piezoelectric energy harvesting systems based on multilayer plates and shells. They employed the finite element method in order to build a coupled electromechanics model of the piezoelectric harvesting structure and a lumped parameter. This was done to develop overarching design principles for piezoelectric energy harvesting devices. [PERSON] et al. [15] studied the topology optimization of the design of piezoelectric plate and shell actuators. The aim of this was to achieve maximum output displacement in a given direction at a given point of the structure, whilst simultaneously minimizing the structural compliance. [PERSON] et al. [16] designed piezoelectric actuators based on the flexible structure actuated by piezoceramics. A simultaneous search was conducted in order to discover the position of the piezoceramic in the design domain and the optimal rotation angles of piezoceramic material axes that maximize output displacements or output forces. [PERSON] et al. [17] et al. developed the design sensitivity analysis and topology design optimization methods for piezoelectric resonators, and computed the necessary design gradients for eigenvalue problems. [PERSON] et al. [18] studied the laminated piezo composite shell structures using topology optimization, with the polarization distributions of the piezoelectric material and the fiber angle of the composite orthotropic layers as the objective. [PERSON] et al. [19] introduced a new method to optimize structure and material simultaneously, by linearizing and formulating in order to solve the macro and microscale design problems, and to gain an overall optimum solution. [PERSON] et al. proved that a topology problem could be reformed into an image segmentation task, and that this could then be solved using a deep neural network [20].