## 2 Problem Formulation of Topology Optimization of a PMPIC
### Binary Topology Optimization of the PMPIC  
In this paper, the goal is to maximize the bandgap width so that more vibration energy can be harvested by the PMPIC. Generally speaking, a topology optimization problem always consists of distributing two material phases in a fixed design domain, and then an objective function is minimized or maximized under a number of constraints. The design domain is discretized into many small elements and material properties are assumed to be constant within each element. The design variable is the material density, which is equal to 0 or 1. If the design variable takes a zero value, we should have the first phase material in the element. Otherwise, we should have the second phase material in the element. In this problem, the physical size, shape, and connectivity of the unit cell are unknown in advance.  
For structural topology optimization, a common method is called the solid isotropic material with penalization (SIMP) approach [31]. Material properties are modeled as the relative material density. In this case, design variables are the element relative densities. However, the drawback of this approach is that no physical material exists with properties described by power-law interpolation. In order to overcome this limitation, the levelset method has been recently introduced in the field of topology optimization [32]. Level set-based topology optimization methods can directly provide clear boundaries in optimal configurations that avoid the presence of intermediate densities. The level set method has been successfully used for topology optimization of metamaterials for the sake of controlling elastic wave propagation. However, the level set method is not feasible for optimizing the PMPIC due to the following two reasons: (i) the optimization function is difficult to be defined by including the interface circuit and the piezoelectric effect and (ii) the optimal result of the level set method may cause the piezoelectric patch to many independent pieces. In this case, the result may be inaccurate due to the piezoelectric effect, and PVEH is also inconvenient to carry out.  
In this paper, a binary-like topology optimization scheme is proposed for the PMPIC, which is inspired by the idea of the SIMP approach. As shown in Figure 2, a square PMPIC unit cell is considered, where the interface circuit is not plotted. Then, the unit cell is divided into small elements and each element is marked by 0 or 1. If an element takes a zero value, we should choose thin plate material in the element. Otherwise, we should choose piezoelectric material in the element. In this way, the PMPIC unit cell can be represented by a 0-1 matrix. Then, the optimization problem is formulated to find the optimal topology of the 0-1 matrix for maximizing the bandgap width. The advantages of the above scheme include (i) compared with the SIMP approach, the above binary-like topology optimization method will not generate nonexistent materials with intermediate densities; (ii) compared with the level set approach, the above binary-like topology optimization method can easily control the connectivity of all \"1\" digits so that the piezoelectric patch forms a whole; (iii) the above binary-like topology optimization problem can be efficiently solved by common 0-1 programming algorithms or evolutionary optimization algorithms. In particular, it should be noted that each isolated host structure in the optimal result will be used as one mass, as shown in Figure 1, instead of mounting additional masses. That is to say, multiple masses may exist in the unit cell.