## 3 Bandgap Algorithm of the PMPIC by Extended Plane Wave Expansion Method
### Bandgap Structures of the PMPIC by an Extended Plane Wave Expansion Method
Thus, the optimization objective function can be defined as  
\[B(\mathbf{Q})=\underset{\mathbf{Q}}{\text{min}}\omega_{2}(\mathbf{Q},\mathbf{k} )-\underset{\mathbf{Q}}{\text{max}}\omega_{1}(\mathbf{Q},\mathbf{k}) \tag{28}\]  
where \(\omega_{1},\omega_{2}\) denote the first-order and second-order dispersion curves calculated by Equation (26), and \(\mathbf{Q}\) is the \(M\times M\) binary matrix representing the topology distribution of the piezoelectric patch. Then, the optimization variables are the elements of \(\mathbf{Q}\), which are defined as  
\[Q_{ij}=\left\{ \begin{array}{ll}0,\text{ }\text{\emph{is the host structure}}\\ 1,\text{ }\text{\emph{is the piezoelectric material}}\end{array} \right.,\text{ }i,j=1,\ldots,M \tag{29}\]  
Different from previous studies on vibration isolations, all elements of \"1\" digits in \(\mathbf{Q}\) should be connected as a whole in order to ensure an integrated piezoelectric patch in this paper. Otherwise, Equation (9) does not hold again. By coincidence, this task can be easily overcome by the algorithm of finding the number of islands (NIs) in the {0, 1} matrix (\(\mathbf{Q}\)). At the same time, the fitness of each \(\mathbf{Q}\) is defined as the width of the first-order bandgap (\(B\)). (2) Initialization: Initialization parameters include geometric and material properties of the thin plate and the piezoelectric patch, the population size, the iterative times, the selection rate, the crossover probability, the mutation probability, and \(M\). Here, it should be noted that each individual in the population must be selected to make its NIs be equal to 1. (3) Selection: For each individual in the population, the first-order and second-order dispersion curves are simulated by Equation (26), and then the first-order band gap is calculated to define its fitness. If the first-order bandgap does not exist, a small number is assigned as its fitness to keep this individual. Based on the roulette wheel selection algorithm, the number (e.g., the population size \(\times\) the selection rate) of individuals in the initialization parent population is selected to construct a new mating pool. (4) Encoding and crossover: Firstly, we choose randomly two individuals in the mating pool as the parents and transform them into a \(M^{2}\)-dimension vector. Then, the selected two individuals are mated to create two offspring for replacing their parents according to a crossover probability (\(P_{cr}\)). Similarly, each offspring will be checked whether its NIs are unequal to 1 or not. If not, the crossover operation needs to be carried out again. In this way, an offspring population will be created. (5) Mutation and decoding: Each individual in the offspring population is mutated according to the mutation probability \(P_{mu}\). The mutation position is chosen randomly, where the element value switches to its opposite. Additionally, the mutated individual will be checked whether its NIs are equal to 1 or not. If not, the mutation operation needs to be carried out again. In this way, a mutated offspring population will be created. Next, each individual in the mutated offspring population is rearranged as a \(M\times M\) binary matrix. The resulted offspring population will be merged with the parent population to create the new parent population. (6) Termination: Repeating steps (3) to (5) until the iterative times are reached. Then, we can obtain the optimal first-order bandgap width and the corresponding optimal binary matrix. Based on the binary matrix, the optimal topology of the unit cell is realized. In summary, the binary-like topology optimization flowchart using the extended GA algorithm is plotted in Figure 4.  
software and here the inductor is chosen as \(L\)= 1 H. Then, the first and second dispersion curves are calculated by the proposed algorithm and the FEM simulation, respectively.