## 3 Bandgap Algorithm of the PMPIC by Extended Plane Wave Expansion Method
### Transverse Motion Equation of the PMPIC  
Based on the [PERSON] plate theory [35], transverse shear strains of a thin plate can be neglected. Then, its middle layer of the thin plate is looked at as the middle surface. In this case, the relationship between the strains (\(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\tau_{xy}\)) and transverse displacement (\(w=w(x,y,t)\)) can be represented as Equation (1). \[\left\{ \begin{array}{l}\varepsilon_{x}=-2\frac{\partial^{2}w}{\partial x^{2 }}\\ \varepsilon_{y}=-2\frac{\partial^{2}w}{\partial y^{2}}\\ \gamma_{xy}=-2z\frac{\partial^{2}w}{\partial x\partial y}\end{array} \right. \tag{1}\]  
According to [PERSON]'s method, the transverse vibration equation of a thin plate is given by  
\[ \begin{array}{l}\rho h\ddot{w}=\frac{\partial}{\partial x}\Big{(}\frac{ \partial H_{11}}{\partial x}+\frac{\partial H_{12}}{\partial y}\Big{)}+\frac{ \partial}{\partial y}\Big{(}\frac{\partial H_{21}}{\partial x}+\frac{\partial H _{22}}{\partial y}\Big{)}\\ H_{11}=\int\sigma_{x}2 dz,H_{22}=\int\sigma_{y}2 dz,H_{12}=H_{21}=\int\tau_{xy}2 dz \end{array}  \tag{2}\]  
where \(H_{ij}\) is the bending moment, \(\rho\), \(h\) are the density and the thickness of the thin plate, respectively, and \(\sigma_{x}\), \(\sigma_{y}\), \(\tau_{xy}\) are the stresses. (i) For the host elastic plate: As a transversely isotropic elastic material, the relationship between the strains and the stresses is as follows [36]:  
\[\left[ \begin{array}{c}\sigma_{x}^{\varepsilon}\\ \sigma_{y}^{s}\\ \tau_{xy}^{s}\end{array} \right]=\frac{E_{s}}{1-\mu_{s}^{2}}\left[ \begin{array} []{ccc}1&\mu_{s}&0\\ \mu_{s}&1&0\\ 0&0&(1-\mu_{s})/2\end{array} \right]\left[ \begin{array}{c}\varepsilon_{x}^{ \varepsilon}\\ \varepsilon_{y}^{s}\\ \gamma_{xy}^{s}\end{array} \right] \tag{3}\]  
Then, its dynamic equation of transverse motion can be expressed as Equation (4) based on Equations (1)-(3). \[-\phi_{s}\frac{\partial^{2}w}{\partial t^{2}}=\frac{\partial^{2}}{\partial x ^{2}}\Big{[}D_{s}\frac{\partial^{2}w}{\partial x^{2}}+v_{s}\frac{\partial^{2} w}{\partial y^{2}}\Big{]}+\frac{\partial^{2}}{\partial y^{2}}\Big{[}D_{s} \frac{\partial^{2}w}{\partial y^{2}}+v_{s}\frac{\partial^{2}w}{\partial x^{2} }\Big{]}+2\frac{\partial^{2}}{\partial x\partial y}\Big{[}\chi_{s}\frac{ \partial^{2}w}{\partial x\partial y}\Big{]} \tag{4}\]  
Figure 2: Binary representative of the PMPIC unit cell. where \(\phi_{s}=\rho_{s}h_{s}\), \(D_{s}=E_{s}h_{s}^{3}/12\big{(}1-\mu_{s}^{2}\big{)}\), \(v_{s}=D_{s}h_{s}\), \(\chi_{s}=D_{s}(1-\mu_{s})\), \(\rho_{s}\), \(E_{s}\), \(\mu_{s}\), \(h_{s}\) are the density, Young's modulus, the Poisson's ratio, and the thickness of the thin plate, respectively [35]. (ii) For the piezoelectric patch: Furthermore, a piezoelectric patch can be assumed to be a transversely isotropic material. Then, the piezoelectric patch is also looked at as a thin plate, and its constitutive equation can be reduced to a 2D form as Equation (5). \[\left[ \begin{array}{c}\varepsilon_{x}^{p}\\ \varepsilon_{y}^{p}\\ \gamma_{xy}^{p}\\ D_{3}\end{array} \right]=\left[ \begin{array}{ccc}s_{11}^{E}&s_{12}^{E}&0&d _{31}\\ s_{12}^{E}&s_{11}^{E}&0&d_{31}\\ 0&0&s_{66}^{E}&0\\ d_{31}&d_{31}&0&e_{33}^{T}\end{array} \right]\left[ \begin{array}{c}\sigma_{x}^ {p}\\ \sigma_{y}^{p}\\ \tau_{xy}^{p}\\ E_{3}\end{array} \right] \tag{5}\]  
where \(s_{11}^{E},s_{12}^{E},s_{66}^{E}=2(s_{11}^{E}-s_{12}^{E})\) are elastic compliance parameters at the constant electric field, \(d_{31}\) is the piezoelectric constant in the 3-1 direction, \(\varepsilon_{33}^{T}\) is the dielectric permittivity atconstant stress, and \(D_{3}\), \(E_{3}\) are the electric displacement and field along the \(z\)-axis, respectively. Moreover, Equation (5) can be rewritten as follows:  
\[\left[ \begin{array}{c}\sigma_{x}^{p}\\ \sigma_{y}^{p}\\ \tau_{xy}^{p}\\ D_{3}\end{array} \right]=\left[